![4 Review of quantum statistical mechanics
unless n = 0, in which case it annihilates the vacuum,
a|0 = 0 (1.9)
Apart from an irrelevant phase, the coefficients appearing in (1.7) and
(1.8) follow from the requirements that a† and a be Hermitian conjugates
and that a†a be the number operator N̂. That is,
N̂|n = a†
a|n = n|n (1.10)
As a consequence the commutator of a with a† is
[a, a†
] = aa†
− a†
a = 1 (1.11)
We can build all states from the vacuum by repeated application of the
creation operator:
|n =
1
√
n!
(a†
)n
|0 (1.12)
Next we need a Hamiltonian. Up to an additive constant, it must be
ω times the number operator. Starting with a wave equation in nonrela-
tivistic or relativistic quantum mechanics the additive constant emerges
naturally. One finds that
H = 1
2 ω
aa† + a†a
= ω
a†a + 1
2
= ω
N̂ + 1
2
(1.13)
The additive term 1
2 ω is the zero-point energy. Usually this term can
be ignored. Exceptions arise when the vacuum changes owing to a back-
ground field, such as the gravitational field or an electric field, as in the
Casimir effect. We shall drop this term in the rest of the chapter and leave
it as an exercise to repeat the following analysis with the inclusion of the
zero-point energy.
The states |n are simultaneous eigenstates of energy and particle num-
ber. We can assign a chemical potential to the particles. This is possible
because there are no interactions to change the particle number. The
partition function is easily computed:
Z = Tr e−β(H−μN̂)
= Tr e−β(ω−μ)N̂
=
∞
n=0
n|e−β(ω−μ)N̂
|n =
∞
n=0
e−β(ω−μ)n
(1.14)
=
1
1 − e−β(ω−μ)
The mean number of particles is found from (1.4) to be
N =
1
eβ(ω−μ) − 1
(1.15)](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-20-320.jpg)
![14 Functional integral representation of the partition function
Δt = tf/N. Then, at each time interval we insert a complete set of states,
alternating between (2.2) and (2.5):
φa|e−iHtf
|φa = lim
N→∞
N
i=1
dπi dφi/2π
× φa|πN πN |e−iHΔt
|φN φN |πN−1
× πN−1|e−iHΔt
|φN−1 · · ·
× φ2|π1 π1|e−iHΔt
|φ1 φ1|φa (2.10)
We know that
φ1|φa = δ(φ1 − φa) (2.11)
(as a shorthand for (2.3)) and that
φi+1|πi = exp
i
d3
x πi(x)φi+1(x)
(2.12)
Since Δt → 0, we can expand as follows, keeping terms up to first order:
πi|e−iHiΔt
|φi ≈ πi| (1 − iHiΔt) |φi
= πi|φi (1 − iHiΔt)
= (1 − iHiΔt) exp
−i
d3
x πi(x)φi(x)
(2.13)
where
Hi =
d3
x H (πi(x), φi(x)) (2.14)
Putting it all together we get
φa|e−iHtf
|φa= lim
N→∞
N
i=1
dπi dφi/2π δ(φ1 − φa)
× exp
⎧
⎨
⎩
−iΔt
N
j=1
d3
x [H(πj, φj) − πj(φj+1 − φj)/Δt]
⎫
⎬
⎭
(2.15)
where φN+1 = φa = φ1. The advantage of alternating between π and φ for
the insertion of a complete set of states is that the Hamiltonian in (2.13)
and (2.15) is evaluated at a single point in time.](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-30-320.jpg)
![2.2 Partition function for bosons 15
Taking the continuum limit of (2.15), we finally arrive at the important
result
φa|e−iHtf
|φa
=
[dπ]
φ(x,tf )=φa(x)
φ(x,0)=φa(x)
[dφ]
× exp
i
tf
0
dt
d3
x
π(x, t)
∂φ(x, t)
∂t
− H (π(x, t), φ(x, t))
(2.16)
The symbols [dπ] and [dφ] denote functional integration as defined in
(2.15). The integration over π(x, t) is unrestricted, but the integration
over φ(x, t) is such that the field starts at φa(x) at t = 0 and ends at
φa(x) at t = tf. Note that all references to operators have gone.
2.2 Partition function for bosons
Recall that
Z = Tr e−β(H−μiN̂i)
=
a
dφa φa|e−β(H−μiN̂i)
|φa (2.17)
where the sum runs over all states. This expression is very similar to that
for the transition amplitude defined in the previous section. In fact we
can express Z as an integral over fields and their conjugate momenta by
making use of (2.16). In order to make that connection, we switch to
an imaginary time variable τ = it. The trace operator in (2.17) simply
means that we must integrate over all φa. Finally, if the system admits a
conserved charge then we must make the replacement
H(π, φ) → H(π, φ) − μN(π, φ) (2.18)
where N(π, φ) is the conserved charge density. We finally arrive at the
fundamental formula
Z =
[dπ]
periodic
[dφ]
× exp
β
0
dτ
d3
x
iπ
∂φ
∂τ
− H(π, φ) + μN(π, φ)
(2.19)
The term “periodic” means that the integration over the field is con-
strained in such a way that φ(x, 0) = φ(x, β). This follows from the trace
operation, setting φa(x) = φ(x, 0) = φ(x, β). There is no restriction over
the π integration. The expression for the partition function (2.19) can
readily be generalized to an arbitrary number of fields and conserved
charges.](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-31-320.jpg)
![2.3 Neutral scalar field 17
for each cube. Thus far we have
Z = lim
M,N→∞
(2π)−M3
N/2
N
i=1
dφi
× exp
Δτ
N
j=1
d3
x
−
1
2
φj+1 − φj
Δτ
2
−
1
2
(∇φj)2
−
1
2
m2
φ2
j − U(φj)
(2.26)
Taking the continuum limit, we obtain
Z = N
periodic
[dφ] exp
β
0
dτ
d3
L
(2.27)
The Lagrangian is expressed as a functional of φ and of its first derivatives.
The formula (2.27) expresses the partition function Z as a functional
integral over φ of the exponential of the action in imaginary time. The
overall normalization constant N is irrelevant, since multiplication of Z
by any constant will not change the thermodynamics.
Next, we turn to the case of noninteracting fields by letting U(φ) = 0.
Interactions will be discussed in a later chapter. We define
S =
β
0
dτ
d3
x L = −
1
2
β
0
dτ
d3
x
∂φ
∂τ
2
+ (∇φ)2
+ m2
φ2
(2.28)
Integrating by parts, and using the periodicity of φ, we obtain
S = −
1
2
β
0
dτ
d3
x φ
−
∂2
∂τ2
− ∇2
+ m2
φ (2.29)
The field admits a Fourier expansion:
φ(x, τ) =
β
V
∞
n=−∞ p
ei(p·x+ωnτ)
φn(p) (2.30)
where ωn = 2πnT, owing to the constraint of periodicity that φ(x, β) =
φ(x, 0) for all x. The normalization in (2.30) is chosen such that each
Fourier amplitude is dimensionless. Substituting (2.30) into (2.29) and
recalling that the field is real, we find that
S = −1
2 β2
n p
(ω2
n + ω2
)φn(p)φ∗
n(p) (2.31)](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-33-320.jpg)
![18 Functional integral representation of the partition function
with ω =
p2 + m2. The integrand depends only on the magnitude of
the field, An(p) = |φn(p)|. Integrating out the phases, we get
Z = N
n
p
∞
−∞
dAn(p) exp
−1
2 β2
(ω2
n + ω2
)A2
n(p)
= N
n
p
(2π)1/2
β2
(ω2
n + ω2
)
−1/2
(2.32)
From the treatment above, we know that a factor of (2π)−1/2 appears
for each momentum integration. Thus, ignoring an overall factor that is
independent of volume and temperature,
Z =
n
p
β2
(ω2
n + ω2
)
−1/2
(2.33)
The partition function can be formally written as
Z = N
[dφ] exp
−1
2 (φ, Dφ)
= N
(det D)−1/2
(2.34)
where N is a constant. Here D equals β2(−∂2/∂τ2 − ∇2 + m2) in (x, τ)
space and β2(ω2
n + ω2) in (p, ωn) space, and (φ, Dφ) is the inner product
on the function space. The expression (2.34) follows from the formula for
Riemann integrals with a constant matrix D:
∞
−∞
dx1 · · · dxn e−xiDij xj
= πn/2
(det D)−1/2
(2.35)
One may also derive (2.33) using (2.34).
We now have
ln Z = −1
2
n p
ln
β2
(ω2
n + ω2
)
(2.36)
Using the following identities,
ln
(2πn)2
+ β2
ω2
=
β2
ω2
1
dθ2
θ2 + (2πn)2
+ ln
1 + (2πn)2
(2.37)
and
∞
n=−∞
1
n2 + (θ/2π)2
=
2π2
θ
1 +
2
eθ − 1
(2.38)
and dropping a temperature-independent term, we can write
ln Z = −
p
βω
1
dθ
1
2
+
1
eθ − 1
(2.39)](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-34-320.jpg)
![20 Functional integral representation of the partition function
α(x) = constant. The conserved current density is then
jμ = i(Φ∗
∂μΦ − Φ∂μΦ∗
) (2.47)
with ∂μjμ = 0. The conservation law may be verified independently using
the equation of motion for Φ. The full current and density are Jμ =
d3x jμ(x) and Q =
d3x j0(x).
It is convenient to decompose Φ into real and imaginary parts using
the real fields φ1 and φ2, Φ = (φ1 + iφ2)/
√
2. In terms of the conjugate
momenta π1 = ∂φ1/∂t, π2 = ∂φ2/∂t, the Hamiltonian density and charge
are
H = 1
2
π2
1 + π2
2 + (∇φ1)2
+ (∇φ2)2
+ m2
φ2
1 + m2
φ2
2
+ 1
4 λ(φ2
1 + φ2
2)2
(2.48)
and
Q =
d3
x(φ2π1 − φ1π2) (2.49)
The partition function is
Z =
[dπ1][dπ2]
periodic
[dφ1][dφ2] × exp
β
0
dτ
d3
x
×
iπ1
∂φ
∂τ
+ iπ2
∂φ2
∂τ
− H(π1, π2, φ1, φ2) + μ(φ2π1 − φ1π2
(2.50)
where we have used a chemical potential associated with the conserved
charge Q. Integrating out the conjugate momenta, we get
Z = (N
)2
periodic
[dφ1][dφ2]
× exp
β
0
dτ
d3
x
−1
2
∂φ1
∂τ
− iμφ2
2
− 1
2
∂φ2
∂τ
+ iμφ1
2
− 1
2 (∇φ1)2
− 1
2 (∇φ2)2
− 1
2 m2
φ2
1 − 1
2 m2
φ2
2 − 1
4 λ(φ2
1 + φ2
2)2
(2.51)
where N is the same divergent normalization factor as before. Notice
that the argument of the exponential in (2.51) differs from one’s naive
expectation of
L(φ1, φ2, ∂μφ1, ∂μφ2; μ = 0) + μj0(φ1, φ2, i∂φ1/∂τ, i∂φ2/∂τ)
by an amount μ2Φ∗Φ, owing to the momentum dependence of j0.
The expression (2.51) cannot be evaluated in closed form unless λ = 0.
In this case, the functional integral becomes Gaussian and can then be
worked out analogously to that for the free scalar field.](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-36-320.jpg)
![2.5 Fermions 23
ρ. Therefore ζ is given by
ζ2
=
ρ − ρ∗(β, μ = m)
2m
(2.60)
when μ = m and T Tc. The critical temperature is determined implic-
itly by the equation
ρ = ρ∗
(βc, μ = m) (2.61)
In the nonrelativistic limit, one obtains
Tc =
2π
m
ρ
ζ(3/2)
2/3
ρ m3
(2.62)
In the ultrarelativistic limit, one finds
Tc =
3ρ
m
1/2
ρ m3
(2.63)
In the limit m → 0, we have |μ| → 0 and Tc → ∞. When m = 0, all the
charge resides in the condensate, at all temperatures, and none is carried
by the thermal excitations.
There is a second-order phase transition at Tc. This can be shown rig-
orously by a careful examination of the behavior of the chemical potential
μ(ρ, T) as a function of T near Tc with ρ fixed. This analysis is left as
an exercise. A more intuitive way to see this involves the general Landau
theory of phase transitions [1]. The order parameter ζ drops continuously
to zero as Tc is approached from below and remains zero above Tc. Phys-
ically, the reason for a phase transition is the following. At T = 0, all the
conserved charge can reside in the zero-momentum mode on account of
the bosonic character of the particles. (This would not be the case for
fermions.) As the temperature is raised, some of the charge is excited out
of the condensate. Eventually, the temperature becomes great enough to
completely melt, or thermally disorder, the condensate. There is no rea-
son for ζ to drop to zero discontinuously; hence the transition is second
order.
2.5 Fermions
We now turn our attention to (Dirac) fermions. In relativistic quantum
mechanics, we know that electrons or muons are described by a four-
component spinor ψ. The components are identified as ψα, with α run-
ning from 1 to 4. The motion of a free electron is characterized by a](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-39-320.jpg)
![24 Functional integral representation of the partition function
wavefunction
ψ(x, t) =
1
√
V p s
M
E
b(p, s)u(p, s) e−ip·x
+ d∗
(p, s)v(p, s) eip·x
(2.64)
Here u and v are positive- and negative-energy plane-wave spinors, respec-
tively. The sum on s runs over the two possible spin orientations for a
spin-1/2 Dirac fermion. The expansion coefficients b(p, s) and d∗(p, s) are
complex functions in relativistic quantum mechanics but become opera-
tors in a field theory. As usual, p · x = pμxμ = Et − p · x. Equation (2.64)
is normalized as
d3
x ψ†
(x, t)ψ(x, t) =
p s
|b(p, s)|2
+ |d(p, s)|2
= 1 (2.65)
In the absence of interactions, the Lagrangian density is
L = ψ̄(i∂ − m)ψ (2.66)
The Dirac matrices γμ, which are defined by the anticommutators
{γμ, γν} = 2gμν, are in the standard convention
γ0
=
1 0
0 −1
(2.67)
γ =
0 σ
−σ 0
Each of these is a 4 × 4 matrix: “1” denotes the unit 2 × 2 matrix and σ
denotes the triplet of Pauli matrices. In (2.66), ψ̄ = ψ†γ0 and ∂ ≡ γμ∂μ =
γμ∂/∂xμ. Written out explicitly,
L = ψ†
γ0
iγ0 ∂
∂t
+ iγ · ∇ − m
ψ (2.68)
The Lagrangian has a global U(1) symmetry, so that ψ → ψe−iα and
ψ† → ψ†eiα. Following Noether’s theorem, there is a conserved current
associated with this symmetry. To find it, we proceed in the same way
as we did for the charged scalar field theory. We allow α to depend on
x, treating it as an independent field. Under the above phase transforma-
tion, L → L + ψ̄ [∂α(x)]ψ. Using the equation of motion for α(x), namely
∂μ(∂L/∂[∂μα(x)]) − ∂L/∂α(x) = 0, we find the conservation law
∂μjμ
= 0
jμ
= ψ̄γμ
ψ (2.69)](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-40-320.jpg)
![2.5 Fermions 25
Now we set α = constant to recover our original theory. The total con-
served charge is
Q =
d3
x j0
=
d3
x ψ†
ψ (2.70)
For relativistic quantum mechanics in the absence of interactions this is
a trivial result because of (2.65).
In the field theory we treat ψ as a basic field. The momentum conjugate
to this field is, from (2.68),
Π =
∂L
∂(∂ψ/∂t)
= iψ†
(2.71)
because γ0γ0 = 1. Thus, somewhat paradoxically, ψ and ψ† must be
treated independently in the Hamiltonian formalism. The Hamiltonian
density is found using the standard procedure:
H = Π
∂ψ
∂t
− L = ψ†
i
∂
∂t
ψ − L = ψ̄(−iγ · ∇ + m)ψ (2.72)
The partition function is
Z = Tr†
e−β(H−μQ̂)
(2.73)
Apart from two differences, which could be lost in the formalism if we are
not careful, we can follow the steps leading up to (2.19) and write
Z =
[idψ†
][dψ] exp
β
0
dτ
d3
x ψ̄
−γ0 ∂
∂τ
+ iγ · ∇ − m + μγ0
ψ
(2.74)
Recall that ψ and ψ† are independent fields, which must be integrated
independently. In contrast with boson fields, there is no advantage in
attempting to integrate the conjugate momentum separately from the
field. The two differences mentioned above have to do with the periodicity
of the field in imaginary time τ and with the nature of the “classical” (in
the path-integral formulation) fields ψ(x, τ) and ψ†(x, τ) over which we
integrate.
