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Progress of Theoretical Physics

Quantum Mechanical Probabilities on a General Spacetime-Surface. II

Norifumi Yamada, Shin Takagi

Published Sep 1, 1991• 30 citations
Publisher

Abstract

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’.

DOI
10.1143/ptp.86.599

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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’.
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 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).
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.
Usual quantum mechanics predicts probabilities for the outcomes of measurements carried out at definite moments of time. However, realistic measurements do not take place in an instant, but are extended over a period of time. The assumption of instantaneous alternatives in usual quantum mechanics is an approximation whose validity can be investigated in the generalized quantum mechanics of closed systems in which probabilities are predicted for spacetime alternatives that extend over time. In this paper we investigate how alternatives extended over time reduce to the usual instantaneous alternatives in the quantum mechanics of a nonrelativistic particle moving in one dimension. Specifically, we consider the coarse-grained alternatives of whether the particle remains to the left, remains to the right, or sometimes crosses the origin in a period of time. We show how the decoherence of this set of spacetime alternatives becomes automatic as the time over which they extend approaches zero and estimate how large this time can be before the interference between the alternatives becomes non-negligible. These results suggest that the time scale over which this kind of coarse graining of quantities such as the center of mass position of a massive body may be extended in time before producing significant interference is much longer than characteristic dynamical time scales. \textcopyright{} 1996 The American Physical Society.
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.)