quantum field theory ( qft ) and quantum optics ( qed ) · 2015-07-28 · quantum field theory (...

Quantum Field Theory ( QFT ) and Quantum Optics (

QED )

Mukul Agrawal

April1 9, 2004

Electrical Engineering, Stanford University, Stanford, CA 94305

Contents

I Conventions/Notations 7

II Background/Introduction 7

1 Quantum Field Theory ( QFT ) - Why We Need It? 7

1.1 Two �Flavors� of QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 QFT as a Special Application of Quantum Mechanics . . . . . . . . . . . . . 8

1.3 QFT as a Fundamentally New Physics � Relativistic Multiparticle Quantum

Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 QFT as convenient alternative language for non-relativistic multipar-

ticle quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 In Summary ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 General Comments on Philosophy of QM 13

2.1 Brief Comment on Linearity of Physical Systems in Quantum Mechanics . . 13

2.2 Brief Comment on Waves and Particles . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Usage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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CONTENTS

2.2.2 Usage 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Usage 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Brief Comments on Canonical Quantization . . . . . . . . . . . . . . . . . . 17

2.4 Brief Comments on Identical Particles . . . . . . . . . . . . . . . . . . . . . 19

III A First Introduction to QFT - Non Relativistic CanonicalQuantization of Classical Wave Phenomenon - Quantization ofDisplacement (Elastic) Fields 20

3 Single Harmonic Oscillator 21

3.1 Quick Classical Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Complex Variable Transformation . . . . . . . . . . . . . . . . . . . . 22

3.2 Quantization of Single Harmonic Oscillator . . . . . . . . . . . . . . . . . . . 24

3.2.1 Consistency Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . 25

3.2.3 Eigen Spectrum of Linear Harmonic Oscillator . . . . . . . . . . . . . 27

3.2.4 Normal Ordered Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Alternative Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Commutation and Bosonic Behavior . . . . . . . . . . . . . . . . . . . . . . . 30

4 Chain of N Coupled Harmonic Oscillators - Concept of Phonons 30

4.1 Quantization in Real Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 k - Space Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Quantization in k - Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3.1 Mnemonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Alternative Interpretation � Phonons . . . . . . . . . . . . . . . . . . . . . . 41

5 More About Phonons 42

5.1 Energy Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Dispersion and Causality for Phonons . . . . . . . . . . . . . . . . . . . . . . 45

6 Continuum - Quantization of Elastic Waves in Continuous Media 47

IV QFT As a Convenient Language for Known Physics 48

Mukul Agrawal

Cite as: Mukul Agrawal, "Quantum Field Theory ( QFT ) and Quantum Optics ( QED )", in Fundamental Physics in

Nano-Structured Materials and Devices (Stanford University, 2008), URL http://www.stanford.edu/~mukul/tutorials.

2


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CONTENTS

7 Introduction 48

8 Identical Particle Hilbert Space Representations 49

8.1 Fock Space and Fock States . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.1.1 Number State Representation . . . . . . . . . . . . . . . . . . . . . . 49

8.1.2 Basis Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.2 Operators in Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.3 Ground State or Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.4 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . 56

8.5 The Wavefunction Operators or The Field Operators . . . . . . . . . . . . . 57

8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

V Non-Relativistic, Massive, Quantum Field Theories ( QFT )60

9 Non-relativistic, Massive, Fermionic Fields 61

9.1 First Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

9.2 Second Scheme � Second Quantization of Schrodinger's Field . . . . . . . . . 67

9.3 Some Special Operators in Quantum Field Theory . . . . . . . . . . . . . . . 69

9.3.1 Note on Single and Multi-Particle Operators . . . . . . . . . . . . . . 69

9.3.2 Particle Number Operator . . . . . . . . . . . . . . . . . . . . . . . . 70

9.3.3 Particle Number Density Operator . . . . . . . . . . . . . . . . . . . 71

9.3.4 Particle Number Current Density Operator . . . . . . . . . . . . . . . 75

9.4 Two Point Correlation Functions/Propagators For Free Fields . . . . . . . . 76

9.4.1 Classical Correlation Functions . . . . . . . . . . . . . . . . . . . . . 76

9.4.2 Quantum Correlation Functions . . . . . . . . . . . . . . . . . . . . . 79

9.4.3 Free Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9.4.4 Green's Function for Non-Interacting Schrodinger Fields in Homoge-

neous Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9.4.5 Green's Function for Non-Interacting but Non-Homogeneous Case . . 88

10 Non-relativistic, Massive, Bosonic Fields 90

VI Non-Relativistic Quantum Electrodynamics ( QED ) 90

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CONTENTS

11 Introduction 90

12 'Free' Electromagnetic Fields in Non-Dispersive Unbounded Medium 91

12.1 Classical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

12.1.1 Classical Equation of Motion . . . . . . . . . . . . . . . . . . . . . . 91

12.1.2 In�nite Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

12.1.3 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

12.1.4 Classical Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

12.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

VII 'Free' Electromagnetic Fields in Non-Dispersive, Inhomoge-neous Medium 98

13 'Free' Electromagnetic Fields in Dispersive, Inhomogeneous Medium 101

13.1 Coherent States / Poisonian Distribution . . . . . . . . . . . . . . . . . . . . 103

13.1.1 Properties of Fock states . . . . . . . . . . . . . . . . . . . . . . . . . 105

13.1.2 Phase Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

13.1.3 Operator Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

13.2 Heisenberg Representation of Field Operators . . . . . . . . . . . . . . . . . 106

14 Optical Coherence Theory 106

14.1 De�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

14.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

14.2.1 Complex Degree of Coherence . . . . . . . . . . . . . . . . . . . . . . 109

14.2.2 Typical Interference Experiments . . . . . . . . . . . . . . . . . . . . 113

14.3 Coherence and Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 113

VIII Relativistic Quantum Field Theories ( QFT ) 114

15 Klein Gordon (KG) Field 114

15.1 Derivation of 'Classical' KG Field . . . . . . . . . . . . . . . . . . . . . . . . 114

15.2 Classical Hamiltonian Theory of Real Valued Scalar KG Field . . . . . . . . 116

15.2.1 Classical Hamiltonian and Lagrangian . . . . . . . . . . . . . . . . . 116

15.2.2 Ψ(x, t) Field in Momentum and Energy Basis . . . . . . . . . . . . . 117

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15.2.3 π(x,t) Field in Momentum and Energy Basis . . . . . . . . . . . . . . 119

15.2.4 Classical Hamiltonian in Momentum and Energy Basis . . . . . . . . 119

15.2.5 Field Expansions in Normalized Variables . . . . . . . . . . . . . . . 120

15.2.6 Inverse Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

15.3 Quantization of Real-Valued Scalar 'Classical' KG Field . . . . . . . . . . . . 121

15.3.1 Quantized Field Operators . . . . . . . . . . . . . . . . . . . . . . . . 121

15.3.2 Quantum Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 123

15.3.3 Number State basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

15.3.4 Coherent and Squeezed States . . . . . . . . . . . . . . . . . . . . . . 124

15.3.5 Heisenberg Representation . . . . . . . . . . . . . . . . . . . . . . . . 125

15.3.6 Two Point Correlation Functions/Propagators For Free Fields . . . . 126

15.4 Classical Hamiltonian Theory of Complex-Valued Scalar KG Field . . . . . . 130

15.4.1 Classical Hamiltonian and Lagrangian . . . . . . . . . . . . . . . . . 130

15.4.2 Classical φ(x, t) and φ?(x, t) Fields in Energy/Momentum Basis . . . 131

15.4.3 Classical π(x, t) and π?(x, t) Fields in Momentum/Energy Basis . . . 132

15.4.4 Classical Hamiltonian in Energy Basis . . . . . . . . . . . . . . . . . 133

15.4.5 Classical Field Expansions in Normalized Variables . . . . . . . . . . 133

15.4.6 Classical Inverse Transforms . . . . . . . . . . . . . . . . . . . . . . . 134

15.5 Quantization of Complex-Valued Scalar KG Field . . . . . . . . . . . . . . . 135

15.5.1 Quantization Field Operators . . . . . . . . . . . . . . . . . . . . . . 135

15.5.2 Quantized Field Operators in Energy/Momentum Basis . . . . . . . . 137

15.5.3 Quantum Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 138

15.5.4 Number State Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

15.5.5 Associated Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

15.6 Quantization of Complex Valued Vector Klein Gordon (KG) Field . . . . . . 140

15.6.1 Quantized Field Operators and Hamiltonian . . . . . . . . . . . . . . 140

15.6.2 Associated Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

16 Dirac Field 142

17 Relativistic Electromagnetic Field Quantization ( QED ) 142

IX Interaction Between Free Fields 142

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18 Di�erent Pictures of Time Evolution 142

18.1 Exponential of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

18.2 Schrodinger's Picture of Time Evolution . . . . . . . . . . . . . . . . . . . . 143

18.3 Heisenberg Picture of Time Evolution . . . . . . . . . . . . . . . . . . . . . . 145

18.4 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

18.5 Inter-Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

18.6 Diagrammatic Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . 149

18.7 Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

18.7.1 Quantum Ensemble Expectations . . . . . . . . . . . . . . . . . . . . 151

X Statistical Quantum Field Theory (Condensed Matter Physics)154

XI Further Resources 154

References 157

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Part I

Conventions/Notations

• Some books use a �hat� or �caret� on top of a symbol to represent an operator as

opposed to a classical variable. I prefer not to use these �hats� but they sometimes

become essential. Usually, the distinction between a classical variable and a quantum

operator is clear from the context. At the same time, I believe, text is more readable

if simple symbols with least numbers of subscripts, superscripts, hats, bars, bold faces

etc. are used! In this article, �hat� on a symbol is used explicitly only in places where

there are chances of confusion. Sometimes in the initial development of the subject we

would use these hats to explicitly di�erentiate between classical and quantum analogues

but would then gradually get rid of it.

• Creation and annihilation operators are always written in Schrodinger picture. Other

�eld operators would have subscripts to explicitly di�erentiate between the Schrodinger,

Heisenberg or interaction picture.

• I would start with SI units initially. But I would also try to gradually move toward

natural units (~ = 1, c = 1) rather than SI units as we develop the subject.

Part II

Background/Introduction

1 Quantum Field Theory ( QFT ) - Why We Need It?

1.1 Two �Flavors� of QFT

Quantum �eld theories (QFT's) have two �avors. On one hand QFT a simple application of

standard non-relativistic quantum mechanics to describe classical wave phenomena in quan-

tum world. For example quantum physics of elastic waves can be described by QFT which,

in this problem, is not a fundamentally new science. Its a very simple application of standard

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1.2 QFT as a Special Application of Quantum Mechanics

non-relativistic quantum mechanics in systems with in�nite degrees of freedom. The name

Quantum Optics is usually considered to represent non-relativistic quantum electrodynamics

(QED). As we would see below, quantum optics is simply obtained by applying standard

quantization rules to the in�nite coupled variables of Maxwell system.

On the other hand QFT also describes quantum physics of classical particles. In non-

relativistic limit, physics involved in QFT is same as physics involved in standard quantum

mechanics. This is a special case. More generally, quantum physics of classical particles

in relativistic limit is a fundamentally new science known as QFT. This is can be called a

non-trivial extension of quantum mechanics (this extension would need new postulates) but

it certainly is not a mere application of standard quantum mechanics.

1.2 QFT as a Special Application of Quantum Mechanics

In its one of the incarnation, QFT is simply a quantum theory of systems that can have

in�nite degrees of freedom. QFT is simply a quantum theory of classical problems which

has many coupled dynamical variables. Notice that a �eld is nothing but a bunch of coupled

variables. For example, a scalar �eld A(x, t) can simply be represented as a bunch of variables

Ax(t). Notice that the time evolution of one of the variable Ax1(t) might depend on the time

evolution of other variable Ax2(t). This second notation, I guess, makes it amply clear

what we mean by multiple coupled variables. Canonical quantization of classical �eld would

now be done in exactly same fashion as the canonical quantization of other many variable

problems, like multi-particle problems, is done. We would work out the details in sections

below.

1.3 QFT as a Fundamentally New Physics � Relativistic Multipar-

ticle Quantum Physics

Fundamentally, Quantum Field Theory (QFT) is required to combine ideas from special

relativity and quantum mechanics. In relativistic regime, quantum physics can not be de-

scribed in any way that would closely resemble standard non-relativistic quantum mechan-

ics. As we would discuss below, there is no consistent theory of single particle relativistic

quantum mechanics (similar to Schrodinger's equation that represents mechanics of single

non-relativistic particle in sub-atomic world). In literature, there exist many attempts at

it (like Pauli's equation, Klein Gordon equation or Dirac equation etc � each of which we

would discuss below, but in more modern formalism). But, as we would see, all of them su�er

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1.3 QFT as a Fundamentally New Physics � Relativistic Multiparticle Quantum Physics

from one or the other di�culties1. Fundamentally, these di�culties arise from the fact that

Einstein mass-energy equivalence removes the possibility of single particles. Since, particles

can dynamically break into smaller mass particles and smaller mass particles can recombine

to give larger mass particles, any relativistic quantum theory needs to be fundamentally

a multi-particle theory. Hence there is no single particle relativistic quantum mechanical

equation. The simplest we can have is a multi-particle equation which describes the motion

of potentially in�nitely many particles with possibility of annihilation and creation. What

follows is called quantum �eld theory. The reason this subject goes by the name of ��eld

theory� is because we start from �eld equations and the concept of particles (in�nitely many

of them) emerges as we develop the formalism.

At this point, let us remind ourselves how the subject of non-relativistic quantum me-

chanics is usually developed. We start with classical single particle equation of motion (e.g.

Newton's equation of motion). We then perform canonical quantization and obtain single

particle quantum equation of motion (which goes by the name of Schrodinger's equation). It

turn out that this equation of motion is actually a �wave equation� and it correctly represents

the dynamics of single non-relativistic particle in sub-atomic world.

Now suppose I want to develop a non-relativistic multi-particle quantum theory. How

do I do it? At least conceptually, it is a very simple extension of single particle quantum

mechanics. All we need to do is to use direct-product spaces and direct-product states (see

below). This approach is usually taught in traditional non-relativistic quantum mechanics

courses. There is an alternative scheme which is extremely bizarre (and hence interesting!) at

�rst look. Just for fun, we can try this alternative route (we know it works even though there

is no reason for it to work � and that's why its fun!). We can take the Schrodinger's single

particle quantum wave equation and �quantize� it again treating it as a classical ��eld�2

(details of how classical �elds are quantized would be discussed below). Why are we trying

this alternative route? This would become clear when we talk about relativistic physics in

a bit. Now, if we try this second approach, what emerges is multi-particle quantum theory!

Yes � a multi-particle theory out of a single particle theory3! Exactly same physics as what

we get from previous route. Surprised? Don't worry, it is very surprising at �rst sight.

First approach usually seems to be more direct approach but remember that the canonical

1The most perturbing problems were existence of negative energy states, non-positive-de�nite probabilityand non-unitary Lorentz transformation.

2Note that term ��eld� is more appropriate than the term �wave�. Entity that we are trying to quantize issimply a function of space and time � it might not necessarily represent anything �propagating� or �moving�.

3Non-relativistic physics is a special case � this does not happen in more general relativistic case.

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1.3 QFT as a Fundamentally New Physics � Relativistic Multiparticle Quantum Physics

quantization itself is a postulation � so saying that this second approach is more arbitrary

than the �rst approach is, strictly speaking, not correct. Both methods are postulations

which we know work.

Now, people tried copying this exact same recipe for relativistic physics. They wanted

to get single-particle relativistic quantum mechanics. So they started with the classical rela-

tivistic equation of motion and performed canonical quantization to obtain a wave equation

that they expected to represent single-particle relativistic quantum mechanics. There was

some bad news and some good news. Bad news �rst. If we start with single particle rel-

ativistic classical equation of motion and quantize it, we don't get a self consistent theory

of single particle relativistic quantum physics. We get a wave equation (or a �eld) which

has many problems4. This equation can not be treated as a valid �single particle� equation

of motion (in fact this equation is an intermediate step in obtaining correct �multi-particle�

physics). Now the good news. If we treat this equation of motion as a classical �eld and

quantize it again, we do get a valid multi-particle relativistic quantum physics. This whole

procedure has no justi�cation. It should, strictly speaking, be treated as postulation.

A couple of side notes. As stressed before, non-relativistic physics was just a special case.

In general, a multi-particle theory can not fall out of a single particle theory without further

postulations. Philosophically, this a bit comforting. Secondly, we were very lucky that canon-

ical quantization of classical particle physics lead us to Schrodinger/Heisenberg/Dirac/Von-

Neumann etc. type of quantum particle physics which worked. Again this is a special case.

Canonical quantization of relativistic classical �particle� physics does not lead to any valid

quantum physics.

So, in conclusion, fundamentally, particle theories actually emerges from �eld theories

with non-relativistic situation being a special case where �eld theory does not take a funda-

mentally important stand (and actually seems very uncomfortable, unnecessary and bizarre).

Second point, that I want to stress again before closing this section, is that this whole part

of science which deals with canonical quantization, is not usually taught in very accurate

fashion. These are just �tricks� that give us physics that experiments have con�rmed to be

true. These tricks can not even be called postulates in mathematically rigorous sense. Just

like in standard non-relativistic quantum mechanics, there is no way to �prove� or �derive�

expressions for quantum Hamiltonians (canonical quantization is just a trick that works for

a few important problem), similarly there is no way to derive Hamiltonians in terms of �eld

4As mentioned before, the most perturbing problems were existence of negative energy states, non-positive-de�nite probability and non-unitary Lorentz transformation. We would discuss these latter.

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1.4 In Summary ...

operators (that we use in QFT). These are supposed to be fundamental postulates. But there

exist a similar canonical quantization scheme, as we would study latter, using which we can

�derive� quantum �eld theory operators starting from a �wrong� supposedly-single-particle

relativistic quantum wave equation (like Dirac equation of KG equation).

1.3.1 QFT as convenient alternative language for non-relativistic multiparticle

quantum physics

As I said above, it is not necessary to quantize a �eld to obtain multi-particle non-relativistic

quantum mechanics. One can obtain the same physics using the concept of direct product

spaces in seemingly more direct approach. So, QFT is not essential for multi-particle non-

relativistic systems. As the methods and tools of QFT gained popularity, we have learned

that they are very useful even in non-relativistic multi-particle situations. Almost all prob-

lems in condensed matter physics (a big class of these problems are non-relativistic) are now

a days conveniently described in terminologies and formalism of QFT.

1.4 In Summary ...

As is clear from above discussion, QFT has two utilities. First, it is a quantum theory of

classical waves. Second, it is a quantum relativistic/non-relativistic theory of multi-particle

(varying number of particle or in�nite number of particles � both can be handled) physics.

As far as �rst usage is concerned, QFT is not a completely new theory. At least if we restrict

ourselves to non-relativistic systems, this formalism has nothing fundamentally new physics

in it (just a few mathematical tools required to handle in�nite degrees). In this case it is a

very simple extension of elementary quantum mechanics.

Now as for the second usage, it becomes a bit tricky. As we would see latter, we treat

single particle quantum mechanics (like Schrodinger's equation or the Dirac equation etc) as

�classical� wave equation and quantize it using same rules as those used for quantizing clas-

sical waves. We would see that this leads to multi-particle quantum theory (relativistically

consistent). This new story is not mathematically as straight forward, though. As we move

to relativistic systems, mathematical formulation becomes very intricate. The reason is that

it was not very simple to enforce Lorentz invariance on classical wave equations. Moreover

representations of Lorentz transformation in Hilbert space has it's own nitty-gritties. So

mathematically speaking, relativistic QFT 5 is far more di�cult than elementary quantum

5Fundamental physicists might insist here that �relativistic� quali�er is not required as QFT in funda-

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1.4 In Summary ...

mechanics (although very little completely new physics as such).

To summarize, here are a couple of fundamental reason why we need QFT :-

• QFT is needed to explain quantization of classical wave phenomenon - with no quantum

description available. One can argue that as far as non-relativistic physics of electrons

(Schrodinger's equation) is considered QFT only gives a new interpretation of same

physics included in Schrodinger's wave equation6. But when we quantize �elds like

electromagnetic �elds (which has no quantum description besides QFT) we get funda-

mentally new concepts using QFT. QFT makes various predictions of non-classical

�states of �elds� � like squeezed light etc. For example if you pick one normal mode

of EM wave then classically all you can change is its amplitude or phase. In classical

light, if you pick one normal mode, there is a Poissonian distribution of probability of

number of photon � that is its a linear superposition of states having di�erent number

of photons in that EM mode with expansion coe�cients being Poissonian distributed.

But QFT predicts that this mode can be in many other states like a superposition of

having one and two photons in one normal mode of EM wave � such a state of light

would not give Poisonian distribution in photon arrival time. There are many more

such phenomena that can not be explained without QFT.

• In relativistic QM, QFT removes many problems associated with negative energies,

in�nite energies and negative probability densities by coherently bringing the concepts

of anti-particles. For example, if we try to solve free-space Klein-Gordon equation

(which is just a simple relativistic version of free-space Schrodinger's equation and is

known to be correct description of a family of particles despite some earlier historical

doubts), 'each' state of the system is associated with two energies - one positive and

one negative. One does not know how to interpret it physically staying within non-

quantized wave equation description. QFT resolves such di�culties by predicting the

presence of anti-particles - each particle has its anti-particle that sits in the negative

energy state (somewhat like holes in semiconductor theory). QFT predicts existence

mentally not required for non-relativistic systems � for them QFT is relativistic QFT!6Technically speaking, second quantization involves no new physics. But �eld quantization do. Second

quantization can be studied in two ways. One can either use concepts of �eld quantization and quantize theSchrodinger's �eld � in such a case it does appear that we are postulating new physics. Alternatively, onecan simply use the anti-symmetry (Pauli's Exclusion) postulate of Fermi particles to obtain exactly samephysics as that by �eld quantization. Such alternative routes are not available if a quantum description(something like Schrodinger's equation) does not already exist - such as in EM waves case. In such case �eldquantization is a postulation of new physics.

Mukul Agrawal



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2: General Comments on Philosophy of QM

of many such particles.

Besides, QFT also provides a very neat and straight forward conceptual formulation under

varied circumstances (that can strictly be studied in conventional QM as well), like :-

• It removes the di�erent notions of single-particle equation or multi-particle equation

and brings in a more complete picture. Its makes it much easier to describe real physical

situation where number of particles is not constant and keeps on changing as a result

of interaction with surroundings. All QFTs are multi-particle theories. One

starts from so called single-particle Schrodinger's equation and gets a multi-particle

QFT. This is no miracle. If we look at conventional quantum mechanics, even with

single particle Schrodinger equation one can study multi-particle systems if one ignores

particle-particle interaction. If you build a QFT of electrons what you get is a non-

interacting multi-electron theory7. In QFT inter-particle interaction is treated as inter-

�eld interaction. Main bene�t of QFT is its conceptually simple and straight forward

formulation.

• It removes the notion of interacting or non-interacting multi-particle theories. Inter-

particle interaction comes from 'inter-�eld interaction'. For example quanti-

zation of Schrodinger's equation leads to multi-particle QFT. For including electron-

electron interaction one needs to include the photons in the picture. So one type of

�eld only brings out one character of a type of particle. One needs to study inter-�eld

interaction to get a complete picture of an electron, for example. This kind of set up

is much more handy - at least conceptually.

2 General Comments on Philosophy of QM

2.1 Brief Comment on Linearity of Physical Systems in Quantum

Mechanics

• One would notice the equations governing the motion (like Schrodinger's equation) in

quantum physics are linear. So when we claim that we understand all the fundamental

7Description remains incomplete unless all other �elds are included as well. For example in QFT electron-electron Coulomb interaction comes from the interaction of electron QFT with photons QFT. So one maywant to call the Schrodinger's quantized �eld (without the presence of other �elds) as 'Quasi-Electrons' �they are strictly eigen energy state as long as other �elds are not present. See the following points to makethings clear.

Mukul Agrawal



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2.2 Brief Comment on Waves and Particles

laws of how universe operate, how are we proposing to handle non-linear processes?

Please check the other tutorial article on linearity of dynamical equations in quantum

physics. Concepts explained there hold even for QFT.


In quantum physics waves/�elds and particles are very closely interleaved � much more than

what we have seen in Schrodinger's type quantum mechanics. One can describe each and

every phenomena through particles only (including static forces like electrostatic attraction

as well as interference/di�raction � we don't need any �eld concept) or using waves/�elds

only. In fact all particles are made up of �elds themselves in QFT. However, while building

these two exactly equivalent formalism we would de�ne very clearly what we actually mean

by 'particle' and what we actually mean by a 'wave'. We would see that these are not exactly

same as our naive understanding of particle and wave concepts. In the prevalent literature

of modern physics people speak of a 'particle' or a 'wave' in several very distinct senses:-

2.2.1 Usage 1

• 'Elementary particles' are often simply called 'particles'. We would understand

these better while studying QFT. These results from the studies of free �elds (no inter-

action). 'Particles' are treated as a 'quanta' of something like spin angular momentum

or mass or electric charge etc. (each elementary particle has a set of associated quantum

numbers). So we claim (this actually falls of naturally from the theory that we would

discuss below, its not treated as a separate postulate) that these quantities like charge

or mass (or it may be something else) can only be incremented or reduced in steps and

not continuously. This is the correct meaning of particle behavior. For, example

when electron gets absorbed from a quantum well by a physically �localized� absorber,

�entire electron� gets absorbed �at once� from entire quantum well irrespective of the

shape of wavefunction. The process is not something like water �ow going into a drain.

How relativistic causality is preserved in this process would be explored in the theory

of QFT. Note that localization in physical space should not be considered as

a proof or guarantee of particle behavior. On the other hand, interference

and di�raction should be associated with correct meaning of wave behavior.

To allow both the e�ects in one physical entity, say an electron, under di�erent exper-

imental setup, the way we explain the physics is to propose that there is 'something'

Mukul Agrawal



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http://www.stanford.edu/~mukul/tutorials/linear.pdf





(like wavefunctions in Schrodinger's mechanics) associated with every system that is

more fundamental than the properties we can measure. So the kind of language we

would be using while studying quantum physics is that � certain �mode� has so many

numbers of �quanta� of certain quantity. The �mode� is usually a state vector in state

space of continuous functions. Various �modes� are allowed to go through interference

which �nally leads to interference e�ects in observable quantities like spatial distribu-

tion etc. Note that simply associating a spatial function is not a proof or

guarantee of wave like behavior. We need to allow these spatial wavefunction to

take negative or complex values (mathematically it should form a vector space) so that

we can add them to get interference e�ects. A positive-de�nite spatial function can

not show interference e�ects. Also, very roughly speaking, the amplitude of �mode�

can only change in discrete fashion. And this would �nally lead to particle behavior.

And this is how wave-particle duality is inbuilt in quantum mechanics. So

this state vector or mode itself can be a function of spatial coordinates that describes

the distribution of quantity concerned in space in some particular fashion. This space

function may or may not accept a straight Born probability interpretation that we

have seen in Schrodinger's quantum mechanics. The mapping between state vector

and actual amplitude/probability distribution (Born like interpretation) depends on

the quantity we are talking about. This interpretation again would fall of naturally as

we develop the theory.

• A comment should be added here about widespread misconception about energy and

momentum of elementary particles. It is widely taught that photons are supposed to be

monochromatic (state of de�nite energy and state of de�nite momentum in freespace).

Remember E = ~ω and p = ~k? This statement is completely incorrect8. We can take

a photon and place it in a linear superposition of two energy eigen state (in free space

for example). So we can have a single photon whose energy and momentum are not

well de�ned.

2.2.2 Usage 2

• In condensed matter physics, when we quantize complicated interacting system, we

would be lead to many other types of 'quanta' that are not elementary particles. Ex-

8Two expressions, known as Einstein and De Broglie relations respectively, are correct depending uponhow you interpret and how you use them. Abuses of these in both teaching and practice are common!

Mukul Agrawal



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amples of these are electron-polarons (quanta of coupled electron phonon system) or

phonon-polariton (quanta of coupled phonon-photon system) etc. Basically these are

elementary excitation of ground state (lowest energy state) of such systems.

By 'elementary excitation' we mean that 'minimal' excitation of system beyond ground

state would have one of these elementary excitation. Some people refer to quanta de-

�ned like this either as normal 'particles' or 'quasi-particles' depending upon the

�lifetime�. Quasi-particles are something that represent energy eigen state as long as

interactions with other particles/�elds are neglected. Interactions give �nite lifetime.

In this sense, actually everything known as 'particles' are just 'quasi-particles' !

• We need to clarify the meaning of last statement a bit further. Let us assume that

we have a single molecule with one extra electron. What happens is that the molecule

goes into a di�erent state of vibrations when extra electron is added. This is what

we call as electron-polaron. Let us assume that there are no other interactions. So

if we place electron-polaron in an energy eigen state of this multi-particle system, it

would remain in that state for ever (life time of the state is in�nite). We can also

place this electron-polaron, if we want, in a linear superposition of two energy eigen

states of this multi-particle system. Then system would keep on oscillating between

two energy eigen state forever. But the electron-polaron would never 'decompose'. In

realistic cases, lifetime is never in�nite but can be long compared to other processes

(electron-polaron would decompose after long time but not within the time scale we

maybe interested in). In such a case people simply refer to this elementary excitation

as a 'particle'. In case this elementary excitation decays faster than the time scale we

are studying, people usually refer to it as 'quasi-particle'.

2.2.3 Usage 3

• The other common usage of the term 'particle' is more intended for �xing analogies

between (loose) classical and quantum interpretations. In a very loose classical sense,

a particle is supposed to have a de�nite position and velocity whereas waves never have

both of them in de�nite quantities at same time. Note that this is not a real indication

of particle behavior even in classical mechanics - lots and lots of people misunderstand

the concept of particles. For example you can take a completely classical acoustic wave

and create wave-packets out of it � and it might seem that you got the wave-particle

duality explained with a theory that is fundamentally a wave theory! Certainly not,

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2.3 Brief Comments on Canonical Quantization

because you would never be able to explain instantaneous and localized absorption of

phonons from everywhere in one go (process would always be like water �owing into

drain as long as you stay in classical mechanics). So the way people very loosely put

it is that � 'wave-particle duality can be resolved/institutionalized by accepting that

actual world obeys �eld/wave mechanics but one can obtain 'particle' like behavior

by creating wave-packets by superposition of �eld/waves'. This interpretation is

not correct. It gives an impression that quantum mechanics is just a simple

classical wave-mechanics. I have even seen books on quantum mechanics with titles

like �Wave Mechanics�. Complete resolution of wave/particle duality comes only when

one understands that wave-like modes or quantum states (which obeys linear algebra)

contains quantized amount of matter. For example when electron would get absorbed,

electron would disappear from 'everywhere' unlike classical wave theories. Even in

Schrodinger's quantum mechanics, you would appreciate the real strength of quantum

mechanics only when you start building multi-particle theory and start studying things

like emission and absorption of particles. One should go back and study some good

text on photoelectric e�ect to really understand that wave-packets of classical waves

can never explain the photon-like (particle-like) behavior observed by the photoelectric

experiments.

• In conclusion, one should understand that if you associate interference/di�raction phe-

nomenon as de�ning properties of waves then the wave-like behavior is inbuilt in quan-

tum mechanics since it associates a more fundamental 'entity' to all system and that

entity has a linear algebraic structure. Phrases like 'wave like' or '�eld like' are often

used to indicate that quantity obeys linear algebra and its the 'amplitude' associated

with the quantity that adds up and not the probabilities which are interpreted as square

of amplitudes. So this can be taken as another usage/interpretation of word 'wave'.

Phrases like - 'particle-like' actually means that when it is absorbed it is absorbed as a

whole from everywhere � physical localization is not real indicator of particle behavior.

Physical localization is easily achievable constructing wave-packets out-o� completely

classical wave mechanics.


Over time, many schemes of quantizing a classical Hamiltonian to obtain a quantum Hamil-

tonian have been developed. All of them are ultimately just �rules�. For any given problem,

Mukul Agrawal



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these �rules� needs to be veri�ed against empirical data. QFT, as a theoretical framework

assumes that Hamiltonian is given. Most popular of these quantization rules is the canon-

ical quantization due to Dirac. In �eld quantization setup, canonical quantization also

goes by the name of second quantization.

• Readers might also want to check other articles on QuantumMechanics and Symmetries

in Physical World.

• There are many ways of obtaining quantum Hamiltonian and other operators from clas-

sical analogues. These are known as quantization rules. Most common of these is the

so called canonical quantization due to Dirac. In �eld theory, canonical quantization

is sometimes also called 'second quantization'.

• It is instructive, and probably interesting as well, to stress that the Dirac's canonical

quantization rule of 'inventing' quantum Hamiltonians from the known classical Hamil-

tonians by converting canonical co-ordinate and momenta into operators and enforcing

a commutation relation between them, is in essence, a claim that one can use the same

force/interaction laws in subatomic world as in classical macroscopic world.

Only the system description (the kinematical part of the story) is di�erent because of

Nature's reluctance to allow us to specify the state of a system beyond certain limit.

This statement is true in many common situations but not in general (for examples

spins have no classical counterparts and can not be invented from classical Hamiltoni-

ans). But it is at the heart of most of the conventional quantum mechanics that we do

in day to day semiconductor physics. When experimentally one come to know that it

does not work then one look around for a new quantum mechanical Hamiltonian that

would describe the new physics. One such famous example is spin.

• Also note that time t has a special and distinct status in non-relativistic

quantum mechanics and is di�erent from the status of spatial position. We

don't replace t with an operator when using Dirac's canonical quantization rule. t

remains a functional parameter whereas spatial co-ordinates and conjugate momenta

are promoted (or may be demoted - whatever you like!) to the status of operators.

This, intuitively speaking, might be related to the fact that we never observe quantized

time (or rather we never explain it that way).

• Since all classical physical quantities can always be represented as a function of Carte-

sian co-ordinates and conjugate momenta, Dirac's canonical momentum, in e�ect, tells

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http://www.stanford.edu/~mukul/tutorials/Quantum_Mechanics.pdf

http://www.stanford.edu/~mukul/tutorials/symetries.pdf




2.4 Brief Comments on Identical Particles

us that all we need to do is to get new de�nitions of co-ordinate and mo-

mentum such that those de�nitions somehow include our inability to specify

co-ordinate and momentum simultaneously. The rest of all the physical quanti-

ties would still hold identical meaning in quantum mechanics and we can use exactly

identical de�nitions. These new de�nitions would be postulates. Or in other words we

postulate a commutation relation between two variables which obey the same classical

physics.

• Note that Dirac's quantization rule and the enforced (postulated) commu-

tation works in Cartesian co-ordinates only. If you change the co-ordinates you

need to obtain new commutation rules using the co-ordinate transformation equations.

One example is the angular momentum. In Hamiltonian formulation angular

position and angular momenta can simply be treated as generalized co-ordinates and

conjugate momenta but they would not obey Dirac's simple commutation rule. One can

obtain commutation relation between any set of generalized co-ordinates and conjugate

momenta starting from the postulated commutation relation of Cartesian co-ordinates

and Cartesian momenta.

• These would also work only as long as quantities involved have integer spins. This we

would explore in more details below.

2.4 Brief Comments on Identical Particles

• QFT would result in di�erent families of identical particles. So its important to under-

stand what happens if particles are identical or in other words indistinguishable from

each other.

