Physics – Quantum Physics
Scientific paper
2005-05-07
Pramana: Journal of Physics, 64 (2005) 593
Physics
Quantum Physics
10 pages, no figures
Scientific paper
10.1007/BF02706206
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.
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