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)
Geometry and physics today. (English) Zbl 1111.83019
This essay represents a kind of meta-theoretical comparison study, between different geometrization methods in theoretical physics: the old ones, belonging to classical differential geometry must (in the author’s opinion) be replaced by some new methods, in the framework of “abstract” differential geometry (described as a general gauge theory on topological manifolds and fiber bundles).
MSC:
83C45Quantization of the gravitational field
81P05General and philosophical topics in quantum theory
83C05Einstein’s equations (general structure, canonical formalism, Cauchy problems)
References:
[1]Auyang, S. Y. (1995). How is Quantum Field Theory Possible? Oxford Univ. Press, Oxford.
[2]Baez, J. C. (Ed.) (1994). Knots and Quantum Gravity. Oxford Univ. Press, Oxford.
[3]Bergmann, P. G. (1979). Unitary Field Theory: Geometrization of Physics or Physicalization of Geometry? In ”The 1979 Berlin Einstein Symposium.” Lecture Notes in Physics, No 100. Springer-Verlag, pp. 84–88.
[4]Bogolubov, N. N., Logunov, A. A. and Todorov, I. T. (1975). Introduction to Axiomatic Quan/-tu/-m Field Theory. W.A. Benjamin, Reading, Mass.
[5]Bourbaki, N. (1970). Algèbre, Chap. 1–3. Hermann, Paris.
[6]Dugundji, J. (1966). Topology. Allyn and Bacon, Boston.
[7]Feynman, R. P. (1992). The Character of Physical Law. Penguin Books, London.
[8]Finkelstein, D. R. (1997). Quantum Relativity. A Synthesis of the Ideas of Einstein and Heisenberg. Springer-Verlag, Berlin (2nd Cor. Print.).
[9]Grauert, H. and Remmert, R. (1984). Coherent Analytic Sheaves. Springer-Verlag, Berlin.
[10]Isham, C. J. (1991). Canonical Groups and the quantization of geometry and topology. In A. Ashtekar and J. Stachel (eds.) ”Conceptual Problems of Quantum Gravity.” Birkhäuser, Basel, pp. 351–400.
[11]Mallios, A. (1986). Topological Algebras. Selected Topics. North-Holland, Amsterdam. [This item is also quoted, for convenience, throughout the text, by TA].
[12]Mallios, A. (1998a). On an axiomatic treatment of differential geometry via vector sheaves. Applications. Math. Japonica (Int. Plaza) 48, 93–180.
[13]Mallios, A. (1998b). Geometry of Vector Sheaves. An Axiomatic Approach to Differential Geometry, Vols. I (Chapts I–V), II (Chapts VI–XI). Kluwer, Dordrecht. [This is still quoted in the text, as VS].
[14]Mallios, A. (2002). Remarks on ”singularities.” gr-qc/0202028.
[15]Mallios, A. (2004). On localizing topological algebras. Contemp. Math. 341, 79–95.
[16]Mallios, A. (2006) Quantum gravity and ”singularities.” Note Mat. 25, 57–76.
[17]Mallios, A. (2005). Modern Differential Geometry in Gauge Theories. Vol.I: Maxwell Fields, Vol.II: Yang-Mills Fields. Birkhäuser, Boston, (2005/2006).
[18]Mallios, A. and Raptis, I. (2001). Finitary spacetime sheaves of quantum causal sets: Curving quantum causality. Int. J. Theor. Phys. 40, 1885–1928. · Zbl 0987.83003 · doi:10.1023/A:1011985002847
[19]Mallios, A. and Raptis, I. (2002). Finitary vCech-de Rham cohomology: much ado without cc-smoothness. Ibid. 41, 1857–1992.
[20]Mallios, A. and Raptis, I. (2003). Finitary, causal, and quantal vacuum Einstein gravity. Ibid. 42, 1479–1620.
[21]Mallios, A. and Raptis, I. (2005). C-smooth singularities exposed: Chimeras of the differential spacetime manifold, research monograph (in preparation); gr-qc/0411121.
[22]Mallios, A. and Rosinger, E. E. (1999). Abstract differential geometry, differential algebras of generalized functions, and de Rham cohomology. Acta Appl. Math. 55, 231–250. · Zbl 0929.18005 · doi:10.1023/A:1006106718337
[23]Mallios, A. and Rosinger, E. E. (2001). Space-time foam dense singularities and de Rham cohomology. ibid. 67, 59–89.
[24]Manin, Yu. I. (1981). Mathematics and Physics. Birkhäuser, Boston.
[25]Papatriantafillou, M. H. (2000). The category of differential triads. Bull. Greek Math. Soc. 44, 129–141.
[26]Papatriantafillou, M. H. (2004). Initial and final differential structures in Proc. Intern. Conf. on ”Topological Algebras and Applications” (ICTAA 2000), Rabat, Maroc. école Normale Supérieure, Takaddoum, Rabat, pp. 115–123.
[27]Papatriantafillou, M. H. (in preparation) Abstract Differential Geometry. A Categorical Perspective (book).
[28]Raptis, I. (2000). Finitary spacetime sheaves. Int. J. Theor. Phys. 39, 1703–1716. · Zbl 0966.83006 · doi:10.1023/A:1003600931892
[29]Rosinger, E. E. (1990) Non-Linear Partial Differential Equations. An Algebraic View of Generalized Solutions. North-Holland, Amsterdam.
[30]Sharpe, R. W. (1997). Differential Geomerty. Cartan’s Generalization of Klein’s Erlangen Program. Springer–Verlag, New York.
[31]Smith, D. E. (1958). History of Mathematics, Vol. I. Dover Publications, New York.
[32]Sorkin, R. D. (1991). Finitary substitute for continuous topology. Int. J. Theor. Phys. 30, 923–947. · Zbl 0733.54001 · doi:10.1007/BF00673986
[33]Stachel, J. (1993). The other Einstein: Einstein contra field theory in M. Beller, R. S. Cohen, and J. Renn (eds.) ”Einstein in Context”: Science in Context 6, 275–290. Cambridge Univ. Press, Cambridge, 1993.
[34]Weinstein, A. (1981). Symplectic geometry. Bull. Amer. Math. Soc. 5, 1–13. · Zbl 0465.58013 · doi:10.1090/S0273-0979-1981-14911-9
[35]Weyl, H. (1949). Philosophy of Mathematics and Natural Sciences. Princeton Univ. Press, Princeton, N.J.
[36]Weyl, H. (1952). Space-Time-Matter. Dover Publications, New York.
[37]Wittgenstein, L. (1997). Tractatus Logico-Philosophicus. Routledge, London.
[38]Wittgenstein, L. (2003). Remarks on Aesthetics, Psychology and Religious Belief. Blackwell, London.