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.