Lagrangian form of stochastic equations and quantum theory

D. Arsenović, Nikola Burić, Dragomir M Davidović +1 more

Hiroshi Hasegawa

The two different forms of stochastic Lagrangian in the literature which stem from a common deterministic model L(x\ifmmode \dot{}\else \.{}\fi{},x,t)=(1/2)mx${\ifmmode \dot{}\else \.{}\fi{}}^{2}$+A (x,t)x\ifmmode \dot{}\else \.{}\fi{}-V(x,t) are analyzed referring to the question of Nelson raised in his recent book Quantum Fluctuations, and a resolution is presented. It is argued that there exists a self-contained framework of stochastic mechanics, different from Nelson's original proposal, which is prescribed by (i) a=(1/2)(DD+DlstarDlstar)x for the stochastic acceleration, (ii) the same procedure as Nelson of connecting both sides of the stochastic differential equation for the process x(t) with the current and osmotic velocities but with entirely real valued wave functions, and (iii) a special relation between V and A which conforms the above L to the Onsager-Machlup Lagrangian.

Sheldon Goldstein

Vasil G. Yordanov

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.

L. de la Peña-Auerbach

Jianhao M. Yang

Stochastic mechanics is regarded as a physical theory to explain quantum mechanics with classical terms such that some of the quantum mechanics paradoxes can be avoided. Here, we propose a new variational principle to uncover more insights on stochastic mechanics. According to this principle, information measures, such as relative entropy and Fisher information, are imposed as constraints on top of the least action principle. This principle not only recovers Nelson’s theory and, consequently, the Schrödinger equation but also clears an unresolved issue in stochastic mechanics on why multiple Lagrangians can be used in the variational method and yield the same theory. The concept of forward and backward paths provides an intuitive physical picture for stochastic mechanics. Each path configuration is considered as a degree of freedom and has its own law of dynamics. Thus, the variation principle proposed here can be a new tool to derive more advanced stochastic theory by including additional degrees of freedom in the theory. The structure of Lagrangian developed here shows that some terms in the Lagrangian are originated from information constraints. This suggests that a Lagrangian may need to include both physical and informational terms in order to have a complete description of the dynamics of a physical system.