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 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 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, ∞]. In this sense, discrete time evolution is nonlocal in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible via the density matrix approach. The interference between nondegenerate quantum mechanical states decays exponentially. This 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.     

Formalism for the individual state of a quantum particle compatible with the bell inequalities,and a dissipative environment conjecture

We suggest a mathematical construction to define an individual state of a quantum particle which enables one to eliminate a contradiction between quantum mechanics of correlated particles and the Relativity Theory.     

Arrow of Time in Rigged Hilbert Space Quantum Mechanics

Arno Bohm and Ilya Prigogine's Brussels–Austin Group have been working on the quantum mechanical arrow of time and irreversibility in rigged Hilbert space quantum mechanics. A crucial notion in Bohm's approach is the so-called preparation/registration arrow. An analysis of this arrow and its role in Bohm's theory of scattering is given. Similarly, the Brussels–Austin Group uses an excitation/de-excitation arrow for ordering events, which is also analyzed. The relationship between the two approaches is initially discussed focusing on their semi-group operators and time arrows. Finally a possible realist interpretation of the rigged Hilbert space formulation of quantum mechanics is considered.     

Discreteness of space from the generalized uncertainty principle

Various approaches to Quantum Gravity (such as String Theory and Doubly Special Relativity), as well as black hole physics predict a minimum measurable length, or a maximum observable momentum, and related modifications of the Heisenberg Uncertainty Principle to a so-called Generalized Uncertainty Principle (GUP). We propose a GUP consistent with String Theory, Doubly Special Relativity and black hole physics, and show that this modifies all quantum mechanical Hamiltonians. When applied to an elementary particle, it implies that the space which confines it must be quantized. This suggests that space itself is discrete, and that all measurable lengths are quantized in units of a fundamental length (which can be the Planck length). On the one hand, this signals the breakdown of the spacetime continuum picture near that scale, and on the other hand, it can predict an upper bound on the quantum gravity parameter in the GUP, from current observations. Furthermore, such fundamental discreteness of space may have observable consequences at length scales much larger than the Planck scale.     

One Electron Atom in Special Relativity with de Sitter Space-Time Symmetry (II)：—Higher Order Contributions

This is a follow of previous work entitled "One Electron Atom in Special Relativity with de Sitter Space-Time Symmetry" [Commun. Theor. Phys. 57 (2012) 930]. In this paper, we consider the higher order calculations and contributions in the previous framework to solve one electron atoms in de Sitter invariant relativistic quantum mechanics. The next-to-leading-order calculations in 1/R²-expansions show that the fine-structure constant α is variant with cosmologic time going by in the de Sitter invariant special relativistic quantum mechanics with standard FRW cosmologic model.