The canonical commutation relations for bosons are
φ̂(x, t), π̂(y, t)
= iδ(x − y)
(2.75)
φ̂(x, t), φ̂(y, t)
= [π̂(x, t), π̂(y, t)] = 0
and for fermions
ψ̂α(x, t), ψ̂†
β(y, t)
= δαβδ(x − y) (2.76)
ψ̂α(x, t), ψ̂β(y, t)
=
ψ̂†
α(x, t), ψ̂†
β(y, t)
= 0](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-41-320.jpg)
![30 Functional integral representation of the partition function
2.6 Remarks on functional integrals
The notation used for functional integration (and differentiation!) is
deceptively simple. It must be kept simple, for if we think back on the
tremendous progress made in mechanics and electromagnetism in the
nineteenth century, it was certainly made easier by the introduction of
compact notation for differentiation, integration, and vectors. This also
seems to be the case with functional methods in modern quantum physics.
However, it is also clear that the mathematical symbols we are using rep-
resent rather exotic entities. For example, (2.6) uses a Dirac delta function
whose argument is a difference between two functions. A less formal and
compact, but more practical, way to view these objects is to start with
a complete orthonormal set of real functions for the physical problem
of interest. Call this set wn(x), with n any positive integer. Then any
function may be written as
a(x) =
∞
n=1
anwn(x) (2.100)
Another function may be expressed as
b(x) =
∞
n=1
bnwn(x) (2.101)
Then
δ (a(x) − b(x)) =
∞
n=1
δ(an − bn) (2.102)
and
[da(x)] =
∞
n=1
∞
−∞
dan (2.103)
and so on. Most physical problems are defined on the space of a continu-
ous variable, such as position. For such problems it is intuitively obvious
that the functional integral ought to be divergent in general since the pos-
sible functional configurations form an uncountably infinite set. Indeed,
it seems that the extent to which mathematical rigor can be applied to
functional integrals is still uncertain. This should be no surprise since they
are just a means of phrasing the physical content of relativistic quantum
field theory. The extent to which mathematical rigor can be applied in
the operator formalism is probably no more certain, because of the highly
singular nature of the products of field operators at a point. For physi-
cal problems defined on a space of discrete variables, some mathematical](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-46-320.jpg)
![3
Interactions and diagrammatic
techniques
Unfortunately it is not possible to carry out the functional integration
in closed form when the Lagrangian contains terms that are more than
quadratic in the fields. The reader is invited to verify this. Thus, it is
important to develop approximation techniques. An approximation that
is expected to be useful when the interactions are weak is found by expand-
ing the partition function in powers of the interaction. The convergence
properties of these perturbation expansions have not been established
with any degree of mathematical rigor, however. An alternative approach
is to evaluate the partition function containing a given Lagrangian on a
spacetime lattice using numerical Monte Carlo methods. This approach
is described in Chapter 10.
3.1 Perturbation expansion
Consider a single scalar field φ. Other, more physical, theories such as
QED, QCD, the Glashow–Weinberg–Salam model, and effective nuclear
models will be considered in later chapters. The reader must be prepared
now to learn some basic techniques before tackling more complicated but
physically relevant theories.
The partition function is
Z = N
[dφ]eS
(3.1)
The action can be decomposed as
S = S0 + SI (3.2)
where S0 is at most quadratic in the field and SI, the part due to inter-
actions, is of higher order. We may expand (3.1) in a power series in
33](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-49-320.jpg)
![34 Interactions and diagrammatic techniques
the part due to interaction, SI:
Z = N
[dφ]eS0
∞
l=0
1
l!
Sl
I (3.3)
Taking the logarithm on both sides we get
ln Z = ln
N
[dφ]eS0
+ ln
1 +
∞
l=1
1
l!
[dφ]eS0
Sl
I
[dφ]eS0
= ln Z0 + ln ZI (3.4)
This explicitly separates the interaction contributions from the ideal gas
contribution, which we have evaluated already. The relevant quantity that
we actually need to compute is
Sl
I0 =
[dφ]eS0
Sl
I
[dφ]eS0
(3.5)
which is the value of SI raised to an arbitrary positive integral power
and averaged over the unperturbed ensemble, represented by S0. The
normalization of the functional integration is now irrelevant, as it cancels
in the expression (3.5).
3.2 Diagrammatic rules for λφ4 theory
The task of actually evaluating (3.4) and (3.5) is significantly more dif-
ficult than our compact notation would suggest. It is in fact useful to
associate diagrams with the mathematical expressions in the expansion.
Diagrams are a common language in particle physics, nuclear physics, sta-
tistical physics and condensed matter physics and allow for the exchange
of ideas and concepts among these different disciplines.
Consider the lowest-order correction to ln Z0 in λφ4 theory. It is
ln Z1 =
−λ
dτ
d3x
[dφ]eS0
φ4(x, τ)
[dφ]eS0
(3.6)
If we express φ(x, τ) as a Fourier series as in (2.30), and insert this into
(3.6) we get
ln Z1 = −λ
dτ
d3
x
n1,...,n4
p1,...,p4
β2
V 2
× exp[i(p1 + · · · + p4) · x] exp [i(ωn1
+ · · · + ωn4
)τ]
A
B
(3.7)](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-50-320.jpg)
![36 Interactions and diagrammatic techniques
The directions of the arrows reflect the signs of the momenta and fre-
quencies. By convention, we draw them pointing towards the vertex, but
we could have chosen a convention in which they all point away. The
functional integration vanishes unless n3 = −n1, p3 = −p1 and n4 = −n2,
p4 = −p2, etc. Thus we connect the ends in pairs. There are three possible
pairings. We then have
ln Z1 = 3 (3.11)
(p1, ωn1
)(p2, ωn2
)
With each closed loop we associate a factor
T
n
d3p
(2π)3
D0(ωn, p)
With the vertex we associate a factor −λ (coming from LI = −λφ4) and
a factor
βδωin,ωout
V δpin,pout
→ βδωin,ωout
(2π)3
δ(pin − pout)
Since the arguments of the frequency–momentum-conserving deltas are
zero we simply get an overall factor βV . The factor V makes ln Z1 a
properly extensive quantity. Pictorially, (3.11) corresponds precisely with
(3.9).
Next we look at order λ2 in ln ZI. From (3.4) it is
ln Z2 = −1
2
[dφ]eS0
SI
[dφ]eS0
2
+ 1
2
[dφ]eS0
S2
I
[dφ]eS0
(3.12)
The first term in (3.12) is simply
−1
2 (ln Z1)2
= −1
2
3 ⊗ 3
(3.13)
The second term in (3.12) may be analyzed algebraically using func-
tional integrals or it may be analyzed diagrammatically. Choosing the lat-
ter approach, we draw two crosses corresponding to the factors φ4(x, τ)
and φ4(x, τ) contained in 1
2 S2
I 0:](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-52-320.jpg)
![3.2 Diagrammatic rules for λφ4 theory 37
We then pair the lines as before. Counting in the factor one-half and all
the possible pairings, we obtain
1
2
× 3 ⊗ 3 +
6 × 6 × 2
2
+
4 × 3 × 2
2
(3.14)
Combining (3.13) and (3.14), we observe that all the disconnected dia-
grams cancel. We are thus left with
ln Z2 = 36 + 12 (3.15)
What is needed at some arbitrary order N in the perturbative expansion
of ln ZI should now be clear. We formally expand in powers of λ:
ln ZI =
∞
N=1
ln ZN (3.16)
where ln ZN is proportional to λN . The “finite-temperature Feynman
rules” at order N are:
1 Draw all connected diagrams.
2 Determine the combinatoric factor for each diagram.
3 Include a factor T n
[d3p/(2π)3]D0(ωn, p) for each line.
4 Include a factor −λ for each vertex.
5 Include a factor (2π)3δ(pin − pout)βδωin,ωout
for each vertex, correspond-
ing to energy(frequency)–momentum conservation. There will be one
factor β(2π)3δ(0) = βV left over.
We now understand why D is called a propagator: it propagates a
particle (or field) from one vertex to the next. We have illustrated the
cancellation mechanism only at second order. However, it is clear why
disconnected diagrams cancel. If, at some order, there existed a contribu-
tion that was the product of K connected diagrams then this contribution
would be proportional to V K. If we have done our job correctly, then ln ZI
is an extensive quantity proportional to V and thus no such contribution
can arise.
The formal proof that in ln ZI the disconnected diagrams cancel goes
as follows. From (3.3) and (3.5) we have
ZI =
∞
l=0
1
l!
Sl
I0 (3.17)
In general, Sl
I0 can be written as a sum of terms, each of which is a
product of connected parts (see (3.14)). Denoting a connected part by a](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-53-320.jpg)
![38 Interactions and diagrammatic techniques
subscript c, we may write
Sl
I0 =
∞
a1,a2,...=0
l!
a1!a2!(2!)a2 a3!(3!)a3 · · ·
SIa1
0cS2
I a2
0c · · · δa1+2a2+3a3+··· ,l
(3.18)
The combinatoric factor takes into account indistinguishability, and the
Kronecker delta picks out the contribution of order λl. Substituting (3.18)
into (3.17) and summing over l eliminates the Kronecker delta:
ZI =
∞
a1,a2,...=0
SIa1
0c
a1!
S2
I a2
0c
a2!(2!)a2
· · · = exp
∞
n=1
1
n!
Sn
I 0c
(3.19)
Hence ln Z1 is simply the sum of the connected diagrams.
As an example, let us apply these rules to the second diagram of (3.15).
We get
= βV (−λ2
)T
n1
d3p1
(2π)3
· · · T
n4
d3p4
(2π)3
× D0(ωn1
, p1) · · · D0(ωn4
, p4)(2π)3
δ(p1 + · · · + p4)βδn1+···+n4,0
(3.20)
The evaluation of expressions such as (3.20) is not simple and will be
discussed in detail in Section 3.4. The diagrammatic technique is a conve-
nient means for keeping track of the combinatoric factors and the order of
the coupling constant in a perturbative expansion of the partition func-
tion. It circumvents much of the tedious algebra associated with the direct
evaluation of functional integrals.
3.3 Propagators
We shall define a finite-temperature propagator in position space by
D(x1, τ1; x2, τ2) = φ(x1, τ1)φ(x2, τ2) (3.21)
where the angle brackets denote an ensemble average. Owing to trans-
lation invariance, D depends only on x1 − x2 and τ1 − τ2. The Fourier
transform is, with x1 = x, x2 = 0, τ1 = τ, τ2 = 0,
D(ωn, p) =
β
0
dτ
d3
x e−i(p·x+ωnτ)
D(x, τ)
=
n1,n2
p1,p2
β
V
φ̃n1
(p1) φ̃n2
(p2)
β
0
dτ
d3
x
× exp[i(p1 − p) · x] exp[i(ωn1
− ωn)τ] (3.22)](https://image.slidesharecdn.com/454923-250519134344-60bbabde/85/Finitetemperature-Field-Theory-Principles-And-Applications-2nd-Ed-Joseph-I-Kapusta-54-320.jpg)
Finitetemperature Field Theory Principles And Applications 2nd Ed Joseph I Kapusta
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- 5. FINITE-TEMPERATURE FIELD THEORY Principles and Applications This book develops the basic formalism and theoretical techniques for study- ing relativistic quantum field theory at high temperature and density. Specific physical theories treated include QED, QCD, electroweak theory, and effective nuclear field theories of hadronic and nuclear matter. Topics include functional integral representation of the partition function, diagrammatic expansions, lin- ear response theory, screening and plasma oscillations, spontaneous symmetry breaking, the Goldstone theorem, resummation and hard thermal loops, lattice gauge theory, phase transitions, nucleation theory, quark–gluon plasma, and color superconductivity. Applications to astrophysics and cosmology include white dwarf and neutron stars, neutrino emissivity, baryon number violation in the early universe, and cosmological phase transitions. Applications to relativistic nucleus–nucleus collisions are also included. JOSEPH I. KAPUSTA is Professor of Physics at the School of Physics and Astron- omy, University of Minnesota, Minneapolis. He received his Ph.D. from the Uni- versity of California, Berkeley, in 1978 and has been a faculty member at the University of Minnesota since 1982. He has authored over 150 articles in refereed journals and conference proceedings. Since 1997 he has been an associate editor for Physical Review C. He is a Fellow of the American Physical Society and of the American Association for the Advancement of Science. The first edition of Finite-Temperature Field Theory was published by Cambridge University Press in 1989; a paperback edition followed in 1994. CHARLES GALE is James McGill Professor at the Department of Physics, McGill University, Montreal. He received his Ph.D. from McGill University in 1986 and joined the faculty there in 1989. He has authored over 100 articles in refereed journals and conference proceedings. Since 2005 he has been the Chair of the Department of Physics at McGill University. He is a Fellow of the American Physical Society.
