Quantum Field Theory in a Nutshell, Second Edition (Instructor Solution Manual, Solutions)

326 Pages·2010·2.576 MB·English

by  A. Zee

Solutions Manual for A. Zee,
Quantum Field Theory in a Nutshell,
nd
2 Ed.
Yoni BenTov
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright (cid:13)c 2012 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540.
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxford-
shire OX20 1TW.
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All Rights Reserved
ISBN (pbk.) 978-0-691-15040-6
The publisher would like to acknowledge the author of this volume for providing the camera-
ready copy from which this book was printed.
Printed in the United States of America.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
I Motivation and Foundation 1
I.2 Path Integral Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I.3 From Mattress to Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
I.4 From Field to Particle to Force . . . . . . . . . . . . . . . . . . . . . . . . . 11
I.5 Coulomb and Newton: Repulsion and Attraction . . . . . . . . . . . . . . . . 14
I.6 Inverse Square Law and the Floating 3-Brane . . . . . . . . . . . . . . . . . 16
I.7 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
I.8 Quantizing Canonically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
I.9 Disturbing the Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
I.10 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
I.11 Field Theory in Curved Spacetime . . . . . . . . . . . . . . . . . . . . . . . 37
II Dirac and the Spinor 39
II.1 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
II.2 Quantizing the Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
II.3 Lorentz Group and Weyl Spinors . . . . . . . . . . . . . . . . . . . . . . . . 42
II.4 Spin-Statistics Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
II.5 Vacuum Energy, Grassmann Integrals, and Feynman Diagrams for Fermions 56
II.6 Scattering and Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 57
II.7 Diagrammatic Proof of Gauge Invariance . . . . . . . . . . . . . . . . . . . . 68
II.8 Photon-Electron Scattering and Crossing . . . . . . . . . . . . . . . . . . . . 71
III Renormalization and Gauge Invariance 74
III.1 Cutting Off Our Ignorance . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
III.3 Counterterms and Physical Perturbation Theory . . . . . . . . . . . . . . . 75
III.5 Field Theory without Relativity . . . . . . . . . . . . . . . . . . . . . . . . 78
III.6 The Magnetic Moment of the Electron . . . . . . . . . . . . . . . . . . . . . 80
III.7 Polarizing the Vacuum and Renormalizing the Charge . . . . . . . . . . . . 81
III.8 Becoming Imaginary and Conserving Probability . . . . . . . . . . . . . . . 90
IV Symmetry and Symmetry Breaking 100
IV.1 Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
IV.3 Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
IV.4 Magnetic Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
IV.5 Nonabelian Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
IV.6 Anderson-Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
IV.7 Chiral Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
V Field Theory and Collective Phenomena 150
V.1 Superfluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
V.2 Euclid, Boltzmann, Hawking, and Field Theory at Finite Temperature . . . 151
V.3 Landau-Ginzburg Theory of Critical Phenomena . . . . . . . . . . . . . . . . 155
V.4 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
V.6 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
V.7 Vortices, Monopoles, and Instantons. . . . . . . . . . . . . . . . . . . . . . . 158
VI Field Theory and Condensed Matter 169
VI.1 Fractional Statistics, Chern-Simons Term, and Topological Field Theory . . 169
VI.2 Quantum Hall Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
VI.4 The σ Models as Effective Field Theories . . . . . . . . . . . . . . . . . . . 185
VI.5 Ferromagnets and Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . 187
VI.6 Surface Growth and Field Theory . . . . . . . . . . . . . . . . . . . . . . . 194
VI.7 Disorder: Replicas and Grassmannian Symmetry . . . . . . . . . . . . . . . 194
VI.8 Renormalization Group Flow as a Natural Concept in High Energy and Con-
densed Matter Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
VII Grand Unification 206
VII.1 Quantizing Yang-Mills Theory and Lattice Gauge Theory . . . . . . . . . . 206
VII.2 Electroweak Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
VII.3 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
VII.4 Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
VII.5 Grand Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
VII.6 Protons Are Not Forever . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
VII.7 SO(10) Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
VIII Gravity and Beyond 271
VIII.1 Gravity as a Field Theory and the Kaluza-Klein Picture . . . . . . . . . . 271
VIII.3 Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
VIII.4 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
IXPart N 302
IX.1 N.2 Gluon Scattering in Pure Yang-Mills Theory . . . . . . . . . . . . . . . . 302
IX.2 N.3 Subterranean Connections in Gauge Theories . . . . . . . . . . . . . . . 311
X Appendix E: Dotted and Undotted Indices 318
Preface
These are the solutions to the problems in Quantum Field Theory in a Nutshell, 2nd Ed.,
that are not already solved in the back of the book. For most of the problems I provide a
detailed solution, while for others I sketch the solution and provide a reference to the litera-
ture for further details. Some problems are intentionally open-ended, serving as more of an
introduction to the literature rather than as a homework assignment. The goal of Zee’s text
is not only to teach quantum field theory but also to facilitate the transition from student to
researcher.
