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

Abstract

We provide a simple geometric meaning for deformations of so-called $$ T\overline{T} $$ T 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 $$ T\overline{T} $$ T 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 $$ T\overline{T} $$ T 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 $$ T\overline{T} $$ T 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.

Measurement catastrophe and ballistic spread of charge density with vanishing current

“…This bears a remarkable resemblance to the scattering phase modification factor in hard-rod deformed models (see, e.g. [29,30]). Such a structure enables exact analytical and numerical analysis of various phenomena that have recently been in the spotlight: Hilbert-space fragmentation [17], pre-relaxation [31], generalised hydrodynamics [9], spin transport phenomena [32], the effect of non-abelian symmetries, time-translation symmetry breaking [10], and the macroscopic effects of local measurements [22].…”

“…This is why a nonzero sum j b j (1 − b j+1 ) is associated with interaction [8]. Since, however, the sum is finite in the state (30) and we are in the thermodynamic limit, the state can be considered weakly interacting. By flipping the spin in position 0 according to the protocol described in section 2, we end up with the state…”

Hard rod hydrodynamics and the Lévy Chentsov field

“…The variable model with a finite number of positive rod sizes was first introduced by Aizenman, Lebowitz, and Marro [2], thereby extending the results for equilibrium spacetime correlations obtained in [17]. Very unexpectedly, hard rods with variable rod length appears as a phenomenological model in an article by Cardy and Doyon [6] in their study of T T deformations in relativistic and nonrelativistic integrable field theories.…”

Completing the bootstrap program for T T ―

“…For short distances we can interpret this behaviour as indicative of an unusual UV theory. Indeed, it is known [52] that for α > 0 particles acquire a typical (positive) length which makes the UV theory ill-defined. For long distances however, there is an interplay between the length scale, the momentum and the parameter α.…”