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Time-Ordered Wick Exponential and Quantum Stochastic Differential Equations

In: Quantum Communication, Computing, and Measurement

Author

Listed:
  • Nobuaki Obata

    (Nagoya University, Graduate School of Polymathematics)

Abstract

Let H = L 2(ℝ,dt) be the real Hilbert space of L 2-functions on ℝ and let Γ(H ℂ) be the Boson Fock space over Hℂ, the complexification of H. Let ℌ be another complex Hilbert space. Given a one-parameter family of operators {L t} acting in Γ(H ℂ) ⊗ ℌ, we consider the initial value problem of the form: 1 $$ \frac{{d{\Xi _t}}}{{dt}}\; = \;{L_t}\;\diamondsuit \;{\Xi _t},\;\;\;\;{\left. \Xi \right|_{t = 0}}\; = \;{\Xi _0}$$ where ◊ is the Wick product. The main purpose of this paper is to prove the unique existence of a solution to equation (1) in the sense of distributions. The main statement will be found in §5.

Suggested Citation

  • Nobuaki Obata, 1997. "Time-Ordered Wick Exponential and Quantum Stochastic Differential Equations," Springer Books, in: O. Hirota & A. S. Holevo & C. M. Caves (ed.), Quantum Communication, Computing, and Measurement, pages 355-363, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-5923-8_37
    DOI: 10.1007/978-1-4615-5923-8_37
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