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

I can't figure out if these are matrices or numbers

  1. Feb 10, 2012 #1

    Matterwave

    User Avatar
    Science Advisor
    Gold Member

    Hi, since Peskin and Schroeder pretty much suppresses the indices in every equation, I am now unable to tell if a lot of these quantities are matrices or numbers. I try to look back, but I still can't seem to figure all of these out. In the Yukawa interaction, for example, the fermion propagator (in momentum space) is a matrix right? While the boson propagator is simply a number?

    Also, I see a lot of expressions like [itex]\overline{u}u[/itex]. Are these numbers or outer product matrices? I am really confused on these. If that's a number, then would [itex]u\overline{u}[/itex] be a matrix? u is the dirac spinor.

    Is the big M a matrix or number? (In spinor space)
     
  2. jcsd
  3. Feb 10, 2012 #2
    Just a quick answer;

    This depends what kind of particles you are trying to describe. For the dirac spinor indeed you are right. Fermions ( spin half guys ) need the SU(2) representation ie the pauli matrices.

    The sum over u\bar{u} will give you a 4x4 matrix. Such as \slash p + m with some normalization.

    What you should do is figure out what u, v are from looking at the dirac equation. Then you can see their Tensor ranks and figure it out from there.

    Hope this helps a bit.
     
  4. Feb 10, 2012 #3

    Matterwave

    User Avatar
    Science Advisor
    Gold Member

    So it seems then that \bar{u} u is a number right? It depends on how the book wants to represent these right. If ubar is a row vector and u is a column vector, then row*column is a number whereas column*row is a matrix right? I do know what a Dirac spinor is, but sometimes I have trouble with the notation. For example, the kronecker delta is usually the identity matrix, but I see sometimes that like ostensibly what I would think would be a number is written as a kronecker delta...so I get confused...
     
  5. Feb 11, 2012 #4
    Yeah.

    To convert \bar{u} = conjugate of U ( ie from column to row or whatever ) times the gamma 0 matrix. So one is 'covariant' and one is 'contravariant'.

    It it should be a number like you say. Yeah I guess it can be a bit confusing.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook