Classical and quantum mechanics via supermetrics in time

TL;DR

This paper explores a supermanifold framework with supermetrics that unifies classical and quantum mechanics through path-integral formulations, revealing how different supermetrics correspond to classical or quantum descriptions.

Contribution

It introduces a supermanifold approach with specific supermetrics that reproduce classical and quantum path-integral formulations of mechanics.

Findings

Supermetrics in supermanifolds distinguish classical from quantum mechanics.

A particular supermetric reproduces the classical path-integral (CPI).

A different supermetric reproduces the quantum path-integral (QPI).

Abstract

Koopman-von Neumann in the 30's gave an operatorial formululation of Classical Mechanics. It was shown later on that this formulation could also be written in a path-integral form. We will label this functional approach as CPI (for classical path-integral) to distinguish it from the quantum mechanical one, which we will indicate with QPI. In the CPI two Grassmannian partners of time make their natural appearance and in this manner time becomes something like a three dimensional supermanifold. Next we introduce a metric in this supermanifold and show that a particular choice of the supermetric reproduces the CPI while a different one gives the QPI.