Two questions on the geometry of gauge fields

Costa, Newton C. A. da; Dória, Francisco Antônio; Furtado-do-Amaral, A. F.; Barros, J. Acácio de

doi:10.1007/bf02054673

Cited by 6 publications

(4 citation statements)

References 17 publications

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“…We believe that it is possible to extend our results while eliminating those restrictions. (One possibility should be to modify the geometry of an irreducible principal fiber bundle in such a way as to handle true copies as false copies (see [6] on this). That is a possible way to define an analytical condition for generic copies through the use of the Atiyah-Singer index theorem.…”

Section: Discussionmentioning

confidence: 99%

“…We are working also in a generalization of such ideas for the case of true gauge field copies. If we modify the geometry of an irreducible principal fiber bundle in a manner to handle true copies as false [4], it seems possible to state topological and analytical conditions for generic gauge field copies.…”

Section: An Analytical Condition For the Existence Of False Gauge Fie...mentioning

confidence: 99%

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The Atiyah - Singer index theorem and the gauge field copy problem

Sant'Anna¹,

Costa²,

Dória³

1997

J. Phys. A: Math. Gen.

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Section: Discussionmentioning

confidence: 99%

Section: An Analytical Condition For the Existence Of False Gauge Fie...mentioning

confidence: 99%

The Atiyah - Singer index theorem and the gauge field copy problem

Sant'Anna¹,

Costa²,

Dória³

1997

J. Phys. A: Math. Gen.

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“…Our results are consequences of general incompleteness theorems that can be found in our papers [4] [17]; other references are [24] [32] [33]. In the present paper we summarize and state without proofs our chief results, with a few comments to clarify their meaning; an Appendix sketches a more formal treatment, which is fully available in the references.…”

mentioning

confidence: 81%

On the Incompleteness of Axiomatized Models for the Empirical Sciences

Costa¹,

Dória²

1992

Philosophica

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“…ics. The starting point was a venerable one: the 1900 list of23 problems that David Hilbert presented to the Second International Congress of Mathematicians in Paris. The sixth problem in Hilbert's list asks for an axiomatic formulation of physics; the tenth problem asks for a decision procedure to verify whether a polynomial Diophantine equation with integral coefficients has any solutions.…”

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confidence: 99%

Gödel incompleteness, explicit expressions for complete arithmetic degrees and applications

Costa

Dória²

1995

Complexity

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We summarize in a n intuitiue vein a few concepts froin recursion theory and fi-om the theory ofjormal systems and then state and comment our recent 1-esults on the incoinpleteness of elementary real analysis and its consequences. Their relation to forcing is also dealt with. 0 1995 John Wiiviley & Sons, Inc. a MOTIVATIONhere are simple noncomputable expressions for functions within mathematics; those expressions creep up T even within languages close to arithmetic, and lead to very simple questions such as, "does the integer-valued function O(n) equal 0 or 12" which turn out to be undecidable in the general case. Analogous nai've looking but intractable expressions for functions can also be found within more elaborate languages, as classical elementary analysis. With their help we can generate infinitely many undecidable sentences with a trivial appearance from arithmetic on and all the way up to the whole of mathematics.Some of those intractable expressions represent the halting function e (m, n), that tells us whether the Turing machine M,,,(n) stops over its input n. Once we have an expression for the halting function, we can obtain explicit expressions for all complete arithmetic degrees and even beyond. Therefore, the associated undecidable predicates represent problems in all the corresponding degrees of unsolvability, both inside and outside the arithmetic hierarchy.We used those undecidable predicates and noncomputable functions to settle several open decision problems, mainly in dynamical systems theory. Two such problems deal with frequently handled questions: 0Is there a decision procedure for chaos? Chaos theory has been a fast growing research area since the early 1970s, a decade after the discovery by E. Lorenz of an apparent, but still unproved chaotic behavior in a deterministic nonlinear dynamical system (for references see [7]). Chaos scientists usually proceed in one of two ways: whenever they wish to know if a given physical process is chaotic the usual starting point is to write down the equations that describe the process; out of them one tries to check whether the process satisfies some of the established mathematical criteria for chaos and randomness.However, those equations are in most cases intractable nonlinear differential equations as they cannot in general be given explicit analytical solutions. Therefore, chaos theorists turn to computer simulations and for most non-

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Two questions on the geometry of gauge fields

Cited by 6 publications

References 17 publications

The Atiyah - Singer index theorem and the gauge field copy problem

The Atiyah - Singer index theorem and the gauge field copy problem

On the Incompleteness of Axiomatized Models for the Empirical Sciences

Gödel incompleteness, explicit expressions for complete arithmetic degrees and applications

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