• Postulate of symmetrization of basic quantum mechanics states that there are only two

types (classes) of identical particles � bosons or fermions. This means that:-

� All states need to be either symmetric or anti-symmetric. For wavefunctions of

two identical particles:-

φ(r1, r2) = ±φ(r2, r1)

� All operators corresponding to physically measurable quantities need to preserve

the Bosonic/Fermionic properties of states. This means that operators corre-

sponding to physical properties need to commute with so called exchange opera-

Mukul Agrawal



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tor. Hence operators need to be completely symmetric with respect to exchange of

particles. For any operator that operates on two identical particle Hilbert space:-

A(r1, r2) = A(r2, r1)

• Further, one can prove that Bosonic and Fermionic behavior is associated with particles

and not with states (this is not a postulate, this statement can be proved). This means

that

� Suppose we have a system of two identical particles. Suppose we know that system

of particles is in an anti-symmetric state. When we add a third identical particle

to the system, it would remain anti-symmetric. One can prove this from physical

arguments.

� If we have system of three identical particles, there does not exist any state of

the system so that it is symmetric with respect to exchange of two of these and

anti-symmetric with respect to exchange of another two. This statement can also

be proved from group theoretic arguments.

Part III

A First Introduction to QFT - Non

Relativistic Canonical Quantization of

Classical Wave Phenomenon -

Quantization of Displacement (Elastic)

Fields

As mentioned above, QFT has two usages. For physical phenomenon that are classically

wave motions (like electromagnetic waves or continuum elastic waves etc), QFT is a simple

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3: Single Harmonic Oscillator

extension of quantum mechanics into systems with in�nite degrees of freedom. As far as

quantization of �elds is concerned, we do not need any new physics or any new postulates.

Standard quantum mechanics is su�cient.

On the other hand QFT also provides us multi-particle theories for those particles which

obey single particle quantum mechanical wave equations. For example quantization of

Schrodinger's wave and the Dirac waves gives us non relativistic and relativistic physics of

multi-electron systems respectively. This part of QFT is a bit tricky. We treat Schrodinger's

equation or the Dirac equation as a �classical� wave equation and quantize it as usual. As

it turns out, this quantization (also known as second quantization) leads to multi-particle

physics.

In this part we would only discuss the quantization classical wave motion. For this part

we would not need any new physics. We would discuss how to apply standard quantum

mechanics in systems with in�nite degrees and we would discuss how to do canonical quan-

tization of classical waves. In next part we would discuss the second �avor of QFT and we

would try to rationalize why we are justi�ed in treating single particle quantum mechanical

equation of motions as �classical� wave equations.

3 Single Harmonic Oscillator

3.1 Quick Classical Recap

Classical Hamiltonian for a harmonic oscillator with natural frequency of oscillation ω and

mass m is written as

H(q, p) =1

2mω2q2 +

p2

2m

Natural frequency is related to the spring-constant and mass as ω =√

Km. q and p are Carte-

sian co-ordinate and momentum respectively. Hamilton's equation of motion (or equivalently

Newtonian equation of motion) can be written as

dp

dt= −mω2q (1)

and

mdq

dt= p

Mukul Agrawal



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We know that the most general solution for these equations is

q(t) = A cos(ωt− φ)

and

p(t) = Amω sin(ωt− φ)

Here A and φ are real valued variables. Any other solution or superpositions of solutions

can always be written in above form 9.

3.1.1 Complex Variable Transformation

One should notice that A and φ are the real valued variables in the above solution. From

above form of the solution for Cartesian co-ordinate q and Cartesian momentum p one can

immediately see that p is 90o out of phase from q. Hence we can, if we want to, write

the solutions as real and imaginary parts of a complex exponential. In other words, if B

is a complex valued number, then we must be able to write such that q ∝ B + B? and

p ∝ i(B − B?). We can write the above written most general solution in a more convenient

and completely alternative form as

q(t) = C exp(−iωt) + C? exp(iωt) ≡ B(t) +B?(t)

and

p(t) = −imωC exp(−iωt) + imωC? exp(iωt) ≡ −imω(B(t)−B?(t))

Notice that, if we let C = A2

exp(iφ) and B(t) = C exp(−iωt), where A and φ are again real

valued, then we retrieve our previous form of the solution q(t) = A cos(ωt−φ). C can now

be a complex valued constant and includes the amplitude and phase information.

B(t) is a complex valued function of time and includes the amplitude, time oscil-

9Here comes a very important point. One should notice that the reason q(t) and p(t) are real valuedfunctions is because the classical equation of motion ( 1 ) has second order time derivative of q(t). If thisderivative had been a �rst order then complex solution would be acceptable. As we would see in the nextsection, quantum mechanical equation only involves a �rst order time derivative (Schrodinger's equation).Hence the solutions are complex exponentials. So we would have to add two solutions to get a real valuedsolution.This a very generic result. A very important di�erence between non-relativistic and relativistic QFT of

elementary particles. It also creates anomalies within non-relativistic QFT. For example, one would alsonotice that the wavefunction operator for a single phonon would include two terms (so called positive energyand negative energy terms) whereas non-relativistic single particle electron wavefunction operator containsonly one positive energy term.

Mukul Agrawal



22




lation and phase information. Remember that the real part of B, within some frequency

independent numerical factor, represents time evolution of Cartesian co-ordinate q and the

imaginary part of B, again within some frequency dependent numerical factors, represents

time evolution of Cartesian momentum.

One can now immediately check that

q2 = CC exp(−i2ωt) + C?C? exp(i2ωt) + 2CC?

and

p2 = −m2ω2CC exp(−i2ωt)−m2ω2C?C? exp(i2ωt) + 2m2ω2CC?

Hence, the Hamiltonian can now be written as

H(q, p) =1

2mω2q2 +

p2

2m= 2mω2CC? ≡ mω2(BB? +B?B)

From here onward we would use this complex exponential form of the solution. We would

call the B ≡ C exp(−iωt) as the �positive energy� part and B? ≡ C? exp(iωt) as

the �negative energy� part of the solution. It sometimes help to remember that

C = A2

exp(iφ). And the solution is q(t) = B +B? and p(t) = −imω(B −B?).

As a general comment, in more complicated systems, its not necessary that q and p be

sinusoidal functions of time and be 90o out of phase. Since a complex valued function of

time is simply a combination of two arbitrary real valued of functions of time, we can always

write q and p as real and imaginary parts of a complex valued function of time. Reversing

the transformation of variables, one can write

B =q

2+ i

p

2mω(2)

B? =q

2− i p

2mω(3)

This transformation can be used to obtain the complex variable B when the classical solution

is not a sinusoidal function.

We would also use normalized variables very often. These normalizations give much

simpler expressions. Suppose we want to write the Hamiltonian as

H = ~ω(bb?)

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3.2 Quantization of Single Harmonic Oscillator

This can easily be done by de�ning �normalized �eld variables� as:-

b =

√2mω

~B

Here are the conclusions

• Most general solution is made up of �positive energy� (B = C exp(−iωt)) and �negativeenergy� (B? = C? exp(jωt)) parts. Coe�cients are complex valued and conjugate of

each other. We would often represent these two parts as B and B?. We would often

refer to these parts as ��eld variables�. These includes phase as well as time information.

• b and b? are just the normalized B and B?.

• When we would quantize the problem in next section, we would �nd that its very

helpful in �nding the eigen energy spectrum of them Hamiltonian to try to factorize

the classical Hamiltonian as above. General procedure would be try and write the

real-valued classical canonical variables q and p in terms of single complex variable b

such that q ∝ b+ b?and p ∝ −i(b− b?).


One can easily get a quantum mechanical Hamiltonian through Dirac's quantization rule by

converting time dependent canonical variables q and p into Hermitian time inde-

pendent Schrodinger's picture operators q and p obeying the canonical commutation

relation

[q, p] = i~

(we experimentally know that it works)

H =p2

2m+

1

2mω2q2

3.2.1 Consistency Check

Whenever a problem is quantized one should do following consistency check. One should

make sure that, in Heisenberg representation, time dependent operators q and p obeys the

Heisenberg's equation of motion. In this particular problem, q and p are Cartesian co-

ordinate and Cartesian momentum respectively. Hence, this is trivially enforced.

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3.2.2 Creation and Annihilation Operators

As we know from elementary quantum mechanics, energy spectrum (eigen values and eigen

vectors) of this Hamiltonian can be obtained using standard di�erential equations methods.

Instead following that straight attack, we would develop a more handy method here. For

this we would �rst de�ne creation and annihilation operators.

We know that the classical q and p were given by

q = B +B?

and

p = −imω(B −B?)

Now when q and p turn into Hermitian time-independent operators then B and B? also into

time-independent operators but non-Hermitian ones. We will call them B and B† (as

would become obvious in a minute, these two operators are indeed Hermitian conjugate of

each other). Transformation of B and B? into operators can easily be seen from classical

inverse transformations.

B(t) =q(t)

2+ i

p(t)

2mω

and

B?(t) =q(t)

2− i p(t)

2mω

So we get

B =q

2+ i

p

2mω

B† =q

2− i p

2mω

If we want, we can call these expression just as their de�nitions in terms of operators q and p.

All we are doing here is to motivate these de�nitions. I want to justify the reader why I want

to de�ne B and B† the way I did. Alternatively, one can see above expressions as natural

consequence of Dirac's quantization of q and p. One can just consider above quantities

as quantized versions of associated classical quantities. Incidentally note that q and p are

Hermitian by de�nition and hence B and B† as de�ned above are Hermitian conjugate of

Mukul Agrawal



25




each other. Inverting these de�nitions, one can write q and p in terms of B and B†

q = B + B†

p = −imω(B − B†)

Now, the de�ned commutation relation between q and p would also enforce a commutation

relation between B and B†. We can see that

[B, B†] = − i

2mω[q, p] =

~2mω

And after some explicit calculations we would see that

H = mω2(BB† + B†B)

Now suppose I want to write this as

H =~ω2

(bb† + b†b)

This can easily be done by de�ning normalized operator b as follows:

b ≡√

2mω

~B

and hence in terms of these new variables

[b, b†] = 1 (4)

So if we want, using the commutator, we also can write the Hamiltonian as

H = ~ω(b†b+1

2) (5)

Also in terms of these new variables

q =

√~

2mω(b+ b†) (6)

p = −imω√

~2mω

(b− b†) (7)

Mukul Agrawal



26




Or

b =

√2mω

~(q

2+ i

p

2mω) (8)

b† =

√2mω

~(q

2− i p

2mω) (9)

3.2.3 Eigen Spectrum of Linear Harmonic Oscillator

After so much of formalism and variable transformations, you may ask what's the point? We

started with a Hamiltonian and ended with Hamiltonian, we haven't solved anything yet?

Well, problem is actually solved by this point. The above �factorized� form of Hamiltonian

is easily diagonalizable. Lets explore this.

The good thing about this scheme is that if |n〉is the normalized energy eigen state with

an eigen value of (n+ 12)~ω then b†|0〉 is also an energy eigen state (may not be normalized)

with energy of (n+ 1 + 12)~ω which can easily be checked by applying Hamiltonian on b†|n〉.

Easy way of doing this is through commutation relations. One should �rst check that

[H, b†] = ~ω[b†b, b†] = ~ω(b†[b, b†] + [b†, b†]b) = ~ωb† (10)

In above we have used expressions 4 and 5. Similarly,

[H, b] = ~ω[b†b, b] = ~ω(b†[b, b] + [b†, b]b) = −~ωb (11)

Now, using expression 10 one can easily check that b†|n〉 is indeed an eigen state of energy

H(b†|n〉) = ([H, b†] + b†H)|n〉

= ~ωb†|n〉+ (n+1

2)~ωb†|n〉

= (n+1

2)~ω(b†|n〉)

Similarly, using expression 11 one can easily check that b|n〉 is also indeed an eigen state of

energy (may not be normalized) with eigen value of (n− 1 + 12)~ω. This is the reason why

these operators are called creation-annihilation operators or the ladder operators.

Speci�cally note that b†|0〉, for example, is normalized even though |0〉 is normalized.Let us try to �gure out the normalization prefactors should be. Let b†|n〉 = An|n + 1〉where states |n〉, ∀n are properly normalized. Taking Hermitian adjoint of this expression

Mukul Agrawal



27




we get 〈n|b = 〈n + 1|A?n. Combining these two one get 〈n|bb†|n〉 = A?nAn= 〈n|12

+ H~ω |n〉 =

12

+ (n+ 12) = n+ 1. Hence, An =

√n+ 1 exp(iφ) where φ is arbitrary real number. Hence,

one can separate out prefactors as

b†|n〉 =√n+ 1|n+ 1〉

and

b|n〉 =√n|n− 1〉

So knowing ground state we can form all the states without explicitly solving the rather

di�cult energy eigen value problem. Even �nding the ground state is pretty simple. One

can show that b|0〉 = 0. Remembering that b =√

2mω~ ( q

2+ i p

2mω) and in q-representation

|0〉 → Ψ0(q), q → q and p→ −i~ ∂∂q

, one gets an identity that

√2mω

~(q

2+

~2mω

∂

∂q)Ψ0(q) = 0

If we de�ne a new variable q′ =√

mω~ q we get

1√2

(q′ +∂

∂q′)Ψ0(q′) = 0

Using the integration factor of exp(q′2/2) one can easily integrate above equation. And

�nally after normalizing the state one can easily see that

Ψ0(q) = (mω

π~)1/4 exp(−mω

~q2

2)

Hence the ground state wavefunction is Gaussian distributed over real space with a normal-

ization factor in front of it. Applying b in q-representation on this state one can then �nd out

all other energy eigen state. Hence we see, as we claimed, that factorizing the Hamiltonian

and �nding creation and annihilation operators is all we had to do to solve the complete

problem.

3.2.4 Normal Ordered Hamiltonian

Second part of the Hamiltonian written above give the zero point energy. Very often we would

shift our zero reference of the energy scale to throw out this part in the energy spectrum. If

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3.3 Alternative Interpretation

we decide to do this, we can simply write the Hamiltonian as

H = ~ω(b†b)

In this simple problem it was obvious how to remove the constant shift from the Hamiltonian.

In more complex problems it might not be as obvious. Following is the formal method of

getting rid o� constant shift. Remember that initially Hamiltonian was written as

H =~ω2

(b†b+ bb†)

Now we de�ne normal ordering of the operators by requiring all creation operators to go

to the left hand and all annihilation operators to go to the right. We claim that doing

this removes the constant zero-point energy. In standard text books normal ordering is

represented by :: symbol. So

: H :=~ω2

: (b†b+ bb†) :=~ω2

(2b†b) = ~ω(b†b)

3.3 Alternative Interpretation

We can interpret it (or talk about it) in a completely equivalent and an alternate language.

This alternate language for the time being would look unnecessary but its usefulness would

become more clear once entire subject of QFT is developed.

An oscillator of natural frequency ω is treated as one single ��eld-mode� which can take

di�erent quanta of energies in steps of ~ω. Depending upon the number of quanta in this

single �eld mode the �wave-function of the �eld � would take di�erent shape. Instead of saying

that an oscillating particle absorbs a photon and goes to higher energy eigen state we say

that the photon is absorbed and a quanta of energy is emitted (or created) in the ��eld mode

ω�. Note that there is still a ��eld-wavefunction� associated. �Field� in this extremely trivial

case has no spatial degree of freedom (in the sense that we don't really have a phonon like

wave in this example). The ��eld-wavefunction� represents the uncertainty associated with

the ��eld amplitude�. Again, in this simple case ��eld amplitude� is simply the displacement

of the oscillating particle from the equilibrium position.

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3.4 Commutation and Bosonic Behavior

3.4 Commutation and Bosonic Behavior

Another important point is that above commutation relation between annihilation and cre-

ation operator �indicates Bosonic identical particles� as we would explore below in more

details. What this means is that the quanta of energy discussed above would follow a ther-

mal equilibrium statistics similar to photons and phonons.

4 Chain of N Coupled Harmonic Oscillators - Concept of

Phonons

4.1 Quantization in Real Space

Let us now look at a multi coupled dynamical variable problem. This would shed some light

on the QFT.

Suppose there are N atoms connected together with identical linear springs. Note that

we can choose the N Cartesian displacements as independent co-ordinates and corresponding

Cartesian momenta to write a simple classical Hamiltonian as:-

H(ql, pl) =∑l

{ p2l

2m+

1

2K(ql+1 − ql)2}

Where l is the atomic site index and K is the spring constant. The equation assumes that

the spring constants between all the identical nearby atoms are exactly same (for simplicity)

and assumes only nearest neighbor coupling10. Both ql and pl are functions of time.

The Hamiltonian leads to a set of 2N separate Hamilton's equations of motion which would

be equivalent to N Newton's equation of motion and N de�nitions of momenta in terms of

co-ordinatesdpldt

= K(ql+1 − ql)−K(ql − ql−1)

pl = mdqldt

Note that the equation of motion for any one Cartesian displacement (or Cartesian momen-

tum) would depend upon the other Cartesian displacements as they are coupled. We can

10Note that as far as order of coupling is �nite, �eld theory developed would be a � local �eld theory� andhence conceptually they are all same � nearest neighbor coupling makes it easier to illustrate the physics.When order of coupling becomes in�nite, then this becomes non-local �eld theory � something that we wouldnot discuss in this article.

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4.1 Quantization in Real Space

go ahead and do the Dirac's quantization and obtain a quantum mechanical Hamiltonian.

Please note that the commutation relations enforced would be

[qn, pl] = i~δn,l

[qn, ql] = 0

[pn, pl] = 0

One should notice how one quantization rule automatically and logically gets replaced by

three in coupled oscillator case � simply because now we have a multi-particle problem at

hand. Note that these are local Cartesian co-ordinates and momenta. Such a quantization

is scheme is usually known as �quantization in real space� (real space in this particular

case is local Cartesian co-ordinate and momenta but in general it can be local generalized

co-ordinate and momenta as in case of EM �eld quantization in real space as we would see

below). The quantum mechanical Hamiltonian can easily be written as :-

H =∑l

{ pl2

2m+

1

2K(ql+1 − ql)2}

The state can be written as

Ψ(t) = ΠlΨl(ql, t)

Note that Ψl depends only on one co-ordinate and Πl represents the direct product over all

l. This would lead to a Schrodinger's equation:-

{∑l

pl2

2m+

1

2K(ql+1 − ql)2}Ψ(t) = i~

∂Ψ(t)

∂t

Which is equivalent to set of N Schrodinger's equations each associated with a single l:-

pl2Ψl(ql, t)

2m+

1

2K(ql+1 − ql)2Ψl+1(ql, t)Ψl(ql, t)

= i~∂

∂tΨl+1(ql, t)Ψl(ql, t)

Similarly one can write Heisenberg equations as well which would be exactly

similar to the classical Hamilton's equations. As a result of direct correspondence

we can easily see that all the N Schrodinger's equations are also coupled. Note that

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4.2 k - Space Description

at each atomic site there is an associated wavefunction of its own Cartesian displacement. So

we have N attached wavefunctions. And the wavefunction evolution at each site is dependent

on the wavefunction evolution attached with other sites. Solving such a equation directly

would not be simple.


There exist a linear transformation of co-ordinates in which the classical Hamilton's equation,

when written as matrix, becomes diagonal. These are simply the classical Normal mode

decomposition. An important e�ect of this change of co-ordinates is that even

the resulting Schrodinger's equations would be decoupled. Mathematically, what

we are doing is that we are taking linear combinations of di�erent Schrodinger's equations

and then de�ning new di�erential operators so that a bunch of new di�erential equations

in terms of these new operators are linearly independent. Which straightaway tells us that

these would lead to the quantum mechanical eigen energy states. Another strange thing

is that, as it turns out, even the Hamiltonian in new co-ordinates have the same

basic form of Harmonic oscillator (we would see how to put it in that form). Which

tells us that the amplitude of each classical normal mode obeys a decoupled

harmonic oscillator Hamiltonian when quantized.

We would now explore in details how this transformation actually works. I believe reader

is well aware of methods to diagonalize matrix equations and �nd eigen solutions and classical

dispersion relations from classical mechanics11. If not, one can easily check by substitution

11Coupled di�erential equation we want to solve are

dpl

dt= mω2(ql+1 − ql)−mω2(ql − ql−1)

which can be written asq1q2...ql...

=

1 2 ... l − 1 l l + 1 ...

1 −2mω2 mω2 0 0 0 0 02 −mω2 −2mω2 mω2 0 0 0 0... ... ... ... ... ... ... ...l 0 0 0 −mω2 −2mω2 mω2 0... ... ... ... ... ... ... ...

q1q2...ql...

Or in matrix notations

¨q = Mq

Let, fl be the eigen vectors and λlbe the corresponding eigen values of matrix M

Mfl = λlfl

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that following are indeed the linearly independent solutions for the simultaneous equations of

motion. For a most general pattern of time evolution one simply takes a linear superposition

over all eigen functions. For unbounded medium the Cartesian displacement (�eld amplitude)

and momentum of any atomic site written in terms of classical normal modes amplitudes

are:-

ql(t) =∑k

Ak cos(kla− ωkt− φk)

and

pl(t) =∑k

mωkAk sin(kla− ωkt− φk)

By substituting such a plane wave solution back into equations of motion one can �nd the

phonon dispersion relation12. We would again �nd it very useful to write these two

Then one can perform a basis transformation so that the matrix M becomes diagonal. Instead of specifyingsystem by specifying vibrational displacement of every lattice site (i.e. ql � hence, basis being (1, 0, 0, ...),(0, 1, 0, ...) etc.), we can specify the amplitude of another set of orthonormal vectors. From elementary linearalgebra we know that if we choose the basis set as the set of eigen vectors fl (column vectors) then aboveequation would become diagonal. So let a new matrix

S = [f1 f2 f3 ... ...]

where we have placed vectors fl along various columns of matrix S. From elementary linear algebra we knowthat this matrix can then be used for basis transformation. Further let

S−1q = α

S−1 ¨q = ¨α

Note that the matrix M and S are time independent. Then above set of coupled di�erential equations canbe written as

¨q = MSα

¨α = S−1MSα

Now we know that

S−1MS =

λ1

λ2

λ3

...

Hence

αl = λlαl

For real valued solutions we get

ql =∑

j

SljAj cos(λlt)

with real valued coe�cients and eigen values.12See Eq 31 in subsection 5.1.

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33




variables as as real and imaginary part of some complex variable. We would again break the

solution into �positive energy part� and �negative energy part�.

ql(t) =1√N

∑k

Ck exp(ikla) exp(−iωkt) + C∗k exp(−ikla) exp(iωkt)

and

pl(t) =1√N

∑k

−imωkCk exp(ikla) exp(−iωkt) + imωkC∗k exp(−ikla) exp(iωkt)

1√N

is normalization for exponentials. For periodic boundary conditions one can see that

k = 2πnNa

where n would be an integer between −N/2 and N/2. Note that we need two

terms in the above equation to make the displacements real. Ck and C∗k contains the phase

and the amplitudes information of the normal modes of oscillations. If we want we can

relate them explicitly to the amplitude and phase as Ck = Ak

2exp(iφk). Usually one de�nes

Ck exp(−iωkt) ≡ Bk and C∗k exp(iωkt) ≡ B∗k. Hence, time, phase and mode oscillation

amplitude information is buried into the �eld co-ordinates de�ned in this way

whereas spatial dependence is taken out. And hence

ql(t) =1√N

∑k

Bk exp(ikla) +B∗k exp(−ikla)

and

pl(t) =1√N

∑k

−imωkBk exp(ikla) + imωkB∗k exp(−ikla)

By direct substitution of above into the classical Hamiltonian and then using the orthonormal

conditions and dispersion relations to enforce relationship between ω and k, after simple but

tedious algebraic manipulations one can show that classical Hamiltonian can be written as

H =∑k

mω2k{B∗kBk +BkB

∗k} (12)

Suppose we want to write this as (it makes things much more handy)

H =∑k

~ωk2{b∗kbk + bkb

∗k} (13)

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This can be done by de�ning normalized �eld variables as

Bk =

√~

2mωkbk

Remember that spatial parts are orthonormalized and �eld variables (positive

energy amplitude and negative energy amplitude) include initial phase, time

evolution and mode oscillation amplitude information and the amplitudes are

further made dimensionless through prefactors. So we can write

ql(t) =1√N

∑k

√~

2mωk{bk exp(ikla) + b∗k exp(−ikla)} (14)

and

pl(t) =1√N

∑k

√~

2mωk{−imωkbk exp(ikla) + imωkb

∗k exp(−ikla)} (15)

and inverse of these can be written as

bk =∑l

[(

√mωk2~N

)pl + i(

√1

2~mωkN)ql] exp(−ikla) (16)

and

b?k =∑l

[(

√mωk2~N

)pl − i(√

1

2~mωkN)ql] exp(ikla) (17)

When we do our standard quantization (I want stress here that we are not doing

any new postulation) by converting the time dependent conjugate variables ql(t) and pl(t)

into time independent operators ql and pl, respectively, (in Schrodinger's picture) and by

enforcing the three commutation relations between ql and pl mentioned in the previous

section, we see that simply from the co-ordinate transformation equations 16 and 17 the

the classical time dependent normalized �eld oscillation amplitudes (negative and positive

energy amplitudes) that is Bk and B∗k become time independent �eld operators bk and b

†k in

Schrodinger's picture and time dependent operator in Heisenberg picture

bk =∑l

[(

√mωk2~N

)pl + i(

√1

2~mωkN)ql] exp(−ikla) (18)

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4.3 Quantization in k - Space

and

bk†

=∑l

[(

√mωk2~N

)pl − i(√

1

2~mωkN)ql] exp(ikla) (19)

One can explicitly check that these follow the boson commutation relation

[bn, b†k] = δn,k

[bn†, b†k] = 0

[bn, bk] = 0

These are called �equal time commutation relation (ETCR)� (this is because when

working in Heisenberg representation value of time variable needs to be kept same in these

relations).

The quantum mechanical Hamiltonian can simply be obtained from 13 by promoting

�eld amplitudes to be operators

H =∑k

~ω{b†kbk +1

2} (20)


In case of some other �elds (for example in the case of EM �elds as we would see latter)

the choice of conjugate variables in real space is not very obvious. In that case one usually

prefers to do quantization in k-space using generalized conjugate variables. We would explain

how this is done by quantizing lattice vibrations in k-space. Let us �rst try to identify the

generalized conjugate variables.

Suppose I want to write the classical Hamiltonian 12 which is repeated below for conve-

nience

H =∑k

mω2k{B∗kBk +BkB

∗k}

as

H =∑k

~ωk2

(b?kbk + bkb?k)

This can easily be done by de�ning normalized variable

Bk =

√~

2mωkbk

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Further, I want to de�ne real valued functions of time which would be proportional to real

and imaginary parts of bk which we soon identify as k-space generalize conjugate variables.

Hence, Let us write

Qk = α(bk + b?k)

Pk = β(bk − b?k)

Which can be inverted as

bk =1

2(Qk

α+Pkβ

) (21)

b?k =1

2(Qk

α− Pk

β) (22)

Hence classical Hamiltonian can be written as

H =∑k

~ωk4

(Q2k

α2− P 2

k

β2)

Now remember that in last section we had identi�ed complex valued positive energy oscil-

lation amplitude BkasAk

2exp(−iωkt + iφk) so that it include information but real valued

amplitude Ak, initial phase φk and time evolution. Hence we can quickly write

Qk = α

√2mωk

~Ak cos(ωkt− φk)

and

Pk = −iβ√

2mωk~

Ak sin(ωkt− φk)

Note that Pk and Qk are not the real space momenta and co-ordinates. Rather

they are real and imaginary part of complex valued mode oscillation amplitude.

Now I want to claim that variables Qk and Pk can be treated as generalized conjugate

variables. This can easily be ensured by checking if these variables satisfy the Hamilton's

equations of motion or not. We can adjust the values of αand β to make sure they do.

Hamilton's equations require that

− ∂H∂Qk

= −~ωk2α2

Qk = Pk =−Qkβωki

α

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and∂H

∂Pk= −~ωk

2β2Pk = Qk = −Pkαωk

−iβHence we get two conditions

~2α

= iβ

~2β

= iα

Both of which are equivalent to

2αβ = −i~ (23)

This condition can easily be satis�ed and hence we can treat Qk and Pk as generalized con-

jugate variables. Now we can do Dirac quantization in these generalized k-space variables

by enforcing standard

[Qk, Pl] = i~δk,l

[Qk, Ql] = 0

[Pk, Pl] = 0

One should notice how one quantization rule automatically and logically gets replaced by

three in coupled oscillator case � simply because now we have a multi-mode problem at

hand. Again note that these are NOT local Cartesian co-ordinates and momenta. Such

a quantization is scheme is usually known as �quantization in k space� or simply the

normal-mode-quantization.

Through expressions 21 and 22 we can see that as a result of quantization the time

dependent classical mode amplitudes bk and b?k get converted to time independent operator

bkand bk, in Schrodinger's picture. Using 21 and 22 and the enforced commutation between

generalized co-ordinates one can easily check that

[bk, b†k] = − [Qk, Pk]

2αβ= − i~

2αβ

If we want to make sure that [bk, b†k] = 1 then we need to enforce

2αβ = −i~ (24)

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It turns out that both 23 and 24 are same. Hence, we can choose one of the two parameters

α and β arbitrarily and everything would still be self consistent. Typically one chooses to

additional condition from dimensional considerations. One may want to make sure that

dimension of Pk is same same dimension Qk times that of mωk (hence, β/α ∝ mω13). Let

us choose (so that expression matches with single oscillator case)

β = −i√mωk~

2(25)

and hence

α =

√~

2mωk(26)

For future convenience, let us write down complete transformations

bk =

√2mωk

~(Qk

2+ i

Pk2mω

) (27)

b?k =

√2mωk

~(Qk

2− i Pk

2mω) (28)

and inverse transformations would be

Qk =

√~

2mωk(bk + b?k) (29)

Pk = −i√mωk~

2(bk − b?k) (30)

So it turns out that even if we do quantization in k-space these operators bkand b†kstill obey the same ETCR boson commutation relations that we obtain in last section.

Hamiltonian also can be written by same expression in terms of these operators.

And hence one can see that real and imaginary parts of the complex-valued pos-

itive energy �eld amplitude of each classical normal mode obeys a single decou-

pled harmonic oscillator Schrodinger's equation. So in these transformed co-ordinates

the ground state would be simply be the properly symmetrized direct product of various

kets representing the ground states on various classical normal modes. The ket corre-

sponding to the ground state of any one of the classical normal mode would be

Gaussian function in �eld amplitudes |Ψk(Q′k)〉 = 1

π4 exp(−Q′2k /2). Note that this is

a wavefunction associated with a �eld mode. The next higher states for any of the

13Remember that we are working with generalized co-ordinates, so this is not necessary as such.

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39




classical normal modes can be build up using raising and lowering operators and would turn

out to be Hermite polynomials. Any general state would be properly symmetrized direct

product of such Hermite polynomials attached with all the classical normal modes. The �eld

operators operate on such a state.

The entire quantum theory of single harmonic oscillator works now for each of the normal

modes. bk†and bk can easily be shown to be creation and annihilation operators respectively

just as we did for a single harmonic oscillator in previous section.

4.3.1 Mnemonics

Here are some memory tricks to quickly write quantized expressions for much more general

problem

• Hamiltonian would always be written as H = ~ω(b†b+ 12)

• Conjugate operators q and p are in general de�ned as q = α(b+ b†) and p = β(b− b†),with [b, b†] = 1 (opposite to normal order)

• Coe�cients α and β are determined by enforcing the commutation [q, p] = i~ and the

Hamilton's equation (in classical analogues ∂H∂p

= q and ∂H∂q

= −p)

� First condition would always lead to a constraint 2αβ = −i~

� Second condition is where quantization becomes problem dependent

∗ We would �rst expand real valued dynamic variables as superposition of all

complex valued solutions

∗ Time dependent coe�cients of expansions (including everything except space

dependence, if any) would be written as B and B?

∗ In order to write Hamiltonian as above, we would de�ne new normalized

variables b and b?

∗ Then de�ne q and p as indicated above, use 2αβ = −i~

∗ Write Hamiltonian in terms of q and p

∗ Find time dependence of q and p using time dependence of b and b?

∗ Determine second coe�cient by making sure that classical Hamilton's equa-

tions are satis�ed

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4.4 Alternative Interpretation � Phonons

4.4 Alternative Interpretation � Phonons

One should now be able to see how QFT language starts making more and more sense. There

are many normal modes. And each mode can go in many di�erent quantum states. We call

that the energy of each of normal �eld modes can be increased in quanta named as phonons.

Each quantum state of any normal mode is represented by a �eld-wavefunction which in

this case are simply the Hermite polynomials just like for a single harmonic oscillator. These

�eld-wavefunctions are functions of �eld oscillation amplitude and represents the uncertainty

in �eld oscillation amplitudes.

One should also notice that QFT (and in this case even standard QM) predicts completely

non-classical states of coherent elastic waves. Closest quantum mechanics can go to coherent

classical elastic wave is only when the �eld oscillation amplitude uncertainty is small and

remains constant as system evolves in time. This would happen only if we put each normal

mode in a quantum state which is linear superposition of states with di�erent number of

quanta in one normal mode such that the expansion coe�cients are Poisonian distributed.

Any other distribution represents a highly non-classical wave.

Second important point is about co-ordinate operators. Note speci�cally that ql does

NOT represent the Cartesian co-ordinate operator of a phonon. That probably

would be represented by some hypothetical operator corresponding to index

number l! Such operator does not exist in QFT! So, I strongly recommend that one

should try to come out of this way of looking at things. One should forget that Cartesian

co-ordinates of QFT particles (that is something equivalent to site index l) can

be operators. In QFT Cartesian co-ordinates of elementary excitations (QFT

particles) are treated as parameters just like time. In QFT its not conventional

to ask a question like � �what is the Cartesian co-ordinate operator for a phonon?�

� since QFT is not written in this language. In fact this is completely in-sync with

relativity as it is required that both time and co-ordinate needs to be treated on equal

footings.

Another important point one should notice is that concept of �eld quantization or concept

of �quasi-particles� can not be generalized to any random problem. Looking at single

harmonic oscillator problem, one might get an impression (this would become more clear

when we quantize Schrodinger's �eld in any random quantum well, say a non-parabolic one)

that one can simply calculate the energy eigen states using Schrodinger's equation and then

start �lling it up one by one and hence once can de�ne creation and annihilation operators.

Remember that entire QFT stands on the concept of identical particles. For example if

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5: More About Phonons

our simple harmonic oscillator had a non-linear restoring force then the energy eigen states

would not be equally spaced and one would not be able to set up a QFT in such a simple

form (in terms of identical phonons) � one would need to introduce the concept of phonon-

phonon interaction and even the concept of phonon would be practically useful only when

phonon-phonon interaction is not two big (so that a concept of quasi-particle has some sort

of experimental validity).

5 More About Phonons

5.1 Energy Continuity Equation

As an interesting example, let us try to write the continuity equation for the �ow of energy

as lattice vibration moves. We would work out details in classical domain as extension into

quantum domain is very straight forward does not provide insight into concept of group

velocity. ∫S

~j(~r, t)d~s = −∂u(t)

∂t

Where u(t) is the total energy inside volume enclosed V by the surface S. Noticing that

Bk =Ak2

exp(iφk) exp(−iωkt)

one can immediately see that the total energy inside in�nitely large volume is constant with

time as expected∂

∂tu(t) =

∂

∂t

∑k

2mω2kB∗kBk = 0

Hence, our major headache is to calculate energy inside �nite volume.

