# Contrary Inferences in Consistent Histories and a Set Selection Criterion

## Abstract

The best developed formulation of closed system quantum theory that handles multiple-time statements, is the consistent (or decoherent) histories approach. The most important weaknesses of the approach is that it gives rise to many different consistent sets, and it has been argued that a complete interpretation should be accompanied with a natural mechanism leading to a (possibly) unique preferred consistent set. The existence of multiple consistent sets becomes more problematic because it allows the existence of contrary inferences [1]. We analyse the conceptual difficulties that arise from the existence of multiple consistent sets and provide a suggestion for a natural set selection criterion. This criterion does not lead to a unique physical consistent set, however it evades the existence of consistent sets with contrary inferences. The criterion is based on the concept of preclusion and the requirement that probability one propositions and their inferences should be non-contextual. The allowed consistent sets turn-out to be compatible with coevents which are the ontology of an alternative, histories based, formulation [2, 3, 4].

## Keywords

Consistent histories Decoherent histories Contrary inferences## Notes

### Acknowledgments

This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) through Grant EP/K022717/1. Partial support from COST Action MP1006 is also gratefully acknowledged. The author is grateful to Robert Griffiths for his remarks concerning the conventional single-framework rule for consistent histories and to the anonymous reviewers for their constructive comments.

## References

- 1.Kent, A.: Consistent Ssets yield contrary inferences in quantum theory. Phys. Rev. Lett.
**78**, 2874–2877 (1997)MathSciNetCrossRefzbMATHADSGoogle Scholar - 2.Sorkin, R.D.: An exercise in “anhomomorphic logic”. J. Phys. Conf. Ser.
**67**, 012018 (2007)CrossRefADSGoogle Scholar - 3.Sorkin, R.D.: Quantum dynamics without the wavefunction. J. Phys. A
**40**, 3207–3222 (2007)MathSciNetCrossRefzbMATHADSGoogle Scholar - 4.Wallden, P.: The coevent formulation of quantum theory. J. Phys. Conf. Ser.
**442**, 012044 (2013)CrossRefADSGoogle Scholar - 5.Dirac, P.A.M.: The Langrangian in quantum mechanics. Physikalische Zeitschrift der Sowjetunion
**3**, 64–72 (1933)Google Scholar - 6.Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys.
**20**, 367–387 (1948)MathSciNetCrossRefADSGoogle Scholar - 7.Griffiths, R.: Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys.
**36**, 219–272 (1984)CrossRefzbMATHADSGoogle Scholar - 8.Omnès, R.: Logical reformulation of quantum mechanics. I. Foundations. J. Stat. Phys.
**53**, 893–932 (1988)CrossRefzbMATHADSGoogle Scholar - 9.Gell-Mann, M., Hartle, J.: Quantum mechanics in the light of quantum cosmology. In: Zurek, W. (ed.) Complexity, Entropy and the Physics of Information, SFI Studies in the Science of Complexity, pp. 425–458. Addison-Wesley, Reading, Redwood City (1990)Google Scholar
- 10.Gell-Mann, M., Hartle, J.: Classical equations for quantum systems. Phys. Rev. D
**47**, 3345–3382 (1993)MathSciNetCrossRefADSGoogle Scholar - 11.Dowker, F., Kent, A.: Properties of consistent histories. Phys. Rev. Lett.
**75**, 3038 (1995)CrossRefADSGoogle Scholar - 12.Dowker, F., Kent, A.: On the consistent histories approach to quantum mechanics. J. Stat. Phys.
**82**, 1575–1646 (1996)MathSciNetCrossRefzbMATHADSGoogle Scholar - 13.Bassi, A., Ghirardi, G.: Decoherent histories and realism. J. Stat. Phys.
**98**, 457–494 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Okon, E., Sudarsky, D.: On the consistency of the consistent histories approach to quantum mechanics. Found. Phys.
**44**, 19–33 (2014)MathSciNetCrossRefzbMATHADSGoogle Scholar - 15.Goldstein, S., Page, D.: Linearly positive histories: probabilities for a robust family of sequences of quantum events. Phys. Rev. Lett.
**74**, 3715–3719 (1995)MathSciNetCrossRefzbMATHADSGoogle Scholar - 16.Hartle, J.: Linear positivity and virtual probability. Phys. Rev. Lett. A
**70**, 022104 (2004)MathSciNetCrossRefADSGoogle Scholar - 17.Sorkin, R.D.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A
**9**, 3119–3128 (1994)MathSciNetCrossRefzbMATHADSGoogle Scholar - 18.Sorkin, R.D.:
*Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability*, Philadelphia, September 8–11, 1994, pp. 229-251. International Press, Cambridge Mass, 1997. In Feng, D.H. and Hu, B-L. (eds.) gr-qc/9507057 - 19.Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. Indiana Univ. Math. J.
