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

arXiv:2607.26431 · quant-ph · Submitted 2026-07-29 · Read on arXiv

Listen

Radio episode about this paper

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "The Klein-Gordon-Fock theory as a one-particle relativistic quantum mechanics".

Mira: The gist: A two-stage formalism is developed that allows one to represent the Klein–Gordon–Fock (KGF) theory (with vector potential) as one-particle relativistic quantum mechanics.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So we’re wrapping up on "The Klein-Gordon-Fock theory as a one-particle relativistic quantum mechanics". The authors essentially showed how you can take the KGF equation and represent it in a standard way.

Mira: They did this by first using a Hamiltonian pseudo-quantum mechanical form in spinor space, then rewriting that into the one component wave function form where they get that standard Hamiltonian quantum mechanical form.

Lev: So what does this actually mean for someone building real hardware? Does it change the practical requirements for simulating or running these kinds of relativistic systems?

Kai: It means if you are working with a scalar boson, you have a clear, standard equation—the one they derived—that you can work with in the space of single-component wave functions.

Mira: The authors provide this general solution for the free KGF equation and show how it relates to that quantum mechanical equation for a free scalar boson.

Lev: It’s less about a new physical theory and more about creating a solid mathematical machinery to model these particles using familiar quantum mechanics tools.

Kai: It’s about showing that we can map the complicated relativistic structure onto something recognizable in standard QM, even if the starting point was tricky.

Mira: The paper is all about establishing this connection between the relativistic wave equations and a standard one-particle quantum mechanical framework for scalar bosons.

Conclusion: Kai: So we’re wrapping up on "The Klein-Gordon-Fock theory as a one-particle relativistic quantum mechanics".

Mira: It basically says you can take that messy KGF equation and put it into a standard one-particle quantum mechanics framework.

Lev: So, if this works, what does it mean for the actual hardware we’re trying to build?

Kai: Well, the paper shows they can find a standard Hamiltonian form for these scalar bosons using just one component wave function.

Mira: The authors built this by starting with a pseudo-quantum mechanical form in spinor space and then mapping it down to the single component. That's the core trick here.

Lev: And what about that mapping? Does it introduce any extra complexity that makes error correction harder than we already thought?

Kai: They find a Hamiltonian operator, H plus, which is bounded from above using an operator eta minus, resulting in a specific form for the energy.

Mira: The result is they get the standard free scalar boson solution in momentum space, connected to the KGF solution by just swapping some functions. It’s a clean link.

Lev: But this is for a free particle, right? What about interacting systems? How does this formalism handle coupling terms when you actually try to build something complex?

Kai: That's what they didn't focus on in the main derivation, but the structure suggests it *could* be extended if you keep that Hamiltonian pseudo-QM idea.

Mira: The authors are setting up a solid foundation for a single particle description, which is a huge step for theorists looking at how to bridge scalar field theory with particle physics.

Lev: It gives us a better target equation to test our error correction codes against, but we still need the full interaction terms defined in this new framework.

Kai: So, the main thing is that they’ve successfully shown the mathematical machinery exists to describe a free scalar boson using standard quantum mechanics language.

Tomsk State Pedagogical University

quant-ph

Submitted: 2026-07-29

Updated: 2026-10-15

Comments: 8 pages

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 79/100

The gist: The gist: A two-stage formalism is developed that allows one to represent the Klein–Gordon–Fock (KGF) theory (with vector potential) as one-particle relativistic quantum mechanics.

Key concepts

Klein-Gordon-Fock (KGF) Theory
This is a relativistic quantum theory describing scalar bosons, which can be either massive or massless. It arises from applying a variational principle to a specific Lagrangian density. The resulting equation governs the behavior of these particles in both space and time.
Hamiltonian Pseudo-QM Form
The authors introduce an intermediate step where the KGF equation is treated using two-component spinors, leading to a Hamiltonian form. This form involves functions $\phi_1$ and $\phi_2$, which simplify the original equation into a standard Schrödinger-type equation of the form $i\hbar\partial\Phi/dt = H\Phi$. This step bridges the gap between relativistic dynamics and quantum mechanics.
One-Component Wave Function
Since scalar bosons have no spin, their state can be described by a single function, $f$. The authors relate the two-component spinor components ($\phi_1$, $\phi_2$) to this single function using angular operators. This mapping allows the complex relativistic solutions to be expressed in terms of this simpler, one-component quantum mechanical wave function.
Operator Mapping
The formalism extends to any self-adjoint operator $A$ by defining a corresponding one-component counterpart, $O\eta$. This shows how operators like momentum and velocity transform when moving from the spinor space to the single-component wave function space. For example, the velocity operator $\vec{v}$ is mapped to $\vec{v}\eta = c\vec{p}/p^2 + m^2c^2$.

Summary

The gist: A two-stage formalism is developed that allows one to represent the Klein–Gordon–Fock (KGF) theory (with vector potential) as one-particle relativistic quantum mechanics.

Introduction and Motivation

The paper aims to refute the view that a consistent one-particle formulation of both the Klein-Gordon-Fock (KGF) equation and the Dirac equation is impossible. The authors establish two generic properties that distinguish Schrödinger and Pauli equations from relativistic ones.

(1) Both linear differential equations have “a Hamiltonian form”: iħ∂ψ/∂t = Hψ, where Hˆ is the Hamiltonian containing all the information about the closed one-particle system under study.

(2) The number of components of the wave function ψ in each of these two equations is equal to the number of internal degrees of freedom of a particle.

Background and KGF Equation

The KGF equation arises from the variational principle applied to a Lagrangian density L = 1/2m (πµΨ)∗(πµΨ) − m 2c 2Ψ∗Ψ. The resulting equation is expressed as (πµπµ − m 2c 2)Ψ ≡ D2t − c2⃗p2 − m2c4Ψ = 0.

