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Journal of Physics A Mathematical and General

Evolution time Klein-Gordon equation and derivation of its nonlinear counterpart

P. N. Kaloyerou, Jean Pierre Vigier

Published Mar 21, 1989• 13 citations
Publisher

Abstract

Recently, relativistic quantum mechanical wave equations involving an extra scalar evolution parameter (analogous to proper time) have been increasingly studied in the literature. Since the physical meaning of this parameter is still very much a matter of discussion the authors aim to give it a clear physical meaning and then to proceed to a stochastic derivation of a nonlinear evolution time Klein-Gordon equation with a view to representing possible vacuum dissipative effects.

DOI
10.1088/0305-4470/22/6/018

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arXiv:1008.4224•Aug 25, 2010•3 citations
Recently, relativistic quantum mechanical wave equations involving an extra scalar evolution parameter (analogous to proper time) have been increasingly studied in the literature. Since the physical meaning of this parameter is still very much a matter of discussion the authors aim to give it a clear physical meaning and then to proceed to a stochastic derivation of a nonlinear evolution time Klein-Gordon equation with a view to representing possible vacuum dissipative effects.
We derive the Schr¨odinger, Dirac, and Klein-Gordon equations from the scalar-time continuity law of Time-Scalar Field Theory (TSFT). By promoting evolution to the scalar-time parameter Θ and introducing a local time-rate field α(x), classical Hamiltonian and quantum operator dynamics unify under the relation dOdΘ = α{O,H} (classical) and dO dΘ = i¯hα[ ˆH ,O] (quantum). Schr¨odinger’s equation follows from ∂Θ = α∂t; the Dirac equation emerges from a Θ-weighted tetrad with a spin-connection term ∝ ∂μ ln α; and the Klein-Gordon equation arises as the relativistic scalar field equation in the effective metric induced by Θ. Standardquantum mechanics is recovered when ∇α = 0, while TSFT predicts falsifiable gradient corrections accessible to precision matter-wave and spinor interferometry. We supply quantitative estimates (e.g. δϕ ∼ 10−3 rad for Q = 109 cavities) and cross-link TSFT’s microphysics to its cosmological continuity, emphasizing multi-scale coherence and experiment-first falsifiability.
The Klein–Gordon equation, written down in 1926 by Oskar Klein and Walter Gordon (and earlier in unpublished form by Schrödinger himself), is the natural relativistic extension of quantum mechanics for spin‑zero particles: (□ + m²c²/ℏ²) Ψ = 0 where □ = (1/c²) ∂ₜ² −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.
A derivation of the Klein-Gordon wave equation without substituting the corresponding quantum mechanical energy and momentum operators in Einstein's relativistic energy-momentum relationship is presented. An additional discussion is made to motivate readers to understand the distinction between the quantum wave equation and the governing law in the context of quantum mechanical formalisms. A resemblance of mathematical structure between the Maxwell wave equation and the deduced Klein-Gordan equation is shown. It is also noted that the wave velocity of the matter wave becomes equal to photon velocity for massless cases. This can be useful to explain why light equally shows matter and particle behaviour.
The Klein-Gordon equation is often created by taking the deterministic relativistic invariant EE = pp+ momo (c=1), replacing E with E-V(x), E=id/dt partial and p with -id/dx partial and having the derivatives act on a function W(x,t). This seems to suggest that the starting point is deterministic classical mechanics (special relativistic form) E,p equation with stochasticity somehow introduced by having average E and p follow from the action of the operators id/dt partial and -id/dx partial on a stochastic function W(x,t). It also makes it somewhat difficult to understand, from first principles, why one would create a linear equation in E,p (compatible with EE=pp+momo), i.e. the Dirac equation. We suggest that the entire approach to Klein-Gordon equation is stochastic, with the EE=pp+momo equation appearing as an invariant expression of average values which describes motion as seen from different frames moving with constant average velocity v. In such a case, one is not limited to the form EE=pp+momo and there is nothing unusual about trying to create an equation linear in E and p. In particular, we argue that the free particle form exp(-iEt) exp(ipx) follows from two separate maximizations of Shannon’s entropy subject to complex constraints (for periodicity). As a result, free motion with a constant E and p is already stochastic as seen from a photon moving in a region with an index of refraction n or a particle moving in a region with constant potential. Changing to a potential V(x) leads to the notion of an average kinetic energy because a constant p is associated with a distribution in x i.e. exp(ipx). One may think in terms of these average values in order to consider frames moving with constant v. This leads to both -Et+px -EE+pp=momo being invariants. If a V(x) is present, one may replace E with E-V(x), but then needs a statistical average Sum over p a(p)exp(ipx) to describe pp.. The time portion may still be exp(-iEt) if one has a bound system characterized by a single average energy. This yields the Klein-Gordon equation.There is, however, no particular reason why one could not use an equation linear in E and p (consistent with -EE+pp=momo), because the latter follows form the an average v (deterministic view) of the stochastic exp(-iEt+ipx) two-dimensional probability.
We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r,t)=ψ(r)e−iEt/ℏΨ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}Ψ(r,t)=ψ(r)e−iEt/ℏ, and define m=E/c2m=E/c^2m=E/c2, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.