C*-algebraic scattering theory and explicitly solvable quantum field theories

Journal Article · Sat Jun 01 00:00:00 EDT 1985 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.526933· OSTI ID:5773373

Warchall, H A

A general theoretical framework is developed for the treatment of a class of quantum field theories that are explicitly exactly solvable, but require the use of C*-algebraic techniques because time-dependent scattering theory cannot be constructed in any one natural representation of the observable algebra. The purpose is to exhibit mechanisms by which inequivalent representations of the observable algebra can arise in quantum field theory, in a setting free of other complications commonly associated with the specification of dynamics. One of two major results is the development of necessary and sufficient conditions for the concurrent unitary implementation of two automorphism groups in a class of quasifree representations of the algebra of the canonical commutation relations (CCR). The automorphism groups considered are induced by one-parameter groups of symplectic transformations on the classical phase space over which the Weyl algebra of the CCR is built; each symplectic group is conjugate by a fixed symplectic transformation to a one-parameter unitary group. The second result, an analog to the Birman--Belopol'skii theorem in two-Hilbert-space scattering theory, gives sufficient conditions for the existence of Moller wave morphisms in theories with time-development automorphism groups of the above type. In a paper which follows, this framework is used to analyze a particular model system for which wave operators fail to exist in any natural representation of the observable algebra, but for which wave morphisms and an associated S matrix are easily constructed.

• Research Organization: Department of Mathematics, The University of Texas, Austin, Texas 78712
• OSTI ID: 5773373
• Journal Information: J. Math. Phys. (N.Y.); (United States), Vol. 26:6
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
QUANTUM FIELD THEORY
ALGEBRA
GROUP THEORY
SCATTERING
ANALYTICAL SOLUTION
CLASSIFICATION
COMMUTATION RELATIONS
DYNAMICS
HAMILTONIANS
HILBERT SPACE
LAGRANGIAN FIELD THEORY
PHASE SPACE
QUANTUM OPERATORS
S MATRIX
SP GROUPS
TIME DEPENDENCE
TRANSFORMATIONS
UNITARITY
BANACH SPACE
FIELD THEORIES
LIE GROUPS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MATHEMATICS
MATRICES
MECHANICS
SPACE
SYMMETRY GROUPS
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