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The quantum Maxwell theory at finite temperature at equilibrium is studied oncompact and closed manifolds in both the functional integral- and Hamiltonianformalism. The aim is to shed some light onto the interrelation between thetopology of the spatial background and the thermodynamic properties of thesystem. The quantization is not unique and gives rise to inequivalent quantumtheories which are classified by {\theta}-vacua. Based on explicitparametrizations of the gauge orbit space in the functional integral approachand of the physical phase space in the canonical quantization scheme, theGribov problem is resolved and the equivalence of both quantization schemes iselucidated. Using zeta-function regularization the free energy is determinedand the effect of the topology of the spatial manifold on the vacuum energy andon the thermal gauge field excitations is clarified. The general results arethen applied to a quantum Maxwell gas on a n-dimensional torus providingexplicit formulae for the main thermodynamic functions in the low- and hightemperature regimes, respectively.

Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds