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The RQM triangle: A paradigm for relativistic quantum mechanics

H. Pierre Noyes

Published Feb 1, 1992• 4 citations
OpenAlex

Abstract

A simple way to relate Lorentz transformations to a finite and discrete model of counter firings is presented. We first abstract from counter firings with finite resolution to rational fraction velocities and a finite step-length. Discrete Lorentz transformations and quantized rotations follow. These are encoded by three integers and a triangle with integer sides: the RQM Triangle. The double slit experiment allows us to observe quantized space steps and measure invariant step lengths for any particle in rational ratio to one such length for any convenient reference particle. A discrete version of the Mandelstam analysis of elastic and anelastic scattering is implied. The results fit naturally into the bit-string formalism used in earlier work.

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arXiv:quant-ph/9901017•Jan 8, 1999•0 citations
A simple way to relate Lorentz transformations to a finite and discrete model of counter firings is presented. We first abstract from counter firings with finite resolution to rational fraction velocities and a finite step-length. Discrete Lorentz transformations and quantized rotations follow. These are encoded by three integers and a triangle with integer sides: the RQM Triangle. The double slit experiment allows us to observe quantized space steps and measure invariant step lengths for any particle in rational ratio to one such length for any convenient reference particle. A discrete version of the Mandelstam analysis of elastic and anelastic scattering is implied. The results fit naturally into the bit-string formalism used in earlier work.
Conventional theories take the structure of relativistic quantum mechanics as given.The two empirical constants c and ft are connected to the arbitrary historical standards of mass, length and time by various, hopefully self-consistent, means.A third fundamental constant such as e 2 , m t , m p , Mpunck has to be taken from experiment before theoretical "predictions" can be attempted.Our theory [H.P.Noyes and D.O.McGoveran, Physics Essays 2, 76 (1989)] is based on the principles of finiteness, discreteness, finite contputability, absolute non-uniqueness, and strict construction.We construct (rather than postulate) the limiting velocity and discrete eventB, and then derive the Lorentz transformations and the non-commutativity of position and velocity.
We base our theory of physics and cosmology on the five principles of finiteness, discreteness, finite computability, absolute non-uniqueness, and strict construction. Our modeling methodology starts from the current practice of physics, constructs a self-consistent representation based on the ordering operator calculus and provides rules of correspondence that allow us to test the theory by experiment. We. use program universe to construct a growing collection of bit strings whose initial portions (labels) provide the quantum numbers that are conserved in the events defined by the construction. The labels are followed by content strings which a,re used to construct event-based finite and discrete coordinates. On general grounds such a theory has a limiting velocity, and positions and velocities do not commute. We therefore reconcile quantum mechanics with relativity at an appropriately fundamental stage in the construction. We show that events in different coordinate systems are connected by the appropriate finite and discrete version of the Lorentz transformation, that S-momentum is conserved in events, and that this conservation
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.
We construct discrete space-time coordinates separated by the Lorentz-invariant intervals h/me in space and h/mc2 in time using discrimination (XOR) between pa?rs of independently generated bit-strings; we prove that if this space is homogeneous and isotropic, it can have only 1, 2 or 3 spatial dimensions once we have related time to a global ordering operator. On this space we construct exact combinatorial expressions for free particle wave functions taking proper account of the interference between indistinguishable alternative paths created by the construction. Because the end-points of the paths are fixed, they specify completed processes; our wave functions are “born collapsed”. A convenient way to represent this model is in terms of complex amplitudes whose squares give the probability for a particular set of observable processes to be completed. For distances much greater than h/me and times much greater than h/mc2 our wave functions can-be approximated by solutions of the free particle Dirac and Klein-Gordon equations. Using an eight-counter paradigm we relate this construction to scattering experiments involving four distinguishable particles, and indicate how this can be used to calculate electromagnetic and weak scattering processes. We derive a non-perturbative formula relating relativistic bound and resonant state energies to mass ratios and coupling constants, equivalent to our earlier derivation of the Bohr relativistic formula for hydrogen. Using the Fermi-Yang model of the pion as a relativistic bound state containing a nucleon-antinucleon pair, we find that (GzN)2 = (2mN/m,)2- 1.
We show that measuring the trajectories of charged particles to finite accuracy leads to the commutation relations needed for the derivation of the free space Maxwell equations using the {\it discrete ordered calculus} (DOC). We note that the finite step length derivation of the discrete difference version of the single particle Dirac equation implies the discrete version of the p,q commutation relations for a free particle. We speculate that a careful operational analysis of the change in momenta occurring in a step-wise continuous solution of the discrete Dirac equation could supply the missing source-sink terms in the DOC derivation of the Maxwell equations, and lead to a finite and discrete (``renormalized'') quantum electrodynamics (QED).