Can we interchange the Dirac Matrices?

[thayes93]

Ok, first off I will admit that I really am pretty much ignorant of proper QM, as I am a first year undergraduate at a UK university.

Today our lecturer, in the final lecture of a Vibrations and Waves course, demonstrated how the Schrödinger equation is derived from applying the Energy and Momentum operators to the non-relativistic relation E = p^2 / 2m. He said he would be impressed if anyone could make this a relativistic equation and would be particularly impressed if anyone could derive the Dirac equation (though he expected this to be impossible for a first year and said that if anyone could they may as well go pick up their PhD now!).

Naturally, a friend and I decided to apply ourselves to the task and within 10 minutes had derived the Klein-Gordon equation. We saw a route to the Dirac equation but found in impossible to think of the object X that satisfies:
X^{2} = \triangledown ^{2} - \frac{1}{2}\frac{\partial ^{2}}{\partial t^{2}}
So after looking up the Dirac equation on wikipedia we came across the Dirac α and β Matrices, and that all made perfect sense.

My question is that, as far as I can tell, matrices alpha 1, 2, and 3 all seem tied to a particular component of the p (momentum) operator. However, from my understanding, the only limitation on each matrix is that its square must be the identity matrix and when multiplied by any of the other matrices, the product must equal zero.
Why then do specific Dirac matrices seem linked with particular components of the momentum operator (i.e alpha 1 is associated with the partial d/dx term)?

Thanks in advance for your help.

 

[thayes93]

Sorry, in the Latex text, the last term should have 1/c not 1/2.

 

[Avodyne]

It's an arbitrary convention, and sometimes different matrices are used. See "other representations" in

http://en.wikipedia.org/wiki/Gamma_matrices

 

[PhilDSP]

As I understand it, the basis for the Dirac matrices follow from the Pauli spin matrices which are normally written to conform to the Condon-Shortley convention. If they (Pauli spin matrices) are written in another configuration (where they are still mutually orthogonal) you lose the most simple way to derive or be synchronized with the coupling to angular momentum and spherical harmonics.