Inequivalence of Coset Constructions for Spacetime Symmetries

TL;DR

The paper investigates whether different coset constructions for spacetime symmetries, exemplified by the Galileon group from the contraction of the conformal coset, are physically equivalent. It analyzes two non-linear realizations, π and q, related by a field–dependent coordinate map and shows that while they share the same S-matrix for asymptotic states, they exhibit different local propagation properties in backgrounds, which is resolved by noting that local couplings map to non-local ones under the transformation. The authors demonstrate, through several backgrounds and models (including DGP and conformal Genesis scenarios), that asymptotic effects cancel only for specific mappings and that non-locality plays a central role in preserving physical causality, thereby signaling inequivalence between representations in local measurements. These results caution against assuming locality and causality transfer unchanged under coset reparameterizations and have implications for AdS/CFT interpretations and the study of spacetime symmetries in EFTs.

Abstract

Non-linear realizations of spacetime symmetries can be obtained by a generalization of the coset construction valid for internal ones. The physical equivalence of different representations for spacetime symmetries is not obvious, since their relation involves not only a redefinition of the fields but also a field-dependent change of coordinates. A simple and relevant spacetime symmetry is obtained by the contraction of the 4D conformal group that leads to the Galileon group. We analyze two non-linear realizations of this group, focusing in particular on the propagation of signals around non-trivial backgrounds. The aperture of the lightcone is in general different in the two representations and in particular a free (luminal) massless scalar is mapped in a Galileon theory which admits superluminal propagation. We show that in this theory, if we consider backgrounds that vanish at infinity, there is no asymptotic effect: the displacement of the trajectory integrates to zero, as can be expected since the S-matrix is trivial. Regarding local measurements, we show that the puzzle is solved taking into account that a local coupling with fixed sources in one theory is mapped into a non-local coupling and we show that this effect compensates the different lightcone. Therefore the two theories have a different notion of locality. The same applies to the different non-linear realizations of the conformal group and we study the particular case of a cosmologically interesting background: the Galilean Genesis scenarios.

Figures (3)

• Figure 1: The deviation from the lightcone averages to zero asymptotically in the theory obtained with a field redefinition of the free theory.
• Figure 2: Superluminality, even at leading order in $\partial^2 \bar{q}/\Lambda^3$, involves the sum of infinitely many diagrams.
• Figure 3: Plot of the speed of propagation as a function of $r$ for $R_\star=0.25 R_V$. The discontinuity in the velocity is produced by the abrupt change in the density field and can be removed by smoothing the density $\rho_\star$.