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Progress of Theoretical Physics

Spacetime Probabilities in Nonrelativistic Quantum Mechanics

Norifumi Yamada, Shuji Takagi

Published Jan 1, 1992• 55 citations
Publisher

Abstract

We demonstrate the existence of spacetime probabilities in nonrelativistic quantum mechanics, that is, quantum mechanical probabilities for a set of alternatives which are associated, not with a surface of constant time, but with spacetime domains with nonzero spatial and temporal width in Newtonian spacetime. We use the criterion that quantum mechanical probabilities can be defined for a set of alternatives if and only if the interference between any two different alternatives vanishes. Although generalized quantum mechanics was formulated on the basis of this criterion, the actual existence of spacetime probabilities has not been known. In this paper we consider a rectangular spacetime domain Ω and introduce a set of spacetime alternatives {Yes, No}: “Yes” is to find a particle in Ω and “No” is the complement to “Yes”. We show that, if the initial amplitude of the particle belongs to a specific class, then the criterion of vanishing interference is met by “Yes” and “No” and spacetime probabilities can be therefore defined for the set {Yes, No}. Owing to the property of the initial amplitude belonging to the class, the resultant probabilities are associated with a clear measurement theoretical meaning.

DOI
10.1143/ptp.87.77

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The science reports of the Tohoku University•Jan 31, 1992•0 citations
We demonstrate the existence of spacetime probabilities in nonrelativistic quantum mechanics, that is, quantum mechanical probabilities for a set of alternatives which are associated, not with a surface of constant time, but with spacetime domains with nonzero spatial and temporal width in Newtonian spacetime. We use the criterion that quantum mechanical probabilities can be defined for a set of alternatives if and only if the interference between any two different alternatives vanishes. Although generalized quantum mechanics was formulated on the basis of this criterion, the actual existence of spacetime probabilities has not been known. In this paper we consider a rectangular spacetime domain Ω and introduce a set of spacetime alternatives {Yes, No}: “Yes” is to find a particle in Ω and “No” is the complement to “Yes”. We show that, if the initial amplitude of the particle belongs to a specific class, then the criterion of vanishing interference is met by “Yes” and “No” and spacetime probabilities can be therefore defined for the set {Yes, No}. Owing to the property of the initial amplitude belonging to the class, the resultant probabilities are associated with a clear measurement theoretical meaning.
A theoretical attempt is made to find a nontrivial example of a set of non-interfering (decohering) alternatives which are not restrictecd to an instant of time and for which quantum mechanical probabilities can be defined. Confining ourselves to nonrelativistic quantum mechanics for a particle, we find an example of such a set. The set consists of two alternatives which formally correspond to two outcomes of a ‘Yes-No’ experiment. ‘Yes’ is the occurrence that a particle is found on a given domain on a simple hypersurface in Newtonian spacetime, and ‘No’ is the complement to ‘Yes’. in this example, absence of interference between the two alternatives is a consequence of a special choice of the symmetry of an initial amplitude. It turns out that the probabilities are 1 for ‘No’ and 0 for ‘Yes’.
A theoretical attempt is made to find a nontrivial example of a set of non-interfering (decohering) alternatives which are not restrictecd to an instant of time and for which quantum mechanical probabilities can be defined. Confining ourselves to nonrelativistic quantum mechanics for a particle, we find an example of such a set. The set consists of two alternatives which formally correspond to two outcomes of a ‘Yes-No’ experiment. ‘Yes’ is the occurrence that a particle is found on a given domain on a simple hypersurface in Newtonian spacetime, and ‘No’ is the complement to ‘Yes’. in this example, absence of interference between the two alternatives is a consequence of a special choice of the symmetry of an initial amplitude. It turns out that the probabilities are 1 for ‘No’ and 0 for ‘Yes’.
A theoretical aspect of quantum mechanical probabilities is studied. In nonrelativistic quantum mechanics (NRQM) for a particle, Schrödinger's wave function assigns probabilities to possible places where the particle is found on a surface (hypersurface in Newtonian spacetime) of constant time. We may then ask if quantum mechanical probabilities can be defined for an exhaustive set of exclusive outcomes of observations made on a surface on which Newtonian time is not constant. Such a question may be said to belong to the study of the role of time in NRQM in a broad sense. Specifically we investigate the theoretical possibility of a probabilistic description of (I) where and how many times the particle intersects the surface, and of (II) where and how many times the particle is found on the surface. For each case we formulate two conditions for the success of a probabilistic description. They are the classifiability condition and the no-interference condition, both of which are conditions on the propagator of the particle. For each case it is proved that the former condition does not hold and that consequently a probabilistic description is theoretically impossible either for (I) or for (II).
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.
The sum-over-histories quantum mechanics of a nonrelativistic particle gives a spacetime formulation of that theory in which the preferred Newtonian time does not enter immediately into the quantum kinematics. We investigate on what spacetime hypersurfaces a Schr\"odinger-Heisenberg formulation of the theory can be recovered and conclude that, in general, this is possible only on the surfaces of constant preferred Newtonian time. The significance of this for a quantum theory of spacetime is discussed.
A theoretical aspect of quantum mechanical probabilities is studied. In particular, it is investigated whether quantum mechanical probabilities can be defined for alternatives which are not restricted to a moment of time. Firstly the meaning and the status of this investigation in quantum mechanics are explained. Secondly a general framework is constructed within which we make the investigation. Thirdly a necessary mathematical tool is reviewed and extended. Lastly the general framework is applied to concrete examples with the help of the mathematical tool. Examples of alternatives are found which are not restricted to a moment of time and for which quantum mechanical probabilities can be defined with clear measurement theoretical meanings if an initial condition of the particle belongs to a specific class. (This is the doctoral thesis of the author.)