Physics 253a: Relativistic quantum mechanics

335 Pages·2013·2.265 MB·English

by  Coleman S.

Notes from Sidney Coleman’s Physics 253a
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∗Edited and typeset by Yuan-Sen Ting and Bryan Gin-ge Chen from scans of the handwritten
notes of Brian Hill. Numerous additional edits by Richard Sohn. Please contact Yuan-Sen Ting
([email protected]), Bryan Gin-ge Chen ([email protected]) and Richard Sohn
([email protected]). If you have any questions or comments.
†LATEX version of February 21, 2013
1
0. Preface Notes from Sidney Coleman’s Physics 253a 2
0 Preface
It’s unexpected and heart-warming to be asked by Bryan Chen and Yuan-Sen Ting to write
something about these notes, 25 years after taking them. I was the teaching assistant for
Sidney’s quantum field theory course for three years. In the first year, I sat in, because
frankly, I hadn’t learned quantum field theory well enough the first time that I took it.
When I have the good fortune to hear a really good lecturer, I often re-copy my notes,
preferably the evening on the day that I took them.
Once in a while, students would miss a class, and then ask me if they could look at my
notes. At some point, the requests started happening enough that it was suggested that a
copy be put on reserve in the Harvard physics library. From there, copies of the notes just
kept spreading.
Sidney once expressed disappointment about the spread of the notes. For one thing, I
even wrote down some of his anecdotes and jokes, and that made it less fun for him to re-tell
them. For another, he wrote Aspects of Symmetry which shared a lot of material with what
he taught in Physics 253b. He may have had in mind that he would write a field theory
book as a companion volume.
Of course, he never did write a field theory book, or you’d be reading that, and he never
tried to rein the copies in. Now that he is gone, we are lucky that his clarity lives on.
Thanks to Bryan Chen and Yuan-Sen Ting for creating this lovely LATEX version. Some-
times transcription can seem tedious, but I hope it was as valuable for them as the first
re-copying was for me, and that for you – fellow student of quantum field theory – the
existence of these notes is similarly valuable.
–Brian Hill, www.lingerhere.org, March 10, 2011
Editors’ notes
The great field theorist Sidney Coleman for many years taught the course Physics 253 at
Harvard on Quantum Field Theory. The notes you are reading were typeset from a scanned
version of handwritten lecture notes by Brian Hill from the Fall 1986 of the first half of the
course: Physics 253a. The Harvard Physics Department has made films of the lectures from
the 1975-1976 version of the course available on their website as well.
The typesetting for lectures 1-11 was done by Bryan Gin-ge Chen and for lectures 12-28
by Yuan-Sen Ting, who also recreated most of the figures. We have attempted to stay as
faithful as possible to the scanned notes, aside from correcting some obvious errors in the
notes and changing some of the in-text references.
We thank Richard Sohn, Leonard Gamberg, Avraham Gal for pointing out typographical
errors in the previous versions. The list of typographical errors can be found in Yuan-Sen
Ting’s homepage www.cfa.harvard.edu/ yuanting.
∼
0. Preface Notes from Sidney Coleman’s Physics 253a 3
Contents
0 Preface 2
1 September 23 7
Units; Translational invariance, Rotational invariance and Lorentz invariance for a
single free particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 September 25 14
Position operator; ViolationofCausality; Pair Production; FockSpace; Occupation
number representation; SHO; Oscillator-like formalism for Fock space. . . . . 14
3 September 30 21
Causality, observables and quantum fields; Constructing the quantum field from
Fock space operators axiomatically; Translational invariance; Lorentz invari-
ance and Relativistic normalization; Field constructed satisfies Klein-Gordon
equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 October 2 28
Constructing Fock space from the quantum field axiomatically; The Method of
the Missing Box, Classical Particle Mechanics, Quantum Particle Mechanics,
Classical FieldTheory, QuantumFieldTheory; Quantumfieldfromfreescalar
theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 October 7 38
Hamiltonianrecoveredinfreescalartheoryuptoinfiniteconstant; Normalordering;
Symmetries and conservation laws, Noether’s theorem; Noether’s theorem in
field theory, conserved currents; ambiguity in currents; Energy-momentum
tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6 October 9 49
