Zee A. Quantum Field Theory in a Nutshell
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V.1 Superfluids
Repulsive bosons
Consider a finite density ¯ρ of nonrelativistic bosons interacting with a short ranged repul-
sion. Return to (III.5.11):
L = iϕ
†
∂
0
ϕ −
1
2m
∂
i
ϕ
†
∂
i
ϕ − g
2
(ϕ
†
ϕ −¯ρ)
2
(1)
The last term is exactly the Mexican well potential of chapter IV.1, forcing the magnitude
of ϕ to be close to
√
¯ρ , thus suggesting that we use polar variables ϕ ≡
√
ρe
iθ
as we did
in (III.5.7). Plugging in and dropping the total derivative (i/2)∂
0
ρ, we obtain
L =−ρ∂
0
θ −
1
2m
1
4ρ
(∂
i
ρ)
2
+ ρ(∂
i
θ)
2
− g
2
(ρ −¯ρ)
2
(2)
Spontaneous symmetry breaking
As in chapter IV.1 write
√
ρ =
√
¯ρ + h (the vacuum expectation value of ϕ is
√
¯ρ), assume
h
√
¯ρ, and expand
1
:
L =−2
¯ρh∂
0
θ −
¯ρ
2m
(∂
i
θ)
2
−
1
2m
(∂
i
h)
2
− 4g
2
¯ρh
2
+
...
(3)
Picking out the terms up to quadratic in h in (3) we use the “central identity of quantum
field theory” (see appendix A) to integrate out h, obtaining
L =¯ρ∂
0
θ
1
4g
2
¯ρ − (1/2m)∂
2
i
∂
0
θ −
¯ρ
2m
(∂
i
θ)
2
+
...
=
1
4g
2
(∂
0
θ)
2
−
¯ρ
2m
(∂
i
θ)
2
+
...
(4)
1
Note that we have dropped the (potentially interesting) term −¯ρ∂
0
θ because it is a total divergence.
284 | V. Field Theory and Collective Phenomena
In the second equality we assumed that we are looking at processes with wave number k
small compared to
8g
2
¯ρm so that (1/2m)∂
2
i
is negligible compared to 4g
2
¯ρ. Thus, we see
that there exists in this fluid of bosons a gapless mode (often referred to as the phonon)
with the dispersion
ω
2
=
2g
2
¯ρ
m
k
2
(5)
The learned among you will have realized that we have obtained Bogoliubov’s classic result
without ever doing a Bogoliubov rotation.
2
Let me briefly remind you of Landau’s idealized argument
3
that a linearly dispersing
mode (that is, ω is linear in k) implies superfluidity. Consider a mass M of fluid flowing
down a tube with velocity v. It could lose momentum and slow down to velocity v
by creating an excitation of momentum k: Mv = Mv
+ k. This is only possible with
sufficient energy to spare if
1
2
Mv
2
≥
1
2
Mv
2
+ ω(k). Eliminating v
we obtain for M
macroscopic v ≥ ω/k. For a linearly dispersing mode this gives a critical velocity v
c
≡ω/k
below which the fluid cannot lose momentum and is hence super. [Thus, from (5) the
idealized v
c
= g
√
2 ¯ρ/m.]
Suitably scaling the distance variable, we can summarize the low energy physics of
superfluidity in the compact Lagrangian
L =
1
4g
2
(∂
μ
θ)
2
(6)
which we recognize as the massless version of the scalar field theory we studied in part I,
but with the important proviso that θ is a phase angle field, that is, θ(x)and θ(x) +2π are
really the same. This gapless mode is evidently the Nambu-Goldstone boson associated
with the spontaneous breaking of the global U(1) symmetry ϕ → e
iα
ϕ.
Linearly dispersing gapless mode
The physics here becomes particularly clear if we think about a gas of free bosons. We can
give a momentum
k to any given boson at the cost of only (
k)
2
/2m in energy. There
exist many low energy excitations in a free boson system. But as soon as a short ranged
repulsion is turned on between the bosons, a boson moving with momentum
k would
affect all the other bosons. A density wave is set up as a result, with energy proportional
to k as we have shown in (5). The gapless mode has gone from quadratically dispersing to
linearly dispersing. There are far fewer low energy excitations. Specifically, recall that the
density of states is given by N(E) ∝ k
D−1
(dk/dE). For example, for D = 2 the density of
states goes from N(E) ∝ constant (in the presence of quadratically dispersing modes) to
N(E) ∝E (in the presence of linearly dispersing modes) at low energies.
