[Paper Review] A discrete approach to the vacuum Maxwell equations and the fine structure constant

Wolfgang Orthuber|ArXiv.org|Dec 23, 2003

advanced mathematical theories7 references3 citations

TL;DR

This paper proposes a discrete finite-difference formulation of the vacuum Maxwell equations using integer lattices and coupling factors derived from the fine structure constant. When the coupling factor is set to √α, the resulting wave packet exhibits nearly equal first two maxima and exponential growth from the sixth peak onward, suggesting a potential discrete origin for α.

ABSTRACT

We recommended consequent discrete combinatorial research in mathematical physics. Here we show an example how discretization of partial differential equations can be done and that quickly unexpected new findings can result from research in this up to now unexplored area. We transformed the vacuum Maxwell equations into finite-difference equations, provided simple initial conditions and studied the development of the electromagnetic fields using special software (see http://www.orthuber.com). The development is wave-like as expected. But it is not trivial, the wave maxima have different heights. If all (by definition minimal) finite differences of the location coordinates are multiplied by numbers (coupling factors) whose squares are equal to the fine structure constant, we noticed: 1. The first two wave maxima have nearly the same height. Of course this can be also coincidental. 2. The following maxima are at first slightly decreasing and then, beginning with the 6th maximum, exponentially increasing.

Motivation & Objective

• To explore the implications of discretizing the vacuum Maxwell equations using finite-difference calculus instead of continuous differential equations.
• To investigate whether the fine structure constant α emerges naturally from a discrete, combinatorial model of electromagnetic field propagation.
• To determine if the observed wave behavior in discrete simulations correlates with physical constants like α.
• To assess the feasibility of using finite-difference algorithms on integer lattices for approximating short-term electromagnetic field dynamics.

Proposed method

• The vacuum Maxwell equations were converted into explicit finite-difference equations using minimal finite differences as units, avoiding analytic methods like Fourier transforms or power series.
• A custom C++ software tool was developed to simulate field evolution on a 4D integer lattice, supporting complex numbers and iterative coupling between lattice points.
• Coupling factors p were applied to shift and scale field values across lattice points, with p set to √α, 1/16, or 1/8 for comparative analysis.
• The algorithm iteratively updates field components (Ex, Bz) over time, tracking amplitude evolution at the origin and spatial distribution after 150 iterations.
• Numerical stability was maintained by discarding low-magnitude lattice points after 150 iterations, confirmed not to affect the first 12 significant digits.
• Simulations were visualized via time-series plots of field amplitudes and spatial distributions, with comparisons across different coupling values.

Experimental results

Research questions

• RQ1 Does a discrete finite-difference formulation of the vacuum Maxwell equations produce wave-like field behavior consistent with classical electrodynamics?
• RQ2 Can the fine structure constant α emerge naturally from the coupling parameters in a discrete lattice model of electromagnetism?
• RQ3 What happens to field amplitude evolution when the coupling factor is set to √α, and does this yield any physically meaningful behavior?
• RQ4 Why does the field amplitude begin to grow exponentially from the sixth maximum in the discrete simulation?
• RQ5 Is the observed near-equality of the first two wave maxima in the √α case a coincidence or indicative of deeper physical significance?

Key findings

• When the coupling factor is set to √α ≈ 0.085424542921, the first two wave maxima in the electric field Ex at the origin are nearly equal, differing by less than 2%.
• For √α, the field amplitude at t=28 (the second maximum) is 0.9507, very close to the initial value of 1, with a relative deviation of less than 2%.
• From the sixth maximum onward, the field amplitudes exhibit exponential growth, with an average growth factor of approximately 1.04 per time step in the range t∈[200,1000].
• The simulation shows a localized wave packet centered at the origin after 150 iterations, indicating coherent field propagation in the discrete model.
• The exponential divergence starting at the sixth peak suggests a positive feedback mechanism from surrounding lattice points, which is not present in standard continuous formulations.
• The discrete model produces a non-unitary, non-conservative evolution unless normalized, raising questions about the physical justification for such corrections.