I thank my colleagues and teachers for helpful discussions in preparing these solutions. In
particular, I have benefitted from talking to Jen Cano, Gavin Hartnett, Kurt Hinterbichler,
Josh Ilany, Yonah Lemonik, Eugeniu Plamadeala, Yinbo Shi, Joe Swearngin, Benson Way
and Chiu-Tien Yu. I am grateful to Joshua Feinberg for providing some of the solutions. I
am also indebted to the faculty at the University of California at Santa Barbara for their pa-
tience in answering my questions. In particular, I thank David Berenstein, Andreas Ludwig,
Ben Monreal, Joe Polchinski, Mark Srednicki, and my thesis adviser, Tony Zee.
Finally, I thank my editor Ingrid Gnerlich at Princeton University Press for providing the
opportunity to undertake this project. I also thank my friends and family for their support
during the completion of this work.
Yoni BenTov
Apr. 9, 2012
v

I Motivation and Foundation
I.2 Path Integral Formulation
1. Verify (5)
(cid:90)
(cid:104)qF|e−iHT|qI(cid:105) = Dqei(cid:82)0Tdt[12mq˙2−V(q)] (5)
Solution:
Start with the Hamiltonian Hˆ = 1 pˆ2 + V(qˆ). Let T = N(cid:15), with N → ∞,(cid:15) → 0,T
2m
fixed. In this way, split up the time evolution operator into N pieces:
e−iHˆT = e−iHˆ(cid:15)e−iHˆ(cid:15)...e−iHˆ(cid:15)
(cid:124) (cid:123)(cid:122) (cid:125)
N copies
Use this decomposition in the transition amplitude (cid:104)q |e−iHˆT|q (cid:105), and insert one copy of the
F 0
identity matrix between each pair of e−iHˆ(cid:15)s:
(cid:104)q |e−iHT|q (cid:105) = (cid:104)q |e−iHˆ(cid:15)e−iHˆ(cid:15)...e−iHˆ(cid:15)|q (cid:105)
F 0 F 0
= (cid:104)q |e−iHˆ(cid:15)1(N−1)e−iHˆ(cid:15)1(N−2)e−iHˆ(cid:15)...e−iHˆ(cid:15)1(1)e−iHˆ(cid:15)|q (cid:105)
F 0
The superscript is just a label to keep track of the fact that we have inserted N −1 identity
matrices. It is convenient to resolve each identity matrix in a complete set of position
eigenstates:
(cid:90) ∞
1(i) = dq |q (cid:105)(cid:104)q |
i i i
−∞
To see why, consider the matrix element of e−iHˆ(cid:15) = 1 − i(cid:0) 1 pˆ2 +V(qˆ)(cid:1)(cid:15) + O((cid:15)2) between
2m
two position eigenstates, |q (cid:105) and |q (cid:105):
i j
(cid:18) (cid:19)
1
(cid:104)q |e−iHˆ(cid:15)|q (cid:105) = (cid:104)q |1−i pˆ2 +V(qˆ) (cid:15)+O((cid:15)2)|q (cid:105)
i j i j
2m
(cid:18) (cid:19)
1
= (cid:104)q |1−i pˆ2 +V(q ) (cid:15)+O((cid:15)2)|q (cid:105)
i j j
2m
= (cid:104)qi|e−i(21mpˆ2+V(qj))(cid:15)|qj(cid:105)
= e−iV(qj)(cid:15)(cid:104)qi|e−i21mpˆ2(cid:15)|qj(cid:105)
The re-exponentiation is justified by the Baker-Campbell-Hausdorff formula:
eAeB = eA+B+1[A,B]+(highercommutators)
2
Set A = −i(cid:15) pˆ2 and B = −i(cid:15)V(xˆ) in the above to get:
2m
e−i(cid:15) pˆ2 e−i(cid:15)V(xˆ) = e−i(cid:15) pˆ2−i(cid:15)V(xˆ)+O((cid:15)2)
2m 2m
1
So to O((cid:15)2) → 0 we can perform the above manipulations.