2
L. D. Landau and E. M. Lifshitz, Statistical Physics, p. 238.
3
Ibid., p. 192.
V.1. Superfluids | 285
As was emphasized by Feynman
4
among others, the physics of superfluidity lies not
in the presence of gapless excitations, but in the paucity of gapless excitations. (After all,
the Fermi liquid has a continuum of gapless modes.) There are too few modes that the
superfluid can lose energy and momentum to.
Relativistic versus nonrelativistic
This is a good place to discuss one subtle difference between spontaneous symmetry
breaking in relativistic and nonrelativistic theories. Consider the relativistic theory studied
in chapter IV.1: L =(∂
†
)(∂) − λ(
†
− v
2
)
2
. It is often convenient to take the λ →∞
limit holding v fixed. In the language used in chapter IV.1 “climbing the wall” costs
infinitely more energy than “rolling along the gutter.” The resulting theory is defined by
L =(∂
†
)(∂) (7)
with the constraint
†
= v
2
. This is known as a nonlinear σ model, about which much
more in chapter VI.4.
The existence of a Nambu-Goldstone boson is particularly easy to see in the nonlinear σ
model. The constraint is solved by =ve
iθ
, which when plugged into L gives L = v
2
(∂θ)
2
.
There it is: the Nambu-Goldstone boson θ .
Let’s repeat this in the nonrelativistic domain. Take the limit g
2
→∞with ¯ρ held fixed
so that (1) becomes
L = iϕ
†
∂
0
ϕ −
1
2m
∂
i
ϕ
†
∂
i
ϕ (8)
with the constraint ϕ
†
ϕ =¯ρ. But now if we plug the solution of the constraint ϕ =
√
¯ρe
iθ
into L (and drop the total derivative −¯ρ∂
0
θ), we get L =−( ¯ρ/2m)(∂
i
θ)
2
with the equation
of motion ∂
2
i
θ = 0. Oops, what is this? It’s not even a propagating degree of freedom?
Where is the Nambu-Goldstone boson?
Knowing what I already told you, you are not going to be puzzled by this apparent
paradox
5
for long, but believe me, I have stumped quite a few excellent relativistic minds
with this one. The Nambu-Goldstone boson is still there, but as we can see from (5) its
propagation velocity ω/k scales to infinity as g and thus it disappears from the spectrum
for any nonzero
k.
Why is it that we are allowed to go to this “nonlinear” limit in the relativistic case?
Because we have Lorentz invariance! The velocity of a linearly dispersing mode, if such a
mode exists, is guaranteed to be equal to 1.
4
R. P. Feynman, Statistical Mechanics.
5
This apparent paradox was discussed by A. Zee, “From Semionics to Topological Fluids” in O. J. P.
´
Ebolic et
al., eds., Particle Physics, p. 415.
286 | V. Field Theory and Collective Phenomena
Exercises
V.1.1 Verify that the approximation used to reach (3) is consistent.
V.1.2 To confine the superfluid in an external potential W(x) we would add the term −W(x)ϕ
†
(x , t)ϕ(x, t)
to (1). Derive the corresponding equation of motion for ϕ. The equation, known as the Gross-Pitaevski
equation, has been much studied in recent years in connection with the Bose-Einstein condensate.
