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

Completeness of classical $\phi^4$ theory on 2D lattices