Abstract
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be jointly measurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton–Isham–Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry à la Connes et al. To each unital C*- algebra \({\mathcal{A}}\) we associate a geometric object—a diagram of topological spaces collating quotient spaces of the noncommutative space underlying \({\mathcal{A}}\)—that performs the role of a generalised Gel'fand spectrum. We show how any functor F from compact Hausdorff spaces to a suitable target category \({\mathsf{C}}\) can be applied directly to these geometric objects to automatically yield an extension \({\tilde{F}}\) acting on all unital C*-algebras. This procedure is used to give a novel formulation of the operator K0-functor via a finitary variant \({\tilde{K}_{ f}}\) of the extension \({\tilde{K}}\) of the topological K-functor. We then delineate a C*-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital C*-algebra the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.
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References
Abramsky S., Brandenburger A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13(11), 113036 (2011)
Acín A., Fritz T., Leverrier A., Sainz A.B.: A combinatorial approach to nonlocality and contextuality. Commun. Math. Phys. 334(2), 533–628 (2015)
Adámek J., Herrlich H., Strecker George E.: Abstract and Concrete Categories: The Joy of Cats. Wiley- Interscience, Hoboken (1990)
Adams J.F.: Vector fields on spheres. Ann. Math. 75(3), 603–632 (1962)
Akemann C.A.: Left ideal structure of C*-algebras. J. Funct. Anal. 6(2), 305–317 (1970)
Alfsen E.M.: On the Dirichlet problem of the Choquet boundary. Acta Math. 120(1), 149–159 (1968)
Alfsen E.M., Shultz F.W.: State Spaces of Operator Algebras: Basic Theory, Orientations, and C*-Products, Mathematics: Theory & Applications. Birkhäuser, Basel (2001)
Atiyah M.F., Anderson D.W.: K-Theory. W. A. Benjamin, New York (1967)
Atiyah M.F., Bott R.: On the periodicity theorem for complex vector bundles. Acta Math. 112(1), 229–247 (1964)
Awodey S., Forssell H.: First-order logical duality. Ann. Pure Appl. Logic 164(3), 319–348 (2013)
Barbosa, R.S.: Contextuality in quantum mechanics and beyond. DPhil Thesis, University of Oxford (2015)
Bell J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38(3), 447–452 (1966)
Bichteler K.: A generalization to the non-separable case of Takesaki’s duality theorem for C*- algebras. Invent. Math. 9(1), 89–98 (1969)
Blackadar B.: Operator Algebras: Theory of C*-Algebras and von Neumann Algebras Encyclopaedia of Mathematical Sciences, vol. 122. Springer, Berlin (2006)
Bohr, N.: Discussion with Einstein on epistemological problems in atomic physics. In: Schilpp, P.A. (ed.) Albert Einstein: Philosopher-Scientist. The Library of Living Philosophers, vol. 7, , pp. 199–241. Northwestern University, Evanston (1949)
Bratteli O.: Inductive limits of finite dimensional C*-algebras. Trans. Am. Math. Soc. 171, 195–234 (1972)
Cabello A., Severini S., Winter A.: Graph-theoretic approach to quantum correlations. Phys. Rev. Lett. 112(4), 040401 (2014)
Christensen E.: Measures on projections and physical states. Commun. Math. Phys. 86(4), 529–538 (1982)
Connes A.: A factor not anti-isomorphic to itself. Ann. Math. 101, 536–554 (1975)
Connes A.: Noncommutative Geometry. Academic Press, New York (1995)
