Courant Snyder Transformation

The transverse motion in phase space in a circular accelerator is elliptical.
One can transform this to a circle using the Twiss parameters.
Let $(x,p)$ be the phase space coordinates. We define the normalized coordinates by

(1)
\begin{align} \pmatrix{\hat x\cr \hat p} = \pmatrix{\frac{1}{\sqrt{\beta}} & 0\cr \frac{\alpha}{\sqrt{\beta}} & \sqrt{\beta}}\pmatrix{x\cr p} \end{align}

The Twiss parameters are related by

(2)
\begin{align} \beta \gamma = 1+\alpha^2 \end{align}

In terms of these coordinates, the Courant Snyder invariant

(3)
\begin{align} 2\epsilon = \gamma x^2 +2\alpha x p + \beta p^2 \end{align}

becomes

(4)
\begin{align} 2\epsilon = \hat x^2 + \hat p^2 \end{align}

Now, let us use this mathematics in a slightly different context. Consider the full beam distribution, and its projection onto the x-y plane.
Suppose we know the second moments in this plane $\sigma_x^2= <x^2>, \sigma_y^2= <y^2>, \sigma_xy= <xy>$. Now, we may define an "emittance" and Twiss parameters in this plane such that:

(5)
\begin{eqnarray} <x^2> &=& \epsilon \beta\cr <y^2> &=& \epsilon \gamma\cr <xy> &=& -\epsilon \alpha \end{eqnarray}

One may then show that the emittance is given by

(6)
\begin{align} \epsilon = \sqrt{<x^2><y^2>-<xy>^2} \end{align}
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