10 Appendix

Abstract

a) $W(t) \; maps \; {\cal S} \times {\cal S} \; into \; {\cal S} \times {\cal S}$

If $u \in {\cal S} ( {\rm I\!R}^{s}) \times {\cal S} ( {\rm I\!R}^{s}) \; \; ( {\cal S} ( {\rm I\!R}^{s})$ is the Schwartz space of C^∞ test functions decreasing at infinity faster than any inverse polynomial), then the solution of the free wave equation is easily obtained by Fourier transform and one has

${W(t) \left( \begin{array}{c} \varphi_{0} (k) \\ \psi_{0} (k) \end{array} \right) = \left( \begin{array}{cc} \cos |k| t & (\sin |k| t)/|k| \\ - |k| \sin |k| t & \cos |k| t \end{array} \right) \left( \begin{array}{c} \varphi_{0} (k) \\ \psi_{0} (k) \end{array} \right). }$     (A.1)

cos |k| t, (sin |k| t) / |k| etc. are multipliers of ${\cal S}$ continuous in t and

${ \frac{d}{dt}W(t) \big|_{\; t=0} = \left(\begin{array}{cc} 0 & 1 \\ |k |^{2} & 0 \end{array} \right) = K. }$     (A.2)

The group property is easily checked.