Zee A. Quantum Field Theory in a Nutshell
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552 | Solutions to Selected Exercises
III.1.3 Under the indicated change, log → log e
ε
= log + ε, and so δM =−iδλ + iCλ
2
3(2ε) + O(λ
3
).
Thus, δM = 0 implies δλ = 6Cλ
2
ε + O(λ
3
) = 6Cλ
2
δ log + O(λ
3
) giving the stated result for
(dλ/d).
III.2.1 For
d
d
x(∂ϕ)
2
to be dimensionless, we need [ϕ] = (d −2)/2. Thus [ϕ
n
] = n(d − 2)/2 and so in order for
d
d
xλ
n
ϕ
n
to be dimensionless, we must have [λ
n
] = n(2 − d)/2 +d.
III.3.3 When we set m = 0, the integrand is manifestly a linear combination of γ matrices. The integral cannot
produce a term independent of the γ matrices, which is what B is. For electrodynamics, the integral is
changed to
(ie)
2
i
2
d
4
k
(2π)
4
1
k
2
[(1 − ξ)
k
μ
k
ν
k
2
− g
μν
]γ
μ
p + k + m
(p + k)
2
− m
2
γ
ν
≡ A(p
2
) p + B(p
2
)
When m = 0, the integrand is a linear combination of the product of three γ matrices, which can only
reduce to one γ matrix, not to none. Incidentally, an alternative way of seeing the results stated here is
to recall from chapter II.1 that with m = 0 the Lagrangian is invariant under the chiral transformation
ψ →e
iθγ
5
ψ .
III.3.4 This essentially follows from D = 4 − B
E
−
3
2
F
E
for B
E
= 0 and F
E
= 2. Then D = 1 but the linear
divergence is reduced to logarithmic divergence by the symmetry argument given in the text.
III.5.2 Basically, you have already done this problem in exercises II.1.3 and II.1.4. You merely have to replace
E and p by ∂/∂t and
∇ (see also chapter III.6).
III.5.3 In nonrelativistic quantum mechanics, the scattering amplitude is given in the Born approximation by
i times the Fourier transform of the potential: i
d
3
xe
i
k
.
x
U(x). The scattering amplitude owing to the
exchange of a scalar meson of mass m is just i/(k
2
− m
2
) −i/(
k
2
+ m
2
). Thus, we just repeat the
calculation in chapter I.4, obtaining
U(x) =−
d
3
k
(2π)
3
e
i
k
.
x
k
2
+ m
2
=−
1
4πr
e
−mr
III.6.1 ¯u(p
)(p
γ
μ
+ γ
μ
p)u(p) = 2m ¯u(p
)γ
μ
u(p) by the equation of motion, but using γ
μ
γ
ν
=
1
2
{γ
μ
, γ
ν
}+
1
2
[γ
μ
, γ
ν
] = η
μν
− iσ
μν
we can also write (p
γ
μ
+ γ
μ
p) = (p
+ p)
μ
+ iσ
μν
(p
− p)
ν
. We thus obtain
the Gordon decomposition.
III.6.2 We compute
q
μ
¯u(p
)[γ
μ
F
1
(q
2
) +
iσ
μν
q
ν
2m
F
2
(q
2
)]u(p) =¯u(p
) qu(p)F
1
(q
2
)
=¯u(p
)(p
− p)u(p)F
1
(q
2
)
=¯u(p
)(m − m)u(p)F
1
(q
2
) = 0
where the first and third equality follows from the antisymmetry of σ and the equation of motion,
respectively.
III.7.1 Proceeding as in the text but living in d−dimensional spacetime, we obtain i
μν
(q) =−i
d
d
l
(2π)
d
N
μν
D
where
1
D
=
1
0
dα
1
D
with D =(l
2
+α(1 −α)q
2
−m
2
+iε)
2
as before but with N
μν
now effectively equal
to −d((1 −
2
d
)g
μν
l
2
+ α(1 − α)(2q
μ
q
ν
− g
μν
q
2
) − m
2
g
μν
). Rotating to Euclidean space we see that we
have to do the integrals (with c
2
≡m
2
−α(1 − α)q
2
)
d
d
E
l
(2π)
d
1
(l
2
+c
2
)
2
and
d
d
E
l
(2π)
d
l
2
(l
2
+c
2
)
2
=
d
d
E
l
(2π)
d
1
(l
2
+c
2
)
−
c
2
d
d
E
l
(2π)
d
1
(l
2
+c
2
)
2
. I did the first of these integrals for you in appendix II in chapter III.1. Generalizing
slightly we have
∞
0
dll
d−1
1
(l
2
+c
2
)
a
=
1
2
c
d−2a
1
0
dx(1 − x)
d
2
−1
x
a−1−
d
2
. I will let you carry on from here.
Solutions to Selected Exercises | 553
Part IV
IV.1.1 Write ϕ = (ϕ
1
, ϕ
2
,...,v + ϕ
N
). We compute
1
2
μ
2
ϕ
2
−(λ/4)( ϕ
2
)
2
up to O(ϕ
3
) and find (upon dropping
the
)
1
2
μ
2
(v
2
+ 2vϕ
N
+ϕ
2
) −
λ
4
(v
4
+ 4v
2
ϕ
2
N
+ 4v
3
ϕ
N
+ 2v
2
ϕ
2
)
The condition of no linear term in ϕ
N
fixes v
2
= μ
2
/λ and so the coefficient of ϕ
2
is equal to
1
2
μ
2
−
λ/4(2v
2
) = 0. The (N − 1) fields ϕ
1
, ϕ
2
,...,ϕ
N−1
are massless.
