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).
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).
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 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’.
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.
The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional spacetime entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation. Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate. In conventional formulations the hyperplane is always spacelike, i.e. is a time = constant hyperplane in the Galilean relativity, and the result is then trivial. A nontrivial result is obtained if the plane is not of this type. When the spacetime coordinates are ( t , x , y , z ), the paper analyses in detail the case that the hyperplane is of the type z = constant. Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface. Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory. A detailed formalism for computing average crossing times on a z = constant hyperplane, and average dwell times and delay times between a pair of z = constant hyperplanes, is presented.
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.