spheres.symplectic.complex_real_symplectic2

spheres.symplectic.complex_real_symplectic2(S, s)

Converts a complex symplectic matrix/vector to a real symplectic matrix/vector. Alternative construction:

If we write \(\textbf{S}\) as \(\begin{pmatrix} E & F \\ F^{*} & E^{*} \end{pmatrix}\), and \(\textbf{s}\) as \(\begin{pmatrix} s \\ s^{*} \end{pmatrix}\) then equivalently:

\[ \begin{align}\begin{aligned}\begin{split}\textbf{R} = \begin{pmatrix}\Re(E+F) & -\Im(E-F) \\ \Im(E+F) & \Re(E-F) \end{pmatrix}\end{split}\\\begin{split}\textbf{r} = \sqrt{2}\begin{pmatrix} \Re(s) \\ \Im(s) \end{pmatrix}\end{split}\end{aligned}\end{align} \]

Parameters

• S (np.array) – Complex symplectic matrix.

• s (np.array) – Complex symplectic displacement vector.

Returns

• R (np.array) – Real symplectic matrix.

• r (np.array) – Real symplectic displacement vector.