Jacobi's Principle and the Disappearance of Time
arXiv:0804.2900 · doi:10.1103/PhysRevD.81.044035
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.
Published version. Significant amendments to presentation. 27 pages
References in corpus (9)
- Quantum Gravity at a Lifshitz Point
- The physical gravitational degrees of freedom
- Scale-Invariant Gravity: Particle Dynamics
- Revisiting observables in generally covariant theories in the light of gauge fixing methods
- Scale-Invariant Gravity: Geometrodynamics
- Constraints and gauge transformations: Dirac's theorem is not always valid
- Relational Particle Models. II. Use as toy models for quantum geometrodynamics
- Classical dynamics on triangleland
- Quantum Machian Time in Toy Models of Gravity
Cited by in corpus (16)
- Problem of Time in Quantum Gravity
- Contact Geometry and Quantum Mechanics
- Reassessing the problem of time of quantum gravity
- Quantum Motion on Shape Space and the Gauge Dependent Emergence of Dynamics and Probability in Absolute Space and Time
- Paths to gravitation via the gauging of parameterized field theories
- Relaxation of first-class constraints and the quantization of gauge theories: from "matter without matter" to the reappearance of time in quantum gravity
- Timeless path integral for relativistic quantum mechanics
- Gravitational Collapse via Wheeler-DeWitt Equation
- GPS Observables in Newtonian Spacetime or Why We Do Not Need 'Physical' Coordinate Systems
- Quantum Cosmological Relational Model of Shape and Scale in 1-d
- Wheeler-DeWitt equation and the late gravitational collapse: effects of factor ordering and the tunneling scenario
- Does Quantum Cosmology Predict the Age of the Universe?
- Time is Order
- Diffeomorphism-Invariant Quantities in Phase Space: More than Correlations
- Relational Quadrilateralland. II. The Quantum Theory
- The Dynamics of Quantum Gravity: the Missing Piece in the Spacetime Emergentist Account