# GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry
## 4 Modified Lorentz Group for Modified Operators
For the arctan modification this results in the time and energy operators  
\[\hat{X}_{0}=\hat{T}=i\hbar\biggl{[}1+\biggl{(}\frac{\pi E}{2E_{M}}\biggr{)}^{ 2}\biggr{]}\partial_{E}\ \ \ \text{and}\ \ \ \hat{P}_{0}=\hat{\mathcal{E}}=\frac{2E_{M}}{\pi}\arctan\biggl{(}\frac{\pi E}{2E _{M}}\biggr{)}. \tag{27}\]  
Now, using the time and energy operators from (26) and (27), we define new boost operators analogously to the new rotation operators of (24) and (25). For the tanh modification, we have  
\[\hat{\mathcal{K}}_{i} \equiv \frac{E}{E_{M}\cosh^{2}(E/E_{M})\tanh(E/E_{M})}\hat{K}_{i} \tag{28}\] \[= \frac{p}{p_{M}\cosh^{2}(p/p_{M})\tanh(p/p_{M})}\hat{K}_{i}=\hat{ k}_{i}\,\]and for the arctan modified operators from (15), one has  
\[\hat{\mathcal{K}}_{i} \equiv \frac{\pi E}{2E_{M}[1+(\pi E/2E_{M})^{2}]\arctan(\pi E/2E_{M})}\hat{ \mathcal{K}}_{i} \tag{29}\] \[[\hat{l}_{i},\hat{l}_{j}]=i\hbar\epsilon_{ijk}\hat{l}_{k}\ \ \ \text{and}\ \ \ [\hat{k}_{i},\hat{k}_{j}]=-i\hbar\epsilon_{ijk}\hat{k}_{k}\ \ \ \text{and}\ \ \ [\hat{l}_{i},\hat{k}_{j}]=i\hbar\epsilon_{ijk}\hat{k}_{k}. \tag{22}\]0  
In the second lines of (28) and (29), we have used the fact that \(E_{M}=p_{M}\) and \(E=|p_{y}|=p\). Looking at the newly defined angular momentum and boost operators in (24), (25) and (28), and (29), we see that they are equivalent to the standard operators. Thus, these new operators will satisfy the same standard Lorentz algebra from (21)  
The modified angular momentum and boost operators defined in (24) and (28) or (25) and (29) trivially satisfy the same Lorentz algebra (22) as the standard operators  
\[[\hat{l}_{i},\hat{l}_{j}]=i\hbar\epsilon_{ijk}\hat{l}_{k}\ \ \ \text{and}\ \ \ [\hat{k}_{i},\hat{k}_{j}]=-i\hbar\epsilon_{ijk}\hat{k}_{k}\ \ \ \text{and}\ \ \ [\hat{l}_{i},\hat{k}_{j}]=i\hbar\epsilon_{ijk}\hat{k}_{k}. \tag{22}\]1  
Thus, although the modifications of the time-space and energy-momentum operators lead to a minimal length--and more specifically a minimum length that is different for directions along the momentum of the system versus orthogonal to the direction of the momentum--they do so without violating the Lorentz algebra, at least in regard to rotations and boosts. This is closely related to the GUP model in [27; 28], which defined modified operators that resulted in a minimal length but still preserved the standard special relativistic relationship between energy and momentum. Retaining the standard special relativistic energy-momentum dispersion relationship for the modified energy and momentum allowed one to have a theory with a minimal length but which did not run afoul of the observational results from short gamma-ray bursts [18; 19], which showed no sign of photon dispersion up to scales shorter than the Planck distance. Generic GUP models usually have photon dispersion, which therefore provides a possible way to test such GUP models. The GUP models in this section do not exhibit photon dispersion and thus are not testable via observations of short gamma-ray bursts. Although the Lorentz algebra is preserved for the modified rotation and boost generators, as shown in (22), one can ask whether there are physical consequences that could distinguish this GUP model with its Lorentz algebra, but modified position and momentum operators from the standard Lorentz algebra with standard position and momentum operators. As mentioned above, the present GUP model does not exhibit photon dispersion, as is the case for the standard Lorentz algebra. However, in references [27; 28] it was suggested that these modified GUPs could offer an explanation for the appearance of ultra-high-energy cosmic rays beyond the GZK [29; 30] cut-off. Additionally, the present GUP models do have some deviation from the usual Lorentz algebra. In the standard Lorentz algebra, position operators commute with other position operators, (i.e., \([\hat{x}_{i},\hat{x}_{j}]=0\)), and momentum operators commute among themselves, (i.e., \([\hat{p}_{i},\hat{p}_{j}]=0\)).