Quantum Scalar Field Theory Based on the Extended Least Action Principle

TL;DR

The paper extends the extended least action principle, or least observability principle, from non-relativistic quantum mechanics to relativistic quantum scalar field theory to derive the Schrödinger equation for the wave functional of a massive scalar field. Using two core assumptions—$\ abla$the Planck constant as a discrete unit of observable action and constant local field fluctuations—the authors define an information metric $I_f$ via relative entropy and minimize the total observable information to recover quantum dynamics. They derive a Gaussian transition density for field fluctuations, establish a field–momentum uncertainty relation, and obtain the Schrödinger equation $i\\hbar\\partial_t\\Psi[\\phi,t] = [ -\\frac{\\hbar^2}{2}\\int d^3x\\frac{\\delta^2}{\\delta\\phi(x)^2} + V(\\phi(x)) ]\\Psi[\\phi,t]$, with a generalized family of equations under Rényi divergences. The framework unifies non-relativistic and relativistic quantum theories within an information-theoretic setting, offering a pathway to curved spacetime extensions and applications to non-scalar fields, while clarifying the Bohm quantum potential as arising from the information metric $I_f$.

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 Schrödinger 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 Schrödinger equation of the wave functional. Furthermore, by defining the information metrics for field fluctuations using general definitions of relative entropy, we obtain a generalized Schrödinger 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.