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A general introduction is presented to the more technical Part II of this book. It is explained how we arrive at our choice of notation, how we attempt and sometimes fail to make the notation completely consistent and straightforward throughout the book.

The prototype of the simplest periodic deterministic systems is a model whose states move on a circle. The prototype of the quantum mechanical periodic system is the harmonic oscillator. In line with the philosophy on which this book it based, these two models can be mapped one onto the other. However, we need an intermediate case: the harmonic . We illustrate the use of the group for this model, and we explain how to take the ‘continuum limit’.

It is shown how the mathematics has to be set up that is needed to relate classical and quantum mechanical periodic worlds, in the case both contain large numbers of states, while, eventually, we wish to introduce perturbations and interactions.

The usual ‘proofs’ that deterministic hidden variable models for quantum mechanics cannot be completely local, are based on rather indirect arguments, using thought experiments, free will, and the like. In this chapter we show how difficulties emerge when we wish to construct mathematical models. We also show how, in principle, the problem can be addressed at the level of model building.

Classical theories starting with continuous degrees of freedom will typically generate bosonic quantum objects; fermionic quantum objects in turn, arise when a classical system contains discrete degrees of freedom—typically: flip-flops. Here, we explain how anti-commuting operators enter the scene, and illustrate the situation using an elegant model: the idealized notion of a ‘neutrino’. What do neutrinos have in common with infinite, at sheets, and where does spin come from? The associated mathematical tool kit is quite delicate.

What is badly needed is an elegant mathematical scheme that replaces position and momentum operators and by discrete, commuting counterparts,  and , in such a way that mappings can be defined between conventional, quantum mechanical systems and deterministic, discrete systems. Here the foundations for such a scheme is suggested. An essential role is played by the Jacobi theta functions.

Sooner or later, we shall have to contemplate multi-particle systems and quantized field theories. These naturally arise whenever quantum mechanics confronts Einstein’s Special Relativity. The prototype is the theory of bosons in one space-, one time dimension. First, we consider the massless, non-interacting case. This allows for a mapping onto a deterministic system: the (continuum limit of a) cellular automaton. Now, both classically and quantum mechanically, these models are trivial and featureless, so we have to ask how mass terms and interactions are to be introduced, and how to go from 1 to 3 space-like dimensions (if not more).

This is where the mathematics becomes hard, but there is something else we can do: (-) is based on a two-dimensional world: the string world sheet. At first sight, our results may seem to be spectacular: , a dimensional lattice with lattice links of length . As yet, however, this only seems to apply to the bulk behavior of strings, not their interaction mechanism, which, in this formalism, is quite complex.

Knowing that strings also generate the gravitational force, this should not have come as a surprise.

In classical, deterministic systems, we may have classical symmetries, transforming ontological states into other ontological states. In line with our general philosophy, we now extend the group of symmetries using quantum symmetries, where ontological states are compared with superimposed states. The groups of symmetry transformations obtained this way may become much larger than that of the classical symmetries. Some of the mathematical procedures, borrowed from ‘ordinary’ quantum mechanics, are explained.

In standard quantum mechanics, the construction of the Schrödinger equation requires a Hamiltonian operator, which must obey some important restrictions. Most of our deterministic models are, to some extent, discrete. How do we carry over the standard Hamiltonian formalism to a discretized world? As in the continuum case, we may assume that conjugate pairs () can be defined, but now and are integers. How can a Hamiltonian function be used to define an evolution law?

Since the finiteness of the speed of light, and other considerations of special relativity, play such an important role in arguments concerning the interpretation of quantum mechanics, a good understanding of quantum field theory is a prerequisite. In this chapter, a brief summary is presented of what Quantum Field Theory is, and what one can conclude from it. The fundamental fields are bosonic fields, accounting for fundamental particles with spin 0 and 1, and fermionic fields describing fundamental particles with spin . Within the framework of Quantum Field Theory, there are no conflicts between quantum mechanics and locality, causality, and positivity of the energy, but there seems to be no ‘underlying real world’. We do note the importance of correlations, even in the vacuum state.