Proof that Half-Harmonic Oscillators become Full-Harmonic Oscillators after the "Wall Slides Away"

Abstract

Normally, the half-harmonic oscillator is active when x > 0 and absent when x < 0. From a canonical quantization perspective, this leads to odd eigenfunctions being present while even eigenfunctions are absent. In that case, only the usual odd eigenfunctions will appear if the wall slides to negative infinity. However, if an affine quantization is used, sliding the wall away shows that all the odd and even eigenfunctions are encountered, exactly like any full-harmonic oscillator..

A Simple Factor in Canonical Quantization yields Affine Quantization Even for Quantum Gravity

“…The classical Hamiltonian, according to ADM [12], uses the metric g ab (x) = g ba (x) and its determinant g(x) ≡ det[g ab (x)], 4 the momentum π cd (x) = π dc (x), or more useful, the momentric 5 π c d (x) ≡ π ce (x) g de (x), all of which leads to…”

“…For example, a traditional harmonic oscillator, with −∞ < p, q < ∞, employs CQ, while a half-harmonic oscillator, with −∞ < p < ∞, but 0 < q < ∞, employs AQ [3]. Partial-harmonic oscillators, with −∞ < p < ∞, while −b < q < ∞ and 0 < b < ∞, also employ AQ [4].…”