IV.3.1 We have
∞
0
dk log[(k
2
+a
2
)/k
2
] = πa and so V
eff
(ϕ) = V(ϕ)+
√
V
(ϕ)/2 +O(
2
).ForL =
1
2
(∂ϕ)
2
−
1
2
ω
2
ϕ
2
, the quantum oscillator with ϕ identified as position, we have V
eff
(0) =
1
2
ω.
IV.3.3 We have
m(ϕ) = fϕ in V
F
(ϕ) = 2i
d
2
p
(2π)
2
log
p
2
−m(ϕ)
2
p
2
which after Wick rotation becomes
−2
d
2
p
E
(2π)
2
log
p
2
E
+ m(ϕ)
2
p
2
E
=−
1
2π
∞
0
dx log
x + m(ϕ)
2
x
After cutting off the integral at
2
and adding a counterterm Bϕ
2
we obtain
V
F
=
1
2π
(f ϕ)
2
log
ϕ
2
M
2
IV.3.4 V
eff
=i
∞
n=1
d
4
k/(2π)
4
(1/2n)[V
(ϕ)/k
2
]
n
.ForV
(ϕ) =
1
2
λϕ
2
the corresponding Feynman diagrams
consist of a circle with nV’s attached to the circumference, where the 2n is the infamous symmetry
factor that I tried to avoid talking about in chapter I.7.
IV.4.1 With H an arbitrary p-form,
ddH =
1
(p + 1)!
1
p!
∂
λ
∂
ν
H
μ
1
μ
2
...μ
p
dx
λ
dx
ν
dx
μ
1
dx
μ
2
...dx
μ
p
=
1
2
1
(p + 1)!
1
p!
[∂
λ
, ∂
ν
]H
μ
1
μ
2
...μ
p
dx
λ
dx
ν
dx
μ
1
dx
μ
2
...dx
μ
p
= 0
IV.5.1 If you have done all the exercises thus far (see exercise I.10.3), you have already made the acquaintance of
the I = 2 scalar field transforming as ϕ
ab
→ R
ac
R
bd
ϕ
cd
= R
ac
ϕ
cd
(R
T
)
db
= (RϕR
T
)
ab
, which thus can
be written as a traceless 3 by 3 symmetric matrix ϕ →RϕR
T
. Now you merely have to write out the
covariant derivative D
μ
ϕ (see IV.5.20) explicitly. [Hint: The action of the generators on ϕ is similar to
what is shown in (B.20).]
IV.5.2 dF = d(dA + A
2
) = dAA − AdA and [A, F ] = AdA − dAA and so dF + [A, F ] = 0. Explicitly with
indices, this reads ε
μνλσ
(∂
ν
F
λσ
+ [A
ν
, F
λσ
]) = 0. In the abelian case, we have, for μ = 0, ε
ij k
∂
i
F
jk
=
∇
.
B = 0 (recall chapter IV.4!), and for μ = i , ε
ij k
(−∂
0
F
jk
+ ∂
j
F
0k
− ∂
j
F
k0
) =−∂
0
B
i
+ (
∇×
E)
i
= 0.
IV.5.4 From the general arguments mentioned in the problem we know that tr F
2
must be the “d of some-
thing.” Now d tr AdA = tr dAdA and d tr
2
3
A
3
=
2
3
tr(dAA
2
− AdAA + A
2
dA) = 2trdAA
2
but on the
other hand tr F
2
= tr(dA + A
2
)(dA + A
2
) = tr(dAdA + 2dAA
2
) since tr A
4
= tr A
3
A =−tr AA
3
=
− tr A
4
= 0. In electromagnetism, tr A
3
= 0, and d tr AdA when written out in elementary notation
is just ∂
μ
(ε
μνλσ
A
ν
∂
λ
A
σ
) =
1
4
ε
μνλσ
F
μν
F
λσ
.
554 | Solutions to Selected Exercises
IV.5.6 We simply plug in the general expression in the text and obtain
L =−
1
4g
2
F
a
μν
F
aμν
+¯q(iγ
μ
D
μ
− m)q
with the covariant derivative D
μ
= ∂
μ
− iA
μ
= ∂
μ
− iA
a
μ
T
a
, where T
a
(a = 1,...,8) are traceless her-
mitean 3 by 3 matrices. Explicitly, (A
μ
q)
α
= A
a
μ
(T
a
)
α
β
q
β
, with α, β = 1, 2, 3 (see chapter VII.3).
IV.6.3 Observe
A
a
μ
τ
a
=
A
3
μ
A
1−i2
μ
A
1+i2
μ
−A
3
μ
with the obvious notation A
1±i2
μ
≡ A
1
μ
± iA
2
μ
. Let ϕ=
0
v
so that
D
μ
ϕ = ∂
μ
ϕ − i(gA
a
μ
τ
a
2
+ g
B
μ
1
2
)ϕ →−
i
2
v
gA
1−i2
μ
−gA
3
μ
+ g
B
μ
Thus, |D
μ
ϕ|
2
contains v
2
[g
2
A
1+i2
μ
A
1−i2
μ
+ (−gA
3
μ
+ g
B
μ
)
2
]. The combinations A
1+i2
μ
, A
1−i2
μ
, and
(−gA
3
μ
+ g
B
μ
) acquire mass while (g
A
3
μ
+ gB
μ
) remain massless.
