# GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry
## 3 Generalizing to \(3+1\) Dimensions
Using (12) and (13), the modified position and momentum operators for the \(\tanh\) modification are  
\[\hat{X}_{\mu}=i\hbar\cosh^{2}\bigg{(}\frac{p}{p_{M}}\bigg{)}\partial_{p_{\mu}} \quad\text{and}\quad\hat{P}_{\mu}=p_{i}\,H\bigg{(}\frac{p}{p_{M}}\bigg{)}=p_{ \mu}\frac{p_{M}}{p}\tanh\bigg{(}\frac{p}{p_{M}}\bigg{)}\, \tag{14}\]  
and for the arctan modification  
\[\hat{X}_{\mu}=i\hbar\bigg{[}1+\bigg{(}\frac{\pi p}{2p_{M}}\bigg{)}^{2}\bigg{]} \partial_{p_{\mu}}\quad\text{and}\quad\hat{P}_{\mu}=p_{\mu}\,H\bigg{(}\frac{p} {p_{M}}\bigg{)}=p_{\mu}\frac{2p_{M}}{\pi p}\arctan\bigg{(}\frac{\pi p}{2p_{M}} \bigg{)}. \tag{15}\]  
In order for the operators in (14) and (15) to satisfy (11), the standard commutator, we require  
\[G\bigg{(}\frac{p}{p_{M}}\bigg{)}\partial_{p_{\mu}}\bigg{[}p_{\
u}\,H\bigg{(} \frac{p}{p_{M}}\bigg{)}\bigg{]}\stackrel{{?}}{{=}}g_{\mu\
u}. \tag{16}\]  
The question mark indicates that it is an open question whether the left-hand side can be equal to the right-hand side. For the modified momentum from (14), condition (16) becomes [26]  
\[G\bigg{(}\frac{p}{p_{M}}\bigg{)}\partial_{p_{\mu}}\bigg{[}p_{ \
u}\,H\bigg{(}\frac{p}{p_{M}}\bigg{)}\bigg{]} = \bigg{[}\frac{p_{M}}{p}\sinh\bigg{(}\frac{p}{p_{M}}\bigg{)} \cosh\bigg{(}\frac{p}{p_{M}}\bigg{)}\bigg{(}g_{\mu\
u}-\frac{p_{\mu}p_{\
u}}{p ^{2}}\bigg{)}+\frac{p_{\mu}p_{\
u}}{p^{2}}\bigg{]} \tag{17}\] \[\hat{X}_{\mu}=i\hbar G\biggl{(}\frac{\hat{x}_{\
u}}{x_{m}},\frac{\hat{p}_{\
u }}{p_{M}}\biggr{)}\partial_{p_{\mu}}\quad\to\quad\hat{X}_{\mu}=i\hbar G\biggl{(} \frac{p}{p_{M}}\biggr{)}\partial_{p_{\mu}}. \tag{9}\]0  
For the modified momentum from (15), condition (16) becomes [26]\[\hat{X}_{\mu}=i\hbar G\biggl{(}\frac{\hat{x}_{\
u}}{x_{m}},\frac{\hat{p}_{\
u }}{p_{M}}\biggr{)}\partial_{p_{\mu}}\quad\to\quad\hat{X}_{\mu}=i\hbar G\biggl{(} \frac{p}{p_{M}}\biggr{)}\partial_{p_{\mu}}. \tag{9}\]1 \[\hat{X}_{\mu}=i\hbar G\biggl{(}\frac{\hat{x}_{\
u}}{x_{m}},\frac{\hat{p}_{\
u }}{p_{M}}\biggr{)}\partial_{p_{\mu}}\quad\to\quad\hat{X}_{\mu}=i\hbar G\biggl{(} \frac{p}{p_{M}}\biggr{)}\partial_{p_{\mu}}. \tag{9}\]2  
In the second lines of (17) and (18), the functions in square brackets on the right-hand side were expanded to the first order in \(|\vec{p}|/p_{M}\). Equations (17) and (18) show that to the first order, one recovers the standard three-dimensional commutators. However, there is a difference at second order. In \(3+1\) dimensions, it is no longer possible to modify the operators and retain the standard commutator. This is due to the fact that the ansatz functions now depend on \(|p_{\mu}|\), which involves all the three spatial and one time component, whereas in the previous section the ansatz functions only depended on one momentum component. Using Equations (14) and (15) and the results of (17), we find that the modified commutator for the \(\tanh\) modification, to the second order, is [26]  
\[\hat{X}_{\mu}=i\hbar G\biggl{(}\frac{\hat{x}_{\
u}}{x_{m}},\frac{\hat{p}_{\
u }}{p_{M}}\biggr{)}\partial_{p_{\mu}}\quad\to\quad\hat{X}_{\mu}=i\hbar G\biggl{(} \frac{p}{p_{M}}\biggr{)}\partial_{p_{\mu}}. \tag{9}\]3  
For the \(\arctan\) modification, there is a subtle point. From (18), we want to expand a term of form \((1+x^{2})\frac{\arctan(x)}{x}\), with \(x\equiv\frac{\pi|\vec{p}|}{2p_{M}}\). Here, it is necessary to expand \(\mathcal{O}(x^{3})\) due to \(x\) in the denominator. For this order, we have \(\arctan(x)\approx x-\frac{x^{3}}{3}\). Thus, for \(\mathcal{O}(x^{2})\), Equation (18) becomes [26]  
\[\hat{X}_{\mu}=i\hbar G\biggl{(}\frac{\hat{x}_{\
u}}{x_{m}},\frac{\hat{p}_{\
u }}{p_{M}}\biggr{)}\partial_{p_{\mu}}\quad\to\quad\hat{X}_{\mu}=i\hbar G\biggl{(} \frac{p}{p_{M}}\biggr{)}\partial_{p_{\mu}}. \tag{9}\]4  
The approximate commutators in (19) and (20) are similar to the modified commutators in [10]--in both cases, the modified commutators are quadratic in \(p\). However, unlike the one-dimensional case, the standard commutator (11) can no longer be recovered with the modified operators from (14) and (15).