Post-Minkowskian scattering angle in Einstein gravity

N.E.J Bjerrum-Bohr,Andrea Cristofoli,Poul H Damgaard

doi:10.1007/jhep08(2020)038

N.E.J Bjerrum-Bohr, Andrea Cristofoli + Show 1 more

Open Access

https://doi.org/10.1007/jhep08(2020)038

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Abstract

Using the implicit function theorem we demonstrate that solutions to the classical part of the relativistic Lippmann-Schwinger equation are in one-to-one correspondence with those of the energy equation of a relativistic two-body system. A corollary is that the scattering angle can be computed from the amplitude itself, without having to introduce a potential. All results are universal and provide for the case of general relativity a very simple formula for the scattering angle in terms of the classical part of the amplitude, to any order in the post-Minkowskian expansion.

Highlights

Using the implicit function theorem we demonstrate that solutions to the classical part of the relativistic Lippmann-Schwinger equation are in one-to-one correspondence with those of the energy equation of a relativistic two-body system
We have unravelled an unexpected equivalence between classical solutions to LippmannSchwinger equations and solutions to the relativistic energy relation of two-body dynamics
We have found that the implicit function theorem applied to the relativistic energy relation is in one-to-one correspondence with the classical part of the solutions to the Lippmann-Schwinger equation of the quantum mechanical scattering problem

Summary

The Lippmann-Schwinger equation in position space

The Lippmann-Schwinger equation is usually expressed as an integral equation involving amplitudes and potentials in momentum space. For the case of non-relativistic systems, its space representation states that the Fourier transform of the classical part of the amplitude is proportional to the potential. We shall here extend this observation by demonstrating that the positionspace representation of the Lippmann-Schwinger equation for fully relativistic systems can be expressed as a differential equation for the potential and the classical part of the amplitude. At linear order in GN the relation is trivial and states that the Fourier transform of the amplitude at tree level is the potential, a textbook observation. At quadratic order things become more interesting and one has This relation reproduces exactly the classical part of the 2PM amplitude in position space. What is far more interesting is the fact that precisely the same series can be understood from an alternative point of view by applying the implicit function theorem to the relativistic energy equation

Dini’s theorem and the Lippmann-Schwinger equation

The scattering angle to all orders

Conclusion

A Classical contributions from the Lippman-Schwinger equation