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0 with the full Hamiltonian H. To adapt this formulation to many channel reactions with "bound states," even in nonrelativistic quantum mechanics one had to find a different separation of free motion and interaction for each channel, a procedure which was not only difficult but highly nontransparent if due account of the Pauli principle was taken (the dispute about "post-prior antisymmetrization"). To our knowledge, the first natural approach to the problem was due to Ekstein (Ref. 9) (compare also Refs. 10 and 11). The results in quantum field theory mentioned above were obtained in full clarity by Ruelle (Ref. 12).
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85037761344
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0|+|x|
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17
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85037760319
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note
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A brief account of some basic notions in the theory of operator algebras is given in the Appendix. For a thorough understanding of this area of mathematics see, for example, Refs. 17 and 18.
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26
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85037783550
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note
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The commutant S′ of a set S of operators consists of all bounded operators commuting with the elements of S; the double commutant 5″ is the commutant of S′. It is assumed that the sets contain with every operator also its adjoint. This must be required correspondingly for the set of fields in (4.2).
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34
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85037760234
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note
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Though one may focus on a system in a finite volume, this does not really change the above statement because one then has to specify the relation to the outside either by introducing a heat bath or by artificial boundary conditions.
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It is contained in the image of the unit ball of H under the mapping by a positive trace class operator. The trace of this operator, the "nuclearity index," is a measure for the size of this set.
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The only physical reason for the appearance of a nontrivial center would be the possibility that superselection rules arising from the charge structure might be recognizable already within a bounded region. But there are good arguments against this. Still, a more careful consideration of this in the regime of local gauge theories might be warranted.
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Similarly to the case of C* algebras, there exists also an abstract version of von Neumann algebras, the W* algebras. In the present context, we do not need to distinguish between those.
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