Jacobi's Principle and the Disappearance of Time

TL;DR

This paper investigates how Jacobi's principle leads to the disappearance of time in classical and quantum models, revealing that superpositions of clocks cause timelessness in quantum theory and boundary conditions in path integrals.

Contribution

It provides a detailed analysis of the mechanism behind the emergence and disappearance of time using path integral quantization in simple particle models.

Findings

Path integral quantization clarifies the physical origin of timelessness.

Superpositions of clocks cause quantum timelessness.

Gauge fixing functions influence boundary conditions and the emergence of duration.

Abstract

Jacobi's action principle is known to lead to a problem of time. For example, the timelessness of the Wheeler-DeWitt equation can be seen as resulting from using Jacobi's principle to define the dynamics of 3-geometries through superspace. In addition, using Jacobi's principle for non-relativistic particles is equivalent classically to Newton's theory but leads to a time-independent Schrodinger equation upon Dirac quantization. In this paper, we study the mechanism for the disappearance of time as a result of using Jacobi's principle in these simple particle models. We find that the path integral quantization very clearly elucidates the physical mechanism for the timeless of the quantum theory as well as the emergence of duration at the classical level. Physically, this is the result of a superposition of clocks which occurs in the quantum theory due to a sum over all histories. Mathematically, the timelessness is related to how the gauge fixing functions impose the boundary conditions in the path integral.