IV.7.4 We have
μν
(k
1
, k
2
) = i
d
4
p
(2π)
4
N
μν
D
+{μ, k
1
↔ ν, k
2
}
where
N
μν
≡ tr γ
5
(p − q + M)γ
ν
(p − k
1
+ M)γ
μ
(p + M)
Only the term linear in M in N
μν
does not vanish, giving N
μν
=4iMε
μνσ τ
k
1σ
k
2τ
. Since we are interested
only in terms of O(k
1
k
2
) we can set D → (p
2
− M
2
)
3
so that
μν
(k
1
, k
2
) =−8Mε
μνσ τ
k
1σ
k
2τ
d
4
p
(2π)
4
1
(p
2
− M
2
)
3
=
i
4π
2
M
ε
μνσ τ
k
1σ
k
2τ
with a dependence on M as stated in the problem. The effect of the regulator, like some unsavory
acquaintance, remains even after we have sent him to infinity.
IV.7.5 We will sketch the solution. The details may be found in the lectures given by S. Adler at the 1970
Brandeis Summer School. The point is to imagine a regularization scheme that preserves the various
relevant symmetries, namely Lorentz invariance, vector current conservation, and Bose statistics. As you
will see, we don’t actually have to specify the regularization. By Lorentz invariance, we have
λμν
(k
1
, k
2
) = ε
λμνσ
k
1σ
A
1
+ ε
λμνσ
k
2σ
A
2
+ ε
λμσ τ
k
1σ
k
2τ
k
ν
1
A
3
+ ε
λμσ τ
k
1σ
k
2τ
k
ν
2
A
4
+ ε
λνσ τ
k
1σ
k
2τ
k
μ
1
A
5
+ ε
λνσ τ
k
1σ
k
2τ
k
μ
2
A
6
+ ε
μνσ τ
k
1σ
k
2τ
k
λ
1
A
7
+ ε
μνσ τ
k
1σ
k
2τ
k
λ
2
A
8
Since the Feynman integral representing
λμν
is superficially linearly divergent, we see that A
3
,...,A
8
are all convergent since we have to pull out three powers of momentum to extract them. In contrast, A
1
and A
2
are logarithmically divergent. But we can relate them to A
3
,...,A
8
by vector current conservation
since 0 = k
1μ
λμν
=ε
λνσ τ
k
1σ
k
2τ
(−A
2
+k
2
1
A
5
+k
1
.
k
2
A
6
) and thus A
2
=k
2
1
A
5
+k
1
.
k
2
A
6
. Similarly for
A
1
. Rationalizing the Feynman integrand and evaluating the trace in the numerator, we can systematically
ignore terms that contribute only to A
1
and A
2
. Furthermore, Bose statistics gives us relations such as
A
3
(k
2
1
, k
2
2
, q
2
) =−A
6
(k
2
2
, k
2
1
, q
2
).
Solutions to Selected Exercises | 555
Part V
V.1.1 We dropped the term h
2
∂
0
θ but kept the term 4g
2
¯ρh
2
. This requires ∂
0
θ g
2
¯ρ, that is, ω g
2
¯ρ, but
since in our solution ω ∼ g
√
¯ρ/mk this requires k g
√
m ¯ρ, which is consistent with what we assumed
about k. Looking at the terms −2
√
¯ρh∂
0
θ − 4g
2
¯ρh
2
in L we see that h ∼ ∂
0
θ/(g
2
√
¯ρ)
√
¯ρ, which is
also consistent.
V.5.1 With γ
5
=σ
3
,
1
2
(I ± γ
5
) clearly projects out the top and bottom component of ψ =
ψ
L
ψ
R
, respectively.
Everything is formally the same as in chapter II.1, but we can also work things out explicitly in the
specific representation given here. Thus,
¯
ψψ = ψ
†
σ
2
ψ = i(ψ
†
R
ψ
L
− ψ
†
L
ψ
R
) and
¯
ψγ
5
ψ = ψ
†
σ
2
σ
3
ψ =
i(ψ
†
R
ψ
L
+ ψ
†
L
ψ
R
). Under the transformation ψ → e
iθγ
5
ψ , ψ
L
→ e
iθ
ψ
L
and ψ
R
→ e
−iθ
ψ
R
, and the
massless Dirac Lagrangian
L = iψ
†
R
(
∂
∂t
+ v
F
∂
∂x
)ψ
R
+ iψ
†
L
(
∂
∂t
− v
F
∂
∂x
)ψ
L
clearly does not change.
V.6.1 This of course just follows from Lorentz invariance. We have
∂
t
ϕ(
x − vt
√
1 − v
2
) =
−v
√
1 − v
2
ϕ
(
x − vt
√
1 − v
2
)
and
∂
x
ϕ(
x − vt
√
1 − v
2
) =
1
√
1 − v
2
ϕ
(
x − vt
√
1 − v
2
)
and thus the equation
(∂
2
t
− ∂
2
x
)ϕ(
x − vt
√
1 − v
2
) + V
[ϕ(
x − vt
√
1 − v
2
)] =0
becomes
ϕ
(
x − vt
√
1 − v
2
) − V
[ϕ(
x − vt
√
1 − v
2
)] =0
Note that this does not depend on the form of V . For any relativistic theory, the soliton moves like a
relativistic particle (obviously!).
