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arXiv:gr-qc/9412013

On the Space-Time Geometry of Quantum Systems

Dirk Graudenz

Published Dec 5, 1994• 6 citations
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Abstract

We describe the time evolution of quantum systems in a classical background space-time by means of a covariant derivative in an infinite dimensional vector bundle. The corresponding parallel transport operator along a timelike curve $\cC$ is interpreted as the time evolution operator of an observer moving along $\cC$. The holonomy group of the connection, which can be interpreted as a group of local symmetry transformations, and the set of observables have to satisfy certain consistency conditions. Two examples related to local $\mbox{SO}(3)$ and $\mbox{U}(1)$-symmetries, respectively, are discussed in detail. The theory developed in this paper may also be useful to analyze situations where the underlying space-time manifold has closed timelike curves.

DOI
10.48550/arxiv.gr-qc/9412013

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We describe the time evolution of quantum systems in a classical background space-time by means of a covariant derivative in an infinite dimensional vector bundle. The corresponding parallel transport operator along a timelike curve \cC\cC\cC is interpreted as the time evolution operator of an observer moving along \cC\cC\cC. The holonomy group of the connection, which can be interpreted as a group of local symmetry transformations, and the set of observables have to satisfy certain consistency conditions. Two examples related to local \mboxSO(3)\mbox{SO}(3)\mboxSO(3) and \mboxU(1)\mbox{U}(1)\mboxU(1)-symmetries, respectively, are discussed in detail. The theory developed in this paper may also be useful to analyze situations where the underlying space-time manifold has closed timelike curves.
A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles ("quantum bundles") over the space–time manifold yield a quantum "formalism" along any 1-dimensional timelike submanifold (a "detector"); in the flat, inertial case this reproduces the basic results of the usual quantum field theory, while in general it could be seen as a local, "linearized" description of the actual physics.
This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.
Quantum connections are defined by parallel transport operators acting on a Hilbert space. They transport tangent operators along paths in parameter space. The metric tensor of a Riemannian manifold is replaced by an inner product of pairs of operator fields, similar to the inner product of the Kubo-Mori formalism of Linear Response Theory. The metric is used to define the dual of a quantum connection.The gradient of the parallel transport operators is the quantum vector potential. It defines the covariant derivative of operator fields. The covariant derivatives are used to quantify the holonomy of the quantum connection. It is shown that a quantum connection is holonomic if and only if its dual is holonomic. If the parallel transport operators are unitary then an alpha-family of quantum connections can be defined in a way similar to Amari's alpha family of connections in Information Geometry. The minus alpha connection is the dual of the alpha connection. In particular, the alpha equal zero connection is self-dual. An operator field can be combined with a path in parameter space to produce a path in operator space. A definition is given for such a path in operator space to be autoparallel. The path in parameter space is then a geodesic for the induced connection.
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 paper consists of two parts. In the first part Schroedinger's equation for a charged quantum particle in a Galilei-Newton curved space-time is derived in a fully geometrical way. Gravitational and electromagnetic fields are coded into space metric and space-time connection. The fundamental geometrical object is a quantum connection in a Hermitian line bundle over the 7-dimensional jet space of 3-velocities. The secondary object is the bundle of Hilbert spaces over absolute time. Time appears as a superselection quantity while Shroedinger equation is interpreted as parallel transport in this bundle. In the second part the problem of measurement in quantum theory is discussed as a part of a more general problem of coupling between quantum and classical systems. The standard framework of quantum theory is extended so as to allow for dynamical central observables within dissipative dynamics. It is shown that within this approach one obtains not only Liouville equation that describes statistical ensembles, but also a piecewise-deterministic random process describing sequences of "events" that can be monitored by a continuous observation of the single, coupled classical system. It also describes "quantum jumps" or "wave packet reductions" that accompany these events. Two example are worked out in some details. The last one deals with the problem oof "how to determine the wave function ?".
A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum system, if we can modify the Hilbert space of state vectors of the system so that it turns into a Hermitian operator. If this operator is time-dependent, the modified Hilbert space is generally time-dependent. This in turn leads to a generic conflict between the condition that the Hamiltonian is an observable of the system and that it generates a unitary time-evolution via the standard Schr\"odinger equation. We propose a geometric framework for addressing this problem. In particular we show that the Hamiltonian operator consists of a geometric part, which is determined by a metric-compatible connection on an underlying Hermitian vector bundle, and a nongeometric part which we identify with the energy observable. The same quantum system can be locally described using a time-dependent Hamiltonian that acts in a time-independent state space and is the sum of a geometric part and the energy operator. The full global description of the system is achieved within the framework of a moderate geometric extension of quantum mechanics where the role of the Hilbert space of state vectors is played by a Hermitian vector bundle E\mathcal{E}E endowed with a metric compatible connection, and observables are given by global sections of a real vector bundle that is determined by E\mathcal{E}E. We examine the utility of our proposal to describe a class of two-level systems where E\mathcal{E}E is a Hermitian vector bundle over a two-dimensional sphere.