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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Spacetime coarse grainings in nonrelativistic quantum mechanics

James B. Hartle

Published Nov 15, 1991• 107 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.44.3173

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arXiv:hep-th/9507107•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Mar 15, 1996•1 citation
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.
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
Hartle's generalized quantum mechanics formalism is used to examine spacetime coarse grainings, i.e., sets of alternatives defined with respect to a region extended in time as well as space, in the quantum mechanics of a free relativistic particle. For a simple coarse graining and suitable initial conditions, tractable formulas are found for branch wave functions. Despite the nonlocality of the positive-definite version of the Klein-Gordon inner product, which means that nonoverlapping branches are not sufficient to imply decoherence, some initial conditions are found to give decoherence and allow the consistent assignment of probabilities.
An effort is continued to define quantum probabilities for continuous histories. The sum-over-paths approach used in Prog. Theor. Phys. 87, 77 (1992) is critically reviewed, improved, and applied again. Consistency between wave nature and particle nature is the criterion used to judge the definability of the probabilities and is formulated as the path classifiability condition (C1) and the no-interference condition (C2). A set of classes of histories satisfying these two conditions is considered as a space-time analog of observables. In particular, Feynman's paths for a nonrelativistic particle are considered as histories and the definability of probabilities for classes of paths is investigated, where classes are defined by classifying paths according to their behavior with respect to a rectangular space-time region \ensuremath{\Omega}\ensuremath{\equiv}\ensuremath{\Delta}X\ifmmode\times\else\texttimes\fi{}\ensuremath{\Delta}T. Confining ourselves to cases where C1 is satisfied, we examine C2 for some sets of classes of paths. Although C2 does not hold in general, some examples are found where C2 holds. In all the examples, the initial wave function is restricted. In some examples, the location of \ensuremath{\Omega} and/or the potential in the region also affect the success of C2. Due to these restrictions, the resultant probabilities for classes of paths take reasonable values. A puzzling example in which probabilities cannot be defined for histories contrary to intuition is resolved by considering an appropriate coarse graining of classes of paths. Considering a rectangular potential barrier, we show that the reflection and the transmission probabilities are special cases of probabilities for histories, and also that probability densities of transmission and reflection times cannot be defined. This study may be taken to be a study of the consistent-histories approach not with discrete histories defined by products of projection operators but with continuous histories defined by Feynman's paths in configuration space. \textcopyright{} 1996 American Institute of Physics.
The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.
A pedagogical introduction is given to the quantum mechanics of closed systems, most generally the universe as a whole. Quantum mechanics aims at predicting the probabilities of alternative coarse-grained time histories of a closed system. Not every set of alternative coarse-grained histories that can be described may be consistently assigned probabilities because of quantum mechanical interference between individual histories of the set. In “Copenhagen ” quantum mechanics, probabilities can be assigned to histories of a subsystem that have been “measured”. In the quantum mechanics of closed systems, containing both observer and observed, probabilities are assigned to those sets of alternative histories for which there is negligible interference between individual histories as a consequence of the system’s initial condition and dynamics. Such sets of histories are said to decohere. We define decoherence for closed systems in the simplified case when quantum gravity can be neglected and the initial state is pure. Typical mechanisms of decoherence that are widespread in our universe are illustrated. Copenhagen quantum mechanics is an approximation to the more general quantum framework of closed subsystems. It is appropriate when there is an approximately isolated subsystem that is a participant in a measurement situation in which (among other things) the decoherence of alternative registrations of the apparatus can be idealized as exact. Since the quantum mechanics of closed systems does not posit the existence of the quasiclassical domain of everyday experience, the domain of the approximate aplicability of classical physics must be explained. We describe how a quasiclassical domain described by averages of densities of approximately conserved quantities could be an emergent feature of an initial condition of the universe that implies the approximate classical behavior of spacetime on accessible scales.
Hartle's generalized quantum mechanics in the sum-over-histories formalism is used to describe a non-Abelian gauge theory. Predictions are made for certain alternatives, with particular attention given to coarse grainings involving the constraint. In this way, the theory is compared to other quantum-mechanical descriptions of gauge theories in which the constraints are imposed by hand. The vanishing of the momentum space constraint is seen to hold, both through a simple formal argument and via a more careful description of the Lorentzian path integral as defined on a spacetime lattice. The configuration space realization of the constraint is shown to behave in a more complicated fashion. For some coarse grainings, we recover the known result from an Abelian theory, that coarse grainings by values of the constraint either predict its vanishing or fail to decohere. However, sets of alternatives defined in terms of a more complicated quantity in the Abelian case are exhibited where definite predictions can be made which disagree with the assumption that the constraints vanish. Finally, the configuration space sum-over-histories theory is exhibited in a manifestly Lorentz-invariant formulation.