These are described in the following sections. Renormalization. In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities. When describing space and time as a continuum, certain statistical and quantum mechanical constructions are ill defined. To define them, the continuum limit has to be taken carefully. Renormalization establishes a relationship between parameters in the theory when the parameters describing large distance scales differ from the parameters describing small distances. Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory. Initially viewed as a suspicious provisional procedure even by some of its originators, renormalization eventually was embraced as an important and self-consistent tool in several fields of physics and mathematics.
Self-interactions in classical physics[edit] Figure 1. . And where. Renormalization. Haag's theorem. Rudolf Haag postulated [1] that the interaction picture does not exist in an interacting, relativistic quantum field theory (QFT), something now commonly known as Haag's Theorem. Haag's original proof was subsequently generalized by a number of authors, notably Hall and Wightman,[2] who reached the conclusion that a single, universal Hilbert space representation does not suffice for describing both free and interacting fields. In 1975, Reed and Simon proved [3] that a Haag-like theorem also applies to free neutral scalar fields of different masses, which implies that the interaction picture cannot exist even under the absence of interactions. Formal description of Haag's theorem[edit] In its modern form, the Haag theorem may be stated as follows:[4] Consider two representations of the canonical commutation relations (CCR), and (where denote the respective Hilbert spaces and the collection of operators in the CCR).
From Hilbert space to Hilbert space such that for each operator . [edit] Haag's theorem. Gauge theory. Multivalued gauge transformations. Supersymmetry.