Dauns J., Hofmann K.H.: Representation of Rings by Sections Memoirs of the American Mathematical Society. vol. 83. American Mathematical Society, Providence (1968)
de Groote, H.F.: Observables IV: the presheaf perspective (2007). arXiv:0708.0677 [math-ph]
de Silva, N.: From topology to noncommutative geometry: K-theory (2014). arXiv:1408.1170 [math.OA]
de Silva, N.: Contextuality and noncommutative geometry in quantum mechanics. DPhil Thesis, University of Oxford (2015)
de Silva, N., Barbosa, R.S.: Partial and total ideals in von Neumann algebras (2014). arXiv:1408.1172 [math.OA]
Dixmier J.: Sur certains espaces considérés par M. H. Stone. Summa Brasiliensis Mathematicae 2, 151–182 (1951)
Döring A.: Kochen–Specker theorem for von Neumann algebras. Int. J. Theor. Phys. 44(2), 139–160 (2005)
Döring, A.: Flows on generalised Gelfand spectra of nonabelian unital C*-algebras and time evolution of quantum systems (2012). arXiv:1212.4882 [math.OA]
Döring, A.: Generalised Gelfand spectra of nonabelian unital C*-algebras (2012). arXiv:1212.2613 [math.OA]
Döring A., Harding J.: Abelian subalgebras and the Jordan structure of a von Neumann algebra. Houston J. Math. 42(2), 559–568 (2010)
Döring A., Isham C.J.: A topos foundation for theories of physics: I Formal languages for physics. J. Math. Phys. 49(5), 053515 (2008)
Döring A., Isham C.J.: A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory. J. Math. Phys. 49(5), 053516 (2008)
Döring A., Isham C.J.: A topos foundation for theories of physics: III Quantum theory and the representation of physical quantities with arrows. J. Math. Phys. 49(5), 053517 (2008)
Döring A., Isham C.J.: A topos foundation for theories of physics: IV Categories of systems. J. Math. Phys. 49(5), 053518 (2008)
Edwards D.A.: The mathematical foundations of quantum mechanics. Synthese 42(1), 1–70 (1979)
Eilenberg S., Steenrod N.E.: Axiomatic approach to homology theory. Proc. Natl. Acad. Sci. USA 31(4), 117–120 (1945)
Einstein A., Podolsky B., Rosen N.: Can quantum-mechanical description of physical reality be considered complete. Phys. Rev. 47(10), 777–780 (1935)
Elliott, G.A.: The classification problem for amenable C*-algebras. In: Chatterji, S.D. (ed.) Proceedings of the International Congress of Mathematicians 1994, pp. 922–932. Birkhäuser, Basel (1995)
Fell J.M.G.: The structure of algebras of operator fields. Acta Math. 106(3), 233–280 (1961)
Fillmore P.A.: A User’s Guide to Operator Algebras Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 14. Wiley-Interscience, Hoboken (1996)
Flori, C., Fritz, T.: (Almost) C*-algebras as sheaves with self-action. J. Noncommut. Geom. (2015). arXiv:1512.01669 [math.OA] (to appear)
Folland G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, Hoboken (1984)
Fujimoto I.: A Gelfand–Naimark theorem for C*-algebras. Pac. J. Math. 184(1), 95–119 (1998)
Gel’fand I.M., Naĭmark M.A.: On the imbedding of normed rings into the ring of operators in Hilbert space. Recueil Mathématique [Matematicheskiĭ Sbornik] Nouvelle Série 12(54), 197–213 (1943)
Giles R., Kummer H.: A non-commutative generalization of topology. Indiana Univ. Math. J. 21(1), 91–102 (1972)
Givant S., Halmos P.: Introduction to Boolean Algebras. Undergraduate Texts in Mathematics. Springer, Berlin (2009)
Gleason A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6(6), 885–893 (1957)
Grothendieck A.: Sur les applications linéaires faiblement compactes d’espaces du type C(K). Can. J. Math. 5, 129–173 (1953)