- 6. CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS General editors: P. V. Landshoff, D. R. Nelson, S. Weinberg S. Carlip Quantum Gravity in 2 + 1 Dimensions† J. C. Collins Renormalization† M. Creutz Quarks, Gluons and Lattices† P. D. D’ Eath Supersymmetric Quantum Cosmology† F. de Felice and C. J. S. Clarke Relativity on Curved Manifolds† B. S. De Witt Supermanifolds, second edition† P. G. O. Freund Introduction to Supersymmetry† J. Fuches Affine Lie Algebras and Quantum Groups† J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists† Y. Fujii and K. Maeda The Scalar–Tensor Theory of Gravitation A. S. Galperin, E. A. Ivanov, V. I. Orievetsky and E. S. Sokatchev Harmonic Superspace† R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravity† M. Göckeler and T. Schücker Differential Geometry, Gauge Theories and Gravity† C. Gómez, M. Ruiz Altaba and G. Sierra Quantum Groups in Two-Dimensional Physics† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, Volume 1: Introduction† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, Volume 1: 2: Loop Amplitudes, Anomalies and Phenomenology† V. N. Gribov The Theory of Complex Angular Momenta S. W. Hawking and G. F. R. Ellis The Large Scale Structure of Space–Time† F. Iachello and A. Arima The Interacting Boson Model F. Iachello and P. van Isacker The Interacting Boson–Fermion Model† C. Itzykson and J.-M. Drouffe Statistical Field Theory, Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory† C. Itzykson and J.-M. Drouffe Statistical Field Theory, Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems† C. Johnson D-Branes J. I. Kapusta and C. Gale, Finite-Temperature Field Theory V. E. Korepin, N. M. Boguliubov and A. G. Izergin The Quantum Inverse Scattering Method and Correlation Functions† M. Le Bellac Thermal Field Theory† Y. Makeenko Methods of Contemporary Gauge Theory† N. Manton and P. Sutcliffe Topological Solitons N. H. March Liquid Metals: Concepts and Theory† I. M. Montvay and G. Münster Quantum Fields on a Lattice† L. O’Raifeartaigh Group Structure of Gauge Theories† T. Ortı́n Gravity and Strings A. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization† R. Penrose and W. Rindler Spinors and Space-Time, Volume 1: Two-Spinor Calculus and Relativistic Fields† R. Penrose and W. Rindler Spinors and Space-Time, Volume 2: Spinor and Twistor Methods in Space-Time Geometry† S. Pokorski Gauge Field Theories, second edition† J. Polchinski String Theory, Volume 1: An Introduction to the Bosonic String† J. Polchinski String Theory, Volume 2: Superstring Theory and Beyond† V. N. Popov Functional Integrals and Collective Excitations† R. J. Rivers Path Integral Methods in Quantum Field Theory† R. G. Roberts The Structure of the Proton† C. Roveli Quantum Gravity W. C. Saslaw Gravitational Physics of Stellar Galactic Systems† H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field Equations, second edition J. M. Stewart Advanced General Relativity† A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects† R. S. Ward and R. O. Wells Jr Twister Geometry and Field Theory† J. R. Wilson and G. J. Mathews Relativistic Numerical Hydrodynamics 1 Issued as a paperback
- 7. Finite-Temperature Field Theory Principles and Applications JOSEPH I. KAPUSTA School of Physics and Astronomy, University of Minnesota CHARLES GALE Department of Physics, McGill University
- 8. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521820820 C J. I. Kapusta and C. Gale 2006 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 First paperback edition 1994 Second edition 2006 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library ISBN-13 978-0-521-82082-0 hardback ISBN-10 0-521-82082-0 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 9. Contents Preface page ix 1 Review of quantum statistical mechanics 1 1.1 Ensembles 1 1.2 One bosonic degree of freedom 3 1.3 One fermionic degree of freedom 5 1.4 Noninteracting gases 6 1.5 Exercises 10 Bibliography 11 2 Functional integral representation of the partition function 12 2.1 Transition amplitude for bosons 12 2.2 Partition function for bosons 15 2.3 Neutral scalar field 16 2.4 Bose–Einstein condensation 19 2.5 Fermions 23 2.6 Remarks on functional integrals 30 2.7 Exercises 31 Reference 31 Bibliography 31 3 Interactions and diagrammatic techniques 33 3.1 Perturbation expansion 33 3.2 Diagrammatic rules for λφ4 theory 34 3.3 Propagators 38 3.4 First-order corrections to Π and ln Z 41 3.5 Summation of infrared divergences 45 3.6 Yukawa theory 47 v
- 10. vi Contents 3.7 Remarks on real time perturbation theory 51 3.8 Exercises 53 References 54 Bibliography 54 4 Renormalization 55 4.1 Renormalizing λφ4 theory 55 4.2 Renormalization group 57 4.3 Regularization schemes 60 4.4 Application to the partition function 61 4.5 Exercises 63 References 63 Bibliography 63 5 Quantum electrodynamics 64 5.1 Quantizing the electromagnetic field 64 5.2 Blackbody radiation 68 5.3 Diagrammatic expansion 70 5.4 Photon self-energy 71 5.5 Loop corrections to ln Z 74 5.6 Exercises 82 References 83 Bibliography 83 6 Linear response theory 84 6.1 Linear response to an external field 84 6.2 Lehmann representation 87 6.3 Screening of static electric fields 90 6.4 Screening of a point charge 94 6.5 Exact formula for screening length in QED 97 6.6 Collective excitations 100 6.7 Photon dispersion relation 101 6.8 Electron dispersion relation 105 6.9 Kubo formulae for viscosities and conductivities 107 6.10 Exercises 114 References 115 Bibliography 115 7 Spontaneous symmetry breaking and restoration 117 7.1 Charged scalar field with negative mass-squared 117 7.2 Goldstone’s theorem 123 7.3 Loop corrections 125 7.4 Higgs model 130
- 11. Contents vii 7.5 Exercises 133 References 133 Bibliography 134 8 Quantum chromodynamics 135 8.1 Quarks and gluons 136 8.2 Asymptotic freedom 139 8.3 Perturbative evaluation of partition function 146 8.4 Higher orders at finite temperature 149 8.5 Gluon propagator and linear response 152 8.6 Instantons 156 8.7 Infrared problems 161 8.8 Strange quark matter 163 8.9 Color superconductivity 166 8.10 Exercises 174 References 175 Bibliography 176 9 Resummation and hard thermal loops 177 9.1 Isolating the hard thermal loop contribution 179 9.2 Hard thermal loops and Ward identities 185 9.3 Hard thermal loops and effective perturbation theory 187 9.4 Spectral densities 188 9.5 Kinetic theory 189 9.6 Transport coefficients 193 9.7 Exercises 194 References 194 10 Lattice gauge theory 195 10.1 Abelian gauge theory 196 10.2 Nonabelian gauge theory 202 10.3 Fermions 203 10.4 Phase transitions in pure gauge theory 206 10.5 Lattice QCD 212 10.6 Exercises 217 References 217 Bibliography 218 11 Dense nuclear matter 219 11.1 Walecka model 220 11.2 Loop corrections 226 11.3 Three- and four-body interactions 232 11.4 Liquid–gas phase transition 233
- 12. viii Contents 11.5 Summary 236 11.6 Exercises 237 References 238 Bibliography 239 12 Hot hadronic matter 240 12.1 Chiral perturbation theory 240 12.2 Self-energy from experimental data 248 12.3 Weinberg sum rules 254 12.4 Linear and nonlinear σ models 265 12.5 Exercises 287 References 287 Bibliography 288 13 Nucleation theory 289 13.1 Quantum nucleation 290 13.2 Classical nucleation 294 13.3 Nonrelativistic thermal nucleation 296 13.4 Relativistic thermal nucleation 298 13.5 Black hole nucleation 313 13.6 Exercises 315 References 315 Bibliography 316 14 Heavy ion collisions 317 14.1 Bjorken model 318 14.2 The statistical model of particle production 324 14.3 The emission of electromagnetic radiation 328 14.4 Photon production in high-energy heavy ion collisions 331 14.5 Dilepton production 339 14.6 J/ψ suppression 345 14.7 Strangeness production 350 14.8 Exercises 356 References 358 Bibliography 359 15 Weak interactions 361 15.1 Glashow–Weinberg–Salam model 361 15.2 Symmetry restoration in mean field approximation 365 15.3 Symmetry restoration in perturbation theory 369 15.4 Symmetry restoration in lattice theory 374 15.5 Exercises 377 References 377 Bibliography 378
- 13. Contents ix 16 Astrophysics and cosmology 379 16.1 White dwarf stars 380 16.2 Neutron stars 382 16.3 Neutrino emissivity 388 16.4 Cosmological QCD phase transition 394 16.5 Electroweak phase transition and baryogenesis 402 16.6 Decay of a heavy particle 408 16.7 Exercises 410 References 411 Bibliography 412 Conclusion 413 Appendix 417 A1.1 Thermodynamic relations 417 A1.2 Microcanonical and canonical ensembles 418 A1.3 High-temperature expansions 421 A1.4 Expansion in the degeneracy 423 References 424 Index 425
- 15. Preface What happens when ordinary matter is so greatly compressed that the electrons form a relativistic degenerate gas, as in a white dwarf star? What happens when the matter is compressed even further so that atomic nuclei overlap to form superdense nuclear matter, as in a neutron star? What happens when nuclear matter is heated to such great temperatures that the nucleons and pions melt into quarks and gluons, as in high-energy nuclear collisions? What happened in the spontaneous symmetry break- ing of the unified theory of the weak and electromagnetic interactions during the big bang? Questions like these have fascinated us for a long time. The purpose of this book is to develop the fundamental principles and mathematical techniques that enable the formulation of answers to these mind-boggling questions. The study of matter under extreme con- ditions has blossomed into a field of intense interdisciplinary activity and global extent. The analysis of the collective behavior of interacting rela- tivistic systems spans a rich palette of physical phenomena. One of the ultimate goals of the whole program is to map out the phase diagram of the standard model and its extensions. This text assumes that the reader has completed graduate level courses in thermal and statistical physics and in relativistic quantum field theory. Our aims are to convey a coherent picture of the field and to prepare the reader to read and understand the original and current literature. The book is not, however, a compendium of all known results; this would have made it prohibitively long. We start from the basic principles of quantum field theory, thermodynamics, and statistical mechanics. This develop- ment is most elegantly accomplished by means of Feynman’s functional integral formalism. Having a functional integral expression for the parti- tion function allows a straightforward derivation of diagrammatic rules for interacting field theories. It also provides a framework for defining gauge theories on finite lattices, which then enables integration by Monte Carlo xi
- 16. xii Preface techniques. The formal aspects are illustrated with applications drawn from fields of research that are close to the authors’ own experience. Each chapter carries its own exercises, reference list, and select bibliography. The book is based on Finite-Temperature Field Theory, written by one of us (JK) and published in 1989. Although the fundamental principles have not changed, there have been many important developments since then, necessitating a new book. We would like to acknowledge the assistance of Frithjof Karsch and Steven Gottlieb in transmitting some of their results of lattice computa- tions, presented in Chapter 10, and Andrew Steiner for performing the numerical calculations used to prepare many of the figures in Chapter 11. We are grateful to a number of friends, colleagues, and students for their helpful comments and suggestions and for their careful reading of the manuscript, especially Peter Arnold, Eric Braaten, Paul Ellis, Philippe de Forcrand, Bengt Friman, Edmond Iancu, Sangyong Jeon, Keijo Kajantie, Frithjof Karsch, Mikko Laine, Stefan Leupold, Guy Moore, Ulrich Mosel, Robert Pisarski, Brian Serot, Andrew Steiner, and Laurence Yaffe.
- 17. 1 Review of quantum statistical mechanics Thermodynamics is used to describe the bulk properties of matter in or near equilibrium. Many scientists, notably Boyle, Carnot, Clausius, Gay- Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the develop- ment of the field over three centuries. Quantities such as mass, pressure, energy, and so on are readily defined and measured. Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system. Great progress in this field was made by physicists like Boltzmann and Maxwell. Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter. Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics. In fact, the ultravio- let catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory. The development of quan- tum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose. The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles. This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. 1.1 Ensembles One normally encounters three types of ensemble in equilibrium statistical mechanics. The microcanonical ensemble is used to describe an isolated system that has a fixed energy E, a fixed particle number N, and a fixed 1
- 18. 2 Review of quantum statistical mechanics volume V . The canonical ensemble is used to describe a system in contact with a heat reservoir at temperature T. The system can freely exchange energy with the reservoir, but the particle number and volume are fixed. In the grand canonical ensemble the system can exchange particles as well as energy with a reservoir. In this ensemble the temperature, volume, and chemical potential μ are fixed quantities. The standard thermodynamic relations are summarized in appendix section A1.1. In the canonical and grand canonical ensembles, T−1 = β may be thought of as a Lagrange multiplier that determines the mean energy of the system. Similarly, μ may be thought of as a Lagrange multiplier that determines the mean number of particles in the system. In a rela- tivistic quantum system, where particles can be created and destroyed, it is most straightforward to compute observables in the grand canonical ensemble. For that reason we use the grand canonical ensemble through- out this book. There is no loss of generality in doing so because one may pass over to either of the other ensembles by performing an inverse Laplace transform on the variable μ and/or the variable β. See appendix section A1.2. Consider a system described by a Hamiltonian H and a set of con- served number operators N̂i. (A hat or caret is used to denote an opera- tor for emphasis or whenever there is the possibility of an ambiguity.) In QED, for example, the number of electrons minus the number of positrons is a conserved quantity, not the number of electrons or positrons sepa- rately, because of reactions like e+e− → e+e+e−e−. These number oper- ators must be Hermitian and must commute with H as well as with each other. They must also be extensive (scale with the volume of the system) in order that the usual macroscopic thermodynamic limit can be taken. The statistical density matrix ρ̂ is the fundamental object in equilibrium statistical mechanics: ρ̂ = exp −β H − μiN̂i (1.1) Here and throughout the book a repeated index is assumed to be summed over. In QED the sum would run over two conserved number operators if one allowed for both electrons and muons. The statistical density matrix is used to compute the ensemble average of any desired observable, rep- resented by the operator Â, via A = Â = Tr Âρ̂ Tr ρ̂ (1.2) where Tr denotes the trace operation. The grand canonical partition function Z = Z(V, T, μ1, μ2, . . .) = Tr ρ̂ (1.3)
- 19. 1.2 One bosonic degree of freedom 3 is the single most important function in thermodynamics. From it all the thermodynamic properties may be determined. For example, the pressure, particle number, entropy, and energy are, in the infinite-volume limit, given by P = ∂(T ln Z) ∂V Ni = ∂(T ln Z) ∂μi (1.4) S = ∂(T ln Z) ∂T E = −PV + TS + μiNi 1.2 One bosonic degree of freedom As a simple example consider a time-independent single-particle quantum mechanical mode that may be occupied by bosons. Each boson in that mode has the same energy ω. There may be 0, 1, 2, or any number of bosons occupying that mode. There are no interactions between the par- ticles. This system may be thought of as a set of noninteracting quantized simple harmonic oscillators. It will serve as a prototype of the relativistic quantum field theory systems to be introduced in later chapters. We are interested in computing the mean particle number, energy, and entropy. Since the system has no volume there is no physical pressure. Denote the state of the system by |n, which means that there are n bosons in the system. The state |0 is called the vacuum. The properties of these states are n|n = δnn orthogonality (1.5) ∞ n=0 |nn| = 1 completeness (1.6) One may think of the bras n| and kets |n as row and column vectors, respectively, in an infinite-dimensional vector space. These vectors form a complete set. The operation in (1.5) is an inner product and the number 1 in (1.6) stands for the infinite-dimensional unit matrix. It is convenient to introduce creation and annihilation operators, a† and a, respectively. The creation operator creates one boson and puts it in the mode under consideration. Its action on a number eigenstate is a† |n = √ n + 1|n + 1 (1.7) Similarly, the annihilation operator annihilates or removes one boson, a|n = √ n|n − 1 (1.8)
- 20. 4 Review of quantum statistical mechanics unless n = 0, in which case it annihilates the vacuum, a|0 = 0 (1.9) Apart from an irrelevant phase, the coefficients appearing in (1.7) and (1.8) follow from the requirements that a† and a be Hermitian conjugates and that a†a be the number operator N̂. That is, N̂|n = a† a|n = n|n (1.10) As a consequence the commutator of a with a† is [a, a† ] = aa† − a† a = 1 (1.11) We can build all states from the vacuum by repeated application of the creation operator: |n = 1 √ n! (a† )n |0 (1.12) Next we need a Hamiltonian. Up to an additive constant, it must be ω times the number operator. Starting with a wave equation in nonrela- tivistic or relativistic quantum mechanics the additive constant emerges naturally. One finds that H = 1 2 ω aa† + a†a = ω a†a + 1 2 = ω N̂ + 1 2 (1.13) The additive term 1 2 ω is the zero-point energy. Usually this term can be ignored. Exceptions arise when the vacuum changes owing to a back- ground field, such as the gravitational field or an electric field, as in the Casimir effect. We shall drop this term in the rest of the chapter and leave it as an exercise to repeat the following analysis with the inclusion of the zero-point energy. The states |n are simultaneous eigenstates of energy and particle num- ber. We can assign a chemical potential to the particles. This is possible because there are no interactions to change the particle number. The partition function is easily computed: Z = Tr e−β(H−μN̂) = Tr e−β(ω−μ)N̂ = ∞ n=0 n|e−β(ω−μ)N̂ |n = ∞ n=0 e−β(ω−μ)n (1.14) = 1 1 − e−β(ω−μ) The mean number of particles is found from (1.4) to be N = 1 eβ(ω−μ) − 1 (1.15)
