# GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry
## 5 3D GUP and Non-Commutative Geometry  
In moving from one-dimensional to three-dimensional GUP models, one can see there is a congruence to the field of non-commutative geometries, which is a different (but in the end hopefully equivalent) approach to a minimal distance scale in phenomenological quantum gravity. A review of non-commutative geometry can be found in [31]. Non-commutative geometry models are most often used to smooth out the singularities in black holes [32], whereas GUP models tend to focus on the high-energy/high-momentum regime of particles. Non-commutative geometries approach quantum gravity from the gravity side, while the GUP approaches quantum gravity from the quantum field theory side. The connection between the 3D GUP and non-commutative geometry models was pointed out in the early work [10]. However, to the best of our knowledge, no one has made a detailed analysis of the connections between minimal lengths achieved through non-commutative geometry and those achieved through 3D GUP models and how consistent the models are with one another. In this section, we take the first step in this direction. One can see that the three-dimensional modified position operators of Equations (14) and (15) imply a non-commutativity between the coordinates. From (14), one finds  
\[ \begin{array}{rcl}[\hat{X}_{i},\hat{X}_{j}]&=&\frac{2i\hbar\cosh^{3}(|\vec{p }|/p_{M})\sinh(|\vec{p}|/p_{M})}{p_{M}|\vec{p}|}(\hat{p}_{i}\hat{x}_{j}-\hat{p}_ {j}\hat{x}_{i})\\ &=&\frac{2i\hbar\cosh(|\vec{p}|/p_{M})\sinh(|\vec{p}|/p_{M})}{p_{M}|\vec{p}|}( \hat{p}_{i}\hat{X}_{j}-\hat{p}_{j}\hat{X}_{i})\\ &=&\frac{2i\hbar\cosh^{2}(|\vec{p}|/p_{M})}{p_{M}^{2}}(\hat{p}_{i} \hat{X}_{j}-\hat{p}_{j}\hat{X}_{i})\\ &=&\frac{2i\hbar}{p_{M}^{2}-|\vec{P}|^{2}}(\hat{P}_{i}\hat{X}_{j}-\hat{P}_{j} \hat{X}_{i})\.\end{array}  \tag{31}\]  
The first line in (31) gives the right hand side of the commutator in terms of the standard position and momentum operators. In the last line, we have given the right hand side of the commutator in terms of the modified position and momentum operators by changing \(|\vec{p}|\) to \(|\vec{P}|\) and by solving the connection between the two momenta given in (14), namely, \(\frac{|\vec{p}|}{p_{M}}=\tanh^{-1}(|\vec{P}|/p_{M})\). Using this relationship in the third line of (31) then gives the last line in (31), which is entirely in terms of the modified operators. For the arctan modified position of (15), we obtain  
\[ \begin{array}{rcl}[\hat{X}_{i},\hat{X}_{j}]&=& 2i\hbar\left[1+\left(\frac{\pi|\vec{p}|}{2p_{M}}\right)^{2}\right]\left(\frac {\pi}{2p_{M}}\right)^{2}(\hat{p}_{i}\hat{x}_{j}-\hat{p}_{j}\hat{x}_{i})\\ &=& 2i\hbar\left(\frac{\pi}{2p_{M}}\right)^{2}(\hat{p}_{i}\hat{X}_{j}- \hat{p}_{j}\hat{X}_{i})\\ &=&\frac{2i\hbar}{\arctan(\pi|\vec{p}|/2p_{M})}\left(\frac{\pi}{2p_{M}} \right)^{2}\left(\frac{\pi|\vec{p}|}{2p_{M}}\right)(\hat{P}_{i}\hat{X}_{j}- \hat{P}_{j}\hat{X}_{i})\\ &=& 2i\hbar\left(\frac{\pi}{2p_{M}|\vec{P}|}\right)\tan\left(\frac{ \pi|\vec{P}|}{2p_{M}}\right)(\hat{P}_{i}\hat{X}_{j}-\hat{P}_{j}\hat{X}_{i})\.\end{array}  \tag{32}\]  
Again, in the last step, we have replaced standard operators with the modified operator and converted \(|\vec{p}|\) to \(|\vec{P}|\) by solving the connection between the two momenta given in (14), namely, \(\frac{\pi|\vec{p}|}{2p_{M}}=\tan(\pi|\vec{P}|/2p_{M})\). When the modified momentum approaches its asymptotic value, \(p_{M}\), the commutators in both (31) and (32) diverge. The right-hand side of the modified position commutators in both (31) and (32) have the angular momentum operator on the right hand side--either the standard angular momentum operators in terms of the standard momentum and standard position (\(\hat{l}_{ij}=\hat{p}_{i}\hat{x}_{j}-\hat{p}_{j}\hat{x}_{i}\)) or modified angular momentum operators in terms of the modified momentum and modified position (\(\hat{L}_{ij}=\hat{P}_{i}\hat{X}_{j}-\hat{P}_{j}\hat{X}_{i}\))--times some multiplicative factors that are momentum-dependent.