Structure and properties of Hughston's stochastic extension of the Schrödinger equation

Journal of Mathematical Physics

Volumn 41, Issue 5, 2000, Pages 2485-2499

  Adler, Stephen L ^a   Horwitz, Lawrence P ^a,b,c  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

^b TEL AVIV UNIVERSITY   (Israel)

^c BAR ILAN UNIVERSITY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034347307     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533255     Document Type: Article

References (37)

1. For a representative, but not exhaustive, survey of the earlier literature, see the papers of Diósi, Ghirardi et al., Gisin, Pearle, and Percival cited by Hughston, Ref. 5 below.
2. N. Gisin, Helv. Phys. Acta 62, 363 (1989).
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4. G. C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A 42, 78 (1990).
5. L. P. Hughston, Proc. R. Soc. London, Ser. A 452, 953 (1996)
6. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
7. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
8. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
9. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
10. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
11. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
12. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
13. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
14. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
15. D. A. Page, Phys. Rev. A 36, 3479 (1987); Y. Aharanov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987); J. Anandan and Y. Aharanov, Phys. Rev. D 38, 1863 (1988); Phys. Rev. Lett. 65, 1697 (1990); G. W. Gibbons, J. Geom. Phys. 8, 147 (1992); L. P. Hughston, "Geometric aspects of quantum mechanics," in Twistor Theory, edited by S. A. Huggett (Dekker, New York, 1995); A. Ashtekar and T. A. Schilling, preprint gr-qc/9706069. For related work, see A. Heslot, Phys. Rev. D 31, 1341 (1985); S. Weinberg, Phys. Rev. Lett. 62, 485 (1989) and Ann. Phys. (N. Y.) 194, 336 (1989).
16. P. Pearle, Phys. Rev. D 13, 857 (1976); 29, 235 (1984); Phys. Rev. A 39, 2277 (1989).
17. P. Pearle, Phys. Rev. D 13, 857 (1976); 29, 235 (1984); Phys. Rev. A 39, 2277 (1989).
18. P. Pearle, Phys. Rev. D 13, 857 (1976); 29, 235 (1984); Phys. Rev. A 39, 2277 (1989).
19. See, e.g., Ref. 4 above and G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470 (1986).
20. α≠0. There is therefore a set of holomorphically overlapping patches, so that the metric of Eq. (4b) is globally defined. See, for example, S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Wiley Interscience, New York, 1969), Vol. II, p. 159.
21. For an excellent exposition of the Itô calculus, see C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1990), Chap. 4.
22. We wish to thank A. Bassi and G. C. Ghirardi for a conversation explaining this connection. Our exposition closely follows theirs.
23. j.
24. j.
25. For a general discussion, see G. Lindblad, Non-Equilibrium Entropy and Irreversibility (Reidel, Dordrecht, 1983), pp. 28-29; for a recent 2×2 matrix derivation and application to K-meson decays, see F. Benatti and R. Floreanini, in Quantum Probability, Banach Center Publications, Vol. 43 (Institute of Mathematics, Polish Academy of Sciences, Warsaw, 1998).
26. For a general discussion, see G. Lindblad, Non-Equilibrium Entropy and Irreversibility (Reidel, Dordrecht, 1983), pp. 28-29; for a recent 2×2 matrix derivation and application to K-meson decays, see F. Benatti and R. Floreanini, in Quantum Probability, Banach Center Publications, Vol. 43 (Institute of Mathematics, Polish Academy of Sciences, Warsaw, 1998).
27. t does not drive the A variance to zero. We wish to thank F. Benatti for calling our attention to the book K. R. Parasarathy, An Introduction to the Quantum Stochastic Calculus (Birkhäuser Verlag, Basel, 1992), Chap. III, which discusses such extensions. 15 For earlier discussions of energy-based reduction, see G. J. Milburn, Phys. Rev. A 44, 5401 (1991); D. Bedford and D. Wang, Nuovo Cimento Soc. Ital. Fis., B 26, 313 (1975); 37, 55 (1977).
28. t does not drive the A variance to zero. We wish to thank F. Benatti for calling our attention to the book K. R. Parasarathy, An Introduction to the Quantum Stochastic Calculus (Birkhäuser Verlag, Basel, 1992), Chap. III, which discusses such extensions. 15 For earlier discussions of energy-based reduction, see G. J. Milburn, Phys. Rev. A 44, 5401 (1991); D. Bedford and D. Wang, Nuovo Cimento Soc. Ital. Fis., B 26, 313 (1975); 37, 55 (1977).
29. t does not drive the A variance to zero. We wish to thank F. Benatti for calling our attention to the book K. R. Parasarathy, An Introduction to the Quantum Stochastic Calculus (Birkhäuser Verlag, Basel, 1992), Chap. III, which discusses such extensions. 15 For earlier discussions of energy-based reduction, see G. J. Milburn, Phys. Rev. A 44, 5401 (1991); D. Bedford and D. Wang, Nuovo Cimento Soc. Ital. Fis., B 26, 313 (1975); 37, 55 (1977).
30. t does not drive the A variance to zero. We wish to thank F. Benatti for calling our attention to the book K. R. Parasarathy, An Introduction to the Quantum Stochastic Calculus (Birkhäuser Verlag, Basel, 1992), Chap. III, which discusses such extensions. 15 For earlier discussions of energy-based reduction, see G. J. Milburn, Phys. Rev. A 44, 5401 (1991); D. Bedford and D. Wang, Nuovo Cimento Soc. Ital. Fis., B 26, 313 (1975); 37, 55 (1977).
31. See L. P. Hughston, Ref. 5. Sec. 11, and earlier work of Diósi, Ghirardi et al., and Penrose cited there; also D. I. Fivel, preprint quant-ph/9710042.
32. See L. P. Hughston, Ref. 5. Sec. 11, and earlier work of Diósi, Ghirardi et al., and Penrose cited there; also D. I. Fivel, preprint quant-ph/9710042.
33. P. A. Redhead, "Vacuum," in Macmillan Encyclopedia of Physics, edited by J. G. Rigden (Simon & Schuster Macmillan, New York, 1996), p. 1657.
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