  We formulate a quantum formalism for the statistical mechanical models of
discretized field theories on lattices and then show that the discrete version
of $\phi^4$ theory on 2D square lattice is complete in the sense that the
partition function of any other discretized scalar field theory on an arbitrary
lattice with arbitrary interactions can be realized as a special case of the
partition function of this model. To achieve this, we extend the recently
proposed quantum formalism for the Ising model \cite{quantum formalism} and its
completeness property \cite{completeness} to the continuous variable case.
