Path Integral Approach to the Nonextensive Canonical Density Matrix

TL;DR

This paper extends Feynman's path integral formulation to nonextensive systems using Tsallis entropy, deriving generalized equations with unnormalized and normalized constraints, and applies these to free particles and scalar fields.

Contribution

It introduces a novel path integral approach for nonextensive canonical density matrices based on Tsallis entropy, including generalized Bloch equations and applications to field theory.

Findings

Derived two equivalent formulations with unnormalized constraints

Formulated generalized Bloch equations, linear and nonlinear forms

Applied methods to free particle and scalar field cases

Abstract

Feynman's path integral is herein generalized to the nonextensive canonical density matrix based on Tsallis entropy. This generalization is done in two ways by using unnormalized and normalized constraints. Firstly, we consider the path integral formulation with unnormalized constraints, and this generalization is worked out through two different ways, which are shown to be equivalent. These formulations with unnormalized constraints are solutions to two generalized Bloch equations proposed in this work. The first form of the generalized Bloch equation is linear, but with a temperature-dependent effective Hamiltonian; the second form is nonlinear and resembles the anomalous correlated diffusion equation (porous medium equation). Furthermore, we can extend these results to the prescription of field theory using integral representations. The second development is dedicated to analyzing the path integral formulation with normalized constraints. To illustrate the methods introduced here, we analyze the free particle case and a non-interacting scalar field. The results herein obtained are expected to be useful in the discussion of generic nonextensive contexts.