Quantum geometry, logic and probability

Main Article Content

Shahn Majid
https://orcid.org/0000-0003-1657-5434

Abstract

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these ‘lattice spacing’ weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form ∂+f = (−Δθ + q − p)f for the graph Laplacian Δθ, potential functions q, p built from the probabilities, and finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a ‘discrete Schrödinger process’ as ∂+ψ = ı(−Δ + V )ψ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced ‘generalised Markov process’ for f = |ψ|2 in which there is an additional source current built from ψ. We also mention our recent work on the quantum geometry of logic in ‘digital’ form over the field F2 = {0, 1}, including de Morgan duality and its possible generalisations.

Article Details

How to Cite
Majid, S. (2020). Quantum geometry, logic and probability. Philosophical Problems in Science (Zagadnienia Filozoficzne W Nauce), (69), 191–236. Retrieved from https://zfn.edu.pl/index.php/zfn/article/view/522
Section
Articles

References

Ambjørn, J., Jurkiewicz, J. and Loll, R., 2001. Dynamically triangulating Lorentzian quantum gravity. Nuclear Physics B [Online], 610(1), pp.347–382. Available at: https://doi.org/10.1016/S0550-3213(01)00297-8.

Argota-Quiroz, J.N. and Majid, S., 2020. Quantum gravity on polygons and R × Zn FLRW model. Classical and Quantum Gravity [Online], 37, p.245001. Available at: <https://iopscience.iop.org/article/10.1088/1361-6382/abbaa8/pdf>.

Ashtekar, A., Pawlowski, T., Singh, P. and Vandersloot, K., 2007. Loop quantum cosmology of k = 1 FRW models. Physical Review D [Online], 75(2), p.024035. Available at: https://doi.org/10.1103/PhysRevD.75.024035.

Bassett, M.E. and Majid, S., 2020. Finite Noncommutative Geometries Related to Fp[x]. Algebras and Representation Theory [Online], 23(2), pp.251–274. Available at: https://doi.org/10.1007/s10468-018-09846-4.

Beggs, E., 2020. Noncommutative geodesics and the KSGNS construction. Journal of Geometry and Physics [Online], 158, p.103851. Available at: https://doi.org/10.1016/j.geomphys.2020.103851.

Beggs, E. and Majid, S., 2017. Spectral triples from bimodule connections and Chern connections. Journal of Noncommutative Geometry [Online], 11(2), pp.669–701. Available at: https://doi.org/10.4171/JNCG/11-2-7.

Beggs, E. and Majid, S., 2019. Quantum geodesics in quantum mechanics. arXiv:1912.13376 [hep-th, physics:math-ph] [Online]. Available at: <http://arxiv.org/abs/1912.13376>.

Beggs, E. and Majid, S., 2020. Quantum Riemannian Geometry [Online]. Vol. 355, Grundlehren der mathematischen Wissenschaften. Springer International Publishing. Available at: https://doi.org/10.1007/978-3-030-30294-8.

Beggs, E.J. and Majid, S., 2014. Gravity induced from quantum spacetime. Classical and Quantum Gravity [Online], 31(3), p.035020. Available at: https://doi.org/10.1088/0264-9381/31/3/035020.

Connes, A., 1994. Noncommutative Geometry (S.K. Berberian, Trans.). San Diego: Academic Press.

Döring, A. and Isham, C., 2011. “What is a Thing?”: Topos Theory in the Foundations of Physics. In: B. Coecke, ed. New Structures for Physics [Online]. Vol. 813, Lecture Notes in Physics. Berlin, Heidelberg: Springer, pp.753–937. Available at: https://doi.org/10.1007/978-3-642-12821-9_13.

Dowker, F., 2013. Introduction to causal sets and their phenomenology. General Relativity and Gravitation [Online], 45(9), pp.1651–1667. Available at: https://doi.org/10.1007/s10714-013-1569-y.

Dubois-Violette, M. and Michor, P.W., 1996. Connections on central bimodules in noncommutative differential geometry. Journal of Geometry and Physics [Online], 20(2), pp.218–232. Available at: https://doi.org/10.1016/0393-0440(95)00057-7.

Fong, B. and Spivak, D.I., 2019. An Invitation to Applied Category Theory: Seven Sketches in Compositionality. Cambridge: Cambridge University Press.

Hale, M., 2002. Path integral quantisation of finite noncommutative geometries. Journal of Geometry and Physics [Online], 44(2), pp.115–128. Available at: https://doi.org/10.1016/S0393-0440(01)00064-X.

