The cluster expansion in statistical mechanics
 David Brydges,
 Paul Federbush
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The GlimmJaffeSpencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by finite range potentials. The HamiltonianH _{0}+V need be stable in the extended sense thatH _{0}+4V+BN≧0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. We define a class of interacting boson and fermion particle theories with a matterlike potential, 1/r suitably truncated at large distance. This system would collapse in the absence of the exclusion principle—the potential is unstable—but the Hamiltonian is stable. This provides an example of a system for which our method proves existence of the infinite volume limit, that is not covered by the classic work of Ginibre, which requires stable potentials.
One key ingredient is a type of Holder inequality for the expectation values of spatially smeared Euclidean densities, a special interpolation theorem. We also obtain a result on the absolute value of the fermion measure, it equals the boson measure.
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 Title
 The cluster expansion in statistical mechanics
 Journal

Communications in Mathematical Physics
Volume 49, Issue 3 , pp 233246
 Cover Date
 19761001
 DOI
 10.1007/BF01608729
 Print ISSN
 00103616
 Online ISSN
 14320916
 Publisher
 SpringerVerlag
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 Authors

 David Brydges ^{(1)}
 Paul Federbush ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Michigan, 48109, Ann Arbor, Michigan, USA