Non-interacting Quantum Systems

This chapter is devoted to the study of quantum statistical system with particular emphasis on non-interacting ones (quantum ideal gases). We begin by reviewing the Spin and Statistics theorem and its implications for quantum thermal physics, then we use techniques from Representation Theory to write down the Grand potential for arbitrary systems of non-interacting bosons and, respectively, fermions. We study the polylogarithm functions and the allied Fermi/Bose integrals; we use these math tools to analyze in detail the physics of the ideal gases of fermions and, respectively, bosons in various regimes (low vs. high temperature, high vs. low density, etc.). We discuss specifically the Bose condensation of the ideal boson gas. Next we consider the black body radiation from two dual points of view: as a quantum ultra-relativistic gas of massless particles (photons) and, respectively, as the quantum statistical mechanics of the Maxwell theory of electromagnetism. In the final two sections we introduce the method of second quantization and path integrals as tools to compute partition functions in the quantum context. In this way we recover the previous results on quantum ideal gases from a higher “dual” perspective shedding light on many subtle issues.