Abstract
We survey several problems related to logical aspects of quantum structures. In particular, we consider problems related to completions, decidability and axiomatizability, and embedding problems. The historical development is described, as well as recent progress and some suggested paths forward.
J. Harding: Partially supported by US Army grant W911NF-21-1-0247. Z. Wang: Partially supported by NSF grants FRG-1664351, CCF 2006463, and DOD MURI grant ARO W911NF-19-S-0008.
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References
Adams, D.H.: The completion by cuts of an orthocomplemented modular lattice. Bull. Austral. Math. Soc. 1, 279–280 (1969)
Amemiya, I., Araki, H.: A remark on Prion’s paper, publ. Res. Inst. Math. Ser. A 2, 423–427 (1966)
Birkhoff, G.: Combinatorial relations in projective geometries. Ann. Math. 36, 743–748 (1935)
Birkhoff, G.: Lattice Theory, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV American Mathematical Society, Providence, R.I. (1967)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. (2) 37(4), 823–843 (1936)
Bruns, G.: Free ortholattices. Canad. J. Math. 5, 977–985 (1976)
Bruns, G.: Varieties of modular ortholattices. Houston J. Math. 9(1), 1–7 (1983)
Bruns, G., Harding, J.: Amalgamation of ortholattices. Order 14, 193–209 (1998)
Bruns, G., Roddy, M.: A finitely generated modular ortholattice. Canad. J. Math. 35(1), 29–33 (1992)
Bugajska, K., Bugajski, S.: The lattice structure of quantum logics. Ann. Inst. Henri Poincaré 19, 333–340 (1973)
Chin, L.H., Tarski, A.: Distributive and modular laws in relation algebras. University of California Publications in Mathematics N.S. 1, pp. 341-383 (1951)
Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice-Hall (1973)
Dunn, J.M., Hagge, T.J., Moss, L.S., Wang, Z.: Quantum logic as motivated by quantum computing. J. Symbol. Logic 70(2), 353–369 (2005)
Dunn, J.M., Moss, L.S., Wang, Z.: Editors introduction: the third life of quantum logic: quantum logic inspired by quantum computing. J. Phil. Logic 42(3), 443–459 (2013)
Dvurečenskij, A.: Gleason’s Theorem and its Applications. Mathematics and its Applications, vol. 60. Springer (1983)
Dye, H.A.: On the geometry of projections in certain operator algebras. Ann. Math. 61, 73–89 (1955)
Evans, T.: Word problems. Bull. Amer. Math. Soc. 84, 789–802 (1978)
Faure, C., Frölicher, A.: Modern Projective Geometry. Springer-Science+Business Media, Dordrecht (2000)
Foulis, D.J.: Baer *-semigroups. Proc. Amer. Math. Soc. 11, 648–654 (1960)
Foulis, D.J., Greechie, R.J., Bennett, M.K.: A transition to unigroups. Int. J. Theoret. Phys. 37(1), 45–63 (1998)
Frayne, T., Morel, A.C., Scott, D.: Reduced direct products. Fund. Math. 51, 195–228 (1962)
Freese, R., Ježek, J., Nation, J.B.: Free Lattices, Mathematical Surveys and Monographs. AMS, Providence, RI (1995)
Fritz, T.: Quantum logic is undecidable. Arch. Math. Logic 60(3–4), 329–341 (2021)
Fussner, W., St. John, G.: Negative translations of orthomodular lattices and their logic (2021). arXiv:2106.03656
Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238(1), 345–371 (2001)
Gehrke, M., Harding, J., Venema, Y.: MacNeille completions and canonical extensions. Trans. Amer. Math. Soc. 358(2), 573–590 (2006)
Godowski, R., Greechie, R.J.: A non-standard quantum logic with a strong set of states. In: Beltrametti, E., van Fraassen, B.C. (eds.) Current Issues in Quantum Logic. Plenum Press, New York (1981)
Godowski, R., Greechie, R.J.: Some equations related to states on orthomodular lattices. Demonstratio Math. 17, 241–250 (1984)
Greechie, R.J.: An orthomodular poset with a full set of states not embeddable in Hilbert space. Carribean J. Sci. Math. 1, 15–26 (1969)
Greechie, R.J.: On generating distributive sublattices of orthomodular lattices. Proc. Am. Math. Soc. 67(1), 17–22 (1977)
Guz, W.: On the lattice structure of quantum logics. Ann. Inst. Henri Poincaré Sec. A 28(1), 1–7 (1978)
Hamhalter, J.: Quantum Measure Theory, Fundamental Theories of Physics. Kluwer Academic Publishers Group, Dordrecht (2003)
Harding, J.: Decompositions in quantum logic. Trans. AMS 348, 1839–1862 (1996)
Harding, J.: Canonical completions of lattices and ortholattices. Tatra Mt. Math. Publ. 15, 85–96 (1998)
Harding, J.: The free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators. Algebra Univ. 48(2), 171–182 (2002)
Harding, J.: Remarks on concrete orthomodular lattices. Int. J. Theor. Phys. 43(10), 2149–2168 (2004)
