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A time-reversible cellular automaton can most easily be described by regarding its evolution law as a two-step process, for instance by first allowing the even sites to update, then the odd sites. The discrete Hamiltonian operator procedure of Chap.  may be used. Subsequently, we apply a perturbative algebra, starting from the Baker Campbell Hausdorff expansion.

Even if the underlying deterministic model may be assumed to obey decent locality requirements, a systematic procedure to define a quantum Hilbert space starting from its ontological states, would in general lead to a quantum system without manifest locality at all. Instead, what we are really after is a procedure that generates a local, relativistic quantum field theory. In this chapter we describe a procedure that follows the construction of relativistic quantum field theory more closely.

We wrap up the main outcomes of our investigations. Even if not all questions and issues could be addressed as precisely and persuasively as one would wish, we do find that the assumption of an underlying theory describing a single, definite world with completely deterministic evolution laws, can be maintained and pursued further. In the more distant future, it is hoped that the Cellular Automaton Theory will enable us to select credible models for grand unification an quantized gravity; in the mean while, the Cellular Automaton Interpretation will be helpful in visualizing what we are actually doing when we are involved with quantum mechanical calculations.