Classical and quantum equivalence principle in terms of the path group

TL;DR

This paper introduces a quantum equivalence principle that maps curved spacetime path integrals onto flat spacetime, leading to covariant equations of motion without scalar curvature terms, and generalizes to external fields.

Contribution

It presents a novel approach to relate curved and flat spacetime path integrals using the path group, improving upon DeWitt's definition and enabling generalization to external fields.

Findings

Derives covariant equations of motion without scalar curvature terms.

Provides a representation of the path group for generalization.

Reduces curved-space path integrals to flat-space path integrals.

Abstract

A natural mapping of paths in a curved space onto the paths in the corresponding (tangent) flat space may be used to reduce the curved-space-time path integral to the flat-space-time path integral. The dynamics of the particle in a curved space-time is expressed then in terms of an integral over paths in the flat (Minkowski) space-time. This may be called quantum equivalence principle. Contrary to the known DeWitt's definition of a curved-space path integral, the present definition leads to the covariant equation of motion without a scalar curvature term. The reduction of a curved-space path integral to the flat-space path integral may be expressed in terms of a representation of the path group. With the help of this representation all the results may be generalized to the case of an arbitrary external field.