Abstract
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 Nelsonâs 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.
Highlights
Quantum mechanics is one of the most successful physical theories and has been experimentally confirmed extensively, there are many fundamental questions still left unanswered
Over the years in the modern history of quantum physics, many more theories and interpretations have been developed.1,2. Among these theories and interpretations, stochastic mechanics is of particular interest because it aims at deriving quantum mechanics from classical physics concepts with an additional assumption that a physical system is constantly undergoing a stochastic process
To search the dynamics equations for the forward and backward path configurations, we first introduce the relative entropy of the forward and backward paths, as this in the end leads to a constraint in the stochastic variational principles we propose here
Summary
Quantum mechanics is one of the most successful physical theories and has been experimentally confirmed extensively, there are many fundamental questions still left unanswered. The origin of probability in quantum mechanics is not clearly understood. The meaning of the wave function, especially the interpretation of wave function collapse in a measurement, has been always a debated topic. These questions were not fully addressed by the traditional Copenhagen interpretation. Over the years in the modern history of quantum physics, many more theories and interpretations have been developed.. There is no need to introduce concepts such as probability amplitude, wave function, or Bornâs rule as fundamental elements for the quantum theory. Instead, they are secondary and can be derived
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