Discretization and Quantum Mechanics

Gerard ’t Hooft

Sean M. Carroll

An-Wei Zhang

In this letter we briefly investigate the mathematical structure of space-time in the framework of discretization. It is shown that the discreteness of space-time may result in a new mechanical system which differ from the usual quantum mechanics (QM).

George Jaroszkiewicz

Quantization We now discuss the quantization of discrete time (DT) classical mechanics (CM). We restrict our attention in this chapter to a system under observation (SUO) consisting of a single point particle of mass m moving in one spatial dimension with Cartesian coordinate x . The generalization to two or more degrees of freedom is as straightforward here as in CT quantum mechanics (QM). The Dirac bra-ket notation will be used for convenience. The standard principles of QM have proven remarkably successful and consistent over the years (Peres, 1993) and we have no reason to alter them apart from changing from CT to DT. This is a significant change. Quantum mechanics became dominant in physics and chemistry because of the success of the Schrödinger equation, which is a first-order-in-time differential equation. Therefore, we should take care to ensure that discretizing time does not undermine the successes of CT QM. There are two reasons why discretizing time in QM might be considered. First, the Schrödinger equation is hard, or even impossible, to solve exactly in many situations and temporal discretization might be seen as a step towards numerical simulation by computer. This motivated the work of Bender (Bender et al. , 1985a, 1985b, 1993) and others on the DT Schrödinger equation. The second reason is one of principle: we may want to explore the properties of DT QM as a self-consistent theory in its own right rather than as an approximation to CT QM.

W. M. Stuckey, Timothy J. McDevitt, Michael Silberstein

We propose a discrete path integral formalism over graphs fundamental to quantum mechanics (QM) based on our interpretation of QM called Relational Blockworld (RBW). In our approach, the transition amplitude is not viewed as a sum over all field configurations, but is a mathematical machine for measuring the symmetry of the discrete differential operator and source vector of the discrete action. Therefore, we restrict the path integral to the row space of the discrete differential operator, which also contains the discrete source vector, in order to avoid singularities. In this fashion we obtain the two-source transition amplitude over a "ladder" graph with N vertices. We interpret this solution in the context of the twin-slit experiment.

James E. House