Abstract
The quantum field-theoretic approach to classical observables due to Kosower, Maybee and O’Connell provides a rigorous pathway from on-shell scattering amplitudes to classical perturbation theory. In this paper, we promote this formalism to describe general classical spinning objects by using coherent spin states. Our approach is fully covariant with respect to the massive little group SU(2) and is therefore completely synergistic with the massive spinor-helicity formalism. We apply this approach to classical two-body scattering due gravitational interaction. Starting from the coherent-spin elastic-scattering amplitude, we derive the classical impulse and spin kick observables to first post-Minkowskian order but to all orders in the angular momenta of the massive spinning objects. From the same amplitude, we also extract an effective two-body Hamiltonian, which can be used beyond the scattering setting. As a cross-check, we rederive the classical observables in the center-of-mass frame by integrating the Hamiltonian equations of motion to the leading order in Newton’s constant.
Highlights
This classical problem in general relativity has recently been offered by methods rooted in quantum field theory (QFT), which profit from a wide variety of on-shell methods developed for the study of quantum scattering amplitudes
Many state-of-the-art post-Minkowskian (PM) computations of the two-body dynamics of Schwarzschild and Kerr black holes have been performed using the philosophy of effective field theory (EFT), either in the genuinely quantum-theoretic sense [3,4,5,6,7,8] or in the form of classical worldline effective theory [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], see refs. [25,26,27,28,29,30,31,32] for concurrent developments using eikonal methods
The coherentstate formalism provides a perfect SU(2) spinor to saturate the little-group indices that were left uncontracted in earlier approaches to the classical limit of quantum scattering with spin [37, 39] — where a heuristic notion of “generalized expectation value” was introduced instead
Summary
We review the KMOC formalism for classical observables [33,34,35,36] focusing on the aspects due to the spin degrees of freedom, which are implemented using coherent states. The starting point of the formalism is to consider the change in the expectation value of a certain quantum operator O due to scattering:. ∆2O where we have used the scattering matrix S = 1 + iT As indicated, this object naturally splits into two parts, linear and quadratic in the scattering transition operator T. This object naturally splits into two parts, linear and quadratic in the scattering transition operator T For this observable to have a well-defined classical interpretation, we need the |in states to behave in a predictable manner in the classical limit. The normalization involves the modified Bessel function of the second kind These wavefunctions produce well-behaved one-particle expectation values pμ ξ = muμ + O(ξ), p2 ξ = m2, p2 ξ uμ=(1,0). In the presence of additional degrees of freedom, one needs to find a way to model them with quantum states in a similar manner [34, 35]
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