[Paper Review] General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function

T Scott, Robert B. Mann|ArXiv.org|Jul 9, 2006

Sports Dynamics and Biomechanics14 references28 citations

TL;DR

This paper introduces a canonical generalization of the Lambert W function, denoted $\Omega_n$, using tetration-based nesting to solve transcendental equations arising in fundamental physics. It provides exact analytical solutions for unequal-mass 2-body and 3-body systems in (1+1)D general relativity and the quantum mechanical hydrogen molecular ion with clamped nuclei, unifying solutions across gravity, quantum mechanics, and delayed differential equations.

ABSTRACT

Herein, we present a canonical form for a natural and necessary generalization of the Lambert W function, natural in that it requires minimal mathematical definitions for this generalization, and necessary in that it provides a means of expressing solutions to a number of physical problems of fundamental nature. In particular, this generalization expresses the exact solutions for general-relativistic self-gravitating 2-body and 3-body systems in one spatial and one time dimension. It also expresses the solution to a previously unknown mathematical link between the lineal gravity problem and the Schroedinger equation.

Motivation & Objective

• To develop a natural and necessary generalization of the Lambert W function that applies universally to fundamental physical problems.
• To resolve the lack of analytical solutions for unequal-mass 2-body and 3-body systems in (1+1)D general relativity.
• To unify solutions across linear gravity and quantum mechanics by identifying a shared mathematical structure.
• To extend the applicability of the Lambert W function to a broader class of transcendental equations, including delayed differential equations.
• To establish a canonical form for the generalized function $\Omega_n$ with transparent correspondence to the standard $W$ function and potential for further generalization.

Proposed method

• Formulate a generalized Lambert W function $\Omega_n$ through iterative nesting of the standard $W$ function, rooted in tetration (iterated exponentiation).
• Re-derive the quantum mechanical solution for the H₂⁺ ion with unequal charges using the generalized $\Omega_n$ function.
• Apply the same framework to solve the (1+1)D Einstein field equations for unequal-mass 2-body and 3-body gravitational systems.
• Demonstrate that the generalized function $\Omega_n$ solves a broad class of transcendental equations, including those arising in delayed differential equations.
• Establish correspondence between the generalized function and known physical models, such as the Schrödinger equation for the hydrogen molecular ion and linear gravity via dilaton theory.
• Use the parameter $\epsilon$ in equation (15) to infer the general form from the standard $W$ function, ensuring consistency and minimal new definitions.

Experimental results

Research questions

• RQ1 Can a natural generalization of the Lambert W function be constructed that unifies solutions in general relativity and quantum mechanics?
• RQ2 How can the generalized function $\Omega_n$ be defined using minimal new mathematical structures while preserving correspondence to the standard $W$ function?
• RQ3 What is the role of tetration in constructing a canonical form for the generalized Lambert W function?
• RQ4 Can the generalized function $\Omega_n$ solve the (1+1)D three-body gravitational problem and the quantum mechanical three-body problem with clamped nuclei?
• RQ5 To what extent does the generalized function $\Omega_n$ unify solutions across delayed differential equations, quantum mechanics, and linear gravity?

Key findings

• The generalized Lambert W function $\Omega_n$ is constructed via tetration-based nesting, satisfying the criteria of minimality, physical applicability, and transparency to the standard $W$ function.
• The case $N=2, M=0$ corresponds to the solution of the unequal-mass 2-body problem in (1+1)D general relativity and the quantum mechanical H₂⁺ ion with unequal charges.
• The generalized function $\Omega_n$ provides exact analytical solutions for the three-body problem in (1+1)D linear gravity and the three-body quantum mechanical system with clamped nuclei.
• The function $\Omega_n$ solves a significant class of delayed differential equations, as shown by its correspondence to equations in neuro-mechanical and physiological models.
• The solution to the Schrödinger equation for a one-dimensional Bose-Fermi mixture in the limit of strong repulsion is a special case of the generalized $\Omega_n$ function.
• The canonical form of $\Omega_n$ is expressed in equation (41), which unifies solutions across gravity, quantum mechanics, and delay differential equations, demonstrating its fundamental physical and mathematical role.