Grothendieck, A.: Classes de faisceaux et théorème de Riemann–Roch. Institut des hautes études scientifiques (1968)
Hamhalter J.: Quantum Measure Theory Fundamental Theories of Physics, vol. 134. Springer, Berlin (2003)
Hamhalter J., Turilova E.: Automorphisms of order structures of abelian parts of operator algebras and their role in quantum theory. Int. J. Theor. Phys. 53(10), 3333–3345 (2014)
Hamilton J., Isham C.J., Butterfield J.: A topos perspective on the Kochen–Specker theorem: III Von Neumann algebras as the base category. Int. J. Theor. Phys. 39(6), 1413–1436 (2000)
Hartshorne R.: Algebraic Geometry Graduate Texts in Mathematics, vol. 52. Springer, Berlin (1977)
Heunen C., Landsman N.P., Spitters B.: A topos for algebraic quantum theory. Commun. Math. Phys. 291(1), 63–110 (2009)
Heunen, C., Landsman, N.P., Spitters, B.: Bohrification. In: Halvorson, H. (ed.) Deep Beauty: Under standing the Quantum World Through Mathematical Innovation, pp. 271–314. Cambridge University Press, Cambridge (2011)
Heunen C., Landsman N.P., Spitters B.: Bohrification of operator algebras and quantum logic. Synthese 186(3), 719–752 (2012)
Heunen C., Landsman N.P., Spitters B., Wolters S.: The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach. J. Austr. Math. Soc. 90(1), 39–52 (2011)
Heunen C., Reyes M.L.: Active lattices determine AW*-algebras. J. Math. Anal. Appl. 416(1), 289–313 (2014)
Howard M., Wallman J., Veitch V., Emerson J.: Contextuality supplies the ‘magic’ for quantum computation. Nature 510(7505), 351–355 (2014)
Isham C.J., Butterfield J.: A topos perspective on the Kochen–Specker theorem: I Quantum states as generalized valuations. Int. J. Theor. Phys. 37, 2669–2733 (1998)
Isham C.J., Butterfield J.: A topos perspective on the Kochen–Specker theorem: II Conceptual aspects and classical analogues. Int. J. Theor. Phys. 38, 827–859 (1999)
Isham C.J., Butterfield J.: A topos perspective on the Kochen–Specker theorem: IV Interval valuations. Int. J. Theor. Phys. 41, 613–639 (2002)
Johnstone P.T.: Stone Spaces Studies in Advanced Mathematics, vol. 3. Cambridge Uni versity Press, Cambridge (1982)
Kadison R.V.: A representation Theory for Commutative Topological Algebra Memoirs of the Amer ican Mathematical Society, vol. 7. American Mathematical Society, Providence (1951)
Kadison R.V., Ringrose, J.R. (1983) Fundamentals of the Theory of Operator Algebras: Volume I. Elementary Theory. Pure and Applied Mathematics, vol. 100I. Academic Press
Kakutani S.: Concrete representation of abstract (m)-spaces (A characterization of the space of continuous functions). Ann. Math. 42(4), 994–1024 (1941)
Khalkhali, M.: Very basic noncommutative geometry (2004). arXiv:math/0408416 [math.KT]
Kochen S., Specker E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17(1), 59–87 (1967)
Kruml D., Pelletier J.W., Resende P., Rosický J., On quantales and spectra of C*-algebras. Appl. Categ. Struct. 11(6), 543–560 (2003)
Kruszyński P., Woronowicz S.: A non-commutative Gelfand–Naimark theorem. J. Oper. Theory 8(2), 361–389 (1982)
Lurie, J.: Math 261y: von Neumann Algebras, Lecture Notes, Harvard University (2011). www.math.harvard.edu/~lurie/261y.html
Mac Lane S., Moerdijk I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory Universitext. Springer, Berlin (1992)
Maeda S.: Probability measures on projections in vonNeumann algebras.Rev.Math. Phys. 1(2–3), 235–290 (1989)
