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arXiv:1805.11731 · Archivum Mathematicum

Contact Quantization: Quantum Mechanics = Parallel transport

Gabriel Herczeg, Emanuele Latini, Andrew K. Waldron

Published Jan 1, 2018• 4 citations

Abstract

Quantization together with quantum dynamics can be simultaneously formulated as the problem of finding an appropriate flat connection on a Hilbert bundle over a contact manifold. Contact geometry treats time, generalized positions and momenta as points on an underlying phase-spacetime and reduces classical mechanics to contact topology. Contact quantization describes quantum dynamics in terms of parallel transport for a flat connection; the ultimate goal being to also handle quantum systems in terms of contact topology. Our main result is a proof of local, formal gauge equivalence for a broad class of quantum dynamical systems—just as classical dynamics depends on choices of clocks, local quantum dynamics can be reduced to a problem of studying gauge transformations. We further show how to write quantum correlators in terms of parallel transport and in turn matrix elements for Hilbert bundle gauge transformations, and give the path integral formulation of these results. Finally, we show how to relate topology of the underlying contact manifold to boundary conditions for quantum wave functions.

DOI: 10.5817/am2018-5-281

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Quantization together with quantum dynamics can be simultaneously formulated as the problem of finding an appropriate flat connection on a Hilbert bundle over a contact manifold. Contact geometry treats time, generalized positions and momenta as points on an underlying phase-spacetime and reduces classical mechanics to contact topology. Contact quantization describes quantum dynamics in terms of parallel transport for a flat connection; the ultimate goal being to also handle quantum systems in terms of contact topology. Our main result is a proof of local, formal gauge equivalence for a broad class of quantum dynamical systems—just as classical dynamics depends on choices of clocks, local quantum dynamics can be reduced to a problem of studying gauge transformations. We further show how to write quantum correlators in terms of parallel transport and in turn matrix elements for Hilbert bundle gauge transformations, and give the path integral formulation of these results. Finally, we show how to relate topology of the underlying contact manifold to boundary conditions for quantum wave functions.

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wave functions. In this picture, the base manifold is an odd-dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a ``phase-spacetime,'' while the fibers are Hilbert spaces. This approach enjoys a ``quantum Darboux theorem'' that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.

We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental “phase-spacetime”. We show that this covariant starting point makes quantization into a purely geometric flatness condition. This makes quantum mechanics purely geometric, and possibly even topological. Our approach is especially useful for time-dependent problems and systems subject to ambiguities in choices of clock or observer. As a byproduct, we give a derivation and generalization of the Wigner functions of standard quantum mechanics.

We study the Hilbert bundle description of stochastic quantum mechanics in curved spacetime developed by Prugovecki, 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 Prugovecki. Furthermore, we show that the quantum-geometric propagator in curved spacetime proposed by Prugovecki, 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.

We point out that the evolution of a quantum system can be considered as a parallel transport of unitary operators in Hilbert spaces along the time with respect to a generalized connection. The different quantum representations of the system are shown to correspond to the choices of cross sections in the principal fiber bundle where the generalized connection is defined. This interpre- tation of time evolution allows us to solve the problem of the formulation of the evolution of a quantum particle in a four-dimensional gauge field.

Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.

Quantum theory of field (extended) objects without a priori space-time geometry has been represented. Intrinsic coordinates in the tangent fibre bundle over complex projective Hilbert state space

CP(N-1)

are used instead of space-time coordinates. The fate of quantum system modeled by the generalized coherent states is rooted in this manifold. Dynamical (state-dependent) space-time arises only at the stage of the quantum "yes/no" measurement. The quantum measurement of the gauge ``field shell'' of the generalized coherent state is described in terms of the affine parallel transport of the local dynamical variables in

CP(N-1)

.