Why Are Currents in Matrix Elements Confusing?

[alsey42147]

i'm a bit confused about the currents in the expression for a matrix element for an interaction...

e.g. you could have a current like (adjoint spinor)x(spinor) which is scalar, this makes sense to me.

or you could have a current like (adjoint spinor)x(gamma matrix)x(spinor) which is vector according to all the books I've looked at. i don't get this - i would have thought that (gamma matrix)x(spinor) is either a vector or another spinor or something with 4 components, but then multiplying that by the adjoint spinor would just leave you with a scalar with 1 component. I'm guessing this is wrong but i can't see why.

also, say the matrix element looks something like:

(number)x[(adjoint)x(gamma-mu)x(spinor)]x[(adjoint)x(gamma-mu)x(spinor)]

with one gamma-mu having the index up, and the other one index down; does this mean that i sum the above expression over mu; i.e. a sum of 4 terms each with a different gamma matrix?

thanks in advance, and apologies for my lack of latex skills.

 

[blechman]

remember that the gamma matrices actually have THREE indices! Writing it out explicity, they are $\gamma^\mu_{\dot{\alpha}\beta}$. The $\mu$ index is the vector index, the undotted lower index is a spinor index and the dotted lower index is an "adjoint spinor" index. So you must contract ALL of these indices together:

$$\bar{\psi}\gamma^\mu\psi\equiv \bar{\psi}^{\dot{\alpha}}\gamma^\mu_{\dot{\alpha}\beta}\psi^{\beta}$$

and so it is a vector. Without the adjoint spinor, it would be this hybrid object with two indices.