# What does couples as the 4th component of a vector mean in the Dirac equation?

1. May 2, 2012

### xahdoom

What does "couples as the 4th component of a vector" mean in the Dirac equation?

I'm doing an exercise regarding the spin-orbit operator and the Dirac equation/particles, and I'm having trouble understanding the link between terminology and mathematics. The particular phrase I'm having trouble with is:

"Investigate how the matrix operator β $\Sigma$ $\cdot$ J commutes with the Hamiltonian for a Dirac particle in a central potential V(r) that couples as the 4th component of a vector"

I've understood the sentence up until "Hamiltonian for a Dirac..."

I didn't consider this a homework question - more of a terminology question - because I've seen this language in textbooks as well. My trouble is that a) I'm not sure what this means in terms of mathematics and how it affects the standard Dirac Hamiltonian and more importantly b) I don't understand what this means conceptually.

If anyone can explain to me what the phrase "couples as the 4th component of a vector" means, or point me towards a piece of literature that would explain, I would be extremely grateful.

2. May 2, 2012

### Bill_K

Re: What does "couples as the 4th component of a vector" mean in the Dirac equation?

For the Schrodinger equation which is nonrelativistic, all you need to give to describe a central potential is V(r). But for coupling to a Dirac particle it matters how the central potential transforms under a Lorentz transformation - whether the potential is a scalar V(r), the time component V0(r) of a vector, or the time-time component V00(r) of a tensor. He wants you to assume it is V0(r).