# GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry
## 6 Summary and Conclusions  
In this paper, we have extended previous work on 1D GUP models [27; 36; 37] to 3D. Our aim in the work on the 1D models was to construct a GUP model with a minimal length, which, while modifying both position and momentum operators, left the commutator the same as in standard quantum mechanics. Further in [27], we wanted to give a GUP model with a minimal length which did not spoil the energy-momentum dispersion relationship, \(E^{2}-p^{2}=m^{2}\), thus avoiding the non-observation of GUP effects like photon dispersion [17; 18; 19]. Section 2, summarized the 1D GUP models. Section 3 extended these 1D models to 3D while retaining rotational invariance. This was accomplished by requiring the modifications to be functions of the total momentum \(|\vec{p}|\). Consequently, we found that the 3D commutator had to take a form different from the standard quantum mechanical commutator in at least one direction. This is seen explicitly in Equations (19) and (20), where the second-order corrections to the position-momentum commutators were of the form \(\propto\delta_{ij}-\frac{p_{i}p_{j}}{|\vec{p}|^{2}}\). In 1D GUP models, the term \(\delta_{ij}-\frac{p_{i}p_{j}}{|\vec{p}|^{2}}\) is zero. We investigated the uncertainty relationships arising from these modified 3D operators and modified commutators. We found that the minimum uncertainty in position was approximately \(\Delta X_{i}^{min}\sim\frac{\hbar}{p_{M}}\)[26]--the same result as in 1D. Next, we examined the minimum distance in two cases. First, we used spherically symmetric wave functions to analyze how small a volume the wave function could be compressed. We found it to be the minimal distance cubed up to a factor of order unity. Second, we examined a system with a large, nonzero momentum in one direction. We found that the minimum distance in the direction of the average momentum was different from the minimum distance in the two orthogonal directions. For the tanh and arctan GUPs, the positional uncertainty in the orthogonal directions was greater than in the direction of nonzero momentum. A similar result occurred in the GUP model in reference [10], i.e., \(\Delta X_{2,3}^{min}>\Delta X_{1}^{min}\). The difference between the minimum position uncertainties in the second case above points to the possibility of a violation of [PERSON] symmetry. The existence of minimal distances already naively implies a violation of [PERSON] symmetry. However, in [27], a 1D GUP model was formulated, which gave a minimal length while preserving the special relativistic energy-momentum relationship, \(E^{2}-|\vec{p}|^{2}=m^{2}\), thus preserving [PERSON] symmetry [26]. A related issue is the connection of the present work to breaking of the isotropy of space, as observed for other GUP models [38]. We plan to study the violation of [PERSON] symmetry of these modified position and momentum operators in future work. The generators of the canonical Lorentz algebra are angular momentum, (i.e., \(\hat{l}_{ij}=\hat{x}_{i}\hat{p}_{j}-\hat{x}_{j}\hat{p}_{i}\)) and boost operators, (i.e., \(\hat{l}_{qi}=-\hat{x}_{0}\hat{p}_{i}+\hat{x}_{i}\hat{p}_{0}\)). The central question is then \"How do the modified position and momentum operators affect the observables associated with modified angular momentum and boost operators?\". Modified boost operators are constructed by defining modified time and energy operators based on the tanh or arctan operators of Equations (14) and (15). Another fascinating consequence of the modifications is the new commutation relations of position operators among themselves. The standard result is \(\{\hat{x}_{i},\hat{x}_{j}\}=0\); the modified position operators from (14) and (15) satisfy \([\hat{X}_{i},\hat{X}_{j}]\propto e_{ijk}\hat{l}_{k}\) so that the commutators are proportional to the angular momentum operators. The non-commutativity of the modified position operators provides a link between GUP models and non-commutative geometry models [31; 32].