Let us consider a volume that has n unit cells and n atoms starting from index l = α

to index l = α + n. Potential energy between �rst atom and the preceding atom would be

shared half-half and similarly the potential energy between last atom and the following atom

would be shared half-half14. Hence the kinetic energy would be

uKE(t) =n∑l=α

p2l

2m

14Half-half splitting might seem arbitrary at �rst. One would expect that if we add up potential energyof neighboring sub unit we should get the total energy of super-unit. One can check that this additivity isensured only if energy is split half-half. Check L. M. Magid, Phys. Rev. 134, A158 (1964) for details.

Mukul Agrawal



42




and the potential energy can be written as

uPE(t) =1

4K(qα − qα−1)2 +

1

4K(qα+n+1 − qα+n)2 +

1

2K

α+n−1∑l=α

(ql+1 − ql)2

Hence∂uKE(t)

∂t=

n∑l=α

plplm

=α+n∑l=α

K

m{pl(ql+1 − ql)− pl(ql − ql−1)}

and

∂uPE(t)

∂t=

K

2m(qα − qα−1)(pα − pα−1)

+K

2m(qα+n−1 − qα+n)(pα+n−1 − pα+n)

+K

m

α+n−1∑l=α

(ql+1 − ql)(pl+1 − pl)

Hence,

∂u(t)

∂t=K

mpα+n(qα+n+1 − qα+n)− K

mpα(qα − qα−1)

+K

2m(qα − qα−1)(pα − pα−1)

+K

2m(qα+n−1 − qα+n)(pα+n−1 − pα+n)

A simple solution can be obtained for ~j if we send n to ∞. Doing so one gets a simple

expression∂u(t)

∂t= − K

2m(qα − qα−1)pα−1

It is interesting to note that only boundary atoms play role in energy �owing into the

volume, as expected. When all interacting atoms are inside the volume then they do not

contribute to the energy �ow into the volume. Now, we can easily check that

〈qαpβ〉t = 0 ∀α = β

〈qαpβ〉t = −∑k

mωkA2k

2Nsin(ka) ∀β = α− 1

Mukul Agrawal



43




Hence

〈j〉t = −〈∂u(t)

∂t〉t =

K

4N

∑k

ωkA2k sin(ka)

〈j〉t =K

2mN

∑k

〈uk〉sin(ka)

ωk

At this point, let us �rst �nd out the dispersion relation, by substituting the plane wave

expansion solutions back into the equation of motions∑k

−mω2kAk cos(kla− ωkt− φk) =

∑k

{2KAk cos(kla− ωkt− φk) cos(ka)

− 2KAk cos(kla− ωkt− φk)}

−mω2k + 2K − 2K cos(ka) = 0

ωk = 2

√K

msin(ka/2) (31)

Hence, group velocity can be written as

vg(k) =

√K

ma cos(ka/2)

Hence one can write

〈j〉t =

√K

m

1

4N

∑k

〈uk〉tsin(ka)

sin(ka/2)

〈j〉t =1

2N

√K

m

∑k

〈uk〉t cos(ka/2)

〈j〉t =1

2Na

∑k

〈uk〉tvg(k)

A factor of 2 seems to be mistaken. This is a very general result with much wider validity15.

It shows that each mode carries energy with group velocity not the phase velocity.

15See L. M. Magid, Phys. Rev. 134, A158 (1964) for generalization for 3D lattices.

Mukul Agrawal



44



5.2 Dispersion and Causality for Phonons


We would discuss phonons in just a minute. Let us �rst take a small diversion and look

into dispersion in electromagnetism with which we are more familiar. We have seen that,

in classical electromagnetism of homogeneous materials, dispersion in ω-k space (other than

linear dispersion) always come with material absorption. This is related to causality. In

classical electromagnetism of homogeneous materials, dispersion in ω-k space originates due

to frequency dependent susceptibility χ = χ(ω) (and hence frequency dependent refractive

index). Frequency dependent susceptibility implies that, in time domain, electric displace-

ment (or electric polarization) and electric �eld do not have a temporally local relation.

From fundamental point of view this happens because electromagnetic �elds are interacting

with massive particles which can not respond at speed of light. Let us �rst discuss a simple

case. For example assume a spatially-uniform time-varying16 electric �eld applied across a

homogeneous material medium. Let us make the time variation really simple to begin with.

A uniform �eld switches on instantaneously across entire sample at t = 0 and maintains a

constant value after that. Electric �eld applies an instantaneous force on the electron cloud

of the atoms and start polarizing them. Since electrons are massive particles they can not

achieve steady state (static) displacement (i.e. polarization) instantaneously. They would

obey Newtonian equation of motion17 and would take �nite time to reach the steady state or

static value18. This tells us that there is no temporally-local correlation between electric �eld

16Here we are ignoring relativistic e�ect for keeping the discussion simple without missing the essentialphysics we want to discuss.

17Or the Schrodinger's equation of motion. Essential physics that we want to discuss remains the same.18Electron cloud might even show some oscillations. If system is damped then oscillation would disappear.

If system is not damped then oscillations would continue. No matter whether we assume damping or not,result that we want to discuss remains the same. That is the polarization at any instance of time dependson the values of electric �eld at previous instance of times as well.To be realistic and for sensible visualization of actual physical events happening, we should include some

amount of damping in the system. If we do not include damping then the response to step-function excitationwould be sinusoidal oscillations and system would never come to equilibrium which is very unrealistic.Similarly for sinusoidal excitation of an undamped system, amplitude of oscillation would shoot to in�nityif frequency of oscillation is equal to natural frequency of excitation. We would actually prove that sinceevery realistic material system is dispersive, it has to have damping as well. Note that this proof does notnecessarily assume existence of damping. All it assumes is that �elds are interacting with massive particlesthat take same time to respond. Damping is introduced only for sensible visualization of what actuallyhappens in reality.It is sometimes instructive to model electron clouds as simple harmonic oscillators. Note that a simple

harmonic oscillator violates causality if damping is not included. Think of a simple system in which twomasses are connected by a spring of �nite length. If we instantaneously move one mass, that second massfeels a modi�ed force instantaneously. Since length of spring is �nite, this implies that information has�own at in�nite speed. Every realistic simple harmonic oscillator would show broadening in the resonance

Mukul Agrawal



45




and polarization. Polarization at �xed space-point at any instance of time would depend on

the electric �eld at all previous instances of time at same spatial location. We can extend this

argument with sinusoidal time variation of spatially uniform electric �eld. In steady state

electron clouds would oscillate in-sync with the external electric �eld. But its amplitude

of oscillation would depend on the external frequency. If the frequency is close to natural

frequency of oscillation of electron cloud then amplitude of oscillations would be very large.

Otherwise they oscillate with very small amplitude19. So this justify our previous claim that

frequency dependent material parameters are actually related to the fact the electromagnetic

�elds are interacting with massive particles and hence time correlation between the two are

not temporally local.

In previous section 5.1 we obtained a dispersion relation for phonons in 1D chain of linear

harmonic oscillators. Does this dispersion imply looses? Does causality play a role here?

Does causality dictates that dispersion would imply losses? Answer is no. This dispersion

is equivalent to the dispersion seen in loss-less photonic crystals for example. This is struc-

ture dispersion and not the dispersion originating from underlying interaction. Free �eld of

phonons does not have any loses. Loses come when one type of �eld interacts with another

type of �eld. Note that in the derivation of dispersion relation for phonons we assumed a

force relation of the type F = kx where k is time-independent spring constant. This means

that stress and strain at the same spatial location have temporally local correlation. This is

approximately correct in most materials but may not be fundamentally correct physics. For

example visco-elastic materials do show time dependent relationship between stress and

strain. In actual material system restoring force originates from electromagnetic interaction

between atoms. If we instantaneously displace one atom, how much restoring force does the

neighboring atom apply? It depends on what kind of interaction we are taking about. If

we are looking at ionic crystals, it might be almost instantaneous application of Coulombic

restoring force. But what if restoring force is mediated through electron clouds. In that case

restoring force would take some time to reach a steady state value. Hence spring constant

frequency and this would in turn be related to losses though causality relations.19Even for an unrealistic undamped oscillator, amplitude is not zero o�-resonance. Although relative

amplitude (compared to the amplitude at resonance) becomes zero. Amplitude of oscillation for an oscillatorthat follows classical equation

mx+ bx+ kx = F exp(jωt)

is

A =F

k −mω2 + iωb=

F/m

ω20 − ω2 + iωb/m

Mukul Agrawal



46



6: Continuum - Quantization of Elastic Waves in Continuous Media

would be a function of time or in other words would be frequency dependent. Hence force

equation needs to be modi�ed like x(t) =∫∞

0g(τ)F (t − τ)dτ where g = 1/k. Such a fre-

quency dependent spring constant would change the dispersion relation and this additional

dispersion can then be related to losses in material through causality type relations.

6 Continuum - Quantization of Elastic Waves in Contin-

uous Media

Suppose the atomic sites come in�nitesimally close and become continuum. Results have in-

tuitive extensions. Probability amplitudes and probabilities both become density functions.

Equation of motion simply use Lagrangian and Hamiltonian density functions. So basically,

instead of having one classical Hamilton equation or one Heisenberg equation

for each spatial co-ordinate we would have operators which are functions of spa-

tial co-ordinates. Similarly all other operators would actually become operator

density. Now again its much more easier to solve the problem by co-ordinate

transformation so that equation at one generalized co-ordinate is independent of

equations at other points. Even the eigen mode/normal modes would be in�nite

and in�nitesimally closely spaced and hence �eld operators would also be oper-

ator densities. Secondly the commutators would become double integrals and Kronecker

deltas would become Dirac-delta functions.

One important point change is that the dynamical variables like local Lagrangian density

L would now be the function of local co-ordinates, time-derivatives of local co-ordinates,

time, as well as local space-derivative of all orders of local co-ordinates. This is needed

to develop the notion of local Lagrange densities and local �eld theories. Note that the

Lagrange's equation then becomes :-

∂

∂t(∂L

∂Ψi

) =∂L

∂Ψi

−∑x,y,z

∂

∂x(∂L

∂ ∂Ψi

∂x

)

and one of the two Hamilton equations (for each �eld) also get modi�ed according to

∂Πi

∂t= − ∂H

∂Ψi

+∑x,y,z

∂

∂x(∂H

∂ ∂Ψi

∂x

)

Everything else follows exactly as above.

Mukul Agrawal



47



Part IV

QFT As a Convenient Language for

Known Physics

7 Introduction

In this section I would only talk about massive non-relativistic particles (like electrons in

crystals). We start with all knowledge about particles physics. For example, in the case

of electrons, we are assuming that we know apriori that they are identical fermions with

spin of 1/2. We also know that they are charged particles with charge e and that they

obey Schrodinger's equation. Once this much of information is known we can actually build

entire subject of QFT without any assumptions. Actually it turns out that QFT now merely

serves a purpose of convenient language. Fundamental utility of QFT is to �prove� (or at

least rationalize) these properties of particles that we have assumed are known apriori. Also

QFT helps combining relativity with quantum mechanics. But for non-relativistic physics

of particles whose basic physics is assumed to be known, QFT is just a new language. This

is the point that I want to expand on in this section.

Even though, in this section, I would only talk about massive non-relativistic particles,

concepts discussed here can easily be extended to relativistic particles as well as to massless

particles like photons (obviously, if we assume that particle characteristics are known apriori).

Only modi�cation one would have to do with respect is to include both positive energy

and negative energy parts (Hermitian conjugates) in various expressions. In non-relativistic

physics classical equation of motion is only �rst order (which is related to the fact that

there are no anti-particles) hence we do not need to include these negative energy terms (or

Hermitian conjugate terms). These would get clear as we discuss this further.

Mukul Agrawal



48



8: Identical Particle Hilbert Space Representations

8 Identical Particle Hilbert Space Representations

Non-relativistic QFT of identical massive particles can be �completely�20 understood if we try

to understand the properties and representation of identical particle Hilbert spaces without

any further postulates. Relativistic case gives some troubles because a particle can not

exist without it's anti-particle21. Moreover ground state consists of in�nite many of anti-

particles. Once these things are understood, it's actually possible to build the Hilbert space

representations in relativistic case and to understand the relativistic QFT �completely� from

there on. One should keep in mind that this is no magic! QFT is in fact developed to

understand these complications with anti-particles, ground states, spin statistics theorem etc.

Once you take these as postulates, obviously everything else is just symbols and notations

and no new physics. So its obvious that with these things taken as proved one can understand

the structure of QFT simply by understanding the structure of Hilbert space.

With this motivation, lets explore the identical particle state and operator representations

in some depth. We would only look into non-relativistic case. Relativistic case would be

discussed latter.

8.1 Fock Space and Fock States

8.1.1 Number State Representation

In basic quantum mechanics, multi-particle Hilbert space is postulated to be the direct

product of single particle Hilbert Spaces. Further, the fact, that identical multi-particle

state space is either a Bosonic subspace (for integer spin particles) or Fermionic subspace

(for half-integer spin particles) of the entire direct product of all single particle spaces, is

treated as one of the basic postulates of non-relativistic quantum theory (see the article on

Quantum Mechanics).

The multi-particle state space can be generated/spanned from in�nitely many choices of

basis sets. One of the obvious choice is the direct product of the basis states that gener-

ate/span single particle state spaces for each of these particles. Most common one is the set

of direct products of single particle energy eigen states. Please note that this choice is

20�Completely� does not mean that we can understand the spin statistics theorem � why half integer spinparticles are Fermions and integer spin particles are Bosons needs to be taken as a postulate in the argumentsin this section.

21This is related to the fact that relativistic quantum mechanical single particle equations involve secondorder time derivatives unlike single particle Schrodinger's equation.

Mukul Agrawal



49





perfectly valid irrespective of whether the particles interact with each other or

not. If they do interact then, for example, the direct product of energy eigen

states of single particles would not be an energy eigen state of the multi-particle

system. But it would still be a valid candidate as a basis state. This postulation

of multi-particle state space as direct product space would directly imply the additivity of

linear momentum, for example. And additivity of energy in case of non-interacting particles.

In case of identical particles, a somewhat simpler way of choosing the basis state might

be to choose occupation number (number of particles in single particle energy states or

maybe any other single particle basis state) as a quantum index. We call it occupation

number state representation. These multi-particle basis states are simply the states

with de�nite number of particles in any of the single particle basis state. We

would show in a minute that this latter scheme works only for identical particles � both

for Bosonic or Fermionic identical particles. These basis states are known as Fock states.

Any general identical multi-particle state can then be written as linear superposition of Fock

states. Fock states are the so called natural basis of Hilbert space of identical particles.

The Hilbert space of identical particles (technically unknown number of them) that is the

symmetric or anti-symmetric subspace of the entire direct product Hilbert space is known

as Fock space.

For the time being assume that particles are not forced to be bosons or fermions. We

would show that this scheme does not work unless particles are fermions or bosons or unless

particles are forced to be in some other kind of symmetric subspace of direct-product space.

Let us choose the single particle energy eigen states as the basis of single particle Hilbert

space. We can decide to symbolically write a generic single particle state

|Ψ〉 =∑m

cm|ΨEm〉

as ∑m

cm|01, 02, ...., 1m, 0m+1, ....0∞〉

Similarly, we might try to write a generic two particle state (which is not required to be

symmetric or anti-symmetric state)

|Ψ〉 =∑i

∑j

cij(|ΨEi〉 ∗ |ΨEj

〉)

Mukul Agrawal



50




as ∑i

∑j

ci,j|01, 02, ....1i, 0i+1, .....1j, 0j+1, ......0∞〉

Here symbol ∗ stands for direct product22. Similarly we might develop notations for many-

particle states. If identical particles are not forced to be either bosons or fermions then what

we are claiming here is that symbolically

|01, 02, ....1i, 0i+1, .....1j, 0j+1, ......0∞〉 ≡ |ΨEi(r1)〉 ∗ |ΨEj

(r2)〉

where r1 is a variable that keeps track of the co-ordinate of the distinguishable particle

labeled as '1' and r2 keeps track of the co-ordinate of another distinguishable particle labeled

as '2'. Note that the number state representation is not complete and is confusing when

particles are not forced to be either boson or fermion. For example how would you distinguish

between|ΨEi(r1)〉 ∗ |ΨEj

(r2)〉 and |ΨEi(r2)〉 ∗ |ΨEj

(r1)〉? One might try to introduce another

labeling scheme to keep track of particle labels. For example one might be inclined to try

|01, 02, ....11i , 0i+1, .....1

2j , 0j+1, ......0∞〉 for above mentioned state. But, as we are going to see,

even this can not be done. For example a valid two particle state

(c1|ΨE1(r1)〉+ c2|ΨE2(r1)〉) ∗ (c3|ΨE1(r2)〉+ c4|ΨE2(r2)〉)

can not be written in terms of number states unless coe�cients are inter-related. Here

particle '1' is in some superposition and particle '2' is also in some other superposition.

Since in both Fermionic and Bosonic sub-spaces of multi-particle Hilbert space such a co-

relation between coe�cients exists, we can use such a representation for identical particle

systems. I hope that by this time reader is convinced that Fock states can only be used as

basis for identical particles.

There are several of visualization bene�ts with number state representation

:-

• Firstly, common processes that we want to study (like optical absorption emission etc)

are non-particle-number-conserving. Number state representation handles this pretty

nicely as we would see below.

• When we would write down the operators in this representation, we would see that

22When it is not imperative to stress the special presence of a direct product we sometime also use thenotation |ΨEi〉|ΨEj 〉 to represent direct product.

Mukul Agrawal



51




it hides a lot of details about the actual force laws that particles are obeying. For

example, irrespective of whether a charged fermion is in electrostatic potential or a

fermion with permanent magnetic dipole is in magnetic �eld, the operators and their

operations on states look exactly the same. The actual details are dumped into the

single particle states over which number state representation is built. So even if you

don't know the exact states you can build an overall theoretical structure.

8.1.2 Basis Conversion

Suppose number state representation of certain generic identical multi-particle state is given.

So the state is given as a linear superposition of Fock states. Fock states are built on top

of certain basis of single particle Hilbert space. For example we might have chosen single

particle energy eigen states as the basis for the single particle Hilbert space and build our

number state representation on top of it. What if I want to change this underlying basis

of the single particle Hilbert space? This would also change the Fock states and hence

the number state representation of the given multi-particle state. We want to �gure out a

formal way of doing this basis change. After a bit of formalism, it would turn out to be

very simple and very useful. We would see that concept of �eld operators (so called

wavefunction operators) fall o� very naturally if try to under this concept of

basis conversion. Similarly we would also learn how any multi-particle operator written in

r-representation can be written in terms of �eld operators (creation annihilation operators

for example).

If the particles are fermions then two identical particles can only occupy an anti-symmetric

subspace of the entire direct product of single particle Hilbert spaces. In this two particle

subspace, any Fock state (that is fermion basis state) can be written a so called Slater

determinant with multiplication of state vectors being interpreted as a direct

product of state vectors. Hence any generic state 2-fermion state can simply be written

as :- ∑i

∑j

cij

∣∣∣∣∣ |1,ΨEi〉 |2,ΨEi

〉|1,ΨEj

〉 |2,ΨEj〉

∣∣∣∣∣ (32)

where |1,ΨEi〉 means particle 1 is in single particle state |ΨEi

〉. And hence a set of following

states

|01, 02, ....1i, 0i+1, .....1j, 0j+1, ......0∞〉 ≡

∣∣∣∣∣ |1,ΨEi〉 |2,ΨEi

〉|1,ΨEj

〉 |2,ΨEj〉

∣∣∣∣∣Mukul Agrawal



52




can be taken as the two particle Fermionic basis states. Statement can easily be generalized

for multiple particles.

Similarly a generic two particle Boson state would be written as

∑i

∑j

cij1√2

∑P

P [|1,ΨEi〉|2,ΨEj

〉] (33)

where P is a linear operator that generates√

2! permutations by exchanging particles among

�xed single particle states. i and j can take all values from 1 to N where N is the number

of single particle states available. Note that√

2! is correct only when i and j are di�erent.

One can easily �gure out a normalization constant when i = j. And hence

|01, 02, ....1i, 0i+1, .....1j, 0j+1, ......0N〉 =1√2

∑P

P [|1,ΨEi〉|2,ΨEj

〉]

are the two particle Bosonic basis states. Statement can easily be generalized for multiple

particles.

8.1.2.1 Co-ordinate Representation to Number State Representation Our �rst

aim is to obtain a number state representation if a completely antisymmetric or symmetric

|r〉 representation of a multi-particle state is given. Suppose φi's are the wavefunctions of thesingle particle basis states. Let us be speci�c and work out the details for two fermions case

�rst. Analogous results can be worked out for Bosons case. Discussion also generalizes to

multi-Boson and multi-Fermion cases. If the wavefunction of a two identical fermion particle

state is given as

Ψ = c1(φ1(r1)φ2(r2)− φ2(r1)φ1(r2)}+ c2(φ3(r1)φ4(r2)− φ4(r1)φ3(r2))

then one can see that this state can be equivalently represented as

c1|11, 12, ....0i, 0i+1, .....0j, 0j+1, ......0N〉+ c2|01, 02, 13, 14, ....0i, 0i+1, .....0j, 0j+1, ......0N〉

But how do we do it more formally so that this is possible for more di�cult cases which are

not that easy to see? The ket representation of the above state in |r1〉 ∗ |r2〉 ≡ |r1〉|r2〉 basis

Mukul Agrawal



53




would be

|Ψ〉 =∑r1,r2

(c1(φ1(r1)φ2(r2)− φ2(r1)φ1(r2)) + c2(φ3(r1)φ4(r2)− φ4(r1)φ3(r2)))|r1〉|r2〉

Even though this basis is capable of spanning entire direct product space, the coe�cients

are such that the particles stay only in Fermionic subspace. We now use the normal linear

algebra trick to change the basis. Following operator when operated over the wavefunction

of the state gives the ket representation of the state in new basis (change from |r1〉|r2〉 →|φm(r1)〉|φn(r2)〉 basis):-

∑m,n

|φm(r1)〉|φn(r2)〉∫dr1dr2{(φ∗m(r1)φ∗n(r2)...}

Note that above expression is an operator. The wavefunction of the state on which it operates

goes inside the integrand on the right hand side. This gives us the representation in terms

of some new basis states |φm(r1)〉|φn(r2)〉. But we actually want the representation in terms

of number states. This is straight forward now. Following is the required operator :-

∑m,n>m

|....0i, 0i+1, 1m...., 1n.....0N〉∫dr1dr2{(φ∗m(r1)φ∗n(r2)− φ∗n(r1)φ∗m(r2))...} (34)

Note that it works properly when you know apriori that particles are for sure Fermions

and wavefunction is properly antisymmetric. Otherwise we have seen that in the complete

direct-product space this number state representation does not make any sense and these

can not span the entire space. We can also write the same operator as :-

∑m,n>m

|....0i, 0i+1, 1m...., 1n.....0N〉∫dr1dr2{φ∗m(r1)φ∗n(r2)...} (35)

8.1.2.2 Number State Representation to Co-ordinate Representation Using

similar linear algebra concepts we can also do the inverse transformation. The inverse

transformation is obtained as :-∑m,n>m

(φm(r1)φn(r2)− φn(r1)φm(r2))〈....0i, 0i+1, 1m...., 1n.....0N | (36)

Mukul Agrawal



54



8.2 Operators in Fock Space

8.1.2.3 General Basis Transformation Using these one can easily do a general basis

transformations of underlying single particle basis set. Using 36 to go to r representation

then do usual basis conversion and then use 35 to get new number state representation.

8.2 Operators in Fock Space

Suppose we are given some operator in r representation A(r1, r2) that operates on two particle

Hilbert space. Let this operator correspond to some physically measurable property of two

identical particle system. Hence, as argued before this operator needs to be symmetrical

with respect to exchange of particles.

A(r1, r2) = A(r2, r1)

We want to know how this operator operates on a state given in number state representation.

E�ectively we want to �nd the equivalent representation of this operator in number state

representation. From above two transformation equations, we can also build a number

state representation of an operator A(r1, r2) given in |r1〉|r2〉 basis as∑m,n>m

|....1m...., 1n.....〉∫dr1dr2(φ∗m(r1)φ∗n(r2)− φ∗n(r1)φ∗m(r2))A(r1, r2)∑

i,j>i

(φi(r1)φj(r2)− φi(r1)φj(r2))〈...., 1i...., 1j.....|

Now we know that the operation of any symmetric linear operator A(r1, r2) on any antisym-

metric state leaves the �nal sate antisymmetric. So we can remove the second antisymmetric

parts from the above expression :-

∑m,n>m

|....1m...., 1n.....〉∫dr1dr2φ

∗m(r1)φ∗n(r2)A(r1, r2)

∑i,j>i

φi(r1)φj(r2)〈...., 1i...., 1j.....|

The same operator can also be written as∫dr1dr2(

1√2

)∑m,n

|....1m...., 1n.....〉φ∗m(r1)φ∗n(r2)A(r1, r2)(1√2

)∑i,j

φi(r1)φj(r2)〈...., 1i...., 1j.....|

(37)

in order to remove the double counting. Note that we do not need factor of 4 as someone

might think without looking too deeply. We have divided the factor of two into two parts

Mukul Agrawal



55



8.3 Ground State or Vacuum

just for symmetry purposes. We have also changed the order of integration and summation.

The process works (except factor of 2 needs to be replaced by factor of N !) for any number

of particles.

These result for are exactly same for both bosons and fermions. The di�erence stays

in the meaning of number states and latter we would bury the di�erence into cre-

ation and annihilation operators that would be used to generate number states.

8.3 Ground State or Vacuum

De�nition, identi�cation and proof of existence of ground state is actually a very tricky part

of the QFT. Things are somewhat simpler at least in free particle non relativistic theories.

We claim that there exist a unique and non-degenerate lowest energy state. We further claim

that this is the state in which there are no particles in the system. We would represent this

state as |0〉. As we build the theory more rigorously, we would discuss in details the issues

with the existence of a ground state in QFT.

8.4 Creation and Annihilation Operators

One can �rst intuitively de�ne creation a†j and annihilation aj operators for fermions giving

them the properties that their names suggest23. By explicitly operating24 these operators

on multiparticle basis states (Fock states), one can then easily prove the anticommutation

23These operators, by de�nition, would operate on any multiparticle basis state (that is Fock state) andthe resultant new state would be another Fock state (that is another basis state) that would either have oneadditional particle or one less particle multiplied by a coe�cient that can only be � +1, −1 or 0. 0 wouldresult when we try to annihilate a particle in an initially empty state. ±1 originates because of state orderingconvention. There are some book-keeping details involved in maintaining the order of states. One needs todecide a convention for ordering the states in the Slater determinant. For example one can decide that lowestenergy single-particle state |E1〉 would form the �rst column and �rst row and this would increase as wechange column and rows. One also needs a second convention regarding creation operators. We can decidethat when we create an operator the new row and new column would go at the end � that is the additionalrow would be last row and additional column would be the last column. Now to obtain the correct operationof a creation operator one needs to combine these two convention. Suppose creation operators creates afermion in states |E3〉. Suppose there was already a particle is |E2〉and another in |E4〉. One �rst adds a rowand a column at end and then �ip the rows and column until one gets the new state in proper ordering asdictated by the �rst convention. So one would get a ± as a result of the operation of the creation operatorand chosen conventions. Note that this arbitrary factor does not a�ect any physics because creation andannihilation operators are not Hermitian operators and they do not represent anything physical. In practiceannihilation and creation operators would always come in pair and this arbitrary factor would cancel o�.

24One does not need explicit construction of these operators. All we need is their de�ning character(annihilation/character) and the book-keeping convention discussed above.

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8.5 The Wavefunction Operators or The Field Operators

relations among them utilizing their pre-assumed Fermionic behavior

{a†j, a†k} = a†ja

†k + a†ka

†j = 0 (38)

{aj, ak} = ajak + akaj = 0

{a†j, ak} = a†jak + aka†j = δjk

Note that these anticommutation rules completely and uniquely de�ne the Fermionic

subspace of the direct product of single identical particle spaces. Hence one can

treat these as completely equivalent statement of anti-symmetrization postulate

of quantum mechanics.

Similarly for bosons, one can �rst intuitively de�ne creation b†j and annihilation bj op-

erators giving them the properties that their names suggest. One can then easily prove the

commutation relations among them utilizing their pre-assumed Bosonic behavior

[b†j, b†k] = b†jb

†k − b

†kb†j = 0 (39)

[bj, bk] = bjbk − bkbj = 0

[bk, b†j] = bkb

†j − b

†jbk = δjk

Note that these commutation rules completely and uniquely de�ne the Bosonic

subspace of the direct product of single identical particle spaces. Hence one can

treat these as completely equivalent statement of symmetrization postulate of

Quantum Mechanics.


As we would see below, the above derivation (8.2) is the logic behind the de�nition and

usage of something called �wavefunction operators� or the ��eld operator�. One should �rst

note that

|0, 0, 1m, 0, 1n, 0, 0〉〈0, 0, 1i, 0, 1j, 0, 0| = a†ma†naiaj

This is self obvious. Readers can convince themselves of the equivalence by just looking at the

operation of both sides on some state given in number state representation. Let us de�ne

so called two particle annihilation (Ψ) and creation (Ψ†) wavefunction operators

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respectively as :-

Ψ(r1, r2) ≡ (1√2

)∑i,j

aiajφi(r1)φj(r2) (40)

Ψ†(r1, r2) ≡ (1√2

)∑m,n

a†ma†nφ∗m(r1)φ∗n(r2) (41)

Exactly analogous de�nitions can be given for boson wavefunction operators by just replacing

fermion creation and annihilation operators inside summation by boson counterparts. These

de�nitions of wavefunction operators can easily be generalized for more numbers of parti-

cles. We simply include more number of operators and wavefunctions inside the summation,

increase the dimension of sums and replace 2 by N !.

Remember that these operators operate on ket/bra vectors and not on wavefunctions. As

we would see below, Ψ†(r1, r2) when operated on ground state |0〉would create two spatially-

well-localized particles � one in state |r1〉 and another on state |r2〉. Created states would

automatically have the required symmetry properties of Boson or Fermion states. Similarly

Ψ(r1, r2) when operated on any multi-particle state would annihilate two particles � one

from state |r1〉 and another from state |r2〉. If state is initially empty, result would be zero.

Similar statements can be made for wavefunction operator corresponding to larger number of

particles. Hence these wavefunction operators are simply basis converted versions

of annihilation and creation operators ai and a†i . While a†icreates particles in the

single particle states∑

j φi(rj)|rj〉, Ψ†(ri) create particles in the single particle state

|ri〉. It should be clear to the reader that |ri〉 are perfectly valid candidate for the basis of

single particle Hilbert space just like states represented by wavefunctions φi(r) are. Hence

one can build a number-state representation for the multi-identical-particle Hilbert space on

top of these single particle basis state |ri〉. When this is done, creation and annihilation

operators would turn out to be Ψ† and Ψ.

One more important point to be stressed is that many authors call (as do I, sometimes)

ai, a†i , Ψ(r) and Ψ†(r) as single particle operators because they annihilate or created single

particles. But remember that they do operate on multi-particle Hilbert spaces.

So in the strict sense of scienti�c terminology they are all multi-particle operators. Another

related and important point is that expressions like Ψ(r1)Ψ(r2) do NOT represent direct

product of operators. Instead they simply represent composition of multiplication of opera-

tors. In other words both operators operate in same multi-particle Hilbert space.

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8.6 Summary

Explicitly

Ψ(r1)Ψ(r2)(|r1〉 ∗ |r2〉 − |r2〉 ∗ |r1〉) 6= Ψ(r1)|r1〉 ∗ Ψ(r2)|r2〉 − Ψ(r1)|r2〉 ∗ Ψ(r2)|r1〉

where ∗ represents direct product.By exchanging the order of multiplication and summation, multi-particle wavefunction

operator can also be written as

Ψ(r1, r2, r3, ....) =1√N !

Ψ(r1)Ψ(r2)Ψ(r3)..... (42)

where

Ψ(rn) =∑i

aiφi(r1) (43)

Annihilation wavefunction operator is simply the Hermitian adjoint of this operator. With

this we can now write down the symmetric multi-particle operators (generalized version of

37) in number state representation for both fermions and bosons in much simpler fashion∫dr1dr2....Ψ

†(r1, r2, ...)A(r1, r2, ...)Ψ(r1, r2, ...) (44)

One should notice that we haven't postulated anything new. The de�nitions of these

wavefunction operators and their importance have just fallen out from the dif-

ferent way (number state representation) we started to represent the Fock space.

8.6 Summary

Lessons we want to learn from this whole exercise are:-

• Field quantization (concept of particles, creation/annihilation operators etc) can be un-

derstood without invoking any postulates provided we know before hand what kind of

particles (spin statistics, existence of anti-particles etc.) are we dealing with. When we

don't know this � we have to properly quantize a wave equation (second quantization).

• Process outlined here is correct for non-relativistic fermions and bosons that satisfy

Schrodinger's equation. It would not work for relativistic elementary particles (since

they coexist with anti-particles). But a similar derivation can be given once spin

statistics theorem as well co-existence of anti-particles are assumed. I would not try to

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give such a derivation. Reason is that the formal procedure of �eld quantization leads

to both the concept of co-existence of anti-particles as well as all the expressions for

wavefunction operators, creation/annihilation operators etc. So a similar derivation

would not provide any new insight.

• Process outlined here would not work even for non-elementary particles (elementary

excitation in condensed matter physics for example) like phonons. Even for these we

need to include the negative energy solutions as we would see latter (equivalent to

inclusion of anti-particles). Although phonons are non-elementary particles and they

do not have anti-particles in strict sense, we still need to include negative energy parts

in our expressions. Reason for this mathematical similarity is the real-valued character

of elastic �elds as opposite to Schrodinger �elds that can be complex valued.

• This exercise is useful in pointing out what is new physics and what is old physics in

QFT. We would see that all this language of creation/annihilation operators etc, which

although seems new, falls out naturally from all the physics that we already know. So

this part does not contain any new physics in it. What is new in QFT are things like

existence of anti-particles, spin-statistics theorem etc. These would fall of naturally

(well ... almost naturally25!) when we take Schrodinger equation or Dirac equation as

a �classical� equation and quantize it. This canonical quantization of �eld equations

(or so called second quantization) is the new physics. And this is what would lead us

to the concepts of anti-particles, spin-statistics etc.

Part V

Non-Relativistic, Massive, Quantum

Field Theories ( QFT )

QM of non-relativistic massive particles is given by Schrodinger's equation. QFT for non-

relativistic massive particles can be very easily understood if we simply try to represent states

25In strict mathematical sense, enormous amount of new mathematical machinery is required to justifyvarious manipulations � like analytic continuation, Euclidean QFT, Gel-Mann Low theorem etc. etc. Ibelieve these things can be rigorously proved. I haven't explore too much into mathematical rigor of these.

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9: Non-relativistic, Massive, Fermionic Fields

and operators in number state representation pretending that they don't interact with each

other. The reason for this is that inter-particle interaction in QFT comes from inter-�eld

interaction. State space representation can be build without knowing the interaction. One

can use either Bosonic or Fermionic symmetrization of multi-particle space.

9 Non-relativistic, Massive, Fermionic Fields

Our aim is to build a non-relativistic version of QFT for fermions. This is basically a

multi-particle QFT equivalent of Schrodinger's quantum mechanics. A non-relativistic

fermion QFT does not need any more postulations beyond those included in the

Schrodinger's formulation. Al we need to do is to learn how to represent states

and operators in number state representation. This part of the job we have already

done in a previous section (8). We just need to learn how to use that scheme now. I would

�rst explain this scheme in next section. I would also give an alternative historical route,

which is more commonly found in text books, to this problem. This would discuss in a latter

section. Latter on we would compare the two routes.