V.6.2 The sine-Gordon theory has an infinite number of vacua occurring at ϕ =(2n + 1)π/β . Thus, there
exists a whole spectrum of solitons, such that ϕ(±∞) = (2n
±
+ 1)π/β . The topological current is
J
μ
= (β/2π)ε
μν
∂
ν
ϕ with the corresponding charge Q = (n
+
− n
−
). The Q = 2 soliton decays into two
Q = 1 solitons.
V.7.4 (i/2π)
S
1
gdg
†
=(i/2π)
S
1
e
iνθ
de
−iνθ
=(i/2π)
S
1
(−iνdθ) =(ν/2π)
2π
0
dθ =ν, which indeed counts
the number of times e
iνθ
winds around the circle. What mathematicians call the winding number is
indeed just the magnetic flux of the physicist.
V.7.5 Within a region small enough so that we can treat ϕ
a
= vδ
a3
as constant, using (D
μ
ϕ)
b
= ∂
μ
ϕ
b
+
eε
bcd
A
c
μ
ϕ
d
we have (D
μ
ϕ)
1
= evA
2
μ
and (D
μ
ϕ)
2
=−evA
1
μ
and thus
F
μν
≡
F
a
μν
ϕ
a
|ϕ|
−
(1/e)ε
abc
ϕ
a
(D
μ
ϕ)
b
(D
ν
ϕ)
c
|ϕ|
3
→ F
3
μν
+ e(A
2
μ
A
1
ν
− A
2
ν
A
1
μ
) = ∂
μ
A
3
ν
− ∂
ν
A
3
μ
precisely the electromagnetic field strength since A
3
μ
is the massless component of the Yang-Mills field.
Let us compute B
k
= ε
ij k
F
ij
far from the magnetic monopole. To calculate the magnetic charge we are
556 | Solutions to Selected Exercises
interested only in the term of order 1/r
2
in
B. Since D
μ
ϕ → O(1/r
2
) by construction we can drop the
second term in F
ij
. Thus, we merely have to compute F
a
ij
≡ ∂
i
A
a
j
− ∂
j
A
a
i
+ eε
abc
A
b
i
A
c
j
. Since F
a
ij
will
eventually be contracted with the unit vector ϕ
a
/|ϕ|=x
a
/r, we can effectively drop some of the terms
in F
a
ij
, thus simplifying the computation. We have
∂
i
A
a
j
= ∂
i
(
1
e
ε
aj l
x
l
r
2
)“ = "
1
e
ε
aj i
1
r
2
and
eε
abc
A
b
i
A
c
j
=
(1/e)ε
abc
ε
bim
ε
cj n
x
m
x
n
r
4
=
(1/e)(δ
ci
δ
am
− δ
cm
δ
ai
)ε
cj n
x
m
x
n
r
4
=
1
er
4
ε
ij n
x
a
x
n
so that
F
a
ij
ϕ
a
|ϕ|
=
F
a
ij
x
a
r
=
1
er
3
(−2 + 1)ε
aij
x
a
=−
1
er
3
ε
aij
x
a
and hence B
k
=−(1/er
2
) ˆx
k
. The magnetic charge g =−4π/e.
Our result appears to differ from Dirac’s quantization condition (IV.4.10) by a factor of 2. The
resolution of this apparent paradox is instructive. In fact, we can always introduce into this theory a
field (which could be a Bose or a Fermi field) transforming in the I =
1
2
representation with the
corresponding covariant derivative D
μ
= ∂
μ
− ie(
1
2
τ
a
)A
a
μ
. The field carries electric charge
1
2
e.
Thus, the fundamental unit of electric charge is actually
1
2
e, not e, and our result g =−4π/e =−2π/(e/2)
is actually nothing but the Dirac quanization condition. (The sign is trivial: just a question of which one
we call the monopole and which the antimonopole.)
V.7.7 Plugging in the Ansatz ϕ
a
=(H (r)/er)(x
a
/r) and A
b
i
=[1 −K(r)]ε
bij
(x
j
/er
2
) [so that H(r) −→
r→∞
evr and
K(r) −→
r→∞
0 in accordance with the asymptotic behavior (V.7.5) and (V.7.6)] into M =
d
3
x{
1
4
(
F
ij
)
2
+
1
2
(D
i
ϕ)
2
+V(ϕ)}we get M as a functional of H and K . Minimizing M gives (with H
=dH /dr etc) the
equations r
2
H
= 2HK
2
+ (λ/e
2
)[H
3
− (ev)
2
r
2
H ] and r
2
K
= K(K
2
− 1) + KH
2
. For help, see M. K.
Prasad and C. M. Sommerfeld, Phys. Rev. Lett. 35: 760, 1975.
V.7.8 The BPS solution corresponds to setting λ = 0 in the two equations in exercise V.7.7, rendering the
equations soluble, with the solution H(r)= evr(coth evr) − 1 and K(r) = evr/(sinh evr). Ask yourself
why H(r)and K(r) approach their asymptotic values exponentially. What determines the length scale?
V.7.9 For help, see B. Julia and A. Zee, Phys. Rev. D11: 2227, 1975.
V.7.11 We derived the lower bound for the mass of the magnetic monopole 4πv|g|∼4 π(ev)/e
2
∼ M
W
/α.
V.7.12 Near the identity element g = e
i
θ
.
σ
1 + i
θ
.
σ and thus gdg
†
−id
θ
.
