I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds = D^{-1}$ and so, it should be possible to get a measure of a line length on a manifold using this ansatz. If the manifold is Minkowski and the Dirac operator is the standard one $D = i\gamma\cdot\partial$ where $\gamma$s belong to a Clifford algebra and are constants, how can I formally derive the variational principle of special relativity $$ {\cal L} = \int \sqrt{dx_0^2-dx_1^2-dx_2^2-dx_3^2} $$ with $\delta{\cal L} = 0$ for the geodesic curves? Is such a question meaningful?
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