Preserving Mass Shell Condition in the Stochastic Optimal Control Derivation of the Dirac Equation

Technical Summary: Preserving Mass Shell Condition in the Stochastic Optimal Control Derivation of the Dirac Equation

Problem Statement
The Dirac equation is the cornerstone of relativistic quantum mechanics for spin-1/2 particles, ensuring that each spinor component satisfies the Klein-Gordon equation and thus the relativistic mass-shell condition ($E^2 = p^2c^2 + m^2c^4$). While stochastic mechanics has successfully recovered the Schrödinger and Klein-Gordon equations, deriving the Dirac equation via Stochastic Optimal Control (SOC) has historically been elusive. Previous attempts, including the authors' prior work [10], relied on linearizing the relativistic kinetic term ($-mc\sqrt{u^\mu u_\mu}$) and the Hamilton-Jacobi-Bellman (HJB) equation. While this linearization facilitated the emergence of a Dirac-like structure, it failed to preserve the mass-shell condition; specifically, the resulting factored equation omitted the necessary $+mc$ term in the conjugate operator, preventing the recovery of the Klein-Gordon equation for individual components.

Methodology
This work reformulates the SOC derivation of the Dirac equation by retaining the standard relativistic square-root kinetic term in its original nonlinear form, rather than linearizing it. The methodology proceeds as follows:

1. Covariant Lagrangian Formulation: The authors define a Lorentz-covariant Lagrangian for a single relativistic charged particle with spin-1/2 in an external electromagnetic field. This Lagrangian includes:

   • The standard nonlinear kinetic term: $-mc\sqrt{w^\mu w_\mu}$.
   • Minimal electromagnetic coupling: $-\epsilon e A_\mu w^\mu$.
   • A covariant spin-field coupling (Pauli term): $-\frac{e\hbar}{2m}\sigma_{\mu\nu}F^{\mu\nu}$.

2. Scalarization for Scalar HJB: Since the spin-field term $\sigma_{\mu\nu}F^{\mu\nu}$ is matrix-valued, it cannot be directly inserted into the scalar HJB equation. To resolve this, the authors introduce a "scalarization spinor" $\chi$, a constant, spacetime-independent eigenspinor of the spin-field matrix ($\sigma_{\mu\nu}F^{\mu\nu}\chi = f\chi$). They define a scalarized Lagrangian $L_\chi$ via the bilinear projection $P_\chi[M] = \frac{\chi^\dagger S M \chi}{\chi^\dagger S \chi}$. This restricts the derivation to electromagnetic backgrounds admitting such an eigenspinor (e.g., uniform magnetic fields) and fixes the system to a specific spin sector.

3. Complex SOC and HJB Derivation: The problem is cast in Complex SOC theory, where the particle moves in four-dimensional complex spacetime. The value function $J$ satisfies the HJB equation. By applying the "weak condition" $w^\mu w_\mu = c^2$ only after differentiation (rather than linearizing the kinetic term), the authors derive an optimal control policy $w_\mu$ dependent on the gradient of the value function.

4. Transformation to Dirac Equation:

   • The HJB equation is transformed using the Hopf-Cole map ($J = i\epsilon\hbar \ln \phi$), converting the nonlinear scalar HJB into a linear second-order equation for the scalar field $\phi$.
   • This linear equation is shown to be equivalent to a "squared-Dirac" equation: $(i\hbar\gamma^\nu\partial_\nu - e\gamma^\nu A_\nu - mc)(i\hbar\gamma^\mu\partial_\mu - e\gamma^\mu A_\mu + mc)\phi = 0$.
   • The Dirac spinor $\psi$ is then constructed as $\psi = (i\hbar\gamma^\mu\partial_\mu - e\gamma^\mu A_\mu + mc)\phi \chi$.
   • In the stationary limit ($\partial_\tau \to 0$), this yields the standard Dirac equation: $(i\hbar\gamma^\mu\partial_\mu - e\gamma^\mu A_\mu - mc)\psi = 0$.

5. Mass-Shell Analysis: The authors analyze the mass-shell condition in both the classical ($\hbar \to 0$) and quantum limits. They demonstrate that the quantum-corrected mass-shell relation includes contributions from the Itô quadratic covariation (quantum potential) and the Pauli term. Crucially, they show that the physical bilinear observable $\langle \pi_\mu \pi^\mu \rangle_{\text{phys}}$ recovers the mass-shell condition exactly when the Itô covariation correction is applied, canceling the quantum potential term and leaving the Pauli invariant.

Key Results

• Restoration of the Mass-Shell Condition: Unlike previous linearized derivations, this approach preserves the relativistic mass-shell condition. The derived equation naturally factors into the Dirac operator and its conjugate, ensuring that solutions satisfy the Klein-Gordon equation.
• Spin-Sector Restriction: The derivation is valid for backgrounds where a spacetime-independent eigenspinor of $\sigma_{\mu\nu}F^{\mu\nu}$ exists. This allows the matrix-valued spin interaction to be treated as a scalar within a fixed spin sector.
• Numerical Validation (Dirac-Landau Problem): The theory was tested numerically using an electron in a uniform magnetic field (Dirac-Landau problem).
  • Action Minimization: Stochastic simulations confirmed that the average stochastic action attains a local minimum at the analytically derived optimal drift $w^\star$.
  • Quantitative Agreement: The average components of the stochastic action (kinetic, electromagnetic, and spin) computed under the optimal drift matched the analytic Dirac-Landau values within 95% confidence intervals.
  • Local Optimality: Perturbation analysis showed that deviations from the optimal drift result in a quadratic increase in the expected action, confirming local minimality.
• Geometric Interpretation: The authors visualized the complex momentum space, showing that the quantum mass shell is a deformation of the classical weak mass shell ($\pi_\mu \pi^\mu = m^2c^2$) by small quantum corrections, which remain bounded and oscillatory.

Significance and Claims
The paper claims to provide a "relativistically consistent SOC derivation" of the Dirac equation that resolves the inconsistency of earlier linearized approaches regarding the mass-shell condition. By retaining the nonlinear kinetic term and utilizing scalarization for the spin-field coupling, the authors bridge stochastic optimal control with relativistic quantum mechanics without sacrificing the fundamental energy-momentum relation.

The work highlights that the Dirac equation can be derived from first principles of stochastic mechanics (specifically Complex SOC) where the $\hbar \to 0$ limit recovers the proper-time relativistic Hamilton-Jacobi equation. The authors position this as a robust framework where the recovery of spectra and wavefunctions is framed as a high-dimensional optimization problem over drift fields, potentially amenable to future variational quantum algorithms (VQE, QAOA). The paper does not propose new experimental setups but offers a theoretical and numerical validation of the SOC framework for relativistic spin-1/2 particles.