# GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry
## 5 3D GUP and Non-Commutative Geometry
All these features are equivalent to the form of modified position commutators and non-commutative geometry that emerged from the GUP model in [10]. However, (31) and (32)are both different from the standard non-commutative geometry discussed in [31], which generally started with a position commutator of the form  
\[[X_{i},X_{j}]=i\Theta_{ij}\, \tag{33}\]  
where \(\Theta_{ij}\) is a constant, anti-symmetric tensor. Generally, the position commutator in standard non-commutative geometry is written with four-vector indices, \([X_{\mu},X_{v}]=i\Theta_{\mu\
u}\), but here we use only three-vector indices to make the comparison with (31) and (32) more transparent. In contrast to the standard right-hand side of a position commutator like (33), the right hand side of our modified position commutator, as in (31) or (32), depends on momentum--either standard momentum or modified momentum. The commutators in (31) and (32) lead to a non-trivial uncertainty relationship between the modified position operators. For (31), the uncertainty relationship is  
\[\Delta X_{i}\Delta X_{j} \geq \frac{\hbar}{p_{M}}\Bigg{\langle}\frac{\cosh^{3}(|\vec{p}|/p_{M}) \sinh(|\vec{p}|/p_{M})}{|\vec{p}|}(\hat{p}_{i}\hat{x}_{j}-\hat{p}_{j}\hat{x}_{ i})\Bigg{\rangle} \tag{34}\] \[\geq \frac{\hbar}{p_{M}^{2}}\langle\left(\hat{p}_{i}\hat{x}_{j}-\hat{p }_{j}\hat{x}_{i}\right)\rangle=\frac{\hbar}{p_{M}^{2}}e_{ijk}\langle\{\hat{k} _{k}\}\,\]  
where in the second line we have approximated the hyperbolic functions to the first order in \(|\vec{p}|/p_{M}\) and written the expectation in terms of the ordinary angular momenta \(e_{ijk}\hat{k}_{k}=\hat{p}_{i}\hat{x}_{j}-\hat{p}_{j}\hat{x}_{i}\). Since \(\langle\hat{k}_{k}\rangle=n\frac{\hbar}{2}\)--an integer multiple of \(\hbar/2\)--we find that the \"area\" uncertainty implied by (34) is \(|\Delta X_{i},\Delta X_{j}|\sim\frac{\hbar\pi^{2}}{2p_{M}^{2}}\). This is consistent, up to factors of order unity, with what we found in Section 4 for the minimum uncertainty of the modified position. From Section 4, we found that for the same order in \(|\vec{p}|/p_{M}\), \(|\Delta X_{i}|\sim\frac{\hbar}{p_{M}}\), which then gave \(|\Delta X_{i}\Delta X_{j}|\sim\frac{\hbar^{2}}{p_{M}^{2}}\) in general agreement with (34). The same results follow from (32), which yields an uncertainty relationship of  
\[\Delta X_{i}\Delta X_{j} \geq \frac{\hbar\pi^{2}}{4p_{M}^{2}}\Bigg{\langle}\bigg{[}1+\bigg{(} \frac{\pi|\vec{p}|}{2p_{M}}\bigg{)}^{2}\bigg{]}(\hat{p}_{i}\hat{x}_{j}-\hat{p} _{j}\hat{x}_{i})\Bigg{\rangle} \tag{35}\] \[\geq \frac{\hbar\pi^{2}}{4p_{M}^{2}}\langle\left(\hat{p}_{i}\hat{x}_{ j}-\hat{p}_{j}\hat{x}_{i}\right)\rangle=\frac{\hbar\pi^{2}}{4p_{M}^{2}}e_{ijk} \langle\hat{k}_{k}\rangle\,\]  
where in the second line we have again approximated the expression to the first order in \(|\vec{p}|/p_{M}\) and inserted the angular momentum operator. Again, the area uncertainty implied by (35), (i.e., \(|\Delta X_{i}\Delta X_{j}|\sim\frac{\hbar\pi^{2}\pi^{2}}{8p_{M}^{2}}\)) is consistent with the implied \"area\" uncertainty from Section 4 obtained via GUP arguments. One final comment is that both (34) and (35) imply a relationship between the area uncertainty and the angular momentum. This connection is also found in spin foam models of quantum gravity [33; 34], where one has that the following relationship between the area operator \(\hat{A}_{S}\) for surface \(S\) is given by  
\[\hat{A}_{S}|\Psi\rangle=8\pi l_{Pl}^{2}\gamma\sqrt{l(l+1)}|\Psi\rangle\, \tag{36}\]  
where \(l_{Pl}\) is the Planck length and \(\gamma\) is the Immirzi parameter [35]. The spin foam relationship between the area operator and the angular momentum in (36) is similar to the relationship in (34) and (35). The area operator on the left hand side of (36) is simply the magnitude of the area, while the \"areas\" in (34) or (35) retain the vector area nature through the indices on \(\Delta X_{i}\).