Hall's transformation via quantum stochastic calculus

• Autorzy: Paula Beazley Cohen , Robin L. Hudson , K. R. Parthasarathy , Sylvia Pulmannová
• Języki publikacji: EN

Abstrakty

EN

It is well known that Hall's transformation factorizes into a composition of two isometric maps to and from a certain completion of the dual of the universal enveloping algebra of the Lie algebra of the initial Lie group. In this paper this fact will be demonstrated by exhibiting each of the maps in turn as the composition of two isometries. For the first map we use classical stochastic calculus, and in particular a stochastic analogue of the Dyson perturbation expansion. For the second map we make use of quantum stochastic calculus, in which the circumambient space is the complexification of the Lie algebra equipped with the ad-invariant inner product.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1998
• Tom: 43
• Numer: 1
• Strony: 147-155

Daty

wydano

1998

Twórcy

autor

Paula Beazley Cohen

• URA Géométrie-Analyse-Topologie, Université des Sciences et Technologies de Lille, F-59 655 Villeneuve d'Ascq Cedex, France

autor

Robin L. Hudson

• Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-81 473 Bratislava, Slovakia

autor

K. R. Parthasarathy

• Indian Statistical Institute, 7, Sansanwal Marg, New Delhi 110016, India

autor

Sylvia Pulmannová

• Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-81 473 Bratislava, Slovakia

Bibliografia

• [Barg] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Part I, Commun. Pure Appl. Math. 24 (1961) 187-214.
• [ChPr] V. Chari and A. Pressley, Quantum Groups, Cambridge 1994.
• [Driv] B. K. Driver, On the Kakutani-Ito-Segal-Gross and the Segal-Bargmann-Hall Isomorphisms, J. Funct. Anal. 133 (1995), 69-128.
• [DrGr] B. K. Driver and L. Gross, Hilbert spaces of holomorphic functions on complex Lie groups, in: New Trends in Stochastic Analysis, ed. K. D. Elworthy et al., World Scientific 1997.
• [Eyre] T. M. W. Eyre, Chaotic expansions of elements of the universal enveloping superalgebra associated with a $ℤ_2$-graded quantum stochastic calculus, preprint, to appear in Commun. Math. Phys.
• [EyHu] T. M. W. Eyre and R. L. Hudson, Generalized Boson Fermion equivalence and representations of Lie superalgebras in quantum stochastic calculus, Commun. Math. Phys. 186 (1997) 87-94.
• [Gros] L. Gross, Uniqueness of ground states for Schrödinger operators over loop groups, J. Funct. Anal. 112 (1993), 373-441.
• [GrMa] L. Gross and P. Malliavin, Hall's transformation and the Segal-Bargmann map, in: Ito's stochastic calculus and probability theory, ed. M. Fukushima et al., Springer 1996.
• [Hall] B. Hall, The Segal-Bargmann 'coherent state' transform for compact Lie groups, J. Funct. Anal. 122 (1994), 103-151.
• [Huds] R. L. Hudson, Translation-invariant quantizations and algebraic structures on phase space, Rep. Math. Phys. 10 (1976), 9-20.
• [HuPa] R. L. Hudson and K. R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Commun. Math. Phys. 93 (1984) 301-322.
• [HuPa2] R. L. Hudson and K. R. Parthasarathy, Unification of Boson and Fermion quantum stochastic calculus, Commun. Math. Phys. 104 (1986) 457-470.
• [HuPu] R. L. Hudson and S. Pulmannová, Chaotic expansions of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus, preprint, to appear in Proc. London Math. Soc.
• [Sega] I. E. Segal, Tensor algebras over Hilbert spaces II, Ann. Math. (2) 63 (1956), 106-134.