The generator $S_3$ can be written as $$\label{dir18}
S_{3} = \pmatrix{0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
  0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & -i \cr 0 & 0 & 0 & i &  0 } .$$ The last five-by-five matrix generates rotations in the two-dimensional space of $(t, s)$. If we introduce these two time variables, the $O(3,2)$ group leads to two coupled Lorentz groups. The particle mass is invariant under Lorentz transformations. Thus, one Lorentz group cannot change the particle mass. However, with two coupled Lorentz groups we can describe the world with variable masses such as the neutrino oscillations.

In Sec. \[2dim\], we used the operators $Q_{3}$ and $K_{3}$ as the generators for the squeezed Gaussian function. For the unitary transformation of Eq.(\[exp01\]), we used $$\label{exp11}
\exp{\left(-i\eta Q_{3}\right)} .$$ However, the exponential form of Eq.(\[exp05\]) can be written as $$\label{exp12}
\exp{\left\{-i(\tanh\eta) \left(Q_{3} + iK_{3}\right)\right\}} ,$$ which is not unitary, as was seen before.

From the space-time point of view, both $K_{3}$ and $Q_{3}$ generate Lorentz boosts along the $z$ direction, with the time 