michael luke relativistic and non-relativistic quantum mechanics

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Michael Luke

QUANTUM MECHANICS

RELATIVISTIC NON-RELATIVISTIC

Lecture Notes

2011

PHY2403F Lecture NotesMichael Luke2011

These notes are perpetually under construction. Please let me know of any typos or errors. I claim little originality; these notes are in large part an abridged and revised version of Sidney Colemans eld theory lectures from Harvard.

2ContentsI. Introduction A. Relativistic Quantum Mechanics B. Conventions and Notation 1. Units 2. Relativistic Notation 3. Fourier Transforms 4. The Dirac Delta Function C. A Na Relativistic Theory ve II. Constructing Quantum Field Theory A. Multi-particle Basis States 1. Fock Space 2. Review of the Simple Harmonic Oscillator 3. An Operator Formalism for Fock Space 4. Relativistically Normalized States B. Canonical Quantization 1. Classical Particle Mechanics 2. Quantum Particle Mechanics 3. Classical Field Theory 4. Quantum Field Theory C. Causality III. Symmetries and Conservation Laws A. Classical Mechanics B. Symmetries in Field Theory 1. Space-Time Translations and the Energy-Momentum Tensor 2. Lorentz Transformations C. Internal Symmetries 1. U (1) Invariance and Antiparticles 2. Non-Abelian Internal Symmetries D. Discrete Symmetries: C, P and T 1. Charge Conjugation, C 2. Parity, P 3. Time Reversal, T IV. Example: Non-Relativistic Quantum Mechanics (Second Quantization) V. Interacting Fields A. Particle Creation by a Classical Source B. More on Green Functions 5 5 9 9 10 14 15 15 20 20 20 22 23 23 25 26 27 29 32 37 40 40 42 43 44 48 49 53 55 55 56 58 60 65 65 69

3C. Mesons Coupled to a Dynamical Source D. The Interaction Picture E. Dysons Formula F. Wicks Theorem G. S matrix elements from Wicks Theorem H. Diagrammatic Perturbation Theory I. More Scattering Processes J. Potentials and Resonances VI. Example (continued): Perturbation Theory for nonrelativistic quantum mechanics VII. Decay Widths, Cross Sections and Phase Space A. Decays B. Cross Sections C. D for Two Body Final States VIII. More on Scattering Theory A. Feynman Diagrams with External Lines o the Mass Shell 1. Answer One: Part of a Larger Diagram 2. Answer Two: The Fourier Transform of a Green Function 3. Answer Three: The VEV of a String of Heisenberg Fields B. Green Functions and Feynman Diagrams C. The LSZ Reduction Formula 1. Proof of the LSZ Reduction Formula IX. Spin 1/2 Fields A. Transformation Properties B. The Weyl Lagrangian C. The Dirac Equation 1. Plane Wave Solutions to the Dirac Equation D. Matrices 1. Bilinear Forms 2. Chirality and 5 E. Summary of Results for the Dirac Equation 1. Dirac Lagrangian, Dirac Equation, Dirac Matrices 2. Space-Time Symmetries 3. Dirac Adjoint, Matrices 4. Bilinear Forms 5. Plane Wave Solutions X. Quantizing the Dirac Lagrangian A. Canonical Commutation Relations 70 71 73 77 80 83 88 92 95 98 101 102 103 106 106 107 108 110 111 115 117 125 125 131 135 137 139 142 144 146 146 146 147 148 149 151 151

4or, How Not to Quantize the Dirac Lagrangian B. Canonical Anticommutation Relations C. Fermi-Dirac Statistics D. Perturbation Theory for Spinors 1. The Fermion Propagator 2. Feynman Rules E. Spin Sums and Cross Sections XI. Vector Fields and Quantum Electrodynamics A. The Classical Theory B. The Quantum Theory C. The Massless Theory 1. Minimal Coupling 2. Gauge Transformations D. The Limit 0 1. Decoupling of the Helicity 0 Mode E. QED F. Renormalizability of Gauge Theories 151 153 155 157 159 161 166 168 168 173 176 179 182 184 186 187 189

5

jet #2

jet #3 jet #1

jet #4 e+

FIG. I.1 The results of a proton-antiproton collision at the Tevatron at Fermilab. The proton and antiproton beams travel perpendicular to the page, colliding at the origin of the tracks. Each of the curved tracks indicates a charged particle in the nal state. The tracks are curved because the detector is placed in a magnetic eld; the radius of the curvature of the path of a particle provides a means to determine its mass, and therefore identify it.

