By Erling Størmer (auth.), Prof. Dr. Ludwig Streit (eds.)
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Extra info for Quantum Dynamics: Models and Mathematics
Definition: 1) Let A be an abelian *- algebra, a representation (11, called standard (Powers [lJ) if (i) If xEA h than 1I(x) is essentially selfadjoint on D) is D. (ii) for x,yeA h the spectral projections of 1I(x) and 1I(y) commute. 2) Let V C Ah be a linear subspace, then we say V generates A if for xeA exists a finite number of elements vi •.. vn V and a polynomial P such that x = P(v 1 ,v 2 , ... v n ). 3) Denote by V* the algebraic dual of V and let Zc V·be a subset then we denote by 30 for all W £ Z}.
4. : Rep. Math. Phys. : Algebraic Aspects of Wightman Field Theory. : Progr. Theor. Phys. : Comm. Math. Phys. l, 49 (1965) Acta Physica Austriaca, Supp!. XVI, 47-58 (1976) © by Springer-Verlag 1976 Structure of Creal-time) PCt)2 Green's Functions J. Dimock* D~partement de Physique Th~orique de Geneve Universit~ By Green's functions in the PCt)2 model we shall mean vacuum expectation values of time-ordered products of field operators. The central role played by such objects in any many-body quantum mechanical problem is well-established.
I. Assumptions: 1) A is an abelian nuclear. - algebra 1 €. A with a separately continuous product. 2) V is a linear subspace of Ah such that A(V) the algebra generated by V U 1 is dense in A. V is a real nuclear space in the topology induced by A. 3) VI denotes the (real) dual space of V and Z c VI a subset P(Z) is the cone of polynomial P(v 1 ... v n ) such that P(~(v1),~(v2) ••. 2. Theorem: Let A, V, Z as above and let w be a state then the following stat3ments are equivalent 1) w is positive on p(Z) and x +w(x·x)~2 is continuous 2) w has a weak integral decomposition 38 where A is a locally compact space and p is a positive normalized measure and the following holds for almost all a) wA is a character on A b) wAtV belongs to the weak closure of Z.