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The function ''H'' is understood to be a real-valued function on the space of states <math>\{X_1,X_2,\cdots\}</math>, while <math>\beta</math> is a real-valued free parameter (conventionally, the [[inverse temperature]]). The sum over the <math>x_i</math> is understood to be a sum over all possible values that each of the random variables <math>X_i</math> may take. Thus, the sum is to be replaced by an [[integral]] when the <math>X_i</math> are continuous, rather than discrete. Thus, one writes
:<math>Z(\beta) = \int \exp \left(-\beta H(x_1,x_2,\dots) \right) \, dx_1 \, dx_2 \cdots</math>
for the case of continuously-varying <math>X_i</math>.
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When ''H'' is an [[observable]], such as a finite-dimensional [[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) = \
When ''H'' is infinite-dimensional, then, for the above notation to be valid, the argument must be [[trace class]], that is, of a form such that the summation exists and is bounded.
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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
:<math>Z = \int \mathcal{D} \
Such is the case for the [[partition function in quantum field theory]].
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