Harding, J.: Decidability of the equational theory of the continuous geometry CG(F). J. Phil. Logic 42(3), 461–465 (2013)
Herrmann, C.: On the equational theory of projection lattices of finite von-Neumann factors. J. Symbolic Logic 75(2), 1102–1110 (2010)
Herrmann, C.: A note on the “Third life of quantum logic” (2019). arXiv:1908.02639
Huhn, A.P.: Schwach distributive Verbände I. Acta Scientiarum Math. 33, 297–305 (1972)
Husimi, K.: Studies on the foundations of quantum mechanics I. Proc. Physico-Math. Soc. Jpn. 9, 766–78 (1937)
Jónsson, B.: Modular lattices and Desargues theorem. Math. Scand. 2, 295–314 (1954)
Jenča, G.: Effect algebras are the Eilenberg-Moore category for the Kalmbach Monad. Order 32(3), 439–448 (2015)
Kalmbach, G.: Orthomodular Lattices. London Mathematical Society Monographs, Academic Press, London (1983)
Kalmbach, G.: The free orthomodular word problem is solvable. Bull. Aust. Math. Soc. 34(2), 219–223 (1986)
Kaplansky, I.: Any orthocomplemented complete modular lattice is a continuous geometry. Ann. Math. 61(3), 524–541 (1955)
Kargaev, P.P.: The Fourier transform of the characteristic function of a set that is vanishing on the interval. (Russian) Mat. Sb. (N.S.) 117(159), 3, 397–411, 432 (1982)
Kharlampovich, O.G., Sapir, M.V.: Algorithmic problems in varieties. Int. J. Algebra Comput. 5(4–5), 379–602 (1995)
Lusin, N.N., Priwaloff, I.I.: Sur l’unicité et la multiplicit des fonctions analytiques. Ann. Sci. Ecole Norm. Sup. 42(3), 143–191 (1925)
MacLaren, M.D.: Atomic orthocomplemented lattices. Pacific J. Math. 14, 597–612 (1964)
Mackey, G.: Mathematical Foundations of Quantum Mechanics. Benjamin, W. A (1963)
Marker, D.: Introduction to the model theory of fields. Lecture Notes Logic 5, 1–37 (1996)
Mayet, R.: Equations holding in Hilbert lattices. Int. J. Theoret. Phys. 45(7), 1257–1287 (2006)
Mayet, R.: Equations and Hilbert lattices, Handbook of quantum logic and quantum structures, pp. 525–554. Elsevier Science BV, Amsterdam (2007)
Megill, N.D., Pavičić, M.: Hilbert lattice equations. Ann. Henri Poincaré 10(7), 1335–1358 (2010)
Megill, N.: Quantum logic explorer home page (2014). us.metamath.org/qlegif/mmql.html
Menger, K.: New foundations of projective and affine geometry. Ann. Math. 37, 456–482 (1936)
Murray, F., Von Neumann, J.: On rings of operators. Ann. Math. 371(2), 116–229 (1936)
Murray, F., von Neumann, J.: On rings of operators. II. Trans. Amer. Math. Soc. 41(2), 208–248 (1937)
Murray, F., von Neumann, J.: On rings of operators. IV. Ann. Math. 44(2), 716–808 (1943),
von Neumann, J.: On rings of operators. III. Ann. Math. 41(2), 94–161 (1940)
von Neumann, J.: Continuous geometry. Foreword by Israel Halperin, Princeton Mathematical Series, No. 25 Princeton University Press, Princeton, NJ (1960)
von Neumann, J.: Collected Works. In: Taub, A.H. (eds), vol. I–IV. Pergamon Press (1962)
Palko, V.: Embeddings of orthoposets into orthocomplete posets. Tatra Mt. Math. Publ. 3, 7–12 (1993)
Redei , M., Stoltzner, M.: John von Neumann and the Foundations of Quantum Physics. Vienna Circle Institute Yearbook (2001)
Roddy, M.S.: On the word problem for orthocomplemented modular lattices. Can. J. Math. 41, 961–1004 (1989)
Rump, W.: Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups. Forum. Math. 30(3), 973–995 (2018)
Slofstra, W.: Tsirelson’s problem and an embedding theorem for groups arising from non-local games. J. Amer. Math. Soc. 33(1), 1–56 (2020). arXiv:1606.03140
Tarski, A.: A Decision Method for Elementary Algebra and Geometry. RAND Corporation, California, Santa Monica (1948)
Titchmarch, E.C.: Introduction to the Theory of Fourier Integrals, 2nd edn. Clarendon Press (1948)
Uhlhorn, U.: Representation of symmetry transformations in quantum mechanics. Ark. Fys. 23, 307–340 (1963)
Varadarajan, V.S.: Geometry of Quantum Theory, 2nd edn. Springer (1985)
Weaver, N.: Mathematical Quantization. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL (2001)
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Harding, J., Wang, Z. (2022). Logical Aspects of Quantum Structures. In: Ngoc Thach, N., Kreinovich, V., Ha, D.T., Trung, N.D. (eds) Financial Econometrics: Bayesian Analysis, Quantum Uncertainty, and Related Topics. ECONVN 2022. Studies in Systems, Decision and Control, vol 427. Springer, Cham. https://doi.org/10.1007/978-3-030-98689-6_6
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