Partition function (mathematics): Difference between revisions

Revision as of 19:10, 10 July 2012

Revision as of 19:20, 10 July 2012

Line 16:

for the case of continuously-varying <math>X_i</math>.

When ''H'' is an [[observable]], such as ina quantumfinite-dimensional mechanics[[matrix (mathematics)|matrix]] or an infinite-dimensional [[Hilbert space]] [[operator (mathematics)|operator]] or element of a [[C-star algebra]], it is common to express the summation as a [[trace (linear algebra)|trace]], so that

:<math>Z(\beta) = \mbox{tr}\left(\exp\left(-\beta H\right)\right)</math>

ForWhen ''H'' is infinite-dimensional, then, for the above notation to be valid, the observableargument must be [[trace class]], that is, of a form such that the summation exists and is bounded.

The number of variables <math>X_i</math> need not be [[countable]], in which case the sums are to be replaced by [[functional integral]]s. Although there are many notations for functional integrals, a common one would be