Note that the Fermionic Hilbert space is a sub-space of full direct product of single particle

Hilbert spaces. And the number states can be used as basis to generate the Fermionic Hilbert

space. So when operators are written explicitly in number state representations

then they would only operate on Fermionic subspace unlike the |r〉 representationof the operator that can mathematically operate on entire direct product. So in

a sense the scheme for writing the operators in number state representation is essentially

equivalent to take the projection of operators on Fermionic subspace. These schemes are not

simple basis change or mathematically speaking these are NOT similarity transformation

of operators because we are putting restrictions on the space these operators can correctly

operate.

The main idea is to obtain a representation of various operators in terms of number

states because that allows us to think in terms of quanta. So we can pretend as if we are

working with classical physics like particles and entire wave-like physics is dumped into the

inter relationship between operators and hence selection rules etc.

There are two major schemes available in the literature to obtain the operators that

operate on multi-particle Fermionic Hilbert space. More common scheme available in most

text books is sometimes known as second quantization. What has been proposed is to

quantize the Schrodinger's equation exactly similar to the procedure followed in quantizing

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9.1 First Scheme

Maxwell's wave equations as we would see below. This scheme sounds more like a

postulate. Whereas it is not supposed to be one. Since all the postulates are already

included in the Schrodinger's equation. The second scheme, which was �rst taught to me by

Prof. D. A. B. Miller26, is more enlightening as it clearly shows us that no more postulates are

needed in order to obtain the Fermionic operators in state space representation. Prof Miller

had introduce the concept of wavefunction-operators in semi-ad hoc fashion. On the other

hand I have already shown how to derive the concept of wavefunction operators rigorously

in section (8). In this section I would attempt to further justify the operators by showing

the complete equivalence of the operators in |r〉 and number state representations in the

sense that the give exactly same states when they operate on exactly same states from within

the Fermionic subspace of direct product space. Another important point is that, from the

very general derivation given in section (8), we can see that it works for Bosonic operators as

well. The reason is pretty simple. As we discussed in the previous section, the number space

representation is simply a projection of operators into smaller subspaces. The di�erence

between bosons and fermions is simply buried into the commutation relationship between

creation and annihilation operators.

On the other hand second quantization picture has its own bene�ts. What if

the quantum mechanical description of single particle physics is not known? For the case of

massive non-relativistic particles physics is known in terms of Schrodinger's equation � so

in this case there is no problem. But this may not be case in other areas of physics. For

example time evolution of a single photon is not known in quantum mechanics so there is

no point of being able to built a multi-particle number state representation of operators. In

such case second quantization proceeds by identifying canonical variables and quantizing a

classical wave equation. In such cases this procedure is known as �eld quantization rather

than second quantization but the procedure would be exactly same.

9.1 First Scheme

• One can �rst intuitively de�ne annihilation a†j and creation aj operators giving them

the properties that their names suggest. One can then easily prove the anticommu-

26In Applied Quantum Mechanics II (EE-223) at Stanford University, CA 94305.

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9.1 First Scheme

tation relations among them utilizing their pre-assumed Fermionic behavior

{a†j, a†k} = a†ja

†k + a†ka

†j = 0 (45)

{aj, ak} = ajak + akaj = 0

{a†j, ak} = a†jak + aka†j = δjk

Note that these anticommutation rules completely and uniquely de�ne

the Fermionic subspace of the direct product of single identical particle

spaces. Hence one can treat these as completely equivalent statement of

anti-symmetrization postulate of Quantum Mechanics.

• Secondly we de�ne a (annihilation) wavefunction operator from annihilation

operators as:-

Ψ =1√N !

∑a,b,c....

aaabac......Ψa(r1)Ψb(r2)Ψc(r3).... (46)

where N is the number of particles and a, b, c... are all the single particle energy eigen

states (if one chooses the number state representation using single particle energy eigen

states as the base). Note that we are multiplying the wavefunctions and not the kets.

� Hermitian conjugate of annihilation wavefunction operator is de�ned as creation

wavefunction operator. Remember that Hermitian conjugate �ips the order of

operators in a product

Ψ† =1√N !

∑a,b,c....

....acabaa....Ψ?c(r1)Ψ?

b(r2)Ψ?a(r3) (47)

� Noting that Ψ† = 1√N !

∑i,j,k.... ai

†aj†ak†......Ψ∗i (r1)Ψ∗j(r2)Ψ∗k(r3)...., ΨΨ† = −Ψ†Ψ

and hence multi-particle wavefunction operator satisfy the anticommu-

tation relation.

• Now comes the most important point. One can easily prove (8) that for any

operator A that operates on entire direct product space of single particle

spaces, following operator: ∫Ψ†AΨdr1dr2...

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9.1 First Scheme

would be the �projection� of A on N identical Fermionic Hilbert space. By

projection what I mean is that this operator only operates on the Fermionic

subspace and gives exactly the same result as that given by A operating on

the same state. The rigorous prove follows from the previous section's (8)

arguments used to build operators in number state representation.

� In this section we would just give a trivial example to satisfy the readers that

what we are claiming is correct. One can easily verify this by testing it on any

generic state given as∑basis cbasis(SlaterDeterminat of order N)basis and with

noninteracting multi-particle Hamiltonian operator. Its easily veri�able for

two particles. The most generic state of two non interacting fermions can be

written as :-

|Ψ〉 =∑m,n

cmn

∣∣∣∣∣ |1,ΨEm(r1)〉 |2,ΨEn(r2)〉|1,ΨEm(r1)〉 |2,ΨEn(r2)〉

∣∣∣∣∣=

∑m,n

cmn(|ΨEm(r1)〉|ΨEn(r2)〉 − |ΨEn(r1)〉|ΨEm(r2)〉)

here multiplication should be read as direct product of kets. Also, if one so wishes,

one can also write it as∑m,n

cmn|01, 02, ....1m, 0m+1, ....1n, 0n+1......〉

Also usingDirac quantization rule for non-interacting multi-particles we would

have the |r〉 representation as :-

H =~2

2m∇2r1

+~2

2m∇2r1

+ V (r1) + V (r2)

If the chose basis states of the single particle space (that is states |ΨEn〉) are sameas the single particle energy eigen state then the operation of Hamiltonian on

most generic state can be written as :-∑m,n

cmn(En + Em)[|ΨEm(r1)〉|ΨEn(r2)〉 − |ΨEm(r1)〉|ΨEn(r2)〉]

On the other hand the wavefunction operator for two particle fermions is given

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9.1 First Scheme

as :-

Ψ =1√2

∑m,n

amanΨm(r1)Ψn(r2)

So we are proposing that the projection of the H in r representation onto the two

particle fermionic sub-space in number state representation is :-

1

2

∑m,n

∑i,j

a†ma†naiaj

∫Ψ?m(r2)Ψ?

n(r1)H(r1, r2)Ψi(r1)Ψj(r2)dr1dr2

Which is same as

1

2

∑m,n

a†ma†nanamEn +

1

2

∑m,n

a†ma†nanamEm

First of these terms can be written as

T1 =1

2

∑m,n

a†ma†nanamEn

Now we use the fact that any two operators with same matrix elements are same.

So we only need to make sure that the operation of two operators on all basis

states (Fock states) give same result. This tells us that in above expression we

can enforce m 6= n without changing the way this operator operates on any Fock

state (basis state). This result is true only for fermions. This is because if m = n

in one of the terms in above summation then that term operated on any Fock

space gives zero. So we can as well drop that term altogether without a�ecting

the operation of this operator on any Fock state. Hence

T1 =1

2

∑m,n 6=m

a†ma†nanamEn

This has an important consequence that all creation operators anti-commute with

all annihilation operators. Let us swap a†m from �rst to third position. This would

give us even number of negative signs � equivalent to no sign change. Hence

T1 =1

2

∑m,n 6=m

Ena†nanNm

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9.1 First Scheme

Where Nm is particle number operator 49 discussed latter. We once again invoke

the similar argument as before. Operation of T1 on any Fock state would remain

intact even if we drop Nm and summation over m from above expression. Hence

T1 =1

2

∑n

Ena†nan =

1

2

∑n

EnNn

Hence

Hnumber-state =∑n

EnNn

So the operation of this Hamiltonian on a most generic two particle Fermionic

state ∑m,n

cmn|01, 02, ....1m, 0m+1, ....1n, 0n+1......〉

gives us ∑m,n

cmn(En + Em)|01, 02, ....1m, 0m+1, ....1n, 0n+1......〉

which is exactly same as what we obtained earlier but using more conventional

Hamiltonian and r representation of the states.

� Similarly one can easily get the Hamiltonian of any number of non interacting

particles in number state representation can be written as

Hnumber-state =∑j

EjNj

where sum is done over single particle basis states.

� A similar simple expression can be obtained for other additive operators. Let

A1 be an operator that operates on single particle space. And A2 be an operator

that operates on two particle space. Let |φj〉 are the single particle basis states.And A1|Φαi

〉 = αi|Φαi〉. By additive operator we mean A2(r1, r2) = A1(r1) +

A1(r2). Or equivalently, A2|Φα1〉|Φα2〉 = (α1 + α2)|Φα1〉|Φα2〉. So the number

state representation of this two particle operator would be

Anumber-state2 =

∫dr1dr2Ψ†(r1, r2)(A1(r1) + A1(r2))Ψ(r1, r2)

=1

2

∑m,n

∑i,j

a†ma†naiaj

∫φ?m(r2)φ?n(r1)(A1(r1) + A1(r2))φi(r1)φj(r2)dr1dr2

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9.2 Second Scheme � Second Quantization of Schrodinger's Field

Let us �rst study only one of the terms from this

T1 =1

2

∑m,n

∑i,j

a†ma†naiaj

∫φ?m(r2)φ?n(r1)A1(r1)φi(r1)φj(r2)dr1dr2

=1

2

∑i,n

∑m

a†ma†naiamA1,ni

We can now invoke similar arguments of equivalency of operators as done in the

previous case (energy operator) discussed above. hence

Anumber-state2

∑i,n

A1,ina†ian (48)

where A1,jkis the matrix element in the chosen basis of single particle operator A1

that operates on single particle space.


Now in this section, we would see that the exactly same physics can be obtained if we treat

Schrodinger's �eld as a classical �eld and quantize it.

We know that Ψ is a complex �eld. Hence we might be tempted to treat it as two in-

dependent �elds (or may be vector �eld having two components) corresponding to its real

and imaginary parts. Or alternatively we might want to treat Ψ and Ψ∗ as two independent

�elds. Having two �elds is similar to having two harmonic oscillators at each spatial point.

So the Hamiltonian or the Lagrangian would contain variables corresponding to two inde-

pendently moving particles. Speci�cally, in continuous variable case, for the latter choice

the Lagrangian 'denisty' would be written as :

L = L(Ψ,Ψ∗,∇Ψ,∇Ψ∗, Ψ, Ψ∗)

But we know that in classical mechanics of two 'unconstrained' particles it is necessary to

provide initial co-ordinates and initial velocities of both the particles to know its future

evolution in time with known forces. This is why one claims that q1 and q2, and q1 and q2

are independent variables and one needs to know the Lagrangian as a function of all these

variables in order to know the time evolution of the system. Such is not the case with

Schrodinger's equation. If we know real and imaginary part of Ψand ∇Ψat t = 0 (at any

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67




single spatial point and its 'neighborhood'), its time evolution is completely known. Hence

all four variables are not independent variables. In other words if write Hamiltonian we do

not need all four variables Ψ, Ψ∗and their associated conjugate momenta.

Secondly, one should note that Lagrangian function (and Hence Hamiltonian function as

well) is never unique. One can add total time derivative of any function of co-ordinates and

time. It simply adds a constant to action and hence its minima still remains same. Similarly

for Lagrangian density one can add a total time derivative (of any function of �led amplitudes

or space or time) at each spatial location and one can also add divergence of any vector �eld

(which could also be any function of �led amplitudes or space or time). Assuming this new

vector �eld goes to zero at in�nity. When we integrate (over in�nite space) this to calculate

total Lagrangian the divergence integrate to zero as there is no �eld at surfaces. Hence,

its no wonder that there are di�erent choices of Lagrangian density available in literature.

Sometimes people choose Ψr as co-ordinate and it turns out that the imaginary part Ψi can

be taken as conjugated momenta associated with it. On the other hand one can also take

Ψ as co-ordinate and Ψ∗ as its conjugate momenta. It depends a lot on how you write your

Lagrangian density which is not a uniquely identi�ed function.

Now we know that the equation of motion for the two independent �elds are

i~∂Ψ

∂t= −~2∇2Ψ

2m+ VΨ

and its complex conjugate. But note that what is more important for us is to know La-

grangian density corresponding to a system whose Euler-Lagrange equation is as given above.

As noted above, the choice of Lagrangian density is not unique. Some choices are preferred

over others but for the time being let us choose to have

L(Ψ,∇Ψ, Ψ,Ψ∗,∇Ψ∗, Ψ∗) = i~Ψ∗Ψ− ~2

2m∇Ψ∗∇Ψ− VΨ∗Ψ

This Lagrangian takes redundant variables and treats Ψand Ψ∗ as independent �elds. Note

that such a Lagrangian is not Hermitian and hence a better choice is one in which the

�rst term is chosen symmetrically as well i~(Ψ∗Ψ − i~Ψ∗Ψ)/2. One can easily verify that

both choices of Lagrangian densities lead to Euler-Lagrange equations which are same as

Schrodinger's Equations for Ψ and Ψ∗. Let us choose the non-Hermitian choice since at the

end we simply need a Hermitian Hamiltonian and we would not care about Lagrangian any

more. One can see that the conjugate momenta associated with Ψis i~Ψ∗ and conjugate

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9.3 Some Special Operators in Quantum Field Theory

momenta associated with Ψ∗ is identically zero. This makes our life much simpler. Suppose

we want to write Hamiltonian. We can immediately see that Hamiltonian is only going

to be a function of Ψ and it conjugate momenta Π (which is just same as Ψ∗). Which

automatically removes redundancy from Hamiltonian. Hence Hamiltonian density

would be

H = (i~Ψ∗)∂Ψ

∂t− (i~Ψ∗Ψ− ~2

2m∇Ψ∗∇Ψ− VΨ∗Ψ)

Which we can write as

H =~2

2m∇Ψ∗∇Ψ + VΨ∗Ψ

This is completely Hermitian. This is the reason why we didn't care much

that we are starting with redundant and non-Hermitian Lagrangian. Note that

occurrence of gradient of conjugate momenta is still a painful experience. But I believe it

can be written as linear superposition of gradient of Ψand Π. Note that both Ψand Πare

non-Hermitian.

Now once we have identi�ed Ψand Π as �canonical variables� we can lift them to

the status of �operators� representing �generalized � co-ordinate and momentum in QFT

picture. The commutation relation to be enforced would either be Bosonic or Fermionic

� both leads to physically correct results. One of them represents Fermionic �elds and

other represents Bosonic �elds. Now wavefunctions can be expanded in linear superposition

of energy eigen wavefunctions of the Schrodinger's equation (single particle). Hence when

wavefunction is lifted to the status of an operator the coe�cient of expansions also become

operators � these are exactly same as the creation and annihilation operators discussed in

previous section. Wave function operators are simply recognized as creation and annihilation

operators at one spatial location. Hence they obey the same commutation relation as creation

and annihilation operators discussed in previous section.


9.3.1 Note on Single and Multi-Particle Operators

• Terminology with respect to single-particle operators and multi-particle operators is

very confusing. Mathematically any operator that operates on multi-particle Hilbert

space should be called multi-particle operator. Physicists sometime use a di�erent

terminology. Often a single particle operator is de�ned as one which can be written

as sum of many di�erent terms where each term only depends upon the variables

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69




associated with only one particle. So two particle Hamiltonian is called a single particle

operator if particles do not interact with each other. Other examples is the �eld

operator Ψ(x) which can operate on Hilbert space of any number of particles but is

sometimes called a single particle operator whereas operator Ψ(x1, x2) is called two-

particle operator. One should remember that both these operators can operate on same

multi-particle Hilbert space. Expression like Ψ(x1, x2) = 1√2Ψ(x1)Ψ(x2) should NOT

be confused with direct products of two operators. In this expression both operators

operate on entire multi-particle Hilbert space. Expression represents �composition� of

operators and not �direct product� of operators.

• In the following we would see some interesting operators that look like single particle

operators but works properly on Hilbert space of any number of particles.

• Sometime there might be some confusion in notations. Symbols like r1 and r2 can

represent two things. 1) They represent coordinates of two di�erent particles. r1 is not

a �xed number but can take all values. 2) They represent two speci�c states of same

particle. In this case they should better be treated as �xed numbers or states and not

as variables at least for visualization purposes.

9.3.2 Particle Number Operator

One can easily prove that

Nk = a†kak (49)

is a multi-particle number operator in one single particle state in the sense that when this

operator operates on any multi-particle basis state (Fock state) it returns a number times

the same state Fock state. For fermions this number would be either 0 or 1. Remember that

every Fock state represent de�nite number (for fermions either 1 or 0) of particles in various

single particle basis states. Hence, for fermions, if system is in a given multiparticle Fock

state then we would either have 1 or 0 particles in a �xed single particle basis state (indexed

by k). It should be stressed that when this operator operates on a linear superposition of

Fock states (then system does not have de�nite number of particles in single particle state)

then resultant state would also be a linear superposition of Fock states (which may not be

same as original Fock state) clearly indicating that state does not have de�nite number of

particles in the single particle basis state indexed by k. One can also show that

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N =∑k

a†kak (50)

is an operator that represents the total number of particles in the system. If the system has

de�nite number of particles (when system is closed and conserves total number of particles in

it) then for such a system when this operator is operated on any arbitrary linear superposition

of Fock states we get number of particles in the system times the exactly same state. System

that do not conserve particle number can go in linear superposition of Fock states where

di�erent Fock states may have di�erent number of particles. When this operator operates

on such a state, resultant state would be di�erent from original state clearly indicating that

we do not have de�nite number of particles in the system.

Just like in fermions case discussed above, one can also prove that

Nk = b†kbk

is an operator representing number of particles in the single particle basis state indexed by k.

Again when this operator operates on Fock state (which are basis states) we get same state

multiplied by a number. When this is operated on any other general linear superposition of

Fock states we would not get same state. Similarly, one can also show that

N =∑k

b†kbk (51)

is an operator representing total number of particles in the system. To clearly understand the

meaning of this operator read the comments provided after the de�nition of similar operator

for fermions.

9.3.3 Particle Number Density Operator

Usually, in QM, we do not treat electron number density as an operator. Treating number

density as an operator would be very useful construct because then we can use the standard

methods of QM to �nd QM �uctuations in electron density even if electron is known to be

in exact QM state.

Let us consider a one dimensional system having one electron in state Ψ1(x1). If we do

a measurement on an ensemble of exact same system to detect a particle exactly at x = a,

what is the probability that we would �nd one there? Zero! We should try to detect a

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particle between x = a and x = a + dx. Sometimes you would �nd one particle there and

sometimes you would �nd zero there. The ensemble average value of number of particles

per unit length at x = a would be Ψ?1(a)Ψ1(a). What if we have system with two non-

interacting distinguishable particles � one in state Ψ1(x1) and other in state Ψ2(x2)? What

is the ensemble average value of number of particles per unit length at x = a in this case?

Is it Ψ?1(a)Ψ1(a)Ψ?

2(a)Ψ2(a)? That is wrong! Note that in this case �probability� (which

can not be greater than 1) and �ensemble average� (which can potentially be >1) of number

density are two di�erent things. Intuitively, one should guess that the average mass density

should be more than that in one particle case. The ensemble average number of particles

per unit length at x = a would be Ψ?1(a)Ψ1(a) + Ψ?

2(a)Ψ2(a).

Now lets say, we want to de�ne a �number density operator� which would be a function

of a space parameter (x) so that we can simply talk about the expectation value of this

operator as a function of x knowing the state of the system. Let us postulate the operator

to be ∑i

δ(x− xi)

where xi is the co-ordinate of ith particle. Note that the variables inside the delta functions

are common functional parameters (basically they are numbers) whereas the whole expression

above is an operator in r-representation (it operates on wavefunctions and not on ket vectors).

So when it operates on a wavefunction it would provide us a number multiplied by another

wavefunction. As we would see in a minute that the �per unit length� normalization is

included in the wavefunction Ψ which is treated as the probability amplitude density. Let

us now explore how this operator works. Let us assume that there is only one electron in

the system (i = 1). Let the system be in a pure quantum state and let the normalized

wavefunction of the system be Ψ1(x1). The density operator would be δ(x − x1). The

expectation value of this operator in the state Ψ1(x1) would be∫

Ψ?1(x1)δ(x−x1)Ψ1(x1)dx1=

Ψ?1(x)Ψ1(x). Note that x is simply a parameter in the operator. Actual space variable is x1

so the integration goes over x1. Now consider the case of two non-interacting distinguishable

particles (i = 2) - one in state Ψ1(x1) and other in state Ψ2(x2). The density operator would

beδ(x − x1) + δ(x − x2). The expectation value of this operator in the state Ψ1(x1)Ψ2(x2)

would be

〈ρ2〉 =

∫ ∫Ψ?

1(x1)Ψ?2(x2)(δ(x− x1) + δ(x− x2))Ψ1(x1)Ψ2(x2)dx1dx2

= AB + CD

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where

A =

∫Ψ?

1(x1)Ψ1(x1)δ(x− x1)dx1 = Ψ?1(x)Ψ1(x)

B =

∫Ψ?

2(x2)Ψ2(x2)dx2 = 1

C =

∫Ψ?

2(x2)Ψ2(x2)δ(x− x2)dx2 = Ψ?2(x)Ψ2(x)

D =

∫Ψ?

1(x1)Ψ1(x1)dx1 = 1

Hence we can conclude that

〈ρ2〉 = Ψ?1(x)Ψ1(x) + Ψ?

2(x2)Ψ2(x2)

Which is what we expected intuitively. Hence, above de�ned operator can really be treated

as a �number density operator�. One can easily convince oneself that this operator would

indeed give correct results even with interacting and/or identical particle systems.

Now suppose we want to write this operator in terms of creation and annihilation oper-

ators. Remember that in general any multi-particle operator A in r-representation can be

written as∫

Ψ†A(r1, r2, ...)Ψdr1dr2... Where Ψ†(r1, r2, ..) = 1√N

∑i,j,... Ψ

?i (r1)Ψ?

j(r2)...a†ia†j...

is the N -particle creation �eld-operator and Ψ(r1, r2, ..) = 1√N

∑i,j,... Ψi(r1)Ψj(r2)...aiaj... is

the N -particle annihilation �eld- operator. Hence the number density operator in number-

state representation can be written as

ρN(x, t) =

∫Ψ†(x1, x2, ...)(

∑i

δ(x− xi))Ψ(x1, x2, ...)dx1dx2..

Using the fact that an N particle wavefunction operator can also be written as

Ψ(x1, x2, ..) =1√N

Ψ(x1)Ψ(x2)...

we can show that the wavefunction operators have many properties identical to those corre-

sponding to wavefunctions themselves. For example∫Ψ†(x1, x2, ...)δ(x− x1)Ψ(x1, x2, ..)dx1 =

1

NΨ†(x)Ψ†(x2)...Ψ(x)Ψ(x2)...

= Ψ†(x, x2, ...)Ψ(x, x2, ..) (52)

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∫Ψ†(x1)Ψ(x1)dx1 =

∑i,j

a†iaj

∫φ?i (x1)φj(x1)dx1

=∑i,j

a†iajδij =∑i

a†iai = N (53)

Note that it may seem that Ψ(x1) is one-particle operator and operates on one-particle

Hilbert space. But this is not true. This operator operates on multiparticle states. Expres-

sion like Ψ(x1, x2, ..) = 1√N

Ψ(x1)Ψ(x2)... helps writing operators for multiparticle systems

but one should not think that Ψ(x1) operates on the �rst particle Hilbert space and Ψ(x2)

operates on second particle Hilbert space. Such a notion is incorrect. Hence, within

N particle space, last expression in Eq. 53∑

i a†iai can be shown to be a particle number

operator N . This can be checked by noting that operation of this operator on any arbi-

trary two-particle state (for example)∑

i 6=,j cij|0, 0, 1i, 0, 1j, ...〉 gives 2 times the same state.

Expression for 53 is a simple justi�cation why Ψ†(x)Ψ(x)can treated as particle

number density operator for a system containing any number of particles. Hence,

it turns out that for any number of particles, we can write

ρ(x) = Ψ†(x)Ψ(x) (54)

This conclusion also follows trivially if one remembers two facts � 1) a†kak give number

of particles in single particle state |k〉 and 2) Ψ(x) represents annihilation operator when

number state representation is built on top of single particle state |x〉. Hence Ψ†(x)Ψ(x)

should give number of particles in single particle state |x〉. This becomes density because

wavefunction represents probability densities.

It should be possible to explicitly show that∫Ψ†(x1, x2, ...){

∑i

δ(x− xi)}Ψ(x1, x2, ..)dx1dx2... =

∫Ψ†(x1)δ(x− x1)Ψ(x1)dx1

= Ψ†(x)Ψ(x)

This can be proved through similar arguments that we used in the previous section to obtain

number state representation of additive operators 48.

Sometimes, it also useful to have the density operator that gives density in Fourier space

(not really reciprocal space) which is simply the Fourier transform of above operator. The

operator expression would greatly depend upon the basis states chosen for number state

representation. In case simple plane waves were chosen, we can take the Fourier transform

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very simply to obtain

ρ(q) =∑k

a†k+qak

Finally the number density operator can be written in terms of NEGF as

ρ(x, t) =i

2π

∫G<(x = x′, E)dE

9.3.4 Particle Number Current Density Operator

Again, usually, in QM current density is not treated as an operator. One can prove that

following expression represents a quantity which obeys the continuity equation for probability

density:

j(x) =i~2m

(Ψ∇Ψ? −Ψ?∇Ψ)

hence it is identi�ed with particle number current density. But such a formulation can not

tell us anything about QM �uctuations in the measurement of current even if the system is

in a well de�ned quantum mechanical state. So let us try to formulate an operator for this

current density.

Classically, the current density should be written as

j(x) =∑i

δ(x− xi)vi

where i is the particle number. One can convert this into an operator using standard canon-

ical quantization techniques. As before, one would treat the δ as an operator and vi = pi/mi

as another operator. Since there is a confusion about the ordering of operators, one would

take combination of both possibilities. Also to make the operator Hermitian the second term

would need to be Hermitian conjugate of �rst term. Hermitian conjugate of momentum op-

erator operates on the left side.

j(x) =1

2

∑i

−δ(x− xi)i~∇i

m+i~∇←im

δ(x− xi)

We notice that this operator looks like a �single particle operator� in the sense that the

multi-particle operator is a sum of di�erent terms each depending upon co-ordinates of only

one particle. Now, to write above operator in terms of creation and annihilation operators,

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9.4 Two Point Correlation Functions/Propagators For Free Fields

one would have to calculate of term:

jij =i~2m

∫−Ψ?

i (x1)(δ(x− x1)∇1 +∇←1 δ(x− x1))Ψj(x1)dx1

jij =i~2m

(Ψi(x)∇xΨ?j(x)−Ψ?

i (x)∇xΨj(x))

Hence the particle current density operator would be written as

j(x) =∑ij

jija†iaj

which can also be written as

j(x) =i~2m

(Ψi(x)∇Ψ†j(x)− (∇Ψ(x))Ψ†(x))

It should again be stressed that even though this operator looks like a single particle operator

it is not so. It is a proper multiparticle operator that operates on multiparticle Hilbert space

and gives correct particle current.


9.4.1 Classical Correlation Functions

Let us consider a classical perfect optical cavity. Let the electric �eld be in certain known

mode pro�le with known amplitude of oscillations. Imagine that we have an ensemble of such

identical systems with �elds in same state. But let the system be slightly noisy. Such systems

can be studied using classical correlation functions. If we do simultaneous measurement at

same time t = t1 and same space point r = r1 on all systems, values of measured electric

�eld might should some noisy distribution. If the mean value of noise is zero then by taking

the ensemble mean of all measured values, we can remove noise from the measurement.

Classical ensemble mean value thus is de�ned as 〈E(r1, t1)〉ensemble.Now suppose we want to study the coherence properties of the noisy �elds. For ex-

ample, suppose we want to check if the time evolution of �elds is coherent and follows

E0 exp(−iωt). This can be done by doing �simultaneous�27 measurements at time t = t1 at

same space points and then again at time t = t2 at same space point and then comparing the

27By �simultaneous� I mean that in the same copy of the ensemble one needs to make measurements att = t1 and t2.

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correlation between data so obtained at two di�erent times. If there was no noise then classi-

cal ensemble autocorrelation function de�ned as〈E(r1, t2)E?(r1, t1)〉ensemble would giveus |E0|2 exp(−iω(t2 − t1)) . If system is time invariant 28 then origin of time is not im-

portant we can write the classical autocorrelation function as 〈E(r1, τ)E?(r1, 0)〉ensemble=|E0|2 exp(−iωτ).

We see that classical autocorrelation function tells us whether there is a de�nite phase re-

lationship between two time measurements. Absolute value of autocorrelation of the system

concerned above is constant for all values of τ indicating that there is de�nite phase relation-

ship between any two time instances. We can also do similar calculation even when noise

is not zero. To keep things simple, let us �rst assume that added �noise� is actually a well

de�ned sinusoidal function of frequency ω1. One can immediately see that autocorrelation

function would be sum of four sinusoidal functions of frequencies ω, ω1, ω−ω1and ω+ω1. If

noise amplitude is small then only sinusoidal term with frequency ω would be large. If noise

is not small then exact τ dependence of the auto-correlation function would actually depend

on the phase of the noise signal. Fourier transform of classical autocorrelation function is

called classical ensemble averaged power spectral density.

Classical systems are often ergodic systems. Ergodic systems are those for which ensem-

ble averaging is same as time averaging. If systems are ergodic then 〈E(r1, t1)〉ensemble =

LimT→0

∫ T/2−T/2E(r1, t1 + t)dt. Similar expression can also be written for classical autocorre-

lation function.

9.4.1.1 Important Points to Note

9.4.1.1.1 Dependent Events Let us explain the meaning of classical two point cor-

relations in a bit more pedagogical fashion. Suppose, when we measure E(r1, 0) on di�erent

copies of the ensemble, we get values equal to Ai with probabilities of αi. Similarly, when we

measure E(r1, τ) on di�erent copies of the ensemble, we get values equal to Bj with prob-

abilities of βj. Does this mean that the probability of obtaining a value of Ai for the �rst

measurement and a value of Bi for the second measurement is is αiβj? In other words, does

this mean that the probability of measuring a pair values AiBj is αiβj? Answer would have

28This can be visualized as our cavity being a closed system in the sense that there is no backgroundtime-varying electromagnetic �elds. If such is the case then origin of time does not matter. Whether we doexperiment today or tomorrow � we should get same results. But what if there is a background �eld thatvaries very slowly from one day to another day. In that case, time evolution of our mode measured today ormeasured tomorrow would be di�erent. We call such systems as time variant systems.

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been yes only if the two measurement events were independent. Take a simple example of

sinusoidal signal with completely random phase (probability of each value of phase occurring

is equal). In other words, among di�erent copies of ensemble, the initial phases of the signal

are di�erent. At both time instances, one would measure all values from −1 to +1 with

equal probabilities. So if we assume that pair of values AiBj occur with a probability of αiβj

then ensemble autocorrelation would be

〈E(r1, τ)E?(r1, 0)〉ensemble ≡∑i,j

BjA?iαiβj =

∑i

A?iαi∑j

Bjβj = 0

But our intuition tells that this is obviously not true. For example if τ = 1/2f where f is

the sinusoidal frequency then whenever you measure Ai at t = 0, then if you do second mea-

surement on the same copy of the ensemble at t = τ , you should measure Bi = −Ai. Hence,we expect non-zero correlation between signals. Hence, two events are not independent and

we can not claim that the probability of that pair of values AiBj occur with a probability of

αiβj.

Actual thing that we want to measure is the joint probability that at time 0

value Ai is measured and, as the same copy of the ensemble evolves in time, at

time τ value Bj would be measured. Hence, if we do measurements on same copy of

ensemble at two times and we �nd that pair of values (A?B)i happens with probability γi

then

〈E(r1, τ)E?(r1, 0)〉ensemble ≡∑i

(BA?)iγi

If certain value of this product is highly likely then we say that �eld is temporally coherent

or equivalently highly correlated at two di�erent time instances.

If the two events are independent then P (Ai∩Bj) = P (Ai)P (Bj). Equivalently P (Bj|Ai) =

P (Bj). And we are allowed to perform two measurements on independent copies of the en-

semble. For the case of dependent events, instead of using P (Ai)P (Bj) = αiβj we should

use the actual joint probability density function P (Ai ∩Bj). In that case one would have to

perform measurements on the same copy of the ensemble.

9.4.1.1.2 E�ects of Measurements Note that classical measurements do not dis-

turb system evolution in time. Hence, fact that we performed a measurement at time t = t1

on certain copy of the ensemble does not change the course of evolution of that copy. Hence

probability of results measured at time t = t2 does not dependent on whether we performed

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a measurement at t = t1or not. Results at t2, although, may depend on the results at t = t1

because of the physics dictating the time evolution links values at t1 and t2.

We would �nd that this is not true in quantum systems. We would be able to develop

similar concepts for quantum systems but we would have to get rid of the notion of actual

performing measurements. We would just talk about what the probability amplitude of

obtaining certain result would have been had a measurement been performed.

9.4.2 Quantum Correlation Functions

9.4.2.1 Correlation Functions in Heisenberg Representation Readers who have

seen two point quantum correlation function before might know that correlation function is

typically de�ned as

gn(τ) ≡ h〈n|x†h(t2)xh(t1)|n〉h (55)

where subscripts represents Heisenberg representations29. In Heisenberg representation, this

expression looks very similar to the classical correlation function. It seems there is nothing

more to discuss about this de�nition. This is not true though. Heisenberg representation

hides some of the crucial details involved in this de�nition. Is Schrodinger representation,

above expression would be written as

gn(τ) ≡ h〈n|U †s (t2)x†sUs(t2) U †s (t1)xsUs(t1)|n〉h= s〈n(t2)|x†s Us(t2)U †s (t1) xs|n(t1)〉s (56)

Here, Us(t) is the time propagation operator discussed. The above expression 56 is not very

intuitive. Naively speaking one would have expected a simpler expression like

gn(τ) = s〈n(t2)|x†s xs|n(t1)〉s (57)

Following may be the reasoning behind the wrong de�nition 57. For simplicity, let us assume

that states |n(t)〉 are the energy eigen states and let |xi〉 be the eigen states of the time

independent operator xs. When operator xs is operated on state |n(t1)〉 = exp(−iEnt1~ )|n(0)〉

we get another state xs|n(t1)〉 =∑

i(xicni exp(−iEnt1~ ))|xi〉 =

∑i,m xicnic

?mi exp(−iEnt1

~ )|m(0)〉.This basically tells us that �if� we had done the measurement of physical quantity of x at

time t = t1 given that system is known to be in state |n〉we would have gotten the results

29 See another article on Statistical Quantum Field Theories for expansion on Heisenberg representations.We would discuss that topic again in this article a bit latter.