I. INTRODUCTION A. Relativistic Quantum Mechanics

Usually, additional symmetries simplify physical problems. For example, in non-relativistic quantum mechanics (NRQM) rotational invariance greatly simplies scattering problems. Why does the addition of Lorentz invariance complicate quantum mechanics? The answer is very simple: in relativistic systems, the number of particles is not conserved. In a quantum system, this has profound implications. Consider, for example, scattering a particle in potential. At low energies, E mc2 where

relativity is unimportant, NRQM provides a perfectly adequate description. The incident particle is in some initial state, and one can fairly simply calculate the amplitude for it to scatter into any nal state. There is only one particle, before and after the scattering process. At higher energies where relativity is important things gets more complicated, because if E mc2 there is enough energy to pop additional particles out of the vacuum (we will discuss how this works at length in the course). For example, in p-p (proton-proton) scattering with a centre of mass energy E > m c2 (where m 140 MeV is the mass of the neutral pion) the process p + p p + p + 0

6 is possible. At higher energies, E > 2mp c2 , one can produce an additional proton-antiproton pair: p+pp+p+p+p and so on. Therefore, what started out as a simple two-body scattering process has turned into a many-body problem, and it is necessary to calculate the amplitude to produce a variety of manybody nal states. The most energetic accelerator today is the Large Hadron Collider at CERN, outside Geneva, which collides protons and antiprotons with energies of several TeV, or several thousands times mp c2 , so typical collisions produce a huge slew of particles (see Fig. I.1). Clearly we will have to construct a many-particle quantum theory to describe such a process. However, the problems with NRQM run much deeper, as a brief contemplation of the uncertainty principle indicates. Consider the familiar problem of a particle in a box. In the nonrelativistic description, we can localize the particle in an arbitrarily small region, as long as we accept an arbitrarily large uncertainty in its momentum. But relativity tells us that this description must break down if the box gets too small. Consider a particle of mass trapped in a container with reecting walls of side L. The uncertainty in the particles momentum is therefore of order /L. h In the relativistic regime, this translates to an uncertainty of order hc/L in the particles energy. For L small enough, L < /c (where h/c c , the Compton wavelength of the particle), the h uncertainty in the energy of the system is large enough for particle creation to occur - particle anti-particle pairs can pop out of the vacuum, making the number of particles in the container uncertain! The physical state of the system is a quantum-mechanical superposition of states with dierent particle number. Even the vacuum state - which in an interacting quantum theory is not the zero-particle state, but rather the state of lowest energy - is complicated. The smaller the distance scale you look at it, the more complex its structure. There is therefore no sense in which it is possible to localize a particle in a region smaller than its Compton wavelength. In atomic physics, where NRQM works very well, this does not introduce any problems. The Compton wavelength of an electron (mass = 0.511 MeV/c2 ), is 1/0.511 MeV 197 MeV fm 4 1011 cm, or about 103 Bohr radii. So there is no problem localizing an electron on atomic scales, and the relativistic corrections due to multi-particle states are small. On the other hand, the up and down quarks which make up the proton have masses of order 10 MeV (c 20 fm) and are conned to a region the size of a proton, or about 1 fm. Clearly

the internal structure of the proton is much more complex than a simple three quark system, and relativistic eects will be huge. Thus, there is no such thing in relativistic quantum mechanics as the two, one, or even zero

7

L L

(a) L> h/c >

(b) L< 02 2

Im i

k+ < 0

2

2

- i

FIG. I.6 Contour integral for evaluating the integral in Eq. (I.48). The original path of integration is along the real axis; it is deformed to the dashed path (where the radius of the semicircle is innite). The only contribution to the integral comes from integrating along the branch cut.

For r > t, i.e. for a point outside the particles forward light cone, we can prove using contour integration that this integral is non-zero. Consider the integral Eq. (I.48) dened in the complex k plane. The integral is along the real axis, and the integrand is analytic everywhere in the plane except for branch cuts at k = i, arising from the square root in k . The contour integral can be deformed as shown in Fig. (I.6). For r > t, the integrand vanishes exponentially on the circle at innity in the upper half plane, so the integral may be rewritten as an integral along the branch cut. Changing variables to z = ik, x |(t) = i (2)2 r i r = e 2 2 r

2 2 2 2 (iz)d(iz)ezr e z t e z t dz ze(z)r sinh z 2 2 t . (I.49)

The integrand is positive denite, so the integral is non-zero. The particle has a small but nonzero probability to be found outside of its forward light-cone, so the theory is acausal. Note the exponential envelope, er in Eq. (I.49) means that for distances r 1/ there is a negligible

chance to nd the particle outside the light-cone, so at distances much greater than the Compton wavelength of a particle, the single-particle theory will not lead to measurable violations of causality. This is in accordance with our earlier arguments based on the uncertainty principle: multi-particle eects become important

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