# Is there a coordinate-free Dirac equation?

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..?

• Always? $/ \!\!\!\partial$ . – Cosmas Zachos Sep 17 at 16:35
• How can you have a differential equation for a function on spacetime without having any coordinates to express where you are in spacetime? – G. Smith Sep 17 at 16:37
• No, you don’t need indices. Just write it out using $t,x,y,z$ derivatives. – G. Smith Sep 17 at 16:38
• There is, although doing it carefully requires introduction of spinor bundles. See ncatlab.org/nlab/show/Dirac+operator for example. – Peter Kravchuk Sep 17 at 16:45
• 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. – 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 $$\nabla \cdot \nabla \phi = -\rho$$.