Markov A.: On mean values and exterior densities. Recueil Mathématique [Matematicheskiĭ Sbornik] Nouvelle Série 4(46), 165–191 (1939)
Marsden J.E., Ratiu T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems Texts in Applied Mathematics, vol. 17. Springer, Berlin (1999)
Mayet R.: Orthosymmetric ortholattices. Proc. Am. Math. Soc. 114(2), 295–306 (1992)
Mulase M.: Category of vector bundles on algebraic curves and infinite dimensional Grassmannians. Int. J. Math. 1(3), 293–342 (1990)
Mulvey C.J.: Anon-commutative Gel’fand–Naimark theorem.Math. Proc. Camb. Philos. Soc. 88, 425–428 (1980)
Pedersen G.K.: SAW*-algebras and corona C* -algebras, contributions to non-commutative topology. J. Oper. Theory 15(1), 15–32 (1986)
Raussendorf R.: Contextuality in measurement-based quantum computation. Phys. Rev. A 88(2), 022322 (2013)
Redhead M.: Incompleteness, Nonlocality, and Realism: AP rolegomenon to the Philosophy of Quantum Mechanics. Oxford University Press, Oxford (1987)
Resende P.: Étale groupoids and their quantales. Adv. Math. 208(1), 147–209 (2007)
Reyes, M.L.: Obstructing extensions of the functor Spec to noncommutative rings (2011). arXiv:1101.2239 [math.RA]
Rieffel M.A.: C*-algebras associated with irrational rotations. Pac. J. Math. 93(2), 415–429 (1981)
Riesz F.: Sur les opérations fonctionnelles linéaires. eComptes Rendus Hebdomadaires Des sánces de l’Académie Des Sciences 149, 974–977 (1909)
Rørdam M., Larsen F., Laustsen N.: An Introduction to K-Theory for C* -Algebras London Mathematical Society Student Texts, vol. 49. Cambridge University Press, Cambridge (2000)
Segal I.E.: Irreducible representations of operator algebras. Bull. Am.Math. Soc. 53(2), 73–88 (1947)
Shulman, M.: nLab: slice 2-category, version 3 (2013). http://ncatlab.org/nlab/revision/slice+2-category/3
Shultz F.W.: Pure states as a dual object for C*-algebras. Commun. Math. Phys. 82(4), 497–509 (1982)
Sikorski R.: Boolean Algebras Ergebnisse der Mathematik und Ihrer Grenzgebiete, 2. Folge, vol. 25. Springer, Berlin (1960)
Spekkens R.W.: Contextuality for preparations, transformations, and unsharp measurements. Phys. Rev. A 71(5), 052108 (2005)
Stone M.H.: The theory of representations for Boolean algebras. Trans. Am. Math. Soc. 40(1), 37–111 (1936)
Stone M.H.: Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc. 41(3), 375–481 (1937)
Swan R.G.: Vector bundles and projective modules. Trans. Am. Math. Soc. 105(2), 264–277 (1962)
Takesaki M.: A duality in the representation theory of C*-algebras. Ann. Math. 85(3), 370–382 (1967)
Takesaki M.: Theory of Operator Algebras I. Springer, Berlin (1979)
Tarski A.: Zur Grundlegung der Boole’schen Algebra I. Fundamenta Mathematicae 24(1), 177–198 (1935)
tom Dieck, T.: Algebraic Topology EMS Textbooks in Mathematics, vol. 7. European Mathematical Society, Zurich (2008)
Wegge-Olsen N.E.: K-Theory and C*-Algebras: A Friendly Approach. Oxford University Press, Oxford (1993)
Yeadon F.J.: Measures on projections in W*-algebras of type II 1. Bull. Lond. Math. Soc. 15(2), 139–145 (1983)
Yeadon F.J.: Finitely additive measures on projections in finite W*-algebras. Bull. Lond. Math. Soc. 16(2), 145–150 (1984)
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de Silva, N., Barbosa, R.S. Contextuality and Noncommutative Geometry in Quantum Mechanics. Commun. Math. Phys. 365, 375–429 (2019). https://doi.org/10.1007/s00220-018-3222-9
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DOI: https://doi.org/10.1007/s00220-018-3222-9