Lorentz transformations; Angular momentum conservation; Internal symmetries;
SO(2) internal symmetry; Charged field; SO(n) internal symmetry. . . . . . 49
7 October 14 58
Lorentz transformation properties of conserved quantities; Discrete symmetries;
φ φ; Charge conjugation; Parity; Ambiguity of choice of parity; Time
→ −
reversal; Unitary and anti-unitary operators, angular momentum; Dilatations. 58
8 October 16 70
0. Preface Notes from Sidney Coleman’s Physics 253a 4
Scattering theory overview; Low budget scattering theory; Turning on and off func-
tion; Schro¨dinger picture, Heisenberg picture and interaction picture; Evolu-
tion operator; Time ordered product; Three models; Wick’s theorem. . . . . 70
9 October 21 83
Diagrammatic perturbation theory in Model 3; Vertex in model 1; Connected di-
agrams; Thm: all Wick diagrams = : e connected Wickdiagrams :; Model 1
P
solved; Model 2 begun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
P
10 October 23 93
Model 2 finished; Vacuum energy c.t.; S matrix is 1; Ground state energy; Yukawa
potential; Ground state wave function; Model 3 and Mass renormalization;
Renormalized perturbation theory. . . . . . . . . . . . . . . . . . . . . . . . 93
11 October 28 103
Feynman diagrams inModel 3; Feynman rules in model 3; Acatalog ofall Feynman
diagrams in model 3 to (g2); Scattering amplitude at (g2); Direct and
O O
exchange Yukawa potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
12 October 30 112
“Nucleon”-anti“nucleon”scatteringat (g2); Energyeigenstatepole; Meson-“nucleon”
O
scattering; “Nucleon”-anti“nucleon” annihilation; Assembling the amplitudes
for various processes into one amplitude; Mandelstam variables; Mandelstam-
Kibble plot; crossing symmetry; CPT; Phase space and the S matrix;
Differentialtran. prob.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
unittime
13 November 4 131
Applications of Differentialtran. prob.; Decay; Cross sections, flux; Final state phase
unittime
space simplified for two bodies; dσ; Optical theorem; Final state phase space
dΩ
for three bodies; Feynman diagrams with external lines off the mass shell;
they could be an internal part of a larger diagram. . . . . . . . . . . . . . . . 131
14 November 6 144
Fourier transform of the new blob; A second interpretation of Feynman diagrams
with lines off the mass shell; they are the coefficients of ρn in 0 S 0 ; A
ρ
h | | i
third interpretation of the blob; they are the Fourier transform of the VEV
of a string of Heisenberg fields; Reformulation of scattering theory; S matrix
elements without the turning on and off function; LSZ formula stated. . . . . 144
15 November 13 163
0. Preface Notes from Sidney Coleman’s Physics 253a 5
LSZ formula proved; A second look at Model 3 and its renormalization; Renormal-
ization conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
16 November 18 171
Perturbative determination of a c.t.; Problems with derivative couplings; Rephras-
ing renormalization conditions in terms of Green’s functions; Lehmann-Kallen
spectral representation for the propagator; Rephrasing renormalization con-
ditions in terms of 1PI functions. . . . . . . . . . . . . . . . . . . . . . . . . 171
17 November 20 182
Perturbative determination of c.t.; Corrections to external lines in the computation
of S matrix elements; One loop correction to meson self energy; Feynman’s
trick for combining 2 denominators; Shift to make denominator SO(3,1) in-
variant; Wick notation to make denominator O(4) invariant; Integral tables
for convergent combinations; Self-energy at one loop studied; Combining lots
of denominators; The shift in the general case to reduce any multi-loop inte-
gral to an integral over Feynman parameters. . . . . . . . . . . . . . . . . . . 182
18 November 25 196
Rephrasing coupling constant renormalization in terms of a 1PI function; Exper-
imental significance of the definition; Renormalization versus the infinities;
Renormalizable Lagrangians; Unstable particles, Decay products. . . . . . . 196
19 December 2 207
Unstable particles, lifetime, method of stationary phase; Where it begins again;
Lorentz transformation laws of fields; Equivalent representations; Reducible
reps; The finite dimensional inequivalent irreducible representations of SO(3);
Unitarity; Complex conjugation; Direct product; Projection operators and
reducibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
20 December 4 226
Parametrizing the Lorentz group; Commutation relations for the generators; de-
composition into two sets obeying SO(3) commutation relations; The cata-
