Quantum Gravitational Corrections to the Real Klein-Gordon Field in the Presence of a Minimal Length

TL;DR

This paper investigates quantum gravitational effects on the Klein-Gordon field within a Lorentz-covariant deformed algebra framework, revealing modified equations that imply a minimal length scale consistent with previous research.

Contribution

It introduces a Lorentz-covariant deformed algebra approach to derive quantum gravitational corrections to the Klein-Gordon equation, identifying conditions for physically acceptable mass states and estimating the minimal length scale.

Findings

Modified Klein-Gordon equation with fourth-order derivatives.

Existence of two massive particles with different masses.

Minimal length estimated between 10^{-17}m and 10^{-15}m.

Abstract

The (D+1)-dimensional $(\beta,\beta')$-two-parameter Lorentz-covariant deformed algebra introduced by Quesne and Tkachuk [C. Quesne and V. M. Tkachuk, J. Phys. A: Math. Gen. \textbf {39}, 10909 (2006).], leads to a nonzero minimal uncertainty in position (minimal length). The Klein-Gordon equation in a (3+1)-dimensional space-time described by Quesne-Tkachuk Lorentz-covariant deformed algebra is studied in the case where $\beta'=2\beta$ up to first order over deformation parameter $\beta$. It is shown that the modified Klein-Gordon equation which contains fourth-order derivative of the wave function describes two massive particles with different masses. We have shown that physically acceptable mass states can only exist for $\beta<\frac{1}{8m^{2}c^{2}}$ which leads to an isotropic minimal length in the interval $10^{-17}m<(\bigtriangleup X^{i})_{0}<10^{-15}m$. Finally, we have shown that the above estimation of minimal length is in good agreement with the results obtained in previous investigations.