Revisiting the algebraic structure of the generalized uncertainty principle

We compare different formulations of the generalized uncertainty principle that have an underlying algebraic structure. We show that the formulation by Kempf, Mangano, and Mann [Phys. Rev. D 52 (1995)], quite popular for phenomenological studies, satisfies the Jacobi identities only for spin zero particles. In contrast, the formulation proposed earlier by one of us (Maggiore) [Phys. Lett. B 319 (1993)] has an underlying algebraic structure valid for particles of all spins and in this sense seems more fundamental. The latter is also much more constrained, resulting in only two possible solutions, one expressing the existence of a minimum length and the other expressing a form of quantum-to-classical transition. We also discuss how this more stringent algebraic formulation has an intriguing physical interpretation in terms of a discretized time at the Planck scale.

Figure 1

Bound on $\mathrm{\Delta }x$ as a function of $\mathrm{\Delta }p$ for different uncertainty principles. Blue: the Heisenberg uncertainty relation allows for arbitrarily small $\mathrm{\Delta }x$ as $\mathrm{\Delta }p$ increases. Orange: the MM GUP Eq. (25) obtained from Eq. (20) predicts a minimum $\mathrm{\Delta }x$ for large $\mathrm{\Delta }p$. Green: the KMM GUP Eq. (31) obtained from Eq. (30) predicts a minimum $\mathrm{\Delta }x$ for a finite $\mathrm{\Delta }p$, after which the position uncertainty increases again. Red: the other MM GUP obtained from Eq. (37), which can be associated to a quantum-to-classical transition where $[x_i,p_j]=0$ when $\mathrm{\Delta }p$ is larger than a critical value. Shaded regions are regions excluded by the corresponding uncertainty relation.