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arXiv:quant-ph/9901017

Thoughts on Commutation Relations and Measurement Accuracy

H. Pierre Noyes

Published Jan 8, 1999• 0 citations
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Abstract

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).

DOI
10.48550/arxiv.quant-ph/9901017

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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).
It is well-known that the coordinate as a continuous variable, consisting of a set of all points between 0 and LLL contradicts the observability of measurement. In other words there might exist a fundamental length in nature, such as the Planck length λPλ_PλP​, so that it is not possible to measure a position coordinate with accuracy smaller than this fundamental length. It is therefore necessary to investigate the formulation of quantum mechanics using only discrete variables as coordinates. To investigate all of quantum mechanics or any branch of physics from this approach is of course a daunting task and thus it is worthwhile to consider a specific simple problem in order to formulate the basic ideas. In this note we compare the solutions of Schrodinger equation for one-dimensional free particle under the usual space-time continuum with those that are obtained when space-time is assumed to be quantized using a simple model. For this purpose, we replace the derivatives occurring in Schrodinger equation with the corresponding discrete derivatives. We compute the probability density (and the probability current) under the two scenarios; they turn out to be quite different in the two cases. We also obtain the operator identity for the commutator [p,x]q[p,x]^q[p,x]q under the assumption of quantized space-time and contrast it with the usual commutator [p,x][p,x][p,x].
This work provides a relativistic, digital quantum simulation scheme for both 2 + 1 and 3 + 1 dimensional quantum electrodynamics (QED), based on a discrete spacetime formulation of theory. It takes the form of a quantum circuit, infinitely repeating across space and time, parametrised by the discretization step Δ t = Δ x . Strict causality at each step is ensured as circuit wires coincide with the lightlike worldlines of QED; simulation time under decoherence is optimized. The construction replays the logic that leads to the QED Lagrangian. Namely, it starts from the Dirac quantum walk, well-known to converge towards free relativistic fermions. It then extends the quantum walk into a multi-particle sector quantum cellular automata in a way which respects the fermionic anti-commutation relations and the discrete gauge invariance symmetry. Both requirements can only be achieved at cost of introducing the gauge field. Lastly the gauge field is given its own electromagnetic dynamics, which can be formulated as a quantum walk at each plaquette.
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.
Standard results of electromagnetic theory are derived from the direct interaction of macroscopic quantum systems; the only assumptions used are the Einstein-deBroglie relations, the discrete nature of charge, the Green’s function for the vector potential, and the continuity of the wave function. No reference is needed to Maxwell’s equations or to traditional quantum formalism. Correspondence limits based on classical mechanics are shown to be inappropriate. “But the real glory of science is that we can find a way of thinking such that the law is evident .” — R. P. Feynman
An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the recently proposed exact discretization of derivatives, which is based on differences that are represented by infinite series. This new mathematical tool allows us to build sensible discrete quantum mechanics based on the suggested differences and includes the correct canonical commutation and uncertainty relations.