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Symmetry of the Dirac operator

We now regard $\Dslash$ as an operator on , defined initially on the dense domain $\sS = \Ga_\smooth(M, S)$ .

$\begin{prop} $\Dslash$ is \emph{symmetric}: that is, whenever $\phi, \psi \in \... ...ket{\Dslash\phi}{\psi} = \braket{\phi}{\Dslash\psi}. \end{displaymath}\end{prop}$

$\begin{proof} We compute the pairings $\pairing{\Dslash\phi}{\psi}$ and $\pairi... ...si} = -i\divg Z_{\phi\psi} \end{displaymath}which has integral zero. \end{proof}$

Pawel Witkowski 2006-03-14