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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

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    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.
     
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