The Poisson structure of the mean-field equations in the Phi^4 theory

TL;DR

This paper demonstrates that the mean-field equations in Phi^4 theory can be formulated within a classical non-canonical Hamiltonian framework using a generalized Poisson structure, revealing structural invariants like the Heisenberg invariant.

Contribution

It introduces a novel Poisson geometric formulation of the mean-field equations in Phi^4 theory, extending the standard Poisson bracket to a non-canonical form.

Findings

Mean-field equations are expressed in a generalized Poisson framework.

The Heisenberg invariant is identified as a structural invariant.

The approach provides a new geometric perspective on Phi^4 theory.

Abstract

We show that the mean-field time dependent equations in the Phi^4 theory can be put into a classical non-canonical hamiltonian framework with a Poisson structure which is a generalization of the standard Poisson bracket. The Heisenberg invariant appears as a structural invariant of the Poisson tensor. (To be pubished in Annals of Physics)