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

- 1999

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topCharpentier Éric. Sur l'élimination des "infinis" en théorie quantique des champs : la régularisation zeta à l'épreuve de l'interprétation de Colombeau ou vice versa. 1999. <http://eudml.org/doc/275831>.

@book{CharpentierÉric1999,

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.},

author = {Charpentier Éric},

language = {eng},

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

url = {http://eudml.org/doc/275831},

year = {1999},

}

TY - BOOK

AU - Charpentier Éric

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

PY - 1999

AB - 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.

LA - eng

UR - http://eudml.org/doc/275831

ER -

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