Minimal length: a source of non-Hermiticity and non-locality in quantum mechanics

TL;DR

The paper investigates how a fundamental minimal length, encoded via the Generalized Uncertainty Principle, alters canonical quantum mechanics by replacing $\hat{p}$ with a generalized momentum $\hat{P}=\hat{p}(1+\beta\hat{p}^2)$, yielding $[\hat{x},\hat{P}]=i\hbar(1+3\beta\hat{P}^2)$ and a minimal position uncertainty $\Delta x_{\min}=\sqrt{3\beta}\,\hbar$. It shows that the generalized momentum basis $|P\rangle$ can be expanded in the canonical-p momentum basis as $|P\rangle=\sum_{k=1}^3\alpha_k|p=\mathcal{P}_k\rangle$, with three roots $\mathcal{P}_k$ leading to a threefold degeneracy and, in some cases, complex roots $\mathcal{P}_2,\mathcal{P}_3$, suggesting complex numbers may be intrinsic under minimal length. The work then demonstrates that a two-particle state with zero total momentum, when analyzed in the $|p\rangle$ basis, yields entangled momentum-space states and explicit constructions like $|-P\rangle=\frac{1}{\sqrt{3}}\sum_k| -\mathcal{P}_k\rangle$, enabling momentum-based non-local correlations. Overall, the findings indicate that minimal length scenarios can generate entanglement and non-locality while potentially underpinning the presence of complex numbers in quantum mechanics, with implications for quantum gravity phenomenology.

Abstract

First, the study tries to shed light on the relationship between purely quantum mechanical momentum measurements (canonical momentum space) and measurements of the generalized momentum operator, including minimal length effects. Additionally, the existence of complex numbers in quantum mechanics seems justifiable as a consequence of minimal length. Finally, a novel method for generating quantum entangled states with complex quantum numbers inspired by the minimal length is also reported. Therefore, theories including a minimal length, like some quantum scenarios of gravity, seem to be able to enrich the current understanding of non-locality.