σ . In a small neighborhood of
the identity element the group manifold is locally Euclidean and so
tr(gdg
†
)
3
= i tr(σ
i
σ
j
σ
k
)dθ
i
dθ
j
dθ
k
=−12dθ
1
dθ
2
dθ
3
is manifestly proportional to the volume element on S
3
.Forg = e
i(θ
1
σ
1
+θ
2
σ
2
+mθ
3
σ
3
)
,tr(gdg
†
)
3
=
−12mdθ
1
dθ
2
dθ
3
.
V.7.13
d
4
x(∂
μ
J
μ
5
) =
d
3
xJ
0
5
|
t=+∞
−
d
3
xJ
0
5
|
t=−∞
. Recalling that J
0
5
=ψ
†
R
ψ
R
−ψ
†
L
ψ
L
, we see that the two
spatial integrals just count the number of right moving fermion quanta minus the number of left moving
fermion quanta at t =±∞respectively. So
d
4
x(∂
μ
J
μ
5
) is an integer. On the other hand, in the text we
proved that
tr F
2
is a topological invariant. In other words, with suitable normalization, evidently
1/(4π)
2
,the integral [1/(4π)
2
]
d
4
xε
μνλσ
tr F
μν
F
λσ
is an integer. Thus, the coefficient 1/(4π)
2
cannot
be shifted even a little bit by quantum fluctuations.
Solutions to Selected Exercises | 557
Part VI
VI.4.2 The quartic interaction term (1/2f
2
)( π
.
∂ π)
2
in L gives the amplitude i(1/2f
2
)i
2
δ
ab
δ
cd
(k
1
k
3
+ k
1
k
4
+
k
2
k
3
+k
2
k
4
) +permutations = (i/2f
2
)δ
ab
δ
cd
(k
1
+k
2
)
2
+permutations for the 4-pion interaction vertex
(where for convenience we have labeled all the momenta as going outward so that k
1
+k
2
+k
3
+k
4
=0).
VI.4.3 After writing σ = v +σ
, we find, as in chapter IV.1, that L =−
1
2
(2μ
2
)σ
2
− λvσ
π
2
−
1
4
λ( π
2
)
2
+
...
, where we have displayed only terms relevant for our purposes. The diagrams contributing to
four-pion interaction are of two types, those involving the λ( π
2
)
2
term and those invoking σ
ex-
change. The former gives for the amplitude (−
i
4
λ)2
.
2(δ
ab
δ
cd
+ δ
ac
δ
bd
+ δ
ad
δ
bc
) while the latter gives
(−iλv)
2
{2i/[(k
1
+ k
2
)
2
− m
2
σ
]}δ
ab
δ
cd
. Thus, expanding to first order in momenta squared we find the
coefficient of δ
ab
δ
cd
:
−iλ − 2iλ
2
v
2
−
1
m
2
σ
(1 +
(k
1
+ k
2
)
2
m
2
σ
) =−iλ +
2iλ
2
v
2
2μ
2
1 +
(k
1
+ k
2
)
2
2μ
2
=
iλ
2μ
2
(k
1
+ k
2
)
2
To compare with exercise VI.4.2 we remember that f
2
= v
2
= μ
2
/λ so that the amplitude here is also
equal to (i/2f
2
)δ
ab
δ
cd
(k
1
+ k
2
)
2
+ permutations, as we had anticipated in the text.
VI.4.4 We will track down factors of 2 carefully but not factors of i and −1. Let us go back to the chiral
transformations ψ →[1 + i
θ
.
(τ/2)γ
5
]ψ and
¯
ψ →
¯
ψ[1 + i
θ
.
(τ/2)γ
5
]. Thus, δ(
¯
ψψ) = θ
a
¯
ψiγ
5
τ
a
ψ and
δ(
¯
ψiγ
5
τ
a
ψ) =−θ
a
¯
ψψ. Hence, for L =
¯
ψ{iγ ∂ + g(σ + i τ
.
πγ
5
)}ψ +L(σ , π) to be invariant we must
have δσ =θ
a
π
a
and δπ
a
=−θ
a
σ . Applying Noether’s theorem J
μ
=(δL/δ∂
μ
ϕ)δϕ, we obtain the current
J
a
μ5
=
¯
ψiγ
μ
γ
5
(τ
a
/2)ψ + π
a
∂
μ
σ − σ∂
μ
π
a
written in the text. Comparing the term ¯piγ
μ
γ
5
n contained in
J
1+i2
μ5
≡J
1
μ5
+iJ
2
μ5
with the current J
5μ
defined in chapter IV.2, we see that J
5μ
=−iJ
1+i2
μ5
. The normal-
ized state |π
−
=(1/
√
2)( |π
1
−i |π
2
) so that 0|π
1+i2
|π
−
=2/
√
2. The current J
1+i2
μ5
contains the
term −v∂
μ
π
1+i2
and thus f =
√
2v. Next, we have to work out the pion-nucleon coupling g
πNN
as de-
fined in chapter IV.2. Here L contains g
¯
ψiτ
.
πγ
5
ψ, which contains
√
2g ¯piγ
5
nπ
−
since π
1−i2
=
√
2π
−
.
Thus, g
πNN
=
√
2g. Putting it together, we see that M =gv translates to 2M =fg
πNN
in agreement
with chapter IV.2.
