Jussi Lindgren1 and Jukka Liukkonen2
1Aalto University
2Nuclear and Radiation Safety Authority, STUK, Helsinki
Stochastic optimal control on Minkowski spacetime leads to quantum mechanics
Abstract
Considering diffusions and coordinate-invariant stochastic control restricted in the Minkowski spacetime, we argue that Minkowski metric is the reason for the imaginary structure in quantum mechanics and therefore quantum mechanics cannot be understood separately from the theory of relativity. By linearizing the Hamilton-Jacobi-Bellman equation, we obtain the famous Stueckelberg covariant wave equation. The Stueckelberg wave equation is the foundation for parameterized relativistic dynamics and could be the right way to understand quantum field theories. From the Stueckelberg wave equation we derive the Wick-rotated Telegrapher’s equation, whose solution has a finite speed of propagation. We demonstrate the link to Dirac’s equation, the Klein-Gordon equation and finally the Schrödinger equation, when passing on to nonrelativistic regimes. The linearity requirement together with the requirement of coordinate invariance lead to the conclusion that ordinary time is actually an entity living in the complex plane and undergoing a Brownian diffusion in the complex plane. The ontology of the model is conjectured as such to be based on the concept of stochastic spacetime as the control problem is based on diffusions in the Minkowski spacetime. Time and space could be seen as stochastic at Planck scales, which could produce the illusion of a random quantum forces or quantum potentials. This is a phenomenological interpretation, as the movement of the test particle is described via stochastic differential equations on the four-dimensional spacetime. The proposed talk is based on our article published in Scientific Reports.