Is it possible to write the Dirac equation without spinors?

[computer]

Is it possible to write the Dirac equation without spinors? Something similar to Maxwell or Schrödinger's equations with simpler tensor algebra elements. Let values not be the same when going from the left coordinate system to the right, they could be valid only in the only system, for example, associated with the geometric center of hydrogen atom. But simple and obvious equation, or several equations.

[Markus Hanke]

10 hours ago, computer said:

Is it possible to write the Dirac equation without spinors? Something similar to Maxwell or Schrödinger's equations with simpler tensor algebra elements.

Yes, it’s possible to do this, since both bispinors and rank-n tensors are possible representations of the Lorentz group. Essentially, you replace the bispinor by an ordinary 4-vector, and replace the gamma matrices by rank-3 tensors. So what you’re really doing is shift some of the transformation properties concerning rotations - encoded in the bispinor - into the gamma matrices themselves. The result can be shown to be physically equivalent to the ordinary Dirac bispinor formalism. For example:

https://www.researchgate.net/publication/1898363_Dirac_Equation_Representation_Independence_and_Tensor_Transformation

But why would you want to do this, I wonder? Bispinors already transform in the same way as tensors, so they are tensor-like objects - unsurprisingly, since both objects are representations of the Lorentz group.

[computer]

Thanks for link to the article. I just wanted to separate the basic physical essence of equation from the mathematics involved in transitions between coordinate systems.

[Markus Hanke]

On 10/17/2022 at 4:17 PM, computer said:

I just wanted to separate the basic physical essence of equation from the mathematics involved in transitions between coordinate systems.

Bispinors are covariant objects, as are the gamma matrices, so the Dirac equation, when written using Dirac notation, has the same form in all reference frames - just like a tensor equation would.