V.2
Euclid, Boltzmann, Hawking, and
Field Theory at Finite Temperature
Statistical mechanics and Euclidean field theory
I mentioned in chapter I.2 that to define the path integral more rigorously we should
perform a Wick rotation t =−it
E
. The scalar field theory, instead of being defined by the
Minkowskian path integral
Z =
Dϕe
(i/)
d
d
x[
1
2
(∂ϕ)
2
−V(ϕ)]
(1)
is then defined by the Euclidean functional integral
Z =
Dϕe
−(1/)
d
d
E
x[
1
2
(∂ϕ)
2
+V(ϕ)]
=
Dϕe
−(1/)E(ϕ)
(2)
where d
d
x =−id
d
E
x, with d
d
E
x ≡ dt
E
d
(d−1)
x.In(1)(∂ϕ)
2
=(∂ϕ/∂t)
2
−(
∇ϕ)
2
, while in (2)
(∂ϕ)
2
=(∂ϕ/∂t
E
)
2
+(
∇ϕ)
2
: The notation is a tad confusing but I am trying not to introduce
too many extraneous symbols. You may or may not find it helpful to think of (
∇ϕ)
2
+V(ϕ)
as one unit, untouched by Wick rotation. I have introduced E(ϕ) ≡
d
d
E
x[
1
2
(∂ϕ)
2
+V(ϕ)],
which may naturally be regarded as a static energy functional of the field ϕ(x). Thus,
given a configuration ϕ(x) in d-dimensional space, the more it varies, the less likely it is
to contribute to the Euclidean functional integral Z.
The Euclidean functional integral (2) may remind you of statistical mechanics. Indeed,
Herr Boltzmann taught us that in thermal equilibrium at temperature T = 1/β, the
probability for a configuration to occur in a classical system or the probability for a state to
occur in a quantum system is just the Boltzmann factor e
−βE
suitably normalized, where E
is to be interpreted as the energy of the configuration in a classical system or as the energy
eigenvalue of the state in a quantum system. In particular, recall the classical statistical
mechanics of an N-particle system for which
E(p, q) =
i
1
2m
p
2
i
+ V(q
1
, q
2
,
...
, q
N
)
288 | V. Field Theory and Collective Phenomena
The partition function is given (up to some overall constant) by
Z =
i
dp
i
dq
i
e
−βE(p, q)
After doing the integrals over p we are left with the (reduced) partition function
Z =
i
dq
i
e
−βV(q
1
, q
2
,
...
, q
N
)
Promoting this to a field theory as in chapter I.3, letting i →x and q
i
→ϕ(x) as before, we
see that the partition function of a classical field theory with the static energy functional
E(ϕ) has precisely the form in (2), upon identifying the symbol as the temperature
T =1/β. Thus,
Euclidean quantum field theory in d-dimensional spacetime
∼ Classical statistical mechanics in d-dimensional space
(3)
Functional integral representation of the quantum partition function
More interestingly, we move on to quantum statistical mechanics. The integration over
phase space {p, q} is replaced by a trace, that is, a sum over states: Thus the partition
function of a quantum mechanical system (say of a single particle to be definite) with the
Hamiltonian H is given by
Z = tr e
−βH
=
n
n|e
−βH
|n
In chapter I.2 we worked out the integral representation of F |e
−iHT
|I . (You should
not confuse the time T with the temperature T of course.) Suppose we want an integral
representation of the partition function. No need to do any more work! We simply replace
the time T by −iβ, set |I =|F =|n and sum over |n to obtain
Z = tr e
−βH
=
PBC
Dqe
−
β
0
dτ L(q)
(4)
Tracing the steps from (I.2.3) to (I.2.5) you can verify that here L(q) =
1
2
(dq/dτ )
2
+ V(q)
is precisely the Lagrangian corresponding to H in the Euclidean time τ . The integral over
τ runs from 0 to β. The trace operation sets the initial and final states equal and so the
functional integral should be done over all paths q(τ) with the boundary condition q(0) =
q(β). The subscript PBC reminds us of this all important periodic boundary condition.
The extension to field theory is immediate. If H is the Hamiltonian of a quantum field
theory in D-dimensional space [and hence d =(D + 1)-dimensional spacetime], then the
partition function (4) is
Z = tr e
−βH
=
PBC
Dϕe
−
β
0
dτ
d
D
xL(ϕ)
(5)
V.2. Finite Temperatures | 289
with the integral evaluated over all paths ϕ(x, τ) such that
ϕ(x,0) = ϕ(x, β) (6)
(Here ϕ represents all the Bose fields in the theory.)
