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In this chapter we will give a complete definition on the lattice of the classical theory of the free scalar field, showing by means of a few simple examples that one recovers in this way the familiar results of the usual approach to the theory. Some concepts and techniques which are important for the subsequent treatment of the quantum theory will also be introduced and explored, such as the treatment of the boundary conditions, the finite difference operators and their eigenvalues, the space of field configurations, the finite Fourier transforms and the transformation to momentum space, and the introduction and treatment of external sources.
It is important to observe that, although from the technical point of view this may be seen as a preliminary exercise for the work in the quantum theory, it is not a prerequisite to the quantum theory on a conceptual level. Conceptually, one should first define the quantum theory and only afterwards derive the classical theory from it, as the classical limit of that quantum theory. Although using the classical theory as an intuitive guide for the construction of the quantum theory may be a very good idea, the conceptual derivation must be from the quantum theory to the classical one, and not the other way around. In other words, while the “quantization” of a classical theory belongs to the realm of imaginative guesswork, the derivation of the classical limit of a quantum theory should be precise deductive work.
The main objective of this chapter, besides introducing useful concepts
and techniques and establishing a standard notation for them, is to
establish that the lattice formalism can be used as a mathematically
complete and precise way to define the familiar structure of the
classical theory of fields, including a careful discussion of the
continuum limit and of the introduction of a physical length scale
leading to the geometry of space-time.