Partition function (mathematics): Difference between revisions

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==Gaussian integrals==

For the case where ''H'' can be written as a [[quadratic form]] involving a [[differential operator]], that is, as

:<math>H = \frac{1}{2} \sum_n x_n D x_n</math>

then thepartition function can be understood to be a sum or [[Gaussian integral|integral]] over Gaussians. The correlation function <math>C(x_j,x_k)</math> can be understood to be the [[Green's function]] for the differential operator (and generally giving rise to [[Fredholm theory]]). In the quantum field theory setting, such functions are referred to as [[propagator]]s; higher order correlators are called n-point functions; working with them defines the [[effective action]] of a theory.

When the random variables are anti-commuting [[Grassman variable]]s, then the partition function can be expressed as a determinant of the operator ''D''. This is done by writing it as a [[Grassmann integral]] or [[Berezin integral]].

==General properties==