Scaling-field representation of Wilson's exact renormalization-group equation

Riedel, Eberhard K.; Golner, Geoffrey R.; Newman, Kathie E.

doi:10.1016/0003-4916(85)90341-0

Cited by 41 publications

(38 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This idea was introduced and developed in references [4,97,98,11] and has generated a large interest that is still ongoing. Progress have been reviewed for instance in references [99,100,8,9,10]. An ERGE can be rewritten as an infinite set of coupled partial differential equations.…”

Section: Polchinski Equation In the Lpamentioning

confidence: 99%

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

Meurice¹

2007

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Abstract.We review recent results concerning the renormalization group (RG) transformation of Dyson's hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigorous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and other RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large-N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review article is to present a configuration space counterpart, suitable for lattice formulations, of other RG equations formulated in momentum space (sometimes abbreviated as ERGE).

show abstract

Section: Polchinski Equation In the Lpamentioning

confidence: 99%

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

Meurice¹

2007

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…(Redundant fields correspond to quasi-local field redefinitions.) It is the existence of this field which causes quasi-local fixed-point theories to divide up into equivalence classes: every fixed-point theory exists as a one-parameter family of physically equivalent theories [15][16][17]24,25]. This is the origin of the quantization of the spectrum of δ.…”

Section: The Energy-momentum Tensormentioning

confidence: 99%

“…It has been appreciated for a long time that every critical fixed-point solution of the ERG equation in fact exists as a line of physically equivalent fixed points [15][16][17]24,25].…”

Section: Quasi-local Representationmentioning

confidence: 99%

On functional representations of the conformal algebra

Rosten¹

2017

Eur. Phys. J. C

View full text Add to dashboard Cite

Starting with conformally covariant correlation functions, a sequence of functional representations of the conformal algebra is constructed. A key step is the introduction of representations which involve an auxiliary functional. It is observed that these functionals are not arbitrary but rather must satisfy a pair of consistency equations corresponding to dilatation and special conformal invariance. In a particular representation, the former corresponds to the canonical form of the exact renormalization group equation specialized to a fixed point whereas the latter is new. This provides a concrete understanding of how conformal invariance is realized as a property of the Wilsonian effective action and the relationship to action-free formulations of conformal field theory. Subsequently, it is argued that the conformal Ward Identities serve to define a particular representation of the energy-momentum tensor. Consistency of this construction implies Polchinski's conditions for improving the energy-momentum tensor of a conformal field theory such that it is traceless. In the Wilsonian approach, the exactly marginal, redundant field which generates lines of physically equivalent fixed points is identified as the trace of the energy-momentum tensor.

show abstract

“…10 Notice that, because one performs a derivative with respect to Λ, the essential contribution to ∂ t Γ comes from the integration over a small range of values of s (corresponding to the rapid decreasing of F Λ (s)), hence an ultraviolet regularization is not needed provided that the resulting RG equation be finite.…”

Section: Effective Actionmentioning

confidence: 99%

Exact Renormalization Group Equations and the Field Theoretical Approach to Critical Phenomena

Bagnuls

Bervillier

2001

Int. J. Mod. Phys. A

View full text Add to dashboard Cite

After a brief presentation of the exact renormalization group equation, we illustrate how the field theoretical (perturbative) approach to critical phenomena takes place in the more general Wilson (nonperturbative) approach. Notions such as the continuum limit and the renormalizability and the presence of singularities in the perturbative series are discussed.This paper has two parts. In the first part we restrict ourselves to the presentation of some selected issues taken from a recent review on the exact renormalization group (RG) equation (ERGE) in the pure scalar case.[1] In the second part we illustrate the Wilson continuum limit of field theory (or, equivalently, what may be understood as the nonperturbative renormalizability). More generally we would like to indicate how the historical first version of the RG, based on a perturbative approach, takes place in the more general nonperturbative framework developed by Wilson. The illustration will be done in the light of actual RG trajectories obtained from a numerical study of the ERGE in the local potential approximation. [2] 1 Selected issues What does "exact" mean ?The word "exact" means: a continuous (i.e. not discrete) realization of the Wilson RG transformation of the action in which no approximation is made and no expansion is involved with respect to some small parameter. We are here only interested in the differential form of the ERGE. One could think that the word exact is not adapted to the Wilson renormalization because, compared to the standard version of renormalization 1 , it involves a finite cutoff and it is only a semi-group. Nevertheless the historical first version is entirely contained in the Wilson theory (see part 2).

show abstract

Scaling-field representation of Wilson's exact renormalization-group equation

Cited by 41 publications

References 36 publications

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

On functional representations of the conformal algebra

Exact Renormalization Group Equations and the Field Theoretical Approach to Critical Phenomena

Contact Info

Product

Resources

About