User:Jheald/sandbox/Bivector

Bivectors as antisymmetric tensors
A bivector B can be represented by an anti-symmetric matrix Bij (in fact, an antisymmetric tensor) when used in a contraction with two vectors to produce a scalar. This is often how bivectors are introduced or defined, especially in older works.
 * $$u_i \mathbf{e}_i \cdot \mathbf{B} \cdot \mathbf{e}_j v_j = u_i \, \langle \mathbf{e}_i \mathbf{B} \mathbf{e}_j \rangle_0 \, v_j = u_i B_{ij} v_j$$

Anti-symmetry is established by noting that, as a scalar, $$\scriptstyle{\langle \mathbf{e}_i \mathbf{B} \mathbf{e}_j \rangle_0}$$ is invariant under the geometric algebra reversal operation, so
 * $$ B_{ij} = \langle \mathbf{e}_j \tilde{\mathbf{B}} \mathbf{e}_i \rangle_0$$

But for a bivector $$\scriptstyle{\tilde{\mathbf{B}} = -\mathbf{B}}$$, and therefore
 * $$ B_{ij} = - \langle \mathbf{e}_j \mathbf{B} \mathbf{e}_i \rangle_0 = -B_{ji}$$

The tensor property follows from the fact that the map $$\scriptstyle{ T: V^* \times V \rightarrow \mathbf{R} }$$ from two vectors to a scalar defined in this way is co-ordinate free, transforming appropriately and continuing to hold under rotations and other transformations.

Electromagnetic tensor

 * Is it worth reflecting the explicit form given in the Electromagnetic tensor article to make the connection more direct:


 * $$F^{\mu\nu} = \begin{bmatrix}

0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{bmatrix} \; \leftrightarrow \; \mathbf{F} $$


 * (Q: raised indices or lowered indices?
 * Should it not be one up and one down, with reversal turning it into the other choice?)

where
 * $$ \mathbf{F} = \frac{1}{c}\overline{E}\mathbf{e}_4 + \overline{B}\mathbf{e}_{123},$$
 * Note (1): 1/c corresponds to the convention in the matrix above, whereas E + i c B from the mathematical descriptions article (cf below) does not
 * Note (2): This formulation has the potential to be a bit confusing at first sight, because the untrained eye sees e4 and e123 it may not think "bivector". This is a bivector formula, because $\scriptstyle{\overline{E}}$ and $\scriptstyle{\overline{B}}$ are vectors, so multiplying them by e4 and e123 gives bivectors, but that may need to be emphasised.|undefined
 * Note (3): It might be worth changing to e0 rather than e4 for the time-like unit vector, if we were going to want to make the closest connection with the matrix up above.


 * c.f. from Mathematical_descriptions_of_the_electromagnetic_field

$$F = \mathbf{E} + ic\mathbf{B} = E^k\gamma_k\gamma_0 -c(B^1\gamma_2\gamma_3 + B^2\gamma_3\gamma_1 + B^3\gamma_1\gamma_2),$$
 * Q1. why the raised indices ? -- I guess perhaps to show summation, but is this really standard notation in GA -- relation to tensor notation too confusing?
 * Q2. apparent difference in sign - minus for the magnetic field part, rather than plus
 * Q3. this is c times the other F -- which is more standard/appropriate ?


 * The different conventions give rise to slightly different forms of Maxwell's equations.
 * From here:

$$ \partial\mathbf{F} = \mathbf{J}.$$
 * whereas there

$$ \nabla F = \mu_0 c J $$

Limitations of representing bivectors as anti-symmetic matrices

 * Is it worth adding a section on the limitations of representing bivectors as anti-symmetic matrices?
 * eg:
 * representing B ei ui as Bij uj loses the trivector component B ∧ ei ui
 * and it doesn't let you represent B u B-1 (even though we do have a section on the exponentiation of anti-symmetric matrices)
 * Because of its group structure, there is a faithful matrix representation of the elements of a GA that reproduces the algebra; but in this representation, to get over issues like the above, vectors (in the sense of spatial vectors) also are represented by matrices -- eg the Pauli matrices, or the Gamma matrices -- and no longer by a column of numbers.
 * In that sort of representation, bivectors (are still represented by anti-symmetric matrices??), but not of the same form Bij.