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American Journal of Physics

A Way of Representing the Motion of a Classical System as Being along a Geodesic in Space-Time

H. L. Armstrong

Published Dec 1, 1954• 0 citations
Publisher

Abstract

Ordinarily, the Jacobi formulation of classical mechanics describes a system by giving the path of a “system point” in “configuration space,” the path being a geodesic if a metric is suitably defined. Nothing is said, however, about the motion in time. It is suggested here that, by adding a degree of freedom in uniform motion to serve as a clock, and identifying the time with the coordinate of that degree of freedom, the configuration space can be extended to a space-time, in which the motion is completely described, both in space and in time, as being along a geodesic. Also, by this representation, the idea of “absolute time” is made unnecessary. A representation such as this is, of course, normal in relativistic theory, but its use here in nonrelativistic mechanics is believed to have some novelty.

DOI
10.1119/1.1933857

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Ordinarily, the Jacobi formulation of classical mechanics describes a system by giving the path of a “system point” in “configuration space,” the path being a geodesic if a metric is suitably defined. Nothing is said, however, about the motion in time. It is suggested here that, by adding a degree of freedom in uniform motion to serve as a clock, and identifying the time with the coordinate of that degree of freedom, the configuration space can be extended to a space-time, in which the motion is completely described, both in space and in time, as being along a geodesic. Also, by this representation, the idea of “absolute time” is made unnecessary. A representation such as this is, of course, normal in relativistic theory, but its use here in nonrelativistic mechanics is believed to have some novelty.
A space-time representation is developed for nonrelativistic wave mechanics as was done for classical mechanics, by adding to the system a rotating body to serve as a clock, and applying the time-independent forms of the equations of motion. The usual Schrödinger equation is obtained in this way, however, only in the limit as the energy of the clock becomes very large. The reason for this restriction is found in an appendix, by comparison with the classical equations. Other appendices deal with certain aspects of the Hamiltonian representation of the problem.
This paper is the first of two papers devoted to formulation of quantum mechanics of a particle in a normal geodesic frame of reference in the general Riemannian space-time. Here canonical quantization of geodesic motion in the 1+3-formalism is considered, the result of which will be compared in the subsequent paper II with that of the field-theoretical approach, see also gr-qc/9807030. The Schrödinger representation of quantum-mechanical kinematics and dynamics is presented in the general-covariant form and a physical interpretation of the state vectors and the position operators is discussed.
In classical physics, the ‘state’ of a particle, at any particular time, is taken to involve a definite position in space and a definite motion; and the state of a system of particles is taken to involve a definite position in space and a definite motion for each of the particles. The situation in relation to quantum mechanics is completely different. As in classical physics, calculations can be made concerning the position and motion of a particle. However, this representation is not by means of symbols for scalar or vector quantities: it is by a more complex mathematical object. This quantum state does not indicate or represent position or motion in any direct way: what it does in substance is to enable calculation of the probability of the particle in question being found to have a position or momentum if an appropriate measurement is made.
Using the essence of Feynman's path integral and the space-time geodesics, an infinity of differentiable paths that follow the geometry of a continuous geodesic are constructed, and a wave function is associated to each path as a probability amplitude identical to the Feynman's probability amplitude for each path. We prove that each probability amplitude obeys to the Schrödinger's equation for a non relativistic physical system moving in a time independent potential, starting from the Jacobi-Hamilton equation.
Classical clocks measure proper time along their worldline, and Riemannian geometry provides tools for predicting the time shown by clocks in both flat and curved spacetimes. Common approaches to time in quantum systems, based for instance on wave functions or density matrices, tend to obscure this geometric property at the quantum level. Here, a new framework is demonstrated for perturbing the classical path-length functional to include quantum degrees of freedom within a modified Riemannian geometry. In this framework, a quantum clock travels on geodesics of a family of spacetimes deformed by parameters specifying the clock's quantum state. Detailed derivations provide potentially testable corrections to gravitational time-dilation in Schwarzschild spacetime that scale with the ratio of the clock's Compton wavelength to its wave packet's spatial extent.
We show that the emergence of time evolution in an otherwise timeless nonrelativistic closed quantum system -- viewed as a poor man's model of generally covariant quantum theory -- can be understood from the perspective of the path integral representation. As often happens in the functional integral approach, this viewpoint offers a more intuitive account of features that become cumbersome in the operator/Hilbert-space formulation. We show how Schrödinger evolution emerges once a clock degree of freedom is identified and placed in a suitable semiclassical `good-clock state'. Our analysis has a consequence that extends to path integral formulations of generally covariant systems with action SSS (including gravity). In such theories certain transition amplitudes take the form exp⁡(iS/ℏ)+exp⁡(−iS/ℏ)\exp(iS/\hbar)+\exp(-iS/\hbar)exp(iS/ℏ)+exp(−iS/ℏ) rather than the expected `forward propagating' exp⁡(iS/ℏ)\exp(iS/\hbar)exp(iS/ℏ). This feature, known as the {\em cosine problem}, appears in concrete regularizations of the path integral, for example in the spin foam representation defining the physical inner product between spin network states in loop quantum gravity. Both formally and in explicit regularizations, this apparent difficulty has led some authors to seek modifications of the basic amplitudes to eliminate backward propagation. Our model shows that the cosine problem is instead a natural consequence of time-reversal invariance of the fundamental dynamics together with the time-neutral boundary states commonly used in transition amplitudes. When a suitable clock system is identified and placed in a semiclassical `good-clock state', it introduces a time arrow selecting the `forward propagating' exp⁡(iS/ℏ)\exp(iS/\hbar)exp(iS/ℏ), without modifying the fundamental dynamics. The analysis clarifies how time emerges under suitable conditions and emphasizes that, in the canonical formulation, quantum gravity is fundamentally timeless.