# Is the following a matrix? (yes/no)

1. Jul 3, 2010

### bjnartowt

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):

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

2. Relevant equations

$$({\gamma ^\mu }{\partial _\mu })({\gamma ^\nu }{A_\nu }) + ({\gamma ^\nu }{A_\nu })({\gamma ^\mu }{\partial _\mu }) =$$ ... 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, $$F^{\mu, \nu}$$, 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).

???

2. Jul 3, 2010

### 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.

3. Jul 4, 2010

### bjnartowt

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

4. Jul 4, 2010

### vela

Staff Emeritus
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

5. Jul 4, 2010

### Phrak

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