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arXiv:2607.26431

The Klein-Gordon-Fock theory as a one-particle relativistic quantum mechanics

N. L. Chuprikov

quant-phSubmitted Jul 29, 2026• Announced Jul 30, 2026• 0 citations• Revised Oct 15, 2026 (v4)
v1 · Jul 29, 2026v2 · Aug 22, 2026v3 · Aug 26, 2026v4 · Oct 15, 2026
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Abstract

The Klein-Gordon-Fock (KGF) theory is presented as a single-particle relativistic quantum theory. A Schrodinger-type equation for a single-component wave function is derived. In addition to the KGF Hamiltonian, operators for momentum and orbital angular momentum for a scalar boson are identified, which act within the space of single-component wave functions. This theory is applicable to both massive and massless scalar bosons.

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8 pages

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From Primitives to the Klein–Gordon Equation

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The Klein–Gordon equation, written down in 1926 by Oskar Klein and Walter Gordon (and earlier in unpublished form by Schrödinger himself), is the natural relativistic extension of quantum mechanics for spin‑zero particles: (□ + m²c²/ℏ²) Ψ = 0 where □ = (1/c²) ∂ₜ² −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.

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ANALYTICAL SOLUTION OF THE KLEIN-FOCK-GORDON EQUATION FOR A FIELD WITH HÜLTEN POTENTIAL IN ORDINARY RELATIVISTIC QUANTUM MECHANICS

Aghaali Ali AHMADOV

In this article, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential domain is investigated within the framework of ordinary quantum mechanics. The Klein-FockGordon equation, which describes the dynamics of relativistic scalar particles, is of particular interest in the case of interactions with short-range potentials of the central type. The Hülten potential, on the other hand, has an exponentially decreasing form and is widely used in modeling atomic, molecular, and nuclear systems due to its approximation to the short-range Coulomb potential and its rapid decay at large distances. In this work, the radial Klein-Fock-Gordon equation is separated in spherical coordinates and the differential equation for the Hülten potential is reduced to a hypergeometric type equation. By applying the appropriate boundary conditions, the quantization condition for the energy spectrum is obtained and the analytical expression of the energy levels for the bound states is derived. The normalized form of the wave functions is expressed by means of special functions. Analysis of the obtained results shows that the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential leads to a clear manifestation of relativistic effects in the energy spectrum of the system and the structure of the wave functions. The energy spectrum depends not only on the parameters and quantum numbers of the potential, but also on relativistic factors introduced through the speed of light c. A comparison of the wave functions shows that the radial solutions obtained by the Klein-Fock-Gordon equation differ somewhat from those obtained by the Schrödinger equation. The shape and maximum of the radial wave function, which characterizes the probability of a particle being at a certain distance from the center, may change slightly compared to the non-relativistic case. These changes are especially noticeable in regions where the potential is deep and the interaction is stronger. Thus, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential field both clarifies the nature of relativistic effects and demonstrates agreement with the results of nonrelativistic quantum mechanics in terms of theoretical consistency and accuracy of the limit transition. This indicates that the model is a suitable theoretical framework for a more fundamental and accurate description of short-range interactions in molecular systems. Keywords: Klein-Fock-Gordon equation, Hülten potential, analytical solutions, energy spectrum, relativistic quantum mechanics.

Scientific Bulletin•Apr 1, 2026•0 citations
The Klein-Gordon-Fock (KGF) theory is presented as a single-particle relativistic quantum theory. A Schrodinger-type equation for a single-component wave function is derived. In addition to the KGF Hamiltonian, operators for momentum and orbital angular momentum for a scalar boson are identified, which act within the space of single-component wave functions. This theory is applicable to both massive and massless scalar bosons.
We show that, in the relativistic regime, scalar bosons satisfy a quantum wave equation which is quite analogous to the Dirac equation. In contrast with the Klein-Gordon equation it is first order with respect to time derivation. It leads in a regular way to the standard Schrödinger equation in the non-relativistic limit. There are two components for the wave function in this representation for the scalar boson, in a way completely analogous to the four components for the spin 1/21/21/2 fermion in the Dirac equation.
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.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.
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.
In this article, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential domain is investigated within the framework of ordinary quantum mechanics. The Klein-FockGordon equation, which describes the dynamics of relativistic scalar particles, is of particular interest in the case of interactions with short-range potentials of the central type. The Hülten potential, on the other hand, has an exponentially decreasing form and is widely used in modeling atomic, molecular, and nuclear systems due to its approximation to the short-range Coulomb potential and its rapid decay at large distances. In this work, the radial Klein-Fock-Gordon equation is separated in spherical coordinates and the differential equation for the Hülten potential is reduced to a hypergeometric type equation. By applying the appropriate boundary conditions, the quantization condition for the energy spectrum is obtained and the analytical expression of the energy levels for the bound states is derived. The normalized form of the wave functions is expressed by means of special functions. Analysis of the obtained results shows that the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential leads to a clear manifestation of relativistic effects in the energy spectrum of the system and the structure of the wave functions. The energy spectrum depends not only on the parameters and quantum numbers of the potential, but also on relativistic factors introduced through the speed of light c. A comparison of the wave functions shows that the radial solutions obtained by the Klein-Fock-Gordon equation differ somewhat from those obtained by the Schrödinger equation. The shape and maximum of the radial wave function, which characterizes the probability of a particle being at a certain distance from the center, may change slightly compared to the non-relativistic case. These changes are especially noticeable in regions where the potential is deep and the interaction is stronger. Thus, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential field both clarifies the nature of relativistic effects and demonstrates agreement with the results of nonrelativistic quantum mechanics in terms of theoretical consistency and accuracy of the limit transition. This indicates that the model is a suitable theoretical framework for a more fundamental and accurate description of short-range interactions in molecular systems. Keywords: Klein-Fock-Gordon equation, Hülten potential, analytical solutions, energy spectrum, relativistic quantum mechanics.