Question

Is it possible to retrieve the matrix elements of the γs by simply knowing their anti-commutation relation: {γμ,γν}=2gμνI4

I'm just trying to reconstruct Dirac's reasoning when he first encountered these relations (previous to his equation). The analogy with the Pauli matrices is evident, which at that time were newly introduced, so it's reasonable to assume some kind of analogy. But it's still too early to guess correctly. He might have recognized the Clifford algebra... but again I think it's not enough.

Answer 1

Yes, it is possible to retrieve the matrix elements of the γs by knowing their anti-commutation relation. By finding the matrix representations that satisfy the anti-commutation relation, we can reconstruct the γ matrices.

Step-by-step explanation:

Yes, it is possible to retrieve the matrix elements of the γs by knowing their anti-commutation relation. The anti-commutation relation {γμ,γν} = 2gμνI4 indicates that the γ matrices satisfy the Clifford algebra. By finding the matrix representations that satisfy the anti-commutation relation, we can reconstruct the γ matrices. Let's take the example of γ^0:

1. Start by defining the γ^0 matrix with unknown elements:
2. γ^0 = [a b c d]
3. Apply the anti-commutation relation:
4. {γ^0, γ^0} = 2I4
5. Calculate the anti-commutator:
6. γ^0 γ^0 + γ^0 γ^0 = 2I4
7. [a b c d][a b c d] + [a b c d][a b c d] = 2I4
8. [a^2+b^2+c^2+d^2 2ab+2cd 2ac-2bd 2ad+2bc] = 2I4
9. Compare the resulting matrix with 2I4:
10. a^2+b^2+c^2+d^2 = 2
11. 2ab+2cd = 0
12. 2ac-2bd = 0
13. 2ad+2bc = 0
14. Solve the equations to find the values of a, b, c, and d:
15. a^2+b^2+c^2+d^2 = 2
16. 2ab+2cd = 0
17. 2ac-2bd = 0
18. 2ad+2bc = 0

By solving these equations, you can find the elements of the γ^0 matrix. Similar steps can be followed to retrieve the elements of other γ matrices.