Simple gamma matrices question

[Ameno]

Hi

I've just read the statement that a matrix that commutes with all four gamma matrices / Dirac matrices has to be a multiple of the identity. I don't see that; can anyone tell me why this is true?

Thanks in advance

 

[Physics Monkey]

You can deduce this yourself using the completeness properties of gamma matrices. By considering (schematically) 1, $$\gamma$$, $$\gamma \gamma$$, etc. one can form a basis of matrices.

If you can decide:
1) that this basis is complete for 4x4 matrices
2) that nothing in this basis besides 1 commutes with all the $$\gamma$$
then you will have what you want.

 

[Ameno]

Thanks.

OK, I also had this answer in mind (which I already found), but I thought that there is a simpler answer because this statement appeared at a point in the script where that basis has not yet been introduced. Perhaps I'll find a simpler answer, but I'm also fine with this one. At least I see that it's not that simple.

 

[Physics Monkey]

Yeah, I wouldn't necessarily claim this is the most elegant method. One could even explicitly compute the commutator of an arbitary 4x4 matrix with the four gamma matrices. This gives 4 4x4 linear matrix equations that will give the same conclusion. This is even less elegant but requires no mention of a basis.

 

[Ameno]

Yes, but wouldn't this require a choice of representation? It would not require a choice of basis, but it wouldn't be canonical in the sense of independence of representation.