- 21. 1.3 One fermionic degree of freedom 5 and the mean energy E is ωN. Note that N ranges continuously from zero to infinity as μ ranges from −∞ to ω. Values of the chemical potential, in this system, are restricted to be less than ω on account of the positivity of the particle number or, equivalently, the Hermiticity of the number operator. There are two interesting limits. One is the classical limit, where the occupancy is small, N 1. This occurs when T ω − μ. In this limit the exponential in (1.15) is large and so N = e−β(ω−μ) classical limit (1.16) The other is the quantum limit, where the occupancy is large, N 1. This occurs when T ω − μ. 1.3 One fermionic degree of freedom Now consider the same problem as in the previous section but for fermions instead of bosons. This is a prototype for a Fermi gas, and later on will help us to formulate the functional integral expression for the partition function involving fermions. These could be electrons and positrons in QED, neutrons and protons in nuclei and nuclear matter, or quarks in QCD. The Pauli exclusion principle forbids the occupation of a single-particle mode by more than one fermion. Thus there are only two states of the system, |0 and |1. The action of the fermion creation and annihilation operators on these states is as follows: α† |0 = |1 α|1 = |0 (1.17) α† |1 = 0 α|0 = 0 Therefore, these operators have the property that their square is zero when acting on any of the states, αα = α† α† = 0 (1.18) Up to an arbitrary phase factor, the coefficients in (1.17) are chosen so that α and α† are Hermitian conjugates and α†α is the number operator N̂: N̂|n = n|n (1.19)
- 22. 6 Review of quantum statistical mechanics It follows that the creation and annihilation operators satisfy the anti- commutation relation {α, α† } = αα† + α† α = 1 (1.20) The Hamiltonian is taken to be H = 1 2 ω α†α − αα† = ω N̂ − 1 2 (1.21) This form follows from the Dirac equation. Notice that the zero-point energy is equal in magnitude but opposite in sign to the bosonic zero- point energy. In this chapter we drop this term for fermions, as we have for bosons. The partition function is computed as in (1.14) except that the sum terminates at n = 1 on account of the Pauli exclusion principle: Z = Tr e−β(H−μN̂) = Tr e−β(ω−μ)N̂ = 1 n=0 n|e−β(ω−μ)N̂ |n = 1 n=0 e−β(ω−μ)n (1.22) = 1 + e−β(ω−μ) The mean number of particles is found from (1.4) to be N = 1 eβ(ω−μ) + 1 (1.23) and the mean energy E is ωN. Note that N ranges continuously from zero to unity as μ ranges from −∞ to ∞. Unlike bosons, for fermions there is no restriction on the chemical potential. As with bosons, there are two interesting limits. One is the classical limit, where the occupancy is small, N 1. This occurs when T ω − μ: N = e−β(ω−μ) classical limit (1.24) which is the same limit as for bosons. The other is the quantum limit. When T → 0 one obtains N → 0 if ω μ and N → 1 if ω μ. 1.4 Noninteracting gases Now let us put particles, either bosons or fermions, into a box with sides of length L. We neglect their mutual interactions, although in principle they must interact in order to come to thermal equilibrium. One can imagine including interactions, waiting until the particles come to equilibrium, and then slowly turning off the interactions. Such a noninteracting gas is often a good description of the atmosphere around us, electrons in a metal or white dwarf star, blackbody photons in a heated cavity or in
- 23. 1.4 Noninteracting gases 7 the cosmic microwave background radiation, phonons in low-temperature materials, neutrons in a neutron star, and many other situations. In the macroscopic limit the boundary condition imposed on the surface of the box is unimportant. For definiteness we impose the condition that the wave function vanishes at the surface of the box. (Also frequently used are periodic boundary conditions.) The vanishing of the wave function on the surface means that an integral number of half-wavelengths must fit in the distance L: λx = 2L/jx λy = 2L/jy λz = 2L/jz (1.25) where jx, jy, jz are all positive integers. The magnitude of the x com- ponent of the momentum is |px| = 2π/λx = πjx/L, and similarly for the y and z components. Amazingly, quantum mechanics tells us that these relations hold for both nonrelativistic and relativistic motion, for both bosons and fermions. The full Hamiltonian is the sum of the Hamiltonians for each mode on account of the assumption that the particles do not interact. We use a shorthand notation in which j represents the triplet of numbers (jx, jy, jz) that uniquely specifies each mode. Thus the Hamiltonian and number operator are H = j Hj (1.26) N̂ = j N̂j Then the partition function is the product of the partition functions for each mode: Z = Tr e−β(H−μN̂) = j Tr e−β(Hj −μN̂j ) = j Zj (1.27) Each mode corresponds to the single bosonic or fermionic degree of free- dom discussed previously. According to (1.4) it is ln Z that is of fundamental interest. From (1.27), ln Z = ∞ jx=1 ∞ j1=1 ∞ jz=1 ln Zjx,jy,jz (1.28) In the macroscopic limit, L → ∞, it is permissible to replace the sum from jx = 1 to ∞ with an integral from jx = 1 to ∞. (The correction to this approximation is proportional to the surface area L2 and the relative contribution is therefore of order 1/L.) We can then use djx = Ld|px|/π
- 24. 8 Review of quantum statistical mechanics to write ln Z = L3 π3 ∞ 0 d|px| ∞ 0 d|py| ∞ 0 d|pz| ln Z(p) (1.29) In all cases to be dealt with in this book the mode partition function depends only on the magnitude of the momentum components. Then the integration over px may be extended from −∞ to ∞ if we divide by 2: ln Z = V d3p (2π)3 ln Z(p) (1.30) Note the natural appearance of the phase-space integral d3xd3p/(2π)3 in this expression. Recalling the mode partition function from the previous sections we have ln Z = V d3p (2π)3 ln 1 ± e−β(ω−μ) ±1 (1.31) where the upper sign (+) refers to fermions and the lower sign (−) refers to bosons. From (1.4) and (1.31) we obtain the pressure, particle number, and energy: P = T V ln Z N = V d3p (2π)3 1 eβ(ω−μ) ± 1 (1.32) E = V d3p (2π)3 ω eβ(ω−μ) ± 1 These formulæ for N and E have the simple interpretation of phase- space integrals over the mean particle number and energy of each mode, respectively. The dispersion relation ω = ω(p) determines the energy for a given momentum. For relativistic particles ω = p2 + m2, where m is the mass. The nonrelativistic limit is ω = m + p2/2m. For phonons the dispersion relation is ω = csp, where cs is the speed of sound in the medium. There are a number of interesting and physically relevant limits. Con- sider the dispersion relation ω = p2 + m2. The classical limit corre- sponds to low occupancy of the modes and is the same for bosons (1.16) and fermions (1.24). The momentum integral for the pressure can be per- formed and written as P = m2T2 2π2 eμ/T K2 m T classical limit (1.33)
- 25. 1.4 Noninteracting gases 9 where K2 is the modified Bessel function. The nonrelativistic limit of this is P = T mT 2π 3/2 e(μ−m)/T classical nonrelativistic limit (1.34) Knowing the pressure as a function of temperature and chemical potential we can obtain all other thermodynamic functions by differentiation or by using thermodynamic identities. The zero-temperature limit for fermions requires that μ m, other- wise the vacuum state is approached. In this limit all states up to the Fermi momentum pF = μ2 − m2 and energy EF = μ are occupied and all states above are empty. The pressure, energy density = E/V , and number density n = N/V are given by P = 1 16π2 2μ3 pF − m2 μpF − m4 ln μ + pF m = 1 16π2 2 3 μp3 F − m2 μpF + m4 ln μ + pF m (1.35) n = p3 F 6π2 In the nonrelativistic limit, P = p5 F 30π2m (1.36) = mn + 3 2 P nonrelativistic limit Electrons and nucleons have spin 1/2 and these expressions need to be multiplied by 2 to take account of that! The low-temperature limit for bosons will be discussed in the next chapter. Massless bosons with zero chemical potential have pressure P = π2 90 T4 (1.37) This is one of the most famous formulae in the thermodynamics of radi- ation fields. If time reversal is a good symmetry, a detailed balance must occur among all possible reactions in equilibrium. For example, if the reac- tion A + B → C + D can occur then not only must the reverse reac- tion, C + D → A + B, occur but it must happen at the same rate. Detailed balance implies relationships between the chemical potentials. It is shown in standard textbooks that, for the reactions just mentioned, the chemical potentials obey μA + μB = μC + μD. For a long-lived resonance
- 26. 10 Review of quantum statistical mechanics that decays according to X → A + B, the formation process A + B → X must happen at the same rate. The chemical potentials are related by μX = μA + μB. Generally any reactions that are allowed by the conserva- tion laws can and will occur. These conservation laws restrict the number of linearly independent chemical potentials. Consider, for example, a sys- tem whose only relevant conservation laws are for baryon number and electric charge. There are only two independent chemical potentials, one for baryon number (μB) and one for electric charge (μQ). Any particle in the system has a chemical potential which is a linear combination of these: μi = biμB + qiμQ (1.38) Here bi is the baryon number and qi the electric charge of the particle of type i. These chemical potentials are all measured with respect to the total particle energy including mass. (The chemical potential μNR i , as cus- tomarily defined in nonrelativistic many-body theory, is related to ours by μNR i = μi − mi.) Bosons that carry no conserved quantum number, such as photons and π0 mesons, have zero chemical potential. Antiparticles have a chemical potential opposite in sign to particles. The electrically charged mesons π+ and π− have electric charges of +1 and −1 and therefore equal and opposite chemical potentials, μQ and −μQ, respectively. The total conserved charge is the number of π+ mesons minus the number of π− mesons: Q = V d3p (2π)3 1 eβ(ω−μQ) − 1 − 1 eβ(ω+μQ) − 1 (1.39) and the total energy is E = V d3p (2π)3 ω eβ(ω−μQ) − 1 + ω eβ(ω+μQ) − 1 (1.40) If the bosons have nonzero spin s, then the phase-space integrals must be multiplied by the spin degeneracy factor 2s + 1. An analogous discussion can be given for fermions. 1.5 Exercises 1.1 Prove that the state |n given in (1.12) is normalized to unity. 1.2 Referring to (1.17), let |0 and |1 be represented by the basis vectors in a two-dimensional vector space. Find an explicit 2 × 2 matrix representation of the abstract operators α and α† in this vector space. 1.3 Calculate the partition function for noninteracting bosons, including the zero-point energy. From it calculate the mean energy, particle number, and entropy. Repeat the calculation for fermions.
- 27. Bibliography 11 1.4 Calculate the average energy per particle of a noninteracting gas of massless bosons with no chemical potential. Repeat the calculation for massless fermions. 1.5 Derive an expression like (1.39) or (1.40) for the entropy. Repeat the calculation for fermions. Bibliography Thermal and statistical physics Reif, F. (1965). Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York). Landau, L. D., and Lifshitz, E. M. (1959). Statistical Physics (Pergamon Press, Oxford). Many-body theory Abrikosov, A. A., Gorkov, L. P., and Dzyaloshinskii, I. E. (1963). Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliffs). Fetter, A. L. and Walecka, J. D. (1971). Quantum Theory of Many-Particle Systems (McGraw-Hill, New York). Negele, J. W. and Orland, H. (1988). Quantum Many-Particle Systems (Addison-Wesley, Redwood City). Numerical evaluation of thermodynamic integrals Johns, S. D., Ellis, P. J., and Lattimer, J. M., Astrophys. J. 473, 1020 (1996).
- 28. 2 Functional integral representation of the partition function The customary approach to nonrelativistic many-body theory is to pro- ceed with the method of second quantization begun in the first chap- ter. There is another approach, the method of functional integrals, which we shall follow here. Of course, what can be done in one formalism can always be done in another. Nevertheless, functional integrals seem to be the method of choice for most elementary particle theorists these days, and they seem to lend themselves more readily to nonperturbative phe- nomena such as tunneling, instantons, lattice gauge theory, etc. For gauge theories they are practically indispensable. However, there is a certain amount of formalism that must be developed before we can start to dis- cuss physical applications. In this chapter, we shall derive the functional integral representation of the partition function for interacting relativistic non-gauge field theories. As a check on the formalism, as well as to obtain some feeling for how functional integrals work, we shall then rederive some well-known results on relativistic ideal gases for bosons and fermions. 2.1 Transition amplitude for bosons Let φ̂(x, 0) be a Schrödinger-picture field operator at time t = 0 and let π̂(x, 0) be its conjugate momentum operator. The eigenstates of the field operator are labeled |φ and satisfy φ̂(x, 0)|φ = φ(x)|φ (2.1) where φ(x) is the eigenvalue, as indicated, a function of x. We also have the usual completeness and orthogonality conditions, dφ(x)|φφ| = 1 (2.2) 12
- 29. 2.1 Transition amplitude for bosons 13 φa|φb = x δ(φa(x) − φb(x)) (2.3) Similarly, the eigenstates of the conjugate momentum field operator satisfy π̂(x, 0)|π = π(x)|π (2.4) The completeness and orthogonality conditions are dπ(x) 2π |ππ| = 1 (2.5) πa|πb = x δ(πa(x) − πb(x)) (2.6) The practical meaning of the formal expressions (2.2), (2.3), (2.5), and (2.6) is elucidated in Section 2.6. Just as in quantum mechanics one may work in coordinate space or in momentum space, one may work here in the field space or in the conjugate momentum space. In quantum mechanics, one goes from one to the other by using x|p = eipx (2.7) In field theory one has the overlap φ|π = exp i d3 x π(x)φ(x) (2.8) In a natural generalization one goes from a denumerably finite number of degrees of freedom N in quantum mechanics to a continuously infi- nite number of degrees of freedom in quantum field theory: N i=1 pixi → d3x π(x)φ(x). For the dynamics one requires a Hamiltonian, which is now a functional of the field and of its conjugate momentum: H = d3 x H(π̂, φ̂) (2.9) Now suppose that a system is in a state |φa at a time t = 0. After a time tf it evolves into e−iHtf |φa, assuming that the Hamiltonian has no explicit time dependence. The transition amplitude for going from a state |φa to a state |φb after a time tf is thus φb|e−iHtf |φa. For statistical mechanical purposes we will be interested in cases where the system returns to its original state after the time tf. To obtain a practical definition of the transition amplitude we use the following pre- scription: we divide the time interval (0, tf) into N equal steps of duration
- 30. 14 Functional integral representation of the partition function Δt = tf/N. Then, at each time interval we insert a complete set of states, alternating between (2.2) and (2.5): φa|e−iHtf |φa = lim N→∞ N i=1 dπi dφi/2π × φa|πN πN |e−iHΔt |φN φN |πN−1 × πN−1|e−iHΔt |φN−1 · · · × φ2|π1 π1|e−iHΔt |φ1 φ1|φa (2.10) We know that φ1|φa = δ(φ1 − φa) (2.11) (as a shorthand for (2.3)) and that φi+1|πi = exp i d3 x πi(x)φi+1(x) (2.12) Since Δt → 0, we can expand as follows, keeping terms up to first order: πi|e−iHiΔt |φi ≈ πi| (1 − iHiΔt) |φi = πi|φi (1 − iHiΔt) = (1 − iHiΔt) exp −i d3 x πi(x)φi(x) (2.13) where Hi = d3 x H (πi(x), φi(x)) (2.14) Putting it all together we get φa|e−iHtf |φa= lim N→∞ N i=1 dπi dφi/2π δ(φ1 − φa) × exp ⎧ ⎨ ⎩ −iΔt N j=1 d3 x [H(πj, φj) − πj(φj+1 − φj)/Δt] ⎫ ⎬ ⎭ (2.15) where φN+1 = φa = φ1. The advantage of alternating between π and φ for the insertion of a complete set of states is that the Hamiltonian in (2.13) and (2.15) is evaluated at a single point in time.
- 31. 2.2 Partition function for bosons 15 Taking the continuum limit of (2.15), we finally arrive at the important result φa|e−iHtf |φa = [dπ] φ(x,tf )=φa(x) φ(x,0)=φa(x) [dφ] × exp i tf 0 dt d3 x π(x, t) ∂φ(x, t) ∂t − H (π(x, t), φ(x, t)) (2.16) The symbols [dπ] and [dφ] denote functional integration as defined in (2.15). The integration over π(x, t) is unrestricted, but the integration over φ(x, t) is such that the field starts at φa(x) at t = 0 and ends at φa(x) at t = tf. Note that all references to operators have gone. 2.2 Partition function for bosons Recall that Z = Tr e−β(H−μiN̂i) = a dφa φa|e−β(H−μiN̂i) |φa (2.17) where the sum runs over all states. This expression is very similar to that for the transition amplitude defined in the previous section. In fact we can express Z as an integral over fields and their conjugate momenta by making use of (2.16). In order to make that connection, we switch to an imaginary time variable τ = it. The trace operator in (2.17) simply means that we must integrate over all φa. Finally, if the system admits a conserved charge then we must make the replacement H(π, φ) → H(π, φ) − μN(π, φ) (2.18) where N(π, φ) is the conserved charge density. We finally arrive at the fundamental formula Z = [dπ] periodic [dφ] × exp β 0 dτ d3 x iπ ∂φ ∂τ − H(π, φ) + μN(π, φ) (2.19) The term “periodic” means that the integration over the field is con- strained in such a way that φ(x, 0) = φ(x, β). This follows from the trace operation, setting φa(x) = φ(x, 0) = φ(x, β). There is no restriction over the π integration. The expression for the partition function (2.19) can readily be generalized to an arbitrary number of fields and conserved charges.
