zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
$\mathcal{A}$-invariance: An axiomatic approach to quantum relativity. (English) Zbl 1161.83353
Summary: The sort of approach claimed by the title of this article is realizable, at least, within the framework of ADG where we do not assume any “spacetime” supplying the dynamics we employ. The latter classical type of argument can naturally be included herewith along with its concomitant impediments that are emanated therefrom and are essentially “absorbed”, technically speaking, by the proposed mechanism. So our approach, being “manifoldless” (thence, no smoothness, in the standard sense) does not contain any such issue, as before, according to the very definitions, being thus “singularities”-free. As a consequence, the equations that one would be able to formulate within the present set-up will be, by the very essence of the matter, already the quantized ones.

MSC:
83C45Quantization of the gravitational field
83C75Space-time singularities, cosmic censorship, etc.
WorldCat.org
Full Text: DOI
References:
[1] Bergman, P.G.: Unitary field theory, geometrization of physics or physicalization of geometry? In: The 1979 Berlin Einstein Symposium. Lecture Notes in Physics, vol. 100, pp. 84--88. Springer, Berlin (1979)
[2] Chern, S.C.: What is Geometry? Am. Math. Mont. 97, 679--686 (1990) · Zbl 0717.51002 · doi:10.2307/2324574
[3] Dirac, P.A.M.: The early years of relativity. In: Holton, G., Elkana, Ye. (eds.) Albert Einstein: Historical and Cultural Perspective: The Centennial Symposium in Jerusalem 1979, pp. 79--90. Dover, New York (1997)
[4] Eddington, A.S.: Report on the Relativity Theory of Gravitation. Fleetway Press, London (1920)
[5] Einstein, A.: The Meaning of Relativity, 5th edn. Princeton Univ. Press, Princeton (1989) · Zbl 0063.01229
[6] Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, New York (1995) · Zbl 0819.13001
[7] Eisenbud, D., Harris, J.: The Geometry of Schemes. Springer, New York (2000) · Zbl 0960.14002
[8] Finkelstein, D.R.: Quantum Relativity. A Synthesis of the Ideas of Einstein and Heisenberg, 2nd edn. Springer, Berlin (1997)
[9] Feynman, R.P.: The Character of Physical Law. Penguin Books, London (1992)
[10] Haag, R.: Local Quantum Physics. Fields, Particles, Algebras, 2nd edn. Springer, Berlin (1996) · Zbl 0857.46057
[11] Isham, C.J.: Canonical groups and quantization of geometry and topology. In: Ashtekar, A., Stachel, J. (eds.) Conceptual Problems of Quantum Gravity, pp. 351--400. Birkhäuser, Basel (1991) · Zbl 0850.83023
[12] MacLane, S.: Categories for the Working Mathematician. Springer, New York (1971) · Zbl 0705.18001
[13] MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic. A First Introduction to Topos Theory. Springer, New York (1992) · Zbl 0822.18001
[14] Mallios, A.: Topological Algebras. Selected Topics. North-Holland, Amsterdam (1986) · Zbl 0597.46046
[15] Mallios, A.: Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry, vols. I--II. Kluwer Academic, Dordrecht (1998). Russian transl., MIR, Moscow, 2000/2001 · Zbl 0904.18001
[16] Mallios, A.: Remarks on ”singularities”. Preprint (2002), arXive: gr-qc/0202028. To appear (in revised form). In: Columbus, F. (ed.) Progress in Mathematical Physics. Nova Science, Hauppauge, New York (in press)
[17] Mallios, A.: On localizing topological algebras. Contemp. Math. 341, 79--95 (2004) · Zbl 1094.46045
[18] Mallios, A.: Quantum gravity and ”singularities”. Note Mat. 25 57--76 (2005/2006). Invited paper · Zbl 1115.83016
[19] Mallios, A.: Geometry and physics today. Int. J. Theor. Phys. 45, 1557--1593 (2006) · Zbl 1111.83019 · doi:10.1007/s10773-006-9130-3
[20] Mallios, A.: Modern Differential Geometry in Gauge Theories: Maxwell Fields, vol. I. Yang-Mills Fields, vol. II. Birkhäuser, Boston (2006/2007) · Zbl 1116.18006