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http://www.stanford.edu/~mukul/tutorials/stat_QFT.pdf




of xi with a probability �amplitude� of ci exp(−iEnt1~ ). Now, when operator xs is operated on

state |n(t2)〉 = exp(−iEnt2~ )|n(0)〉 we get another state xs|n(t2)〉 =

∑i(xici exp(−iEnt2

~ ))|xi〉=∑

i,m xicnic?mi exp(−iEnt2

~ )|m(0)〉. This tells us that if we had done the measurement of

physical quantity of x at time t = t2 given that system is known to be in state |n〉we wouldhave gotten the results of xi with a probability �amplitude� of ci exp(−iEnt2

~ ). With analogy

from classical physics we may then de�ne a quantum autocorrelation function as

gn(τ) ≡∑i,j

x?ixjc?i cj exp(

−iEnτ~

)

≡ 〈n(t2)|x†s | xs|n(t1)〉

Let me stress that above expression is incorrect. The reason above expression is incorrect

because we assumed the two events to be independent. We have not make sure that two

events can happen in the same copy of the ensemble as state evolves coherently. Correct

way of thinking about this is discussed in the following section.

9.4.2.2 Correlation Functions in Schrodinger's Picture Let us consider a closed

system. Hence, Hamiltonian is time-independent. Let xs be a time independent operator

(may not be Hermitian and may not represent any physical property of the system) in

Schrodinger representation. Let t1 = 0 and t2 = τ . Suppose we use energy eigen states (at

time t = 0) |n(0)〉 of our closed system as the basis states.

Let |n(0)〉 =∑

i cni|xi〉where |xi〉 are the eigen states of time independent operator xs.

Equivalently, |xi〉 =∑

n c?ni|n(0)〉. For simplicity, let system be in state |n(0)〉at time t =

t1 = 0. Probability amplitude of system being in state |xi〉 is cni. Equivalently, probabilityamplitude that this state would be mapped to the state |xi〉 by the operation of operator

xs is cnixi. Given that xs has mapped the state |n(0)〉to cnixi|xi〉 = cnixi∑

m c?mi|m(0)〉, at

time t = τ , this state would evolve to a new state

cnixi∑m

c?mi exp(−iEmτ

~)|m(0)〉 (58)

Now, suppose |n(0)〉 has evolved to become |n(τ)〉. Probability amplitude of this state

being in state |xj〉 is cnj exp(−iEnτ~ ) . Or equivalently, probability amplitude that operation

of xs would map the state |n(τ)〉 to state |xj〉 is cnjxj exp(−iEnτ~ ). Given that xs has mapped

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the state |n(τ)〉 to state

cnjxj exp(−iEnτ

~)|xj〉 = cnjxj

∑p

c?pj exp(−iEnτ

~)|p(0)〉 (59)

what is the joint probability amplitude of the two events 58 and 59 taking place? This

amplitude simply is

gn(τ) =∑i,j

∑p

c?njcpjcnic?pi(x

?jxi) exp(

i(En − Ep)τ~

)

One can easily see that this is same as

gn(τ) = 〈n(τ)|xs | Us(τ)xs|n(0)〉 (60)

Here is an alternative way of thinking. What the two point quantum correlation function

tells us is this. Given that the system is in state |n(0)〉, what is the probability amplitude

of following things happening in the coherent time evolution path of the state as the state

evolves to |n(τ)〉 at time t = τ? We want to know the probability amplitude of following

events taking place. At time t = 0 system is in state |xi〉 and at time t = τ system is in

state |xj〉. Probability amplitude of system being in state |xi〉 at t = 0 is simply cni. At

time t = τ , cni|xi〉 state would evolve to another state cni∑

m c?mi exp(−iEmτ

~ )|m(0)〉. Prob-ability amplitude that the system is in state |xj〉 is cni

∑m c

?micmj exp(−iEmτ

~ ). The state

of the system can also be written as cni∑

m,p c?micmj exp(−iEmτ

~ )c?pj|p(0)〉. Hence probabil-

ity amplitude of the system being in state |n(τ)〉 = |n(0)〉 exp(−iEnτ~ ) at the same time is

cni∑

m c?micmj exp(−iEmτ

~ )c?nj exp(−iEnτ~ ). Hence autocorrelation function would be

gn(τ) =∑i,j

∑p

c?nicpjcnic?pi(x

?jxi) exp(

i(En − Ep)τ~

)

which same as that obtained above.

9.4.2.3 Some More Comments We would often evaluate correlation terms of the form

〈n|x†h(t2) | xh(t1)|n〉 = 〈n(t2)|x†sUs(t2) | U †s (t1)xs|n(t1)〉

Again, since system is closed, we should have time translation symmetry. Hence, we can

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simply evaluate 〈n|xh(τ)xh(0)|n〉. We would see that this is an interesting parameter and

we would get interesting information once this or a similar parameter is calculated. This

parameter is in fact similar to classical time correlation function discussed above

g(τ) = LimT→∞1

T

∫ T/2

−T/2x(t)x(t− τ)dt

As discussed above, if the system is ergodic, one can replace classical time averaging by

classical ensemble averaging.

In many situations correct quantum analog that we might want to look at is a more

symmetric parameter that does not depend on the ordering of time dependent operators.

We would often try try to calculate terms like

gn(τ) = 〈n|12

(xh(τ)xh(0) + xh(0)xh(τ))|n〉

Or terms like

fn(τ) =〈n|(xh(τ)xh(0)− xh(0)xh(τ))|n〉

0

when τ > 0

when τ < 0

We would de�ne many other such correlation functions which are generally known as Green's

functions. In classical physics, Green's functions are useful tool for solving inhomogeneous

di�erential equations that is for equations that contain a source term. Quantum mechanical

Green's function that we would de�ne below would initially seem very di�erent from those

seen in theory of di�erential equations. Latter on as we develop the subject further you

would start seeing the similarities between the two.

9.4.3 Free Fields

In this article I would only discuss non-interacting particles. In QFT we call it free �eld.

Interacting �elds are discussed in the article on Statistical QFT.

We choose single particle energy eigen states as basis on top of which we build our number

state representation. One of the greatest simpli�cation in free �elds case is that free �eld

ground state is a very simple state with no particles in the system. Operation of creation

and annihilation operators on this state is also trivially known. In following one additional

simpli�cation in case of free �elds.

I want to the Heisenberg representation of creation and annihilation operators. This is

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given as

ai,h(t) ≡ U †(t)ai,sUs(t)

here ai,s is the Schrodinger representation of annihilation operator that annihilates a particle

in the ithsingle particle state with single particle energy of Ei. By de�nition 〈n|ai,s|m〉isnon-zero only when state |m〉 and |n〉 represent almost same state of the system with only

di�erence being that in state |m〉 system has one additional particle in the ithsingle particle

state as compared to the system in state |n〉. Now the action of Us(t) on any basis state

(direct product of single particle energy eigen states) is trivially known for free �elds. Hence

we can immediately see that for free �elds

ai,h(t) = ai,s exp(−iEit/~) (61)

and similarly

a†i,h = a†i,s exp(iEit/~) (62)

9.4.4 Green's Function for Non-Interacting Schrodinger Fields in Homogeneous

Space

Propagators are related to two point correlation functions. Sometime names might be used

interchangeably. Two point correlation functions are also related to Green's functions of

Schrodinger Hamiltonian operator. Many di�erent types of Green's functions can be de�ned.

First two we would de�ne below are not really Green's function. But they are usually referred

to as Green's function because these two are the building blocks for the 4 di�erent types of

Green's functions of the Schrodinger's operator. The most basic one isParticle Propagator

or a Greater Green's Function de�ned as

G>(x2 − x1, t2 − t1) ≡ G>(x2, t2 ← x1, t1) ≡ 〈0h|Ψh(x2, t2)Ψ†h(x1, t1)|0h〉 (63)

In the above and all of the following discussion x stands for all three coordinates. I am using

simpler notations to make sure that equations don't look scary but convey the message. Also

note that we are assuming that conventional electrostatic potential V (x) in Schrodinger's

equation of motion is zero. hence space is homogeneous and we have space translation

symmetry in addition to the time translation symmetry. Hence G>(x2, t2 ← x1, t1) can

be written as G>(x2 − x1, t2 − t1).

Note that what we are doing here is that we are creating a single particle at x1 at time t1

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and �nding the amplitude of probability that system propagates to become particle at x2 at

time t2. But above expression can be further generalized by creating a single particle in, say,

single particle classical momentum mode p (that is state |p〉) and then looking an amplitude

probability that state evolves into a state |q〉. Using this Greater Green's function one can

study the causality of the theory. Note that it is not necessary that t1 < t2.

Another fundamentally important Green's function is what is known as Anti-Particle

(or Hole) Propagator or a Lesser Green's Function. This is de�ned as follows

G<(x2 − x1, t2 − t1) ≡ G<(x2, t2 ← x1, t1) ≡ 〈0h|Ψ†h(x2, t2)Ψh(x1, t1)|0h〉 (64)

One very important point is that the commutation relation between Ψh and Ψ†hare the equal time commutation relation (ETCR). There is no de�ned relation-

ship between �eld operators at di�erent times. So there would be no obvious

correlation between lesser and greater Green's function other than that discussed

above.

Let us try to obtain the Fourier transform of these two correlation function (these are not

really the Green's functions of Schrodinger's operator). Substituting single particle creation

wavefunction operator 43 and its adjoint in the de�nition of Greater Green's function 63 we

can immediately write

G>(x2 − x1, t2 − t1) =∑i

exp(−iEi(t2 − t2)/~)φi(x2)φ?i (x1) (65)

where φi(x) are the single particle energy eigen states with single particle energy Ei. For free

particles φi(x) = 1√V

exp(ikix) and Ei = ~2k2i /(2m). Here

√V is included for normalization.

In continuum limit we get

G>(x2 − x1, t2 − t1) =1

V

∫d3k exp(−i~

2k2

2m(t2 − t2)/~) exp(ik(x2 − x1)) (66)

Similarly

G<(x2 − x1, t2 − t1) = 0 (67)

Integral in the expression for the greater Green's function 66 is actually very simple Gaussian

integral and can be evaluated analytically if desired as

G>(x2 − x1, t2 − t1) =1

V(

m

2πi~(t2 − t1))3/2 exp(

im(x2 − x1)2

2~(t2 − t1))

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The trivial result for lesser Green's function 67 is because non-relativistic Schrodinger

equation does not accept anti-particles, �elds are non-interacting and quantum ensemble

averaging is done on ground state of the system30. This is not a general rule as we would see

in the case of Klein Gordon �elds (which accepts anti-particles) latter on in this article and

in case when averaging is done over equilibrium and no-equilibrium statistical ensembles as

we would see in a di�erent article on Statistical QFT. We would �nd these expressions useful

in the following discussion.

Above two functions are the basic building blocks for other types of correlation functions

which have better physical and mathematical properties. Using these two functions we

de�ne following four other correlation functions which are more commonly encountered in

text books. A set of functions that are sometime useful are time ordered and anti time

ordered Green's functions

Gt(x2 − x1, t2 − t1) ≡ Gt(x2, t2 ← x1, t1) ≡ 〈0h|T Ψh(x2, t2)Ψ†h(x1, t1)|0h〉

Gt(x2 − x1, t2 − t1) ≡ Gt(x2, t2 ← x1, t1) ≡ 〈0h|T Ψh(x2, t2)Ψ†h(x1, t1)|0h〉

Time ordered Green's function is also sometime (in the case of free/non-interacting �eld

operators propagating ground state31) called the Feynman propagator. Note that

Gt(x2 − x1, t2 − t1) =

{G> if t2 > t1

G< = 0 if t2 < t1

}(68)

Gt(x2 − x1, t2 − t1) =

{G< = 0 if t2 > t1

G> if t2 < t1

}(69)

Two other functions are retarded and advanced Green's functions. Retarded Green's function

is de�ned as

GR(x2 − x1, t2 − t1) ≡ −iΘ(t2 − t1)(〈[0h|Ψ†h(x2, t2), Ψh(x1, t1)]|0h〉)

= −iΘ(t2 − t1)(G>(x2 − x1, t2 − t1)−G<(x2 − x1, t2 − t1))

= −iΘ(t2 − t1)G>(x2 − x1, t2 − t1) (70)

30That basically means that we trying to apply non-interacting �eld operators on ground state of thesystem. So we trying to see how a single particle created in an empty system propagates.

31That is, quantum ensemble average is done on ground state of the system.

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Note that Θ is a Heaviside step function32. Note that some books leave out the −i factor.Similarly advanced Green's Function is de�ned as

GA(x2 − x1, t2 − t1) ≡ −iΘ(t1 − t2)(〈[0h|Ψ†h(x2, t2), Ψh(x1, t1)]|0h〉)

= −iΘ(t1 − t2)(G>(x2 − x1, t2 − t1)−G<(x2 − x1, t2 − t1))

= −iΘ(t1 − t2)G>(x2 − x1, t2 − t1) (71)

For the free Schrodinger �eld (and averaged over ground state), it turns out that retarded

and time ordered Green's functions are same thing and advanced and the anti-time-ordered

are the same thing (apart from conventional −i factor). This is not a general rule as we

would see in the case of Klein Gordon �elds (which accepts anti-particles) latter on in this

article and in case when averaging is done over equilibrium and no-equilibrium statistical

ensembles as we would see in a di�erent article on Statistical QFT.

In classical physics Green's function is a popular tool for solving inhomogeneous dif-

ferential equations with a position and/or time dependent source term. Green's functions

discussed above do not seem to have any resemblance to the Green's functions discussed in

classical physics. This not true though. In the following we would see that Green's function

just de�ned are related to the classical Green's function of the Schrodinger's equation. These

four correlation functions just de�ned are actually very closely related to each other. We

would show that Gt, Gt, GR and GA are all Green's function of time dependent Schrodinger

operator. That is, if we write a time dependent Schrodinger equation with a 4-dimensional

delta function on right hand side as a source term then above Green's functions satisfy the

equation. This gives a prescription of actually calculating these functions. One can Fourier

transform the whole equation. Which then get converted to a simple algebraic equation.

From this one can calculate the Fourier transform of the Green's function which can then be

inverse transformed to calculated GF in real space.

Suppose G(x2, t2;x1, t1) is a Green's function of Schrodinger operator (free �eld case)

{ ~2

2m

∂2

∂x2+ i~

∂

∂t}G(x2, t2;x1, t1) ≡ −iδ(x2 − x1, t2, t1) (72)

i on the right hand side included because of convention. Without loss of generality, let x1 = 0

and t1 = 0. Let 4D Fourier transform of G(x, t) be G(k, ω). In order to remind ourselves

32Usual de�nition assumes that Θ(t) = 12 for t = 0 (see for example Bracewell). Some books de�ne Θ(0)=0.

This values does not alter our results here.

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about multi-dimensional Fourier transforms, let us consider a simple 2D function f(x1, x2)

with a 2D Fourier transform of f(kx1, kx2)

f(kx1, kx2) ≡∫ ∞−∞

dx1dx2f(x1, x2) exp(−i(kx1x1 + kx2x2))

33Di�erentiating this equation with respect to x1 we get

0 =

∫ ∞−∞

dx1dx2{df(x1, x2)

dx1

exp(−i(kx1x1 + kx2x2))− ikx1f(x1, x2) exp(−i(kx1x1 + kx2x2))}

Hence∫ ∞−∞

dx1dx2df(x1, x2)

dx1

exp(−i(kx1x1+kx2x2)) = ikx1

∫ ∞−∞

dx1dx2f(x1, x2) exp(−i(kx1x1+kx2x2))

Same rule can be extended for 4D Fourier transforms. Lets remember one more fact∫ ∞−∞

dxδ(x) exp(ikx) = 1

Hence if we Fourier transform the de�ning equation for the Green's function 72 of the

Schrodinger operator, by de�nition then, G(k, ω) would satisfy the following algebraic equa-

tion

G(k, ω) =i

~ω − ~2k2

2m

(73)

Now taking its 4D Fourier inverse

G(x2 − x1, t2 − t1) =

∫d3kdω

(2π)4

i

~ω − ~2k2

2m

exp(−iω(t2 − t1) + ik(x2 − x1)) (74)

Let us look at the ωintegral a bit more closely. On ωaxis we have a pole at ω = ~k2

2m.

Remember that from residue theorem∫C

dω

2πexp(iωt)f(ω) = ±2πi

∑Res{f(ω)}

Sign is positive if contour integral is done anti-clockwise and is negative if it is done in

33As a conventional de�nition of Fourier transforms, time domain forward Fourier transform would havepositive sign in exponential and space domain forward Fourier transform would have negative sign in expo-nential. This is conventional in quantum mechanics but is opposite of standard engineering convention.

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clockwise direction. If t2 − t1 < 0 then we should close the contour in the upper half and

if t2 − t1 > 0 then we should close the contour on the lower half. We can also choose the

contour to go either below (equivalent to adding negative imaginary part to the frequency)

the pole or above the pole (equivalent to adding positive imaginary part to the frequency).

Suppose we decide to go above the pole. Then for t2 − t1 < 0 we don't have any pole

inside the contour and hence G(x2 − x1, t2 − t1) = 0. And for t2 − t1 > 0 we have a pole

inside the contour hence pole contributes a residue of exp(−i~k2

2m(t2 − t1)) and

G(x2 − x1, t2 − t1) = Θ(t2 − t1)

∫d3k

(2π)3exp(−i~k

2

2m(t2 − t1)) exp(ik(x2 − x1)) (75)

This is a simple Gaussian integral and can be performed easily. Comparing with the expres-

sion for retarded Green's function 70 and that of greater Green's function 66 we see that this

is exactly same as the retarded Green's function (apart from the conventional −i factor).Similarly we can choose to go below the pole. In that case, we would get an expression

which is exactly (within conventional factor of −i) as the advanced Green's function 71 (readwith the help of expression �r greater Green's function 66).

Hence we conclude that quantum mechanically de�ned retarded Green func-

tion 70 and advanced Green's function (and hence time ordered and anti time or-

dered Green's functions as well) are indeed Green's functions for the Schrodinger's

di�erential equation.

9.4.5 Green's Function for Non-Interacting but Non-Homogeneous Case

In this section we consider a practically important case of non-interacting Schrodinger �elds.

We assume that particles are no-interacting but there may be spatially inhomogeneous non-

zero background potential.

Note that the basic expressions for the greater Green's function 65, lesser Green's function

67 and, hence, the four other Green's function derived from those remain intact. We just need

to be careful not to assume the single particle energy wavefunction to be spatial plane waves.

Also note that we do not have the space translation symmetry anymore. But since our

system is closed we still have time translation symmetry. Hence G>(x2, t2 ← x1, t1) can

be written as G>(x2, x1, t2− t1). Note that Green's function now would depend on

two space coordinates and not just di�erence of those two.

We start with the basic expression for greater and lesser Green's functions - expressions

65 and 67 respectively. Based on these two, retarded Green's function 70 (or equivalently

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the time ordered Green's function 68 apart from the conventional −i factor) can be written

as

GR(x2, x1, t2 − t1) = −iΘ(t2 − t1)G>(x2, x1, t2 − t1)

= −iΘ(t2 − t1)∑i

exp(−iEi(t2 − t2)/~)φi(x2)φ?i (x1) (76)

It seems useful to stress that expression 76 is not a function of energy for temporal frequency

as that has been summed over. In case we want to represent retarded Green's function in

terms of temporal frequency, we just perform a temporal Fourier transform. We know that

the Fourier transform of Heaviside step function Θ(t) is πδ(ω) + iω(see Bracewell but with

signs of ω �ipped, for example). Hence we conclude following Fourier transform pair

Θ(τ) exp(−iEiτ/~)→ πδ(ω − Ei/~) +i~

~ω − Ei

Hence Fourier transformed (now only time domain) retarded Green's function can be written

as

GR(x2, x1, ω) = −i∑i

{πδ(ω − Ei/~) +i~

~ω − Ei}φi(x2)φ?i (x1)

= −i∑i

~πδ(~ω − Ei)φi(x2)φ?i (x1) +∑i

~~ω − Ei

φi(x2)φ?i (x1)

Which can equivalently be written as

GR(x2, x1, E) = −i∑i

πδ(E − Ei)φi(x2)φ?i (x1) +∑i

1

E − Eiφi(x2)φ?i (x1) (77)

Hence single-particle local density of states (LDOS) de�ned as

ρ(x,E) ≡∑i

δ(E − Ei)φi(x)φ?i (x)

can be written as

ρ(x,E) = − 1

πIm{GR(x, x, E)}

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10: Non-relativistic, Massive, Bosonic Fields

10 Non-relativistic, Massive, Bosonic Fields

Follows exactly same two alternative schemes given above. Only di�erence is in the commu-

tation relations among �eld operators or creation/annihilation operators. Simpler expression

obtained for additive operators 48 does not hold true for bosons though.

Part VI

Non-Relativistic Quantum

Electrodynamics ( QED )

11 Introduction

Non-relativistic electromagnetic �eld quantization is better known as Quantum Optics. The

reason is because non-relativistic quantization of Maxwell equations are mostly useful for

explaining the properties of light and its interaction with materials. A more elaborate

relativistic version of the subject is more commonly known as Quantum Electrodynamics

(QED) and comes under the bigger subject area of QFT of Gauge Fields. QED is mostly

useful for describing electromagnetic interaction between elementary particles. Hence the

name QED is popular among particle physicists. Laser and device people mostly need non-

relativistic version and in this community subject is better known as Quantum Optics (or

sometimes term cavity-QED is also used).

The idea is to get a classical wave equation. Form a Hamiltonian density function and

identify the generalized co-ordinate and conjugate momentum so that Hamilton's equation

of motion gives the classical wave equation (generally its easier to do this in Fourier space).

Elevate the generalized co-ordinate and conjugate momentum to the status of operators and

force a commutation relationship between them. It turns out that only if you force

Bosonic commutation relation you get the correct physics. This equivalent to

postulate that EM �elds are build of Bosons. Obtain quantum mechanical Hamilto-

nian.

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12: 'Free' Electromagnetic Fields in Non-Dispersive Unbounded Medium

12 'Free' Electromagnetic Fields in Non-Dispersive Un-

bounded Medium

Strictly speaking �elds in material medium can not be called 'free' �elds because �elds do

interact with material atoms and molecules. But in this section we would not call these as

'interactions'. We would use classical Maxwell equations in material medium and quantize

those �elds. Quantized �elds so obtained would be treated as 'free' �elds.

To begin with let us focus on a homogeneous, isotropic, non-dispersive (loss-less), linear

medium. Medium in unbounded and has no sources.

12.1 Classical Fields

12.1.1 Classical Equation of Motion

Maxwell's equation in SI units in such a medium are

~∇. ~E = 0

~∇. ~B = 0

~∇× ~E = −∂~B

∂t

and

~∇× ~B = εµ∂ ~E

∂t

Combining these we get the wave equation

~∇× ~∇× ~E = −εµ∂2 ~E

∂t2

Now using the identity~∇× ~∇× ~E = ~∇(~∇. ~E)− ~∇2 ~E

~∇2 ~E = εµ∂2 ~E

∂t2(78)

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Assuming harmonic time oscillations ~E ∼ ~E exp(−iωt) 34we get

~∇2 ~E = −εµω2 ~E

Further assuming harmonic space oscillations ~E ∼ ~E exp(i~k.~r) we get the dispersion relation

~k.~k = ω2µε (79)

Since we have assumed non-dispersive lossless medium ~k.~k is real and phase velocity and

group velocity are same.

Note that most general solution can be obtained as spatial and temporal Fourier inverse

(linear superposition of above solutions). Due to the dispersion relation obtained above,

integral reduce to be just one dimensional. Because of the assumed non-dispersion, even the

most general pulse form would travel at phase velocity without distortion.

In non-dispersive loss-less medium plane waves can be both homogeneous or inhomoge-

neous plane waves. Which basically means that the �eld amplitudes in a constant phase

plane may or may not be constant. In concise language, even though ~k.~k is real ~k can have

complex valued components. Note that most general solution is Fourier superposition of

solutions like ~E ∼ ~E exp(i~k.~r − iωt). Only constraint that we have is 79.

Let me use this opportunity and take a little diversion to explain the importance of

inhomogeneous waves in the Fourier integrals, even though they are non-existent for lossless

unbounded media being considered in this section. In an homogeneous plane wave, �eld

vectors ~E and ~B are necessarily perpendicular to the direction of propagation. Further the

two these �eld components are also perpendicular to each other. But both these conditions

are not necessarily obeyed by inhomogeneous plane waves. We can substitute the assumed

plane wave solution in the divergence Maxwell equations to obtain

~k. ~E = 0

~k. ~B = 0 (80)

34Physical �elds are real valued. While this is does not bother us much in classical electrodynamics aswe know how to obtain physically signi�cant results at the end of the calculations, we would have to worryabout the real valued-ness of the �elds before we quantize them. This is very important for us in presentcontext as we are developing a new subject. Quantization complex valued �elds would give nonsense. At thispoint we don't have to worry about this too much. We would talk about it again when we start quantizing�elds.

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These expressions are indeed true even for complex valued ~k. Even though ~k.~k is real, there

is possibility of complex ~k. For example consider a homogeneous plane wave incident from a

higher refractive index (but non-dispersive, lossless) medium onto a lower index medium at

an oblique incidence greater than the total internal re�ection angle. In such a case, inside

low index medium, kxy would be real but kz would be imaginary (Z axis being perpendicular

to the interface). Now since there are components of ~k that are complex, in which direction

does ~k points? This is determined only by real part of such a component. Hence ~k points

towards kxy. Hence above scalar product equations (divergence Maxwell equations) do not

lead to conclusion that �eld vectors are perpendicular to the direction of propagation. In

fact in the example discussed above, if the incident wave has perpendicular polarization (in

conventional optics language this is also called the s polarization i.e. ~E is perpendicular to

plane of incidence) then two components of ~B �eld in refractive medium are out of phase

and would rotate in a circular path around ~E. This can be show through use of the curl

Maxwell equation. Let us assume that plane of incidence is the XY plane (ky = 0) and ~E

points towards positive Y axis and the incident wave is traveling towards positive Z and X

axises. Then in the low index material, kxBx + kzBz = 0. Also one can easily show that

inside low index material Ex = Ez = 0. Further, the curl equation in the Maxwell equation

gives us~k × ~E = ω ~B (81)

Above curl expression is also correct even when ~k is complex valued. Hence Bx = −kzEy/ωand Bz = kxEy/ω. If kz and kx are real valued then ~B is linear polarized and is perpendicular

to both ~E and ~k. On the other hand, if kz is purely imaginary then Bx would be 90 degrees

out of phase with Ey. Hence ~B would be perpendicular to ~E but NOT perpendicular to ~k.

In fact ~B would revolve around ~E can have nonzero component along ~k. In a general case

these three equation together would decide direction, amplitude and phases of the two �elds.

If ~k is real then we would have homogeneous plane wave. In this case ~E, ~Band ~k are all

perpendicular to each other. In this section we would concentrated on unbounded lossless

media. So ~k is real and superposition of only homogeneous waves provide most general

solution.

12.1.2 In�nite Volume

Let us assume that space is actually periodic with arbitrary cubic unit cell size ax = ay =

az = a in three Cartesian directions. Further let us assume a �nite but very large crystal

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size that includes Nx = Ny = Nz = N unit cells in three directions. Let the length of this

crystal be L. Hence aN = L. Also volume V = L3. Now, Bloch theorem dictates that for

the eigen solutions of the wave equation:

~E(~r + ~R, t) = ~E(~r, t) exp(i~k. ~R) (82)

here~R = niaxx+ njayy + nkaz z

It is nice to check the self consistency. Since we can enforce arti�cial periodicity by choosing

arbitrary ~R, Bloch theorem should be obeyed for arbitrary primitive vectors and hence

arbitrary ~R. We know that for system we are considering, plane waves are the eigen solutions.

And they do obey 82. Hence enforcing arti�cial periodicity should not create any problems.

Now we would also enforce periodic boundary conditions because, for very large vol-

ume, system properties should be independent of boundary conditions. Enforcing periodic

boundary conditions we get

exp(i~k.~L) = 1

Hence~kj =

2πnxL

x+2πnyL

y +2πnzL

z

where nx, ny nz are integers which can take values up to N and j → {nx, ny, nz}. Also notethat ∫

V

exp(i~kl.~r) exp(−i~km.~r)d3~r = V δlm

Hence instead of exp(i~kl.~r) we would use 1√V

exp(i~kl.~r) as expansion functions so that they

are orthonormal within the volume considered.

12.1.3 General Solution

Now let us explicitly write the real valued physical �elds. Adding all eigen solutions of

wave equation 78, one can write the general solution as

E(r, t) =1√V

1

2

∑j,p

{Cj exp(−iωjt) exp(ikj.r) + C?j exp(iωjt) exp(−ikj.r)}ep

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and

H(r, t) =1√V

√ε

µ

1

2

∑j,p

{Cj exp(−iωjt) exp(ikj.r) + C?j exp(iωjt) exp(−ikj.r)}hp

Note that above �elds are real valued. We are simply taking real part, as done in conventional

classical electrodynamics, by adding the complex conjugate to each expansion term. Here

Cj are complex valued constants. We can also write solutions explicitly in terms of real

valued parameters:

E(r, t) =1√V

∑j,p

Aj cos(kj.r− ωjt+ φ)ep (83)

H(r, t) =1√V

√ε

µ

∑j,p

Aj cos(kj.r− ωjt+ φ)hp

Here, Cj = Aj exp(iφ) where Aj is real valued constant. If we de�ne

Bj(t) = Cj exp(−iωjt)

then we can also write

E(r, t) =1√V

1

2

∑j,p

{Bj exp(ikj.r) +B?j exp(−ikj.r)}ep

and

H(r, t) =1√V

1

2

√ε

µ

∑j,p

{Bj exp(ikj.r) +B?j exp(−ikj.r)}hp

Here Bj are complex valued functions of time (don't confuse with magnetic �elds!).

12.1.4 Classical Hamiltonian

Electromagnetic energy stored in the system is

W =1

2

∫V

d3~r{εE(r, t).E?(r, t) + µH(r, t).H?(r, t)}

Please note that E and H in above expression are real valued (even though we are sticking

with ? notations for future convenience). Hence in the energy expression we have prefactor12and not 1

4as readers might be more familiar with.

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Now, ∫V

d3~rE(r, t).E?(r, t) =1

2

∑j,p

BjB?j

and ∫V

d3~rH(r, t).H?(r, t) =1

2

ε

µ

∑j,p

BjB?j

Hence

W =1

4ε∑j,p

BjB?j +B?

jBj (84)

Problem of EM �eld quantization is slightly di�erent from other �elds that we have

discussed so far in the sense that in other �elds choice of conjugate variables q and p was

obvious - either because they exactly corresponded to classical co-ordinate and momentum

(as in the case of phonons) or because we knew the Lagrangian density (as in the case of

Schrodinger's equation). With EM �elds, choice is not obvious. We take guide from the

harmonic oscillator problem. Suppose, I want to write the classical Hamiltonian 84 as

W =∑j,p

~ωj2

(b?jbj + bjb?j)

where p indicates polarization. This can easily be done by choosing

Bj =

√2~ωjε

bj

I am interested in de�ning two real valued functions of time Qj and Pj which are proportional

to real and imaginary parts of bj. So let35

Qj =1√2

(bj + b?j)

Pj =− i~√2

(bj − b?j)

bj = (Qj√

2+ i

Pj√2~

)

b?j = (Qj√

2− i Pj√

2~)

35For choices of proportionality constants, check the mnemonics subsection under discussion of phonons.

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12.2 Quantization

Hence classical Hamiltonian can be written as

W =∑j,p

~ωj(Q2j

2+P 2j

2~2)

Remembering that Bj = Aj exp(iφj) exp(−iωjt), we can write

Qj(t) =

√2ε

~ωjAj cos(ωjt− φ) (85)

Pj(t) = −~

√2ε

~ωjAj sin(ωjt− φ) (86)

One should compare 85 and 83 and appreciate that Qj(t) basically represents how the real

valued electric �eld oscillates with time. Whereas Pj(t) represents a function which is 90

degrees out of phase.

Lets check if Hamilton's equations are obeyed

−∂W∂Qj

= −~ωjQj = Pj

and∂W

∂Pj=ωjPj

~= qj

Hence, Qj and Pj would satisfy the Hamilton's equations.

12.2 Quantization

In EM �eld case, we are actually doing k space quantization.

[Qj, Pj] = i~

Hence

[bn, b†k] = δn,k

[bn†, b†k] = 0

[bn, bk] = 0

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The quantum mechanical Hamiltonian can simply be obtained by promoting �eld amplitudes

to be operators

H =∑k

~ω{b†kbk +1

2}

Also

E(r, t) =∑j,p

√~ωj2εV{bj exp(ikj.r) + b†j exp(−ikj.r)}ep

and

H(r, t) =

√ε

µ

∑j,p

√~ωj2εV{bj exp(ikj.r) + b†j exp(−ikj.r)}hp

Part VII

'Free' Electromagnetic Fields in

Non-Dispersive, Inhomogeneous Medium

Strictly speaking �elds in material medium can not be called 'free' �elds because �elds do

interact with material atoms and molecules. But in this section we would not call these as

'interactions'. We would use classical Maxwell equations in material medium and quantize

those �elds. Quantized �elds so obtained would be treated as 'free' �elds.

Let us �rst concentrate on non-dispersive and hence loss-less media. Again all material

media are dispersive (through Kramer's Kronig relations) but if we are concentrating on

frequency ranges far away from material resonances, one can assume material to be loss-

less non-dispersive medium with refractive index di�erent from 1. Maxwell's equation in SI

units in such a medium (further assumed source free and non-magnetic) are

∇.E = 0

∇.B = 0

∇× E = −µ0∂H

∂t

and

∇×H = ε∂E

∂t

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Let us say that the classical normal mode expansion gives us (explicitly real valued

�elds)

E(r, t) =1

2

∑j

{Cjηj(t)fj(r) + C∗j η∗j (t)f

∗j (r)}

where

ηj(t) = exp(−iωjt)

and

fj(r) =gj(r)√ε(r)

where gj(r) forms a complete set of orthonormal basis but not fj(r). Hence, mode orthonor-

mality is enforced as ∫ε(r)fj(r).f

∗j′(r)dr = δjj′

Enforcing Maxwell equation

∇× E+j = −µ0

∂H+j

∂t1

2{Cjηj(t)∇× fj(r)} = iωµ0H

+j

Superscript + denotes that we have not made the �elds real valued yet and we are doing

operations on one �component� of real valued �elds. Hence we can write36

H(r, t) =1

2

∑j

−iµ0ωj

{Cjηj(t)∇× fj(r)− C∗j η∗j (t)∇× f∗j (r)}

Now if we de�ne mode oscillation amplitude variable Bj = Cjηj(t) and B∗j = C∗j ηj(t) one

36Incidentally, one should note that enforcing the other curl Maxwell equation one can prove the gj(r)forms a complete set of orthonormal set.

∇×H+j = ε

∂E+j

∂t12−iµ0ωj

Cjηj(t)∇×∇× fj(r) = −iωjεE+j

Cjηj(t)∇×∇× fj(r)2µ0ω2

j ε= E+

j =12Cjηj(t)fj(r)

Hence∇×∇× fj(r)µ0ω2

j ε(r)= fj(r)

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can write

E(r, t) =1

2

∑j

{Bjfj(r) +B?j f∗j (r)}

H(r, t) =1

2

∑j

−iµ0ωj

{Bj∇× fj(r)−B?j∇× f∗j (r)}

One should check that if fj(r) = exp(ikj.r)/√ε0 then ∇×fj(r) = (ikj exp(ikj.r)/

√ε0)h hence

we get the same expression as obtained previously if treat coe�cients here are scaled by√ε0.