VI.6.1 See figure VI.6.1. From h = (d/cos θ) d(1 +
1
2
θ
2
) and (∂h/∂x) = tan θ θ , we have (∂h/∂t ) ∝ θ
2
∝
(∂h/∂x)
2
, thus giving rise to the term (λ/2)(
∇h)
2
. It all goes back to Mr. Pythagoras.
VI.6.2 We integrate the term
1
2
d
D
xdt[((∂/∂t) −
∇
2
)h]
2
in S(h) by parts to obtain −
1
2
d
D
xdt[h((∂/∂t) +
∇
2
)((∂/∂t) −
∇
2
)h]. Thus, the propagator is the inverse of the operator (∂/∂t +
∇
2
)(∂/∂t −
∇
2
) =
∂
2
/∂t
2
− (
∇
2
)
2
, the Fourier transform of which is −(ω
2
+ k
4
).
VI.8.3 With h
(x , t) =h
x + g ut , t
+u
.
x +
g
2
u
2
t, we have ∂h
/∂t = ∂h/∂t + g u
.
∇h + (g/2)u
2
and
∇h
=
∇h +u. Thus the combination (∂h/∂t ) −
g
2
(
∇h)
2
is invariant, as is (obviously)
∇
2
h. In other words,
˜
S(h) must be constructed out of these two invariant combinations.
VI.8.5 Look at the action S(h) =
1
2
d
D
xdt[(∂h/∂t) −∇
2
h −g(∇h)
2
/2]
2
. Comparing ∂h/∂t −∇
2
h we see that
time has the dimension of length squared: T ∼L
2
. From the term
d
D
x dt (∂h/∂t)
2
and the fact that
S is dimensionless, we have [h]
2
∼ T
2
/(L
D
T)∼ 1/L
D−2
and so h has the dimension of (1/L
D−2
)
1
2
.
Comparing
∇
2
h with g(
∇h)
2
we see that g has the dimension of 1/h, that is, L
(D−2)/2
.
VI.8.7 We are told that L(dg/dL) = (2 − D)g/2 + (2D − 3)f
D
g
3
+
...
. We are assuming that the terms
(
...
) can be neglected. Thus (in what follows a
2
and b
2
are two generic positive numbers) for D = 1,
L(dg/dL) = a
2
g − b
2
g
3
and g flows toward the fixed point g
∗
= a/b. (Incidentally, the KPZ equation
is soluble for D = 1 by methods not explained in this text and both z and χ are known exactly.) For
D = 2, L(dg/dL) = b
2
g
3
and g flows toward some unknown strong (presumably) coupling fixed point.
For D = 3, L(dg/dL) =−a
2
g + b
2
g
3
. The fixed point g
∗
= a/b is unstable. For g<g
∗
, g flows toward
the trivial (i.e., free, or Gaussian) fixed point. Since the theory at the fixed point is free we know the
558 | Solutions to Selected Exercises
critical exponents exactly: z = 2 and χ = (2 − D)/2. For g>g
∗
, g flows toward some unknown strong
(presumably) coupling fixed point.
Part VII
VII.1.1. Set n
μ
A
μ
(x) = 0 with A
μ
= U
†
A
μ
U + iU
†
∂
μ
U, so that n
.
∂U (x) = in
.
A(x)U (x). Define λ(x) =
r
.
x/(r
.
n) for any 4-vector r and write x = λ(x)n + x
⊥
, so that r
.
x
⊥
= 0. Then
U(x)= Pe
i
λ(x)
0
dσn
.
A(σ n+x
⊥
)
(with a path ordering) solves the differential equation, since n
.
∂λ =1 by construction.
VII.1.2 Using the BHC formula given, we have (the V ’s are clearly irrelevant)
U
ij
U
jk
= e
iaA
μ
e
iaA
ν
= e
ia(A
μ
+A
ν
)−
1
2
a
2
[A
μ
, A
ν
]+a
3
C+a
4
D+O(a
5
)
Similarly,
U
kl
U
li
= e
−iaA
μ
e
−iaA
ν
= e
−ia(A
μ
+A
ν
)−
1
2
a
2
[A
μ
, A
ν
]+a
3
E+a
4
F +O(a
5
)
:
the prime reminding us that the A
μ
and A
ν
in this expression is to be evaluated on the “north” and
“west” side of the plaquette in figure VII.1.2, respectively, in contrast to the A
μ
and A
ν
in U
ij
U
jk
which are
evaluated on the “south” and “east” side, respectively. Here C, D, E, and F denote various commutators,
which we drag along merely to show that they eventually drop out in what interests us. (Note how the
different terms are associated with different powers of a, as indicated. Note also that in some places we
have dropped the prime on A and absorbed the “error” in doing so into terms of higher order in a.) Thus,
U
kl
U
li
= e
−ia(A
μ
+A
ν
)−ia
2
(∂
ν
A
μ
−∂
μ
A
ν
−
1
2
i[A
μ
, A
ν
])+a
3
G+a
4
H +O(a
5
)
where G and H denote sums of commutators and terms such as ∂
ν
∂
ν
A
μ
and ∂
ν
∂
ν
∂
ν
A
μ
. Applying the
BHC formula again to the order indicated we have
U
ij
U
jk
U
kl
U
li
= e
ia
2
(∂
μ
A
ν
−∂
ν
A
μ
)−a
2
[A
μ
, A
ν
]+O(a
4
)
= e
ia
2
F
μν
+a
3
I +a
4
J +O(a
5
)
with F
μν
= ∂
μ
A
ν
− ∂
ν
A
μ
+ i[A
μ
, A
ν
]. The same remarks on G and H apply to I and J . The Yang-Mills
field strength emerges naturally, as we would anticipate. Since the traces of commutators and of A vanish,
when we apply the trace all the junk drops out to O(a
5
) and we have
S(P ) =Re tr[1 −
1
2
a
4
F
μν
F
μν
+ O(a
5
)]
By gauge invariance, the corrections must be of even order in a but for our purposes we don’t care about
them anyway. Evidently, f and g are related by some uninteresting factors of a.
