Is the following a matrix? (yes/no)

[bjnartowt]

Homework Statement

Is the following a matrix?

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

$$\begin{array}{c}<br /> ({\gamma _\mu }{a^\mu }){\gamma _\nu }{b^\nu } + {\gamma _\nu }{b^\nu }({\gamma _\mu }{a^\mu }) \equiv 2{a_\mu }{b^\mu } \\ <br /> ({\gamma _\mu }{a^\mu })({\gamma _\mu }{a^\mu }) \equiv {a_\mu }{a^\mu } \\ <br /> \end{array}$$

Homework Equations

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

The Attempt at a Solution

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

Maybe it isn't a matrix. The operator $$\partial _\mu$$ 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).

?

 

[Phrak]

\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.

 

[bjnartowt]

\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.

Oh dear...back to the drawing board...

 

[vela]

\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.

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

http://en.wikipedia.org/wiki/Gamma_matrices

 

[Phrak]

In which case the contraction yields the same type as the Dirac matrix.