Chemical algebra. VI:G-weighted metrics of non-compact groups: Group of translations in the Euclidean space

Abstract

The generalized definition equation of a G-weighted metricds ² from the datum of any groupG acting onto a vector space mapped by a continuous numerical functionμ is applied whenE =R ⁿ andG = the group of translations inR ⁿ. Here,G does not act linearly inR ⁿ andR ⁿ is considered as an affine space. The solution readsds ² = −d ²(InI)/(Bp),I = (4iπx ⁰)^n/2 ·ψ, wherex ⁰ = −i/(2p), ψ is a solution of the Schrödinger-type equation Δψ +iϖψ/ϖx ⁰ = 0, andB is a uniform term depending onx ⁰. Whenn = 3,p is interpreted as the reciprocal of a time variable. Attempts to identifyds ² with the spatial part of a space-time metric of general relativity failed except for the flat Robertson and Walker spaces. In the simplest case,B = 1/R ²(t) and$\Psi (p,r) = e^{ - pr^2 /2}$. A uniform but non-constant “imaginary potential energy” of the space can be formally derived: V(x ⁰) = 3i/(2x ⁰). Despite a striking formal link with tools of physical mathematics, no physical validation of the propositions of chemical algebra is claimed.