Stochastic quantization of relativistic theories

Abstract

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Derivation of Dirac equation from the stochastic optimal control principles of quantum mechanics

“…There are a variety of approaches 18 to the derivation of the Dirac equation, but none of them is based on stochastic optimal control theory. Although there have been other investigations that aim to construct a stochastic optimization theory for relativistic quantum mechanics 12,[19][20][21][22] , none of them have succeeded in deriving the Dirac equation or in explaining the nature of the Dirac spinor. In this paper, we extend the stochastic optimal control theory of quantum mechanics to also derive the Dirac equation 23 and to provide new insights into the nature of Dirac's four-component spinor ψ.…”

Stochastic Mechanics and the Unification of Quantum Mechanics with Brownian Motion

“…The theory of stochastic mechanics has been extended in various ways. In particular, the extension of stochastic mechanics to a single relativistic particle is now well understood [53][54][55][56][57][58][59][60][61]. Also, the generalization of stochastic mechanics to particles moving on (pseudo-)Riemannian manifolds [23,[62][63][64][65][66] is well established.…”

Spacetime Stochasticity and Second Order Geometry

“…Extensions of second order geometry to Lorentzian manifolds are straightforward, as the framework is developed for any smooth manifold with a connection [5]. Furthermore, similar to a classical relativistic theory, the formulation of a relativistic theory on Lorentzian manifolds introduces a relativistic constraint equation [11,12]. The velocity field is then a solution of the system…”

“…, on these segments using a Wick rotation. The stochastic motion is then given by the solution of the Itô system [4,11]…”