Consistency, amplitudes, and probabilities in quantum theory

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 57, Issue 3, 1998, Pages 1572-1582

  Caticha, Ariel ^a  

^a STATE UNIVERSITY OF NEW YORK   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000293533     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.57.1572     Document Type: Article

References (43)

1. R. T. Cox, Am. J. Phys. 14, 1 (1946).AJPIAS
2. E. T. Jayne, Papers on Probability, Statistics and Statistical Physics, edited by R. D. Rosenkrantz (Reidel, Dordrecht, 1983).
3. See, for example, J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987);
4. D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990), and references therein.AJPIAS
5. L. de Broglie, Non-Linear Wave Mechanics—A Causal Interpretation (Elsevier, Amsterdam, 1950);
6. P. Pearle, Phys. Rev. D 13, 857 (1976).PRVDAQ
7. I. Bialynicki-Birula and J. Mycielski, Ann. Phys. (N.Y.) 100, 62 (1976); APNYA6
8. A. Shimony, Phys. Rev. A 20, 394 (1979).PLRAAN
9. S. Weinberg, Phys. Rev. Lett. 62, 485 (1989); PRLTAO
10. S. WeinbergAnn. Phys. (N.Y.) 194, 336 (1989).
11. N. Herbert, Found. Phys. 12, 1171 (1982); FNDPA4
12. W. K. Wootters and W. Zurek, Nature (London) 299, 802 (1982); NATUAS
13. N. Gisin, Helv. Phys. Acta 62, 363 (1989); HPACAK
14. N. GisinPhys. Lett. A 143, 1 (1990);
15. J. Polchinski, Phys. Rev. Lett. 66, 397 (1991); PRLTAO
16. H. Scherer and P. Busch, Phys. Rev. A 47, 1647 (1993).PLRAAN
17. A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49, 91 (1982).PRLTAO
18. C. G. Shull, D. K. Atwood, J. Arthur, and M. A. Horne, Phys. Rev. Lett. 44, 765 (1980); PRLTAO
19. R. Gahler, A. G. Klein, and A. Zeilinger, Phys. Rev. A 23, 1611 (1981); PLRAAN
20. J. J. Bollinger et al, Phys. Rev. Lett. 63, 1031 (1989).PRLTAO
21. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, 1958).
22. R. P. Feynman, Rev. Mod. Phys. 20, 267 (1948); RMPHAT
23. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
24. J. M. Jauch, Foundations of Quantum Mechanics (Addison-Wesley, New York, 1968).
25. The Logico-Algebraic Approach to Quantum Mechanics, edited by C. A. Hooker (Reidel, Dordrecht, 1979), Vols. I and II.
26. A. M. Gleason, J. Rat. Mech. Anal. 6, 885 (1957), reprinted in Ref. 13, Vol. I.
27. D. Finkelstein, Trans. NY Acad. Sci. 25, 621 (1963).TNYAAE
28. J. B. Hartle, Am. J. Phys. 36, 704 (1968).AJPIAS
29. N. Graham, in The Many-Worlds Interpretation of Quantum Mechanics, edited by B. S. DeWitt and N. Graham (Princeton University Press, Princeton, 1973).
30. S. Youssef, Mod. Phys. Lett. A 6, 225 (1991); MPLAEQ
31. Mod. Phys. Lett. AS. Youssef9, 2571 (1994);
32. S. YoussefPhys. Lett. A 204, 181 (1995).
33. This result is briefly described without proof in A. Caticha (unpublished).
34. These propositions are special instances of what are usually called histories. Since the word “consistency” also appears in this paper the reader might be led to suspect a connection with Griffith’s “consistent histories.” See, e.g., R. B. Griffiths, Phys. Rev. A 54, 2759 (1996); PLRAAN
35. R. Omnes, Rev. Mod. Phys. 64, 339 (1992) and references therein. We hasten to point out that there is no connection. The “consistent histories” approach seeks to clarify issues of interpretation by introducing rules (the consistency conditions) which restrict discourse to a subset of histories within which probabilities can be assigned. Our goal is different, we seek to justify the formalism itself; the propositions we consider are those to which amplitudes can be assigned. Our use of special histories does not reflect a lack of generality.RMPHAT
36. The two conditions implicit in the notion of independence address the two ways in which the wave function of two systems can become entangled: either during the preparation or through interaction. The challenge here is to implement these conditions without using tools that are available in the usual formulations (ideas of entanglement, interaction potentials, etc.) but are not yet available to us.
37. See, e.g., S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1995).
38. See, e.g., D. Finkelstein, J. M. Jauch, S. Schiminovich, and D. Speiser, J. Math. Phys. 3, 207 (1962); JMAPAQ
39. J. Math. Phys.D. FinkelsteinJ. M. JauchS. SchiminovichD. Speiser4, 788 (1963);
40. S. L. Adler, Phys. Rev. Lett. 55, 783 (1985); PRLTAO
41. S. L. AdlerCommun. Math. Phys. 104, 611 (1986);
42. D. Hestenes, Spacetime Algebra (Gordon and Breach, New York, 1966);
43. Clifford (Geometric) Algebras, edited by W. E. Baylis (Birkhäuser, Boston, 1996).