Abstract
We develop an approach to the representations theory of the algebra of the square of white noise based on the construction of Hilbert modules. We find the unique Fock representation and show that the representation space is the usual symmetric Fock space. Although we started with one degree of freedom we end up with countably many degrees of freedom. Surprisingly, our representation turns out to have a close relation to Feinsilver's finite difference algebra. In fact, there exists a holomorphic image of the finite difference algebra in the algebra of square of white noise. Our representation restricted to this image is the Boukas representation on the finite difference Fock space. Thus we extend the Boukas representation to a bigger algebra, which is generated by creators, annihilators, and number operators.
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L. Accardi, Y. G. Lu, and I. V. Volovich, White Noise Approach to Classical and Quantum Stochastic Calculi, Preprint, Rome, to appear in the lecture notes of the Volterra International School of the same title, held in Trento (1999).
P. Sniady, Quadratic Bosonic and Free White Noise, Preprint (1999).
L. Accardi and M. Skeide, Interacting Fock Space Versus Full Fock Module, Preprint, Rome, 1998, Revised in January 2000.
P. J. Feinsilver, “Discrete analogues of the Heisenberg-Weyl algebra,” Mh. Math., 104, 89-108 (1987).
A. Boukas, “Stochastic calculus on the finite-difference Fock space,” in: Quantum Probability and Related Topics (Accardi L., editor), Vol. VI, World Scientific (1991).
K. R. Parthasarathy and K. B. Sinha, “Unification of quantum noise processes in Fock spaces,” in: Quantum Probability and Related Topics (Accardi L., editor), Vol. VI, World Scientific (1991), pp. 371-384.
M. Skeide, “Hilbert modules in quantum electrodynamics and quantum probability,” Comm. Math. Phys., 192, 569-604 (1998).
L. Accardi and Y. G. Lu, “From the weak coupling limit to a new type of quantum stochastic calculus,” in: Quantum Probability and Related Topics (Accardi L., editor), Vol. VII, World Scientific (1992), pp. 1-14.
L. Accardi and Y. G. Lu, “The Wigner semi-circle law in quantum electrodynamics,” Comm. Math. Phys., 180, 605-632 (1996).
M. V. Pimsner, “A class of C?-algebras generalizing both Cuntz-Krieger algebras and crossed products by Z,” in: Free Probability Theory (Voiculescu D. V., editor), Vol. 12, Fields Institute Communications (1997), pp. 189-212.
R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-valued Free Probability Theory, Vol. 627, Mem. Amer. Math. Soc, Amer. Math. Soc., Providence, R.I. (1998).
L. Accardi, Y. G. Lu, and I. V. Volovich, Interacting Fock Spaces and Hilbert Module Extensions of the Heisenberg Commutation Relations, IIAS Publications, Kyoto (1997).
B. V. R. Bhat and M. Skeide, Tensor Product Systems of Hilbert Modules and Dilations of Completely Positive Semigroups, Preprint, Cottbus, Revised version, Rome, Submitted to Infinite-Dimensional Analysis, Quantum Probability and Related Topics, 1999.
W. Arveson, Continuous Analogues of Fock Space Vol. 409, Mem. Amer. Math. Soc, Amer. Math. Soc., Providence, R.I. (1989).
A. Boukas, Quantum Stochastic Analysis: a Non-Brownian Case, PhD thesis, Southern Illinois University (1988).
A. Boukas, “An example of quantum exponential process,” Mh. Math., 112, 209-215 (1991).
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Accardi, L., Skeide, M. Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras. Mathematical Notes 68, 683–694 (2000). https://doi.org/10.1023/A:1026644229489
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DOI: https://doi.org/10.1023/A:1026644229489