Commun. in Mathematical Phys., 12, 226-232, 1969.
We construct a Hilbert space H, spanned by vectors |O>, where O is a bounded measurable set in nu-dimensional real space, and we interpret |O> as a state where all the points x in O are occupied by an incompressible fluid, and x not in O is unoccupied. H is generated by applying unitary "filling" operators U(O) to a cyclic vector |phi>, the completely unoccupied state. The operators U(O) generate a commutative C*-algebra, of which the hermitian elements are interpreted as the observables of the theory. All the infinitely divisible representations of the symmetric group of order 2 are found. We give a generalisation to a theory with any number of particle types.
This paper was the first to give the definition of an infinitely divisible cyclic representation of a group, and to prove that a continuous tensor product of cyclic group representations exists if and only if, almost everywhere, the representation is infinitely divisible, after the outline of the theory was given by the author in "Local Quantum Theory", (Ed. R. Jost), Academic Press, 1969=Proc of the Varenna Summer School, 1968. Because of some similarities with an earlier paper by Araki and Woods, "Complete Boolean Lattices of Type I von Neumann Algebras", I called my theorem "the Araki-Woods embedding theorem". In fact, my hypotheses are different from those of Araki and Woods, but the conclusion, the embedding of the theory into a Fock space, is similar.
A more detailed and general version of the theorem was derived independently by H. Araki in his paper "Factorisable Representations of Current Algebra", Publ Res Inst. Math. Sci.,(KYOTO), 1970/71.
This paper was the start of a large amount of work by K. R. Parthasarathy and K. Schmidt, A. Guichardet, G. C. Hegerfeldt, and more recently, by Schurmann and Quaegebure.
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© by Ray Streater, 16/6/00.