Justification of c-number substitutions in bosonic hamiltonians

The validity of substituting a c-number z for the k = 0 mode operator a₀ is established rigorously in full generality, thereby verifying one aspect of Bogoliubov's 1947 theory. This substitution not only yields the correct value of thermodynamic quantities such as the pressure or ground state energy, but also the value of |z|² that maximizes the partition function equals the true amount of condensation in the presence of a gauge-symmetry-breaking term. This point had previously been elusive.