  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.
