The Wightman Formalism
In Chapter 6 we took as the definition of a physical system a pair (A, S), where 3 is an abstract C*-algebra (algebra of observables) and S is the set of “physical” states, one of its properties being the capability of distinguishing the elements of A. The Hermitian elements of A play the role of generalized variables which, in principle, can be measured experimentally (hence the terminology “algebra of observables”). We now single out the class of field systems. Intuitively we can imagine the observables of a field system to be certain functionals of a collection of “fundamental fields” which are functions on Minkowski space satisfying specified (field) equations. However, in the most interesting cases, the value of the quantum field at a point in space-time is devoid of meaning as will be shown presently, but the somewhat more general idea of localization comes to our aid for the correct definition of quantum field systems.
KeywordsGauge Transformation Spinor Field Vacuum Vector Wightman Function Physical Hilbert Space
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