Title: Quantum Scalar Field Theory Based On Principle of Least Observability

Abstract: Recently it is shown that the non-relativistic quantum formulations can be derived from a principle of least observability(arXiv:2302.14619). 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 can be considered as an extension of 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. This enables us to calculate the degree of observability from the dynamics of a classical field. Second, there are constant random field fluctuations. A novel method is introduced to define the information metrics to measure additional observability due to the field fluctuations. Applying the variation principle to minimize the total degree of observability 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 least observability 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.