# Classical or quantum spinor

#### RedX

The Dirac field is an anticommutating field. But what part of it is anticommutating? Is it the spinors, or the coefficient in front of the spinors? In quantum field theory I think it's the coefficients that anticommute, so that the spinors should commute, but not their coefficients. In classical field theory, the coefficients are just regular numbers, so then it must be that the spinors anticommute?

#### Fredrik

Staff Emeritus
Science Advisor
Gold Member
It's the creation and annihilation operators. The gamma matrices also satisfy anticommutation relations, but they are a lot less interesting. The anticommutation relations satisfied by the creation/annihilation operators are the reason why the particles that correspond to the Dirac field must be fermions.

#### RedX

So if you have $$u_{\alpha}(p) u_{\beta}(k)$$, then this equals $$u_{\beta}(k) u_{\alpha}(p)$$, and not the negative? u(p) is the usual basis solution to the Dirac equation in momentum space.

#### Bob_for_short

Exactly. There is no sign change. u(p) are not matrices, so they commute, just like vector components.

Bob.

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