One can also write the classical Hamiltonian as :-

W =∑

j

1

4B∗jBj +BjB

∗j

But, if want to write it as :-

W =∑

j

~ωj2b∗jbj + bjb

∗j

then we normalize the mode oscillation amplitudes to de�ne normalized mode oscillation

amplitude variable as

bj =

√1

2~ωjBj

and we get

E(r, t) =∑j

√~ωj2{bjfj(r) + b∗j f

∗j (r)}

H(r, t) =∑j

−iµ0

√~

2ωj{bj∇× fj(r)− b∗j∇× f∗j (r)}

The classical Hamiltonian can now be written as

W =∑

j

~ωj2b∗jbj + bjb

∗j

If we write bj = qj + ipj and b∗j = qj − ipj (q and p being real numbers) then we can see

that qj and pj turns out to be generalized co-ordinates and conjugate momentum in the

sense that they satisfy the Hamilton's equation of motion. Note that qj turns out to be

the normalized real mode oscillation electric �eld amplitude. Whereas pj turns

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13: 'Free' Electromagnetic Fields in Dispersive, Inhomogeneous Medium

out to be normalized real mode oscillation magnetic �eld amplitude. When we

quantize it and elevate qj and pj to the status of operators satisfying Bosonic

commutation relation then one can prove that b†j and bj behaves exactly as the

creation and annihilation operators of a decoupled harmonic oscillator. One can

easily show that these operators obey the Bosonic commutation relations as well.

The fact that this scheme works in which we treat �elds as conjugate variables

should be treated as a postulate of quantum optics.

• H =∑

j ~ωj(b†jbj + 12), Nj = b†jbj

• [b†i , b†j] = 0, [bi, bj] = 0 and [bi, b

†j] = δij

• bj →√nj , b

†j →

√nj + 1

13 'Free' Electromagnetic Fields in Dispersive, Inhomo-

geneous Medium

Let us say that the classical normal mode expansion gives us :-

E(r, t) =1

2

∑j

Ajηj(t)fj(r) + A∗jη∗j (t)f

∗j (r)

H(r, t) =1

2

∑j

1

µ0

Ajωjκj(t)∇xfj(r) +

A∗jωjκ∗j(t)∇xf ∗j (r)

where

ηj(t) = exp(−iωjt) exp(−ωjt/2Qj)

κ(t) =1

ωj

dηj(t)

dt

Hence approximately,

κj(t) = −i exp(−iωjt) exp(−ωjt/2Qj) = −iηj(t)

Note that the mode orthonormalization is enforced as:-∫ε(r)fj(r)f

∗j′(r)dr = δjj′

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13: 'Free' Electromagnetic Fields in Dispersive, Inhomogeneous Medium

Usually fj =gj√ε. Now if we de�ne mode oscillation amplitude variable Bj = Ajηj(t) and

B∗j = A∗jηj(t) one can write the classical Hamiltonian as :-

W =∑

j

1

4B∗jBj +BjB

∗j

But, if want to write it as :-

W =∑

j

~ωj2b∗jbj + bjb

∗j

then we normalize the mode oscillation amplitudes to de�ne normalized mode oscillation

amplitude variable as

bj = −i

√1

2~ωjBj

and we get

E(r, t) =∑j

i

√~ωj2bjfj(r) +

√~ωj2b∗jf∗j (r)

H(r, t) =∑j

√~

2µ0ωjbj∇× fj(r) +

√~

2µ0ωjb∗j

1

ωj∇× f ∗j (r)

The classical Hamiltonian can now be written as

W =∑

j

~ωj2b∗jbj + bjb

∗j

If we write bj = qj + ipj and b∗j = qj − ipj (q and p being real numbers) then we can see

that qj and pj turns out to be generalized co-ordinates and conjugate momentum in the

sense that they satisfy the Hamilton's equation of motion. Note that qj turns out to be

the normalized real mode oscillation electric �eld amplitude. Whereas pj turns

out to be normalized real mode oscillation magnetic �eld amplitude. When we

quantize it and elevate qj and pj to the status of operators satisfying Bosonic

commutation relation then one can prove that b†j and bj behaves exactly as the

creation and annihilation operators of a decoupled harmonic oscillator. One can

easily show that these operators obey the Bosonic commutation relations as well.

The fact that this scheme works in which we treat �elds as conjugate variables

should be treated as a postulate of quantum optics.

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13.1 Coherent States / Poisonian Distribution

• H =∑

j ~ωj(b†jbj + 12), Nj = b†jbj

• [b†i , b†j] = 0, [bi, bj] = 0 and [bi, b

†j] = δij

• bj →√nj , b

†j →

√nj + 1


Coherent state is de�ned as an eigen state of annihilation operator (non Hermitian)b. Eigen

values would be complex valued. Apriori we don't even know that these eigen states exist

but this proved below. Let us work with single optical mode. Suppose eigen state in

Fock basis is written as

|v〉 =∞∑n=0

cn|n〉

and

b|v〉 ≡ v|v〉 (87)

where v can be a complex number. One can easily get an equation that says

∞∑n=1

cn√n|n− 1〉 = v

∞∑n=0

cn|n〉

we know that |n〉 forms a complete set of orthogonal states. Hence one get a recursion

relation between cn. So one can �nally write every coe�cient in terms of one coe�cient c0

and get

|v〉 = c0

n=∞∑n=0

vn√n!|n〉

We can also eliminate c0 by normalizing the state

|v〉 = exp(−|v|2/2)n=∞∑n=0

vn√n!|n〉

Interesting thing to note is that |v = 0〉 ≡ |n = 0〉. Hence the vacuum state is same as

the coherent state with associated eigen value of v = 0. hence vacuum state can either be

treated as coherent state or the Fock state. Also note that

P (n) = |〈n|v〉|2 =|v|2n

n!exp(−|v|2/2)

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and

〈v|n|v〉 =∑n

nP (n) = |v|2

Even for very small mean number of photons (except for zero) in the single mode, there is

always �nite probability of �nding very large number of photons in this mode.

Also not that according to the de�nition 87, application of annihilation operator on

coherent state leaves the state intact (apart from a factor). Hence even if photons are

absorbed from coherent state, state remains coherent.

Since almost all photo detection and measurement process involves absorption of photons,

annihilation operator has a special signi�cance. And so does the coherent state.

One can also write coherent state as

|v〉 = exp(−|v|2/2)n=∞∑n=0

vnb†n√n!|0〉 = exp(−|v|2/2) exp(vb†)|0〉

One should note that

exp(−v?b)|0〉 = |0〉

hence we can write

|v〉 = exp(−|v|2/2) exp(vb†) exp(−v?b)|0〉

Using Campbell-Baker-Hausdor� theorem one can show that

exp(−|v|2/2) exp(vb†) exp(−v?b) = exp(vb† − v?b) = D(v)

hence

|v〉 = D(v)|0〉

For q representation of coherent state

Ψv(q) ={

(ω

π~)1/4 exp(Im(v))

}exp

{− ω

2~[q − (

2~ω

)1/2v]

}Hence we see that the coherent state is similar to Gaussian function where the peak is shifted

by complex distance (2~ω

)1/2v. Since v = 0 is vacuum state, we can see why we call coherent

state as displaced vacuum state.

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Coherent state evolves in time as

|Ψ(t)〉 = exp(−iωt/2) exp(−iωtn)|v〉

using Fock representation for coherent state one can immediately see that

|Ψ(t)〉 = exp(−iωt/2)|v exp(−iωt)〉

which is just another coherent state.

Glauber-Sudarshan representation of arbitrary EM �eld state says that any state can

represented as

ρ =

∫φ(v)|v〉〈v|d2v

where ρis the density matrix. Because ρis Hermitian , φ(v) would be real valued. One should

remember that |v〉 are not orthogonal and hence such an expansion should be interpreted

carefully. It is generally said that when φ(v) behaves like a probability distribution (non-

negative) then light can be considered in classical state. Obviously for a coherent state φ(v)

would be 2D delta function. For random phase single mode laser light (generally intensity

is stable and does not vary, but phase shows random �uctuations and drifts in time). Then

one can write φ(v) = 12πr0

δ(|v| − r0). Since phase of v does not appear in this equation it is

understood to be distributed uniformly from 0 to 2π. Similarly for single mode of thermal

light φ(v) = 1π〈n〉 exp(−|v|2/〈n〉).

13.1.1 Properties of Fock states

One can easily show

〈n|b|n〉 = 〈n|b†|n〉 = 0

〈n|q|n〉 = 〈n|p|n〉 = 0

〈n|q2|n〉 = (~/ω)(n+1

2)

〈n|p2|n〉 = ~ω(n+1

2)

Hence product of standard deviations from mean values {〈(q − 〈q〉)2〉}1/2= {〈q2〉}1/2

and

{〈p2〉}1/2would be equal to ~(n+ 1

2). So product of uncertainties increases as the excitation

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13.2 Heisenberg Representation of Field Operators

is increased.

Fock states in q representation can be written as Hermit-Gaussian polynomials.

13.1.2 Phase Operator

We try to design an operator that can give us phase of oscillation. We de�ne the phase

operator as

ζ ≡ ˆexp(iφ) ≡ 1

(n+ 1/2)1/2b

and then we de�ne cosine and sine operators

C =1

2

{ζ + ζ†

}S =

1

2i

{ζ − ζ†

}13.1.3 Operator Theorems

One can easily prove

exp(ipx)q exp(ipx) = q + x

Suppose we de�ne displacement operator as

D = vb† − v?b

then

exp(−D)b exp(D) = b+ v

and

exp(−D)b† exp(D) = b† + v?

13.2 Heisenberg Representation of Field Operators

14 Optical Coherence Theory

14.1 De�nitions

Stochastic or Random Process In non mathematical terms a stochastic or a random

process is a set (or an ensemble) of space-time functions. Suppose we are looking at a beam

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14.1 De�nitions

of light. Suppose we concentrate on electric �eld oscillations with time at one space point

only. Let this time evolution be written as fi(t). Now if we do the same experiment again,

we may not get the same time evolution again � we may get fj(t). Collection or ensemble

of all such functions would be called a stochastic or a random process.

One can also call random process a collection of random variables. So a set of values for

a �xed j {fi(tj)} are di�erent possible outcomes of electric �eld measurement at �xed time

and �xed space. In this language we can treat f(t = tj) as a random variable.

Stationary Random Process We might be able to de�ne statistical properties like mean,

variance etc. of a random variable f(t = tj). Now these same quantities can be de�ned for

all random variables in a random process � that means we can de�ne mean variance etc.

at each instance of time. To be more speci�c, I measure electric �eld amplitude at �xed

space and time point by repeating the experiment many times. I take average of all values

received. Now if I do the same same thing for a di�erent instance of time, what would be

the result? Would the statistical values be equal? May be or may not be. If they are then

we call it a stationary random process.

In fact, all statistical properties should remain independent of time. That means prob-

ability density function P (fi(t = tj)) ∀i should be independent of tj (this is actually called

�rst order stationary random process).

Now let us look at a little bit more complicated situation. Suppose we look at joint

probability density function P (fi(t = tk), fj(t = tl)) ∀i, j. If it turns out that this joint

probability density function is same as P (fi(t = tk + τ), fj(t = tl + τ)) ∀i, j for any value of

τ then it would mean that absolute value of time does not matter. Joint probability density

function only depends on the di�erence between times at which two random variables are

measured. Such a process is called second order stationary random process. For a

second order stationary process, the auto-correlation function becomes an important quantity

to study.

A random process is called strict sense stationary (SSS) random process if all types of

probability density functions are independent of absolute value of time and only depends on

relative values of time.

One also de�nes a wide sense stationary (WSS) random process, for which we de�ne that

mean of a random variable is independent at which instance of time it is measured and the

auto-correlation function is only dependent on the relative time instances. Note that, by

de�nition, all second order stationary processes are WSS but reverse is not true.

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14.2 Coherence

Macroscopically Stationary or Steady Light Macroscopically stationary or steady

light is one which does not exhibit any �uctuations in �elds at macroscopic time scales.

Basically �uctuations can be represented as stationary random process whose mean period

and correlation time are much shorter than averaging interval needed to make an observa-

tion. Generally speaking, if you look at one possible time evolution of light at one space

time (basically one signal out of ensemble of signals in the random process), then there

would be very fast oscillating noise (much much faster than the mean frequency of light).

In a macroscopically stationary light, this noise averages to zero. Moreover this noise is

completely second-order uncorrelated. So if fi(t) is one evolution in the ensemble then we

can write this as fi(t) = gi(t) + ni(t) where ni(t) is the microscopic fast oscillating noise.

Now we are claiming that the ensemble mean〈ni(t)〉i is zero and is independent of t and also

〈ni(t)ni(t + τ)〉i is zero for all τ (greater than a very small value) and is independent of t.

Please note that light can still have macroscopically detectable noise which is part of gi(t).

Concept can be understood more easily if we assume ergodicity. In that case we can

replace ensemble averages with time averages. So if look at one possible time evolution fi(t)

out of the ensemble, then very fast oscillating ni(t) averages to zero within a very small

interval of time. Also time average ni(t)ni(t + τ) is zero as soon as τ is greater than an

extremely small number.

Quasi-monochromatic Light If the spectral bandwidth of the light is much smaller than

the average frequency than we that light is quasi-monochromatic.

14.2 Coherence

In standard books, authors separately de�ne concepts of longitudinal coherence (also known

as temporal coherence) and transverse coherence (also known as spatial coherence). Strictly

speaking, these concepts are interlinked and can not be de�ned separately. Maxwell's equa-

tions links the longitudinal and transverse propagation of light and hence coherence proper-

ties in both direction are interleaved.

Temporal coherence is manifestation of two-time correlation (ensemble averaged) between

�eld values measured at the same space point. For stationary �elds this two time correlation

function only depends on the di�erence between two time instances. Hence its a function

of only one time variable. Basic idea is that if are interested in time oscillation correlations

(ensemble averaged) at one spatial point only, then just by Fourier transform, this can be

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14.2 Coherence

related to the temporal spectrum of the �eld at the same space point37.

On the other hand, spatial coherence is manifestation of two-space point correlation

(ensemble averaged) between �eld values measured at two di�erent space points but at same

instance of time.

From practical point of view, we do not actually measure ensemble averages but we

measure time averages. All detectors do some sort of time averaging (time averaging period

is usually much larger than the mean time period of oscillation of the light �eld). For ergodic

�elds ensemble averages can be replaced by time averages. So assuming that optical �elds

are ergodic, when interference patterns are recorded by detectors, we are in fact recording

correlation functions.

In most simple interference experiments, we always detect net �elds at single spatial

points only. So in that sense we are mostly recording two time correlation functions. But

in some experiments (like Young's double slit experiment) this two time correlation can

be traced back to two-space-point correlation functions. So we say that these experiments

manifest the spatial coherence of the light. Whereas in some experiments (like thin �lm inter-

ference or Michelson interference) we are actually measuring two time correlation function.

So we say that these experiments manifests temporal coherence of the light.

14.2.1 Complex Degree of Coherence

Let V (r, t) is a complex valued analytic function whose real part represents certain light �eld

component. One can de�ne the cross-correlation (known as mutual coherence function)

as

Γ(r1, r2; t1, t2) = 〈V ?(r1, t1)V (r2, t2)〉ensemble

For stationary processes one can write

Γ(r1, r2; τ) = 〈V ?(r1, t)V (r2, t+ τ)〉ensemble

And moreover, for ergodic �elds, one can replace the ensemble average by time averaging

Γ(r1, r2; τ) = 〈V ?(r1, t)V (r2, t+ τ)〉time = LimT→∞1

2T

∫ +T

−TdtV ?(r1, t)V (r2, t+ τ)

37Generally speaking, one should also be able to relate this to the temporal spectrum of the extendeduniform homogeneous source itself.

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14.2 Coherence

One can normalize the mutual coherence function to de�ne a quantity γ(r1, r2; τ) known as

complex degree of coherence

γ(r1, r2; τ) =Γ(r1, r2; τ)

(Γ(r1, r1; 0))1/2(Γ(r2, r2; 0))1/2

From interference measurements, one can determine the real part of γ. But since γ is an

analytic function, its real part is related to its imaginary part through Hilbert transform pair.

In typical interference experiments, the visibility of fringes is related to the absolute value

of γ. One can also de�ne equal time correlation (ETCR) functions J(r1, r2) = Γ(r1, r2; 0)

and j(r1, r2) = γ(r1, r2; 0). These terms are typically known as mutual intensities.

One can also de�ne cross-spectral density function W (r1, r2; ν) as

W (r1, r2; ν)δ(ν − ν ′) = 〈V ?(r1, ν)V (r2, ν′)〉ensemble

where V is the spectral Fourier transform of V . It turns out that cross-spectral density

function is simply the Fourier transform of mutual coherence function Γ(r1, r2; τ). That

is W = FΓ. This is known as Wiener-Khintchine theorem. One can also de�ne a

normalized cross-spectral density function µ(r1, r2; ν) which is known as spectral degree

of coherence at frequency ν

µ(r1, r2; ν) =W (r1, r2, ν)

W 1/2(r1, r, ν)W 1/2(r2, r2, ν)

It turns out that µ 6= Fγ.

One can easily show that

∇21V (r1, t1) =

1

c2

∂2V (r1, t1)

dt21

∇22V (r2, t2) =

1

c2

∂2V (r2, t2)

dt22

Hence, it is straightforward to show that

∇21Γ(r1, r2; t1, t2) =

1

c2

∂2Γ(r1, r2; t1, t2)

dt21

∇22Γ(r1, r2; t1, t2) =

1

c2

∂2Γ(r1, r2; t1, t2)

dt22

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14.2 Coherence

For stationary �elds then

∇21Γ(r1, r2; τ) =

1

c2

∂2Γ(r1, r2; τ)

d2τ

∇22Γ(r1, r2; τ) =

1

c2

∂2Γ(r1, r2; τ)

d2τ

And with temporal Fourier transforms

∇21W (r1, r2; ν) = −k2W (r1, r2; ν)

∇22W (r1, r2; ν) = −k2W (r1, r2; ν)

where k = 2πν/c. This clearly shows the coupling between spatial and temporal coherence.

Light emitted by a spatial incoherent quasi-monochromatic source, would acquire some spa-

tial coherence in far �eld as it propagates.

Longitudinal Coherence (Temporal coherence related toγ(r1, r1; τ)) Let us us say

we considering a beam of light emitted by a very small source. Light is quasi-monochromatic

and is macroscopically steady.

Formation of fringes in a Michelson interferometer is considered as manifestation of tem-

poral coherence. This is because existence of fringes means that �elds at same space point but

at two di�erent times are correlated. Experimentally one would see fringes only if ∆t∆ν ≤ 1

where ∆t is relative delay introduce by the interferometer between two beams. We de�ne

1/∆ν as the coherence time and c/∆ν as longitudinal coherence length.

If Young's double slit experiment is done with a point source or a line source, this would

also manifest the temporal coherence e�ects. Field oscillations at two slits would now always

be in sync. But as we look at higher order maxima on the screen, the path length di�erence

from slits to screen may exceed the longitudinal coherence length. This e�ect can be removed

if we place a narrow band �lter after the double slits.

As the bandwidth of the source increases (decreasing temporal coherence), interference

fringes would seem to be lost. This is simply because eye can not di�erentiate colors very

well. Temporal coherence e�ects can be rigorously studied in Fourier domain. If modal

decomposition is done, and each frequency interference pattern is simulated we would not

see any coherence e�ects. Each spectral mode is completely coherent and hence generates

perfect fringes. In white light, fringes seem to be lost because of superposition of these

perfect colored fringes. If a spectrometer is used to resolve the �eld pattern generated by

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14.2 Coherence

light from a point white source passing through double slit on the screen, one can see these

perfect fringes. This is called Edser-Butler bands and spectrum is known as channeled

spectrum(author?) [1].

Incandescent sources (e.g. heated �lament) or gas discharge sources with higher spectral

purity typically have longitudinal coherence length of the order of 3m whereas well stabilized

laser its can be as large as 30km. (typically HeNe laser 10m, red line of Cadmium lamp

0.2m, and ~100um for white light). Longitudinal coherence length of solar radiation is about

250nm.

Transverse Coherence (Spatial coherence related to γ(r1, r2, τ ∼ τ0)) Spatial co-

herence is described through another kind of interference experiment - Young's double slit

interference experiment. In this we consider an extended thermal source. Spatial coherence

deals with correlation between the light �elds reaching at one point from two di�erent space

points. In Young's double slit experiment, appearance of fringes would manifest that there

is spatial coherence between two light �elds coming from two pinholes at one point of the

observation screen. This fact then gets related to fact that the �elds reaching at two spatially

separated slits from an extended source should be co-related.

We claim that �elds reaching two slits of the experiment from an extended source would

be co-related with each other. Consider two point sources within the extended light source.

And consider two slits of the experiment. Now light reaching at two slits from one of the

point sources would be co-related (because of assumed temporal coherence) if the path length

di�erence is smaller than the longitudinal coherence length. Similarly light reaching from

the second point source at two slits would be correlated. Key concept here is that light

�elds reaching at one slit from two point source are NOT correlated because point

sources in extended source are supposed to be statistically independent. Despite this fact

the total �eld at two slits would be correlated.

Now if the size of the extended source is increased then path length di�erence would

increase and fringes would dis-appear (due to quasi-monochromaticity). This can also be

understood alternatively. One can think each point source generating an intensity fringe

pattern on the screen. Due to symmetry consideration, di�erent point sources would generate

interference patterns that are spatially separated from each other. They averages out and

fringe patterns disappear.

If source is of squared shape with one length ∆s and if two slits subtends and angle ∆θ at

the center of the source then fringes would appear if ∆θ∆s ≤ λ where λ is mean wavelength.

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14.3 Coherence and Power Spectrum

Note that one can try to do this double slit experiment with solar light provided one uses

a bandpass �lter (before the double slits38) to remove the temporal coherence e�ects. One

would �nd that, based on the angle sun subtends at earth, area in which fringes would be

visible (hence, one can de�ne a parameter called transverse coherence length) on the screen

would be extremely small. Whereas one can see better fringes from stars that are much

farther because they would behave more like single point sources.

14.2.2 Typical Interference Experiments

• Wavefront division

• Amplitude division

14.3 Coherence and Power Spectrum

Coherence and power spectrum are related terms. In quantum optics, spectrum becomes a

very confusing term at �rst sight. Let me explain this a bit further. Suppose I am looking

at single mode of �free� EM �eld. Basically it's a vacuum mode. What I mean by �free�

�eld is that we are also looking at a closed system of only optical �elds, so Hamiltonian is

time independent. There might be a cavity, so that free �eld mode pro�le can be di�erent

from a plane wave. But there is no interaction with anything and closed system consists of

only photons in this single mode. So this single mode is actually energy eigen state of the

complete system. Keep in mind that we looking at a single mode (for example all other modes

might have very di�erent frequencies). Now quantum mechanically we can have di�erent

number of photons in this mode. We may even have quantum mechanical uncertainties in

number of photons in this mode. So if we measure a power spectrum what would we get?

Classically one may try to imagine a sequence of pulses with di�erent amplitude (to simulate

di�erent number of photons) but same local frequency (to simulate single monochromatic

mode behavior). But if we Fourier transform such a train of pulses, signal would have a

broad spectrum. So does signal really have a broad spectrum? Remember we are looking

at monochromatic mode. Answer is no, signal would not have broad spectrum (assuming

we are looking at a closed system). Right way to picture the situation is to imagine many

38If �lter is put after the slits, temporal coherence e�ect can be removed but not the spatial coherencee�ects. So if the size of the source is extended, interference pattern would indeed be lost. On the other handif monochromaticity of the light is reduced (accordingly size of the source is reduce to achieve same spatialcoherence), interference pattern would remain same.

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monochromatic plane wave with di�erent amplitudes. And system is in linear superposition

of all these. Hence spectrum is really a delta function. Quantum mechanically, best way

to describe signal spectrum is to de�ne it as Fourier transform of auto correlation function

between �eld operators. One can explicitly show that for a closed system, magnitude of auto

correlation function is constant and hence spectrum is a delta function.

Now let us consider a slightly more complicated system. Suppose our closed system

consists of photons and a few interacting atoms. Total Hamiltonian is still time independent.

But now the single mode of EM �eld does not represent the energy eigen state of the system.

But we only want to look at photon states. System states might actually be entangled. So

how do we separate photon states only? We can do this by using reduced density matrices.

Basic idea is that now correlation would be function of t as well as τ . In other words

spectrum would be time dependent. One may be able to approximate this some how as time

independent spectrum but even then we would have broad power spectrum.

Almost all optical detectors work by absorbing photons. Hence any power spectrum

analyzer would have to have absorbers.

Part VIII

Relativistic Quantum Field Theories (

QFT )

15 Klein Gordon (KG) Field

15.1 Derivation of 'Classical' KG Field

Klein-Gordon equation is the most simple relativistic quantum equation. Let us see how one

can obtain KG equation following standard quantization rule. We know that ~p.~p+m2 = E2.

In 4-vector notations

pµpµ = m2

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15.1 Derivation of 'Classical' KG Field

where momentum 4-vector is de�ned as

pµ ≡ {E, p1, p2, p3}

Remember that corresponding contravariant 4-vector would have negative signs with last

three entries

pµ ≡ {E,−p1,−p2,−p3}

Hence

pµpµ = E2 − ~p.~p = m2

One then does the standard quantization (need the energy operator as well, see below) and

obtains

−~2∂µ∂µ = m2

In natural units (~ = 1,c = 1) above equation is written as

−∂µ∂µ = m2

This is the KG equation. Writing it in standard form

−∇2 +m2 = − ∂2

∂t2

Alternative way of obtaining the above equation (which follows the standard quantization

more closely) is as follows.

H = i~∂

∂t

Hence

H2 = −~2 ∂2

∂t2

Now in relativistic quantum mechanics H2 = p2 +m2. Converting p to operator and plugging

in the above equation one can again get the KG equation. Only di�erence from standard

quantization technique is that one needs square of energy as there is square root involved.

This is the starting points of all troubles in relativistic quantum mechanics.

There were lots of problems with the KG equation. That's the reason for which initially

it was discarded. We now know that it is correct equation for spin zero bosons like pions39.

39Scalar real valued KG �eld would represent neutral pions. Charged pions would be represented by scalarcomplex valued KG �eld. Note that charge comes out as a conserved quantity due to a symmetry of complex

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15.2 Classical Hamiltonian Theory of Real Valued Scalar KG Field

We now know how to interpret the peculiarities of this equation. Major problem with this

equation is that it can accept both positive and negative energies. We would see that this

can be interpreted as anti-particle states. If one interprets 2 as probabilities then one gets

into the trouble of negative probabilities. We will discuss the Noether's theorem latter.

Basically its a continuity equation. From the continuity equation one obtains the correct

interpretation of probabilities and densities associated with the Ψ. We would then interpret2 as charge density.


15.2.1 Classical Hamiltonian and Lagrangian

A free-space KG equation is written as

(−∇2 +m2)Ψ(x, t) = −∂2Ψ(x, t)

∂t2

First of all, we want to �nd the Lagrangian density function L(Ψ(x), ˙Ψ(x), ∂Ψ(x), t) so that

we can �nd the conjugate momentum variable π(x) which is conjugate to Ψ(x). Note that

we are already working in generalized co-ordinate system. I would sometime drop the

explicit time dependence from Ψ(x, t) for smaller notations, spatial dependence is usually

kept to speci�cally indicate that we dealing with continuum version of in�nite set of general-

ized variables. We start by selecting Ψ(x) as in�nite set of generalized co-ordinate variable.

This choice is never unique. The choice of Lagrangian density is never unique as well.

We should be able to �nd a function so that the associated Lagrange's equation is the above

KG �eld equation. Let us try (we would justify that this is one of the many possible valid

choices) :-

L(Ψ(x), Ψ(x), ∂Ψ(x), t) =1

2Ψ2 − 1

2(∇Ψ)2 − 1

2m2Ψ2 (88)

values scalar KG �eld which is known as gauge invariance. Why this conserved quantity is identi�ed withelectric charge is a bit involved issue. See Chapter 3 in the book �Quantum Field Theory� by Lewis H.Ryder. One may also try to introduce a vector (a Euclidean vector in real 3D space) complex valued �eld.But this equation has problems under Lorentz transformations and one needs to move to Dirac equation.One then realizes that vector nature of �eld represents spin. One should note that even the complex scalar�eld can be treated as two-component real valued scalar �elds. And one may want to call it a vector �eld.But note that this vector space has noting to do with rotational symmetry in real physical 3D space.

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The Euler-Lagrange's equation of motion would be :-

d

dt(

∂L

∂ ˙(Ψ(x))) +∇(

∂L

∂(∇Ψ(x))))− ∂L

∂Ψ(x)= (

∂2

∂t2−∇2 +m2)Ψ(x) = 0

Which is exactly same as the KG equation. Hence, above expression for Lagrange density is

one of the many possible acceptable expressions. From L we can easily obtain π(x) (variable

conjugate to Ψ(x)) as follows :-

π(x) ≡ ∂L

∂ ˙(Ψ(x))= Ψ(x) (89)

And we can now construct the Hamiltonian density :-

H(Ψ(x), π(x),∇(Ψ(x))t) ≡ π(x) ˙Ψ(x)− L =1

2π2 +

1

2(∇Ψ)2 +

1

2m2Ψ2 (90)

If we move to energy eigen basis, we can �nd a more convenient expression for Hamiltonian

density. So let try that �rst.

15.2.2 Ψ(x, t) Field in Momentum and Energy Basis

One can easily check that the energy eigen states of free-space KG equation are (we still

have to add solutions to make them real valued) :-

ΨE(x, t) = A exp(ipx− iEt)

where,

E = ±√m2 + p2 = ±Ep

Here Ep is always a positive number. Note that for one value of p, there are two

values of E possible - one positive and one negative (classical KG �eld theory does not have

any acceptable justi�cation for negative energies). Hence we note that classical �eld can be

written as a general superposition of all possible modes :-

Ψ(x, t) =1

2π

∫dp{Ψ+(p) exp(ipx) exp(−iEpt)}+

1

2π

∫dp{Ψ−(p) exp(ipx) exp(iEpt)}

(91)

The factors of 2π have been included just because of conventions. These can be dumped

into Ψ+ and Ψ− which we anyway would-re normalize latter on. Above can also be written

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as:

Ψ(x, t) =1

2π

∫dp{f(p, t) exp(ipx)} (92)

where,

f(p, t) = Ψ+(p) exp(−iEpt) + Ψ−(p) exp(iEpt) (93)

Note that 92 is exactly a spatial Fourier transform. Which is also same as expanding Ψ(x, t)

in momentum basis. Note that energy basis expansion needs two expansion coe�cient for

one |E| (or for one p) but momentum space expansion needs only one coe�cient. Ψ+(p)

and Ψ−(p) would be recognized as energy mode amplitudes and f(p, t) as momentum mode

amplitude of �eld Ψ(x).

Let us restrict our analysis for the time being to only those solutions which are real-

valued. More generic complex valued solutions and their interpretations would be seen

latter. Hence:

1

2π


1

2π


=1

2π

∫dp{Ψ∗+(p) exp(−ipx) exp(iEpt)}+

1

2π

∫dp{Ψ∗−(p) exp(−ipx) exp(−iEpt)}

This can also be re-written as

1

2π


1

2π

∫dp{Ψ−(−p) exp(−ipx) exp(iEpt)}

=1

2π

∫dp{Ψ∗+(p) exp(−ipx) exp(iEpt)}+

1

2π

∫dp{Ψ∗−(−p) exp(ipx) exp(−iEpt)}

Hence:

1

2π

∫dp{Ψ+(p)−Ψ∗−(−p)} exp(ipx) exp(−iEpt)

=1

2π

∫dp{Ψ∗+(p)−Ψ−(−p)} exp(−ipx) exp(iEpt)

This would be true only if

Ψ+(p)−Ψ∗−(−p) = 0

or (and)

Ψ∗+(p)−Ψ−(−p) = 0 (94)

This is an important result. It shows that the restriction that the �eld Ψ(x, t) is

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real ensures that the mode amplitude of negative energy state is always related

to the mode amplitude of positive energy state. With this, one can write 91 as:

Ψ(x, t) =1

2π


1

2π

∫dp{Ψ∗+(p) exp(−ipx) exp(iEpt)}

(95)

15.2.3 π(x,t) Field in Momentum and Energy Basis

Di�erentiating 95 with respect to time and using 89:

π(x, t) =−iEp

2π{∫dp{Ψ+(p) exp(ipx) exp(−iEpt)} −


(96)

Which can also be written as:

π(x, t) =−iEp

2π{∫dp{Ψ+(p) exp(ipx) exp(−iEpt)}−

∫dp{Ψ−(p) exp(ipx) exp(iEpt)} (97)

And analogy to 92, we can also write

π(x, t) =−iEp

2π

∫dp{g(p, t) exp(ipx)} (98)

where,

g(p, t) = Ψ+(p) exp(−iEpt)−Ψ−(p) exp(iEpt) (99)

With this, we have been able to write the generalized co-ordinate and its conjugate

momentum as B+B? and B−B? just as in the case of single Harmonic oscillator case. We

would see latter how these two parts of solution gets converted to creation and annihilation

operators and how Hamiltonian gets 'factorized' and subsequently diagonalized when we do

the quantization in the next section.

15.2.4 Classical Hamiltonian in Momentum and Energy Basis

One should notice that the classical Hamiltonian (not density) can be written as :

H =

∫dx{1

2π2 +

1

2(∇Ψ)2 +

1

2m2Ψ2} =

∫dp

1

4π{E2 +p2 +m2}{Ψ+(p)Ψ∗+(p)+Ψ∗+(p)Ψ+(p)}

(100)

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Or,

H =

∫dp

1

2πE2{Ψ+(p)Ψ∗+(p) + Ψ∗+(p)Ψ+(p)} (101)

Suppose we want to write the classical Hamiltonian as

H =

∫dp

1

2π

Ep2{Ψ+(p)Ψ∗+(p) + Ψ∗+(p)Ψ+(p)} (102)

This can easily be done by de�ning new normalized:

Ψ+(p)|old →Ψ+(p)|new√

2Ep(103)

15.2.5 Field Expansions in Normalized Variables

Ψ(x, t) =1

2π√

2Ep{∫dp{Ψ+(p) exp(ipx) exp(−iEpt)}+


=1

2π√

2Ep{∫dp{Ψ+(p) exp(ipx) exp(−iEpt)}+


=1

2π


where,

f(p, t) =1√2Ep{Ψ+(p) exp(−iEpt) + Ψ−(p) exp(iEpt)} (105)

π(x, t) =−i√Ep/2



=−i√Ep/2



=1

2π


where,

g(p, t) = −i√Ep/2{Ψ+(p) exp(−iEpt)−Ψ−(p) exp(iEpt)} (107)

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15.3 Quantization of Real-Valued Scalar 'Classical' KG Field

15.2.6 Inverse Transforms

One can also try to �nd the inverse expansions by exploiting the orthonormality of exponen-

tials: ∫dxΨ(x, t) exp(−iqx) =

1√2Eq{Ψ+(q) exp(−iEqt) + Ψ∗+(−q) exp(iEqt)}

∫dxπ(x, t) exp(−iqx) = −i

√Eq2{Ψ+(q) exp(−iEqt)−Ψ∗+(−q) exp(iEqt)}

Adding and subtracting above two expressions:

Ψ+(q) exp(−iEqt) =

∫dx{√Eq2

Ψ(x, t) +i√2Eq

π(x, t)} exp(−iqx) (108)

Ψ∗+(−q) exp(iEqt) =

∫dx{√Eq2

Ψ(x, t)− i√2Eq

π(x, t)} exp(−iqx)

This last expression can also be written as:

Ψ∗+(q) exp(iEqt) =

∫dx{√Eq2

Ψ(x, t)− i√2Eq

π(x, t)} exp(iqx) (109)


15.3.1 Quantized Field Operators

With this much of background preparations we are now ready to do the standard quan-

tization. One can do the standard quantization in real space (obviously with generalized

co-ordinates though) by converting the in�nite set of time dependent real space co-ordinates

and corresponding conjugate momenta � Ψ(x, t) and π(x, t) � into time independent (in

Schrodinger's picture) Hermitian operators Ψ(x) and π(x) and enforcing standard com-

mutation relation between these:

[Ψ(x), Ψ(y)] = 0

[π(x), π(y)] = 0

[Ψ(x), π(y)] = iδ(x− y) (110)

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Note that the right hand side on the last expression a delta function (not a Kronecker delta).

