Addenda and corrections to work done on the path-integral approach to classical mechanics

TL;DR

This paper refines the geometric understanding of the path-integral formulation of classical mechanics, corrects previous transformation rules, and explores alternative formulations using purely bosonic variables.

Contribution

It corrects the transformation rules for auxiliary variables and clarifies the geometric structure of the path-integral space, introducing a bosonic alternative formulation.

Findings

The Hamiltonian is an exact scalar under the new transformations.

The variable space is the cotangent bundle of the reversed-parity tangent bundle.

A bosonic variable-based path-integral formulation is constructed.

Abstract

In this paper we continue the study of the path-integral formulation of classical mechanics and in particular we better clarify, with respect to previous papers, the geometrical meaning of the variables entering this formulation. With respect to the first paper with the same title, we {\it correct} here the set of transformations for the auxiliary variables $\lambda_{a}$. We prove that under this new set of transformations the Hamiltonian ${\widetilde{\cal H}}$, appearing in our path-integral, is an exact scalar and the same for the Lagrangian. Despite this different transformation, the variables $\lambda_{a}$ maintain the same operatorial meaning as before but on a different functional space. Cleared up this point we then show that the space spanned by the whole set of variables ($\phi, c, \lambda,\bar c$) of our path-integral is the cotangent bundle to the {\it reversed-parity} tangent bundle of the phase space ${\cal M}$ of our system and it is indicated as $T^{\star}(\Pi T{\cal M})$. In case the reader feel uneasy with this strange {\it Grassmannian} double bundle, we show in this paper that it is possible to build a different path-integral made only of {\it bosonic} variables. These turn out to be the coordinates of $T^{\star}(T^{\star}{\cal M})$ which is the double cotangent bundle of phase-space.