Field Algebras and Their Applications
After the theory of superselection rules and the method of morphisms had clarified the differences between fields and observables, there appeared many investigations studying quantum field systems on the basis of field algebras. This approach presented some advantages over formalism of observable algebras: a concrete quantum field system is usually given in terms of some fields so that algebras generated by them provide the Shortest way to their algebraic description (whereas to obtain observable algebras we need some additional operations, like forming gauge-invariant combinations of fields). Moreover, if our starting objects are quantized fields, it gives us richer information than that available in local quantum theory: above all, the relativistic covariance properties are now more detailed (see Section 1.3.3,1.3.5 and 3.2.2). On the other hand, field algebras are not given beforehand but must be constructed from fields. The way of such a construction is not immediately obvious since all boson fields are represented by unbounded operators while all the apparatus above has been based so far on normed algebras. Hence the alternative arises: either to develop a new version of the algebraic approach able to include unbounded operators or to rid of the latter finding some way to describe systems of field operators by means of (local nets of) von Neumann algebras. We shall discuss the former in the present section and the latter in the next section.
KeywordsDouble Cone Local Algebra Cyclic Vector Superselection Rule Wightman Function
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