Interacting Bosons at Finite Temperature: How Bogolubov Visited a Black Hole and Came Home Again

TL;DR

This paper explores the thermal equilibrium state of a weakly interacting Bose gas using two methods, linking quantum statistical mechanics with black hole thermodynamics and revealing a symmetrical Bogolubov transformation.

Contribution

It introduces a novel approach combining Bogolubov transformations to analyze the thermal state of Bose gases and connects it with black hole thermodynamics.

Findings

Explicit density matrix expressions for the thermal state.

Efficient calculation of expectation values in the doubled system.

Connection between Bose gas thermal states and black hole quantum states.

Abstract

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.