Article

# Jacobi's Principle and the Disappearance of Time

04/2008; DOI:doi:10.1103/PhysRevD.81.044035
Source: arXiv

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. Comment: Published version. Significant amendments to presentation. 27 pages

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### Keywords

boundary conditions

classical level

Dirac quantization

dynamics

gauge

Jacobi's action principle

Jacobi's principle

Mathematically

Newton's theory

non-relativistic particles

path integral

path integral quantization

physical mechanism

Published version

quantum theory

time-independent Schrodinger equation

timeless

timelessness

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