# Re-Writing the Dirac Equation in True Covariant Form

This is a rather brief inquiry, but to get to the point it's always frustrated me that in non-relativistic and relativistic quantum mechanics spin matrices are written as a "vector of matrices" instead of just using covariant notation to represent the the quantity as a rank-3 tensor (triad) or just a general 3D array in the case that a certain scenario doesn't provide that the quantity is strictly a tensor. Anyways, it is acceptable to write the Dirac equation as $$i\hbar \gamma^{\nu \:\: \: \mu}_{\: \: \alpha} \partial_{\mu} \psi^{\alpha} - mc \delta^{\nu}_{\:\alpha} \psi^{\alpha}=0$$ If this isn't acceptable, could you please explain why? If it is, why is the so called covariant form of the Dirac equation not written this way, why is there such a convention of writing these triads as vectors of matrices?

• The Dirac equation is already covariant. Note that $\psi$ transforms as a spinor not a vector under Lorentz transformations. cheng.physics.ucdavis.edu/teaching/230A-s07/rqm4_rev.pdf – Luke Burns Mar 22 at 4:30
• I do realize this, the question is in regards to writing the $\gamma$-matrices more true to how covariant equations are typically written. Is there anything incorrect about viewing the equation as a contraction of the components of the wavefunction with the lower index of the $\gamma$ "triad" – David G. Mar 22 at 4:32
• Furthermore your choice of Greek indices for both representations blurs the distinction between the two kinds of indices. It is extremely confusing. – G. Smith Mar 22 at 4:54
• My point of view is that it is fine to make the spinor indices explicit, and I believe some books do for clarity. However, it is not acceptable to make the two kinds of indices look the same. And it is not acceptable to refer to it as a tensor of rank 3. It is simultaneously a tensor of rank 2 under the spinor representation and a tensor of rank 1 under the vector representation. – G. Smith Mar 22 at 5:05
• You may be able to do this, but it's not clear to me that there's any utility in doing so, as it obscures the basic structure of the gammas. Dirac chose these matrices precisely for their relativistic structure: the gammas are effectively an orthonormal basis for a four-dimensional vector space with Minkowski metric. The algebraic structure is important. – Luke Burns Mar 22 at 5:05