Now insert an identity matrix to the right of e−i 1 pˆ2(cid:15), but this time resolve it in a com-
2m
plete set of momentum eigenstates:
(cid:90) ∞ dp
1 = |p(cid:105)(cid:104)p|
2π
−∞
This gives, for the above matrix element, the following:
(cid:104)qi|e−iHˆ(cid:15)|qj(cid:105) = e−iV(qj)(cid:15)(cid:104)qi|e−i21mpˆ2(cid:15)1|qj(cid:105)
(cid:18)(cid:90) ∞ dp (cid:19)
= e−iV(qj)(cid:15)(cid:104)qi|e−i21mpˆ2(cid:15) |p(cid:105)(cid:104)p| |qj(cid:105)
2π
−∞
(cid:90) ∞ dp
= e−iV(qj)(cid:15) (cid:104)qi|e−i21mpˆ2(cid:15)|p(cid:105)(cid:104)p|qj(cid:105)
2π
−∞
(cid:90) ∞ dp
= e−iV(qj)(cid:15) e−i21mp2(cid:15)(cid:104)qi|p(cid:105)(cid:104)p|qj(cid:105)
2π
−∞ (cid:124)(cid:123)(cid:122)(cid:125)(cid:124) (cid:123)(cid:122) (cid:125)
eiqip e−iqjp
(cid:90) ∞ dp
= e−iV(qj)(cid:15) e−i21mp2(cid:15)+ip(qi−qj)
2π
−∞
Now do all of this for the original transition amplitude. This will require N −1 resolutions
of the identity in position eigenstates, as already indicated, and it will require N resolutions
of the identity in momentum eigenstates.
(cid:104)q |e−iHˆT|q (cid:105) =
F 0
(cid:104)q |e−iHˆ(cid:15)1(N−1)e−iHˆ(cid:15)1(N−2)...1(1)e−iHˆ(cid:15)|q (cid:105) =
F 0
(cid:20)(cid:90) ∞ (cid:21) (cid:20)(cid:90) ∞ (cid:21) (cid:20)(cid:90) ∞ (cid:21)
(cid:104)q |e−iHˆ(cid:15) dq |q (cid:105)(cid:104)q | e−iHˆ(cid:15) dq |q (cid:105)(cid:104)q | ... dq |q (cid:105)(cid:104)q | e−iHˆ(cid:15)|q (cid:105)
F N−1 N−1 N−1 N−2 N−2 N−2 1 1 1 0
−∞ −∞ −∞
(cid:90) ∞
= dq ...dq (cid:104)q |e−iHˆ(cid:15)|q (cid:105)(cid:104)q |e−iHˆ(cid:15)|q (cid:105)(cid:104)q |...|q (cid:105)(cid:104)q |e−iHˆ(cid:15)|q (cid:105)(cid:104)q |e−iHˆ(cid:15)|q (cid:105)
N−1 1 F N−1 N−1 N−2 N−2 2 2 1 1 0
−∞
2
• (cid:104)qF|e−iHˆ(cid:15)|qN−1(cid:105) = e−iV(qN−1)(cid:15)(cid:90) ∞ dpN−1 e−ip2N2m−1(cid:15)+ipN−1(qF−qN−1)
2π
−∞
• (cid:104)qN−1|e−iHˆ(cid:15)|qN−2(cid:105) = e−iV(qN−2)(cid:15)(cid:90) ∞ dpN−2 e−ip2N2m−2(cid:15)+ipN−2(qN−1−qN−2)
2π
−∞
...
