Memory Correlators and Ward Identities in the 'in-in' Formalism

TL;DR

This work extends soft-gravity Ward identities to the in-in formalism, deriving an in-in supertranslation Ward identity and showing that soft charges map to ANEC insertions. Consequently, the connected memory correlators in in-in observables are nonzero and computable from ANEC correlators using celestial CFT data, via the celestial diamond framework. The approach combines the KMOC in-in formalism with celestial energy detectors and provides a concrete route to express memory and related correlators in terms of ANEC data, including clear relations for two-point functions. The results bridge celestial holography and detector-based collider observables, suggesting measurable infrared structures and guiding future work on higher-point correlators and subleading soft theorems.

Abstract

The symmetries of asymptotically flat spacetimes impose constraints on observables at infinity. The consequences of this have been extensively explored for S-matrix elements, where soft theorems are known to be equivalent to Ward identities for asymptotic symmetries. However, recently there has been interest in broader classes of asymptotic observables. Here, we consider soft graviton insertions in the 'in-in' formalism. We derive a Ward identity for supertranslations and compute two point functions for the soft charges for 'in-in' correlators. We find that the connected memory correlators are non-trivial in this set up and can be straightforwardly inferred from the average null energy (ANEC) correlators using observations from celestial Conformal Field Theory (cCFT).

Figures (4)

• Figure 1: In contrast to $\mathcal{S}$-matrix elements (left), 'in-in' correlators (right) involve inserting operators in the out state and summing over a complete set of intermediate states ($|\rm{out}\rangle\langle \rm{out}|$) on the positive-energy cut (dotted line).
• Figure 2: Diagrams in the 'in-in' formalism that we use to compute the one-point and two-point functions. The red circles denote detectors that are at the cut and are represented by vertex functions $V(q,q')$ in the Feynman diagrams.
• Figure 3: Relationship between crossing and type-I and II contributions to the soft factors. The red contributions to the correlators are due to the soft insertion. In diagram C, the state $|\rm{out}'\rangle$ contains an implicit sum over a soft particle.
• Figure 4: Memory celestial diamond Pasterski:2021dqe. The descendant of the radiative memory modes $\mathcal{N}_{zz}$ and $\mathcal{N}_{\bar{z}\bar{z}}$ is the soft charge since it is given by two derivatives, notated by the two arrows, acting on the memory modes.