Simple gamma matrices question

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?

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Physics Monkey
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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.

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