Discrete Quantum Mechanics on Finite Fields

Sean M. Carroll

John Gardiner

Anadijiban Das, Sourav Haldar

The discrete phase space representation of quantum mechanics involving a characteristic length is investigated. The continuous (1 + 1)-dimensional phase space is first discussed for the sake of simplicity. It is discretized into denumerable infinite number of concentric circles. These circles, endowed with “unit area”, are degenerate phase cells resembling closed strings.Next, Schrödinger wave equation for one particle in the three dimensional space under the influence of a static potential is studied in the discrete phase space representation of wave mechanics. The Schrödinger equation in the arena of discrete phase space is a partial difference equation. The energy eigenvalue problem for a three dimensional oscillator is exactly solved.Next, relativistic wave equations in the scenario of three dimensional discrete phase space and continuous time are explored. Specially, the partial finite difference-differential equation for a scalar field is investigated for the sake of simplicity. The exact relativistic invariance of the partial finite difference-differential version of the Klein-Gordon equation is rigorously proved. Moreover, it is shown that all nine important Green’s functions of the partial finite difference-differential wave equation for the scalar field are non-singular.In the appendix, a possible physical interpretation for the discrete orbits in the phase space as degenerate, string-like phase cells is provided in a mathematically rigorous way.

William Sulis

The process algebra has been used successfully to provide a novel formulation of quantum mechanics in which non-relativistic quantum mechanics (NRQM) emerges as an effective theory asymptotically. The process algebra is applied here to the formulation of quantum field theory. The resulting QFT is intuitive, free from divergences and eliminates the distinction between particle, field and wave. There is a finite, discrete emergent space-time on which arise emergent entities which transfer information like discrete waves and interact with measurement processes like particles. The need for second quantization is eliminated and the particle and field theories rest on a common foundation, clarifying and simplifying the relationship between the two.

Stanley Gudder

Johan Noldus

We start by rewriting classical mechanics in a quantum mechanical fashion and point out that the only difference with quantum theory re- sides at one point. There is a classical analogon of the collapse of the wavefunction and an extension of the usual Born rule is proposed which might solve this problem. We work only with algebra’s over the real (com- plex) numbers, general number fields of finite characteristic allowing for finite dimensional representations of the commutation relations are not considered given that such fields are not well ordered and do not give rise to a well defined probability interpretation. Our theory generalizes however to discrete spacetimes and finite dimensional algebra’s. Looking at physics this way, spacetime itself distinguishes itself algebraically by means of well chosen commutation relations and there is further nothing special about it meaning it has also a particle interpretation just like any other dynamical variable. Likewise, there is no reason for the dynamics to be Hamiltonian and therefore we have a nonconservative formulation of quantum physics at hand. The harmonic oscillator (amongst few oth- ers) distinguishes itself because the algebra forms a finite dimensional Lie algebra; the classical and quantum (discrete) harmonic oscillator are stud- ied in a some more generality and examples are given which are neither classical, nor quantum.