The continuity equation derived from this system involves the charge density ρ and current density ⃗j, where ρ can take negative values and is interpreted as a charge density rather than a probability density. Furthermore, the energy-momentum tensor Tµν contains components whose time components are interpreted as the energy density of the scalar boson field and its momentum density.

Hamiltonian Pseudo-QM Form

The authors introduce a Hamiltonian pseudo-QM form for the KGF equation by assuming φ = 0 and A⃗ = A⃗(⃗r). This leads to the introduction of functions ϕ1 and ϕ2, which reduce solving Eq. (4) to solving the system iħ∂Φ/∂t = HΦ, where H = mc2 −cπ−cπ−mc2.

This system yields a continuity equation ∂/∂t(Φ†Φ) ≡ Ns, which is shown to be preserved over time for states in L2(R3), leading to the probability density being defined by ⟨ΦΦ⟩ = 1/2ϵ2 ∫R3 (DtΨ)∗(DtΨ) + c2(πΨ∗)(πΨ) + m2c4Ψ∗Ψ d3⃗r. This norm is then equated to mc2/ϵ2 ∫R3 T00(⃗r)d3⃗r = 1.

Transition to a One-Component Wave Function

Since a particle with spin 0 has only one internal degree of freedom, the state must be specified by a one-component wave function f. The authors assume the existence of angular operator η and normalized function f such that ϕ1 = cos ηf and ϕ2 = sin ηf. Substituting these into the system derived from Eq. (8) leads to compatibility conditions, specifically tan η = -Dt − mc2/cπ = -cπDt + mc2. This implies that the operator Dt must satisfy the equation (Dt)2 = c2π2/2 + m2c4.

Hamiltonian Quantum-Mechanical Form

The search for a Hamiltonian quantum-mechanical form of the KGF equation is completed by relating the spinor components back to the one-component function f. This results in the operator H(+) being found, which is derived from H(+) = cos η H11 cos η + cos η H12 sin η + sin η H21 cos η + sin η H22 sin η. Using the specific values for the components, the final form obtained is H(+) = cπ2/2 + m2c2. This operator is bounded from above when using the operator η(−), resulting in H(−) = -cπ2/2 + m2c2.

Observables’ Operators

The formalism extends to any selfadjoint operator A, which can be written as a 2 × 2 matrix O. The condition for the one-component counterpart Oη is given by Oη = cos η ν1 cos η + cos η ν2 sin η + sin η ν†2 cos η + sin η ν3 sin η. For the velocity operator ⃗v = ∂H/∂p, its one-component counterpart takes the form ⃗vη = c⃗p/p2 + m2/c2. In the ultrarelativistic limit, the velocity operator is shown to be ⃗vη ≈ c⃗p/p.

Solution of the Quantum Mechanical Equation

The general solution for a free relativistic KGF particle in momentum space is found to be f(k, t) = A(k)e−iE(k)t/ħ, where E(k) = c√ħ2/2k2 + m2c2. The corresponding solution for the quantum mechanical equation is f(r, t) = c2/ϵ (2π3/2) ∫R3 pM + µ (√M + i√µ) B(k)e−iE(k)t/ħ. This solution is connected to the KGF solution by the relationship A(k) = c2/ϵ (√M + i√µ) pM + µ B(k), and only f(r, t) represents a wave function describing the quantum mechanical state of a free scalar boson.

Conclusion

The KGF theory for φ = 0 and A⃗(⃗r) is presented as a single-particle relativistic quantum theory by first writing the KGF equation in a Hamiltonian pseudoquantum mechanical form in the space of two-component spinors. Subsequently, a quantum-mechanical equation is constructed in the space of single-component wave functions. The paper provides a general solution for the free KGF equation and shows how it relates to the quantum mechanical equation for a free scalar boson. This formalism successfully connects the relativistic wave equations to a standard one-particle quantum mechanical framework.

References

[1] Greiner W. Relativistic quantum mechanics. Wave equations. Third Edition. Springer. 1987 [2] Berestetsky V.B., Lifshits E.M., Pitaevsky L.P., Quantum electrodynamics. 1979 (in Russian) [3] N. L. Chuprikov, Dirac Theory as a One-Particle Relativistic Quantum Mechanics in the Space of Unit Two-Component Spinors. Brazilian Journal of Physics (2026) 56:178 https://doi.org/10.1007/s13538-026-02101-y [4] H. Feshbach and F. Villars, Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles. Ref. of Mod. Phys. V.30, No.

Improvements for AI systems

  1. The AI system can accurately represent relativistic scalar boson states by solving a Hamiltonian QM form derived from a two-stage formalism, specifically finding A wave function describing the quantum-mechanical state of a free (massive) scalar boson.

  2. The system can model the evolution of a free KGF particle by using the solution derived in Section 7, which is given by Ψ(r, t) = 1/(2π)(3/2) ∫R cubed B(k)e(-iE(k)t/ħ)d 3k, where E(k) is the relativistic energy.

  3. The system can calculate observables like velocity and position in the nonrelativistic limit using Hψ ≈ π 2/(8m2c2) for position, demonstrating how it transitions from a relativistic description to a familiar quantum mechanical one.

  4. The AI can compute the one-component quantum mechanical counterpart of operators, such as the velocity operator where ⃗vη ≈ c⃗p/p, allowing it to generate physically meaningful predictions in the ultrarelativistic limit.

  5. The system can handle external vector fields by utilizing the formalism derived for KGF theory with a vector potential, specifically by solving equations involving H(+) = c 2π/2 + m 2c 2 for an external field scenario.

Related papers