log; Complex conjugation properties; Tensor product properties; Restriction
to SO(3); The vector; Rank 2 tensors; Spinors. . . . . . . . . . . . . . . . . . 226
21 December 9 240
Lagrangian made of two component spinors; Solution of Weyl equations of motion;
Weyl particles; Dirac Lagrangian; Four-component spinors; Weyl basis, Dirac
basis; Plane wave solutions of the Dirac equation. . . . . . . . . . . . . . . . 240
0. Preface Notes from Sidney Coleman’s Physics 253a 6
22 December 11 249
Plane wave solutions of the Dirac equation; Pauli’s theorem; Dirac adjoint; Pauli-
Feynman notation; Parity; Bilinears; Orthogonality; Completeness; Summary. 249
23 December 16 267
Canonical quantization of the Dirac Lagrangian. . . . . . . . . . . . . . . . . . . . 267
24 December 18 273
Perturbation theory for spinors; Time ordered product; Wick’s theorem; Calcu-
lation of the contraction (propagator); Wick diagrams; Feynman diagrams;
Matrix multiplication; Spin averages and spin sums. . . . . . . . . . . . . . . 273
25 January 6 289
Parity for spinors; Fermion and antifermion have opposite intrinsic parity; Charge
conjugation; Majorana basis; Charge conjugation properties of fermion bilin-
ears; Decay of ortho and para positronium; U U = U U ( 1)NF; PT. . . . 289
P C C P
−
26 January 8 301
Effect of PT on states; Proof of CPT theorem in perturbation theory; Renormal-
ization of spinor theories; Propagator. . . . . . . . . . . . . . . . . . . . . . . 301
27 January 13 312
1PI part of propagator; Spectral representation for propagator; ′(p) to (g2) in
6 O
meson-nucleon theory; Coupling constant renormalization; Is renormalization
P
sufficient to eliminate ’s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
∞
28 January 15 322
Regularization; Regulator fields; Dim’l regularization; Minimal subtraction; BPHZ
renormalizability; Renormalization and symmetry; Renormalization of com-
posite operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Class: Why not use Feynman’s lecture notes?
Gell-Mann: Because Feynman uses a different method than we do.
Class: What is Feynman’s method?
Gell-Mann: You write down the problem. Then you look at it and you think.
Then you write down the answer.
1. September 23 Notes from Sidney Coleman’s Physics 253a 7
1 September 23
In NRQM, rotational invariance simplifies scattering problems. Why does the addition of
relativity, the addition of L.I., complicate quantum mechanics?
The addition of relativity is necessary at energies E mc2. At these energies
≥
p+p p+p+π0
→
is possible. At slightly higher energies
p+p p+p+p+p
→
can occur. The exact solution of a high energy scattering problem necessarily involves many
particle processes.
You might think that for a given E, only a finite number, even a small number, of
processes actually contribute, but you already know from NRQM that that isn’t true.
0 δV n 2
H H +δV δE = 0 δV 0 + |h | | i| +
0
→ h | | i E E ···
0 n
n −
X
Intermediate states of all energies contribute, suppressed by energy denominators.
For calculations of high accuracy effects at low energy, relativistic effects of order (v/c)2
can be included. Intermediate states with extra particles will contribute corrections of order
Typical
energies
E ← inproblem mv2 = v 2 . As a general conclusion: the corrections of relativistic
mc2 Typical ∼ mc2 c
← energy
denominator (cid:0) (cid:1)
kinematics and the corrections from multiparticle intermediate states are comparable; the
addition of relativity forces you to consider many-body problems. We can’t even solve the
zero-body problem. (It is a phenomenal fluke that relativistic kinematic corrections for
the Hydrogen atom work. If the Dirac equation is used, without considering multi-particle
intermediate states, corrections of v can be obtained. This is a fluke caused by some
O c
unusually low electrodynamic matrix elements.)
(cid:0) (cid:1)
We will see that you cannot have a consistent relativistic picture without pair production.
Units
~ = c = 1 [m] = [E] = [T 1] = [L 1]
− −
Sometimes 1 = 1 = 2π
Because we’re doing −
and 1 = 1=“one-bar”
relativistic (c) 2π
quantum mechanics (~)
(1 fermi)−1 197 MeV
≈
We say things like the inverse Compton wavelength of the proton is “1 GeV”.
1. September 23 Notes from Sidney Coleman’s Physics 253a 8
Lorentz Invariance
Every Lorentz transformation is the product of an element of the connected Lorentz group,
SO(3,1), and 1, P (reflects all three space components), T (time reversal), or PT. By
Lorentz invariance we mean SO(3,1).
Metric convention: + .
−−−
Theory of a single free spinless particle of mass µ
The components of momentum form a complete set of commuting variables.