• (cid:104)q2|e−iHˆ(cid:15)|q1(cid:105) = e−iV(q1)(cid:15)(cid:90) ∞ dp1 e−i2pm21(cid:15)+ip1(q2−q1)
2π
−∞
• (cid:104)q1|e−iHˆ(cid:15)|q0(cid:105) = e−iV(q0)(cid:15)(cid:90) ∞ dp0 e−i2pm20(cid:15)+ip0(q1−q0)
2π
−∞
Note that we need N resolutions into momentum eigenstates instead of just N − 1: after
inserting N − 1 resolutions of the identity into position eigenstates, we need one set of
momentum eigenstates for each position “ket,” |q (cid:105),...,|q (cid:105), including the initial position
0 N−1
q over which we do not integrate. In any case, the amplitude is now:
0
(cid:104)q |e−iHˆT|q (cid:105) =
F 0
(cid:90) ∞ dq ...dq dp0...dpN−1 e−i(cid:20)p2N2m−1(cid:15)−pN−1(qF−qN−1)(cid:21)...e−i(cid:20)2pm20(cid:15)−p0(q1−q0)(cid:21)e−iV(qN−1)...e−iV(q0)
1 N−1
2π 2π
−∞
Completingthesquareforeachterminbracketswillyieldone-dimensionalGaussianintegrals:
p2 (cid:15) (cid:20) 2m (cid:21)
i (cid:15)−p (q −q ) = p2 − p (q −q )
2m i i+1 i 2m i (cid:15) i i+1 i
(cid:15) (cid:20)(cid:16) m (cid:17)2 m2 (cid:21)
= p − (q −q ) − (q −q )2
2m i (cid:15) i+1 i (cid:15)2 i+1 i
(cid:15) (cid:20)(cid:16) m (cid:17)2(cid:21) m (cid:18)q −q (cid:19)2
i+1 i
= p − (q −q ) −(cid:15)
i i+1 i
2m (cid:15) 2 (cid:15)
Therefore, each of the momentum integrals gives:
(cid:90) ∞ dpi e−i(cid:20)2pm2i (cid:15)−pi(qi+1−qi)(cid:21) = 1 e+im2(cid:16)qi+1(cid:15)−qi(cid:17)2(cid:90) ∞ dpie−i2(cid:15)m(pi−m(cid:15)(qi+1−qi))2
2π 2π
−∞ −∞
1 +im(cid:16)qi+1−qi(cid:17)2(cid:114) π
= e 2 (cid:15)
2π (i(cid:15)/(2m))
(cid:114)
m +im(cid:16)qi+1−qi(cid:17)2
= e 2 (cid:15)
2πi(cid:15)
Again, we have N copies of this integral, so the transition amplitude is:
(cid:104)qF|e−iHˆT|q0(cid:105) = (cid:90) ∞ dq1...dqN−1(cid:18)(cid:114) m (cid:19)N e+im2(cid:16)qF−q(cid:15)N−1(cid:17)2...e+im2(q1−(cid:15)q0)2e−iV(qN−1)...e−iV(q0)
2πi(cid:15)
−∞
= (cid:90) dN−1q(cid:18)(cid:114) m (cid:19)N ei(cid:80)Nj=−01(cid:15)(cid:20)m2(cid:16)qj+1(cid:15)−qj(cid:17)2−V(qj)(cid:21)
2πi(cid:15)
3
Remember that the limits N → ∞ and (cid:15) → 0 with T fixed are supposed to be implied
throughout. Withthisinmind, definetheintegraloverpathsandgotoacontinuumnotation:
(cid:90) (cid:90) (cid:18)(cid:114) (cid:19)N
m
Dq ≡ lim dN−1q
q(0) = q0 N → ∞ 2πi(cid:15)
q(T) = q
F (cid:15) → 0
T fixed
(cid:34) (cid:35) (cid:34) (cid:35)
N(cid:88)−1 m (cid:18)q −q (cid:19)2 (cid:90) T m (cid:18)dq(cid:19)2
j+1 j
(cid:15) −V(q ) → dt −V(q)
j
2 (cid:15) 2 dt
0
j=0
In the above, q ≡ q . This yields the desired result:
N F
(cid:90)
(cid:104)qF|e−iHT|q0(cid:105) = Dqei(cid:82)0Tdt(12mq˙2−V(q)) with q(t = 0) = q0 , q(t = T) = qF
2. Derive (24)
(cid:88)
(cid:104)x x ...x x (cid:105) = (A−1) ...(A−1) (24)
i j k (cid:96) ab cd
Wick
Solution:
Define the following N-dimensional integral:
(cid:90)
Z[J(cid:126)] ≡ dNxe−1(cid:126)xTM(cid:126)x+J(cid:126)·(cid:126)x = Ce+1J(cid:126)T(M−1)J(cid:126)
2 2
(cid:126)
In the above, (cid:126)x and J are N-dimensional vectors, M is an N × N symmetric matrix, the
superscript T denotes the transpose, and C is a constant that we set equal to 1. If you don’t
like that, then assume that all expectation values in what follows are implicitly divided by
the constant C. It is just a statistical normalization factor, the equivalent of ensuring that
the sum of all probabilities equals 1 rather than some other number.
With this in mind, the expectation value of x is:
1
(cid:90)
(cid:104)x (cid:105) = dNxe−1(cid:126)xTM(cid:126)xx
1 2 1
(cid:90)
∂
= dNxe−1(cid:126)xTM(cid:126)x+J(cid:126)·(cid:126)x|
∂J 2 J(cid:126)=0
1
∂
(cid:126)
= Z[J]|
∂J J(cid:126)=0
1
4