Affine Quantization on the Half Line

TL;DR

This paper explores affine quantization on the half line, demonstrating its effectiveness through models like the free particle and harmonic oscillator, especially where canonical quantization faces challenges.

Contribution

It revisits and formulates affine quantization on the half line, providing solutions to simple models and highlighting its advantages over canonical quantization in certain phase spaces.

Findings

Affine quantization successfully applied to the half line.

Solutions obtained for free particle and harmonic oscillator models.

Potential advantages over canonical quantization in non-trivial phase spaces.

Abstract

The similarity between classical and quantum physics is large enough to make an investigation of quantization methods a worthwhile endeavour. As history has shown, Dirac's canonical quantization method works reasonably well in the case of conventional quantum mechanics over $\mathbb{R}^n$ but it may fail in non-trivial phase spaces and also suffer from ordering problems. Affine quantization is an alternative method, similar to the canonical quantization, that may offer a positive result in situations for which canonical quantization fails. In this paper we revisit the affine quantization method on the half line. We formulate and solve some simple models, the free particle and the harmonic oscillator.