Chaotic decompositions in ℤ 2 -graded quantum stochastic calculus

Timothy Eyre

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 167-174
  • ISSN: 0137-6934

Abstract

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A brief introduction to ℤ 2 -graded quantum stochastic calculus is given. By inducing a superalgebraic structure on the space of iterated integrals and using the heuristic classical relation df(Λ) = f(Λ + dΛ) - f(Λ) we provide an explicit formula for chaotic expansions of polynomials of the integrator processes of ℤ 2 -graded quantum stochastic calculus.

How to cite

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Eyre, Timothy. "Chaotic decompositions in $ℤ_2$-graded quantum stochastic calculus." Banach Center Publications 43.1 (1998): 167-174. <http://eudml.org/doc/208835>.

@article{Eyre1998,
abstract = {A brief introduction to $ℤ_2$-graded quantum stochastic calculus is given. By inducing a superalgebraic structure on the space of iterated integrals and using the heuristic classical relation df(Λ) = f(Λ + dΛ) - f(Λ) we provide an explicit formula for chaotic expansions of polynomials of the integrator processes of $ℤ_2$-graded quantum stochastic calculus.},
author = {Eyre, Timothy},
journal = {Banach Center Publications},
keywords = {-graded quantum stochastic calculus; superalgebraic structure; space of iterated integrals; chaotic expansions of polynomials},
language = {eng},
number = {1},
pages = {167-174},
title = {Chaotic decompositions in $ℤ_2$-graded quantum stochastic calculus},
url = {http://eudml.org/doc/208835},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Eyre, Timothy
TI - Chaotic decompositions in $ℤ_2$-graded quantum stochastic calculus
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 167
EP - 174
AB - A brief introduction to $ℤ_2$-graded quantum stochastic calculus is given. By inducing a superalgebraic structure on the space of iterated integrals and using the heuristic classical relation df(Λ) = f(Λ + dΛ) - f(Λ) we provide an explicit formula for chaotic expansions of polynomials of the integrator processes of $ℤ_2$-graded quantum stochastic calculus.
LA - eng
KW - -graded quantum stochastic calculus; superalgebraic structure; space of iterated integrals; chaotic expansions of polynomials
UR - http://eudml.org/doc/208835
ER -

References

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  1. [C] C. Chevalley, The Construction and Study of Certain Important Algebras, Publ. Math. Soc. Japan I, Princeton University Press, Princeton, NJ, 1955. Zbl0065.01901
  2. [E1] T. M. W. Eyre, Chaotic Expansions of Elements of the Universal Enveloping Superalgebra Associated with a ℤ 2 -Graded Quantum Stochastic Calculus, Commun. Math. Phys. 192 (1998), 9-28. Zbl0910.60094
  3. [E2] T. M. W. Eyre, Graded Quantum Stochastic Calculus and Representations of Lie Superalgebras, Lecture Notes in Mathematics, Springer, Berlin, to appear; earlier form exists as Nottingham preprint. 
  4. [EH] T. M. W. Eyre and R. L. Hudson, Representations of Lie Superalgebras and Generalised Boson-Fermion Equivalence in Quantum Stochastic Calculus, Commun. Math. Phys. 186 (1997), 87-94. Zbl0882.60097
  5. [HP1] R. L. Hudson and K. R. Parthasarathy, Quantum Ito's Formula and Stochastic Evolutions, Commun. Math. Phys. 93 (1984), 301-323. Zbl0546.60058
  6. [HP2] R. L. Hudson and K. R. Parthasarathy, Unification of Fermion and Boson Stochastic Calculus, Commun. Math. Phys. 104 (1986), 457-470. Zbl0604.60063
  7. [HPu] R. L. Hudson and S. Pulmannova, Chaotic Expansions of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus, Proc. LMS, to appear. Zbl0904.60084
  8. [K] V. G. Kac, Lie Superalgebras, Advances in Mathematics 26 (1977), 8-96. Zbl0366.17012
  9. [L] J. M. Lindsay, Independence for Quantum Stochastic Integrators, Quantum Probability and Related Topics Vol. VI, 1991, 325-332. Zbl0937.60046
  10. [P] K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel (1992). Zbl0751.60046
  11. [S] M. Scheunert, The Theory of Lie Superalgebras, Lecture Notes in Mathematics, Vol. 716, Springer, Berlin, (1979). Zbl0407.17001

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