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By Paul A. Meyer
In recent times, the classical idea of stochastic integration and stochastic differential equations has been prolonged to a non-commutative set-up to strengthen types for quantum noises. the writer, a consultant of classical stochastic calculus and martingale conception, attempts to supply an creation to this swiftly increasing box in a manner which will be available to probabilists accustomed to the Ito imperative. it might probably additionally, nevertheless, offer a method of entry to the tools of stochastic calculus for physicists conversant in Fock area research. For this moment version, the writer has extra approximately 30 pages of recent fabric, totally on quantum stochastic integrals.
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30 II. Spin These vectors arealso studied in a paper by Radcliffe in the same book. There are other sets of coherent and exponential vectors of physical interest. b) For large values of S the ladder representation is close to the quantum harmonic oscillator. To see this let us perform a change of operators and basis vectors : if we put S = k + 1 and a+ = A+ /Vk, a- = A- /Vk, lf . (1- it 1 tHI- t) ... /). )hj-l For fixed j and large k this tends to the description of the harmonic oscillator. c) The case k = 1 was the analogue of simple Bernoulli space in quantum probability.
The concrete * -algebra we have thus constructed will be called the complex Clifford algebra C(N) with N generators. We are going to widen this definition, characterizing this algebra up to isomorphism, first using a system of generators, and then in an intrinsic way. 5 Let v be an integer, and let A be an algebra over CC, generated (as an algebra) by its unit 1 and by generators 'Yk ( k = 1, ... 1) /j 'Yk +'Yk 'Yi = 0 (j =/: k) . Given a subset A = {it < ... To determine the structure of the ladder space we must also know the scalar product : the basis vectors are mutually orthogonal, and to compute their norms we use for the firsttime the fact that A+ and A- are mutually adjoint. 4) < Xi+l• Xi+I > = < A+xi,A+xi > = < Xi,A- A+xi > = (i + l)(k- i)