The book starts by defining quantum field theory in a mathematically simple but solid way, using a lattice. The definition is constructive, based on the familiar real numbers, and algorithmic, establishing a clear path for obtaining answers to questions posed within the theory. The mathematics needed to deal with the lattice and with the space of field configurations defined over it is developed in detail, in a simple and straightforward manner. The preliminary mathematical background needed by the reader is no more than the usual one present in the undergraduate physics curriculum.
The case of non-relativistic quantum mechanics is used throughout the book as a basic example of the successful use of the mathematical structure. That mathematical structure is used first to define the classical theory of fields, then the quantum theory, both based on a given fundamental action functional. The Gaussian model is used as the simple paradigmatic model for the development of the ideas. All the development is analytical, with no reliance on large-scale computer simulations. It is all made on finite lattices of arbitrary size, to be followed by the explicit examination of the continuum limit afterwards, as well as by the process of rotation from Euclidean space to Minkowski space.
The nature of the correlation and propagation structure of the resulting object is examined in detail, thus establishing the groundwork for the subsequent analysis of the theory. The relation of the generation of finite physical masses and correlation lengths with the critical behavior of the models in the continuum limit is examined in detail. The mechanism that generates the physical length scales, which in quantum field theory must be intrinsic to the models, is illustrated and clarified.
External sources are introduced and their important role in the theory made clear. The theory is re-interpreted as the generation of a functional mapping between given external sources and the average field configurations. The functional generators of the correlation functions are introduced, leading to the concept of the effective action. The physical interpretation of the effective action is made clear, as a classical action which encodes the properties of the quantum theory.
Block variables are introduced and examined in detail. Their role in establishing the physical interpretation of the theory is made clear. The blocked correlation function of the Gaussian model is calculated explicitly. In doing this further insight into the rather singular correlation structure of the theory is obtained.
The concept of energy is introduced, as well as the Hamiltonian functional. The usual results are obtained in the case of non-relativistic quantum mechanics. States of particles in the quantum theory of fields are given by an explicit construction. It is shown that these particles present the usual relativistic relations between momenta and energy in the continuum limit, and that the virtual states, which are not on-shell, have energies that collapse to the vacuum in that limit. The interpretation of physical particles as the exchange of packets of energy with the modes of the field within a space-time cavity is emphasized.
Contact with the usual canonical formalism involving vectors and operators in a Hilbert space is attempted. The usual results are obtained in the case of non-relativistic quantum mechanics, and the problems which arise in the case of quantum field theory are clearly exposed. The book closes with a critical review of the status of the theory, and a new tentative proposal for temporal evolution, involving block variables.
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