- 32. 16 Functional integral representation of the partition function 2.3 Neutral scalar field The most general renormalizable Lagrangian for a neutral scalar field φ is L = 1 2 ∂μφ ∂μ φ − 1 2 m2 φ2 − U(φ) (2.20) where the potential is U(φ) = gφ3 + λφ4 (2.21) and λ ≥ 0 for the stability of the vacuum. The momentum conjugate to this field is π = ∂L ∂(∂0φ) = ∂φ ∂t (2.22) and the Hamiltonian is obtained through the usual Legendre transforma- tion H = π ∂φ ∂t − L = 1 2 π2 + 1 2 (∇φ)2 + 1 2 m2 φ2 + U(φ) (2.23) There is no conserved charge. We shall evaluate the partition function by returning to the discretized version: Z = lim N→∞ N i=1 ∞ −∞ dπi 2π periodic dφi × exp N j=1 d3 x iπj(φj+1 − φj) − Δτ 1 2 π2 j + 1 2 (∇φj)2 + 1 2 m2 φ2 j + U(φ) (2.24) The momentum integrals can be evaluated immediately since they are simply products of Gaussian integrals. We divide position space into M3 small cubes with V = L3, L = aM, a → 0, M → ∞, M being an integer. For convenience and to make sure that Z remains explicitly dimensionless at each stage of the calculation, we write πj = Aj/(a3Δτ)1/2 and integrate Aj from −∞ to ∞. We get ∞ −∞ dAj 2π exp −1 2 A2 j + i a3 Δτ 1/2 (φj+1 − φj)Aj = (2π)−1/2 exp −a3(φj+1 − φj)2 2Δτ (2.25)
- 33. 2.3 Neutral scalar field 17 for each cube. Thus far we have Z = lim M,N→∞ (2π)−M3 N/2 N i=1 dφi × exp Δτ N j=1 d3 x − 1 2 φj+1 − φj Δτ 2 − 1 2 (∇φj)2 − 1 2 m2 φ2 j − U(φj) (2.26) Taking the continuum limit, we obtain Z = N periodic [dφ] exp β 0 dτ d3 L (2.27) The Lagrangian is expressed as a functional of φ and of its first derivatives. The formula (2.27) expresses the partition function Z as a functional integral over φ of the exponential of the action in imaginary time. The overall normalization constant N is irrelevant, since multiplication of Z by any constant will not change the thermodynamics. Next, we turn to the case of noninteracting fields by letting U(φ) = 0. Interactions will be discussed in a later chapter. We define S = β 0 dτ d3 x L = − 1 2 β 0 dτ d3 x ∂φ ∂τ 2 + (∇φ)2 + m2 φ2 (2.28) Integrating by parts, and using the periodicity of φ, we obtain S = − 1 2 β 0 dτ d3 x φ − ∂2 ∂τ2 − ∇2 + m2 φ (2.29) The field admits a Fourier expansion: φ(x, τ) = β V ∞ n=−∞ p ei(p·x+ωnτ) φn(p) (2.30) where ωn = 2πnT, owing to the constraint of periodicity that φ(x, β) = φ(x, 0) for all x. The normalization in (2.30) is chosen such that each Fourier amplitude is dimensionless. Substituting (2.30) into (2.29) and recalling that the field is real, we find that S = −1 2 β2 n p (ω2 n + ω2 )φn(p)φ∗ n(p) (2.31)
- 34. 18 Functional integral representation of the partition function with ω = p2 + m2. The integrand depends only on the magnitude of the field, An(p) = |φn(p)|. Integrating out the phases, we get Z = N n p ∞ −∞ dAn(p) exp −1 2 β2 (ω2 n + ω2 )A2 n(p) = N n p (2π)1/2 β2 (ω2 n + ω2 ) −1/2 (2.32) From the treatment above, we know that a factor of (2π)−1/2 appears for each momentum integration. Thus, ignoring an overall factor that is independent of volume and temperature, Z = n p β2 (ω2 n + ω2 ) −1/2 (2.33) The partition function can be formally written as Z = N [dφ] exp −1 2 (φ, Dφ) = N (det D)−1/2 (2.34) where N is a constant. Here D equals β2(−∂2/∂τ2 − ∇2 + m2) in (x, τ) space and β2(ω2 n + ω2) in (p, ωn) space, and (φ, Dφ) is the inner product on the function space. The expression (2.34) follows from the formula for Riemann integrals with a constant matrix D: ∞ −∞ dx1 · · · dxn e−xiDij xj = πn/2 (det D)−1/2 (2.35) One may also derive (2.33) using (2.34). We now have ln Z = −1 2 n p ln β2 (ω2 n + ω2 ) (2.36) Using the following identities, ln (2πn)2 + β2 ω2 = β2 ω2 1 dθ2 θ2 + (2πn)2 + ln 1 + (2πn)2 (2.37) and ∞ n=−∞ 1 n2 + (θ/2π)2 = 2π2 θ 1 + 2 eθ − 1 (2.38) and dropping a temperature-independent term, we can write ln Z = − p βω 1 dθ 1 2 + 1 eθ − 1 (2.39)
- 35. 2.4 Bose–Einstein condensation 19 Carrying out the integral and dropping terms that are independent of temperature and volume, we finally get ln Z = V d3p (2π)3 −1 2 βω − ln(1 − e−βω ) (2.40) This expression is identical to the bosonic version of (1.31) with μ = 0, except that (2.40) includes the zero-point energy. Both E0 = − ∂ ∂β ln Z0 = 1 2 V d3p (2π)3 ω (2.41) and P0 = T ∂ ∂V ln Z0 = − E0 V (2.42) should be subtracted, since the vacuum is a state with zero energy and pressure. 2.4 Bose–Einstein condensation An interesting system is obtained by considering a theory with a charged scalar field Φ. The field Φ is then complex and describes bosons of pos- itive and negative charge, i.e., they are each other’s antiparticle. The Lagrangian density in this case is L = ∂μΦ∗ ∂μ Φ − m2 Φ∗ Φ − λ(Φ∗ Φ)2 (2.43) This expression has an obvious U(1) symmetry: Φ → Φ = Φe−iα (2.44) where α is a real constant. This is a global symmetry since the multiplying phase factor is independent of spacetime location. By Noether’s theorem, there is a conserved current associated with each continuous symmetry of the Lagrangian. We can find this current by letting the phase factor α depend on the spacetime coordinate for a moment. In this case the U(1) transformation is L → L = ∂μ(Φ∗ eiα(x) )∂μ (Φe−iα(x) ) − m2 Φ∗ Φ − λ(Φ∗ Φ)2 = L + Φ∗ Φ∂μα ∂μ α + i∂μα(Φ∗ ∂μ Φ − Φ∂μ Φ∗ ) (2.45) The equation of motion for the “field” α(x) is ∂μ ∂L ∂(∂μα) = ∂L ∂α (2.46) Since ∂L/∂α = 0, it follows that the “current” ∂L/∂(∂μα) = Φ∗Φ∂μα + iΦ∗∂μΦ − iΦ∂μΦ∗ is conserved. We recover our original theory by letting
- 36. 20 Functional integral representation of the partition function α(x) = constant. The conserved current density is then jμ = i(Φ∗ ∂μΦ − Φ∂μΦ∗ ) (2.47) with ∂μjμ = 0. The conservation law may be verified independently using the equation of motion for Φ. The full current and density are Jμ = d3x jμ(x) and Q = d3x j0(x). It is convenient to decompose Φ into real and imaginary parts using the real fields φ1 and φ2, Φ = (φ1 + iφ2)/ √ 2. In terms of the conjugate momenta π1 = ∂φ1/∂t, π2 = ∂φ2/∂t, the Hamiltonian density and charge are H = 1 2 π2 1 + π2 2 + (∇φ1)2 + (∇φ2)2 + m2 φ2 1 + m2 φ2 2 + 1 4 λ(φ2 1 + φ2 2)2 (2.48) and Q = d3 x(φ2π1 − φ1π2) (2.49) The partition function is Z = [dπ1][dπ2] periodic [dφ1][dφ2] × exp β 0 dτ d3 x × iπ1 ∂φ ∂τ + iπ2 ∂φ2 ∂τ − H(π1, π2, φ1, φ2) + μ(φ2π1 − φ1π2 (2.50) where we have used a chemical potential associated with the conserved charge Q. Integrating out the conjugate momenta, we get Z = (N )2 periodic [dφ1][dφ2] × exp β 0 dτ d3 x −1 2 ∂φ1 ∂τ − iμφ2 2 − 1 2 ∂φ2 ∂τ + iμφ1 2 − 1 2 (∇φ1)2 − 1 2 (∇φ2)2 − 1 2 m2 φ2 1 − 1 2 m2 φ2 2 − 1 4 λ(φ2 1 + φ2 2)2 (2.51) where N is the same divergent normalization factor as before. Notice that the argument of the exponential in (2.51) differs from one’s naive expectation of L(φ1, φ2, ∂μφ1, ∂μφ2; μ = 0) + μj0(φ1, φ2, i∂φ1/∂τ, i∂φ2/∂τ) by an amount μ2Φ∗Φ, owing to the momentum dependence of j0. The expression (2.51) cannot be evaluated in closed form unless λ = 0. In this case, the functional integral becomes Gaussian and can then be worked out analogously to that for the free scalar field.
- 37. 2.4 Bose–Einstein condensation 21 The components of Φ can be Fourier-expanded: φ1 = √ 2ζ cos θ + β V n p ei(p·x+ωnτ) φ1;n(p) (2.52) φ2 = √ 2ζ sin θ + β V n p ei(p·x+ωnτ) φ2;n(p) Here ζ and θ are independent of (x, τ) and determine the full infrared behavior of the field; that is, φ1;0(p = 0) = φ2;0(p = 0) = 0. This allows for the possibility of condensation of the bosons into the zero-momentum state. Condensation means that in the infinite-volume limit a finite frac- tion of the particles resides in the n = 0, p = 0 state. Setting λ = 0 and substituting (2.52) into (2.51) after an integration by parts, see (2.29), we find Z = (N )2 n p dφ1;n(p) dφ2;n(p) eS (2.53) where S = βV (μ2 − m2 )ζ2 − 1 2 n p ! φ1;−n(−p), φ2;−n(−p) D φ1;n(p) φ2;n(p) and D = β2 ω2 n + ω2 − μ2 −2μωn 2μωn ω2 n + ω2 − μ2 Carrying out the integrations, ln Z = βV (μ2 − m2 )ζ2 + ln(det D)−1/2 (2.54) The second term can be handled as follows: ln det D = ln n p β4 (ω2 n + ω2 − μ2 )2 + 4μ2 ω2 n = ln n p β2 ω2 n + (ω − μ)2 + ln n p β2 ω2 n + (ω + μ)2 (2.55)
- 38. 22 Functional integral representation of the partition function Putting all this together, ln Z = βV (μ2 − m2 )ζ2 − 1 2 n p ln # β2 ω2 n + (ω − μ)2 $ − 1 2 n p ln # β2 ω2 n + (ω + μ)2 $ (2.56) The last two terms in (2.56) are precisely of the form (2.36). All we need to do is recall (2.40) and make the substitutions ω → ω − μ and ω → ω + μ, respectively, for the two terms in (2.56). We obtain ln Z = βV (μ2 − m2 )ζ2 − V d3p (2π)3 × βω + ln ! 1 − e−β(ω−μ) + ln ! 1 − e−β(ω+μ) (2.57) There are several observations we can make about (2.57). The momentum integral is convergent only if |μ| ≤ m. The parameter ζ appears in the final expression but θ does not, as expected from the U(1) symmetry of the Lagrangian. In this context, since the parameter ζ is not determined a priori, it should be treated as a variational parameter that is related to the charge carried by the condensed particles. At fixed β and μ, ln Z is an extremum with respect to variations of such a free parameter. Thus ∂ ln Z ∂ζ = 2βV (μ2 − m2 )ζ = 0 (2.58) which implies that ζ = 0 unless |μ| = m, in which case ζ is undetermined by this variational condition. When |μ| m we simply recover the results obtained in Chapter 1, namely (1.31). To determine ζ when |μ| = m, note that the charge density ρ = Q/V is given by ρ = T V ∂ ln Z ∂μ μ=m = 2mζ2 + ρ∗ (β, μ = m) (2.59) where ρ∗ = d3p (2π)3 1 eβ(ω−m) − 1 − 1 eβ(ω+m) − 1 (The case μ = −m is handled analogously.) Here the separate contribu- tions from the condensate (the zero-momentum mode) and the thermal excitations are manifest. If the density ρ is kept fixed and the tempera- ture is lowered, μ will decrease until the point μ = m is reached. If the temperature is lowered even further then ρ∗(β, μ = m) will be less than
- 39. 2.5 Fermions 23 ρ. Therefore ζ is given by ζ2 = ρ − ρ∗(β, μ = m) 2m (2.60) when μ = m and T Tc. The critical temperature is determined implic- itly by the equation ρ = ρ∗ (βc, μ = m) (2.61) In the nonrelativistic limit, one obtains Tc = 2π m ρ ζ(3/2) 2/3 ρ m3 (2.62) In the ultrarelativistic limit, one finds Tc = 3ρ m 1/2 ρ m3 (2.63) In the limit m → 0, we have |μ| → 0 and Tc → ∞. When m = 0, all the charge resides in the condensate, at all temperatures, and none is carried by the thermal excitations. There is a second-order phase transition at Tc. This can be shown rig- orously by a careful examination of the behavior of the chemical potential μ(ρ, T) as a function of T near Tc with ρ fixed. This analysis is left as an exercise. A more intuitive way to see this involves the general Landau theory of phase transitions [1]. The order parameter ζ drops continuously to zero as Tc is approached from below and remains zero above Tc. Phys- ically, the reason for a phase transition is the following. At T = 0, all the conserved charge can reside in the zero-momentum mode on account of the bosonic character of the particles. (This would not be the case for fermions.) As the temperature is raised, some of the charge is excited out of the condensate. Eventually, the temperature becomes great enough to completely melt, or thermally disorder, the condensate. There is no rea- son for ζ to drop to zero discontinuously; hence the transition is second order. 2.5 Fermions We now turn our attention to (Dirac) fermions. In relativistic quantum mechanics, we know that electrons or muons are described by a four- component spinor ψ. The components are identified as ψα, with α run- ning from 1 to 4. The motion of a free electron is characterized by a
- 40. 24 Functional integral representation of the partition function wavefunction ψ(x, t) = 1 √ V p s M E b(p, s)u(p, s) e−ip·x + d∗ (p, s)v(p, s) eip·x (2.64) Here u and v are positive- and negative-energy plane-wave spinors, respec- tively. The sum on s runs over the two possible spin orientations for a spin-1/2 Dirac fermion. The expansion coefficients b(p, s) and d∗(p, s) are complex functions in relativistic quantum mechanics but become opera- tors in a field theory. As usual, p · x = pμxμ = Et − p · x. Equation (2.64) is normalized as d3 x ψ† (x, t)ψ(x, t) = p s |b(p, s)|2 + |d(p, s)|2 = 1 (2.65) In the absence of interactions, the Lagrangian density is L = ψ̄(i∂ − m)ψ (2.66) The Dirac matrices γμ, which are defined by the anticommutators {γμ, γν} = 2gμν, are in the standard convention γ0 = 1 0 0 −1 (2.67) γ = 0 σ −σ 0 Each of these is a 4 × 4 matrix: “1” denotes the unit 2 × 2 matrix and σ denotes the triplet of Pauli matrices. In (2.66), ψ̄ = ψ†γ0 and ∂ ≡ γμ∂μ = γμ∂/∂xμ. Written out explicitly, L = ψ† γ0 iγ0 ∂ ∂t + iγ · ∇ − m ψ (2.68) The Lagrangian has a global U(1) symmetry, so that ψ → ψe−iα and ψ† → ψ†eiα. Following Noether’s theorem, there is a conserved current associated with this symmetry. To find it, we proceed in the same way as we did for the charged scalar field theory. We allow α to depend on x, treating it as an independent field. Under the above phase transforma- tion, L → L + ψ̄ [∂α(x)]ψ. Using the equation of motion for α(x), namely ∂μ(∂L/∂[∂μα(x)]) − ∂L/∂α(x) = 0, we find the conservation law ∂μjμ = 0 jμ = ψ̄γμ ψ (2.69)
