Interaction of classical fermions with bosons (Q1282544)

The author starts with the second quantized Hamiltonian \[ {\widehat H}={\widehat H}({\widehat C}^\pm (x,p,s),{\widehat b}^\pm_m) \] describing a system of interacting fermions and bosons (\({\widehat C}^\pm (x,p,s)\) - fermionic creation/annihilation operators, \({\widehat b}^\pm_m\) - bosonic ones): there is a non-local four-fermionic self-interaction \(\sim {\widehat C}^+ {\widehat C}^+ {\widehat C}^- {\widehat C}^-\) and a boson-fermion term \(\sim ({\widehat b}^+ {\widehat C}^+ {\widehat C}^- + h.c.)\). The eigenfunctions of \({\widehat H}\) corresponding to an even number \(N\) of fermions can be expressed via eigenfunctions of a particular bosonized Hamiltonian \[ {\widehat H}={\widehat H}_N({\widehat B}^\pm (x,p,s;x',p',s'), {\widehat b}^\pm_m) \] with fermions replaced by bilocal bosonic fields. The author investigates the semiclassical limit by summing up over \(N\). Switching off the boson-fermion interaction he obtains (under particular conditions) a temperature state of a system of classical ``single'' fermions. It is argued that the phase transition to the coupled state occurs at the temperature at which the investigated system of equations bifurcates from the ``single'' fermion solution.