Influence Phase via Feynman Functional Integral in The Density Matrix Representation

The interaction of the electron with lattice vibrations is a system coupled to a large number of environmental degrees of freedom. In most cases, the environment mimics a large heat bath characterized by a temperature T and so the environment state will therefore be described by an equilibrium density matrix. Sometimes, interest is also in this equilibrium density matrix of the system itself where such a state may be achieved after equilibration resulting from a weak coupling with a heat bath. To describe such thermal equilibrium states and dynamics of the system simultaneously, it is necessary to express the equilibrium density matrices via path integrals. Certainly, this is possible since one can write the equilibrium density operator in position representation.

We examine the motion of an electron in a polar crystal where we also consider its interaction with lattice vibrations. The interaction of the electron with its polarization in the phonon cloud forms the quasi-particle called the polaron defined earlier and may be characterized by its mass, energy, mobility, and so on. The interaction of the electron with lattice vibrations seems a complex problem and may involve two different effects: this renormalizes the energy spectrum as well as the mass of the electron. Different approaches may give different values of these renormalized quantities. In the polaron theory, it is necessary to have one unique approach and, in particular, the method of the density matrix in the language of Feynman functional integration is one of them [2,50]. The kinetic equation in principle may be helpful. The presentation of the problem for this case may not be sequential.

Interest may first be on the Schrodinger equation for which the problem may be approximately solvable. Further, we write absolutely the independent Boltzmann equation. If the first part of the problem is exactly solvable then we may be left with the scattering problem. Consequently, the problem is left to be solved by the density matrix that seems to be more appropriate than via the Boltzmann equation.