Bogolubov–Parasiuk theorem in the α-parametric representation

Bergère, M. C.; Lam, Yuk‐Ming P.

doi:10.1063/1.523078

Cited by 55 publications

(73 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A complete proof of this renormalizability property goes well beyond the scope of this Letter and is given elsewhere [15]. the analysis being inspired from the direct proof by Bergère and Lam of the renormalizability in field theory of Feynman amplitudes in the α-representation [17].…”

Section: Fig 1: Factorization Property (9)mentioning

confidence: 99%

Renormalization of crumpled manifolds

1993

View full text Add to dashboard Cite

We consider a model of D-dimensional tethered manifold interacting by excluded volume in IR d with a single point. By use of intrinsic distance geometry, we first provide a rigorous definition of the analytic continuation of its perturbative expansion for arbitrary D, 0 < D < 2. We then construct explicitly a renormalization operation R, ensuring renormalizability to all orders. This is the first example of mathematical construction and renormalization for an interacting extended object with continuous internal dimension, encompassing field theory.

show abstract

Section: Fig 1: Factorization Property (9)mentioning

confidence: 99%

Renormalization of crumpled manifolds

1993

View full text Add to dashboard Cite

show abstract

“…Such a reorganization is presented in section 7, and requires an elaborate "equivalence classes of nests" construction, inspired from [23].…”

Section: They Correspond To An Upper Critical Dimension D ⋆ = 2d/(2 −mentioning

confidence: 99%

“…Our construction is inspired by a construction by Bergère and Lam in [23] in the context of local field theories in the Schwinger representation. Extensive modifications are however necessary in order to make this construction applicable in our context.…”

Section: Equivalence Classes Of Nests: General Constructionmentioning

confidence: 99%

Renormalization theory for interacting crumpled manifolds

David¹,

Duplantier²,

Guitter³

1993

Nuclear Physics B

100

View full text Add to dashboard Cite

We consider a continuous model of D-dimensional elastic (polymerized) manifold fluctuating in d-dimensional Euclidean space, interacting with a single impurity via an attractive or repulsive δ-potential (but without self-avoidance interactions). Except for D = 1 (the polymer case), this model cannot be mapped onto a local field theory. We show that the use of intrinsic distance geometry allows for a rigorous construction of the high-temperature perturbative expansion and for analytic continuation in the manifold dimension D. We study the renormalization properties of the model for 0 < D < 2, and show that for bulk space dimension d smaller that the upper critical, the perturbative expansion is ultra-violet finite, while ultraviolet divergences occur as poles at d = d ⋆ . The standard proof of perturbative renormalizability for local field theories (the Bogoliubov Parasiuk Hepp theorem) does not apply to this model. We prove perturbative renormalizability to all orders by constructing a subtraction operator R based on a generalization of the Zimmermann forests formalism, and which makes the theory finite at d = d ⋆ . This subtraction operation corresponds to a renormalization of the coupling constant of the model (strength of the interaction with the impurity). The existence of a Wilson function, of an ǫ-expansionà la Wilson Fisher around the critical dimension, of scaling laws for d < d ⋆ in the repulsive case, and of non-trivial critical exponents of the delocalization transition for d > d ⋆ in the attractive case is thus established. To our knowledge, this study provides the first proof of renormalizability for a model of extended objects, and should be applicable to the study of self-avoidance interactions for random manifolds.

show abstract

“…It can be used as a starting point to work out the renormalization of the model directly in parametric space, as can be done in the commutative case [14]. It is also a good starting point to define the regularization and minimal dimensional renormalization scheme of NCΦ 4 4 .…”

Section: Introductionmentioning

confidence: 99%

Parametric Representation of Noncommutative Field Theory

Gurău

Rivasseau

2007

Commun. Math. Phys.

157

View full text Add to dashboard Cite

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable φ 4 4 quantum field theory on the Moyal non commutative R 4 space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual "Kirchoff" or "Symanzik" polynomials of commutative field theory, but contain richer topological information. I IntroductionNon-commutative field theories (for a general review see [1]) deserve a thorough and systematic investigation. Indeed they may be relevant for physics beyond the standard model. They are certainly effective models for certain limits of string theory [2]-[3]. What is often less emphasized is that they can also describe effective physics in our ordinary standard world but with non-local interactions, such as the physics of the quantum Hall effect [4].In joint work with J. , is renormalizable to all orders in perturbation theory using direct space multiscale analysis. *

show abstract

Bogolubov–Parasiuk theorem in the α-parametric representation

Cited by 55 publications

References 10 publications

Renormalization of crumpled manifolds

Renormalization of crumpled manifolds

Renormalization theory for interacting crumpled manifolds

Parametric Representation of Noncommutative Field Theory

Contact Info

Product

Resources

About