Part VIII
VIII.1.7 R
12
= dω
12
= d(−cos θdϕ) = sin θdθdϕ = 2R
12
θϕ
dθdϕ ⇒ R
12
θϕ
=
1
2
sin θ . Since e
θ
1
= 1, e
ϕ
2
= 1/ sin θ ,
we obtain R ≡ R
ab
μν
e
μ
a
e
ν
b
= 2R
12
θϕ
e
θ
1
e
ϕ
2
= 1, independent of θ and ϕ as expected.
Part N
N.1.1. The effective action for an electrically neutral system is given in the point particle limit by S =
dτ(−m +
b
E
E
μ
E
μ
+b
B
B
μ
B
μ
+
...
), with E
μ
and B
μ
defined in the text. The interaction terms involve two powers
of derivatives, which translate into two powers of ω in the scattering amplitude and hence four powers
of ω in the scattering cross section. (Note that a possible term like
dτF
μν
F
μν
can be absorbed into the
two terms already displayed.)
N.3.2. As in (III.3.7) we have 3V
3
+ 4V
4
= 2I + n where I denotes the number of internal lines. The number
of loops (III.3.6) L = I − (V
3
+ V
4
− 1) is 0 in a tree diagram. Thus V
3
= n − 2 − 2V
4
≤ n − 2.
Further Reading
Books on field theory
This is a list of field theory textbooks that I know about. I do not necessarily recommend them all.
In food as in books, each has his or her own taste.
T. Banks, Modern Quantum Field Theory, Cambridge University Press, New York, 2008.
J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964.
———, Relativistic Quantum Fields, McGraw-Hill, New York, 1965.
L. S. Brown, Quantum Field Theory, Cambridge University Press, New York, 1992.
S. J. Chang, Introduction to Quantum Field Theory, World Scientific, Singapore, 1990.
T. P. Cheng and L. F. Li, Gauge Theory of Elementary Particle Physics, Clarendon Press, Oxford, 1984.
F. Dyson and D. Derbes, Advanced Quantum Mechanics, World Scientific, Singapore, 2007.
R. P. Feynman, Quantum Electrodynamics, W. A. Benjamin, New York, 1962.
K. Huang, Quantum Field Theory, John Wiley & Sons, New York, 1998.
C. Itzykson and J-B. Zuber, Quantum Field Theory, McGraw-Hill, New York,1980.
T. D. Lee, Particle Physics and Introduction to Field Theory, Taylor & Francis, New York, 1981.
V. P. Nair, Quantum Field Theory, Springer, New York, 2005.
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Perseus, Reading MA,
1995.
L. H. Ryder, Quantum Field Theory, 2nd Ed., Cambridge University Press, New York, 1996.
M. Stednicki, Quantum Field Theory, Cambridge University Press, New York, 2007.
G. Sterman, An Introduction to Quantum Field Theory, Cambridge University Press, New York, 1993.
S. Weinberg, Quantum Theory of Fields,Vols.1&2,Cambridge University Press, New York, 1996.
X. G. Wen, Quantum Field Theory of Many-Body Systems, Oxford University Press, New York, 2007.
and finally, of course,
F. Mandl, Introduction to Quantum Field Theory, Interscience, New York, 1959.
560 | Further Reading
Books on various topics mentioned
A. A. Abrikosov, L. Gorkov, and A. Dzyaloshinski, Methods of Quantum Field Theory in Statistical
Physics, Prentice Hall, Englewood Cliffs, NJ, 1963.
S. L. Adler, “Perturbation Theory Anomalies,” in: Lectures on Elementary Particles and Quantum Field
Theory, 1970, Brandeis University Summer Institute in Theoretical Physics, S. Deser et al, ed.,
MIT Press, Cambridge, 1970.
P. Anderson, Basic Notions of Condensed Matter Physics, Benjamin-Cummings, Menlo Park, CA 1984.
D. Bailin and A. Love, Supersymmetric Gauge Field Theory and String Theory, IOP Publishing, Bristol
and Philadelphia, 1994.
R. Balian and J. Zinn-Justin, eds., Methods in Field Theory, North Holland Publishing, Amsterdam,
and World Scientific, Singapore, 1981.
A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press,
Cambridge, 1995.
D. Budker, S. J. Freedman, and P. H. Bucksbaum, eds., Art and Symmetry in Experimental Physics:
Festschrift for Eugene D. Commins, American Institute of Physics, New York, 2001.
J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, New York,
1996.
S. Coleman, Aspects of Symmetry, Cambridge University Press, Cambridge, 1985.
J. Collins, Renormalization, Cambridge University Press, Cambridge, 1985.
E. D. Commins, Weak Interactions, McGraw-Hill, New York, 1973.
E. D. Commins and P. H. Bucksbaum, Weak Interactions of Leptons and Quarks, Cambridge University
Press, Cambridge, 2000.
