# GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry
## 3 Generalizing to \(3+1\) Dimensions  
We now generalize the results of the previous section to \(3+1\) space-time dimensions. To make this transition, the modified commutator from (1) now becomes  
\[[\hat{X}_{\mu},\hat{P}_{\
u}]=ig_{\mu\
u}\hbar F\biggl{(}\frac{\hat{x}_{\mu}}{ x_{m}},\frac{\hat{p}_{\mu}}{p_{M}}\biggr{)}. \tag{8}\]  
The indices \(\mu,\
u\) take on the values \(0,1,2,3\), and \(g_{\mu\
u}=diag(1,-1,-1,-1)\) is the \(3+1\) flat space-time metric. A more general version of the commutator from (8) would be a modified commutator of the form \([\hat{X}_{\mu},\hat{P}_{\
u}]=i\hbar F_{\mu\
u}\biggl{(}\frac{\hat{x}_{\
u}}{ x_{m}},\frac{\hat{p}_{\mu}}{p_{M}}\biggr{)}\), where we modify each component of the metric separately rather than having an overall scalar modification of the commutator. In the limit limit limit of large distance/small momentum, one would need to have the limit \(F_{\mu\
u}\to g_{\mu\
u}\). We made the choice of a scalar modification to the commutator given in (8) primarily for simplicity. It would be worth considering a more general modification, such as where one modifies each component of the metric separately. Next, we define modified space-time operators, \(\hat{X}_{\mu}\), and energy-momentum operators, \(\hat{P}_{\mu}\), which will yield (8). The modified position operator is written as  
\[\hat{X}_{\mu}=i\hbar G\biggl{(}\frac{\hat{x}_{\
u}}{x_{m}},\frac{\hat{p}_{\
u }}{p_{M}}\biggr{)}\partial_{p_{\mu}}\quad\to\quad\hat{X}_{\mu}=i\hbar G\biggl{(} \frac{p}{p_{M}}\biggr{)}\partial_{p_{\mu}}. \tag{9}\]  
In the last step, we assumed that the ansatz \(G\) depends only on the magnitude of the canonical momentum, i.e., \(p=|p_{\mu}|\). Next, we take the modified momentum operator to be  
\[\hat{P}_{\mu}=H\biggl{(}\frac{\hat{x}_{\
u}}{x_{m}},\frac{\hat{p}_{\
u}}{p_{M} }\biggr{)}\hat{p}_{\mu}\quad\to\quad\hat{P}_{\mu}=H\biggl{(}\frac{p}{p_{M}} \biggr{)}\ \hat{p}_{\mu}\, \tag{10}\]  
where again have taken the ansatz function \(H\) to depend only on the magnitude of the canonical momentum. Now, \(G\) and \(H\) must to be chosen to yield (8). For both the modified position (9) and modified momentum (10) we assumed that the modification ansatz functions only depended on the magnitude of the momentum, (i.e., \(p=|p_{\mu}|\)) rather than on the momentum, (i.e., \(p_{\mu}\)). This was done to help ensure rotational symmetry. With a direct dependence on \(p_{\mu}\), we were unable to easily preserve rotational symmetry. Following reference [27], we want to select the ansatz functions \(G\) and \(H\) in (9) and (10) so that \(F=1\) in (8), i.e., so that \(\hat{X}_{\mu}\) and \(\hat{P}_{\mu}\) satisfy the standard commutator relationship  
\[[\hat{X}_{\mu},\hat{P}_{\
u}]=i\hbar g_{\mu\
u}. \tag{11}\]  
The ansatz function \(H\) is chosen to be the \(3+1\)-dimensional generalization of the 1-dimensional ansatz function from (3), namely [26],  
\[H\bigg{(}\frac{p}{p_{M}}\bigg{)}=\frac{2p_{M}}{\pi p}\arctan\bigg{(}\frac{\pi p }{2p_{M}}\bigg{)}\quad\text{and}\quad H\bigg{(}\frac{p}{p_{M}}\bigg{)}=\frac{p _{M}}{p}\tanh\bigg{(}\frac{p}{p_{M}}\bigg{)}. \tag{12}\]  
Although the ansatz function in (12) appears identical to the one in (3), it should be noted that the function \(H\) in (12) depends on \(p=|p_{\mu}|=\sqrt{p_{0}^{2}-p_{i}^{2}}\), not on the one-dimensional momentum as in (3). This distinction will come into play shortly when we examine the commutator of the modified position with the modified momentum. The \(G\) ansatz functions which generalize the 1-dimensional version from (5) are written as  
\[G\bigg{(}\frac{p}{p_{M}}\bigg{)}=\bigg{[}1+\bigg{(}\frac{\pi p}{2p_{M}}\bigg{)} ^{2}\bigg{]}\quad\text{and}\quad G\bigg{(}\frac{p}{p_{M}}\bigg{)}=\cosh^{2} \bigg{(}\frac{p}{p_{M}}\bigg{)}. \tag{13}\]  
Again, while this appears identical to the ansatz function \(G\) from (13), the ansatz function \(G\) in (13) depends on the magnitude of the four-momentum rather than the magnitude of the one-dimensional momentum.