Sur l'élimination des "infinis" en théorie quantique des champs : la régularisation zeta à l'épreuve de l'interprétation de Colombeau ou vice versa

Charpentier Éric

GDML_Books, (1999), p.

Abstract

Zeta regularization is one of the heuristic techniques used, specifically, in quantum field theory (QFT) in order to extract from divergent expressions some finite and well-defined values. Colombeau’s theory of generalized functions (containing the distributions and allowing their multiplication) provides a mathematically rigorous setting for QFT, where only convergent expressions appear. The aim of this paper is to compare these two points of view, both in their underlying logic and in their results. After an account of the general idea proposed here (a link between zeta regularization and Colombeau constraints), this idea is applied and put to test in some instances of the Casimir effect for the conformal Klein-Gordon (CKG) equation (the meaning of all these terms is recalled):   • first, on a sphere $S_R^{d-1}$: for d=2 one recovers the Casimir energy given by zeta regularization; for d odd too, if some constraints imposed to make the result finite are consistent; for d even ≥4 a term which seems arbitrary remains - it seems that in order to give a specific value to this term, one ought to complete the standard Colombeau interpretation;   • then, on the cylindrical space $S_R^{1} × ℝ^{d-2}$: Colombeau’s interpretation gives the same result as zeta regularization; here the unique constraint does have solutions. We also consider some variants (a space confined between two hyperplanes - here, there are two constraints - and the historical Casimir effect, where there is only one constraint, and it does have solutions). One also considers the diagonal values of the CKG Green function on $S_R^{1}$, where zeta regularization does not give a finite result: the Colombeau formalism provides a (Colombeau) number which is not associated with an ordinary number, and whose dependence on a “resolution” ε corresponds to the usual renormalization (semi-) group. Finally, applying the method to the embedding of the string worldsheet into the Minkowski space-time, one recovers the correct Virasoro algebra, in the sense of Colombeau’s weak equality, and obtains a mathematical formulation of some heuristic arguments by Susskind about the quantum spread of the string. The paper is to a large extent self-contained, and is accessible to a wide audience of mathematicians.

2000 Mathematics Subject Classification (version of July 31, 1998): 46F10 Operations with distributions (generalized functions), 46F30 Generalized functions for nonlinear analysis [Colombeau's generalized functions], 40G99 [Zeta-function (Ramanujan) method of summability], 81S05 Commutation relations [canonical quantization], 81T20 Quantum field theory on curved space backgrounds, 81T30 String and superstring theories, also 33C55, 81Q10.

Résumé

Table des matières 0. Introduction......................................................................................................................................................................................5  0.1. Le problème.................................................................................................................................................................................5  0.2. Les solutions................................................................................................................................................................................6  0.3. Ce que propose le présent article................................................................................................................................................7 1. Présentation heuristique de la méthode: la substance intuitive de la régularisation zeta.................................................................8  1.1. Fonction theta “anomale”.............................................................................................................................................................8  1.2. Régularisation zeta et contraintes de Colombeau........................................................................................................................9 2. Effet Casimir et fonction de Green sur le cercle.............................................................................................................................11  2.1. Rappel de la théorie classique, notations...................................................................................................................................11  2.2. Quantification.............................................................................................................................................................................11 3. Effet Casimir dans un espace sphérique $S_R^{d-1}$: dimension d ≥ 3........................................................................................16  3.1. Fonctions test, zooms.................................................................................................................................................................17  3.2. Définition des ensembles $_q$ de fonctions test....................................................................................................................17  3.3. Définition des fonctions et nombres généralisés........................................................................................................................18  3.4. Rappels de la théorie de Klein-Gordon classique......................................................................................................................20  3.5. Quantification.............................................................................................................................................................................22  3.6. “L’effet Casimir cosmologique”...................................................................................................................................................26  3.7. Méthode de la fonction génératrice............................................................................................................................................31 4. L’effet Casimir dans un espace cylindrique $S_R^1 × ℝ^{d-2}$.....................................................................................................32  4.1. Notations et théorème................................................................................................................................................................32  4.2. Preuve du théorème 4.1: d pair (≥4)..........................................................................................................................................33  4.3. Preuve du théorème 4.1: d impair..............................................................................................................................................35 5. Application à la quantification canonique de la corde....................................................................................................................36  5.1. Théorie classique: rappels.........................................................................................................................................................36  5.2. Quantification par la méthode de Colombeau............................................................................................................................37 6. Appendice I: Introduction à la théorie de Colombeau.....................................................................................................................42  6.1. Fonctions généralisées..............................................................................................................................................................42  6.2. Valeurs des fonctions généralisées et intégration......................................................................................................................44  6.3. Généralisations..........................................................................................................................................................................44  6.4. Une nouvelle sorte de géométrie................................................................................................................................................45 7. Appendice II: Preuve que ∀n, ∀t, $∂ⁿₜφ(ψ_{ε,θ},t)$ est modéré......................................................................................................45 8. Appendice III: Preuve que $F_{ε,ψ}$ est une fonction de Schwartz................................................................................................46 9. Appendice IV: Expression des “pré-régulateurs” comme combinaisons linéaires de fonctions de Clifford d’argument t₁x²/2..........48 10. Appendice V: Comment la forme du “cut-off” lève un “paradoxe”.................................................................................................50 Références.........................................................................................................................................................................................53

Détail

EUDML-ID : urn:eudml:doc:275831