lorentz invariance - iu bdermisek/qft_09/qft-i-2-4p.pdf · lorentz invariance based on s-2 lorentz...

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Lorentz invariance based on S-2 Lorentz transformation (linear, homogeneous change of coordinates): that preserves the interval : All Lorentz transformations form a group: product of 2 LT is another LT identity transformation: inverse: for inverse can be used to prove: 20 Infinitesimal Lorentz transformation: thus there are 6 independent ILTs: 3 rotations and 3 boosts not all LT can be obtained by compounding ILTs! +1 proper -1 improper proper LTs form a subgroup of Lorentz group; ILTs are proper! Another subgroup - orthochronous LTs, ILTs are orthochronous! 21 When we say theory is Lorentz invariant we mean it is invariant under proper orthochronous subgroup only (those that can be obtained by compounding ILTs) Transformations that take us out of proper orthochronous subgroup are parity and time reversal : orthochronous but improper nonorthochronous and improper A quantum field theory doesn’t have to be invariant under P or T. 22 How do operators and quantum fields transform? Lorentz transformation (proper, orthochronous) is represented by a unitary operator that must obey the composition rule: infinitesimal transformation can be written as: from using and expanding both sides, keeping only linear terms in we get: since are arbitrary general rule: each vector index undergoes its own Lorentz transformation! are hermitian operators = generators of the Lorentz group 23

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Page 1: Lorentz invariance - IU Bdermisek/QFT_09/qft-I-2-4p.pdf · Lorentz invariance based on S-2 Lorentz transformation (linear, homogeneous change of coordinates): that preserves the interval

Lorentz invariancebased on S-2

Lorentz transformation (linear, homogeneous change of coordinates):

that preserves the interval :

All Lorentz transformations form a group:

product of 2 LT is another LT

identity transformation:

inverse:

for inverse can be used to prove:

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Infinitesimal Lorentz transformation:

thus there are 6 independent ILTs: 3 rotations and 3 boosts

not all LT can be obtained by compounding ILTs!+1 proper

-1 improper

proper LTs form a subgroup of Lorentz group; ILTs are proper!

Another subgroup - orthochronous LTs,

ILTs are orthochronous!

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When we say theory is Lorentz invariant we mean it is invariant under proper orthochronous subgroup only (those that can be obtained by compounding ILTs)

Transformations that take us out of proper orthochronous subgroup are parity and time reversal:

orthochronous but improper

nonorthochronous and improper

A quantum field theory doesn’t have to be invariant under P or T.

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How do operators and quantum fields transform?

Lorentz transformation (proper, orthochronous) is represented by a unitary operator that must obey the composition rule:

infinitesimal transformation can be written as:

from using

and expanding both sides, keeping only linear terms in we get:

since are arbitrary

general rule: each vector index undergoes its own Lorentz transformation!

are hermitian operators = generators of the Lorentz group

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Page 2: Lorentz invariance - IU Bdermisek/QFT_09/qft-I-2-4p.pdf · Lorentz invariance based on S-2 Lorentz transformation (linear, homogeneous change of coordinates): that preserves the interval

using and expanding to linear order in we get:

These comm. relations specify the Lie algebra of the Lorentz group.

We can identify components of the angular momentum and boost operators:

and find:

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in a similar way for the energy-momentum four vector we find: Pµ = (H/c, P i)

using and expanding to linear order in we get:

or in components:

in addition:

Comm. relations for J, K, P, H form the Lie algebra of the Poincare group.

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Finally, let’s look at transformation of a quantum scalar field:

Recall time evolution in Heisenberg picture:

this is generalized to:

Px = Pµxµ = P · x!Ht x is just a label

we can write the same formula for x-a:

e+iPa/!e!iPx/!!(0)e+iPx/!e!iPa/! = !(x! a)

we define space-time translation operator:

and obtain:

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Similarly:

Derivatives carry vector indices:

is Lorentz invariant

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Page 3: Lorentz invariance - IU Bdermisek/QFT_09/qft-I-2-4p.pdf · Lorentz invariance based on S-2 Lorentz transformation (linear, homogeneous change of coordinates): that preserves the interval

Canonical quantization of scalar fieldsbased on S-3

Hamiltonian for free nonrelativistic particles:

Furier transform:

a(x) =!

d3p

(2!)3/2eip·x a(p)

we get:

can go back to x using:

!d3x

(2!)3eip·x = "3(p)

!d3p

(2!)3eip·x = "3(x)

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Canonical quantization of scalar fields

(Anti)commutation relations:

[A,B]! = AB !BA

Vacuum is annihilated by :

is a state of momentum , eigenstate of with

is eigenstate of with energy eigenvalue:

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Relativistic generalization

Hamiltonian for free relativistic particles:

spin zero, but can be either bosons or fermions

Is this theory Lorentz invariant?

