1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Is the following a matrix? (yes/no)

  1. Jul 3, 2010 #1
    1. The problem statement, all variables and given/known data

    Is the following a matrix?

    Given: these identities (ordinarily in "Feynman slash notation": not sure how to do slashes in LaTeX):

    [tex]\begin{array}{c}
    ({\gamma _\mu }{a^\mu }){\gamma _\nu }{b^\nu } + {\gamma _\nu }{b^\nu }({\gamma _\mu }{a^\mu }) \equiv 2{a_\mu }{b^\mu } \\
    ({\gamma _\mu }{a^\mu })({\gamma _\mu }{a^\mu }) \equiv {a_\mu }{a^\mu } \\
    \end{array}[/tex]

    2. Relevant equations

    [tex]({\gamma ^\mu }{\partial _\mu })({\gamma ^\nu }{A_\nu }) + ({\gamma ^\nu }{A_\nu })({\gamma ^\mu }{\partial _\mu }) = [/tex] ... a matrix or a vector?

    3. The attempt at a solution

    Maybe it is a matrix. My final answer needs a matrix answer (specifically: I need the field-strength tensor, [tex] F^{\mu, \nu} [/tex], to pop up eventually.

    Maybe it isn't a matrix. The operator [tex] \partial _\mu [/tex] is a differential operator, and A is the four-vector-potential. that suggests I should wind up with a differentiated version of the field-strenght tensor, which would be awfully-boring in a certain gauge I forget the name of (it'd be zero/divergenceless, a la Griffiths Intro Elementary Particles, p. 239-240).

    ???
     
  2. jcsd
  3. Jul 3, 2010 #2
    \partial_{mu} is a one-form. Contracting a one-form with a vector yields a scalar. In fact, all your terms are one-form vector contractions.
     
  4. Jul 4, 2010 #3
    Oh dear...back to the drawing board...
     
  5. Jul 4, 2010 #4

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    But [itex]\gamma^\mu[/itex] are the Dirac matrices, so the contraction [itex]\gamma^\mu A_\mu[/itex] is actually a linear combination of matrices.

    http://en.wikipedia.org/wiki/Gamma_matrices
     
  6. Jul 4, 2010 #5
    In which case the contraction yields the same type as the Dirac matrix.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook