Variational Approximation of Density Matrix on the Basis of Ground State: Thermalization of Vacuum

Abstract

We formulate the Feynman's variational principle for a density matrix by means of Bohm-Madelung representation using Lagrange variables, which appear to be dynamical invariants of phase flux with sources on the ends of trajectories. With a linear parametrization in invariants for the flux the variational problem can be formulated as the extended principle of least action in which the functional variables appear to be trajectories of the center and the dispersion of the renormalized vacuum wave packet, which is reduced at the ends of the trajectories. Approximating the extended action in terms of bilocal frequency, we derive an explicit expression which provides the exact result for the linear systems and, in general case, proper semi-classical correction and correct asymptotics for the ground state. We examine the effect of spontaneous symmetry breaking on the coordinate distribution as well.