Geometric Algebra: Signs of electromagnetic field tensor components?
Here's a question that may look like an E&M question, but is really just a geometric algebra question. In particular, I've got a sign off by 1 somewhere I think and I wonder if somebody can spot it.
Doran/Lasenby (Geometric Algebra for Physicists) writes for the electromagnetic field:
[tex]
F = \mathbf{E} + I\mathbf{B}
[/tex]
[tex]
\mathbf{E} = \sum \sigma_i E_i
[/tex]
[tex]
\mathbf{B} = \sum \sigma_i B_i
[/tex]
[tex]
\sigma_i = \gamma_i \gamma_0
[/tex]
[tex]
I = \gamma_0 \gamma_1 \gamma_2 \gamma_3
[/tex]
Using a [itex](+,-,-,-)[/itex] metric [itex](c=1)[/itex].
Components of this tensor F are
[tex]
F^{\mu\nu} = (\gamma^\nu \wedge \gamma^\mu) \cdot F
[/tex]
They write this out in matrix form for comparision to other relativistic electrodynamics texts, but if I try calculating this I get different signs for the [itex]B_i[/itex] terms. I've written out
my steps for this. Can somebody spot an error in the algebra?
Calculation of the [itex]E[/itex] components is simpler. [itex]\mathbf{E}[/itex] explicitly is:
[tex]
\mathbf{E} = E_1 \gamma_{10} + E_2 \gamma_{20} + E_3 \gamma_{30}
[/tex]
Calculation of the [itex]\nu = 0[/itex] terms is:
[tex]
\begin{align*}
F^{\mu 0}
&= (\gamma^0 \wedge \gamma^\mu) \cdot F \\
&= (\gamma_0 \wedge (-\gamma_\mu)) \cdot (\sum E_i \gamma_{i0}) \\
&= -E_\mu (\gamma_0 \wedge \gamma_\mu) \cdot (\gamma_\mu \wedge \gamma_{0}) \\
&= -E_\mu \gamma_0 (\gamma_\mu \cdot (\gamma_\mu \wedge \gamma_{0})) \\
&= -E_\mu \gamma_0 ( \underbrace{\gamma_\mu \cdot \gamma_\mu}_{=-1} \gamma_{0} - \underbrace{\gamma_\mu \cdot \gamma_{0}}_{=0} \gamma_\mu) \\
&= E_\mu \gamma_0 \cdot \gamma_0 \\
&= E_\mu \\
\end{align*}
[/tex]
This is consistent with column zero of their matrix of tensor components:
[tex]
F^{\mu\nu} =
\begin{bmatrix}
0 & -E_1 & -E_2 & -E_3 \\
E_1 & 0 & -B_3 & B_2 \\
E_2 & B_3 & 0 & -B_1 \\
E_3 & -B_2 & B_1 & 0 \\
\end{bmatrix}
[/tex]
For the [itex]B[/itex] components, first I expanded out [itex]I\mathbf{B}[/itex] explicitly:
[tex]
\begin{align*}
I\mathbf{B}
&= \sum \gamma_0 \gamma_1 \gamma_2 \gamma_3 \gamma_i \gamma_0 B_i \\
&= \sum \gamma_1 \gamma_2 \gamma_3 \gamma_i B_i \\
&= B_1 \gamma_{23} + B_2 \gamma_{31} + B_3 \gamma_{12}
\end{align*}
[/tex]
This calculation is easier for just one pair of index, and for example, calcuating the [itex]12[/itex] component I get:
[tex]
\begin{align*}
F^{12}
&= (\gamma^2 \wedge \gamma^1) \cdot ( \gamma_1 \wedge \gamma_2 ) B_3 \\
&= \gamma^2 \cdot (\gamma^1) \cdot ( \gamma_1 \wedge \gamma_2 )) B_3 \\
&= \gamma^2 \cdot ( \gamma^1 \cdot \gamma_1 \gamma_2 - \gamma^1 \cdot \gamma_2 \gamma_1 ) B_3 \\
&= \gamma^2 \cdot ( -\gamma_1 \cdot \gamma_1 \gamma_2 ) B_3 \\
&= \gamma^2 \cdot \gamma_2 B_3 \\
&= - \gamma_2 \cdot \gamma_2 B_3 \\
&= - (-1) B_3 \\
&= B_3 \\
\end{align*}
[/tex]
Observe that the sign is opposite for this compared to what's in the matrix above. I don't see a mistake in my calculation, but this isn't listed in the errata even after two editions
so I'm assuming I have one hiding in there somewhere.