By Paul A. Meyer

In fresh years, the classical thought of stochastic integration and stochastic differential equations has been prolonged to a non-commutative set-up to improve versions for quantum noises. the writer, a consultant of classical stochastic calculus and martingale concept, attempts to supply an advent to this quickly increasing box in a manner which may be obtainable to probabilists acquainted with the Ito essential. it will probably additionally, nevertheless, supply a method of entry to the equipment of stochastic calculus for physicists acquainted with Fock area research. For this moment variation, the writer has further approximately 30 pages of recent fabric, totally on quantum stochastic integrals.

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4. Angular momentum 27 After raising to the power n and passing to the limit, the matrix element is the exponential of [vl2 2 lul 2 2 z~ i ~ -~ ~ ( l - e i~sin,~, T)+(fiz-¢v)e i),sin~ T + five2i~ . This looks complicated, and needs some comments. First of all, what we are supposed to have computed is <¢(u),n(z,¢,2;~)¢(v)> where n(z,¢,2A)=e za+-~a-+2iAa° . The language of future chapters will replace coherent vectors ¢ ( u ) ¢ ( v ) by e x p o n e n t i a l v e c t o r s C ( u ) E(v) proportional to the preceding ones, but with a different normalization so that < ~ ( u ) , E(v) > = e < u,v>.

Since S . commutes with Z . it keeps Cp stable, and decomposing the representation amounts to finding the eigenvectors of S in each Cp. Recalling that Z = S - I - 2N = 2J - 2 N , we see that for u even (odd) all components have integer (half-integer) spin. 7) the greatest allowed value of spin is exactly u/2, and corresponds to the chaos Co of order 0. e. 's, conceniently normalized. On the first chaos Ca, the eigenvalue of Z is u/2 - 1, and therefore any ladder built on an eigenvector of J in Ca has at least u rungs.

Weyl operators 3 Weyl suggested the following approach to the quantization problem of associating a q u a n t u m observable F(P, Q) with a function F(p, q) on classical phase space. First define the u n i t a r y operators e - i (rP+sQ) for real r , s. Then take the Fourier transform of the classical observable F ( r , s) = f e -i (~P+sq) r ( p , q) dpdq , and transform it bazkwards into the operator, F(P, Q) = f e i (rP+sQ) F(r, s)drds For a discussion of this method, which is extremely interesting as a version of pseudodifferential calculus, see Folland [Fol].

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