Scale invariance vs conformal invariance

Abstract

In this review article, we discuss the distinction and possible equivalence between scale invariance and conformal invariance in relativistic quantum field theories. Under some technical assumptions, we can prove that scale invariant quantum field theories in $d=2$ dimension necessarily possess the enhanced conformal symmetry. The use of the conformal symmetry is well appreciated in the literature, but the fact that all the scale invariant phenomena in $d=2$ dimension enjoy the conformal property relies on the deep structure of the renormalization group. The outstanding question is whether this feature is specific to $d=2$ dimension or it holds in higher dimensions, too. As of January 2014, our consensus is that there is no known example of scale invariant but non-conformal field theories in $d=4$ dimension under the assumptions of (1) unitarity, (2) Poincaré invariance (causality), (3) discrete spectrum in scaling dimension, (4) existence of scale current and (5) unbroken scale invariance in the vacuum. We have a perturbative proof of the enhancement of conformal invariance from scale invariance based on the higher dimensional analogue of Zamolodchikov's $c$-theorem, but the non-perturbative proof is yet to come. As a reference we have tried to collect as many interesting examples of scale invariance in relativistic quantum field theories as possible in this article. We give a complementary holographic argument based on the energy-condition of the gravitational system and the space-time diffeomorphism in order to support the claim of the symmetry enhancement. We believe that the possible enhancement of conformal invariance from scale invariance reveals the sublime nature of the renormalization group and space-time with holography.

145 pages.This review is based on a lecture note on scale invariance vs conformal invariance, on which the author gave lectures at Taiwan Central University for the 5th Taiwan School on Strings and Fields, and I welcome your comments. v2,v3: minor corrections and reference added, v4: Substantial anniversary update. If you prefer the lecture style (with more typos) please fetch the older version

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• On Scale and Conformal Invariance in Four Dimensions
• Consistency of local renormalization group in d=3
• Aspects of the zero $\Lambda$ limit in the AdS/CFT correspondence
• Minimal Distances Between SCFTs
• Two-loop evolution equations for light-ray operators
• Conformal scaling and the size of $m$-hadrons
• Improvement of energy-momentum tensor and non-Gaussianities in holographic cosmology
• The a-theorem and entanglement entropy
• Comments on scale and conformal invariance in four dimensions
• General Covariance from the Quantum Renormalization Group
• Perturbative search for dead-end CFTs
• Scale Invariance plus Unitarity Implies Conformal Invariance in Four Dimensions
• IR fixed points in $SU(3)$ gauge Theories
• $a-c$ test of holography vs quantum renormalization group
• Lectures on 4d N=1 dynamics and related topics
• Imaginary supergravity or Virial supergravity?
• Clusters of galaxies in a Weyl geometric approach to gravity
• Conformal Behavior in QCD
• Zoology of heterotic supercurrent supermultiplets in d=2
• Superspace formulation of the local RG equation
• Massless Wigner particles in conformal field theory are free
• Conformal Invariance for a Class of Scale Invariant Theories in Four Dimensions
• Strong gauging or decoupling ghost matter
• Local renormalization group functions from quantum renormalization group and holographic bulk locality
• Conformal Quivers and Melting Molecules