Delta function is de�ned such that δ(x−y) = 0 ∀x 6= y and such that∫δ(x−y)dx = 1. After

doing this, we notice from 108 and 109 that even the of energy mode expansion coe�cients

Ψ+(p) exp(−iEpt) and Ψ∗+(p) exp(iEpt) get converted into operators :

a(p) =

∫dx{√Eq2

Ψ(x) +i√2Eq

π(x)} exp(−ipx) (111)

a†(p) =

∫dx{√Eq2

Ψ(x)− i√2Eq

π(x)} exp(ipx) (112)

One can easily check that indeed 112 is the adjoint of 111 as direct consequence of ˆΨ(x)

and π(x) being the Hermitian operators by de�nition. We would see below that these new

operators actually behave as creation and annihilation operators and the Hamiltonian can

easily be written in diagonal forms with their help. Using 111 and 112 one can easily check

that

[a(p), a(q)] = 0

[a†(p), a†(q)] = 0

[a(p), a†(q)] = δ(p− q) (113)

Note that the right hand side on the last expression a Kronecker delta function (not a

delta function). Kronecker delta function is de�ned such that δ(x − y) = 0 ∀x 6= y and

δ(x− y) = 1 for x = y. Also, from 105 and 107, the momentum mode expansion coe�cients

also get converted to operators:

f(p) =1√2Ep{a(p) + a†(−p)} (114)

g(p) = −i√Ep/2{a(p)− a†(−p)} (115)

Please note that these operators are not Hermitian. The commutation relation between

these operators can also be easily evaluated:

[f(p), g(q)] =−i2{−[a(p), a†(−q)] + [a†(p), a(−q)]} =

−i2{−(δ(p+ q)) + (−δ(p+ q))}

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[f(p), g(q)] = iδ(p+ q)

Also inverse transformations can readily be calculated as well:

Ψ(x) =1

2π

∫dp{ 1√

2Ep}{a(p) exp(ipx) + a†(p) exp(−ipx)} (116)

π(x) =1

2π

∫dp{−i

√Ep/2}{a(p) exp(ipx)− a†(p) exp(−ipx)} (117)

Its good to see that these operators are certainly Hermitian as we initially de�ned them to

be.

15.3.2 Quantum Hamiltonian

Now the quantum version of Hamiltonian can be found starting from 141:

H =

∫dp

1

2π

Ep2{a(p)a†(p) + a†(p)a(p)}

=

∫dp

1

2πEp{a†(p)a(p) +

1

2[a(p), a†(p)]} (118)

=

∫dp

1

2πEp{a†(p)a(p) +

1

2δ(0)} (119)

In the last expression we have used the 113 derived above. Note that the second term on

the right hand sign sums to in�nity. Since its a constant energy term, we would throw out

this term. Note that in most physical problems only energy di�erence is of concern. There

are some places where in�nite energy of ground state can lead to problems but we would not

encounter any such situation in this article. Hence we would write:

H =

∫dp

1

2πEp{a†(p)a(p)} (120)

15.3.3 Number State basis

15.3.3.1 Ground State

15.3.3.2 Higher States of De�nite Particle Number Now let us try to explore some

properties of a†(p) and a(p) operators. Suppose |φ〉 is a known eigen state of Hamiltonian

de�ned in 168. So that H|φ〉 = E|φ〉. a†(q)|φ〉 would be some state in the same Hilbert

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space. Let us operate Hamiltonian on this state. So∫dp

1

2πEpa

†(p)a(p)a†(q)|φ〉 =

∫dp

1

2πEpa

†(p)(δ(p− q) + a†(q)a(p))|φ〉

=

∫dp

1

2πEp(δ(p− q)a†(p) + a†(p)a†(q)a(p))|φ〉

=

∫dp

1

2πEp(δ(p− q)a†(p) + a†(q)a†(p)a(p))|φ〉

=

∫dp{ 1

2πEp(δ(p− q)a†(p)|φ〉}+ a†(q)E|φ〉

= Eqa†(q)|φ〉+ a†(q)E|φ〉

= (E + Eq)a†(q)|φ〉

Hence, a†(q)|φ〉 is also an energy eigen state (with energy of E + Eq) of the Hamiltonian.

Similarly we can show that a(q)|φ〉 is also an energy eigen state (with energy of E − Eq) ofthe Hamiltonian. More concise mathematical prove is to show that:

[H, a†(p)] = Epa†(p) (121)

and

[H, a(p)] = −Epa(p) (122)

Above two properties guarantee that proposed a†(p) and a(p) indeed behave like creation

and annihilation operators respectively.

15.3.3.3 Orthonormality and Completeness

15.3.4 Coherent and Squeezed States

Let us look at following state (this actually Poisonian is a coherent state as we would show)

:

|ψc〉 = exp(ca†(p))|0〉 =∞∑i=0

{ci(a†(p))i

i!}|0〉

Here c is a complex number. Now, we also know that

|n(p)〉 =1√n(p)!

(a†(p))n(p)|0〉

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Hence state |Ψc〉 can be written as

|Ψc〉 =∞∑i=0

cn(p)√n(p)!

|n(p)〉

We notice that in one single classical momentum mode we have an inde�nite number of

particles. The probability amplitude of �nding certain number of particles has Poisonian

distribution. We would then like to study U(t ← 0)|Ψc〉. This would justify that, indeed,

the above constructed state is a coherent state.

15.3.5 Heisenberg Representation

Using 169, one can generalize that relation and obtain

Hna†(p) = a†(p)(H − Ep)n

Note that the creation and annihilation operators encountered in previous sections are all

in Schrodinger pictures. Now using the Taylor expansion of exponential operators, one can

easily check that the Heisenberg representation of these operators is given by

a†h(p) = U †s (t← 0)a†s(p)Us(t← 0) = a†s(p) exp(−iEpt)

and

ah(p) = U †s (t← 0)as(p)Us(t← 0) = as(p) exp(iEpt)

Hence, �eld operators in Heisenberg representation in relativistic notations (4-vectors) can

be written as

Ψh(x) =1

2π

∫dp{ 1√

2Ep}{as(p) exp(−ip.x) + a†s(p) exp(ip.x)}|p0=Ep

(123)

πh(x) =1

2π

∫dp{−i

√Ep/2}{as(p) exp(−ip.x)− a†s(p) exp(ip.x)}|p0=Ep

(124)

Notice that the signs in the exponentials have reversed. One should notice that p.x = pµxµ =

pµxµ = Ept− p1x1− p2x2− p3x3. This justi�es the sign reversal. Spatial part automatically

get the previous sign. These are very important and very generic results. Time and space

would always have opposite signs. Time exponential with negative sign is called �positive

energy part of the solution�. Time exponential with positive sign is called �negative energy

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125




part of the solution�. Creation operator always goes with negative energy part

of the solution and annihilation operator always goes with the positive energy

solution part of the solution.

15.3.6 Two Point Correlation Functions/Propagators For Free Fields

Propagators are related to two point correlation functions. Sometime names might be used

interchangeably. Two point correlation functions are also related to Green's functions of KG

operator. Many di�erent types of Green's functions can be de�ned. First two we would are

not really Green's function. But they are usually referred to as Green's function because

these two are the building blocks for the 4 di�erent types of Green's functions of the KG

operator. The most basic one is Particle Propagator or a Greater Green's Function.

In relativistic notations this is de�ned as

G>(x− y) ≡ G>(x← y) ≡ 〈0h|Ψh(x)Ψ†h(y)|0h〉

Note that what we are doing here is that we are creating a single particle at y (a 4-vector)

and �nding the amplitude of probability that system propagates to become particle at x

(again a 4-vector). Also note that for real valued scalar KG �eld Ψh(x) = Ψ†h(x)

as the �eld operator is Hermitian. But above expression can be further generalized by

creating a single particle in, say, single particle classical momentum mode p (that is state

|p〉) and then looking an amplitude probability that state evolves into a state |q〉. Using

this Greater Green's function one can study the causality of the theory. Note that it is not

necessary that y0 < x0.

Another fundamentally important Green's function is what is known as Anti-Particle

(or Hole) Propagator or a Lesser Green's Function. This is de�ned as follows

G<(x− y) ≡ G<(x← y) ≡ 〈0h|Ψ†h(x)Ψh(y)|0h〉

So basically,

G>(x− y) = (G<(y − x))?

Also for real valued scalar KG �eldG>(x−y) = G<(x−y). Note that x and y are the 4-vectors

in above equation. So we are not only �ipping the time but also the space co-ordinates. One

very important point is that the commutation relation between Ψ and Ψ† are the

equal time commutation relation. There is no de�ned relationship between �eld

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operators at di�erent times. So there would be no obvious correlation between

lesser and greater Green's function other than that discussed above.

Let us try to obtain the Fourier transform of these two correlation function (these are not

really the Green's functions of KG operator). Substituting 123 in the de�nition of Greater

Green's function we can immediately write

G>(x− y) =

∫d3p

(2π)3

1

2Epexp(−ip.(x− y))|p0=Ep

Similarly,

G<(x− y) =

∫d3p

(2π)3

1

2Epexp(−ip.(y − x))|p0=−Ep

We would �nd these expressions useful in the following discussion. One should notice that

the coe�cients of exponentials in right hand side are NOT really 4D Fourier transform.

One can consider this coe�cient multiplied by time exponential as spatial Fourier transform

which is a function of time.

Above two functions are the basic building blocks for other types of correlation functions

which have better physical and mathematical properties. Using these two functions we

de�ne following four other correlation functions which are more commonly encountered in

text books. A set of functions that are sometime useful are time ordered and anti time

ordered Green's functions. Time ordered Green's function is also sometime (in the case

of free/non-interacting �elds) called the Feynman propagator.

Gt(x− y) ≡ Gt(x← y) ≡ 〈0h|T Ψh(x)Ψ†h(y)|0h〉

Gt(x− y) ≡ Gt(x← y) ≡ 〈0h|T Ψh(x)Ψ†h(y)|0h〉

Note that

Gt(x− y) =

{G>if x0 > y0

G<if x0 < y0

}

Gt(x− y) =

{G<if x0 > y0

G>if x0 < y0

}Two other functions are retarded and advanced Green's functions. Retarded Green's function

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is de�ned as

GR(x− y) ≡ −iΘ(x0 − y0)(〈[0h|Ψ†h(x), Ψh(y)]|0h〉) = −iΘ(x0 − y0)(G>(x− y)−G<(x− y)

Note that Θ is a Heaviside step function. Note that some books leave out the −i factor.Similarly advanced Green's Function is de�ned as

GA(x− y) ≡ −iΘ(y0 − x0)(〈[0h|Ψ†h(x), Ψh(y)]|0h〉) = −iΘ(y0 − x0)(G>(x− y)−G<(x− y)

These four correlation functions are actually very closely related to each other. We would

show that Gt, Gt, GR and GA are all Green's function of KG equation. That is, if we write

a inhomogeneous KG equation with a 4-dimensional delta function on right hand side then

above Green's functions satisfy the equation. This gives a prescription of actually calculating

these functions. One can Fourier transform the whole equation. Which then get converted

to a simple algebraic equation. From this one can calculate the Fourier transform of the

Green's function which can then be inverse transformed to calculated GF in real space.

Suppose G(x− y) is a Green's function of KG operator

{( ∂2

∂t2−∇2) +m2}G(~x− ~y, tx − ty) ≡ −iδ(~x− ~y, tx − ty)

Or equivalently in 4-vector notation

{∂µ∂µ +m2}G(x− y) ≡ −iδ(x− y)

Let 4D Fourier transform of G(x) be G(p).

In order to remind ourselves about multi-dimensional Fourier transforms, let us consider

a simple 2D function f(x1, x2) with a 2D Fourier transform of f(kx1, kx2)

f(kx1, kx2) ≡∫ ∞−∞

dx1dx2f(x1, x2) exp(−i2π(kx1x1 + kx2x2))

Di�erentiating this equation with respect to x1 we get

0 =

∫ ∞−∞

dx1dx2{df(x1, x2)

dx1

exp(−i2π(kx1x1+kx2x2))−i2πkx1f(x1, x2) exp(−i2π(kx1x1+kx2x2))}

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Hence∫ ∞−∞

dx1dx2df(x1, x2)

dx1

exp(−i2π(kx1x1+kx2x2)) = i2πkx1

∫ ∞−∞

dx1dx2f(x1, x2) exp(−i2π(kx1x1+kx2x2))

Same rule can be extended for 4D Fourier transforms. Lets remember one more fact∫ ∞−∞

d4xδ(x) exp(−i2πkx) = 1

Hence if we Fourier transform the de�ning equation for the Green's function of the KG

operator (with y = 0), by de�nition then, G(p) would satisfy the following algebraic equation

G(p) =i

p2 −m2

Now taking its 4D Fourier inverse

G(x− y) =

∫d4p

(2π)4

i

p2 −m2exp(−ip.(x− y))

=

∫d4p

(2π)4(i/2Epp0 − Ep

− i/2Epp0 + Ep

) exp(−ip.(x− y))

Note that p2 − m2 = (p0)2 − E2p = E2 − E2

p . Now, if we look at the integral over energy

scale more closely, we would realize that the integrand has two poles at p0 = ±Ep. First

thing is that we need to choose a contour along the real axis so that poles do not fall exactly

on the contour. There are four ways of selecting this part of contour. Depending upon

how we decide to integrate we get above de�ned four di�erent GFs. Secondly, we need to

close the contour either in upper or lower plane depending upon x0 > y0 or y0 > x0. Let

us do the integration on energy scale. If we choose to go �above� both the poles then for

x0 < y0 (close contour in lower half) G(x − y) = 0. For x0 > y0 (close contour in lower

half) G(x− y) = G>(x− y)−G<(x− y). Hence this is retarded Green's function. Similarly

if we go �below� both poles we would get advanced GF. If we below negative energy below

and above positive energy pole (this is known as Feynman prescription) we would get time

ordered GF. If go above negative energy pole but below positive energy pole then we would

get anti-time ordered GF.

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15.4 Classical Hamiltonian Theory of Complex-Valued Scalar KG Field

15.4 Classical Hamiltonian Theory of Complex-Valued Scalar KG

Field

15.4.1 Classical Hamiltonian and Lagrangian

Action for the classical complex valued �eld is given as

S ≡∫Ldt =

∫d4x(∂µφ

?∂µφ−m2φ?φ)

Hence, Lagrangian density function can be written as

L(φ(x), φ?(x), ∂µφ(x), ∂µφ?(x), t) = ∂µφ

?∂µφ−m2φ?φ

=∂φ?

∂t

∂φ

∂t− ~∇φ?.~∇φ−m2φ?φ (125)

Note that we don't have explicit time dependence in this problem. We know from Lagrange

Principle of least action that the extremum would give the Euler-Lagrange equation of mo-

tion. The Euler-Lagrange's equation of motions would be :-

d

dt(

∂L

∂ ˙(φ(x))) +∇(

∂L

∂(∇φ(x))))− ∂L

∂φ(x)= (

∂2

∂t2−∇2 +m2)φ?(x) = 0

andd

dt(

∂L

∂ ˙(φ?(x))

) +∇(∂L

∂(∇φ?(x))))− ∂L

∂φ?(x)= (

∂2

∂t2−∇2 +m2)φ(x) = 0

Both of these are exactly same as the KG equation. Hence both φ(x) and φ?(x) satis�es KG

equation.

From the expression for Lagrangian density 125, the conjugate momentum density func-

tion (conjugate to φ(x)) would be:-

π1(x) ≡ ∂L

∂φ(x)=∂φ?

∂t

and that conjugate to φ?(x) is

π2(x) ≡ ∂L

∂φ?(x)=∂φ

∂t

We notice that

π1(x) = π?2(x)

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So we can use more simple notations:

π?(x) = π1(x) ≡ ∂L

∂φ(x)=∂φ?

∂t(126)

π(x) = π2(x) ≡ ∂L

∂φ?(x)=∂φ

∂t(127)

Now, once we know the conjugate momenta density, one can easily obtain the Hamiltonian

density function H(φ(x), φ?(x), π?(x), π(x), ~∇φ(x)~∇φ?(x), t) as

H ≡ π?(x)φ(x) + π(x)φ?(x)− L

= π?(x)φ(x) + π(x)φ?(x)− ∂φ?

∂t

∂φ

∂t+ ~∇φ?.~∇φ+m2φ?φ

Now using 126 and 127 we can write the classical Hamiltonian density function as:

H(φ(x), φ?(x), π?(x), π(x), ~∇φ(x)~∇φ?(x), t) = π(x)π?(x) + ~∇φ?.~∇φ+m2φ?φ (128)

Note that the order of π?π and another similar term is not determined of clas-

sical mechanics. Applying integration operator∫d3x one can then get the Classical

Hamiltonian function.

15.4.2 Classical φ(x, t) and φ?(x, t) Fields in Energy/Momentum Basis

A free-space KG equation is written as

(−∇2 +m2)Ψ(x, t) = −∂2Ψ(x, t)

∂t2

One can easily check that the classical energy eigen states of free-space KG equation are :-

ΨE(x, t) = A exp(ipx− iEt)

where,

E = ±√m2 + p2 = ±Ep

Here Ep is always a positive number. Note that for one value of p, there are two values

of E possible - one positive and one negative . Hence we note that arbitrary classical �eld

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can be written as a general superposition of all possible solutions :-

φ(x, t) =1

2π

∫dp{φ+(p) exp(ipx) exp(−iEpt)}+

1

2π

∫dp{φ−(p) exp(ipx) exp(iEpt)} (129)

The factors of 2π have been included just because of conventions. These can be dumped

into φ+ and φ− which we anyway would-re normalize latter on. One should note that above

expansion is simply spatial Fourier expansion. This can be understood by noting that above

can also be written as:

φ(x, t) =1

2π


where,

f(p, t) = φ+(p) exp(−iEpt) + φ−(p) exp(iEpt) (131)

Note that 130 is exactly a spatial Fourier transform. Which is also same as expanding φ(x, t)

in momentum basis. Note that energy basis expansion needs two expansion coe�cient for

one |E| (or for one p) but momentum space expansion needs only one coe�cient. φ+(p)

and φ−(p) would be recognized energy mode amplitudes and f(p, t) as momentum mode

amplitude of �eld φ(x, t). Now 129 can also be written as

φ(x, t) =1

2π

∫dp{φ+(p) exp(ipx) exp(−iEpt)}+

1

2π

∫dp{φ−(−p) exp(−ipx) exp(iEpt)}

(132)

Now taking the complex conjugate of 132:

φ?(x, t) =1

2π

∫dp{φ?−(−p) exp(ipx) exp(−iEpt)}+

1

2π

∫dp{φ?+(p) exp(−ipx) exp(iEpt)}

(133)

15.4.3 Classical π(x, t) and π?(x, t) Fields in Momentum/Energy Basis

Di�erentiating 132 with respect to time and using 127:

π(x, t) =−iEp

2π{∫dp{φ+(p) exp(ipx) exp(−iEpt)} −


(134)

And analogy to 130, we can also write

π(x, t) =−iEp

2π


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where,

g(p, t) = φ+(p) exp(−iEpt)− φ−(p) exp(iEpt) (136)

Now taking the complex conjugate of 134:

π?(x, t) =−iEp

2π[

∫dp{φ?−(−p) exp(ipx) exp(−iEpt)} − {

∫dp{φ?+(p) exp(−ipx) exp(iEpt)}]

(137)

Note that the condition 126 is automatically enforced.

15.4.4 Classical Hamiltonian in Energy Basis

One should notice that the classical Hamiltonian (not density) can be written as :

H =

∫dx{π(x)π?(x) + ~∇φ?.~∇φ+m2φ?φ} (138)

=

∫dp

1

2π{E2

p + p2 +m2}{φ+(p)φ∗+(p) + φ−(−p)φ?−(−p)} (139)

Or,

H =

∫dp

2

2πE2p{φ+(p)φ∗+(p) + φ−(−p)φ?−(−p)} (140)

Suppose we want to write the classical Hamiltonian as

H =

∫dp

1

2π

Ep2{φ+(p)φ∗+(p) + φ−(−p)φ?−(−p)} (141)

This can easily be done by de�ning new normalized �eld variables such that:

Ψ+(p)|old →Ψ+(p)|new

2√Ep

(142)

15.4.5 Classical Field Expansions in Normalized Variables

φ(x, t) =1

4π√Ep{∫dp{φ+(p) exp(ipx) exp(−iEpt) +

∫dpφ−(−p) exp(−ipx) exp(iEpt)}

(143)

φ?(x, t) =1

4π√Ep

∫dp{φ?−(−p) exp(ipx) exp(−iEpt) +

∫dpφ?+(p) exp(−ipx) exp(iEpt)}

(144)

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π(x, t) =−i√Ep

4π{∫dp{φ+(p) exp(ipx) exp(−iEpt)} −


(145)

π?(x, t) =−i√Ep

4π[

∫dp{φ?−(−p) exp(ipx) exp(−iEpt)}−{

∫dp{φ?+(p) exp(−ipx) exp(iEpt)}]

(146)

15.4.6 Classical Inverse Transforms

One can also try to �nd the inverse expansions by exploiting the orthonormality of exponen-

tials: ∫dxφ(x, t) exp(−iqx) =

1

2√Eq{φ+(q) exp(−iEqt) + φ−(q) exp(iEqt)} (147)

∫dxπ(x, t) exp(−iqx) =

−i√Eq

2{φ+(q) exp(−iEqt)− φ−(q) exp(iEqt)} (148)∫

dxφ?(x, t) exp(−iqx) =1

2√Eq{φ?−(−q) exp(−iEqt) + φ?+(−q) exp(iEqt)} (149)

∫dxπ?(x, t) exp(−iqx) =

−i√Eq

2{φ?−(−q) exp(−iEqt)− φ?+(−q) exp(iEqt)} (150)

Adding and subtracting 147 and 148 one gets:

φ+(q) exp(−iEqt) =

∫dx{√Eqφ(x, t) +

i√Eqπ(x, t)} exp(−iqx) (151)

φ−(q) exp(iEqt) =

∫dx{√Eqφ(x, t)− i√

Eqπ(x, t)} exp(−iqx)


φ−(−q) exp(iEqt) =

∫dx{√Eqφ(x, t)− i√

Eqπ(x, t)} exp(iqx) (152)

Similarly, adding and subtracting 149 and 150, one gets:

φ?−(−q) exp(−iEqt) =

∫dx{√Eqφ

?(x, t) +i√Eqπ?(x, t)} exp(−iqx) (153)

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15.5 Quantization of Complex-Valued Scalar KG Field

φ?+(−q) exp(iEqt) =

∫dx{√Eqφ

?(x, t)− i√Eqπ?(x, t)} exp(−iqx)


φ?+(q) exp(iEqt) =

∫dx{√Eqφ

?(x, t)− i√Eqπ?(x, t)} exp(iqx) (154)


15.5.1 Quantization Field Operators

Once we have the classical conjugate variables and classical Hamiltonian, we are now ready to

do the standard quantization. One can do the standard quantization in real space (obviously

with generalized co-ordinates though) by converting the in�nite set of time dependent real

space co-ordinates and corresponding conjugate momenta � φ(x, t), π?(x, t) and φ?(x, t),

π(x, t) � into time independent (in Schrodinger's picture) operators φ(x) , π†(x) and φ†(x) ,

π(x) respectively40 and enforcing standard commutation relation between these:

[φ(~x), φ(~y)] = 0

[π†(~x), π†(~y)] = 0

[φ(~x), π†(~y)] = iδ(~x− ~y) (155)

[φ†(~x), φ†(~y)] = 0

[π(~x), π(~y)] = 0

[φ†(~x), π(~y)] = iδ(~x− ~y) (156)

Also since φ(~x) and φ†(~x) as well as π(~x) and π†(~x) are independent variables, we postulate

that all other pairs commute with each other. Note that the right hand side of above

two expression are Dirac delta functions (not the Kronecker delta). In Heisenberg represen-

tation these all are equal time commutation relations (ETCR). Now the corresponding

quantized Hamiltonian density operator can be written as:

H =

∫d3x{π(~x)π†(~x) + ~∇φ†(~x). ~∇φ(~x) +m2φ†(~x)φ(~x)} (157)

40Suppose we convert real and imaginary part of φ into Hermitian operators, that is Re{φ} → φr and

Im{φ} → φi so that φ→ φ. From this one can at once see that φ? → φ†.

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We notice that due to postulated commutation relations, there is no confusion

about ordering of the operators while going from classical to quantum represen-

tation.

Now, we can convert these operators into time depending operators by moving to Heisen-

berg representation. Noting that the Hamiltonian is explicitly time independent, we can

write, for φ1(~x) �eld operator for example, in Heisenberg representation

φ(~x, t) = exp(iHt)φ(~x) exp(−iHt)

Similarly other operators are also converted into Heisenberg representation. Note that in this

representation all above commutation relations still hold but they are equal time relations.

Now Heisenberg equation of motion for �eld φ(~x) is:-

i∂φ(~x, t)

∂t= [φ(~x, t),

∫d3xH]

= [φ(~x, t),

∫d3y{π(~y, t)π†(~y, t) + ~∇φ†(~y, t). ~∇φ(~y, t) +m2φ†(~y, t)φ(~y, t)}]

=

∫d3y{π(~y, t)[φ(x, t), π†(~y, t)]}

=

∫d3yπ(~y, t)iδ(~x− ~y) = iπ(~x, t)

Also,

i∂π(~x, t)

∂t= [π(~x, t),

∫d3xH]

= [π(~x, t),

∫d3y{π(~y, t)π†(~y, t) + ~∇φ†(~y, t). ~∇φ(~y, t) +m2φ†(~y, t)φ(~y, t)}]

=

∫d3y{−∇2φ(~y, t)[π(x, t), φ†(~y, t)] +m2φ(~y, t)[π(x, t), φ†(~y, t)]}

= −∫d3y{−∇2φ(~y, t) +m2φ(~y, t)}iδ(~x− ~y) = −i{−∇2φ(x, t) +m2φ(x, t)}

Combining the two equations:

∂ 2φ(x, t)

∂t2= ∇2φ(x, t)−m2φ(x, t)

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Repeating exactly same problem one can also show that:

∂2φ†(x, t)

∂t2= ∇2φ†(x, t)−m2φ†(x, t)

This proves that the Heisenberg equation of motion for the quantized �eld operators is just

the KG equation written for �eld operators.

15.5.2 Quantized Field Operators in Energy/Momentum Basis

Once we do the standard quantization by converting φ(x, t), π?(x, t) and φ?(x, t), π(x, t)

into time independent (in Schrodinger's picture) operators φ(x) , π†(x) and φ†(x) , π(x)

respectively and by enforcing standard commutation relation 155 and 156 (and all other

pairs commuting) between these operators, we immediately notice from 151, 152 , 154 and

153 that even the of energy mode expansion coe�cients φ+(p) exp(−iEpt), φ−(−q) exp(iEqt),

φ?−(−q) exp(−iEqt) and φ∗+(p) exp(iEpt) get converted respectively into operators :

φ+(p) exp(−iEpt)→ a(p) =

∫dx{√Epφ(x) +

i√Epπ(x)} exp(−ipx) (158)

φ−(−q) exp(iEqt)→ b†(p) =

∫dx{√Epφ(x)− i√

Epπ(x)} exp(ipx) (159)

φ?−(−q) exp(−iEqt)→ b(p) =

∫dx{√Epφ

†(x) +i√Epπ†(x)} exp(−ipx) (160)

φ∗+(p) exp(iEpt)→ a†(p) =

∫dx{√Epφ

†(x)− i√Epπ†(x)} exp(ipx) (161)

One can easily check that indeed 161 and 159 are the adjoint of 158 and 160

respectively and hence the choice of symbols is well justi�ed. We would see below

that these new operators actually behave as creation and annihilation operators and the

Hamiltonian can easily be written in diagonal forms with their help. Using 158 and 161 one

can easily check that:

[a(p), a(q)] = 0

[a†(p), a†(q)] = 0

[a(p), a†(q)] = δ(p− q) (162)

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Similarly, using 159 and 160 one can easily check that:

[b(p), b(q)] = 0

[b†(p), b†(q)] = 0

[b(p), b†(q)] = δ(p− q) (163)

And all other pairs commute. Note that the right hand sides on the last expressions

in above two sets of commutation relations are Kronecker deltas (not a delta functions).

Kronecker deltas de�ned such that δ(x− y) = 0 ∀x 6= y and δ(x− y) = 1 for x = y.

Also inverse transformations can readily be calculated as well:

φ(x) =1

4π√Ep{∫dp{a(p) exp(ipx) +

∫dpb†(p) exp(−ipx)} (164)

φ†(x) =1

4π√Ep

∫dp{b(p) exp(ipx) +

∫dpa† exp(−ipx)} (165)

π(x) =−i√Ep

4π{∫dp{a(p) exp(ipx)} −

∫dp{b†(p) exp(−ipx)} (166)

π†(x) =−i√Ep

4π{∫dp{b(p) exp(ipx)−

∫dp{a†(p) exp(−ipx)} (167)

15.5.3 Quantum Hamiltonian

Now the quantum version of Hamiltonian can be found starting from 141 and using 158-154:

H =

∫dp

1

2π

Ep2{a†(p)a(p) + b†(p)b(p)} (168)

Factor 1/2 is there because of the normalization of the �eld variables I chose. I encounter

the problem of operator ordering which basically originates from the ordering of π?and π in

classical Lagrangian. I have straighten up the order and threw away the in�nite term.

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15.5.4 Number State Basis

Now let us try to explore some properties of a†(p) and a(p) operators. From 168 we can

immediately see that:

[H, a†(p)] = Epa†(p) (169)

and

[H, a(p)] = −Epa(p) (170)

Exactly same relations hold for other set of creation and annihilation operators as well.

Above two properties guarantee that proposed a†(p) and a(p) (and b†and b) indeed behave

like creation and annihilation operators respectively. Also note that the state operator φ(x)

simultaneously creates one and destroys other so they do behave like particle and antiparticle.

15.5.5 Associated Charge

Q = −i∫d3x(π?φ− φ?π)

Hence,

Q = −i∫d3x(π†φ− φ†π)

Hence,

Q =−1

4(2π)3

∫d3p(a†(p)a(p)− b†(p)b(p))

Note that factor of 1/4 is result of the de�nition normalized �eld variables. This we can get

rid of by properly normalizing variables. Also I have again straighten up the ordering of the

operators.

Now, let us evaluate the following commutation relation:

[a†(p)a(p) + b†(p)b(p), a†(q)a(q)− b†(q)b(q)] = [a†(p)a(p), a†(q)a(q)]

− [a†(p)a(p), b†(q)b(q)]

+ [b†(p)b(p), a†(q)a(q)]

− [b†(p)b(p), b†(q)b(q)]

= 0

Since H and Q commute, the eigen states of Q are also the eigen states of H and vice

versa. More explicitly, let a single eigen state ofH be |1p; 0〉where �rst number represents that

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15.6 Quantization of Complex Valued Vector Klein Gordon (KG) Field

there is one particle of type 1 in state p and zero particle of type 2. Hence H|1p; 0〉 = |1p; 0〉.Whereas Q|1p; 0〉 = |1p; 0〉. On the other hand H|0; 1p〉 = |0; 1p〉 and Q|0; 1p〉 = −|0; 1p〉.Hence Q can be interpreted as some sort of quantized �charge� carried my massive particles.

Antiparticle carries negative and equal amount of charge as does a particle.

15.6 Quantization of Complex Valued Vector Klein Gordon (KG)

Field

15.6.1 Quantized Field Operators and Hamiltonian

The Lagrangian density function generalizes to

L(φi(x), φ?i (x), ∂µφi(x), ∂µφ?i (x), t) = ∂µφ

?i∂

µφi −m2φ?iφi

=∂φ?i∂t

∂φi∂t− ~∇φ?i .~∇φi −m2φ?iφi (171)

Where in this problem i ∈ {1, 2} and sum convention is assumed. From the expression

for Lagrangian density 171, the conjugate momentum density function (conjugate to φ1(x))

would be:-

π?1(x) ≡ ∂L

∂φ(x)=∂φ?1∂t

and that conjugate to φ?1(x) is

π?1(x) ≡ ∂L

∂φ?(x)=∂φ1

∂t

And exactly analogous results exist for i = 2.

Now, once we know the conjugate momenta density, one can easily obtain the Hamiltonian

density function H(φi(x), φ?i (x), π?i (x), πi(x), ~∇φi(x)~∇φ?i (x), t) as

H ≡ π?i (x)φi(x) + πi(x)φi?(x)− L

= π?i (x)φi(x) + πi(x)φi?(x)− ∂φ?i

∂t

∂φi∂t

+ ~∇φ?i .~∇φi +m2φ?iφi

So we can write the classical Hamiltonian density function as:

H(φi(x), φ?i (x), π?i (x), πi(x), ~∇φi(x)~∇φ?i (x), t) = πi(x)π?i (x) + ~∇φ?i .~∇φi +m2φ?iφi (172)

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15.6 Quantization of Complex Valued Vector Klein Gordon (KG) Field

Again note that the order of π?π and another similar term is not determined of

classical mechanics. Applying integration operator∫d3x one can then get the Classical

Hamiltonian function.

Commutation relations generalizes in a straight forward manner. Considering two �elds

to be �independent� we �rst of all postulate exactly same set of commutation rules as 155 ,

156 and other pairs independently for both i = 1 and i = 2. Taking clue from the theory

of quantization of multi-particle systems we can postulate that all other pairs with i 6= j

commutes.

Since both the �elds behave independently and due to the postulated commutation re-

lations the proof that φ1and φ2 has the Heisenberg equation of motions as Klein Gordon

equation is exactly same as the that in problem 1. I don't feel like repeating the derivation

un-necessary. Proof is step by step exactly same.We move energy basis and denies the oper-

ators φi, φ†i and their conjugate momenta in terms of two independent set of creation and

annihilation operators of particles and antiparticles (hence we would have 8 such operators

in this problem). Let us call these ai, a†i , biand b†i with i ∈ {1, 2}. Derivations again are

exactly same. I am not repeating those. Hamiltonian generalizes to :

H =

∫dp

1

2π

Ep2{a†i (p)ai(p) + b†i (p)bi(p)} (173)

15.6.2 Associated Charge

Expression for Q also generalizes in straight forward fashion as well :

Q = −i∫d3x(π†i φi − φ

†i πi)

=−1

4(2π)3

∫d3p(a†i (p)ai(p)− b

†i (p)bi(p))

This commutes with H and, again, the proof is exactly same as before

Qp = − i2

∫d3x(π†i σ

pijφi − φ†i σpijπj)

I could not work out the commutations. Somehow above expression does not look right to

me.

