Correlators of Worldline Proper Length

A classical observer can measure elapsed proper time along their worldline. When observers are coupled to a system with internal correlations, measurements of elapsed time may inherit these correlations. We show that derivatives of the on-shell action with respect to worldline mass compute correlation functions of worldline proper length at tree level. We study worldlines coupled to a scalar field. We calculate the length-length two-point function and find it arises from correlated path fluctuations. As an application, we propose that the logarithm of local correlators serves as a generating function of length correlators, which generalizes the on-shell action prescription. Using this proposal, we extract AdS worldline observables from local CFT correlators as computed by Witten diagrams. We briefly discuss extensions to gravity, interferometers, and the holographic encoding of observer time.