Solutions of the Klein–Fock–Gordon Equation and Coherent States on the Horosphere of the Lobachevsky Momentum Space

Solutions that satisfy the Klein–Fock–Gordon equation in quasi-Cartesian coordinates of the three-dimensional relativistic Lobachevsky momentum space associated with the horospheres of this space are found. The Euclidean plane geometry is realized on these horospheres. This representation of solutions to the Klein–Fock–Gordon equation is closely related to invariant geometric images (horosphere, parallel bundle, and parallel bundle axis) of the three-dimensional Lobachevsky space, in which a part naturally arises associated with the selected direction: the parallel bundle axis orthogonal to horospheres. Such a representation is appropriate for the kinematics of an incident particle in a laboratory system in the processes of particle collisions. The connection of the solutions with the usual coherent states on the horosphere of the Lobachevsky momentum space is established.