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

Quantum Scalar Field Theory Based On Principle of Least Observability