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==Normalization==
The values taken by <math>\beta</math> depend on the [[mathematical space]] over which the random field varies. Thus, real-valued random fields take values on a [[simplex]]: this is the geometrical way of saying that the sum of probabilities must total to one. For quantum mechanics, the random variables range over [[complex projective space]] (or complex-valued [[projective Hilbert space]]), where the random variables are interpreted as [[probability amplitude]]s. The emphasis here is on the word ''projective'', as the amplitudes are still normalized to one. The normalization for the potential function is the [[Jacobian matrix and determinant|Jacobian]] for the appropriate mathematical space: it is 1 for ordinary probabilities, and ''i'' for Hilbert space; thus, in [[quantum field theory]], one sees <math>it H</math> in the exponential, rather than <math>\beta H</math>. The partition function is very heavily exploited in the [[path integral formulation]] of quantum field theory, to great effect. The theory there is very nearly identical to that presented here, aside from this difference, and the fact that it is usually formulated on four-dimensional space-time, rather than in a general way.
==Expectation values==
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