The structure of the thermal equilibrium state of a weakly interacting Bose
gas is of current interest. We calculate the density matrix of that state in
two ways. The most effective method, in terms of yielding a simple, explicit
answer, is to construct a generating function within the traditional framework
of quantum statistical mechanics. The alternative method, arguably more
interesting, is to construct the thermal state as a vector state in an
artificial system with twice as many degrees of freedom. It is well known that
this construction has an actual physical realization in the quantum
thermodynamics of black holes, where the added degrees of freedom correspond to
the second sheet of the Kruskal manifold and the thermal vector state is a
state of the Unruh or the Hartle-Hawking type. What is unusual about the
present work is that the Bogolubov transformation used to construct the thermal
state combines in a rather symmetrical way with Bogolubov's original
transformation of the same form, used to implement the interaction of the
nonideal gas in linear approximation. In addition to providing a density
matrix, the method makes it possible to calculate efficiently certain
expectation values directly in terms of the thermal vector state of the doubled
system.Comment: 25 pages, LaTeX. To appear in a special issue of Foundations of
Physics in honor of Jacob Bekenstei
