Justification of c-Number Substitutions in Bosonic Hamiltonians

TL;DR

This paper rigorously justifies the substitution of a c-number for the zero mode operator in bosonic Hamiltonians, confirming its validity for calculating thermodynamic quantities and the condensation amount, thus solidifying part of Bogoliubov's 1947 theory.

Contribution

It provides a rigorous proof for the c-number substitution in bosonic Hamiltonians, validating its use in thermodynamic calculations and condensation analysis.

Findings

The c-number substitution yields correct thermodynamic quantities.

The value of |z|^2 maximizes the partition function and equals the true condensation.

The proof confirms a key aspect of Bogoliubov's 1947 theory.

Abstract

The validity of substituting a c-number $z$ for the $k=0$ mode operator $a_0$ 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 like 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 -- a point that had previously been elusive.