Hall's transformation via quantum stochastic calculus

Paula Cohen; Robin Hudson; K. Parthasarathy; Sylvia Pulmannová

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 147-155
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Cohen, Paula, et al. "Hall's transformation via quantum stochastic calculus." Banach Center Publications 43.1 (1998): 147-155. <http://eudml.org/doc/208833>.

@article{Cohen1998,
abstract = {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.},
author = {Cohen, Paula, Hudson, Robin, Parthasarathy, K., Pulmannová, Sylvia},
journal = {Banach Center Publications},
keywords = {Hall's transformation; Bargmann-Fock representation; heat kernel; Lie group; Lie algebra; universal enveloping algebra; deformation quantization; stochastic flow; quantum stochastic calculus; Dyson perturbation expansion},
language = {eng},
number = {1},
pages = {147-155},
title = {Hall's transformation via quantum stochastic calculus},
url = {http://eudml.org/doc/208833},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Cohen, Paula
AU - Hudson, Robin
AU - Parthasarathy, K.
AU - Pulmannová, Sylvia
TI - Hall's transformation via quantum stochastic calculus
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 147
EP - 155
AB - 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.
LA - eng
KW - Hall's transformation; Bargmann-Fock representation; heat kernel; Lie group; Lie algebra; universal enveloping algebra; deformation quantization; stochastic flow; quantum stochastic calculus; Dyson perturbation expansion
UR - http://eudml.org/doc/208833
ER -

References

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  1. [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. 
  2. [ChPr] V. Chari and A. Pressley, Quantum Groups, Cambridge 1994. Zbl0839.17009
  3. [Driv] B. K. Driver, On the Kakutani-Ito-Segal-Gross and the Segal-Bargmann-Hall Isomorphisms, J. Funct. Anal. 133 (1995), 69-128. Zbl0846.43001
  4. [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. 
  5. [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. 
  6. [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. Zbl0882.60097
  7. [Gros] L. Gross, Uniqueness of ground states for Schrödinger operators over loop groups, J. Funct. Anal. 112 (1993), 373-441. Zbl0774.60059
  8. [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. 
  9. [Hall] B. Hall, The Segal-Bargmann 'coherent state' transform for compact Lie groups, J. Funct. Anal. 122 (1994), 103-151. Zbl0838.22004
  10. [Huds] R. L. Hudson, Translation-invariant quantizations and algebraic structures on phase space, Rep. Math. Phys. 10 (1976), 9-20. Zbl0362.46050
  11. [HuPa] R. L. Hudson and K. R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Commun. Math. Phys. 93 (1984) 301-322. Zbl0546.60058
  12. [HuPa2] R. L. Hudson and K. R. Parthasarathy, Unification of Boson and Fermion quantum stochastic calculus, Commun. Math. Phys. 104 (1986) 457-470. Zbl0604.60063
  13. [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. 
  14. [Sega] I. E. Segal, Tensor algebras over Hilbert spaces II, Ann. Math. (2) 63 (1956), 106-134. 

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