Quantization of discrete deterministic theories by Hilbert space extension

Quantization of a theory usually implies that it is being replaced by a physically different system. In this paper it is pointed out that if a deterministic theory is completely discrete, such as a classical gauge theory on a lattice, with discrete gauge group, then there is an essentially trivial procedure to quantize it. The equations for the evolution of the physical variables are kept unchanged, but are reformulated in terms of the evolution of the transition turns a system into a conventional quantum theory, which may have more symmetries that can be seen in the original classical theory. This is illustrated in a cellular automaton, of which only the quantum version is time-reversal symmetric. Another automation shows self-duality only after Hilbert space extension. We discuss the importance of such observations for physics. The procedure can also be used to construct a completely finite and soluble quantum gravity model in 2 + 1 dimensions