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arXiv:quant-ph/0503182 · Physical Review A

Representations of spacetime alternatives and their classical limits

August W. Bosse, James B. Hartle

Published Aug 12, 2005• 9 citations
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Abstract

Different quantum mechanical operators can correspond to the same classical quantity. Hermitian operators differing only by operator ordering of the canonical coordinates and momenta at one moment of time are the most familiar example. Classical spacetime alternatives that extend over time can also be represented by different quantum operators. For example, operators representing a particular value of the time average of a dynamical variable can be constructed in two ways: first, as a projection onto the value of the time-averaged Heisenberg picture operator for the dynamical variable and, second, as the class operator defined by a sum over those histories of the dynamical variable that have the specified time-averaged value. We show by both explicit example and general argument that the predictions of these different representations agree in the classical limit and that sets of histories represented by them decohere in that limit.

DOI
10.1103/physreva.72.022105

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Different quantum mechanical operators can correspond to the same classical quantity. Hermitian operators differing only by operator ordering of the canonical coordinates and momenta at one moment of time are the most familiar example. Classical spacetime alternatives that extend over time can also be represented by different quantum operators. For example, operators representing a particular value of the time average of a dynamical variable can be constructed in two ways: first, as a projection onto the value of the time-averaged Heisenberg picture operator for the dynamical variable and, second, as the class operator defined by a sum over those histories of the dynamical variable that have the specified time-averaged value. We show by both explicit example and general argument that the predictions of these different representations agree in the classical limit and that sets of histories represented by them decohere in that limit.
These are the author's lectures at the 1992 Les Houches Summer School, "Gravitation and Quantizations". They develop a generalized sum-over-histories quantum mechanics for quantum cosmology that does not require either a preferred notion of time or a definition of measurement. The "post-Everett" quantum mechanics of closed systems is reviewed. Generalized quantum theories are defined by three elements (1) the set of fine-grained histories of the closed system which are its most refined possible description, (2) the allowed coarse grainings which are partitions of the fine-grained histories into classes, and (3) a decoherence functional which measures interference between coarse grained histories. Probabilities are assigned to sets of alternative coarse-grained histories that decohere as a consequence of the closed system's dynamics and initial condition. Generalized sum-over histories quantum theories are constructed for non-relativistic quantum mechanics, abelian gauge theories, a single relativistic world line, and for general relativity. For relativity the fine-grained histories are four-metrics and matter fields. Coarse grainings are four-dimensional diffeomorphism invariant partitions of these. The decoherence function is expressed in sum-over-histories form. The quantum mechanics of spacetime is thus expressed in fully spacetime form. The coarse-grainings are most general notion of alternative for quantum theory expressible in spacetime terms. Hamiltonian quantum mechanics of matter fields with its notion of unitarily evolving state on a spacelike surface is recovered as an approximation to this generalized quantum mechanics appropriate for those initial conditions and coarse-grainings such that spacetime geometry
The approximations of classical mechanics resulting from quantum mechanics are richer than a correspondence of classical dynamical variables with self-adjoint Hilbert space operators. Assertion that classical dynamic variables correspond to self-adjoint Hilbert space operators is disputable and sets unnatural limits on quantum mechanics. Well known examples of classical dynamical variables not associated with self-adjoint Hilbert space operators are discussed as a motivation for the realizations of quantum field theory that lack Hermitian field operators but exhibit interaction.
We study the probability assignment for the outcomes of time-extended measurements. We construct the class operator that incorporates the information about a generic time-smeared quantity. These class operators are employed for the construction of positive-operator-valued measures for the time-averaged quantities. The scheme highlights the distinction between velocity and momentum in quantum theory. Propositions about velocity and momentum are represented by different class operators, hence they define different probability measures. We provide some examples, we study the classical limit, and we construct probabilities for generalized time-extended phase space variables.
In classical mechanics the state of a system is defined by the numerical values of a set of dynamical variables such as the position coordinates and the velocities. In quantum mechanics the state of a system at an instant is represented by a unit vector in a Hilbert space, where a set of axes are the eigenvectors of a complete set of observables. Here the observable are not the state vectors but the expectation values of a set of self-adjoint or Hermitian operators corresponding to the dynamical variables. The variation of 〈A〉 with time is seen as arising in any one of the following ways: 1. The state vector changes with time, but the operator A remains unchanged.
Sum-over-histories generalizations of nonrelativistic quantum mechanics are explored in which probabilities are predicted, not just for alternatives defined on spacelike surfaces, but for alternatives defined by the behavior of spacetime histories with respect to spacetime regions. Closed, nonrelativistic systems are discussed whose histories are paths in a given configuration space. The action and the initial quantum state are assumed fixed and given. A formulation of quantum mechanics is used which assigns probabilities to members of sets of alternative coarse-grained histories of the system, that is, to the individual classes of a partition of its paths into exhaustive and exclusive classes. Probabilities are assigned to those sets which decohere, that is, whose probabilities are consistent with the sum rules of probability theory. Coarse graining by the behavior of paths with respect to regions of spacetime is described. For example, given a single region, the set of all paths may be partitioned into those which never pass through the region and those which pass through the region at least once. A sum-over-histories decoherence functional is defined for sets of alternative histories coarse-grained by spacetime regions. Techniques for the definition and effective computation of the relevant sums over histories by operator-product formulas are described and illustrated by examples. Methods based on Euclidean stochastic processes are also discussed and illustrated. Models of decoherence and measurement for spacetime coarse grainings are described. Issues of causality are investigated. Such spacetime generalizations of nonrelativistic quantum mechanics may be useful models for a generalized quantum mechanics of spacetime geometry.