[Paper Review] Scale Invariance plus Unitarity Implies Conformal Invariance in Four Dimensions

Kara Farnsworth, Markus A. Luty+1 more|arXiv (Cornell University)|Sep 16, 2013

Black Holes and Theoretical Physics13 references14 citations

TL;DR

This paper proves non-perturbatively that any four-dimensional unitary and Lorentz-invariant quantum field theory with a conserved scale current is conformally invariant. By analyzing the scale anomaly and ruling out spin-2, dimension-2 operators in unitary theories, it establishes that scale invariance implies conformal invariance in 4D under these conditions.

ABSTRACT

We give a non-perturbative proof that any 4D unitary and Lorentz-invariant quantum field theory with a conserved scale current is in fact conformally invariant. We show that any scale invariant theory (unitary or not) must have either a vanishing anomaly for global scale transformations or an operator of spin 2 and dimension 2. Neither of these possibilities is allowed for unitary theories, proving the result. This is also a strong constraint on non-unitary Euclidean theories with scale but not conformal invariance, suggesting the conjecture that all such theories are free field theories.

Motivation & Objective

• To establish that scale invariance implies conformal invariance in four-dimensional unitary quantum field theories.
• To analyze the role of the scale anomaly and its implications for the existence of operators with spin 2 and dimension 2.
• To constrain non-unitary Euclidean theories with scale but not conformal invariance, suggesting they may be free field theories.
• To provide a non-perturbative argument based on unitarity and Lorentz invariance, avoiding assumptions from perturbation theory.

Proposed method

• Analyzes the structure of the scale current and its conservation in 4D quantum field theories.
• Examines the scale anomaly in the context of global scale symmetry and its implications for operator content.
• Applies unitarity and Lorentz invariance to rule out the existence of spin-2, dimension-2 operators in unitary theories.
• Uses the absence of such operators to conclude that the scale anomaly must vanish in unitary theories.
• Applies the standard argument that a vanishing scale anomaly implies conformal invariance.
• Extends the analysis to non-unitary Euclidean theories, suggesting they must be free if scale-invariant but not conformally invariant.

Experimental results

Research questions

• RQ1 Does scale invariance imply conformal invariance in four-dimensional unitary quantum field theories?
• RQ2 What constraints does unitarity impose on the existence of operators with spin 2 and dimension 2?
• RQ3 Can a non-vanishing scale anomaly coexist with unitarity in 4D Lorentz-invariant quantum field theories?
• RQ4 Are there non-unitary Euclidean theories with scale invariance but not conformal invariance, and what are their properties?
• RQ5 What is the role of the scale current's conservation in enforcing conformal invariance under unitarity?

Key findings

• Any 4D unitary and Lorentz-invariant quantum field theory with a conserved scale current is conformally invariant.
• In unitary theories, the scale anomaly must vanish, which is a necessary condition for conformal invariance.
• The existence of a spin-2, dimension-2 operator is ruled out in unitary theories, eliminating a potential obstruction to conformal invariance.
• The absence of such operators implies that scale invariance leads to conformal invariance in the unitary case.
• Non-unitary Euclidean theories with scale but not conformal invariance are strongly constrained and likely correspond to free field theories.
• The result holds non-perturbatively, relying only on unitarity, Lorentz invariance, and the conservation of the scale current.