**6**, 885–893 (1957)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech.
**17**, 59–87 (1967)MathSciNetzbMATHGoogle Scholar - 21.Shimony, A.: Contextual hidden variables theories and bell’s inequalities. Br. J. Philos. Sci.
**35**, 25–45 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Bohm, D., Hiley, B.J.: The Undivided Universe: An Ontological Interpretation of Quantum Theory. Routledge, London (1993)Google Scholar
- 23.Gudder, S.: On hidden variable theories. J. Math. Phys.
**11**, 431–436 (1970)MathSciNetCrossRefzbMATHADSGoogle Scholar - 24.Sorkin, R. D.: in G.F.R. Ellis, J. Murugan and A. Weltman (eds),
*Foundations of Space and Time*(Cambridge University Press). preprint arxiv:1004.1226 - 25.Einstein, A., Podolsky, B., Rosen, N.: Can quantum mechanical description of physical reality be considered complete? Phys. Rev.
**47**, 777–780 (1935)CrossRefzbMATHADSGoogle Scholar - 26.Martin, X., O’Connor, D., Sorkin, R.: Random walk in generalized quantum theory. Phys. Rev. D
**71**, 024029 (2005)MathSciNetCrossRefADSGoogle Scholar - 27.Isham, C.: Topos theory and consistent histories: the internal logic of the set of all consistent sets. Int. J. Theor. Phys.
**36**, 785–814 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Flori, C.: preprint arXiv:0812.1290 (2008)
- 29.Anastopoulos, C.: On the selection of preferred consistent sets. Int. J. Theor. Phys.
**37**, 2261–2272 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 30.Gell-Mann, M., Hartle, J.: Quasiclassical coarse graining and thermodynamic entropy. Phys. Rev. A
**76**, 022104 (2007)CrossRefADSGoogle Scholar - 31.Riedel, C.J., Zurek, W. and Zwolak, M.: preprint arXiv:1312.0331 (2013)
- 32.Kent, A.: Quantum histories and their implications. Lect. Notes Phys.
**559**, 93–115 (2000)CrossRefADSGoogle Scholar - 33.Griffiths, R.B.: Choice of consistent family, and quantum incompatibility. Phys. Rev. A
**57**, 1604–1618 (1998)CrossRefADSGoogle Scholar - 34.Griffiths, R.B.: Consistent Quantum Theory. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
- 35.Griffiths, R.B.: The new quantum logic. Found. Phys.
**44**, 610–640 (2014)MathSciNetCrossRefADSGoogle Scholar - 36.Griffiths, R.B., Hartle, J.: Comment on “consistent sets yield contrary inferences in quantum theory”. Phys. Rev. Lett.
**81**, 1981 (1998)MathSciNetCrossRefzbMATHADSGoogle Scholar - 37.Kent, A.: Kent replies. Phys. Rev. Lett.
**81**, 1982 (1998)MathSciNetCrossRefzbMATHADSGoogle Scholar - 38.Dowker, F., Ghazi-Tabatabai, Y.: The Kochen-Specker theorem revisited in quantum measure theory. J. Phys. A
**41**, 105301 (2008)MathSciNetCrossRefADSGoogle Scholar - 39.Surya, S., Wallden, P.: Quantum covers in quantum measure theory. Found. Phys.
**40**, 585–606 (2010)MathSciNetCrossRefzbMATHADSGoogle Scholar - 40.Wallden, P.: Spacetime coarse grainings and the problem of time in the decoherent histories approach to quantum theory. Int. J. Theor. Phys.
**47**, 1512–1532 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 41.Aharonov, Y., Vaidman, L.: Complete description of a quantum system at a given time. J. Phys. A
**24**, 2315–2328 (1991)MathSciNetCrossRefADSGoogle Scholar - 42.Sorkin, R.: Does a quantum particle know its own energy? J. Phys. Conf. Ser.
**442**, 012014 (2013)CrossRefADSGoogle Scholar - 43.Ghazi-Tabatabai, Y., Wallden, P.: Dynamics and predictions in the co-event interpretation. J. Phys. A Math. Theor.
**42**, 235303 (2009)MathSciNetCrossRefADSGoogle Scholar - 44.Dowker, F., Johnston, S., Sorkin, R.: Hilbert spaces from path integrals. J. Phys. A Math. Theor.
**43**, 275302 (2010)MathSciNetCrossRefADSGoogle Scholar - 45.Dowker, F., Henson, J., Wallden, P.: A histories perspective on characterising quantum non-locality. New J. Phys.
**16**, 033033 (2014)MathSciNetCrossRefADSGoogle Scholar