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arXiv:2503.08110 · Journal of Physics A Mathematical and Theoretical

Preserving mass shell condition in the stochastic optimal control derivation of the Dirac equation

Vasil G. Yordanov

Published Jun 25, 2026• 0 citations
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Abstract

Abstract The Dirac equation, central to relativistic quantum mechanics, governs spin- 1 2 particles and their antiparticles, with each spinor component satisfying a Klein–Gordon-type equation—the quantum counterpart of the relativistic mass-shell condition. In our prior work (Yordanov 2024 Sci. Rep. 14 6507), we derived the Dirac equation using stochastic optimal control (SOC) theory by linearizing both the Lagrangian’s kinetic term and the Hamilton–Jacobi–Bellman (HJB) equation, but that earlier linearized derivation did not preserve the relativistic mass-shell condition. Here, we formulate the SOC problem starting from a specified covariant Lagrangian for a single relativistic charged particle in an external electromagnetic field, in which the standard relativistic square-root kinetic term and the minimal electromagnetic coupling are retained in their original form. In addition, the Lagrangian is supplemented by a covariant spin-field coupling term describing the interaction of the particle’s intrinsic spin with the electromagnetic field. Because this spin-field term is matrix-valued, it is incorporated through scalarization so that the HJB problem remains scalar. To retain a scalar HJB formulation, the present construction is restricted to electromagnetic backgrounds for which σ μ ν F μ ν admits a spacetime-independent eigenspinor; the Dirac equation is then obtained in the corresponding fixed spin sector. This yields a relativistically consistent SOC derivation in which the ℏ → 0 limit of the HJB equation recovers the proper-time relativistic Hamilton–Jacobi equation, whose stationary form yields the classical mass-shell condition, while the stationary HJB equation yields the quantum-corrected mass-shell relation. We illustrate the theory with stochastic simulations of the Dirac–Landau problem (electron in a uniform magnetic field): at the analytic optimal drift the average stochastic action attains a local minimum, and the average action components are consistent with the Dirac–Landau values within statistical uncertainty.

DOI
10.1088/1751-8121/ae82cf

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Abstract The Dirac equation, central to relativistic quantum mechanics, governs spin- 1 2 particles and their antiparticles, with each spinor component satisfying a Klein–Gordon-type equation—the quantum counterpart of the relativistic mass-shell condition. In our prior work (Yordanov 2024 Sci. Rep. 14 6507), we derived the Dirac equation using stochastic optimal control (SOC) theory by linearizing both the Lagrangian’s kinetic term and the Hamilton–Jacobi–Bellman (HJB) equation, but that earlier linearized derivation did not preserve the relativistic mass-shell condition. Here, we formulate the SOC problem starting from a specified covariant Lagrangian for a single relativistic charged particle in an external electromagnetic field, in which the standard relativistic square-root kinetic term and the minimal electromagnetic coupling are retained in their original form. In addition, the Lagrangian is supplemented by a covariant spin-field coupling term describing the interaction of the particle’s intrinsic spin with the electromagnetic field. Because this spin-field term is matrix-valued, it is incorporated through scalarization so that the HJB problem remains scalar. To retain a scalar HJB formulation, the present construction is restricted to electromagnetic backgrounds for which σ μ ν F μ ν admits a spacetime-independent eigenspinor; the Dirac equation is then obtained in the corresponding fixed spin sector. This yields a relativistically consistent SOC derivation in which the ℏ → 0 limit of the HJB equation recovers the proper-time relativistic Hamilton–Jacobi equation, whose stationary form yields the classical mass-shell condition, while the stationary HJB equation yields the quantum-corrected mass-shell relation. We illustrate the theory with stochastic simulations of the Dirac–Landau problem (electron in a uniform magnetic field): at the analytic optimal drift the average stochastic action attains a local minimum, and the average action components are consistent with the Dirac–Landau values within statistical uncertainty.
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Abstract Recent studies have extended the use of the stochastic Hamilton-Jacobi-Bellman (HJB) equation to include complex variables for deriving quantum mechanical equations. However, these studies often assume that it is valid to apply the HJB equation directly to complex numbers, an approach that overlooks the fundamental problem of comparing complex numbers when finding optimal controls. This paper explores the application of the HJB equation in the context of complex variables. It provides an in-depth investigation of the stochastic movement of quantum particles within the framework of stochastic optimal control theory. We obtain the complex diffusion coefficient in the stochastic equation of motion using the Cauchy-Riemann theorem, considering that the particle’s stochastic movement is described by two perfectly correlated real and imaginary stochastic processes. During the development of the covariant form of the HJB equation, we demonstrate that if the temporal stochastic increments of the two processes are perfectly correlated, then the spatial stochastic increments must be perfectly anti-correlated, and vice versa. The diffusion coefficient we derive has a form that enables the linearization of the HJB equation. The method for linearizing the HJB equation, along with the subsequent derivation of the Dirac equation, was developed in our previous work [V. Yordanov, Scientific Reports 14, 6507 (2024)]. These insights deepen our understanding of quantum dynamics and enhance the application of stochastic optimal control theory to quantum mechanics.
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Spinor structure is usually introduced as part of relativistic quantum theory. We show that its underlying algebra can instead emerge without quantisation from a covariant statistical description of a massive relativistic particle subject to an explicit finite-dimensional linear factorisation of the complete massive mass-shell constraint motivated by requiring the theory retain both sheets of the massive mass shell. A finite-dimensional factorisation of the complete mass-shell constraint, linear in all four momentum components, forces a Clifford algebra. In (3+1) dimensions its minimal complex representation is four-dimensional, while each on-shell factor selects a rank-two sector; in this representation statistical completeness requires an arbitrary 2×22\times22×2 matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with ℏ\hbarℏ, supplies the scale of physical spin-12\tfrac1221​. Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy within the stated deformation class leads to the Weyl--Moyal phase-space formulation of quantum mechanics.
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Here we argue that spinor structure arises naturally if relativistic statistical mechanics is formulated directly on phase spacetime. Requiring a first-order phase-spacetime description that retains both mass-shell branches leads to a Clifford factorisation of the relativistic constraint and hence to a 4×44\times44×4 spinor-matrix distribution function. We show that deformation quantisation leads to a phase-space formulation of spin quantum mechanics. We argue that projection onto positive- and negative-energy sectors recovers the standard relativistic classical transport equations in the appropriate scalar limits, while the corresponding left- and right- stargenvalue equations reproduce the constraint structure of the Dirac-Wigner formulation. The result is a phase-space route from relativistic statistical mechanics to spinor quantum mechanics, in which spin algebra emerges as the internal structure required by any relativistic statistical theory containing both mass-shell branches and the dimensions of angular momentum from quantum non-locality.