Path reparametrization in functional integrals on vector bundles

H. Kleinert, A. Chervyakov

S. N. Storchak

S. N. Storchak

S. N. Storchak

Kleinert, H, Chervyakov, A

We demonstrate the reparametrization invariance of perturbatively defined one-dimensional functional integrals up to the three-loop level for a path integral of a quantum-mechanical point particle in a box. We exhibit the origin of the failure of earlier authors to establish reparametrization invariance which led them to introduce, superfluously, a compensating potential depending on the connection of the coordinate system. We show that problems with invariance are absent by defining path integrals as the $\\epsilon \\to 0$ limits of $1+ \\epsilon$-dimensional functional integrals.

Bozhidar Z Iliev

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper. In the third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, we find the state liftings and observable liftings corresponding to state vectors and observables respectively in the different pictures of motion. The equations of motion for these quantities are derived. Using the results obtained, problems concerning the integrals of motion are considered from the bundle viewpoint. Necessary and sufficient invariant bundle conditions for a dynamical variable to be an integral of motion are found.