- 41. 2.5 Fermions 25 Now we set α = constant to recover our original theory. The total con- served charge is Q = d3 x j0 = d3 x ψ† ψ (2.70) For relativistic quantum mechanics in the absence of interactions this is a trivial result because of (2.65). In the field theory we treat ψ as a basic field. The momentum conjugate to this field is, from (2.68), Π = ∂L ∂(∂ψ/∂t) = iψ† (2.71) because γ0γ0 = 1. Thus, somewhat paradoxically, ψ and ψ† must be treated independently in the Hamiltonian formalism. The Hamiltonian density is found using the standard procedure: H = Π ∂ψ ∂t − L = ψ† i ∂ ∂t ψ − L = ψ̄(−iγ · ∇ + m)ψ (2.72) The partition function is Z = Tr† e−β(H−μQ̂) (2.73) Apart from two differences, which could be lost in the formalism if we are not careful, we can follow the steps leading up to (2.19) and write Z = [idψ† ][dψ] exp β 0 dτ d3 x ψ̄ −γ0 ∂ ∂τ + iγ · ∇ − m + μγ0 ψ (2.74) Recall that ψ and ψ† are independent fields, which must be integrated independently. In contrast with boson fields, there is no advantage in attempting to integrate the conjugate momentum separately from the field. The two differences mentioned above have to do with the periodicity of the field in imaginary time τ and with the nature of the “classical” (in the path-integral formulation) fields ψ(x, τ) and ψ†(x, τ) over which we integrate. The canonical commutation relations for bosons are φ̂(x, t), π̂(y, t) = iδ(x − y) (2.75) φ̂(x, t), φ̂(y, t) = [π̂(x, t), π̂(y, t)] = 0 and for fermions ψ̂α(x, t), ψ̂† β(y, t) = δαβδ(x − y) (2.76) ψ̂α(x, t), ψ̂β(y, t) = ψ̂† α(x, t), ψ̂† β(y, t) = 0
- 42. 26 Functional integral representation of the partition function These commutation relations are the only ones allowed by the fundamen- tal spin-statistics theorem in relativistic quantum field theory. In the limit → 0 the field operators are replaced by their eigenvalues. For the case of bosons, those eigenvalues are actually c-number functions, as illustrated in (2.1). We have expressed the partition function as a functional inte- gral over these c-number functions, or “classical fields”. For the case of fermions, the → 0 limit is rather peculiar since the eigenvalues replac- ing the field operators anticommute with each other! This is of course connected with the Pauli exclusion principle and with the famous spin- statistics theorem. Note that (2.74) instructs us to integrate over these “classical” but anticommuting functions. The mathematics necessary to handle this situation was studied by Grassmann. There are Grassmann variables, Grassmann algebra, and Grassmann calculus. For a single Grassmann variable η, there is only one anticommutator to define the algebra, {η, η} = 0 (2.77) Because of this, the most general function of η is (using a Taylor series expansion) f(η) = a + bη, where a and b are c-numbers. Integration is defined by dη = 0 (2.78) dη η = 1 The first of these says that the integral is invariant under the shift η → η + a, and the second is just a convenient normalization. In a more general setting, we may have a set of Grassmann variables ηi, i = 1, 2, . . . N, and a paired set η† i . The algebra is defined by {ηi, ηj} = {ηi, η† j } = {η† i , η† j } = 0 (2.79) The most general function of these variables may be written as f = a + i aiηi + i biη† i + i,j aijηiηj + i,j bijη† i η† j + i,j cijη† i ηj + · · · + dη† 1η1η† 2η2 · · · η† N ηN (2.80) Integration over all variables of (2.80) is defined by dη† 1dη1 · · · dη† N dηN f = d (2.81) Integrals over Grassmann variables were introduced for the explicit pur- pose of dealing with path integrals over fermionic coordinates. The
- 43. 2.5 Fermions 27 bibliography at the end of this chapter refers the interested reader to more detailed treatments. For our purposes, the only integral we need is dη† 1dη1 · · · dη† N dηN eη† Dη = det D (2.82) where D is an N × N matrix. This formula is simple to prove if N = 1 or 2. The general case is left as an exercise for the reader. As with bosons, it is most convenient to work in (p, ωn) space instead of (x, τ) space. In imaginary time we can write ψα(x, τ) = 1 √ V n p ei(p·x+ωnτ) ψ̃α;n(p) (2.83) where both n and p run over negative and positive values. For an arbitrary function defined over the interval 0 ≤ τ ≤ β, the discrete frequency ωn can take on the values nπT. For bosons we argued that we must take ωn = 2πnT in order that φ(x, τ) be periodic, which followed from the trace operation in the partition function. This can be verified by examining the properties of the thermal Green’s function for bosons defined by GB(x, y; τ1, τ2) = Z−1 Tr ρ̂Tτ φ̂(x, τ1)φ̂(y, τ2) (2.84) Here Tτ is the imaginary time ordering operator, which for bosons acts as follows: Tτ φ̂(x, τ1)φ̂(y, τ2) = φ̂(τ1)φ̂(τ2)θ(τ1 − τ2) + φ̂(τ2)φ̂(τ1)θ(τ2 − τ1) (2.85) where θ is the step-function. Using the fact that Tτ commutes with ρ̂ = e−βK, where K ≡ H − μQ̂, and the cyclic property of the trace we find that GB(x, y; τ, 0) = Z−1 Tr e−βK φ̂(x, τ)φ̂(y, 0) = Z−1 Tr φ̂(y, 0) e−βK φ̂(x, τ) = Z−1 Tr e−βK eβK φ̂(y, 0) e−βK φ̂(x, τ) = Z−1 Tr e−βK φ̂(y, β)φ̂(x, τ) = Z−1 Tr ρ̂Tτ φ̂(x, τ)φ̂(y, β) = GB(x, y; τ, β) (2.86) (Notice that φ̂(y, β) = eβKφ̂(y, 0) e−βK, in analogy with the real time Heisenberg time-evolution expression φ̂(y, t) = eiHtφ̂(y, 0) e−iHt.) The result (2.86) implies that φ(y, 0) = φ(y, β) and hence ωn = 2πnT.
- 44. 28 Functional integral representation of the partition function For fermions, however, instead of (2.85) one has (in direct analogy with the real time Green’s functions) Tτ ψ̂(τ1)ψ̂(τ2) = ψ̂(τ1)ψ̂(τ2)θ(τ1 − τ2) − ψ̂(τ2)ψ̂(τ1)θ(τ2 − τ1) (2.87) Following the same steps as in (2.86), one is led to GF(x, y; τ, 0) = −GF(x, y; τ, β) (2.88) This implies that ψ(x, 0) = −ψ(x, β) (2.89) and hence ωn = (2n + 1)πT (2.90) This antiperiodicity required of fermion fields is in no way inconsistent with the trace operation in the partition function. The trace only means that the system returns to its original state after a “time” β. Since the sign of ψ is just an overall phase and hence is not observable, the right-hand side of (2.89) describes the same physical state as the left-hand side. Now we are ready to evaluate (2.74). Inserting (2.83) and using (2.82) we get Z = n p α idψ̃† α;n(p)dψ̃α;n(p) eS (2.91) where S = n p iψ̃† α;n(p)Dαρψ̃ρ;n(p) D = −iβ (−iωn + μ) − γ0 γ · p − mγ0 and so Z = det D (2.92) In (2.92) the determinantal operation is carried out over both Dirac indices (thus with 4 × 4 matrices) and in frequency–momentum space. Using ln det D = Tr ln D (2.93) and (2.67), one finds that ln Z = 2 n p ln # β2 (ωn + iμ)2 + ω2 $ (2.94)
- 45. 2.5 Fermions 29 Since the summation is over both negative and positive frequencies (2.94) can be put into a form analogous to (2.55), ln Z = n p ln β2 (ω2 n(ω − μ)2 ) + ln β2 (ω2 n(ω + μ)2 ) (2.95) Following (2.37), we write ln (2n + 1)2 π2 + β2 (ω ± μ)2 = β2 (ω±μ)2 1 dθ2 θ2 + (2n + 1)2π2 + ln 1 + (2n + 1)2 π2 (2.96) The sum over n may be carried out by using the summation formula ∞ n=−∞ 1 (n − x)(n − y) = π(cot πx − cot πy) y − x (2.97) This gives ∞ n=−∞ 1 (2n + 1)2π2 + θ2 = 1 θ 1 2 − 1 eθ + 1 (2.98) Integrating over θ and dropping terms that are independent of β and μ, we finally obtain ln Z = 2V d3p (2π)3 βω + ln ! 1 + e−β(ω−μ) + ln ! 1 + e−β(ω+μ) (2.99) This result agrees with that derived in Chapter 1 using completely differ- ent methods. Notice the factor 2 in (2.99). This factor comes out automatically and owes its existence to the spin-1/2 nature of the fermions. Separate con- tributions from particles (μ) and antiparticles (−μ) are evident. Finally, this formula also contains a contribution from the zero-point energy. To recapitulate, the difference between fermions and bosons in the func- tional integral approach to the partition function is essentially twofold. First, for fermions we must integrate over Grassmann variables instead of c-number variables. Contrast the result (2.92), Z = det D, for fermions with the result (2.34), Z = (det D)−1/2, for bosons. Integration over c- number variables would have led to a factor −1 in (2.99) instead of the factor 2. Second, and this is related to the first, is the fact that the fermion fields are actually antiperiodic in imaginary time, with period β, instead of periodic as is the case for bosons. The consequence is that ωn = (2n + 1)πT for fermions whereas ωn = 2πnT for bosons. These two points account for the difference between (2.57) (with ζ = 0, of course) and (2.99).
- 46. 30 Functional integral representation of the partition function 2.6 Remarks on functional integrals The notation used for functional integration (and differentiation!) is deceptively simple. It must be kept simple, for if we think back on the tremendous progress made in mechanics and electromagnetism in the nineteenth century, it was certainly made easier by the introduction of compact notation for differentiation, integration, and vectors. This also seems to be the case with functional methods in modern quantum physics. However, it is also clear that the mathematical symbols we are using rep- resent rather exotic entities. For example, (2.6) uses a Dirac delta function whose argument is a difference between two functions. A less formal and compact, but more practical, way to view these objects is to start with a complete orthonormal set of real functions for the physical problem of interest. Call this set wn(x), with n any positive integer. Then any function may be written as a(x) = ∞ n=1 anwn(x) (2.100) Another function may be expressed as b(x) = ∞ n=1 bnwn(x) (2.101) Then δ (a(x) − b(x)) = ∞ n=1 δ(an − bn) (2.102) and [da(x)] = ∞ n=1 ∞ −∞ dan (2.103) and so on. Most physical problems are defined on the space of a continu- ous variable, such as position. For such problems it is intuitively obvious that the functional integral ought to be divergent in general since the pos- sible functional configurations form an uncountably infinite set. Indeed, it seems that the extent to which mathematical rigor can be applied to functional integrals is still uncertain. This should be no surprise since they are just a means of phrasing the physical content of relativistic quantum field theory. The extent to which mathematical rigor can be applied in the operator formalism is probably no more certain, because of the highly singular nature of the products of field operators at a point. For physi- cal problems defined on a space of discrete variables, some mathematical
- 47. Bibliography 31 rigor can be applied. This is one reason why certain spacetime theories are defined on a spacetime lattice. This will be studied in Chapter 10. 2.7 Exercises 2.1 For the charged scalar field show, by direct application of the equa- tion of motion for Φ, that jμ = i(Φ∗∂μΦ − Φ∂μΦ∗) is conserved. 2.2 If jμ is conserved show that Q̇ = d dt d3 x j0(x, t) = 0 2.3 Obtain (2.62) and (2.63), starting from (2.59) to (2.61). 2.4 For Bose–Einstein condensation, consider μ as a function of ρ and T. If ρ is held fixed, show that μ and ∂μ/∂T are continuous but ∂2μ/∂T2 is discontinuous at Tc. 2.5 Prove (2.82). 2.6 Fill in the steps leading from (2.91)–(2.93) to (2.94). 2.7 When m = 0 show that (2.99) can be evaluated in closed form, lead- ing to P = T ln Z/V = μ4/12π2 + μ2T2/6 + 7π2T4/180. Reference 1. Landau, L. D., and Lifshitz, E. M. (1959). Statistical Physics (Pergamon Press, Oxford). Bibliography Path integrals in quantum mechanics Feynman, R. P., Phys. Rev. 91, 1291 (1953). Feynman, R. P. and Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals (McGraw-Hill, New York). Functional integrals in field theory Lee, T. D. (1982). Particle Physics and Introduction to Field Theory (Harwood Academic, London). Nash, C. (1978). Relativistic Quantum Fields (Academic Press, New York). Ramond, P. (1989). Field Theory: A Modern Primer (Addison-Wesley, Redwood City). Functional integrals in field theory at finite temperature Bernard, C., Phys. Rev. D 9, 3312 (1974).
- 48. 32 Functional integral representation of the partition function Thermal phase transitions Landau, L. D., and Lifshitz, E. M. (1959). Statistical Physics (Pergamon Press, Oxford). Relativistic Bose–Einstein condensation Kapusta, J. I., Phys. Rev. D 24, 426 (1981). Haber, H. E., and Weldon, H. A., J. Math. Phys. 23, 1852. (1982). Grassmann variables Chandlin, D. J., Nuovo Cim. 4, 231 (1956). Berezin, F. A. (1966). Method of Second Quantization (Academic Press, New York). McLerran, L. D., Rev. Mod. Phys. 58, 1021 (1986).
