We know that ${\gamma_{mu},\gamma_{nu}}$= 2 $g_{\mu\nu}$, where $g_{\mu\nu}$ is the flat metric . That shows spinors are someway related to the metric of space. Can anyone please explain how?
This relation means that the gamma matrices realize a Clifford algebra and this can be shown to be a requirement for the Lorentz covariance of the Dirac equation, i.e. the definition of target space spinors. The metric need not be flat,in fact.
In a curved space metric becomes position dependent gμν→gμν(x).
In a MInkowski space, which is a flat space, Lorentz transformations are global transformations. In a curved space-time Lorentz transformations act differently at each point where one uses the flat tangent space at each point. Therefore one sees that here Lorentz transformations become local. In a curved space-time if a theory contains spinors, then the theory should be invariant under local Lorentz transformations. They are gauge theories of the Lorentz group.
One has to study how Dirac spinors transform locally under the Lorentz group, and the Lorentz covariant derivative which acts on spinors.
Dear Basabendu Barman,
May be because of our Space is really flat...
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