Communications in Mathematical Physics - Volume 226 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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Another benefit of using this asymptotic measure-theoretic approach is operational in nature. Experimental data from position and momentum measurements on an elementary quantum system (via visibility data from interference experiments) is collected which is then used to construct the associated POVM. This method [S] is based on using frame manuals for the instrument state space and Sakai operators associated to localization operators on rectangles in the classical phase space X. The POVM {Ah¯ } depends on Planck’s constant, of course, and generally satisfies the (unsharp) separation property Ah¯ ( 1 ∩ 2) = Ah¯ ( 1 )Ah¯ ( 2 ).

In this example, we give a geometric classification of certain asymptotic spectral measures associated to pure spin- 21 particles. Recall that pure spin systems are represented by the Hilbert space H = C2 [BGL, S]. We then have B(H) ∼ = M2 (C). The Pauli spin operators σ1 , σ2 , σ3 are the 2 × 2 matrices σ1 = 01 , 10 σ2 = 0 −i , i 0 σ3 = 1 0 , 0 −1 which satisfy the relations: • σi∗ = σi , σi2 = I , • σi σj = −σi σj for i = j , • σi σj = i ij k σk for i = j , where I denotes the identity operator.

For example, Guentner [G1] showed that Wick quantization on the Fock space F of C defines a positive asymptotic morphism ¯ {QW h¯ } : C0 (C) → K(F), whose E-theory class is equal to the class of the ∂-operator W ¯ = [[Q ]] ∈ E(C0 (C), C). ) See the papers [N1, N2, Ro] and the books [C, GVF] for more on the connections between operator algebra K-theory, E-theory, and quantization. We show that there is a fundamental quantum-E-theory relationship by introducing the concept of an asymptotic spectral measure (ASM or asymptotic PVM) {Ah¯ }h¯ >0 : → B(H) associated to a measurable space (X, ).

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Communications in Mathematical Physics - Volume 226 by M. Aizenman (Chief Editor)

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