A Discrete Time Presentation of Quantum Dynamics

Inspired by the discrete evolution implied by the recent work on loop quantum cosmology, we obtain a discrete time description of usual quantum mechanics viewing it as a constrained system. This description, obtained without any approximation or explicit discretization, mimics features of the discrete time evolution of loop quantum cosmology. We discuss the continuum limit, physical inner product and matrix elements of physical observables to bring out various issues regarding viability of a discrete evolution. We also point out how a continuous time could emerge without appealing to any continuum limit.Comment: 20 pages, RevTex, no figures. Additional Clarifications added. Version accepted for publication in Class. Quant. Gra

Stochasticity, decoherence and an arrow of time from the discretization of time?

Certain intriguing consequences of the discreteness of time on the time evolution of dynamical systems are discussed. In the discrete-time classical mechanics proposed here, there is an {\it arrow of time} that follows from the fact that the replacement of the time derivative by the backward difference operator alone can preserve the non-negativity of the phase space density. It is seen that, even for free particles, all the degrees of freedom are {\it correlated} in principle. The forward evolution of functions of phase space variables by a finite number of time steps, in this discrete-time mechanics, depends on the entire continuous-time history in the interval $[0, \infty]$. In this sense, discrete time evolution is {\it nonlocal} in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible {\it via} the density matrix approach. The interference between non-degenerate quantum mechanical states decays exponentially. This {\it decoherence} is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.Comment: 10 pages, no figure

Consistency Conditions for Fundamentally Discrete Theories

The dynamics of physical theories is usually described by differential equations. Difference equations then appear mainly as an approximation which can be used for a numerical analysis. As such, they have to fulfill certain conditions to ensure that the numerical solutions can reliably be used as approximations to solutions of the differential equation. There are, however, also systems where a difference equation is deemed to be fundamental, mainly in the context of quantum gravity. Since difference equations in general are harder to solve analytically than differential equations, it can be helpful to introduce an approximating differential equation as a continuum approximation. In this paper implications of this change in view point are analyzed to derive the conditions that the difference equation should satisfy. The difference equation in such a situation cannot be chosen freely but must be derived from a fundamental theory. Thus, the conditions for a discrete formulation can be translated into conditions for acceptable quantizations. In the main example, loop quantum cosmology, we show that the conditions are restrictive and serve as a selection criterion among possible quantization choices.Comment: 33 page

Exclusive electroproduction of J/psi mesons at HERA

The exclusive electroproduction of J/psi mesons, ep->epJ/psi, has been studied with the ZEUS detector at HERA for virtualities of the exchanged photon in the ranges 0.15<Q^2<0.8 GeV^2 and 2<Q^2<100 GeV^2 using integrated luminosities of 69 pb^-1 and 83 pb^-1, respectively.The photon-proton centre-of-mass energy was in the range 30<W<220 GeV and the squared four-momentum transfer at the proton vertex |t|<1.The cross sections and decay angular distributions are presented as functions of Q^2, W and t. The effective parameters of the Pomeron trajectory are in agreement with those found in J/psi photoproduction. The spin-density matrix elements, calculated from the decay angular distributions, are consistent with the hypothesis of s-channel helicity conservation. The ratio of the longitudinal to transverse cross sections, sigma_L/sigma_T, grows with Q^2, whilst no dependence on W or t is observed. The results are in agreement with perturbative QCD calculations and exhibit a strong sensitivity to the gluon distribution in the proton.Comment: 33 pages, 10 figures. Submitted to Nuclear Physics