Physics – Quantum Physics
Scientific paper
1996-12-17
Physics
Quantum Physics
Lectures presented at the 1996 Cargese Summer School on FUNCTIONAL INTEGRATION: BASICS AND APPLICATIONS. Author's institution
Scientific paper
A simple mapping procedure is presented by which classical orbits and path integrals for the motion of a point particle in flat space can be transformed directly into those in curved space with torsion. Our procedure evolved from well-established methods in the theory of plastic deformations, where crystals with defects are described mathematically as images of ideal crystals under active nonholonomic coordinate transformations. Our mapping procedure may be viewed as a natural extension of Einstein's famous equivalence principle. When applied to time-sliced path integrals, it gives rise to a new "quantum equivalence principle" which determines short-time action and measure of fluctuating orbits in spaces with curvature and torsion. The nonholonomic transformations possesses a nontrivial Jacobian in the path measure which produces in a curved space an additional term proportional to the curvature scalar R, thus canceling a similar term found earlier by DeWitt. This cancelation is important for correctly describing semiclassically and quantum mechanically various systems such as the hydrogen atom, a particle on the surface of a sphere, and a spinning top. It is also indispensable for the process of bosonization, by which Fermi particles are redescribed by those fields.
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