Let’s prove it from a different direction, direction that we will use for any quantum field theory from now:

start from a Lorentz invariant lagrangian or action

derive equation of motion (for scalar fields it is K.-G. equation)

find solutions of equation of motion

show the Hamiltonian is the same as the one above

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A theory is described by an action:

where is the lagrangian.

Equations of motion should be local, and so

Thus:

where is the lagrangian density.

is Lorentz invariant:

For the action to be invariant we need:

the lagrangian density must be a Lorentz scalar!

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Page 4: Lorentz invariance - IU Bdermisek/QFT_09/qft-I-2-4p.pdf · Lorentz invariance based on S-2 Lorentz transformation (linear, homogeneous change of coordinates): that preserves the interval

Let’s consider:

Any polynomial of a scalar field is a Lorentz scalar and so are products of derivatives with all indices contracted.

and let’s find the equation of motion, Euler-Lagrange equation:

(we find eq. of motion from variation of an action: making an infinitesimal variation in and requiring the variation of the action to vanish)

arbitrary constant

integration by parts, and at infinity in any direction (including time)

!"(x) = 0

is arbitrary function of x and so the equation of motion is Klein-Gordon equation

! = 1, c = 1

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Solutions of the Klein-Gordon equation:

one classical solution is a plane wave:

is arbitrary real wave vector and

The general classical solution of K-G equation:

where and are arbitrary functions of , and

is a function of (introduced for later convenience) |k|

if we tried to interpret as a quantum wave function, the second term would represent contributions with negative energy to the wave function!

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real solutions:

k !" !k

thus we get:

(such a is said to be on the mass shell)kµ

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Finally let’s choose so that is Lorentz invariant:

manifestly invariant under orthochronous Lorentz transformations

on the other hand

sum over zeros of g, in our case the only zero is k0 = !

for any the differential is Lorentz invariant

it is convenient to take for which the Lorentz invariant differential is:

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Page 5: Lorentz invariance - IU Bdermisek/QFT_09/qft-I-2-4p.pdf · Lorentz invariance based on S-2 Lorentz transformation (linear, homogeneous change of coordinates): that preserves the interval

Finally we have a real classical solution of the K.-G. equation:

where again: , ,

For later use we can express in terms of :

where and we will call .

Note, is time independent.

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Constructing the hamiltonian:

Recall, in classical mechanics, starting with lagrangian as a function of coordinates and their time derivatives we define conjugate momenta and the hamiltonian is then given as:

In field theory:

hamiltonian density

and the hamiltonian is given as:

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In our case:

Inserting we get:

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!d3x

(2!)3eip·x = "3(p)

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Page 6: Lorentz invariance - IU Bdermisek/QFT_09/qft-I-2-4p.pdf · Lorentz invariance based on S-2 Lorentz transformation (linear, homogeneous change of coordinates): that preserves the interval

From classical to quantum (canonical quantization):“coordinates” and “momenta” are promoted to operators satisfying canonical commutation relations:

operators are taken at equal times in the Heisenberg picture

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We have derived the classical hamiltonian:

We kept ordering of a’s unchanged, so that we can easily generalize it to quantum theory where classical functions will become operators that may not commute.

The hamiltonian of the quantum theory:

(2!)3"3(0) = Vsee the formula for delta function

is the total zero point energy per unit volume

we are free to choose:

the ground state has zero energy eigenvalue.

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Summary:

We have rederived the hamiltonian of free relativistic bosons by quantization of a scalar field whose equation of motion is the Klein-Gordon equation (starting with manifestly Lorentz invariant lagrangian).

is equivalent to:

for:

does not work for fermions, anticommutators lead to trivial hamiltonian!

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