Logic is to the quantum as geometry is to gravity

doi:10.1017/CBO9780511920998.015

15 - Logic is to the quantum as geometry is to gravity

Published online by Cambridge University Press: 05 August 2012

Edited by

Amanda Weltman and

Rafael Sorkin: Affiliation:
Perimeter Institute
Jeff Murugan: Affiliation:
University of Cape Town
Amanda Weltman: Affiliation:
University of Cape Town
George F. R. Ellis: Affiliation:
University of Cape Town

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Summary

I will propose that the reality to which the quantum formalism implicitly refers is a kind of generalized history, the word history having here the same meaning as in the phrase sum-over-histories. This proposal confers a certain independence on the concept of event, and it modifies the rules of inference concerning events in order to resolve a contradiction between the idea of reality as a single history and the principle that events of zero measure cannot happen (the Kochen-Specker paradox being a classic expression of this contradiction). The so-called measurement problem is then solved if macroscopic events satisfy classical rules of inference, and this can in principle be decided by a calculation. The resulting conception of reality involves neither multiple worlds nor external observers. It is therefore suitable for quantum gravity in general and causal sets in particular.

Quantum gravity and quantal reality

Why, in our attempts to unify our theories of gravity and the quantum, has progress been so slow? One reason, no doubt, is that it's simply a very hard problem. Another is that we lack clear guidance from experiments or astronomical observations. But I believe that a third thing holding us back is that we haven't learned how to think clearly about the quantum world in itself, without reference to “observers” and other external agents.

Because of this we don't really know how to think about the Planckian regime where quantum gravity is expected to be most relevant. We don’t know how to think about the vacuum on small scales, or about the inside of a black hole, or about the early universe. Nor do we have a way to pose questions about relativistic causality in such situations.

Type: Chapter
Information: Foundations of Space and Time
Reflections on Quantum Gravity
, pp. 363 - 384

DOI: https://doi.org/10.1017/CBO9780511920998.015 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2012

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References

[1] L., Bombelli, J., Lee, D., Meyer, and R. D., Sorkin, Spacetime as a causal set, Phys. Rev. Lett. 59: 521–4 (1987);Google Scholar

R.D., Sorkin, Causal sets: discrete gravity (Notes for the Valdivia Summer School), in Lectures on Quantum Gravity (Series of the Centro De Estudios Cient).

[2] D. P., Rideout and R. D., Sorkin, Aclassical sequential growth dynamics for causal sets, Phys. Rev.D 61: 024002 (2000), http://arXiv.org/abs/gr-qc/9904062.Google Scholar

[3] R. D., Sorkin, Quantum dynamics without the wave function, J. Phys. A: Math. Theor. 40: 3207–21 (2007) (http://stacks.iop.org/1751-8121/40/3207), quant-ph/0610204, http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.Google Scholar

[4] R. D., Sorkin, Quantum mechanics as quantum measure theory, Mod. Phys. Lett.A 9 (No. 33): 3119–27 (1994), gr-qc/9401003, http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.Google Scholar

[5] R. B., Salgado, Some identities for the quantum measure and its generalizations, Mod. Phys. Lett. A17: 711–28 (2002), gr-qc/9903015.Google Scholar

[6] R. D., Sorkin, Quantum measure theory and its interpretation, in Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Non-integrability, held Philadelphia, September 8–11, 1994, edited by D.H., Feng and B.-L., Hu, pp. 229–51 (International Press, Cambridge, MA, 1997), gr-qc/9507057, http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.

[7] L., Hardy, Quantum theory from five reasonable axioms, http://arXiv.org/abs/quantph/0101012v4.

[8] J. B., Hartle, Spacetime quantum mechanics and the quantum mechanics of spacetime, in B., Julia and J., Zinn-Justin (eds), Gravitation et Quantifications: Les Houches Summer School, session LVII, 1992 (Elsevier Science B.V., 1995), http://arXiv.org/abs/grqc/9304006.

[9] U., Sinha, C., Couteau, Z., Medendorp, I., Söllner, R., Laflamme, R. D., Sorkin, and G., Weihs, Testing Born's rule in quantum mechanics with a triple slit experiment, in Foundations of Probability and Physics-5, edited by L., Accardi, G., Adenier, C., Fuchs, G., Jaeger, A. Yu., Khrennikov, J.A., Larsson, S., Stenholm, American Institute of Physics Conference Proceedings, Vol. 1101, pp. 200–207 (New York, 2009) (e-print: arXiv: 0811.2068 [quant-ph]).

[10] Y., Ghazi-Tabatabai and P., Wallden, Dynamics & predictions in the co-event interpretation, J. Phys. A: Math. Theor. 42: 235303 (2009), http://arXiv.org/abs/0901.3675.Google Scholar

[11] A. N., Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, von A.Kolmogoroff (Berlin, Springer, 1933).

Translated as: A. N., Kolmogorov, Foundations of the Theory of Probability (New York, Chelsea Publ. Co., 1956).

[12] A., Stairs, Phil. Sci. 50: 578 (1983); and private communication.

[13] S., Sinha and R. D., Sorkin, A sum-over-histories account of an EPR(B) experiment, Found. Phys. Lett. 4: 303–35 (1991).Google Scholar

[14] D., Craig, F., Dowker, J., Henson, S., Major, D., Rideout, and R. D., Sorkin, A Bell inequality analog in quantum measure theory, J. Phys. A: Math. Theor. 40: 501–23 (2007), quant-ph/0605008, http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.Google Scholar

[15] S., Surya and P., Wallden, Quantum covers in quantum measure theory, http://arXiv.org/abs/0809.1951.

[16] Y., Ghazi-Tabatabai, Quantum Measure Theory: A New Interpretation, http://arXiv.org/abs/0906.0294 (quant-ph).

[17] R. D., Sorkin, An exercise in ‘anhomomorphic logic’, J. Phys.: Conf. Ser. 67: 012018 (2007), a special volume edited by L. Diosi, H.-T. Elze, and G. Vitiello, and devoted to the Proceedings of the DICE-2006 meeting, held September 2006, in Piombino, Italia, arxiv quant-ph/0703276, http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.Google Scholar

[18] S., Gudder, An anhomomorphic logic for quantum mechanics, http://arXiv.org/abs/0910.3253 (quant-ph).

[19] R. D., Sorkin, To what type of logic does the ‘tetralemma’ belong?, http://arXiv.org/abs/10035735 (math:logic), http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/.

[20] F., Dowker and P., Wallden, Modus ponens and the interpretation of quantum mechanics (in preparation).

[21] S., Gudder, Quantum measure and integration theory, J. Math. Phys. 50: 123509 (2009), http://arXiv.org/abs/0909.2203 (quant-ph).Google Scholar

[22] F., Dowker and Y., Ghazi-Tabatabai, The Kochen–Specker theorem revisited in quantum measure theory, J. Phys.A 41: 105301 (2008), http://arXiv.org/abs/0711.0894 (quant-ph).Google Scholar

[23] C., Furey and R. D., Sorkin, Anhomomorphic co-events and the Hardy thought experiment (in preparation).

[24] F., Dowker and I., Herbauts, Simulating causal collapse models, Class. Quant. Grav. 21: 2963–80 (2004), http://arXiv.org/abs/quant-ph/0401075.Google Scholar

[25] Y., Ghazi-Tabatabai and P., Wallden, The emergence of probabilities in anhomomorphic logic, J. Phys.: Conf. Ser. 174: 012054 (2009), http://arXiv.org/abs/0907.0754 (quant-ph).Google Scholar