# GUP, Lorentz Invariance (Non)-Violation, and Non-Commutative Geometry
## 4 Modified Lorentz Group for Modified Operators  
In this section, we investigate what happens to the Lorentz symmetry with the modification of the operators given in (14) and (15). In general, one expects that introducing a minimal length and/or modifying the space-time and energy-momentum operators will alter/break the original Lorentz symmetry. However, we find that it is possible to retain Lorentz symmetry despite the modification of the operators. First, we review the standard Lorentz symmetry for standard space-time and energy-momentum operators. One can construct the generators for the Lorentz group from the standard space-time and energy-momentum operators as  
\[\hat{l}_{a\beta}=\hat{x}_{a}\hat{p}_{\beta}-\hat{x}_{\beta}\hat{p}_{a}\;. \tag{21}\]For \(\alpha=0\) and \(\beta=i\), we have boost generators \(\hat{k}_{i}=\hat{I}_{0i}\). For \(\alpha=i\) and \(\beta=j\), one has rotation generators, namely, angular momentum \(\hat{l}_{i}=\frac{1}{2}\epsilon_{ijk}\hat{l}_{jk}=\epsilon_{ijk}\hat{x}_{j}\hat{p }_{k}\). These generators satisfy the standard Lorentz algebra, namely,  
\[[\hat{l}_{i},\hat{l}_{j}]=i\hbar\epsilon_{ijk}\hat{l}_{k}\ \ \ \text{and}\ \ \ [\hat{k}_{i},\hat{k}_{j}]=-i\hbar\epsilon_{ijk}\hat{k}_{k}\ \ \ \text{and}\ \ \ [\hat{l}_{i},\hat{k}_{j}]=i\hbar\epsilon_{ijk}\hat{k}_{k}. \tag{22}\]  
Now, with the modified space-time and energy-momentum operators from (14) and (15), one can construct modified generators of the Lorentz group as  
\[\hat{L}_{\alpha\beta}=\hat{X}_{\alpha}\hat{P}_{\beta}-\hat{X}_{\beta}\hat{P}_{ \alpha}. \tag{23}\]  
These give new rotation generators of the form \(\hat{L}_{ij}=\hat{X}_{i}\hat{P}_{j}-\hat{X}_{j}\hat{P}_{i}\), which can be written in vector form as \(\hat{L}_{i}=\frac{1}{2}\epsilon_{ijk}\hat{L}_{jk}=\epsilon_{ijk}\hat{X}_{j} \hat{p}_{k}\). The new boost generators in vector form are \(\hat{K}_{i}=\hat{L}_{0i}=\hat{X}_{0}\hat{P}_{i}-\hat{X}_{i}\hat{P}_{0}\). It is easy to verify that these modified angular momentum operators and modified boost operators _do not_ satisfy the standard algebra in (21) for the arctan modified operators or the tanh modified operators. However, following reference [10], it is possible to make simple modifications to the operators defined in (23) that satisfy the Lorentz algebra of (21). For the tanh modified operators, one can define a modified angular momentum operator as  
\[\hat{\mathcal{L}}_{i}\equiv\frac{p}{p_{M}\cosh^{2}(p/p_{M})\tanh(p/p_{M})}\hat {L}_{i}=\epsilon_{ijk}\hat{x}_{j}\hat{p}_{k}=\hat{l}_{i}\, \tag{24}\]  
and for the arctan modified operators from (15), one has  
\[\hat{\mathcal{L}}_{i}=\frac{\pi p}{2p_{M}[1+(\pi p/2p_{M})^{2}]\arctan(\pi p/2 p_{M})}\hat{L}_{i}=\epsilon_{ijk}\hat{x}_{j}\hat{p}_{k}=\hat{l}_{i}. \tag{25}\]  
Next, we turn to the modified boost generators which are the time-space components of \(\hat{L}_{\alpha\beta}\), namely, \(\hat{L}_{0i}=\hat{X}_{0}\hat{P}_{i}-\hat{X}_{i}\hat{P}_{0}\). We define modified \"time\" and \"energy\" operators \(\hat{X}_{0}=\hat{T}\) and \(\hat{P}_{0}=\hat{\mathcal{E}}\) (note that we have set \(c=1\) here and throughout the paper). To maintain the structure, the time operator needs to be modified via the same factors as the position operator, and the energy operator needs to be modified via the same factors as the momentum operator. For the tanh modification of (14), this results in modified time and energy operators of the form  
\[\hat{X}_{0}=\hat{T}=i\hbar\cosh^{2}\biggl{(}\frac{E}{E_{M}}\biggr{)}\partial_{ E}\ \ \ \ \text{and}\ \ \ \hat{P}_{0}=\hat{\mathcal{E}}=E_{M}\tanh\biggl{(}\frac{E}{E_{M}}\biggr{)}\, \tag{26}\]  
where \(E_{M}=p_{M}\), and \(E=|p_{\mu}|=p\) is the usual energy operator.