Meta-conformal invariance and the boundedness of two-point correlation functions

TL;DR

This paper investigates the boundedness of two-point correlation functions under various extended dynamical symmetries, proposing a dual space approach for meta-conformal invariance in 1+1 dimensions to ensure their boundedness.

Contribution

It introduces a dual space extension for meta-conformal invariance in 1+1 dimensions, resolving unboundedness issues of two-point functions and providing a canonical interpretation.

Findings

Meta-conformal two-point functions can be made bounded via dual space extension.

Galilei-conformal correlators derive from meta-conformal invariance through contraction.

Schr"odinger-covariant functions are causal response functions.

Abstract

The covariant two-point functions, derived from Ward identities in direct space, can be affected by consistency problems and can become unbounded for large time- or space-separations. This difficulty arises for several extensions of dynamical scaling, for example Schr\"odinger-invariance, conformal Galilei invariance or meta-conformal invariance, but not for standard ortho-conformal invariance. For meta-conformal invariance in 1+1 dimensions, these difficulties can be cured by going over to a dual space and an extension of these dynamical symmetries through the construction of a new generator in the Cartan sub-algebra. This provides a canonical interpretation of meta-conformally covariant two-point functions as correlators. Galilei-conformal correlators can be obtained from meta-conformal invariance through a simple contraction. In contrast, by an analogus construction, Schr\"odinger-covariant two-point functions are causal response functions. All these two-point functions are bounded at large separations, for sufficiently positive values of the scaling exponents.