Feynman Path Integral Formulation

5.6 Complex Periodic Time 165
with ˙φ = i∂φ/∂τ, N a normalization factor, and complex time t = −iτ. The functional
integration [dφ] only contain those paths which are periodic in the complex
time τ, i.e. φ(τ = β)=φ(τ) (for Fermions they need to be anti-periodic). Integrating
over the momenta π gives
tre −β H = N ′ ∫
periodic
{ ∫ β ∫
[dφ]exp dτ 1¯h
0
}
d 3 xL (φ,i ˙φ)
(5.111)
where N ′ (β) is a second β-dependent constant that comes from the Gaussian π
integration. The latter has to be defined, as in the original non-relativistic path integral,
by introducing a lattice spacing and doing the π integration carefully. Note
that the finite temperature formalism has automatically achieved a Wick rotation to
the Euclidean theory, and the Feynman iε prescription is no longer needed. Furthermore,
because of the periodicity in complex time, all energy integrals are converted
into finite frequency sums [see, for example, Abrikosov, Gorkov and Dzyaloshinski,
1963; Fetter and Walecka, 1971].
In the gravitational case, a similar functional integral needs to be evaluated at
finite temperatures. Formally it is given by
Z(β) =tre −β H = N ′ ∫
periodic
{ ∫ β
[dg μν ]exp 1¯h
0
∫
dτ
d 3 xL ( g μν ,iġ μν
) } .
(5.112)
Here the path integral is over all gravitational fields g μν which are periodic in imaginary
time τ, with period β; only in the semi classical limit these are restricted to
the saddle points of L , corresponding to suitable solutions of the classical field
equation of general relativity. But the introduction of a finite temperature β does
not of course alleviate the short distance problem of ultraviolet divergences, which
still remains and needs to somehow be addressed in order to obtain finite quantum
corrections to Z(β).
The periodicity in imaginary time of the Schwarzschild solution has in fact been
used to provide an alternative path integral derivation of black hole radiance (Hartle
and Hawking, 1976). There the amplitude for a black hole to emit a scalar particle in
a particular mode is expressed as a sum over paths connecting the future singularity
and infinity. By analytic continuation in the complexified Schwarzschild space this
amplitude is then related to that for a particle to propagate from the past singularity
to infinity and hence, by time reversal, to the amplitude for the black hole to absorb
a particle in the same mode. The form of the connection between the emission and
absorption probabilities then shows that the black hole will emit scalar particles with
a thermal spectrum characterized by the temperature of Eq. (5.65).
The thermodynamic analogy, applied to a black hole with a temperature T =
β −1 = 1/8πM with (average) energy E = M and entropy S = 4πGM 2 , gives a free
energy F ≡ E − TS= 1 2M and therefore a partition function Z (for a single state!)
Z = ∑ e −βE n
= e −βF → e − β2
16πG . (5.113)
n