Lawvere, F.W., 1991. Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes. In: A. Carboni, M.C. Pedicchio and G. Rosolini, eds. Category Theory [Online], Lecture Notes in Mathematics. Berlin, Heidelberg: Springer, pp.279–281. Available at: https://doi.org/10.1007/BFb0084226.

Majid, S., 1988. Hopf algebras for physics at the Planck scale. Classical and Quantum Gravity [Online], 5(12), p.1587. Available at: https://doi.org/10.1088/0264-9381/5/12/010.

Majid, S., 1991. Principle of Representation-Theoretic Self-Duality. Physics Essays [Online], 4(3), pp.395–405. Available at: https://doi.org/10.4006/1.3028923.

Majid, S., 1993. Quantum random walks and time reversal. International Journal of Modern Physics A [Online], 08(25), pp.4521–4545. Available at: https://doi.org/10.1142/S0217751X93001818.

Majid, S., ed., 2012. On Space and Time [Online], Canto Classics. Cambridge: Cambridge University Press. Available at: https://doi.org/10.1017/CBO9781139197069.

Majid, S., 2013. Noncommutative Riemannian geometry on graphs. Journal of Geometry and Physics [Online], 69, pp.74–93. Available at: https://doi.org/10.1016/j.geomphys.2013.02.004.

Majid, S., 2014. The self-representing Universe. In: B. Carr, M. Eckstein, M. Heller and S.J. Szybka, eds. Mathematical structures of the universe. Kraków: Copernicus Center Press, pp.357–387.

Majid, S., 2015. Algebraic approach to quantum gravity I: relative realism. In: J. Ladyman et al., eds. Road to reality with Roger Penrose. Kraków: Copernicus Center Press, pp.117–177.

Majid, S., 2018. On the emergence of the structure of physics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences [Online], 376(2118), p.20170231. Available at: https://doi.org/10.1098/rsta.2017.0231.

Majid, S., 2019a. Quantum gravity on a square graph. Classical and Quantum Gravity [Online], 36(24), p.245009. Available at: https://doi.org/10.1088/1361-6382/ab4975.

Majid, S., 2019b. Quantum Riemannian geometry and particle creation on the integer line. Classical and Quantum Gravity [Online], 36(13), p.135011. Available at: https://doi.org/10.1088/1361-6382/ab2424.

Majid, S., 2020a. Quantum geometry of Boolean algebras and de Morgan duality. Journal of Noncommutative Geometry, in press. arXiv: 1911.12127 [math.QA].

Majid, S., 2020b. Reconstruction and quantization of Riemannian structures. Journal of Mathematical Physics [Online], 61(2), p.022501. Available at: https://doi.org/10.1063/1.5123258.

Majid, S. and Pachoł, A., 2018. Classification of digital affine noncommutative geometries. Journal of Mathematical Physics [Online], 59(3), p.033505. Available at: https://doi.org/10.1063/1.5025815.

Majid, S. and Pachoł, A., 2020a. Digital finite quantum Riemannian geometries. Journal of Physics A: Mathematical and Theoretical [Online], 53(11), p.115202. Available at: https://doi.org/10.1088/1751-8121/ab1cf2.

Majid, S. and Pachoł, A., 2020b. Digital quantum groups. Journal of Mathematical Physics [Online], 61(10), p.103510. Available at: https://doi.org/10.1063/5.0020958.

Majid, S. and Ruegg, H., 1994. Bicrossproduct structure of κ-Poincare group and non-commutative geometry. Physics Letters B [Online], 334(3), pp.348–354. Available at: https://doi.org/10.1016/0370-2693(94)90699-8.

Majid, S. and Tao, W.-Q., 2019. Generalised noncommutative geometry on finite groups and Hopf quivers. Journal of Noncommutative Geometry [Online], 13(3), pp.1055–1116. Available at: https://doi.org/10.4171/JNCG/345.

Pachter, L. and Sturmfels, B., 2004. Tropical geometry of statistical models. Proceedings of the National Academy of Sciences [Online], 101(46), pp.16132–16137. Available at: https://doi.org/10.1073/pnas.0406010101.

Reyes, G.E. and Zolfaghari, H., 1996. Bi-Heyting algebras, toposes and modalities. Journal of Philosophical Logic [Online], 25(1), pp.25–43. Available at: https://doi.org/10.1007/BF00357841.