Dirac equation is always written with indices. Is there any way to write it down without any indices ABSTRACT or not, and without coordinates,basis vectors etc..?

  • 2
    $\begingroup$ Always? $/ \!\!\!\partial$ . $\endgroup$ – Cosmas Zachos Sep 17 at 16:35
  • $\begingroup$ How can you have a differential equation for a function on spacetime without having any coordinates to express where you are in spacetime? $\endgroup$ – G. Smith Sep 17 at 16:37
  • $\begingroup$ No, you don’t need indices. Just write it out using $t,x,y,z$ derivatives. $\endgroup$ – G. Smith Sep 17 at 16:38
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    $\begingroup$ There is, although doing it carefully requires introduction of spinor bundles. See ncatlab.org/nlab/show/Dirac+operator for example. $\endgroup$ – Peter Kravchuk Sep 17 at 16:45
  • $\begingroup$ This is still in component form exactly the opposite of what I want. It's not about being invariant under component of frame transformations , it's about being completely coordinateless as I specified in the question. $\endgroup$ – Leo Kovacic Sep 18 at 8:18

Writing out $\gamma^\mu \partial_\mu$ is really no different than writing anything that looks like an inner product (for example, the Laplacian). If you're concerned that the indices don't make the frame-independence obvious, then just write it as a four-vector dot product: $$ (i\gamma\cdot\partial - m)\psi = 0. $$

I'm not saying that this notation is usual, or even that you will find it anywhere, but it seems no different to me than writing, say, $\nabla \times \nabla f = 0$ or $\nabla \cdot \nabla \phi = -\rho$.

  • $\begingroup$ Ok so why don't you write it down exactly as you did the last line if it's all the same.. $\endgroup$ – Leo Kovacic Sep 18 at 7:56

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