Quantum mechanics

Pure mathematics

Mathematical physics

Equivalence (formal languages)

Materials science

Scattering amplitude

Particle physics

Scattering

Acoustics

S-matrix

Lattice (music)

Matrix (chemical analysis)

Physics

Mathematics

Geometry

Composite material

Rotation formalisms in three dimensions

Quantum chromodynamics

Unitarity

Physical review. D/Physical review. D.

In recent years, different on-shell $\mathbf{3}\to\mathbf{3}$ scattering formalisms have been proposed to be applied to both lattice QCD and infinite volume scattering processes. We prove that the formulation in the infinite volume presented by Hansen and Sharpe in Phys.~Rev.~D92, 114509 (2015) and subsequently Brice\~no, Hansen, and Sharpe in Phys.~Rev.~D95, 074510 (2017) can be recovered from the $B$-matrix representation, derived on the basis of $S$-matrix unitarity, presented by Mai {\em et al.} in Eur.~Phys.~J.~A53, 177 (2017) and Jackura {\em et al.} in Eur.~Phys.~J.~C79, 56 (2019). Therefore, both formalisms in the infinite volume are equivalent and the physical content is identical. Additionally, the Faddeev equations are recovered in the non-relativistic limit of both representations.

Equivalence of three-particle scattering formalisms