Abstract
We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of Gödel's first incompleteness theorem within an (axiomatized) theory of gauge fields.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. F. Furtado do Amaral, PhD Thesis, Institute of Physics, Federal University of Rio de Janeiro (1983).
A. F. Furtado do Amaral, F. A. Doria, and M. Gleiser,J. Math. Phys. 24, 1888 (1983).
R. D. Ball,Phys. Rep. 182, 1 (1989).
N. C. A. da Costa and R. Chuaqui,Erkenntnis 29, 95 (1988).
N. C. A. da Costa and F. A. Doria, “Suppes Predicates for Classical Physics,” in J. Echeverriaet al., The Space of Mathematics, de Gruyter (1992).
N. C. A. da Costa and F. A. Doria,Int. J. Theor. Phys. 30, 1041 (1991).
N. C. A. da Costa and F. A. Doria, “Classical Physics and Penrose's Thesis,” inFound. Phys. Lett. 4, 363 (1991).
N. C. A. da Costa and F. A. Doria, “Sur l'Incomplétude formelle de la mécanique classique,”Int. J. Theor. Phys.,32, 2187 (1993).
N. C. A. da Costa, F. A. Doria, and A. F. F. Amaral, “A dynamical system where proving chaos is equivalent to proving fermat's conjecture,”Int. J. Theor. Phys. 32, 2187 (1993).
N. C. A. da Costa, F. A. Doria, and J. A. de Barros,Int. J. Theor. Phys. 29, 935 (1990).
F. A. Doria,J. Math. Phys. 20, 1464 (1979).
F. A. Doria, “A bifurcation set associated to the copy phenomenon in the space of gauge fields,” in G. I. Zapata, ed.,Functional Analysis, Holomorphy and Approximation Theory II (North-Holland, Amsterdam, 1984), p. 69.
F. A. Doria, M. Ribeiro da Silva, and A. F. Furtado-do-Amaral,Lett. Nuovo Cimento 40, 509 (1984).
F. A. Doria and S. M. Abrahão,J. Math. Phys. 19, 1650 (1978).
F. A. Doria, S. M. Abrahão, and A. F. Furtado do Amaral,Prog. Theor. Phys. 75, 1440 (1986).
H. Flanders,Differential Forms (Academic Press, New York, 1963).
D. Feed and K. Uhlenbeck,Instantons and Four-Manifolds (Springer, New York, 1984).
K. Gödel,Monatsh. Math. & Phyk. 38, 173 (1931).
L. Jantscher,Distributionen (de Gruyter, 1971).
F. W. Kamber and P. Tondeur,Foliated Bundles and Characteristic Classes (Lectures Notes in Math., No. 493) (Springer, New York, 1975).
S. Kobayashi and K. Nomizu,Foundations of Differential Geometry (Wiley, New York, 1963).
R. Penrose,Ann. Phys. (N.Y.) 40, 171 (1960).
D. Richardson,J. Symbol. Logic 33, 514 (1968).
J. Shoenfield,Mathematical Logic (Addison-Wesley, Reading, Massachusetts, 1967).
S. Sternberg,Lectures on Differential Geometry (Prentice-Hall, New York, 1964).
I. Stewart,Nature (London) 352, 664 (1991).
P. Suppes,Scientific Structures and Their Representation, preliminary version, Standord University (1988).
J. Thierry-Mieg, “Classical geometrical interpretations of ghost fields and anomalies in Yang-Mills theory and quantum gravity,” in W. A. Bardeen and A. R. While, eds.,Symposium on Anomalies, Geometry, Topology (World Scientific, Singapore, 1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
da Costa, N.C.A., Doria, F.A., Furtado-do-Amaral, A.F. et al. Two questions on the geometry of gauge fields. Found Phys 24, 783–800 (1994). https://doi.org/10.1007/BF02054673
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02054673