- 49. 3 Interactions and diagrammatic techniques Unfortunately it is not possible to carry out the functional integration in closed form when the Lagrangian contains terms that are more than quadratic in the fields. The reader is invited to verify this. Thus, it is important to develop approximation techniques. An approximation that is expected to be useful when the interactions are weak is found by expand- ing the partition function in powers of the interaction. The convergence properties of these perturbation expansions have not been established with any degree of mathematical rigor, however. An alternative approach is to evaluate the partition function containing a given Lagrangian on a spacetime lattice using numerical Monte Carlo methods. This approach is described in Chapter 10. 3.1 Perturbation expansion Consider a single scalar field φ. Other, more physical, theories such as QED, QCD, the Glashow–Weinberg–Salam model, and effective nuclear models will be considered in later chapters. The reader must be prepared now to learn some basic techniques before tackling more complicated but physically relevant theories. The partition function is Z = N [dφ]eS (3.1) The action can be decomposed as S = S0 + SI (3.2) where S0 is at most quadratic in the field and SI, the part due to inter- actions, is of higher order. We may expand (3.1) in a power series in 33
- 50. 34 Interactions and diagrammatic techniques the part due to interaction, SI: Z = N [dφ]eS0 ∞ l=0 1 l! Sl I (3.3) Taking the logarithm on both sides we get ln Z = ln N [dφ]eS0 + ln 1 + ∞ l=1 1 l! [dφ]eS0 Sl I [dφ]eS0 = ln Z0 + ln ZI (3.4) This explicitly separates the interaction contributions from the ideal gas contribution, which we have evaluated already. The relevant quantity that we actually need to compute is Sl I0 = [dφ]eS0 Sl I [dφ]eS0 (3.5) which is the value of SI raised to an arbitrary positive integral power and averaged over the unperturbed ensemble, represented by S0. The normalization of the functional integration is now irrelevant, as it cancels in the expression (3.5). 3.2 Diagrammatic rules for λφ4 theory The task of actually evaluating (3.4) and (3.5) is significantly more dif- ficult than our compact notation would suggest. It is in fact useful to associate diagrams with the mathematical expressions in the expansion. Diagrams are a common language in particle physics, nuclear physics, sta- tistical physics and condensed matter physics and allow for the exchange of ideas and concepts among these different disciplines. Consider the lowest-order correction to ln Z0 in λφ4 theory. It is ln Z1 = −λ dτ d3x [dφ]eS0 φ4(x, τ) [dφ]eS0 (3.6) If we express φ(x, τ) as a Fourier series as in (2.30), and insert this into (3.6) we get ln Z1 = −λ dτ d3 x n1,...,n4 p1,...,p4 β2 V 2 × exp[i(p1 + · · · + p4) · x] exp [i(ωn1 + · · · + ωn4 )τ] A B (3.7)
- 51. 3.2 Diagrammatic rules for λφ4 theory 35 where A = l q dφ̃l(q) exp − 1 2 β2 (ωl 2 + q2 + m2 )φ̃l(q)φ̃−l(−q) × φ̃n1 (p1)φ̃n2 (p2)φ̃n3 (p3)φ̃n4 (p4) and B = l q dφ̃l(q) exp − 1 2 β2 (ω2 l + q2 + m2 )φ̃l(q)φ̃−l(−q) The integrations over x and τ yield a factor βV δn1+···+n4,0 δp1+···+p4,0 . The numerator of the whole expression for ln Z1 will be zero by symmet- ric integration unless n3 = −n1, p3 = −p1 and n4 = −n2, p4 = −p2, or the other two permutations thereof. This will satisfy the constraints of the Kronecker deltas and the integrals will factorize. The integrals in the numerator are canceled by those in the denominator except for the two corresponding to l = n1, q = p1 and l = n2, q = p2, and the other two permutations. Using ∞ −∞ dx x2e−ax2 /2 ∞ −∞ dx e−ax2/2 = 1 a (3.8) we obtain ln Z1 = −3λβV T n d3p (2π)3 D0(ωn, p) 2 (3.9) Here we have defined the propagator in frequency–momentum space as D0(ωn, p) = 1 ω2 n + p2 + m2 (3.10) The expression (3.9) can be associated with a diagram in the following way. Remember that we are calculating ln Z1 to first order in λ. With φ4(x, τ) we associate a cross with four arms (because of the fourth power of φ), with the vertex located at (x, τ): φ4 (x, τ): (x, τ) After expressing each field φ(x, τ) as a Fourier series we draw the figure (p2, ωn2 ) (p3, ωn3 ) (p1, ωn1 ) (p4, ωn4 )
- 52. 36 Interactions and diagrammatic techniques The directions of the arrows reflect the signs of the momenta and fre- quencies. By convention, we draw them pointing towards the vertex, but we could have chosen a convention in which they all point away. The functional integration vanishes unless n3 = −n1, p3 = −p1 and n4 = −n2, p4 = −p2, etc. Thus we connect the ends in pairs. There are three possible pairings. We then have ln Z1 = 3 (3.11) (p1, ωn1 )(p2, ωn2 ) With each closed loop we associate a factor T n d3p (2π)3 D0(ωn, p) With the vertex we associate a factor −λ (coming from LI = −λφ4) and a factor βδωin,ωout V δpin,pout → βδωin,ωout (2π)3 δ(pin − pout) Since the arguments of the frequency–momentum-conserving deltas are zero we simply get an overall factor βV . The factor V makes ln Z1 a properly extensive quantity. Pictorially, (3.11) corresponds precisely with (3.9). Next we look at order λ2 in ln ZI. From (3.4) it is ln Z2 = −1 2 [dφ]eS0 SI [dφ]eS0 2 + 1 2 [dφ]eS0 S2 I [dφ]eS0 (3.12) The first term in (3.12) is simply −1 2 (ln Z1)2 = −1 2 3 ⊗ 3 (3.13) The second term in (3.12) may be analyzed algebraically using func- tional integrals or it may be analyzed diagrammatically. Choosing the lat- ter approach, we draw two crosses corresponding to the factors φ4(x, τ) and φ4(x, τ) contained in 1 2 S2 I 0:
- 53. 3.2 Diagrammatic rules for λφ4 theory 37 We then pair the lines as before. Counting in the factor one-half and all the possible pairings, we obtain 1 2 × 3 ⊗ 3 + 6 × 6 × 2 2 + 4 × 3 × 2 2 (3.14) Combining (3.13) and (3.14), we observe that all the disconnected dia- grams cancel. We are thus left with ln Z2 = 36 + 12 (3.15) What is needed at some arbitrary order N in the perturbative expansion of ln ZI should now be clear. We formally expand in powers of λ: ln ZI = ∞ N=1 ln ZN (3.16) where ln ZN is proportional to λN . The “finite-temperature Feynman rules” at order N are: 1 Draw all connected diagrams. 2 Determine the combinatoric factor for each diagram. 3 Include a factor T n [d3p/(2π)3]D0(ωn, p) for each line. 4 Include a factor −λ for each vertex. 5 Include a factor (2π)3δ(pin − pout)βδωin,ωout for each vertex, correspond- ing to energy(frequency)–momentum conservation. There will be one factor β(2π)3δ(0) = βV left over. We now understand why D is called a propagator: it propagates a particle (or field) from one vertex to the next. We have illustrated the cancellation mechanism only at second order. However, it is clear why disconnected diagrams cancel. If, at some order, there existed a contribu- tion that was the product of K connected diagrams then this contribution would be proportional to V K. If we have done our job correctly, then ln ZI is an extensive quantity proportional to V and thus no such contribution can arise. The formal proof that in ln ZI the disconnected diagrams cancel goes as follows. From (3.3) and (3.5) we have ZI = ∞ l=0 1 l! Sl I0 (3.17) In general, Sl I0 can be written as a sum of terms, each of which is a product of connected parts (see (3.14)). Denoting a connected part by a
- 54. 38 Interactions and diagrammatic techniques subscript c, we may write Sl I0 = ∞ a1,a2,...=0 l! a1!a2!(2!)a2 a3!(3!)a3 · · · SIa1 0cS2 I a2 0c · · · δa1+2a2+3a3+··· ,l (3.18) The combinatoric factor takes into account indistinguishability, and the Kronecker delta picks out the contribution of order λl. Substituting (3.18) into (3.17) and summing over l eliminates the Kronecker delta: ZI = ∞ a1,a2,...=0 SIa1 0c a1! S2 I a2 0c a2!(2!)a2 · · · = exp ∞ n=1 1 n! Sn I 0c (3.19) Hence ln Z1 is simply the sum of the connected diagrams. As an example, let us apply these rules to the second diagram of (3.15). We get = βV (−λ2 )T n1 d3p1 (2π)3 · · · T n4 d3p4 (2π)3 × D0(ωn1 , p1) · · · D0(ωn4 , p4)(2π)3 δ(p1 + · · · + p4)βδn1+···+n4,0 (3.20) The evaluation of expressions such as (3.20) is not simple and will be discussed in detail in Section 3.4. The diagrammatic technique is a conve- nient means for keeping track of the combinatoric factors and the order of the coupling constant in a perturbative expansion of the partition func- tion. It circumvents much of the tedious algebra associated with the direct evaluation of functional integrals. 3.3 Propagators We shall define a finite-temperature propagator in position space by D(x1, τ1; x2, τ2) = φ(x1, τ1)φ(x2, τ2) (3.21) where the angle brackets denote an ensemble average. Owing to trans- lation invariance, D depends only on x1 − x2 and τ1 − τ2. The Fourier transform is, with x1 = x, x2 = 0, τ1 = τ, τ2 = 0, D(ωn, p) = β 0 dτ d3 x e−i(p·x+ωnτ) D(x, τ) = n1,n2 p1,p2 β V φ̃n1 (p1) φ̃n2 (p2) β 0 dτ d3 x × exp[i(p1 − p) · x] exp[i(ωn1 − ωn)τ] (3.22)
- 55. Exploring the Variety of Random Documents with Different Content
- 56. thirty-eight—a young thirty-eight. There’s an old thirty-eight which applies to greedy school-teachers, gangrenous woman government-clerks, fading hard-hearted stenographers, over- righteous woman doctors; to all whose virtue is ever indecently on guard. But there’s a glory-tinted sun-kissed young thirty-eight which applies to sensitive high-strung generously-emotional women like Bella Lot. She had smooth hands with supple tapering fingers, an irregular expressive-lipped mouth like a pimpernel-bloom, firm slim feet and the quivering suggestive white knees of a wood- nymph. From any angle-of-view can she be blamed for hating to take that equipment away from the city-de-luxe which was its so proper setting and hiding it in the sage-brush? Furthermore Bella had a lover in Sodom. It is beyond a sane effort of the imagination that she could have loved that unpleasing old man Lot. The best and worst that can be said of him is that he was a fit addition to the company of the old Patriarchs who were for the most part an exceeding craven crew. The martyrs, the sages and especially the prophets had their splendors. But the lean old patriarchs—The sporting blood of all of them—in the sense of merest simplest courage—from Adam down, would hardly aggregate one drop. There are any number of reasons—as many as Bella had charms—to account for Lot’s having married her. But what she could have seen in him to make her wish or even willing to be married to him is a deep mystery to me. It may have been his family. I believe Bella lacked family: she was just a person. And was he not nephew to Abraham? But even being niece-in-law to Abraham himself seems insufficient compensation for being Lot’s Wife. The Lots had two young daughters, one fifteen and one seventeen, it might be. I do not know their names—call them Ethel and Agnes. But they were of a recalcitrant temper and absorbed in their own
- 57. racy pastimes among the younger youth of Sodom and they had no need of their mother. Besides, they ‘took after’ their father. So Bella was fain to turn outward in search of nurturing matter whereon to feed her humanness. Had it been expected of her to play fair with the patriarch she would have played fair. But it was not expected of her by anyone in Sodom—far from it, and least of all by the patriarch. She was eight-and-thirty, and Lot—he was doubtless eight or nine hundred years old, after the surprising long-lived fashion of the period. So Bella found a lover ready and awaiting her. She would have found a lover in the circumstances even without caring to. But she quite cared to, I think. Everything points that way, and when one remembers that good old man her husband one can not censure her but only pity her. Be it as it may she had one—one as real as anything could be in that town of sparkling froth. Of the lover’s identity—little is known, as the historians say. My fancy as I filed my fingernails failed me on the point. Suffice it to state that ever and anon as time passed in Sodom the gray-green eyes of Bella were gazed into with fondness, affection, adoration and desire: the white eyelids of Bella had showers of light kisses bestowed on them, soft-falling as rose-petals shaken loose in summer winds: the tapering white hands of Bella were caressed and caressing with the oddly intense tenderness of physical love: the pale red hair of Bella was ruffled and fluffed and disarrayed by the fingers of love: the red-pimpernel mouth of Bella was touched, bruised, clung to by the lips of love: the svelte whiteness and nymph-knees of Bella glowed as she broached love’s arms:—and all went much merrier than marriage bells. In short, Bella paid herself with usury for the deadliness of being Lot’s Wife. And there we have the crux of Bella’s dread of leaving Sodom and its tempered sweetness for the arid sage-brush hills and the
- 58. respectively cold and hectic companionship of the good old patriarch and the recalcitrant daughters. It can not be claimed for Bella that any white poetic fires gleamed across her soul, that any limning beauty shone palely from within her. The air of Sodom was not conducive to suchlike matters and Bella was no finer than her breeding and generation. But she was gentle and wistful and kind of heart. She was lovely to look at and ingenuously lovable in her clinging affection and disarming naturalness. She was all one could want to imagine in the word charming. Came the night set for destruction and the Lot family fled according to schedule. They fled away in the early damps of an autumn evening through the outer city gates and along a rough road faintly lit by a dying moon. They had three separate reasons for fleeing. Lot fled because he was a patriarch and was given to doing craven Old-Testamentish things of that sort: Bella fled because she was Lot’s Wife and obliged to act out the rôle: and Ethel and Agnes fled because they had true patriarchal blood in their veins and had therefore no marked inclination to remain in Sodom to be annihilated—‘safety first’ was one of their watchwords. They fled in the van. Lot came after them, being less swift of foot. Bella lagged behind. She didn’t want to go. Every way she looked at it she didn’t want to go. She hated that flight for a thousand reasons. The ghastly moon shed a terror on her with its dim rays. The ground was hard and rutted with frosty mud and bruised her slender feet through her white buckskin sandals. She wore a loose ninon gown of white silk and linen with a gold girdle around her narrow loins and a gold clasp at the left shoulder. Binding her long hair, so palely red in the moon, was a white-and- gold fillet. In one hand she carried a gold-and-enamel link bracelet, a gift but that afternoon from the lover. Suddenly she stopped and
- 59. cried to herself, ‘I’m too lovely for this fate—I’m too lovely and beloved—the cruelty of God—: I’ll not go on!’ She thought of the gleams and colorings of Sodom. She quickly reckoned the cost and decided to pay it. She was a rare good sport, and a quaint. She looked back at the doomed city blazing in brimstone—‘But his wife looked back from behind him, and she became a pillar of salt.’— As I put away my chamois-skin buffer and glass paste-jar through my mind floated the pensive burden of a by-gone French song— ‘Oh, the poor, oh, the poor, oh, the poor—dear—girl’— She must have made a beautiful statue, all in glistening salt. I wish I had a glistening little salty replica of it to set on my desk: a so unusual, a so dainty conceit, Lot’s Wife!
- 60. W My echoing footsteps To-morrow HILE I live so still in this life-space, while I muse and meditate and analyze everything I touch, while I walk, while I work, while I change from one plain frock to the other: in quiet hours roiled tumbling storms of vicarious unhopeful Passion whirl, whirl in me: Passion of Soul, Passion of Mind, Passion of living, Passion of this mixed world: in terror, in wild unease, in reasonless mournful joy. I never knew real Passion, Passion-meanings, till I reached thirty. It is now I’m at life’s storm-center, youth’s climax, the high-pulsed orgasmic moment of being alive. At twenty the woman’s chrysalis soul and aching pulses awaken in crude chaste Spring-cold beauty. At forty her fires either have subsided to dim-glowing coals or leaped to too-positive, too- searing, too-obvious flames—her bones and the filigrees of her spirit may be alike dry, brittle-ish. But at thirty her Spring has but changed to midsummer. Poesy still waits upon her Passions. My Spring has changed, bloomed, burst to midsummer. Soft electrical heat-currents of being swing and sweep around me. They touch me and enter my veins. But the liquid essences of youth still quell and compass them. I am at youth’s climax—a half- sullen, half-smouldering youth which still is youth.
- 61. My rose of life is fragrant and aglow. Its sweet pink petals are uncurled and conscious in the wavering light. Winds flutter and stir and rumple and twist those petals— To-day is a To-morrow of countless unrests. Large and little Passions beat at me all the blue-and-copper day. I walked my floor with irregular lagging steps. I felt menacing, dangerous to myself, dynamic as nitro-glycerine: and smoothly drearily sane as a bar of white soap. I stood at my window and looked long at the circling range of mountains which skirt this Butte. Nothing else I have looked at, of sea or plain or hill, affected me like that chain of barren peaks. They are arid splendor and pale purple witchery and grief and lasting sadness and deathlike beauty and woe and wonder. Their color quietly stormed my eyes and blurred them with tears. It was a mood in which any color or gleam or thought or strain of music or note of sad world-laughter or any un-sane loveliness of poetry could enchant or flay or transport me to my frayed last nerve. There is terror in facing death on battlefields, on sinking ships, in black ice-floes, in blazing buildings. But to me no death, for I fear no death, could be so dreadfully pregnant with in-turning woe and frenzy and all intolerable feeling as facing starkly my futile life. My life is a vast stone bastile of many little Rooms in which I am a prisoner. I am locked there in solitude on bread and water and let to roam in it at will. And each Room is tenanted by invisible garbled furies and dubious ecstasies. I run with echoing footsteps from Room to Room to escape them: but each Room is more unhabitable than the last. There are scores of little Rooms, each with its ghosts, each different. In one Room silent voices in the air accuse my tired Spirit of wanton vacillations and barren lack of purpose and utter waste,
- 62. waste, waste of itself. And they threaten death and destruction. I know that accusation and I hate it: I hate it the more for that it’s wholly just. To escape it I run from that Room along a dim passage into another one. In it unseen fingers clutch my Heart. In their touch also is an accusation: of selfishness and waste and want of something to beat for: and in their touch is the savor of wild wishes and human longings and passionate prayers for something warm and simple and real to rest against: and in their pressing clutching turbulent touch is a tormenting half-promise, chance-promise, no- promise: and the hovering inevitable threat of death and destruction. That too I know and hate and half-love: and I can’t bear it. So I run out of that Room along a passage and into another. I hear my footsteps echoing as I run. —as a child when I ran in the early night through a dark leaf-lined tunnel-like driveway the sound of my own flying footsteps on the hardened gravel was the only thing that frightened me. I quite believed there were bears in the brushwood on either side, but fear of them never struck to the core of my child-being like the unknown thing in my echoing steps. And it is fear I feel now from the ghost-sound of my ghost-footsteps running, running away from the little Rooms. It is realer to me now than were my child footsteps to my child-self long ago: it is more definite than my hand which writes this: it is hideous— Out of a dim passage I run into another little Room. In it some gray filmy threads, like strands of loose cobwebs caught on ceilings, float about. They sweep gently against my cheeks and hands and neck, and cling and twine and lightly hold with the half-felt feeling peculiar to bits of cobwebs on the skin. And it torments my woman- flesh with calefacient thrills fierce and goading and sweet. There also is the accusation, now against my Body; for tissues and strength wasted: for useless fires meant to warm human seeds to
- 63. life, meant to make me fruitful, meant to make me bear dear race- burdens: accusation for the cosmic waste of hot objectless desire, for the subtle guilt of a Lesbian tendency, for an unleashed over- positive sex-fancy. With it too is the lowering promise of death and destruction. It also is just. But out of my borne-along helplessness in it comes no culpable emotion because of cobweb thrills and their arraignment but only a wearing wearying despair. I rush out of that Room in shrugging impatience, with only scorn for a threat of death, for a threat of destruction—but with a wild fear of my own flying steps. I hurry and hurry on from door to door: but it’s no good. In some other Room my brain is anathematized from frowning walls as an impish demoniac power which I use with no good intent and therefore with bad intent: and again I shrink and run away. In another Room are all the lies I have ever told: I have told legions—my own peculiar lies, gentler on me than truths: they dart around me in the Room like black heavy-winged moths, clouds of them fluttering at my forehead. They drive me out shivering. In another Room four times when I was a not-good-sport confront me in a row like pictures and sting me and make me hide my eyes: I’d rather be a leper, a beast, a maniac than a not-good-sport (for my own precious reasons)—and I rush away again. In some other Room— —the same galling torment in all the Rooms. Wherever I run with the echo-echo of steps there are Accusing voices and half-formed Prayer and uncertain Yearning and violent yet dumb and inexpectant Protest and the unfailing Threat of death and destruction: not earth-death but universe-death: death and death and death everywhere coming on and on: myself knowing the just note in it all and from it grown numb with some cold and restless terror. Also I know no door I run through with my panic-feet will
- 64. ever set me free of the bastile except a death door: the earthly death of this tired life— But it’s from this maelstrom that the flashing burning sparkling mad magic of being alive leaps out brilliant and barbarous—and throbbing and splendid and sweet. A merely human hunger comes back on me. Then I want all I ever wanted with a hundredfold more voltage of wanting than I have ever yet known. I am all unhopeful, all unpeaceful, all a desperate Languor and a tragic Futileness: I am an unspeakably untoward thing. And already I have been seared and scarred trivially from standing foolishly near some foolish human melting-pots. No matter for any of it. I want to plunge headlong into life—not imitation life which is all I’ve yet known, but honest worldly life at its biggest and humanest and cruelest and damnedest: to be blistered and scorched by it if it be so ordered—so that only it’s realness—from the outside of my skin to the deeps of my spirit. It is not happiness I want—nothing like it: its like never existed since this world began. I want to feel one big hot red bloody Kiss-of-Life placed square and strong on my mouth and shot straight into me to the back wall of my Heart. I write this book for my own reading. It is my postulate to myself. As I read it it makes me clench my teeth savagely: and coldly tranquilly close my eyelids: it makes me love and loathe Me, Soul and bones. Clench and close as I will the winds flutter and stir and crumple and twist my petals as they will:—as I sit here tiredly, tiredly sane.
