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In physics equations are often expressed in index notation, which allows Lorentz covariance to be checked by implementing the Lorentz transformation on the components. An example is the Dirac equation [itex](\mathrm{i}\gamma^\mu \partial_\mu - \bar{m})\psi(x)[/itex].

I prefer to think of tensors as multilinear functions from the tangent space and its dual to the real numbers for two reasons. Firstly it avoids the need to explicitly check for Lorentz covariance and secondly it generalizes to be generally covariant when the spacetime is curved.

How would one express the Dirac equation in coordinate independent fashion so as to make it generally covariant (or even just Lorentz covariant?). The first thing one would have to deal with is the abuse of notation [itex]\gamma^\mu \partial_\mu[/itex]. Secondly one would have to think about where the [itex]\psi(x)[/itex] actually lives on the manifold (it's clearly not a smooth section of the tangent bundle like [itex]\partial_\mu[/itex]).

Would anyone be able to shed some light on this?