\pmatrix{1 & 2\alpha \cr  0 & 1} \pmatrix{x \cr y} ,$$ which leads to $$\label{shr02}
 \pmatrix{x' \cr y'} =  \pmatrix{x + 2\alpha y \cr y} .$$ This shear is one of the basic transformations in engineering sciences. In physics, this transformation plays the key role in understanding the internal space-time symmetry of massless particles [@wig39; @wein64; @kiwi90jmp]. This matrix plays the pivotal role during the transition from the oscillator mode to the damping mode in classical damped harmonic oscillators [@bkn14; @bkn15].

Under this transformation, the Gaussian form becomes $$\label{gau33}
 \psi_{shr}(x,y) = \frac{1}{\sqrt{\pi}}
 \exp{\left\{-\frac{1}{2}\left[(x - 2\alpha y)^2 + y^2\right]\right\}}.$$ It is possible to expand this into a series of the form of Eq.(\[seri22\]) [@kimyeh92].

The transformation applicable to the Gaussian form of Eq.(\[gau00\]) is $$\label{shr05}
\exp{\left(-2\alpha y \frac{\partial}{\partial x} \right)}  ,$$ and the generator is $$\label{shr07}
   - i y \frac{\partial}{\partial x}  .$$ It is of interest to see where this generator stands among the ten generators of Dirac.

However, the most pressing problem is whether the sheared Gaussian form can be regarded as a rotated squeezed state. The basic mathematical issue is that the shear matrix of Eq.(\[shr01\]) is triangular and cannot be diagonalized. Therefore, it cannot be a squeezed state. 