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==The sign problem in field theory==
{{efn|Sources for this section include Chandrasekharan & Wiese (1999)<ref name='Wiese-cluster'/> and Kieu & Griffin (1994)<ref name='Kieu'/>, in addition to those cited.}}In a field theory approach to multi-particle systems, the fermion density is controlled by the value of the fermion [[chemical potential]] <math>\mu</math>. One evaluates the [[Partition function (quantum field theory)|partition function]] <math>Z</math> by summing over all classical field configurations, weighted by <math>\exp(-S)</math> where <math>S</math> is the [[Action (physics)|action]] of the configuration. The sum over fermion fields can be performed analytically, and one is left with a sum over the [[boson]]ic fields <math>\sigma</math> (which may have been originally part of the theory, or have been produced by a [[Hubbard–Stratonovich transformation]] to make the fermion action quadratic)
:<math>Z = \int D \sigma \; \rho[\sigma]</math>
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