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Scale invariance vs conformal invariance

Yu Nakayama

TL;DR

The review analyzes when scale invariance implies conformal invariance in relativistic quantum field theories, detailing a rigorous proof in d=2 via Zamolodchikov’s c-theorem and discussing higher-dimensional progress. It develops a local renormalization group framework to relate the trace of the energy–momentum tensor to RG data, including the a-theorem in d=4 and its perturbative proofs, as well as holographic arguments based on energy conditions. While no non-conformal scale-invariant counterexample is known in d=4 under standard assumptions, the nonperturbative status remains open, with holography offering both supportive and countervailing perspectives. The work highlights how conformal invariance, RG irreversibility, and holography collectively illuminate the deep structure of quantum field theories and spacetime geometry, suggesting a profound link between symmetry enhancement and the renormalization group flow.

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.

Scale invariance vs conformal invariance

TL;DR

The review analyzes when scale invariance implies conformal invariance in relativistic quantum field theories, detailing a rigorous proof in d=2 via Zamolodchikov’s c-theorem and discussing higher-dimensional progress. It develops a local renormalization group framework to relate the trace of the energy–momentum tensor to RG data, including the a-theorem in d=4 and its perturbative proofs, as well as holographic arguments based on energy conditions. While no non-conformal scale-invariant counterexample is known in d=4 under standard assumptions, the nonperturbative status remains open, with holography offering both supportive and countervailing perspectives. The work highlights how conformal invariance, RG irreversibility, and holography collectively illuminate the deep structure of quantum field theories and spacetime geometry, suggesting a profound link between symmetry enhancement and the renormalization group flow.

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 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 dimension enjoy the conformal property relies on the deep structure of the renormalization group. The outstanding question is whether this feature is specific to 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 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 -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.

Paper Structure

This paper contains 76 sections, 317 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Physical applications we have in mind
  3. Organization of the review and conventions
  4. Statement of the problem
  5. Conformal algebra as maximal bosonic space-time symmetry
  6. Local field theory realization
  7. Space-time symmetry and energy-momentum tensor
  8. Unitarity, causality and discreteness of spectrum
  9. Consequence of scale invariance alone
  10. Consequence of conformal invariance
  11. Formulation of the problem
  12. Energy-momentum tensor and renormalization group
  13. Schwinger functional
  14. Promoting coupling constants to background fields
  15. Curved background
  16. ...and 61 more sections

Figures (4)

  • Figure 1: We see a graphical distinction between scale invariance and conformal invariance in $d=2$ dimension. Our perception is approximately invariant under scale transformation but not invariant under conformal transformation. Do you think conformal transformation keeps the "same shape"?
  • Figure 2: We show artificially generated examples of (possible?) renormalization group flow. The left hand side contains UV fixed point as well as IR fixed point. The right hand side shows a cyclic behavior with UV fixed point.
  • Figure 3: The $s$ channel scattering amplitude shows positivity of $a_{\mathrm{UV}}-a_{\mathrm{IR}}$.
  • Figure 4: (A) The author's family name is written alphabet. (B) Same but in Japanese (or Chinese) characters. (C) This is the traditional form which Sun Yat-sen must have encountered in Tokyo.