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16: Dirac Field

16 Dirac Field

17 Relativistic Electromagnetic Field Quantization ( QED

)

Part IX

Interaction Between Free Fields

18 Di�erent Pictures of Time Evolution

18.1 Exponential of Operators

Suppose we have an equation which we want to solve:

i~∂Ψs(t)

∂t= H0,sΨs(t)

where H0,s is time independent. Let us try to solve it in following fashion. Let us assume

�rst that Ψs(t) = Ψs(0). This is our zeroth order approximation. First order correction term

can be found by substituting this into the right hand side of above equation. That gives us

Ψ(1)s (t) = (− i

~

∫ t

0

dtH0,s)Ψs(0) = (−it~H0,s)Ψs(0)

Substituting this again in the right hand side of original equation:

Ψ(2)s (t) =

1

2(−itH0,s

~)2Ψ(0)

Similarly, after calculating the nth order correction term and adding them all we can write

Ψs(t) = {∞∑n=0

1

n!(−itH0,s

~)n}Ψ(0)

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18.2 Schrodinger's Picture of Time Evolution

Please note that H0,s is an operator and not a simple number. Still, we note that the terms

under summation sign resembles a Taylor expansion. We de�ne a �symbolic notation�

for such a series as

Ψs(t) = {∞∑n=0

1

n!(−itH0,s

~)n}Ψ(0) ≡ exp(

−iH0,st

~)Ψ(0)

Note that this notation is valid only when H0,s is time independent. When operator

becomes time dependent we can follow a similar route but would have to be more careful.

We do this in the following section 18.6.


Suppose we break the Hamiltonian involved in the time dependent Schrodinger's equation

(from here on we would call it Hamiltonian operator in Schrodinger's picture) in two parts

� Hs(t) = H0,s + Vs(t). Here, H0,s is time independent part of the Hamiltonian. Whereas

Vs(t) may or may not be time independent. In cases where Vs is time independent, separation

of Hamiltonian into two time independent parts is simply for mathematical convenience.

Its helpful to take out that part of Hamiltonian whose eigen states are trivial. Then one

can build a time-independent perturbation theory to calculate the eigen states of the total

Hamiltonian. In any case, we would keep our discussions very general and would keep the

possibility of Vs being time dependent. We would consider many di�erent situations involving

equilibrium and non-equilibrium. Usually in equilibrium Vs would be time independent. We

would discuss these cases in details latter on.

The actual time evolution of system would be given by the Schrodinger's equation of

motion:

i~∂Ψs(t)

∂t= (H0,s + Vs(t))Ψs(t) (174)

Let us assume, for the time being, that Vs(t) = 0. We need to consider this case to de�ne

of some of the terms that we would use latter. In this case the Schrodinger's equation (174)

simpli�es to:

i~∂Ψs(t)

∂t= H0,sΨs(t)

Now, since, H0,s is time independent, one can symbolically write the solution of the above

equation as:

Ψs(t) = exp(−iH0,st/~)Ψs(0)

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Or equivalently, by de�ning unperturbed time-propagation operator in Schrodinger's

picture as:

U0,s(t) ≡ exp(−iH0,st/~) (175)

the same equation can also be written as

i~∂U0,s(t)

∂t= H0,sU0,s(t) (176)

Please note two speci�c points. First of all, if you have seen Heisenberg equation of motion

before, its instructive to make it clear that we are not writing the time evolution of the system

in Heisenberg picture (equation is strikingly similar). Moreover, U0,s(t) is not Hermitian

(U0,s(t) 6= U †0,s(t)), rather it is Unitary (U−10,s (t) = U †0,s(t)). That is, its eigen values are

complex.

Moving back to actual case where Vs(t) 6= 0, we de�ne time propagator in Schrodinger's

picture as:

Ψs(t) ≡ Us(t)Ψs(0) (177)

And with this de�nition, the Schrodinger equation (174) becomes:

i~∂Us(t)

∂t= Hs(t)Us(t)

Note that only in a very special case when Hs 6= Hs(t) (usually equilibrium), one can write:

Us(t) = exp((−iH0,st− iVst)/~)

Moreover, even in this special case, its not correct to break exponentials into two parts as Vs

and H0,s might not commute. We would try to avoid such exponentials as much as possible

and develop a general theory.

Key point to remember is that in Schrodinger picture states have a very

complicated time dependence. Operators can have a time dependence but this

dependence is preknown. So the whole problem is to solve the time dependence

of the states.

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18.3 Heisenberg Picture of Time Evolution

18.3 Heisenberg Picture of Time Evolution

In Heisenberg representation, states have no time dependence whereas operators have all the

time dependence. One de�nes the state of the system in Heisenberg representation as:

Ψh = U †s (t)Ψs(t) (178)

Hence, comparing with (177), one can easily conclude that:

Ψh = Ψs(0) (179)

One similarly de�nes operator in Heisenberg picture41 as:

Ah(t) = U †s (t)As(t)Us(t) (180)

I would stress again that only in the special case of time independent total Hamiltonian

(Hs 6= Hs(t)), one can write:

Us(t) = exp(−iHst/~)

We would try to avoid such exponentials as much as possible and develop a general theory.

Key point to remember is that in Heisenberg picture, states have absolutely

no time dependence whereas operators have all the time dependence. We take

out time dependence from Schrodinger states by multiplying it by U †s from

left side. And we deposit this time dependence into operators by multiplying

Schrodinger operators by U †s from left side and by Us from right side.

Language of Heisenberg representation sometimes become very confusing. One needs a

little bit of experience to make some sense out of it. Basically, Heisenberg representation uses

a time dependent basis to represent states of the system (something like moving or rotating

reference frames). Hence basis states ARE time dependent. When state of the system

is represented as linear superposition of basis states, the coe�cients of expansion are time

independent and that's why we say that states are time independent. Here is another way

of looking at the same thing. We have seen that operators are time dependent in Heisenberg

41Schrodinger's equation gets converted into time evolution equation for operators in Heisenberg represen-tation. One can easily show that any Hermitian operator would obey following equation

i~∂Ah(t)∂t

= [Ah(t), H]

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18.4 Interaction Picture

representation. Therefore if we calculate the eigen states of any operators, the would be time

dependent. We can use these eigen states as the basis so the basis would be time dependent.

One should note that this kind of confusion can arise even in Schrodinger's picture. For

example when say that we are using eigen states as basis in Schrodinger's picture, do we

include exp(−iωt) factors in states or not? Answer is no. In Schrodinger's picture basis

states are time-independent. In other words we use the eigen states of energy at t = 0 as

the basis.

Let us take a trivial example to make things absolutely clear. Let us consider the prob-

lem of a single free particle. We would choose our basis as a co-ordinate basis. In strict

sense they are eigen states of co-ordinate operator calculated at certain time say t = 0. We

call these to be in Heisenberg representation, which means they don't have any time de-

pendence. To make things absolutely clear xh(t = 0)|x〉h = x|x〉h. Hence whenever you see

the symbol |x〉h � one thing should be clear that they are eigen states of position operator

constructed at t = 0. In this basis our Hamiltonian can be written as Hs = H0,s = −~ ∂2

∂x2 .

Momentum operator in Schrodinger picture would be Ps = P0,s = −i~ ∂∂x. Eigen states of

this operator in Schrodinger picture would be time dependent. States left to themselves

evolve in time:- |p(t)〉s = exp(−ip2t/2m~) exp(ipx). Momentum operator itself is time in-

dependent. In more complicated systems states would evolve in much more complicated

fashion. A momentum eigen state t = 0 might not remain a momentum eigen state at

latter time. In Heisenberg representation momentum operator become time dependent:

Ph(t) = P0,h(t) = exp(iH0,st/~)(−i~ ∂∂x

) exp(−iH0,st/~). Eigen states constructed at di�er-

ent times would be di�erent. After that, states left to themselves would not evolve with

time.


We de�ne the state of the system in interaction picture as:

Ψs(t) = exp(−iH0,st/~)Ψi(t) = U0,s(t)Ψi(t) (181)

Basically what we are doing here is that we are removing the free-system time evolution as

this part of the evolution is pre-known and we want to concentrate on the evolution due to

time dependent perturbation part of Hamiltonian. One can invert above equation and write:

Ψi(t) = U †0,s(t)Ψs(t) (182)

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Substituting t = 0 one can easily see that:

Ψi(0) = Ψs(0) (183)

Substituting (181) into the complete time evolution equation in Schrodinger's picture (174):

i~ exp(−iH0,st/~)∂Ψi(t)

∂t+H0,sΨs(t) = H0,sΨs(t) + (Vs(t) exp(−iH0,st/~)Ψi(t))

∂Ψi(t)

∂t= (exp(iH0,st/~)Vs(t) exp(−iH0,st/~))Ψi(t)

Now if we de�ne any general Hermitian (not Unitary like Ui) operators in interaction

picture as:

Vi(t) = exp(iH0,st/~)Vs(t) exp(−iH0,st/~) (184)

which can also be written in notational format as:

Vi(t) = U †0,s(t)Vs(t)U0,s(t) (185)

we can then write the time evolution of wavefunction in interaction picture as (this is basically

Schrodinger's equation in interaction picture):

∂Ψi(t)

∂t= Vi(t)Ψi(t) (186)

One can also de�ne the time propagation operator in interaction picture as

Ψi(t) = Ui(t)Ψi(0) (187)

Hence one can write

i~∂Ui(t)

∂t= Vi(t)Ui(t) (188)

This is a very important equation that we would use for generating perturbation se-

quence42.

Key point to remember is that in Interaction picture, states have only non-

42Note that we can write down the time evolution of any general operator in interaction picture as well (ininteraction picture operators might become explicit functions of time). Let us assume that an operator inSchrodinger's picture As does not have any explicit time dependence. Following argument can be generalizedif needed. Using the de�nition of operators in interaction picture (184) and using the equation describing

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18.5 Inter-Relation

trivial time dependence whereas operators have only trivial time dependence

(interaction picture is closer to Schrodinger picture then Heisenberg picture).

We take out trivial time dependence from Schrodinger states by multiplying it

by U †0,s from left side. And we deposit this trivial time dependence into operators

by multiplying Schrodinger operators by U †0,s from left side and by U0,s from right

side.

18.5 Inter-Relation

Using de�nition of state (181) and (187) propagator in interaction picture one can see that:

Ψs(t) = U0,s(t)Ψi(t) = U0,s(t)Ui(t)Ψi(0)

Moreover, from the boundary condition (183) and the time propagator in Schrodinger's

picture (177) :

Us(t) = U0,s(t)Ui(t) (190)

This is a pretty important relationship. After developing the perturbation in interaction

picture, this equation is used to move back to Schrodinger's picture.

Secondly, using the de�nition of operator representation in Heisenberg picture (180) and

above expression (190):

Ah(t) = U †i (t)U †0,s(t)As(t)U0,s(t)Ui(t)

Which in turn means:

Ah(t) = U †i (t)Ai(t)Ui(t) (191)

the equation of motion of U0,s(176):

i~∂Ai(t)∂t

= i~∂U†0,s(t)∂t

AsU0,s(t) + i~U†0,sAs(t)∂U0,s(t)∂t

i~∂Ai(t)∂t

= −H†0,sU†0,s(t)AsU0,s(t) + U†0,sAs(t)H0,sU0,s(t)

now, simply by looking at the de�nition, one can �gure out that U0,s(t) and H0,s commute (using this sameproperty one can show that H0,s = H0,i). Hence:

i~∂Ai(t)∂t

= U†0,s(t)[As, H0,s]U0,s(t) = [Ai, H0,i] (189)

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18.6 Diagrammatic Perturbation Theory

This another, very important relationship that we would use time and again.


The time evolution equation of Ui(t) derived above (188) can be used to obtain a perturbation

solution for Ui(t). Only important point to remember is that the Hamiltonian at di�erent

times do not necessarily commute. So one needs to include the time ordering operator when

converting equation into integral equation in terms of exponentials. Let us explore this

explicitly. Integrating (188), symbolically one can write:

Ui(t)− Ui(0) = −i∫ t

0

Vi(t1)Ui(t1)dt1

Noting that Ψi(0) = Ψs(0) one can see that Ui(0) = 1 (an identity operator). Hence

Ui(t)− 1 = −i∫ t

0

Vi(t1)Ui(t1)dt1

Now a zeroth order approximation term would obtained if we assume Vi(t) = 0. So

U(0)i (t) = 1

Substituting this back into above integral, the �rst order correction term would be:

U(1)i (t) = −i

∫ t

0

Vi(t1)dt1 (192)

Similarly, second order correction term would be:

U(2)i (t) = (−i)2

∫ t

0

Vi(t1){∫ t1

0

Vi(t2)dt2}dt1 (193)

= (−i)2

∫ t

0

dt1

∫ t1

0

dt2Vi(t1)Vi(t2) (194)

First thing to be stressed is to point out that this is an integral of time dependent operators �

which might not commute at di�erent times. Hence, in general, Vi(t1)Vi(t2) 6= Vi(t2)Vi(t1). So

one should not try to shu�e the operators in a product like this. Secondly, one should notice

that one has to keep the order of integrations. For example, in above integration scheme,

one needs to evaluate the integral in curly braces before doing the second integral. Another

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way of saying the same thing is that for any given value of the variable t1, the variable t2 can

only takes values from 0 up to t1. Hence, t2 ≤ t1. Combining above two peculiarities, one

should make sure that in the second form of the above equation (193), operators with latter

times as variables should go to the left. Or in other words time variable of any operator

should never be allowed to become greater than the time variable of an operator sitting to

its left.

Now these di�erent time limits on two integral make things very di�cult. If we can �nd

a way of making the two limits on two integrals independent of each other it would make

things much simpler. Let us de�ne a new operator which shu�es the operators in a product

so that latter always time sits on the left. We call it a time ordering operator. That is

T [Vi(t1)Vi(t2)] ≡

{Vi(t1)Vi(t2) if t1 > t2

Vi(t2)Vi(t1) if t2 > t1

}(195)

One would notice that W (t1, t2) = T [Vi(t1)Vi(t2)] is a symmetric operator in t1 and t2 plane

about the line t1 = t2. In other words W (1, 2) = W (2, 1). Looking at the equation 193, one

would realize that U(2)i (t) is proportional to the upper triangular integration of a two variable

functionW (t1, t2) on t1 and t2 plane. Moreover since, W (t1, t2) is symmetrical about t1 = t2,

we can also write above expression as

U(2)i (t) =

(−i)2

2

∫ t

0

dt1

∫ t

0

dt2T [Vi(t1)Vi(t2)]

Hence we were able to get rid of the di�erent limits on the integrals by introducing a

concept of time ordering operator. Same way, one can write an nth order correction term

as well. Summing all terms, an in�nite order estimate of Ui(t) can be written as:

Ui(t) = I +∞∑n=1

(−i)n

n!

∫dt1

∫dt2

∫dt3.....

∫dtnT [Vi(t1)Vi(t2)Vi(t3).....Vi(tn)] (196)

With T being the time-ordering operator such that left most operator has biggest time. That

is (tn < tn−1.... < t1) if above written order is already time-ordered. All integrals go from

ti = 0 to ti = t. We now de�ne a symbolic representation for the above series. We

would write in short:

Ui(t) = T [exp(−i∫ t

0

dt′Vi(t′))]

One should compare this with the time-independent version de�ned in previous section 18.1.

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18.7 Expectation Values


18.7.1 Quantum Ensemble Expectations

18.7.1.1 Gell-Mann and Low Theorem First of all, suppose instead of propagating

a state from t = 0 to t = t, I want to propagate from t = t1 to t = t2. This can easily be

achieved as:

Ui(t2 ← t1) = Ui(t2)U †i (t1) (197)

Similarly,

Us(t2 ← t1) = Us(t2)U †s (t1) (198)

We can also generalize the relation between above two propagators (186) as follows. We �rst

note that:

Ψi(t2) = Ui(t2 ← t1)Ψi(t1)

Now using (186) we get Ψs(t) = U0,s(t)Ψi(t) and its inverse Ψi(t) = U †0,s(t)Ψs(t). Hence

Ψs(t2) = U0,s(t2)Ui(t2 ← t1)Ψi(t1) = U0,s(t2)Ui(t2 ← t1)U †0,s(t1)Ψs(t1)

Hence

Us(t2 ← t1) = U0,s(t2)Ui(t2 ← t1)U †0,s(t1) (199)

Similarly other relations can also be generalized to two-time operators.

Now let us �rst de�ne the zero reference level of our energy spectrum. Let |0〉0,h be thelowest energy eigen state of the unperturbed Hamiltonian. We write it in Heisenberg repre-

sentation to explicitly specify that this state does not has any time dependence whatsoever.

Or in other words |0〉0,h represents |0(t = 0)〉0,s or any other time that is taken as a reference

time. We de�ne the reference level of energies to be such that

Hs,0|0〉0,h = 0

Now, let |n〉h be the energy eigen state at (t = 0) of perturbed system. Again Heisenberg

subscript has been added to explicitly say that these states do not have any time dependence.

Hs|n〉h = En|n〉h

One should speci�cally note that Hs in above equation can certainly be time dependent.

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This is just an eigen value equation (which certainly can have time dependent operators).

To make things clear, in Schrodinger's picture we would have

|n〉s = exp(−iEnt/~)|n〉h

We are speci�cally interested in |0〉h. This state is unknown and we hope that this can be

calculated by evolving |0〉0,h from −∞ to present time. In general one can always write

Us(t2 ← t1)|0〉0,h =∞∑n=0

αn exp(−iEn(t2 − t1)/~)|n〉h

where αn = h〈n|0〉0,h. One of the important assumptions that we have to make

here is that α0 6= 0. As far as concept of �perturbation� is valid, we claim that this has

to be true. There are formal problems in this assertion. It can be shown in QFT that, in

in�nite volumes, any two states are orthogonal. We can get rid of this problem by working

with �nite volume and taking limits at the end. Moreover we would assume that the

state |0〉h is non-degenerate. This can not be proved mathematically. In fact there are

interesting counter examples like those of spontaneously broken symmetry. But for the time

being we would exclude those exotic problems and assume that perturbed ground state is

unique. Additionally, we would make another claim without proving it. We would assume

that we are allowed to send the time variable to complex values in the above

expression. This can be rigorously justi�ed but I am skipping that part. Procedure we

would follow is known as analytic continuation. This type of QFT is known as �Euclidean

QFT�. Basically, if real time QFT is Lorentz invariant then analytic continued theory would

be O(4) invariant. We would keep the inner product independent of rotation in complex t

plane.

So let us send t1 → −∞(1 − iε) where ε is a small real number. Also choose t2 = 0.

Hence lowest energy term would be slowest decaying term. Hence,

|0〉h = limt1→−∞(1−iε)

1

α0 exp(iE0t1)Us(0← t1)|0〉0,h

Now from 199 we know that Us(0 ← t1) = Ui(t2 ← t1)U †0,s(t1). And moreover U †0,s(t1)|0〉0,h= exp(i0(t1)/~)|0〉0,h = |0〉0,h from the de�nition of the zero level of energy scale. Hence, we

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can write the above expression as

|0〉h = limt1→−∞(1−iε)

1

α0 exp(iE0t1)Ui(0← t1)|0〉0,h (200)

From here onward, we would simply write this expression in shorthand notation as

|0〉h = βUi(0← −∞)|0〉0,h (201)

Here β is a complex number. Limit is understood. On similar lines, one can also derive

〈0|h = βUi(+∞← 0)〈0|0,h

For the bra state one can get an alternative expression simply by taking adjoint of 201.

〈0|h = β?Ui(−∞← 0)〈0|0,h

This is known as Gell-Mann and Low Theorem.

18.7.1.2 Quantum Ensemble Expectation One interesting point, and that is the

most important reason why we are studying it in statistical mechanics context, is that even

evaluation of expectation values of �instantaneous time operators� can be included in the

perturbation expansion. What I mean by instantaneous time operators is that operators

that perturb the state of the system instantaneously. Something like a creation or anni-

hilation operators, for example. Let cλ,i(t1) and cλ,h(t1)be the interaction and Heisenberg

representation of one such operator. Suppose, I want to evaluate something like

〈Ψh|cλ,h(t1)|Ψh〉

Where |Ψh〉 is the Heisenberg representation of an exact energy eigen state of the system

(proper of visualizing such an expression is discussed in Green's function section). We

are simply evaluating a quantum mechanical expectation value. Note that, usually, even

Ψs(0) = Ψi(0) is unknown since this state includes the e�ects of perturbation (interaction

among particles of system).

Now, from Gel-Mann and Low Theorem discussed above, one can write above expression

as

〈Ψh,0|Ui(−∞← 0)Ui(0← t1)cλ,i(t1)Ui(t1 ← 0)Ui(0← −∞)|Ψh,0〉 (202)

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Where I have used the fact that Ui(0 ← t1) = U †i (t1 ← 0). Here states are now non-

interacting ground states. In certain cases above expression can also be written as:

〈Ψh,0|Ui(∞← 0)Ui(0← t1)cλ,i(t1)Ui(t1 ← 0)Ui(0← −∞)|Ψh,0〉〈Ψh,0|Ui(+∞← −∞)|Ψh,0〉

(203)

This second expression can be justi�ed provided 〈Ψs,0|Ui(+∞ ← 0) is a well de�ned

state. In equilibrium this can be justi�ed. But not in non-equilibrium. Now if we include

time ordering operator, above can be written as:

〈Ψh,0|T Ui(−∞← −∞)cλ,i(t1)|Ψh,0〉 (204)

or, in some cases:〈Ψh,0|T Ui(+∞← −∞)cλ,i(t1)|Ψh,0〉〈Ψh,0|Ui(+∞← −∞)|Ψh,0〉

(205)

After expanding Ui in perturbation, cλ,i is properly placed in time sequence and above

expectation can easily be evaluated. This is the way one extract equilibrium information

about complicated interacting systems using perturbation.

Part X

Statistical Quantum Field Theory

(Condensed Matter Physics)

Please refer to a separate tutorial article (author?) [2] on Non-Equilibrium Green's Func-

tions (NEGF) and Statistical QFT.

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Part XI

Further Resources

Reading

• Background Material

� Some useful mathematical background can be found in one of the articles [3]

written by the author or [4] or the wikipedia pages [5].

� For a brief review of postulatory nature of quantum mechanics see another article

[6] or following references [7, 8, 9]. Symmetries in physical laws and linearity of

quantummechanics are discussed in related articles [10, 11]. References [12, 13, 14]

also provides good discussion about symmetries. Review of quantum �eld theory

(QFT) can be found in the reference [15] or [16, 17, 18] while an introductory

treatment of quantum measurements can be found here [19] or here [20, 8].

� A good discussion on equilibrium quantum statistical mechanics can be found in

references [21, 22, 23] while an introductory treatment of statistical quantum �eld

theory (QFT) and details of density matrix formalism can be found in the refer-

ences [2, 24]. A brief discussion of irreversible or non-equilibrium thermodynamics

can be found here [25, 26].

� To understand relationship between magnetism, relativity, angular momentum

and spins, readers may want to check references [27, 28] on magnetics and spins.

Some details of electron spin resonance (ESR) measurement setup can be found

here [29, 30].

� Electronic aspects of device physics and bipolar devices are discussed in [31,

32, 33, 34, 35]. Details of electronic band structure calculations are discussed in

references [36, 37, 38] and semiclassical transport theory and quantum transport

theory are discussed in references [39, 40, 41, 42].

� List of all related articles from author can be found at author's homepage.

• Quantum Field Theory

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http://www.stanford.edu/~mukul/tutorials/math.pdf


http://www.stanford.edu/~mukul/tutorials/symmetries.pdf



http://www.stanford.edu/~mukul/tutorials/Quantum_Optics.pdf


http://www.stanford.edu/~mukul/tutorials/Quantum_Measurements.pdf

http://www.stanford.edu/~mukul/tutorials/stat_mech.pdf



http://www.stanford.edu/~mukul/tutorials/Irreversible.pdf

http://www.stanford.edu/~mukul/tutorials/magnetic.pdf

http://www.stanford.edu/~mukul/tutorials/esr.pdf

http://www.stanford.edu/~mukul/tutorials/device_physics.pdf

http://www.stanford.edu/~mukul/tutorials/bipolar.pdf

http://www.stanford.edu/~mukul/tutorials/valance band.pdf

http://www.stanford.edu/~mukul/tutorials/scattering.pdf

http://www.stanford.edu/~mukul/tutorials/mesoscopic_transport.pdf





� Peskin's book ((author?) [17]) on quantum �eld theory is nowadays becoming a

standard text book. This book is mostly written from elementary particle physics

point of view. It's mostly useful in learning how to solve real world �eld theory

problems. It's fairly mathematically sound but gives very little insight into the

physical aspects of the subject itself.

� Weinberg's book ((author?) [18]) is another modern standard text book. As far

as physical insight into the subject is concerned, it's better than Peskin's book,

but still not very satisfying.

� Three related texts books by Greiner ([43, 44, 45]) on � QFT , QED, and rela-

tivistic QM are also useful. They are not very readable. Organization of these

books is also very erratic.

• Quantum Optics

� Quantum optics ([46]) text book authored by M. O. Scully and M. S. Zubairy is

one of the most easy to read book in quantum optics and quantum coherence. It

provides insight into basic physics (for example, why only E+ part is included in

coherence correlation functions or why position operator for photon can not be

de�ned or what is the classical limit of QFT etc.). This book does not deal with

cavity QED. For cavity QED see the next reference.

� Vogel et al. ([47]) wrote a nice text book introduction to cavity QED. It in-

cludes �eld quantization in inhomogeneous dielectric media (basically concept of

orthonormal mode quantization as opposed to simple k-space quantization) as

well as quantization in inhomogeneous lossy media.

� Recently, there has been some nice research on much more neat quantization

in lossy media. In lossy media, Maxwell operator is not Hermitian and hence

it's eigen modes are not orthogonal. So one has to give up orthogonal mode

quantization scheme. But one can develop an analogous scheme that proceeds by

studying classical Green's function of the wave equation.

� L. Mandel and E. Wolf, Optical Coherence and Quantum Optics ([48]) is fairly

detailed, really good easy to read book on classical and quantum coherence.

• Classical Coherence

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REFERENCES

� John Strong, Concepts of Classical Optics ([1]) explains coherence in interference

�lters in details.

� L. Mandel and E. Wolf, Optical Coherence and Quantum Optics ([48]) is really

good easy to read book.

� W. Wnag, H. Kozaki, J. Rosen and M. Takeda, �Synthesis of longitudinal coher-

ence functions by spatial modulation of an extended light source: a new interpre-

tation and experimental veri�cations,� Appl. Opt. 41, 1962 (2002).

• Phonons

� L. M. Magid, Phys. Rev. 134, A158 (1964), energy velocity generalization for

3D lattices.

• Quantum Field Theory for Solid State Physics

� Haken's book ([16]) is a nice elementary introduction to �eld theory as used in

solid state physics (it's mostly QFT of quasi-particles).

� David Pines's book ([49]) is another elementary readable book in this subject.

• Statistical Quantum Field Theory

� Abrikosov's ([24]) book is a masterly written classic on this subject.

� Fetter's book ([50]) is somewhat more readable than Abrikosov's book.

� Mahan's book ([51]) is another popular text book in this subject though I per-

sonally don't like it much. The latest edition is actually much much better than

older versions.

� Doniach's book ([52]) on Green's functions as used in solid state physics can also

be useful in learning how to solve practical problems.

References

[1] J. Strong, Concepts of Classical Optics (Dover Publications, 2004). (Cited on pages 112

and 157.)

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REFERENCES

[2] M. Agrawal, �Non-Equilibrium Statistical Quantum Field Theory,� (2005). URL http:

//www.stanford.edu/~mukul/tutorials/stat_QFT.pdf. (Cited on pages 154 and 155.)

[3] M. Agrawal, �Abstract Mathematics,� (2002). URL http://www.stanford.edu/~mukul/

tutorials/math.pdf. (Cited on page 155.)

[4] E. Kreyszig, �Advanced engineering mathematics,� (1988). (Cited on page 155.)

[5] C. authored, �Wikipedia,� URL http://www.wikipedia.org. (Cited on page 155.)

[6] M. Agrawal, �Axiomatic/Postulatory Quantum Mechanics,� (2002). URL http://www.

stanford.edu/~mukul/tutorials/Quantum_Mechanics.pdf. (Cited on page 155.)

[7] A. Bohm, Quantum Mechanics/Springer Study Edition (Springer, 2001). (Cited on

page 155.)

[8] J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univer-

sity Press, 1996). (Cited on page 155.)

[9] D. Bohm, Quantum Theory (Dover Publications, 1989). (Cited on page 155.)

[10] M. Agrawal, �Symmetries in Physical World,� (2002). URL http://www.stanford.edu/

~mukul/tutorials/symetries.pdf. (Cited on page 155.)

[11] M. Agrawal, �Linearity in Quantum Mechanics,� (2003). URL http://www.stanford.

edu/~mukul/tutorials/linear.pdf. (Cited on page 155.)

[12] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, The

De�nitive Edition Volume 3 (2nd Edition) (Feynman Lectures on Physics (Hardcover))

(Addison Wesley, 2005). (Cited on page 155.)

[13] H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics (3rd Edition) (Addison

Wesley, 2002). (Cited on page 155.)

[14] R. Shankar, Principles of Quantum Mechanics (Plenum US, 1994). (Cited on page 155.)

[15] M. Agrawal, �Quantum Field Theory (QFT) and Quantum Optics (QED),� (2004).

URL http://www.stanford.edu/~mukul/tutorials/Quantum_Optics.pdf. (Cited on

page 155.)

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http://www.wikipedia.org










REFERENCES

[16] H. Haken, Quantum Field Theory of Solids: An Introduction (Elsevier Science Publish-

ing Company, 1983). (Cited on pages 155 and 157.)

[17] M. E. Peskin, An Introduction to Quantum Field Theory (HarperCollins Publishers,

1995). (Cited on pages 155 and 156.)

[18] S. Weinberg, The Quantum Theory of Fields, Vol. 1: Foundations (Cambridge Univer-

sity Press, 1995). (Cited on pages 155 and 156.)

[19] M. Agrawal, �Quantum Measurements,� (2004). URL http://www.stanford.edu/

~mukul/tutorials/Quantum_Measurements.pdf. (Cited on page 155.)

[20] Y. Yamamoto and A. Imamoglu, �Mesoscopic Quantum Optics,� Mesoscopic Quantum

Optics, published by John Wiley & Sons, Inc., New York, 1999. (1999). (Cited on

page 155.)

[21] M. Agrawal, �Statistical Quantum Mechanics,� (2003). URL http://www.stanford.edu/

~mukul/tutorials/stat_mech.pdf. (Cited on page 155.)

[22] C. Kittel and H. Kroemer, Thermal Physics (2nd Edition) (W. H. Freeman, 1980).

(Cited on page 155.)

[23] W. Greiner, L. Neise, H. Stöcker, and D. Rischke, Thermodynamics and Statistical

Mechanics (Classical Theoretical Physics) (Springer, 2001). (Cited on page 155.)

[24] A. A. Abrikosov, Methods of Quantum Field Theory in Statistical Physics (Selected

Russian Publications in the Mathematical Sciences.) (Dover Publications, 1977). (Cited

on pages 155 and 157.)

[25] M. Agrawal, �Basics of Irreversible Thermodynamics,� (2005). URL http://www.

stanford.edu/~mukul/tutorials/Irreversible.pdf. (Cited on page 155.)

[26] N. Tschoegl, Fundamentals of equilibrium and steady-state thermodynamics (Elsevier

Science Ltd, 2000). (Cited on page 155.)

[27] M. Agrawal, �Magnetic Properties of Materials, Dilute Magnetic Semiconductors, Mag-

netic Resonances (NMR and ESR) and Spintronics,� (2003). URL http://www.stanford.

edu/~mukul/tutorials/magnetic.pdf. (Cited on page 155.)

[28] S. Blundell, �Magnetism in condensed matter,� (2001). (Cited on page 155.)

Mukul Agrawal



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REFERENCES

[29] M. Agrawal, �Bruker ESR System,� (2005). URL http://www.stanford.edu/~mukul/

tutorials/esr.pdf. (Cited on page 155.)

[30] C. Slitcher, �Principles of Magnetic Resonance,� Springer Series in Solid State Sciences

1 (1978). (Cited on page 155.)

[31] M. Agrawal, �Device Physics,� (2002). URL http://www.stanford.edu/~mukul/

tutorials/device_physics.pdf. (Cited on page 155.)

[32] M. Agrawal, �Bipolar Devices,� (2001). URL http://www.stanford.edu/~mukul/

tutorials/bipolar.pdf. (Cited on page 155.)

[33] R. F. Pierret, Semiconductor Device Fundamentals (Addison Wesley, 1996). (Cited on

page 155.)

[34] R. F. Pierret, Advanced Semiconductor Fundamentals (2nd Edition) (Modular Series on

Solid State Devices, V. 6) (Prentice Hall, 2002). (Cited on page 155.)

[35] S. Sze, Physics of Semiconductor Devices (John Wiley and Sons (WIE), 1981). (Cited

on page 155.)

[36] M. Agrawal, �Electronic Band Structures in Nano-Structured Devices and Materials,�

(2003). URL http://www.stanford.edu/~mukul/tutorials/valanceband.pdf. (Cited on

page 155.)

[37] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks Cole, 1976). (Cited on

page 155.)

[38] S. L. Chuang, Physics of Optoelectronic Devices (Wiley Series in Pure and Applied

Optics) (Wiley-Interscience, 1995). (Cited on page 155.)

[39] M. Agrawal, �Classical and Semiclassical Carrier Transport and Scattering Theory,�

(2003). URL http://www.stanford.edu/~mukul/tutorials/scattering.pdf. (Cited on

page 155.)

[40] M. Agrawal, �Mesoscopic Transport,� (2005). URL http://www.stanford.edu/~mukul/

tutorials/mesoscopic_transport.pdf. (Cited on page 155.)

[41] M. Lundstrom, Fundamentals of carrier transport (Cambridge Univ Pr, 2000). (Cited

on page 155.)

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http://www.stanford.edu/~mukul/tutorials/valance band.pdf

http://www.stanford.edu/~mukul/tutorials/scattering.pdf





REFERENCES

[42] S. Datta, Electronic transport in mesoscopic systems (Cambridge Univ Pr, 1997). (Cited

on page 155.)

[43] D. Bromley and W. Greiner, Relativistic Quantum Mechanics. Wave Equations

(Springer, 2000). (Cited on page 156.)

[44] W. Greiner and J. Reinhardt, Quantum Electrodynamics (Springer, 2002). (Cited on

page 156.)

[45] D. Bromley, W. Greiner, and J. Reinhardt, Field Quantization (Springer, 1996). (Cited

on page 156.)

[46] M. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, 1997). (Cited

on page 156.)

[47] W. Vogel, D. Welsch, and S. Wallentowitz, Quantum optics: an introduction (Wiley-

VCH, Berlin, 2001). (Cited on page 156.)

[48] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University

Press, 1995). (Cited on pages 156 and 157.)

[49] D. Pines, Elementary Excitations in Solids : Lectures on Phonons, Electrons, and Plas-

mons (Advanced Book Classics) (Perseus Books Group, 1999). (Cited on page 157.)

[50] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover

Publications, 2003). (Cited on page 157.)

[51] G. D. Mahan, Many Particle Physics (Physics of Solids and Liquids) (Springer, 2000).

(Cited on page 157.)

[52] S. Doniach, Green's functions for solid state physicists (Frontiers in physics, 44) (W.

A. Benjamin, 1974). (Cited on page 157.)

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