- 65. T A comfortably vicious person To-morrow HE blue-and-copper of yesterday is dead and buried this To- morrow in a maroon twilight. I this moment saw darkly from my window the somber hills in their heavy spell of pale-purple and grief and splendor and sadness and beauty and wonder and woe. But their color brings no tears to my wicked gray eyes. The passion-edged mood is burnt out. Gone, gone, gone. I listlessly change into the other black dress for listless dinnertime and all my thought is that my abdomen is beautifully flat and that I must purchase a new petticoat. I rub a little rouge on my pale mouth and I idlingly recall a clever and filthy story I once heard. I laugh languidly at it and feel myself a comfortably vicious person. I pronounce a damn on the familiar ache in my beloved left foot and turn away from myself. I stick out the tip of my forked-feeling tongue at the bastard clock on the stairs. I note the hour on it with a fainness in my spirit- gizzard to dedicate Me from that time forth to a big blue god of Nastiness: Nastiness so restful, humorous, appetizing, reckless, sure-of-itself.
- 66. —these hellish To-morrows creeping in their petty pace: they bring in weak-kneed niceness, and they bring in doubts, and they bring in meditation and imagery and all-around humanness, till I’m a mere heavy-heeled dubious complicated jade.
- 67. I In my black dress and my still room To-morrow HAVE fits of Laughter all to myself. The world is full of funny things. All to myself I Laugh at them. I lounge at my desk in the small night hours, and I finger a pencil or a box or a rubber or a knife and rest my chin on my hand, and sit on my right foot, and Laugh intermittently at this or that. Ha! ha! ha! I say inwardly: with all my Heart: relishingly. I laugh at the thought of a mouse I once encountered lying dead— so neat, so virtuous—though soft and o’er-long dead—with its tail folded around it—in a porcelain tea-pot: a strong inimical anomaly to all who viewed it. It had a look of a saint in effigy in a whited sepulcher. Looked at as a mouse it seemed out of place. Looked at as a saint it was perfect. I Laugh at the recollection of a lady I once met who had thick black furry eyebrows incongruous to her face, which she took off at night and laid on her bureau. They were at once ‘detached’ and detachable: itself a subtle phenomenon. She referred to her mind as her ‘intellects’ and talked with a quaint bogus learnedness, and in remarkable grammar, of the Swedenborgian doctrines. Looked at as a person she was inadequate. Looked at as a conundrum she was gifted and profound. I Laugh at that extraordinary tailor in the Mother Goose rhyme— him ‘whose name was Stout,’ who cut off the petticoats of the little
- 68. old woman ‘round about,’ herself having recklessly fallen asleep on the public highway. The tale leaves me the impression that such were the straitly economic ideas of the tailor that he obtained all his cloth by wandering about with his shears until he happened upon persons slumbering thus publicly and vulnerably. Looked at in any light that tailor is ever surprising, ever original, ever rarely delectable. I Laugh at William Jennings Bryan. How William Jennings Bryan may look to the country and world-at- large I have never much considered. It is all in the angle of view: St. Simeon Stilites may seem rousingly funny to some: Old King Cole may have been a frosty dullard to those who knew him best. To me William Jennings Bryan means bits of my relishingest brand of gay mournful Laughter. The ensemble and detail of William Jennings Bryan and his career as a public man, viewed impersonally—as one looks at the moon— is something hectic as hell’s-bells. I remember William Jennings Bryan when his star first rose. It was before Theodore Roosevelt was more than a name: before the battleship Maine was sunk at Havana: before Lanky Bob wrested the heavyweight title from Gentleman Jim at Carson: before aëroplanes were and automobiles were more than rare thin- wheeled restless buggies: before the song ‘My Gal She’s a High- born Lady’ had yet waned: before one Carrie Nation had hewn her way to fame with a hatchet. I was a short-skirted little girl devouringly reading and observing everything, and I took note of all those. So I took note of William Jennings Bryan nominated for president by the Democratic convention in eighteen-ninety-six. The zealous Democratic newspapers referred to him, though he was then thirty-six, as the Boy Orator of the Platte. Looked at as a
- 69. grown man, advocating free coinage of silver at sixteen-to-one—a daring dashing Democrat, he was a plausible thing and even romantic. Looked at as a Boy Orator he turned at once into a bald and aged lad oddly flavored with an essence of Dare-devil Dick, of the boy on the burning deck, of a kind of political Fauntleroy madly matured. Long years later with the top of his hair and his waistline buried deep in his past he became Secretary of State: and at the same time a Chautauqua Circuit lecturer—entertaining placid satisfied audiences alternately with a troupe of Swiss Yodlers. Of all things, yodlers. Politics makes strange bedfellows and always did. But never before has the American Department of State combined and vied with the yodler’s art to entertain and instruct. Looked at as a monologist he might pass if sufficiently interpolated with ah-le-ee! and ah-le-o-o! Looked at as Secretary of State he is grilling and gruelling to the senses: a frightful figure quite surpassing a mouse softly dead in a tea-pot, a pair of detachable fuzzy Swedenborg- addicted eyebrows, a presumptuously economical tailor. And he entertained the foreign ministers at a state dinner, did this unusual man, and he gave them to drink—what but grape-juice, grape-juice in its virginity. Plain water might have seemed the crystalline expression of a rigid puritanic spirit. Budweiser Beer, bitter and bourgeois, might have been possible though surprising. But grape-juice, served to seasoned Latin Titles and Graybeards and Gold-Braid, long tamely familiar with the Widow Clicquot: that in truth seems, after all the years, boyishly oratorical, wildly and darkly Nebraskan. Looked at as an appetizing wash for a children’s white-collared and pink-sashed party, or for anybody on a summer afternoon, grape-juice is satisfactory. In the careless hands of William Jennings Bryan with his soul so unscrupulously at peace,
- 70. the virgin grape-juice becomes a vitriolic thing: a defluent purple river crushing one’s helpless spirit among its rocks and rapids. —a terrible American, William Jennings Bryan. He is for ‘peace at any price.’ There were some, long and long ago, who suffered and endured one starveling winter in camp at Valley Forge that William Jennings Bryan might wax Nebraskanly fat: and he is valiantly for peace: at any price— For that my Laughter is tinged with fulfilling hatred. Rich hot-livered Laughter must have in it essential love or hatred. To William Jennings Bryan everything he has done in his political career must seem all right. It is all right, undoubtedly. Just that. —that Silver-tongued Boy Orator those Yodlers that Peerless Leader that Grape-juice— They come breaking into my melancholy night-hours with an odd high-seasoned abruptness. I wonder what God thinks of him. It might be God thinks well of him. But I—in my black dress and my still room—I say inwardly and willy-nilly, and with all my Heart and relishingly: Ha! ha! ha!
- 71. O Their little shoes To-morrow FTEN in windy autumn nights I lie awake in my shadowy bed and think of the children, the Drab-eyed thousands of children in this America who work in coal mines and factories. Whenever I’m wakeful and the night is windy and my room is dark and I lie in aloneness—a long aloneness: centuries—then shadows come from far-off world-wildnesses and float and flutter dimly unhappy around my bed. They tell me tales of shame and tame petty hopelessness and trifling despair. And the one that comes oftenest is the one that tells of those Drab- Eyed children distances from here, but very immediate, who work in coal mines and factories. I read about them in magazines and newspapers, but they aren’t then one one-hundredth so real as when their shadow floats as close to me in the windy autumn night. Once in Pennsylvania I saw a group of children, very Drab in the Eyes and very thin in the necks and legs, who worked in a mill. Their look made its imprint in my memory and more in my flesh. And it comes back as if it were the only thing that mattered as I lie wakeful in the windy night. The children—unconscious and smiling their small decayed smiles— they are living and being crushed between greed and need as between two murderous millstones. Their frail flesh and their little brittle bones, their voices and their pinched insides, the sweet
- 72. vague childish looks which belong in their faces are squeezed and crunched by two millstones—squeezed, squeezed till their scrawny fledgeling bodies are dry, breathless, and are gasping, strangling, striving frightfully for life: and still are slowly, all too slowly, dying between two millstones. If it were their own greed or their own need—but it’s the greed of fat people and the need of their own warped gaunt parents. Betwixt the two the children meet homelike hideous ruin. Placidly they are cheated and blighted and blasted, placidly and with the utmost domesticness. The most darkling-luminous thing about the Drab-Eyed children is that they never weep. They talk among themselves and smile their little dreadful decayed smiles, but they don’t weep. When they walk it’s with a middle-aged gait: when they eat their noontime food it’s as grown people do, with half-conscious economic and gastronomic consideration. They count their Tuesdays and Wednesdays with calculation as work-days, which should be childishly wind-sweptly free. Which is all of less weight than the heavy fact that they never weep. They reckon themselves fairly fortunate with their bits of silver in yellow envelopes every Saturday. They are permitted to keep a bit of it, each child a bit for herself or himself, so that on Sunday afternoons they lose themselves for precious hours watching Charlie Chaplin. Many pink-faced inconsequent children whose parents nurture them and guard them and eternally misunderstand them are less worldlily lucky. But the pink-faced children often weep—loudly, foolishly like puppies and snarling furry cubs—and wet sweet salt tears of proper childishness are round and bright on their cheeks and lashes. It’s a sun-washed blestness for them: they’re impelled and allowed to weep. But the Drab Eyes shed no tears—they know no reason why they should. There’s no impulse
- 73. for soft liquid grief in the murderous philosophy of two grinding millstones. And there’s no time—the lives of the work-children move on fast. Their very shoes are ground between the millstones. —their little shoes are heartbreaking. The millstones grind many things along with little-little shoes of children: germs of potent splendid humanness that might grow bigly American in heroic ways or in sane round honesty: germs that might grow into brave barbaric beauty or warm wistful sweetness: germs that would grow into lips blooming tender and fragrant as jonquils or into minds swimming with lyrics:—what is strongly lasting and glorified in the forlorn divine human thing—crumpled—twisted forever when millstones grind children’s little poor shoes— The young Drab Eyes are endlessly betrayed: their very color thieved. There’s no reason why they should weep. But there’s a far-blown sound as if ten thousand bad and good worldly eyes were weeping in their stead: with a note in it careless, compassionate and jadedly menacing. I seem to hear it in the wakeful windy night. And I hear no world- music pouring out of small throats of work-children shrill with woe- and-joy. The sound they make is a dumb sound, for they never weep: a ghost-wail of partly-dead children borne lowly across this mixed world on a stale hellish breeze.
- 74. W The sleep of the dead To-morrow HEN I’m dead I want to Rest awhile in my grave: for I’m Tired, Tired always. My Soul must go on as it has gone on up to now. It has a long way to go, and it has come a long way. My Soul first started on its journey somewhere in Asia before the dawn of this civilization. And it has gone on since through the centuries and through strange phases of Body, terrors of flesh and blood, suffering long. But it has gone someway on, each space of the journey taking it nearer to the journey’s-End. It is the dim-felt memory of those journeys that heaps the Tiredness on me now. Not only is my spirit Tired. Through my spirit my hands are Tired: my knees are Tired: my drooping shoulders: my thin feet: my sensitive backbone. When I lift my hand in the sunshine the weight of the yellow honeyed air bears down and down on it because I’m so Tired. When I start to walk on stone pavements the ache of them is in my feet before I set a foot on them because I’m so Tired. The pulse in my veins Tires my blood as it beats. My low voice, though I speak but rarely—it Tires my throat. My breath Tires my chest. The weight of my hair Tires my forehead and temples. My plain frocks Tire my Body to wear. My swift trenchant thoughts Tire my Mind.
- 75. It is not the Tiredness of effort though I strive to the limits of my strength every day. It is not pain, Restful pain. It is Tired Tiredness. So when I’m dead I want to Rest awhile in my grave. It would Rest me. In the Episcopal Church they use a ritual of poetic beauty, full of Restful things. One of them is the sleep of the dead. The crucified Nazarene slept three days. But all others of us when we go down into our graves are to sleep until a Judgment Day. ‘Judgment Day’ is preposterous and evilly crude: there’s no judgment till each can judge himself simply and cruelly in the morning light. But the sleep of the dead— —the sleep of the dead. Its sound by itself without the thought is Restful— And the thought is Restful. I imagine me wrapped in a shroud of soft thin wool cloth of a pale color, laid in a plain wood coffin: and my eyelids are closed, and my Tired feet are dead feet, and my hands are folded on my breast. And the coffin is nine feet down in the ground and the earth covers it. Upon that some green sod: and above, the ancient blue deep sheltering sky: and the clouds and the winds and the suns and moons, and the days and nights and circling horizons—those above my grave. And my Body laid at its length, eyes closed, hands folded, down there Resting: my Soul not yet gone but laid beside my Body in the coffin Resting. —might we lie like that—Resting, Resting, for weeks, months, ages— Year after long year, Resting.
- 76. I Stickily mad To-morrow T is damn-the-Smell-of-Turpentine! Here I happen on a damn in me which is not desultory but bloodily strong and alive and alone. The wood in my blue-white room has been newly painted. For a day and a night I intermittently encounter and go to bed in a spirit of Turpentine. It bears a cruel obscure abortive message to my nerves. I lie wakeful in the dark and try to reason out a logicalness or poetry in a thing so artfully pestilential. But I am hysterically lost in it and my heart beats hysterically in it. I remember the inexpressible ingenuity of man: of white man as against bone-brained savage races. Every invented usefulness feels like divine witchcraft. A pen and a bottle of perfume and a door- knob and a granite kettle and an electric light: I have the use of each since white man is so ingenious. Were I a red Indian I should have only the awkward barbarous stupid tools my race had used a thousand years. I contrast the two as I lie wakeful, with a sense of richness and of detailed repletion and of material blestness. But at once comes the Smell of Turpentine and announces itself something outside that and different, something stronger, something masterfuler than ingenuity and savagery together. It tortures my nerves: it burns my eyes: it lames my flesh: it jerks
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