Exploring Dirac Matrices in the Context of the Dirac Equation

[paweld]

I wonder if I can chose any 4x4 matrices $$\gamma^\mu$$ which fullfil anticommutationn relations
$$\{\gamma^\mu,\gamma^\nu \}=2g^{\mu\nu}$$ as a matricies
in Dirac equation:
$$i \gamma^\mu \partial_\mu \psi= m \psi$$.
What changes in the theory if I chose different matricies?
(of course I have to consistently use this different matricies)
What if this matricies has explicit time dependence and I'm
looking for solutions evolving in time as $$\exp (-i\omega t)$$.

 

[dextercioby]

In the derivation by Dirac of his famous equation the "gamma's" are constant matrices. If you want to, you can pass the time dependency of the spinor wave functions onto the "gamma's" and keep the spinor functions depending on p/x only and not on time anymore.

 

[paweld]

Thanks for answer. I'm interested in slightly more complicated tansformation.
For example instead of traditional $$\gamma$$ matrices, let's chose the following:
$$\tilde{\gamma}^0=\cosh at \gamma^0 + \sinh at \gamma^1, <br /> \tilde{\gamma}^1=\sinh at \gamma^0 + \cosh at \gamma^1 ,<br /> \tilde{\gamma}^2=\gamma^2,\tilde{\gamma}^3=\gamma^3$$.
Is it true that the Dirac equation is still
$$i\tilde{\gamma}^\mu \partial_\mu \psi = m\psi$$
but I have to use this different matrices everywhere
(i.e. the coupling with electromagnetic filed would be $$A_\mu \psi^\dagger \tilde{\gamma}^\mu \psi$$)

 

[Ben Niehoff]

If you want to make the gammas coordinate-dependent, I suspect you may have to change Dirac's equation to

$i\partial_\mu (\gamma^\mu \psi) = m\psi$

However, this is a guess, not based on any rigorous derivation. Try deriving the equation from scratch to be sure.

 

[tom.stoer]

Gamma matrices become coordinate dependent in GR.