Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Currents in matrix elements

  1. Mar 28, 2008 #1
    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:


    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.
  2. jcsd
  3. Mar 28, 2008 #2


    User Avatar
    Science Advisor

    remember that the gamma matrices actually have THREE indices! Writing it out explicity, they are [itex]\gamma^\mu_{\dot{\alpha}\beta}[/itex]. The [itex]\mu[/itex] 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:

    [tex]\bar{\psi}\gamma^\mu\psi\equiv \bar{\psi}^{\dot{\alpha}}\gamma^\mu_{\dot{\alpha}\beta}\psi^{\beta}[/tex]

    and so it is a vector. Without the adjoint spinor, it would be this hybrid object with two indices.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Currents in matrix elements