$T{\overline T}$ deformations and the width of fundamental particles

TL;DR

This paper offers a geometric interpretation of $T{ar T}$ deformations as a universal finite width of particles, connecting classical and quantum systems, and relating to state-dependent metric changes and generalized hydrodynamics.

Contribution

It introduces a geometric perspective on $T{ar T}$ deformations, linking them to particle widths and state-dependent metrics in both relativistic and non-relativistic systems.

Findings

$T{ar T}$ deformations induce a finite width in particles.

The width corresponds to a state-dependent change in the metric.

The approach applies to both integrable and non-integrable systems.

Abstract

We provide a simple geometric meaning for deformations of so-called $T{\overline T}$ type in relativistic and non-relativistic systems. Deformations by the cross products of energy and momentum currents in integrable quantum field theories are known to modify the thermodynamic Bethe ansatz equations by a "CDD factor". In turn, CDD factors may be interpreted as additional, fixed shifts incurred in scattering processes: a finite width added to the fundamental particles (or, if negative, to the free space between them). We suggest that this physical effect is a universal way of understanding $T{\overline T}$ deformations, both in classical and quantum systems. We first show this in non-relativistic systems, with particle conservation and translation invariance, using the deformation formed out of the densities and currents of particles and momentum. This holds at the level of the equations of motion, and for any interaction potential, integrable or not. We then argue, and show by similar techniques in free relativistic particle systems, that $T\overline T$ deformations of relativistic systems produce the equivalent phenomenon, accounting for length contractions. We also show that, in both the relativistic and non-relativistic cases, the width of particles is equivalent to a state-dependent change of metric, where the distance function discounts the particles' widths, or counts the additional free space. This generalises and explains the known field-dependent coordinate change describing $T\overline T$ deformations. The results connect such deformations with generalised hydrodynamics, where the relations between scattering shifts, widths of particles and state-dependent changes of metric have been established.