On quantum and parallel transport in a Hilbert bundle over spacetime

Abstract

We study the Hilbert bundle description of stochastic quantum mechanics in curved spacetime developed by Prugovečki, which gives a powerful new framework for exploring the quantum mechanical propagation of states in curved spacetime. We concentrate on the quantum transport law in the bundle, specifically on the information which can be obtained from the flat space limit. We give a detailed proof that quantum transport coincides with parallel transport in the bundle in this limit, confirming statements of Prugovečki. We furthermore show that the quantum-geometric propagator in curved spacetime proposed by Prugovečki, yielding a Feynman path integral-like formula involving integrations over intermediate phase space variables, is Poincaré gauge covariant (i.e. is gauge invariant except for transformations at the endpoints of the path) provided the integration measure is interpreted as a ``contact point measure'' in the soldered stochastic phase space bundle raised over curved spacetime.

25 pages, Plain TeX, harvmac/lanlmac

References in corpus (2)

• A Proposal for Solving the ``Problem of Time" in Canonical Quantum Gravity
• On the Space-Time Geometry of Quantum Systems

Cited by in corpus (15)

• Fibre bundle formulation of nonrelativistic quantum mechanics: I. Introduction. The evolution transport
• Local description of S-matrix in quantum field theory in curved spacetime using Riemann-normal coordinate
• Geometro-Stochastically Quantized Fields with Internal Spin Variables
• Symplectic Dirac-Kähler Fields
• Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
• Fibre bundle formulation of nonrelativistic quantum mechanics (full version)
• Fibre bundle formulation of nonrelativistic quantum mechanics. 0. Preliminary considerations: Quantum mechanics from a geometric-observer's viewpoint
• On Quantum-Geometric Connections and Propagators in Curved Spacetime
• Pseudo-forces in quantum mechanics
• Hodge decomposition for elliptic complexes over unital $C^{\ast }$-algebras
• Understanding the gravitational and cosmological redshifts as Doppler shifts by gravitational phase factors
• Elliptic complexes over $C^{\ast }$-algebras of compact operators
• Local quantum information dynamics
• Issues in Quantum-Geometric Propagation
• Hilbert bodies as quantum-classical continua