# GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry
## 2 GUP Models in One Spatial Dimension  
In this section, we construct a GUP in one spatial dimension along the lines sketched out in the introduction, i.e., we build modified position and momentum operators which preserve the standard quantum mechanical commutation relationships. In the rest of this work, capital letters indicate modified operators, and small letters are the standard/canonical operators.  
We begin by writing a general commutation relationship in one spatial dimension as  
\[[\hat{X},\hat{P}]=i\hbar F\bigg{(}\frac{\hat{x}}{x_{m}},\frac{\hat{p}}{p_{M}} \bigg{)}\;. \tag{1}\]  
The capital \(\hat{X}\) and \(\hat{P}\) are the _modified_ position and momentum, and the general function \(F\Big{(}\frac{\hat{x}}{x_{m}},\frac{\hat{p}}{p_{M}}\Big{)}\) depends on the standard position and momentum, \(\hat{x}\) and \(\hat{p}\), which satisfy the standard commutator \([\hat{x},\hat{p}]=i\hbar\). The constants \(x_{m}/p_{M}\) set the small distance/large momentum cut-off scale. When \(\Delta x\gg x_{m}\) and/or \(\Delta p\ll p_{M}\), the ansatz function should have the limit \(F\Big{(}\frac{\hat{x}}{x_{m}},\frac{\hat{p}}{p_{M}}\Big{)}\to 1\), i.e., the modified commutator becomes the standard one for large distances and/or small momenta [26].  
Next, we need to choose \(\hat{X}\) and \(\hat{P}\) to obtain (1). A general form for the modified position and momentum operator are  
\[\hat{X}=i\hbar G\bigg{(}\frac{\hat{x}}{x_{m}},\frac{\hat{p}}{p_{M}}\bigg{)} \partial_{p}\quad\text{and}\quad\hat{P}=H\bigg{(}\frac{\hat{x}}{x_{m}},\frac{ \hat{p}}{p_{M}}\bigg{)}\;\hat{p}\;, \tag{2}\]  
where the ansatz functions \(G\) and \(H\) are chosen to give (1). We want to select \(G\) and \(H\) so that \(F=1\) while at the same time still giving a minimum distance. We simplify the analysis by taking all the ansatz functions to depend only on the momentum, i.e., \(G\Big{(}\frac{\hat{x}}{x_{m}},\frac{\hat{p}}{p_{M}}\Big{)}\to G\Big{(}\frac{p} {p_{M}}\Big{)}\) and \(H\Big{(}\frac{\hat{x}}{x_{m}},\frac{\hat{p}}{p_{M}}\Big{)}\to H\Big{(}\frac{p} {p_{M}}\Big{)}\), where we have let \(\hat{p}\to p\) since in momentum space the momentum operator is simply multiplication by \(p\).  
For the ansatz function \(H\), we want \(\hat{P}\approx p\) for \(p\ll p_{M}\) and \(\hat{P}\approx p_{M}\) for \(p\gg p_{M}\). In references [27; 28], two specific forms of ansatz function \(H\) which satisfy these conditions were given as  
\[H\bigg{(}\frac{p}{p_{M}}\bigg{)}=\frac{2p_{M}}{\pi p}\arctan\bigg{(}\frac{\pi p }{2p_{M}}\bigg{)}\quad\text{and}\quad H\bigg{(}\frac{p}{p_{M}}\bigg{)}=\frac{ p_{M}}{p}\tanh\bigg{(}\frac{p}{p_{M}}\bigg{)}\;. \tag{3}\]  
Once the ansatz function \(H\) is defined, the ansatz function \(G\) is determined by requiring that  
Using the two sets of ansatz functions from (3) and (5) in (2), we find that the modified position and momentum operators are  
\[\hat{X}=i\hbar\biggl{[}1+\biggl{(}\frac{\pi p}{2p_{M}}\biggr{)}^{2}\biggr{]} \partial_{p}\quad\text{and}\quad\hat{P}=\frac{2p_{M}}{\pi}\arctan\biggl{(} \frac{\pi p}{2p_{M}}\biggr{)}\, \tag{6}\]  
and  
\[\hat{X}=i\hbar\cosh^{2}\biggl{(}\frac{p}{p_{M}}\biggr{)}\partial_{p}\quad\text {and}\quad\hat{P}=p_{M}\tanh\biggl{(}\frac{p}{p_{M}}\biggr{)}\, \tag{7}\]  
The modified operators in (6) and (7) both lead to the standard commutator \([\hat{X},\hat{P}]=i\hbar\)[27; 28]. Therefore, the standard uncertainty also remains the same \(\Delta X\Delta P\geq\frac{\hbar}{2}\), but now in terms of the modified position and momentum. Since the momentum operator is capped at \(p_{M}\) for both (6) and (7), we find that \(\Delta P\sim p_{M}\) for both cases in (6) and (7). Thus, the minimum distance for both versions of the modified operators is \(\Delta X\Delta P\geq\frac{\hbar}{2}\to\Delta X\geq\frac{\hbar}{2p_{M}}\).