User:Jheald/sandbox/Bivector

Bivectors as antisymmetric tensors

A bivector B can be represented by an anti-symmetric matrix B_ij (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.

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#Geometric_algebra_formulations

$F=\mathbf {E} +ic\mathbf {B} =E^{k}\gamma _{k}\gamma _{0}-c(B^{1}\gamma _{2}\gamma _{3}+B^{2}\gamma _{3}\gamma _{1}+B^{3}\gamma _{1}\gamma _{2}),$

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

$\nabla F=\mu _{0}cJ$

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 e_i u_i as B_ij u_j loses the trivector component B ∧ e_i u_i

* 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 B_ij.