The Uncertainty Principle and Classical Amplitudes

Abstract

We study the variance in the measurement of observables during scattering events, as computed using amplitudes. The classical regime, characterised by negligible uncertainty, emerges as a consequence of an infinite set of relationships among multileg, multiloop amplitudes in a momentum-transfer expansion. We discuss two non-trivial examples in detail: the six-point tree and the five-point one-loop amplitudes in scalar QED. We interpret these relationships in terms or a coherent exponentiation of radiative effects in the classical limit which generalises the eikonal formula, and show how to recover the impulse, including radiation reaction, from this generalised eikonal. Finally, we incorporate the physics of spin into our framework.

Angular momentum of zero-frequency gravitons

“…Let us emphasize again that, although we focus here on the soft limit, Eq. (3.27) only relies on the transversality of F µν , and thus easily extends to the case where F µν is replaced by the more general b-space version of the 2 → 3 amplitude [11,61,20,12,48], a.k.a. the stress-energy pseudotensor T µν like in [26].…”

“…A prominent tool that has been helpful to extract classical information from the elastic 2 → 2 amplitude is the eikonal exponentiation [30,31,32,33,34,35,36,37,19,15,20,38], whose inception actually dates back to the late eighties [39,40,41,42,43,44]. Important endeavours have been also devoted to generalizing the eikonal framework to allow for the presence of additional outgoing graviton states [45,46,47,48], which are actually unavoidable in any physical process.…”

The SAGEX Review on Scattering Amplitudes, Chapter 14: Classical Gravity from Scattering Amplitudes

“…This amplitude describes the emission of one photon, while clearly the classical field must contain many photons. The resolution to this puzzle [23] is that the classical radiation is a coherent state, parameterised by the five-point amplitude, which indeed has a very large occupation number. The expectation value is sensitive to the parameter of the state, and is given by equation (51).…”

“…The expression for the waveform in terms of amplitudes [12] establishes the double copy's utility for computing the waveform to all orders in perturbation theory. Other researchers have applied and extended these ideas in a number of important directions [13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Radiation reaction for spinning black-hole scattering

“…The increasing sensitivity of the detectors demands for an effort to develop new theoretical tools capable of high-precision predictions of GW waveform templates. For these reasons, recently there has been considerable interest in the use of amplitude-based techniques for studying the inspiral phase of binary non-spinning massive objects coalescence, extracting the conservative part of the Effective-One-Body (EOB) potential [8][9][10][11][12][13][14][15][16][17][18][19][20][21], classical observables [13,[22][23][24][25], waveforms and radiation [20,21,[26][27][28][29][30]. Astrophysical objects are also characterized by their spin.…”