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 TMF, 1997, Volume 111, Number 2, Pages 218–233 (Mi tmf1002)

Causal structure of quantum stochastic integrators

J. Gough

St. Patrick's College

Abstract: A class of concrete representations of a non-commutative Stratonovich calculus is defined and its relationship with the quantum Ito calculus of Hudson and Parthasarathy is made explicit. Given a quantum field interacting with a quantum mechanical system, it is possible to extract a quantum noise description for the field using a suitable scaling limit (here the weak coupling limit). The motivation for our construction is to discuss the relationship between the micro-causality of a quantum field and the notion of macro-causality of the quantum noise which replaces it. We derive the Stratonovich quantum stochastic differential equation for the limit evolution operator and show that it agrees with the quantum stochastic limit theory of Accardi, Frigerio and Lu when we convert to the Ito form. The Stratonovich approach, being inherently closer to the physical microscopic equations, leads to an overwhelmingly simplified derivation of the quantum stochastic limit equations of motion. The unification of the two quantum stochastic calculi is given and their physical origins explained.

DOI: https://doi.org/10.4213/tmf1002

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English version:
Theoretical and Mathematical Physics, 1997, 111:2, 563–575

Bibliographic databases:

Citation: J. Gough, “Causal structure of quantum stochastic integrators”, TMF, 111:2 (1997), 218–233; Theoret. and Math. Phys., 111:2 (1997), 563–575

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf1002
• https://doi.org/10.4213/tmf1002
• http://mi.mathnet.ru/eng/tmf/v111/i2/p218

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. J. Gough, “Non-commutative Ito and Stratonovich noise and stochastic evolutions”, Theoret. and Math. Phys., 113:2 (1997), 1431–1437
2. Gough, J, “A new approach to non-commutative white noise analysis”, Comptes Rendus de l Academie Des Sciences Serie i-Mathematique, 326:8 (1998), 981
3. Gough, J, “Asymptotic stochastic transformations for nonlinear quantum dynamical systems”, Reports on Mathematical Physics, 44:3 (1999), 313
4. Gough, J, “The Stratonovich interpretation of quantum stochastic approximations”, Potential Analysis, 11:3 (1999), 213
5. Gough, J, “Dissipative canonical flows in classical and quantum mechanics”, Journal of Mathematical Physics, 40:6 (1999), 2805
6. J. Gough, “Bosonic and fermionic white noises and the reflection process”, Theoret. and Math. Phys., 124:1 (2000), 887–896
7. Gough, J, “Noncommutative Markov approximations”, Doklady Mathematics, 64:1 (2001), 112
8. Gough, J, “Quantum white noises and the master equation for Gaussian reference states”, Russian Journal of Mathematical Physics, 10:2 (2003), 142
9. Von Waldenfels, W, “Symmetric differentiation and Hamiltonian of a quantum stochastic”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 8:1 (2005), 73
10. Von Waldenfels W., “The Hamiltonian of a simple pure number process”, Quantum Probability and Infinite Dimensional Analysis, Qp-Pq Quantum Probability and White Noise Analysis, 18, 2005, 518–524
11. von Waldenfels W., “The Singular Coupling Limit for a Simple Pure Number Process”, Stochastics, 84:2-3, SI (2012), 417–423
12. Gough J.E., “Symplectic Noise and the Classical Analog of the Lindblad Generator”, J. Stat. Phys., 160:6 (2015), 1709–1720
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