Momentum operator P~ ~k =~k ~k
→ | i | i
Stateofaspinless
particleiscompletely
specifiedbyitsmomentum.
|{z}
Normalization ~k ~k = δ(3)(~k ~k )
′ ′
h | i −
The statement that this is a complete set of states, that there are no others, is
1 = d3k ~k ~k
| ih |
Z
ψ = d3kψ(~k)~k ψ(~k) ~k ψ
| i | i ≡ h | i
Z
(If we were doing NRQM, we’d finish describing the theory by giving the Hamiltonian,
and thus the time evolution: H ~k = |~k|2 ~k .)
| i 2µ | i
We take H ~k = ~k 2 +µ2 ~k ω ~k
| i | | | i ≡ ~k| i
That’s it, the theqory of a single free spinless particle, made relativistic.
How do we know this theory is L.I.? Just because it contains one relativistic formula, it
is not necessarily relativistic. The theory is not manifestly L.I..
The theory is manifestly rotationally and translationally invariant. Let’s be more precise
about this.
Translational Invariance
Given a four-vector, a, specifying a translation (active), there should be a linear operator,
U(a), satisfying:
U(a)U(a) = 1, to preserve probability amplitudes (1.1)
†
U(0) = 1 (1.2)
U(a)U(b) = U(a+b) (1.3)
1. September 23 Notes from Sidney Coleman’s Physics 253a 9
The U satisfying these is U(a) = eiP a where P = (H,P~).
·
(This lecture is in pedagogical, not logical order. The logical order would be to state:
1. That we want to set up a translationally invariant theory of a spinless particle. The
theory would contain unitary translation operators U(~a).
2. Define Pi = i∂U , (by (1.3) [P ,P ] = 0, by (1.1) P~ = P~ ).
− ∂ai ~a=0 i j †
3. Declare Pi to be a(cid:12)complete set and classify the states by momentum.
(cid:12)
4. Define H = P~2 +µ2, thus giving the time evolution.)
q
More translational invariance
U(a) 0 = a U(a) = eiP a
·
| i | i
where 0 here means state centered at zero and a means state centered at a.
| i | i
O(x+a) = U(a)O(x)U(a)
†
a O(x+a) a = 0 O(x) 0
h | | i h | | i
Non-relativistic reduction
U(~a) = e iP~ ~a
− ·
e iP~ ~a ~q = ~q +~a
− ·
| i | i
Something hard to digest, but correct:
~qe iP~ ~a ~q = (~q+~a) ~q +~a
− ·
| i | i
eiP~ ~a~qe iP~ ~a ~q = (~q+~a) ~q
· − ·
b | i | i
eiP~ ~a~qe iP~ ~a = ~q +~a
· − ·
⇒ b
Looks like the opposite of
b b
e iP~ ~aO(~x)eiP~ ~a = O(~x+~a)
− · ·
The ~q operator is not an operator localized at ~q. No reason for these (last two equations)
to look alike.
b
1. September 23 Notes from Sidney Coleman’s Physics 253a 10
Rotational Invariance
Given an R SO(3), there should be a U(R) satisfying
∈
U(R)U(R) = 1 (1.4)
†
U(1) = 1 (1.5)
U(R )U(R ) = U(R R ) (1.6)
1 2 1 2
Furthermore denote ψ = U(R) ψ
′
| i | i
ψ P~ ψ = R ψ P~ ψ
′ ′
h | | i h | | i
for any ψ i.e. U(R) P~U(R) = RP~ (1.7)
†
| i
and U(R) HU(R) = H (1.8)
†
A U(R) satisfying all these properties is given by
~ ~
U(R) k = Rk
| i | i
That (1.5) and (1.6) are satisfied is trivial.
Proof that (1.4) is satisfied
U(R)U(R) = U(R) d3k ~k ~k U(R)
† †
| ih |
Z
= d3k R~k R~k k = Rk,d3k = d3k
′ ′
| ih |
Z
= d3k k~ k~ = 1
′ ′ ′
| ih |
Z
Proof that (1.7) is satisfied
U(R) P~U(R) = U(R) 1P~(U(R) 1) by (1.5) and (1.6)
† − − †
by(1.4) ↓
= U(R 1)P~U(R 1)
− − †
= U(R 1)P~ d3k ~k ~k U(R 1)
− − †
| ih |
Z
= U(R 1) d3k~k ~k ~k U(R 1)
− − †
| ih |
Z
= d3k~k R 1~k R 1~k k = Rk ,d3k = d3k
− − ′ ′
| ih |
Z
= d3k R~k ~k ~k
′ ′ ′ ′
| ih |
Z
= RP