Vacuum Energy Densities and Multiplicative Anomalies in a Free Bose Gas

The vacuum energy density or free energy of a free charged Bose gas at nonzero densities is studied in the context of the debate about Multiplicative Anomalies. Some ζ-function regularised calculations of the free energy in the literature are reexamined, clarified and extended. A range of apparently distinct answers can obtained. Equivalent dimensional regularisation results are also presented for comparison. I conclude that operator ordering and normal ordering are not responsible for these differences. Rather it is an undesirable but unavoidable property of ζ-function regularisation which leads to these different results, making it a bad scheme in general. By comparison I show how dimensional regularisation calculations give a consistent result without any complications, making this a good scheme in this context.