M. Creutz, Quarks, Gluons and Lattices, Cambridge University Press, Cambridge, 1983.
P. A. M. Dirac, The Principles of Quantum Mechanics , Oxford University Press, Oxford, 1935. (On
p. 253 he explained why he wanted the equation of motion for the electron to be first order in time
derivative.)
A. Dobado et al., Effective Lagrangians for the Standard Model, Springer-Verlag, Berlin, 1997.
O.J.P.
´
Eboli et al., Particle Physics, World Scientific, Singapore 1992.
R. P. Feynman, Statistical Mechanics, Perseus Publishing, Reading, MA, 1998.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York,
1965.
J. M. Figueroa-O’ Farrill, Electromagnetic Duality for Children, on the World Wide Web 1998.
V. Fitch et al., eds., Critical Problems in Physics, Princeton University Press, Princeton, 1997.
M. Gell-Mann and Y. Ne’eman, The Eightfold Way, W. A. Benjamin, New York, 1964.
H. B. Geyer, ed., Field Theory, Topology and Condensed Matter Physics, Springer, 1995 (A. Zee,
“Quantum Hall Fluids.”)
M. L. Goldberger and K. M. Watson, Collision Theory, Dover, New York, 2004.
N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, Read-
ing, MA, 1992.
F. Guerra and N. Robotti, Ettore Majorana: Aspects of His Scientific and Academic Activity, Springer,
New York, 2008.
C. Itzykson and J-M. Drouffe, Statistical Field Theory, Cambridge University Press, Cambridge, 1989.
S. Iyanaga and Y. Kawada, eds., Encyclopedic Dictionary of Mathematics, MIT Press, Cambridge, 1980.
L. Kadanoff, Statistical Physics, World Scientific, Singapore, 2000.
G. Kane and M. Shifman, eds., The Supersymmetric World: The Beginning of the Theory,
World
Scientific, Singapore, 2000.
J. I. Kapusta, Finite-Temperature Field Theory, Cambridge University Press, Cambridge, 1989.
L. D. Landau and E. M. Lifschitz, Statistical Physics, Addison-Wesley, Reading, MA, 1974.
S. K. Ma, Modern Theory of Critical Phenomena, Benjamin/Cummings, Reading, MA, 1976.
Further Reading | 561
H. J. W. M
¨
uller-Kirsten and A. Wiedemann, Supersymmetry, World Scientific, Singapore 1987.
T. Muta, Foundations of Quantum Chromodynamics, World Scientific, Singapore, 1998.
D. I. Olive and P. C. West, eds., Duality and Supersymmetric Theories, Cambridge University Press,
Cambridge, 1999.
J. Polchinski, String Theory, Cambridge University Press, Cambridge, 1998.
J. J. Sakurai, Invariance Principles and Elementary Particles, Princeton University Press, Princeton,
1964.
L. Schulman, Techniques and Applications of Path Integrals, John Wiley & Sons, New York, 1981.
R. F. Streater and A. S. Wightman, PCT, Spin Statistics, and All That, W. B. Benjamin, New York,
1968.
G. ’t Hooft, Under the Spell of the Gauge Principle, Word Scientific, Singapore, 1994.
G. ’t Hooft et al., eds. Recent Developments in Gauge Theories, Plenum, New York, 1980.
D. Voiculescu, ed., Free Probability Theory, American Mathematical Society, Providence, R.I., 1997.
S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, New York, 1972.
C. N. Yang, Selected Papers 1945–1980 with Commentary, W. H. Freeman, San Francisco, 1983.
A. Zee, Unity of Forces in the Universe, World Scientific, Singapore, 1982.
J.-B. Zuber, ed., Mathematical Beauty of Physics, World Scientific, Singapore, 1997.
Some popular books and books on the history of quantum field theory
M. Bartusiak, Einstein’s Unfinished Symphony, Joseph Henry Press, Washington, D.C., 2000.
I. Duck and E. C. G. Sudarshan, Pauli and the Spin-Statistics Theorem, World Scientific, Singapore
1997.
R. P. Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press, Princeton,
2006.
D. Kaiser, Drawing Theories Apart, University of Chicago Press, Chicago, 2005.
A. I. Miller, Early Quantum Electrodynamics, Cambridge University Press, Cambridge, 1994.
L. O’Raifeartaigh, The Dawning of Gauge Theory, Princeton University Press, Princeton, 1997.
S. S. Schweber, QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga, Princeton
University Press, Princeton, 1994.
A. Zee, Fearful Symmetry, Princeton University Press, Princeton, 1999.
———, Einstein’s Universe, Oxford University Press, New York, 2001.
———, Swallowing Clouds, University of Washington Press, Seattle, 2002.
Further Reading for Part N
In writing a textbook, the author has the luxury of not preparing a detailed scholarly bibliography
(unless he or she chooses to follow the example of S. Weinberg, who is, in my opinion, most
admirable in this regard). Even more extravagant is the freedom accorded to authors of popular
books who in most cases give their unsuspecting and gullible readers the impression that the physics
of an entire era was done by two or three greats, individuals worthy of their own personality cults.
Presenting recent developments still in flux, I am faced with the dilemma of whether to give proper
credit. In scholarly publications, conscientious referencing is of course ethically mandated, but this
is a textbook. Fortunately, in this age of omniscient search engines, the reader could easily compile
a bibliography more exhaustive than even a myopic humanist used to be able to muster in half a
lifetime. I could do the same, but it is of little help to you for me to merely list the names of those