A remarkable result indeed! To study a field theory at finite temperature all we have to
do is rotate it to Euclidean space and impose the boundary condition (6). Thus,
Euclidean quantum field theory in (D + 1)-dimensional
spacetime, 0 ≤ τ<β
∼ Quantum statistical mechanics in D-dimensional space
(7)
In the zero temperature limit β →∞we recover from (5) the standard Wick-rotated
quantum field theory over an infinite spacetime, as we should.
Surely you would hit it big with mystical types if you were to tell them that temperature
is equivalent to cyclic imaginary time. At the arithmetic level this connection comes merely
from the fact that the central objects in quantum physics e
−iHT
and in thermal physics
e
−βH
are formally related by analytic continuation. Some physicists, myself included, feel
that there may be something profound here that we have not quite understood.
Finite temperature Feynman diagrams
If we so desire, we can develop the finite temperature perturbation theory of (5), working
out the Feynman rules and so forth. Everything goes through as before with one major
difference stemming from the condition (6) ϕ(x , τ = 0) = ϕ(x , τ = β). Clearly, when
we Fourier transform with the factor e
iωτ
, the Euclidean frequency ω can take on only
discrete values ω
n
≡(2π/β)n, with n an integer. The propagator of the scalar field becomes
1/(k
2
4
+
k
2
) → 1/(ω
2
n
+
k
2
). Thus, to evaluate the partition function, we simply write the
relevant Euclidean Feynman diagrams and instead of integrating over frequency we sum
over a discrete set of frequencies ω
n
= (2πT)n, n =−∞,
...
, +∞. In other words, after
you beat a Feynman integral down to the form
d
d
E
kF(k
2
E
), all you have to do is replace it
by 2πT
n
d
D
kF[(2πT)
2
n
2
+
k
2
].
It is instructive to see what happens in the high-temperature T →∞limit. In summing
over ω
n
, the n = 0 term dominates since the combination (2πT)
2
n
2
+
k
2
occurs in the
denominator. Hence, the diagrams are evaluated effectively in D-dimensional space. We
lose a dimension! Thus,
Euclidean quantum field theory in D-dimensional spacetime
∼ High-temperature quantum statistical mechanics in
D-dimensional space
(8)
This is just the statement that at high-temperature quantum statistical mechanics goes
classical [compare (3)].
290 | V. Field Theory and Collective Phenomena
An important application of quantum field theory at finite temperature is to cosmology:
The early universe may be described as a soup of elementary particles at some high
temperature.
Hawking radiation
Hawking radiation from black holes is surely the most striking prediction of gravitational
physics of the last few decades. The notion of black holes goes all the way back to Michell
and Laplace, who noted that the escape velocity from a sufficiently massive object may
exceed the speed of light. Classically, things fall into black holes and that’s that. But with
quantum physics a black hole can in fact radiate like a black body at a characteristic
temperature T .
Remarkably, with what little we learned in chapter I.11 and here, we can actually
determine the black hole temperature. I hasten to add that a systematic development would
be quite involved and fraught with subtleties; indeed, entire books are devoted to this
subject. However, what we need to do is more or less clear. Starting with chapter I.11, we
would have to develop quantum field theory (for instance, that of a scalar field ϕ) in curved
spacetime, in particular in the presence of a black hole, and ask what a vacuum state (i.e.,
a state devoid of ϕ quanta) in the far past evolves into in the far future. We would find a
state filled with a thermal distribution of ϕ quanta. We will not do this here.
In hindsight, people have given numerous heuristic arguments for Hawking radiation.
Here is one. Let us look at the Schwarzschild solution (see chapter I.11)
ds
2
=
1 −
2GM
r
dt
2
−
1 −
2GM
r
−1
dr
2
− r
2
dθ
2
− r
2
sin
2
θdφ
2
(9)
At the horizon r = 2GM , the coefficients of dt
2
and dr
2
change sign, indicating that
time and space, and hence energy and momentum, are interchanged. Clearly, something
strange must occur. With quantum fluctuations, particle and antiparticle pairs are always
popping in and out of the vacuum, but normally, as we had discussed earlier, the uncer-
tainty principle limits the amount of time t the pairs can exist to ∼ 1/E. Near the black
hole horizon, the situation is different. A pair can fluctuate out of the vacuum right at the
horizon, with the particle just outside the horizon and the antiparticle just inside; heuris-
tically the Heisenberg restriction on t may be evaded since what is meant by energy
changes as we cross the horizon. The antiparticle falls in while the particle escapes to spa-
tial infinity. Of course, a hand-waving argument like this has to be backed up by detailed
calculations.
