zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Quantum observables algebras and abstract differential geometry: the topos-theoretic dynamics of diagrams of commutative algebraic localizations. (English) Zbl 1113.81071
Summary: We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection.
81R15Operator algebra methods (quantum theory)
18F20Categorical methods in sheaf theory
18D30Fibered categories
14F05Sheaves, derived categories of sheaves, etc.
53B50Applications of local differential geometry to physics
58A03Topos-theoretic approach to differentiable manifolds
[1]Artin, M., Grothendieck, A., and Verdier, J. L. (1972). Theorie de topos et cohomologie etale des schemas, Springer LNM 269 and 270, Springer-Verlag, Berlin.
[2]Bell, J. L. (1986). From absolute to local mathematics. Synthese 69.
[3]Bell, J. L. (1988). Toposes and Local Set Theories, Oxford University Press, Oxford.
[4]Bell, J. L. (2001). Observations on category theory. Axiomathes 12.
[5]Bell, J. L. (1982). Categories, toposes and sets. Synthese 51(3).
[6]Bohr, N. (1958). Atomic Physics and Human Knowledge, John Wiley, New York.
[7]Borceaux, F. (1994). Handbook of Categorical Algebra, Vols. 1–3, Cambridge U. P., Cambridge.
[8]Bub, J. (1997). Interpreting the Quantum World, Cambridge University Press, Cambridge.
[9]Butterfield, J. and Isham, C. J. (1998). A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations. International Journal of Theoretical Physics 37, 2669. · Zbl 0979.81018 · doi:10.1023/A:1026680806775
[10]Butterfield, J. and Isham, C. J. (1999). A topos perspective on the Kochen–Specker theorem: II. Conceptual aspects and classical analogues. International Journal of Theoretical Physics 38, 827. · Zbl 1007.81009 · doi:10.1023/A:1026652817988
[11]Butterfield, J. and Isham, C. J. (2000). Some possible roles for topos theory in quantum theory and quantum gravity. Foundations of Physics 30, 1707. · doi:10.1023/A:1026406502316
[12]Davis, M. (1977). A relativity principle in quantum mechanics. International Journal of Theoretical Physics 16, 867. · Zbl 0392.03040 · doi:10.1007/BF01807619
[13]Dieks, D. J. (1993). Quantum mechanics and experience. Studies in History and Philosophy of Modern Physics, 23.
[14]Folse, H. J. (1985). The Philosophy of Niels Bohr. The Framework of Complementarity, North-Holland, New York.
[15]Kelly, G. M. (1971). Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Notes Series 64, Cambridge U. P., Cambridge.
[16]Lawvere, F. W. and Schanuel, S. H. (1997). Conceptual Mathematics, Cambridge University Press, Cambridge.
[17]MacLane, S. (1971). Categories for the Working Mathematician, Springer-Verlag, New York.
[18]MacLane, S. and Moerdijk, I. (1992). Sheaves in Geometry and Logic, Springer-Verlag, New York.
[19]Mallios, A. (1998). Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry, vols. 1–2, Kluwer Academic Publishers, Dordrecht.
[20]Mallios, A. (2003). Remarks on ”singularities.” Progress in Mathematical Physics, Columbus, F. (Ed.), Nova Science Publishers, Hauppauge, New York (invited paper), gr-qc/0202028.
[21]Mallios, A. (2004a). On localizing topological algebras. Contemporary Mathematics 341, gr-qc/0211032.
[22]Mallios, A. (2004b) Geometry and Physics Today, physics/0405112.
[23]Mallios, A. (2005a). On Algebra Spaces, preprint.
[24]Mallios, A. (2005b). Modern Differential Geometry in Gauge Theories: Vol. 1. Maxwell Fields, Birkhäuser, Boston.
[25]Mallios, A. (2006a). Quantum gravity and ”singularities.” Note di Matematica, in press (invited paper), physics/0405111.
[26]Mallios, A. (2006b). Modern Differential Geometry in Gauge Theories: Vol. 2. Yang-Mills Fields, forthcoming by Birkhäuser, Boston.
[27]Mallios, A. and Raptis, I. (2004). CSmooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold, gr-qc/0411121.
[28]Mallios, A. and Rosinger, E. E. (1999). Abstract differential geometry, differential algebras of generalized functions, and de Rham cohomology. Acta Appl. Math. 55, 231. · Zbl 0929.18005 · doi:10.1023/A:1006106718337
[29]Mallios, A. and Rosinger, E. E. (2001). Space-time foam dense singularities and de Rham cohomology. Acta Appl. Math. 67, 59. · Zbl 1005.46020 · doi:10.1023/A:1010663502915
[30]Raptis, I. (2001). Presheaves, Sheaves, and their Topoi in Quantum Gravity and Quantum Logic, gr-qc/0110064.
[31]Rawling, J. P. and Selesnick, S. A. (2000). Orthologic and quantum logic. Models and computational elements. Journal of the Association for Computing Machinery 47, 721.
[32]Selesnick, S. A. (2004). Quanta, Logic and Spacetime (2nd. ed), World Scientific.
[33]Takeuti, G. (1978). Two applications of logic to mathematics. Mathematical Society of Japan 13, Kano Memorial Lectures 3.
[34]Varadarajan, V. S. (1968). Geometry of Quantum Theory, Vol. 1, Van Nostrand, Princeton, New Jersey.
[35]Zafiris, E. (2001). Probing quantum structure through Boolean localization systems for measurement of observables. International Journal of Theoretical Physics 39(12).
[36]Zafiris, E. (2004a). Boolean coverings of quantum observable structure: A setting for an abstract differential geometric mechanism. Journal of Geometry and Physics 50, 99. · Zbl 1068.18013 · doi:10.1016/j.geomphys.2003.11.010
[37]Zafiris, E. (2004b). Quantum Event Structures from the perspective of Grothendieck Topoi. Foundations of Physics 34(7).
[38]Zafiris, E. (2004c). Interpreting observables in a quantum world from the categorial standpoint. International Journal of Theoretical Physics 43(1).
[39]Zafiris, E. (2006). Generalized topological covering systems on quantum events structures. Journal of Physics A: Mathematical and General 39.