Simple criterion for the occurrence of Bose-Einstein condensation

TL;DR

This paper establishes a simple criterion for Bose-Einstein condensation, showing it occurs only if the spectral dimension is at least three, applicable to both nonrelativistic and relativistic systems without interactions.

Contribution

The paper introduces a general condition based on spectral dimension for the occurrence of Bose-Einstein condensation in various quantum systems.

Findings

Condensation occurs only if spectral dimension q ≥ 3.

The criterion applies to both nonrelativistic and relativistic systems.

Uses generalized ζ-functions to define the quantum theory.

Abstract

We examine the occurrence of Bose-Einstein condensation in both nonrelativistic and relativistic systems with no self-interactions in a general setting. A simple condition for the occurrence of Bose-Einstein condensation can be given if we adopt generalized $\zeta$-functions to define the quantum theory. We show that the crucial feature governing Bose-Einstein condensation is the dimension $q$ associated with the continuous part of the eigenvalue spectrum of the Hamiltonian for nonrelativistic systems or the spatial part of the Klein-Gordon operator for relativistic systems. In either case Bose-Einstein condensation can only occur if $q\ge3$.