Page 29 - Nature Of Space And Time

period
b

t 2 − t 1 = −i ; 2 = 1

X
Z = < n | exp(− H) | n >
Z
^
= D[ ]exp(−iI[ ])

sums over all eld con gurations n. This means that e ectively one is doing the path
integral over all elds on a spacetime that is identi ed periodically in the imaginary
time direction with period . Thus the partition function for the eld at temperature
T is given by a path integral over all elds on a Euclidean spacetime. This spacetime is

periodic in the imaginary time direction with period = T −1 .
If one does the path integral in
at spacetime identi ed with period in the imaginary
time direction one gets the usual result for the partition function of black body radiation.
However, as we have just seen, the Euclidean- Schwarzschild solution is also periodic in

imaginary time with period 2 . This means that elds on the Schwarzschild background

will behave as if they were in a thermal state with temperature .
2
The periodicity in imaginary time explained why the messy calculation of frequency
mixing led to radiation that was exactly thermal. However, this derivation avoided the
problem of the very high frequencies that take part in the frequency mixing approach.
It can also be applied when there are interactions between the quantum elds on the
background. The fact that the path integral is on a periodic background implies that all

physical quantities like expectation values will be thermal. This would have been very
di cult to establish in the frequency mixing approach.
One can extend these interactions to include interactions with the gravitational eld

itself. One starts with a background metric g 0 such as the Euclidean-Schwarzschild metric
that is a solution of the classical eld equations. One can then expand the action I in a
power series in the perturbations g about g 0 .

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