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arXiv:1504.07582

Relativistic generalization of the Born rule

Kazemi, Mohammad Javad, Mohammad Hosein Barati, Rokni, Yahya, Jafar Khodagholizadeh

Published Apr 28, 2015• 0 citations
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Abstract

We have shown that the Born rule $(ρ=|ψ|^{2})$ is inconsistent with Lorentz symmetry of the Salpeter equation (square root Klein-Gordon equation). So we find relativistic modification of the Born rule as $ρ=\frac{1}{2mc^{2}}(|\sqrt{\widehat{H}-mc^{2}}ψ|^{2}+|\sqrt{\widehat{H}+mc^{2}}ψ|^{2})$ , which is consistent with Lorentz symmetry of this equation

DOI
10.48550/arxiv.1504.07582

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In this paper, we investigate the Lorentz invariance of the square root Klein-Gordon equation. This equation with interpreting ∣ψ∣2|\psi|^{2}∣ψ∣2 as position probability density is not Lorentz invariance but we can make this equation Lorentz invariant with the help of new proper interpretation of the wave function. In this regard we consider the relation between the probability density and the wave function asρ=ψψ∗+(D^ψ)∗D^ψ\rho=\psi \psi^{\ast}+ (\hat{D}\psi)^{\ast}\hat{D}\psiρ=ψψ∗+(D^ψ)∗D^ψ and show appropriate selection of the linear operatorD^\hat{D}D^the square root Klein-Gordon equation will be Lorentz invariant. In fact the solutions of this equation is equivalent to the positive energy solutions of the Dirac equation. \end{abstract}
The Klein–Gordon equation, written down in 1926 by Oskar Klein and Walter Gordon (and earlier in unpublished form by Schrödinger himself), is the natural relativistic extension of quantum mechanics for spin‑zero particles: (□ + m²c²/ℏ²) Ψ = 0 where □ = (1/c²) ∂ₜ² − ∇². The equation is second‑order in both time and space, treats them symmetrically, and reduces to the free‑particle Schrödinger equation in the non‑relativistic limit. Its plane‑wave solutions satisfy the relativistic mass‑shell condition E² = p²c² + m²c⁴. Why is a relativistic version needed at all? The Schrödinger equation describes how quantum systems evolve when they’re moving slowly compared to the speed of light — fine for atoms, molecules, and everyday chemistry, where electrons move at less than 1% of c. But the Schrödinger equation is first‑order in time and second‑order in space, treating the two on different footings. Special relativity says space and time are unified into a single four‑dimensional structure where Lorentz transformations mix them, and any equation that privileges time over space breaks Lorentz invariance. A truly fundamental quantum equation must respect this symmetry. Klein–Gordon is the minimal scalar‑field equation that does. The standard derivation runs by analogy: start with the relativistic energy–momentum relation E² = p²c² + m²c⁴, replace E by iℏ∂ₜ and p by −iℏ∇, apply both sides to a wavefunction. This produces the right equation but it’s not really a derivation — it’s a guess motivated by the Schrödinger operator‑substitution procedure. This walkthrough shows how the Event Density framework derives the Klein–Gordon equation from substrate primitives — the same primitives that gave the Born rule, the Schrödinger equation, and the Bell–Tsirelson bound — plus the commitment to Lorentz covariance. The participation measure becomes Lorentz‑covariant. The unique second‑order Lorentz‑scalar differential operator on a scalar field is the d’Alembertian; the minimum‑order Lorentz‑scalar linear equation is therefore (□ + M²) Ψ = 0. Dimensional analysis fixes M² = m²c²/ℏ². Local U(1) gauge invariance forces the covariant derivative and the conserved four‑current. The non‑relativistic limit, via rest‑energy factorization, recovers the free‑particle Schrödinger equation exactly. The structural payoff: the Klein–Gordon equation is what you get when the substrate ontology is pushed through a Lorentz‑covariant reformulation. No new primitives are required. Relativistic and non‑relativistic quantum mechanics are not two theories glued together by analogy; they are the same substrate ontology applied at different velocity regimes. The walkthrough is honest about the known caveats — negative‑energy solutions and the non‑positive‑definite j₀ current — both of which resolve at the Dirac level and at QFT.