Quantum Scalar Field Theory Based on the Extended Least Action Principle

TL;DR

This paper introduces an extended least action principle incorporating quantum fluctuations and information metrics, deriving the Schrödinger equation for scalar fields and generalizing quantum field theory.

Contribution

It extends the least action principle to quantum scalar fields by incorporating information metrics and fluctuations, deriving key quantum equations from a unified variational approach.

Findings

Derived Schrödinger equation for scalar fields from the extended principle

Established a link between information metrics and quantum fluctuations

Generalized the Schrödinger equation using relative entropy

Abstract

Recently it is shown that the non-relativistic quantum formulations can be derived from a least observability principle [36]. In this paper, we apply the principle to massive scalar fields, and derive the Schr\"{o}dinger equation of the wave functional for the scalar fields. The principle extends the least action principle in classical field theory by factoring in two assumptions. First, the Planck constant defines the minimal amount of action a field needs to exhibit in order to be observable. Second, there are constant random field fluctuations. A novel method is introduced to define the information metrics to measure additional observable information due to the field fluctuations, \added{which is then converted to the additional action through the first assumption.} Applying the variation principle to minimize the total actions allows us to elegantly derive the transition probability of field fluctuations, the uncertainty relation, and the Schr\"{o}dinger equation of the wave functional. Furthermore, by defining the information metrics for field fluctuations using general definitions of relative entropy, we obtain a generalized Schr\"{o}dinger equation of the wave functional that depends on the order of relative entropy. Our results demonstrate that the extended least action principle can be applied to derive both non-relativistic quantum mechanics and relativistic quantum scalar field theory. We expect it can be further used to obtain quantum theory for non-scalar fields.