Abstract
The main contribution of this paper is to explain where the imaginary structure comes from in quantum mechanics. It is shown how the demand of relativistic invariance is key and how the geometric structure of the spacetime together with the demand of linearity are fundamental in understanding the foundations of quantum mechanics. We derive the Stueckelberg covariant wave equation from first principles via a stochastic control scheme. From the Stueckelberg wave equation a Telegrapher’s equation is deduced, from which the classical relativistic and nonrelativistic equations of quantum mechanics can be derived in a straightforward manner. We therefore provide meaningful insight into quantum mechanics by deriving the concepts from a coordinate invariant stochastic optimization problem, instead of just stating postulates.
Highlights
The main contribution of this paper is to explain where the imaginary structure comes from in quantum mechanics
Even though the Schrödinger equation is adopted as a postulate of Quantum Mechanics in the literature, we argue that it can be derived in a meaningful manner and from a didactical and pedagogical point of view, the postulate approach is not totally satisfying
As we argue that relativistic invariance is one of the building blocks in understanding quantum physics, we need to consider diffusions not in just three spatial dimensions, but we need to allow the time variable to obey a diffusion process as well
Summary
We take classical mechanics as the starting point. In line with the existing literature, we assume that nature tries to minimize the classical action, so that. Before we consider the HJB equation any further, we need to make sure that when performing stochastic optimal control on spacetimes, the expectation over spacetime is invariant with respect to coordinate transformations between reference frames. This demand means that the laws of physics are to be the same in any coordinate system, cf general relativity and special theory of relativity. The Hamiltonian is the kinetic energy (four-vectors instead of ordinary vectors) plus the potential energy, analytically continued due to the imaginary unit from the invariant volume form on Minkowski spacetime
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