Scale invariance vs conformal invariance (1302.0884v4)

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) Poincar\'e 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.

• The paper shows that under specific technical conditions, scale invariance in QFTs can lead to conformal invariance.
• It employs rigorous renormalization group analysis and holographic approaches to substantiate the symmetry enhancement.
• The study highlights constraints like the a-theorem and the c-theorem as key to understanding invariance in various dimensions.

Essay on "Scale Invariance vs Conformal Invariance" by Yu Nakayama

Yu Nakayama's paper "Scale Invariance vs Conformal Invariance" addresses a profound question in the field of quantum field theories (QFTs): under what conditions does scale invariance imply conformal invariance? The motivation for this inquiry stems from the observation that although scale invariance is a weaker condition than conformal invariance, in many known cases, scale-invariance in a QFT is enhanced to conformal invariance.

Overview

Nakayama's paper meticulously reviews the conditions and distinctions between scale and conformal invariance within the framework of relativistic QFTs. He asserts that scale invariant quantum field theories exhibit conformal invariance under certain technical assumptions, particularly when reduction to two dimensions is considered. This is established via rigorous examination of the renormalization group and perturbative arguments, supported by holographic considerations.

Key Concepts and Arguments

The paper clarifies that both scale invariance and conformal invariance can be described via symmetries of the energy-momentum tensor: scale invariance demands the trace of the energy-momentum tensor to be the divergence of a current, while conformal invariance demands it to vanish entirely.

In two dimensions, the celebrated Zamolodchikov’s c-theorem provides a powerful tool to establish that scale invariance does indeed imply conformal invariance. It involves a c-function that is monotonically decreasing along the renormalization group flow, serving as a measure of degrees of freedom. The paper reviews Polchinski’s extension of this theorem, demonstrating how under assumptions like unitarity and a discrete spectrum, scale invariance in two-dimensional QFTs implies conformal invariance.

Higher Dimensions and Perturbative Proof

In four dimensions, the situation gets more intricate. Nakayama reviews the local renormalization group flow approach championed by Jack and Osborn, which uses a technique involving space-time-dependent coupling constants to reveal deep insights into scale vs conformal invariance. The central piece is the so-called "a-theorem" which suggests that the Euler anomaly coefficient (a) monotonically decreases along RG flows and is conjectured to behave like the c-theorem in two dimensions. This perspective is strengthened by Komargodski and Schwimmer’s argument using dilaton scattering which provides non-perturbative evidence for the a-theorem.

Nakayama further explores perturbative proofs of the enhancement of conformal symmetry from scale invariance, by demonstrating the absence of consistent scale-invariant but non-conformal field theories at the perturbative level. The constraints posed by the gradient flow of the a-function are crucial here, particularly highlighting that β functions along certain redundant directions vanish, reinforcing the absence of non-trivial scale without conformal invariant solutions.

Holographic Approach and Implications

The paper also ventures into the domain of holography, particularly AdS/CFT correspondence, as a tool for understanding this scale to conformal transition. Through a holographic RG flow, which involves analyzing the geometric flow of metrics in a higher-dimensional space (AdS), the null energy condition translates into monotonicity conditions akin to the c-theorem. The conditions derived holographically yield insights parallel to those derived from QFT approaches, thus providing strong support for the conjecture in higher dimensions.

Conclusion and Future Directions

Nakayama's analysis posits that while scale invariance frequently implies conformal invariance, the theoretical landscape is far more nuanced, especially in dimensions greater than two. The integration of renormalization group theory, perturbation theory, and holographic principles affords a comprehensive framework to address this. Future research directions may include specific examples in less explored dimensions, or within theories with less typical symmetries, to further test these conclusions.

Nakayama's paper is a critical resource for researchers in theoretical physics, particularly those investigating the profound connections between symmetries in quantum field theories and their implications through the vast lens of holography. It elegantly ties together seemingly disparate concepts under the unifying theme of symmetry enhancement, offering both rigorous mathematical treatment and insightful physical interpretation.