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Journal of Physics A Mathematical and General

Conformal covariance in the framework of Wilson's renormalization group approach

Lothar Schäfer

Published Mar 1, 1976• 25 citations
Publisher

Abstract

The author constructs a conformal operator in analogy to the generating operator of Wilson's incomplete-integration renormalization group. The invariance of the partition function with respect to that conformal operation yields identities among the cumulants. Evaluating these identities he finds a generalized and corrected form of the selection rule which determines those two-point cumulants which show a long-range tail. A general equation which governs the asymptotic form of the three-point cumulants is established. It is solved for several examples which involve operators of vector- or tensor-type. It is found that surface effects cannot be excluded a priori. However, the asymptotic expressions for the cumulants are consistent with a neglect of surface effects.

DOI
10.1088/0305-4470/9/3/008

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The author constructs a conformal operator in analogy to the generating operator of Wilson's incomplete-integration renormalization group. The invariance of the partition function with respect to that conformal operation yields identities among the cumulants. Evaluating these identities he finds a generalized and corrected form of the selection rule which determines those two-point cumulants which show a long-range tail. A general equation which governs the asymptotic form of the three-point cumulants is established. It is solved for several examples which involve operators of vector- or tensor-type. It is found that surface effects cannot be excluded a priori. However, the asymptotic expressions for the cumulants are consistent with a neglect of surface effects.
We derive an identity which determines the infinitesimal response of the cumulant average of the product of mmm arbitrary local fluctuating quantities (connected mmm-point functions) to changes in its arguments engendered by a member of an arbitrary continuous group of point transformations. We illustrate the use of the identity in a variety of cases, and in particular we show that there is an intimate connection between the covariance of the connected mmm-point functions under the special conformal group and under the group of dilations. By combining our identity with the assumptions of the operator algebra, we find that at the critical point the connected density mmm-point functions are covariant under both groups. We also give another derivation for an expression for the exponent $x=(\frac{1}{2})(5\ensuremath{-}\ensuremath{\eta})$, which was previously found by Green and Gunton.
We derive an asymptotic formula for operator product expansion coefficients of heavy operators in two dimensional conformal field theory. This follows from modular invariance of the genus two partition function, and generalises the asymptotic formula for the density of states from torus modular invariance. The resulting formula is universal, depending only on the central charge, but involves the asymptotic behaviour of genus two conformal blocks. We use monodromy techniques to compute the asymptotics of the relevant blocks at large central charge to determine the behaviour explicitly.
A generalization of Wilson's renormalization group equation is discussed. A large class of equivalent fixed points exists for each fixed point. The eigen perturbations fall into two classes: (i) scaling operators whose scaling exponents are independent of the fixed point within the class of equivalent fixed points, and (ii) redundant operators whose exponents depend on the choice of the renormalization group equation. The exponents of the redundant operators are spurious since the free energy depends only on the scaling fields of the scaling operators but not on the scaling fields of the redundant operators.
The $\ensuremath{\epsilon}$-expansion formula for the critical exponent $\ensuremath{\eta}$ derived by Shukla and Green from Wilson's exact renormalization-group equation is found to yield the standard result $\ensuremath{\eta}=\frac{1}{54}{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, independent of the choice of the incomplete-integration function. The result is obtained analytically in the limit $a\ensuremath{\rightarrow}\ensuremath{\infty}$ and numerically for arbitrary aaa, where aaa denotes the normalization constant of the Gaussian fixed-point functional.
A class of renormalization groups characterized as the general Gaussian renormalization group is defined. It includes the sharp cutoff renormalization group of Wilson, the linear renormalization group of Bell and Wilson, and the incomplete‐integration renormalization group of Wilson. The renormalization group approach to critical phenomena and the expansion around four dimensions is reviewed. The fixed point of the general Gaussian renormalization group is determined to order ε2, where ε=4−d (d is the dimensionality of the system), and the critical exponents are determined to the same order. The main conclusion is that the exponents do not depend on the cutoff functions of the renormalization group.