The subject of the present book is traditionally referred to as the algebraic approach to quantum field theory. This name indicates only the mathematical formalism employed, however, and does not, therefore, express the essence of the method completely. The algebraic approach to quantum field theory is not just a special mathematical formalism; it is a particular school of thought in physics. It can be thought of as a trend which aims at describing relativistic quantum systems (i.e. systems of quantized fields, as we understand them today) by means of the theory of algebras and in terms of observables and states treated as fundamental physical objects. In its underlying principles, this approach originates from and is a direct development of, the algebraic formulations of nonrelativistic quantum mechanics proposed in the late 1920s by von Neumann, Dirac, Jordan, and others. Initially, it was stated as an axiomatic formalism. Later on, as already mentioned, it ceased to be purely axiomatic. Today the algebraic approach is no longer an abstract axiomatic construction, but a modern dynamical method applied to description of concrete quantum field systems. However, since the results of the preceding stage were mainly general and rigorous theorems, they are incorporated as a firm basis in new investigations. And observables and states are invariably the basic physical concepts.
KeywordsDouble Cone Local Algebra Cyclic Vector Poincare Group Axiomatic Formalism
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