The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional space-time 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 Galilean relativity, and the result is then trivial. A nontrivial result is obtained if the plane is not of this type. When the space-time coordinates are (t, x, y, z), the paper analyzes 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 for a zone of interaction contained between a pair of z=constant hyperplanes, is presented.

The conventional time-dependent Schrödinger equation describes only unidirectional time evolution of the state of a physical system, i.e., forward, or, less commonly, backward. This paper proposes a generalized quantum dynamics for the description of joint, and interactive, forward and backward time evolution within a physical system. The principal mathematical assumption for bidirectional evolution in general is that the space of states should be taken to be not merely a Hilbert space, but a more restricted entity known as a Kreȋn space, which is a complex Hilbert space with a Hermitean operator that has eigenvalues +1 and −1 only, and that therefore gives rise to an indefinite metric. The vector subspaces of states with positive or negative norm with respect to the indefinite metric will-for open channels-be construed to be states in forward or, respectively, backward evolution along the time axis. The quantum dynamics is generated by a pseudo-Hermitean Hamiltonian operator and conserves inner products with respect to the indefinite metric. Input and output states are defined in physically plausible ways such that the output comprises both reflected and transmitted states from a zone of interaction in time; a unitary transformation between input and output states is obtained from the pseudounitary transformation between the initial and final states. Three applications are studied: (1) a formal theory of collisions in terms of perturbation theory; (2) a relativistically invariant quantum field theory for a system that kinematically comprises the direct sum of two quantized real scalar fields, such that one subfield evolves forward and the other backward in time, and such that there is dynamical coupling between the subfields; (3) an argument that in the latter field theory, the dynamics predicts that in a range of values of the coupling constants, the expectation value of the vacuum energy of the universe is forced to be zero to high accuracy. [Added in arXiv version: It is also speculated that * email: ghahne@mail.arc.nasa.gov the ideas presented contain a kernel of explanation for the existence of a negative average energy density in the cosmos.]

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The infinite series, absolutely convergent if |x| + |y| < 1, for Appell's F2 (α, β, β′, γ, γ′; x, y) is analytically continued into a linear combination of four infinite series in powers of (x - 1) and (y - 1); each of the latter four series is absolutely convergent if |x − 1| + |y − 1| < 1. The analytic continuation is carried out by manipulation of the Mellin-Barnes integral representations for the hypergeometric functions appearing in the course of the calculation.

Muñoz, Seidel, and Muga [Phys. Rev. A 79, 012108 (2009)], following an earlier proposal by Pollak and Miller [Phys. Rev. Lett. 53, 115 (1984)] in the context of a theory of a collinear chemical reaction, showed that suitable moments of a two-flux correlation function could be manipulated to yield expressions for the mean quantum dwell time and mean square quantum dwell time for a structureless particle scattering from a timeindependent potential energy field between two parallel lines in a twodimensional spacetime. The present work proposes a generalization to a charged, nonrelativistic particle scattering from a transient, spatially confined electromagnetic vector potential in four-dimensional spacetime. The geometry of the spacetime domain is that of the slab between a pair of parallel planes, in particular those defined by constant values of the third (z) spatial coordinate. The mean N th power, N = 1, 2, 3, . . ., of the quantum dwell time in the slab is given by an expression involving an Nflux-correlation function. All these means are shown to be nonnegative. The N = 1 formula reduces to an S-matrix result published previously [G. E. Hahne, J. Phys. A 36, 7149 (2003)]; an explicit formula for N = 2, and of the variance of the dwell time in terms of the S-matrix, is worked out. A formula representing an incommensurability principle between variances of the output-minus-input flux of a pair of dynamical variables (such as the particle's time flux and others) is derived.