Partition function (quantum field theory) について

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Partition function (quantum field theory) ：ウィキペディア英語版

Partition function (quantum field theory)

In quantum field theory, the partition function Z() is the generating functional of correlation function. It is usually expressed by something like the following functional integral:
:

) = \int \mathcal\phi e^

where S is the action functional.
The partition function in quantum field theory is a special case of the mathematical partition function, and is related to the statistical partition function in statistical mechanics. The primary difference is that the countable collection of random variables seen in the definition of such simpler partition functions has been replaced by an uncountable set, thus necessitating the use of functional integrals over a field

\phi

.
== Uses ==

The n-point correlation functions

G_n

can be expressed using the path integral formalism as
:

G_n(x_1,...,x_n)\equiv \langle \Omega | T \ | \Omega \rangle= \frac

where the left-hand side is the time-ordered product used to calculate S-matrix elements. The

\mathcal \phi

on the right-hand side means integrate over all possible classical field configurations

\phi(x)

with a phase given by the classical action

S()

evaluated in that field configuration.〔http://www.amazon.com/Quantum-Field-Theory-Standard-Model/dp/1107034736, Ch.14〕
The generating functional

Z()

can be used to calculate the above path integrals using an auxiliary function

J

(called ''current'' in this context).
From the definition (in a 4D context)
:

) = \int \mathcal\phi \exp \left\ \left \right\}

it can be seen using functional derivatives that the n-point correlation functions

G_n(x_1,...,,x_n)

are given by
:

G_n(x_1,...,x_n) = (-i \hbar)^n \frac \left. \frac \right|_

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