Gibbs States of a Lattice System of Quantum Anharmonic Oscillators

Abstract

Gibbs states of interacting quantum lattice systems are constructed as positive functionals on von Neumann algebras whose elements (observables) represent physical quantities [8], [13]. For the systems, the algebra of observables of every subsystem in a finite subset of the lattice may be represented as the C*-algebra of bounded operators on a Hilbert space, the theory of Gibbs states is quite well elaborated [8]. But if one needs to include into consideration also unbounded operators, the situation becomes much more complicated. In 1975 an approach to the construction of Gibbs states, which uses the integration theory in path spaces, has been initiated [1] (see also [5], [6], [7], [11], [13], [15]). Here the state at a temperature T = β-¹ is defined by means of a probability measure μβ on a certain infinite-dimensional space, analogously to the Euclidean quantum field theory. That is the reason why μβ is known as the Euclidean Gibbs state.

Supported in part by the Polish Scientific Research Committee under the Grant 2 PO3A 02915