Scale and conformal invariance in field theory: a physical counterexample

TL;DR

Addressing whether scale invariance implies conformal invariance in two dimensions, the paper presents a physical counterexample: a two-dimensional elasticity model with vector displacement fields that is scale-invariant but not conformally invariant. The argument centers on the trace Tμ^μ = -∂_μ K^μ with K_μ not reducible to a gradient, and explicit expressions show that conformal invariance fails except in the unphysical limit k+g=0 (where c=2). Although an improved tensor can make the trace two-point function vanish, the theory is not reflection-positive and is non-unitary, consistent with a non-positive-definite Hamiltonian. The result sharpens the understanding of when scale invariance can fail to imply conformal invariance in low dimensions and underscores the role of positivity and field representations.

Abstract

In this note, we illustrate how the two-dimensional theory of elasticity provides a physical example of field theory displaying scale but not conformal invariance.