N-body gravity and the Schrödinger equation

Abstract

We consider the problem of the motion of N bodies in a self-gravitating system in two spacetime dimensions. We point out that this system can be mapped onto the quantum-mechanical problem of an N-body generalization of the problem of the H + 2 molecular ion in one dimension. The canonical gravitational N-body formalism can be extended to include electromagnetic charges. We derive a general algorithm for solving this problem, and show how it reduces to known results for the 2-body and 3-body systems.

Canonical reduction for dilatonic gravity in3+1dimensions

“…Regarding the first issue, in lower dimensions, namely 1 + 1, a normalizable quantum theory combining gravity, quantum mechanics and even an electromagnetic interaction was found through the addition of a dilaton [4]. It also reduces to the Newtonian N -body gravitational action in the nonrelativistic limit [5][6][7].…”

“…It also reduces to the Newtonian N -body gravitational action in the nonrelativistic limit [5][6][7]. The action for the gravitational scalar-tensor formulation in 1 + 1 dimensions [1,3,4,8] coupled to N particles is, in the presence of a cosmological constant Λ …”

“…Although GRT yields trivial field equations in 1 + 1 dimensions, incorporating a dilaton in the manner shown in (1) ensures a nontrivial set of field equations with the correct Newtonian limit [1]. It was later found that in 1 + 1 dimensions the above N -body problem could be mapped onto the quantummechanical problem of an N -body generalization of the problem of the H + 2 molecular ion in one dimension, combining into a normalizable theory represented by the Schrödinger equation [4]. The formalism could also be extended to include electromagnetic charges.…”

Asymptotic series of generalized Lambert W function

“…The Lambert W function appears in a myriad number of applications. In particular, it appears in the "lineal" gravity two-body problem [1,2] as a solution to the Einstein Field equations in (1 + 1) dimensions. The Lambert W function appears as a solution for the case when the two-bodies have exactly the same mass.…”

“…This generalization originally appeared from the (quantum-mechanical) Double Well Dirac Delta Potential model [3], a one-dimensional version of a special case of the quantum-mechanical three-body system known as the Hydrogen Molecular Ion (and also appears in quantum gravity [2]). For this problem, specifically r 1 = 1, r 2 = λ, c = 2 R where R is the internuclear distance.…”