Finite-temperature correlations in the trapped Bose-Einstein gas

There is a large literature (cf. eg. [1, 2]) which, under conditions of translational invariance, has used functional integral methods to calculate, ab initio, the equilibrium finite temperature 2-point correlation functions (Green ’s functions) \[\left\langle {\hat \psi (r,\tau ){{\hat \psi }^\dag }(r',\tau ')} \right\rangle \] \(G\left( {r,r'} \right) \equiv \left\langle {\psi \left( {r,\tau } \right){{{\hat{\psi }}}^{\dag }}\left( {r',\tau '} \right)} \right\rangle \) for a Bose gas in each of d=3, d=2, d=1 space dimensions: (…) means thermal average and τ, τ′ are ‘thermal times’ for which 0<τ,<τ′β and β−1=k B T, T the temperature. These functional integral methods [1, 2] solve the many-…