If black holes do indeed radiate at a definite temperature T , and that is far from obvious
a priori, we can estimate T easily by dimensional analysis. From (9) we see that only the
combination GM , which evidently has the dimension of a length, can come in. Since T
has the dimension of mass, that is, length inverse, we can only have T ∝1/GM.
To determine T precisely, we resort to a rather slick argument. I warn you from the
outset that the argument will be slick and should be taken with a grain of salt. It is only
meant to whet your appetite for a more correct treatment.
V.2. Finite Temperatures | 291
Imagine quantizing a scalar field theory in the Schwarzschild metric, along the line
described in chapter I.11. If upon Wick rotation the field “feels” that time is periodic with
period β , then according to what we have learned in this chapter the quanta of the scalar
field would think that they are living in a heat bath with temperature T =1/β.
Setting t →−iτ, we rotate the metric to
ds
2
=−
1 −
2GM
r
dτ
2
+
1 −
2GM
r
−1
dr
2
+ r
2
dθ
2
+ r
2
sin
2
θdφ
2
(10)
In the region just outside the horizon r
>
∼
2GM, we perform the general coordinate trans-
formation (τ , r) →(α, R) so that the first two terms in ds
2
become R
2
dα
2
+dR
2
, namely
the length element squared of flat 2-dimensional Euclidean space in polar coordinates.
To leading order, we can write the Schwarzchild factor (1 − 2GM/r) as
(r − 2GM)/(2GM) ≡ γ
2
R
2
with the constant γ to be determined. Then the second term
becomes dr
2
/(γ
2
R
2
) = (4GM)
2
γ
2
dR
2
, and thus we set γ = 1/(4GM) to get the desired
dR
2
. The first two terms in −ds
2
are then given by R
2
(dτ /(4 GM))
2
+ dR
2
. Thus the Eu-
clidean time is related to the polar angle by τ = 4GMα and so has a period of 8πGM = β.
We obtain thus the Hawking temperature
T =
1
8πGM
=
c
3
8πGM
(11)
Restoring by dimensional analysis, we see that Hawking radiation is indeed a quantum
effect.
It is interesting to note that the Wick rotated geometry just outside the horizon is given
by the direct product of a plane with a 2-sphere of radius 2GM, although, this observation
is not needed for the calculation we just did.
Exercises
V.2.1 Study the free field theory L =
1
2
(∂ϕ)
2
−
1
2
m
2
ϕ
2
at finite temperature and derive the Bose-Einstein
distribution.
V.2.2 It probably does not surprise you that for fermionic fields the periodic boundary condition (6) is replaced
by an antiperiodic boundary condition ψ(x ,0) =−ψ(x, β) in order to reproduce the results of chap-
ter II.5. Prove this by looking at the simplest fermionic functional integral. [Hint: The clearest exposition
of this satisfying fact may be found in appendix A of R. Dashen, B. Hasslacher, and A. Neveu, Phys. Rev.
D12: 2443, 1975.]
V.2.3 It is interesting to consider quantum field theory at finite density, as may occur in dense astrophysical
objects or in heavy ion collisions. (In the previous chapter we studied a system of bosons at finite density
and zero temperature.) In statistical mechanics we learned to go from the partition function to the grand
partition function Z = tr e
−β(H−μN )
, where a chemical potential μ is introduced for every conserved
particle number N . For example, for noninteracting relativistic fermions, the Lagrangian is modified
to L =
¯
ψ(i ∂ −m)ψ +μ
¯
ψγ
0
ψ . Note that finite density, as well as finite temperature, breaks Lorentz
invariance. Develop the subject of quantum field theory at finite density as far as you can.