# Quantum Dynamics: Models and Mathematics by Erling Størmer (auth.), Prof. Dr. Ludwig Streit (eds.)

By Erling Størmer (auth.), Prof. Dr. Ludwig Streit (eds.)

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Extra info for Quantum Dynamics: Models and Mathematics

Example text

The fi,rst expression is only formal. but the second can be made exact by a modification of Nelson's technique. The crucial points are that nv ~ o. and that each line connects at least one vertex which is fully contracted. h) is analytic in Ime < 0 and continuous in Ime ~ o. Theorem II [D3] There exists a partition of unity X. J 1 for any N. A o ) and {e:lme

Definition: 1) Let (11,:1)) be a P-positive representation and X€(ll,fJ)~ then we define x » 0 if (f,n(p)xf) ::. D and all pE P. J,P ) the linear span of C~(n, V,P ). , A an induced P-positive extension then we define D ,P) A Replacing now the set {(11, D)~}~ resp. D)~}1 by the convex + "A resp. C1 (1i,){, D,p) we can now weakly compact sets c~ (11, :D,P) proceed as in the last section. 3. Theorem: 1) Every P-positive representation (11, D) of A is majorized by a maximal regular induced P-positive extension.

If this is done, then the result is similar to the last theorem. 33 IV. Integral Decomposition of States In the following let us assume, that A is a _ - algebra. J). Moreover, if '1, ( D) was a separable Hilbert space then the same is true J\ for 'Ie ( (). M,. ) .. and such that ~ consists of all bounded diagonal operators with respect to this decomposition. Now it is natural to ask whether one can decompose also the representation 1i'(A). Since 1i'(x) is generally an unbounded operator, such a decomposition is not always possible.