[21] Mallios, A.: On algebra spaces. Contemp. Math. 427, 263--283 (2007) · Zbl 1122.46034
[22] Mallios, A.: Relational mathematics: A response to quantum relativity. Invited paper (to appear) · Zbl 1161.83353
[23] Mallios, A., Raptis, I.: $\mathcal{C}^{\infty}-$ smooth singularities exposed: Chimeras of the differential spacetime manifold. Preprint (Mar. 2005), arXiv: gr-qc/0411121
[24] Mallios, A., Rosinger, E.E.: Abstract differential geometry, differential algebras of generalized functions and de Rham cohomology. Acta Appl. Math. 55, 231--250 (1999) · Zbl 0929.18005 · doi:10.1023/A:1006106718337
[25] Mallios, A., Rosinger, E.E.: Space-time foam dense singularities and de Rham cohomology. Acta Appl. Math. 67, 59--89 (2001) · Zbl 1005.46020 · doi:10.1023/A:1010663502915
[26] Mallios, A., Rosiger, E.E.: Dense singularities and de Rham cohomology. In: Strantzalos, P., Fragoulopoulou, M. (eds.) Topological Algebras with Applications to Differential Geometry and Mathematical Physics. Proc. Fest-Col. in Honour of Professor Anastasios Mallios, 16--18/9/1999, pp. 54--71. Dept. Math. Univ. Athens Publs. (2002)
[27] Mallios, A., Zafiris, E.: Topos-theoretic relativization of physical representability and quantum gravity (manuscript) · Zbl 1238.81103
[28] Manin, Yu.I.: Mathematics and Physics, Birkhäuser, Boston (1981)
[29] Morgan, F.: Geometric Measure Theory, 3rd edn. Academic, San Diego (2000) · Zbl 0974.49025
[30] Nakahara, M.: Geometry, Topology and Physics. Adam Hilger, Bristol (1990) · Zbl 0764.53001
[31] Papatriantafillou, M.: Abstract Differential Geometry. A Categorical Perspective. Monograph (in preparation)
[32] Raptis, I.: Categorical quantum gravity. Int. J. Theor. Phys. 45, 1499--1527 (2006) · Zbl 1153.83336
[33] Raptis, I.: Finitary topos for locally finite, causal and quantal vacuum Einstein gravity. Int. J. Theor. Phys. 46, 688--739 (2007) · Zbl 1118.83314 · doi:10.1007/s10773-006-9240-y
[34] Raptis, I.: A dodecalogue of basic didactics from applications of abstract differential geometry to quantum gravity. Int. J. Theor. Phys. 46 (2007), gr-gc/0607038 · Zbl 1188.83076
[35] Shafarevich, I.R.: Basic Notions of Algebra. Springer, Berlin (1997) · Zbl 0860.16001
[36] Sorkin, R.D.: Finitary substitute for continuous topology. Int. J. Theor. Phys. 30, 923--947 (1991) · Zbl 0733.54001 · doi:10.1007/BF00673986
[37] Stachel, J.: The other Einstein: Einstein contra field theory. In: Beller, M., Cohen, R.S., Renn, J. (eds.) Einstein in Context. Science in Context, vol. 6, pp. 275--290. Cambridge Univ. Press, Cambridge (1933)
[38] Torretti, R.: Relativity and Geometry, Dover, New York (1983/1996) · Zbl 0515.53001
[39] Zafiris, E.: Probing quantum structure through Boolean localization systems. Int. J. Theor. Phys. 39, 2761--2768 (2000) · Zbl 0986.81005 · doi:10.1023/A:1026409131643
[40] Zafiris, E.: Quantum event structures from the perspective of Grothendieck topoi. Found. Phys. 34, 1063--1090 (2004) · Zbl 1072.81009 · doi:10.1023/B:FOOP.0000037623.08379.df
[41] Zafiris, E.: Boolean coverings for quantum observables structures: A setting for an abstract differential geometric mechanism. J. Geom. Phys. 50, 99--114 (2004) · Zbl 1068.18013 · doi:10.1016/j.geomphys.2003.11.010
[42] Zafiris, E.: Generalized topological covering systems on quantum events structures. J. Phys. A Math. Gen. 39, 1485--1505 (2006) · Zbl 1100.81002 · doi:10.1088/0305-4470/39/6/020
[43] Zafiris, E.: Quantum observables algebras and abstract differential geometry: The topos-theoretic dynamics of diagrams of commutative algebraic localizations. Int. J. Theor. Phys. 46, 319--382 (2007) · Zbl 1113.81071 · doi:10.1007/s10773-006-9223-z