Are there any geometrically meaningful/useful mixed-grade objects in geometric algebra other than rotors?

Question

While reading about geometric algebra, I have seen variables that are meant to represent blades, and variables that are meant to represent rotors, i.e. multivectors with a scalar and bivector component. But I have not seen any applications where a variable represents a mixed-grade object that is not the sum of a scalar and bivector. Are there examples of such objects being geometrically meaningful or useful?

Answer 1

As mentioned in @kieranor's answer, electromagnetism provides many examples of multivectors that have more structure than 0,2 multivectors that can represent complex numbers. Here are a few specific examples from electromagnetism in it's $\mathbb{R}^3$ representation

1. Maxwell's equation: $ \left( { \boldsymbol{\nabla} + \frac{1}{{c}} \partial_t} \right) F = J $, where $ F = \mathbf{E} + I c \mathbf{B} $ is the electromagnetic field (vector plus bivector), and $ J = \eta \left( { c \rho - \mathbf{J} } \right) + I \left( { c \rho_\textrm{m} - \mathbf{M} } \right) $ is the current density multivector. In the latter magnetic sources $ \rho_\textrm{m}, \mathbf{M} $ are included for antenna theory applications, but can be dropped for conventional electromagnetism. Without the magnetic sources the multivector current density has scalar and vector components. The magnetic sources add bivector and pseudoscalar terms.
2. The Green's function for the spacetime gradient $ \boldsymbol{\nabla} + (1/c) \partial_t $ (i.e. Green's function for Maxwell's equation for infinite boundary value conditions) satisfies $$\left( { \boldsymbol{\nabla} + (1/c) \partial_t } \right) G(\mathbf{x} - \mathbf{x}', t - t') = \delta(\mathbf{x} - \mathbf{x}') \delta(t - t'),$$ and has the value $$G(\mathbf{x} - \mathbf{x}', t - t')=\frac{1}{{4\pi}} \left( {- \frac{\hat{\mathbf{r}}}{r^2} \frac{\partial {}}{\partial {r}}+ \frac{\hat{\mathbf{r}}}{r}+ \frac{1}{{c r}} \partial_t} \right)\delta( -r/c + t - t' ),$$ where $ \mathbf{r} = \mathbf{x} - \mathbf{x}', r = \left\lVert {\mathbf{r}} \right\rVert $ and $ \hat{\mathbf{r}} = \mathbf{r}/r $. This Green's function is a multivector with scalar and vector components.
3. Plane wave solutions to Maxwell's equation have multivector factors like $ 1 + \hat{\mathbf{k}} $ that include scalar and vector components. Example: $$F(\mathbf{x}, t)=\text{Real} \left( {\left( { 1 + \hat{\mathbf{k}} } \right)\mathbf{E}\,e^{-j \mathbf{k} \cdot \mathbf{x} + j \omega t}} \right),$$ where $ \left\lVert {\mathbf{k}} \right\rVert = \omega/c $, $ \hat{\mathbf{k}} = \mathbf{k}/\left\lVert {\mathbf{k}} \right\rVert $ is the unit vector pointing along the propagation direction, and $ \mathbf{E} $ is any complex-valued vector variable, such that $ \mathbf{E} \cdot \mathbf{k} = 0 $.

   It is common to find scalar+vector factors of this form in field solutions. For example the field for an infinite line charge has the form $$F \propto \hat{\boldsymbol{\rho}} \left( { 1 - \mathbf{v}/c} \right).$$ Many of the solutions that can be found analytically have a multivector $ 1 - \mathbf{v}/c $ factor like this (circular line charge, ...).

   Another example of such multivector factors can be found in a representation of plane, circular, and elliptically polarized field solutions of the form: $$F = \left( { 1 + \mathbf{e}_3 } \right) \mathbf{e}_1 e^{i\psi} f(\phi).$$ Here the pseudoscalar of the transverse plane $ i = \mathbf{e}_1 \mathbf{e}_2 $, has been used as the imaginary, and $ f(\phi) $ is a complex valued function with respect to such an imaginary representation.

4. The statics solution to Maxwell's equation selects grades 1 and 2 from a multivector product: $$F(\mathbf{x})= \frac{1}{{4\pi}} \int_V dV' \frac{{\left\langle{{(\mathbf{x} - \mathbf{x}') J(\mathbf{x}')}}\right\rangle}_{{1,2}}}{\left\lVert {\mathbf{x} - \mathbf{x}'} \right\rVert^3} + F_0,$$ where $ F_0 $ is any function for which $ \boldsymbol{\nabla} F_0 = 0 $. This solution incorporates both Coloumb's law and the Biot-Savart law, and follows from the Green's function given above.
5. The energy momentum tensor (conventionally written as $T^{\mu\nu}$) is a multivector with scalar and vector components $$T(a) = \frac{1}{{2}} \epsilon F a F^\dagger,$$ where $ a $ a multivector parameter with scalar and vector components.
6. The electromagnetic field can be written in terms of a multivector potential $ A $ as follows $$ F = {\left\langle{{\left( { \boldsymbol{\nabla} -(1/c) \partial_t } \right) A}}\right\rangle}_{{1,2}},$$ where $$ A = - \phi + c \mathbf{A} + \eta I \left( { -\phi_m + c \mathbf{F} } \right).$$ Here, as before, the magnetic sources $ \phi_m $, and $ \mathbf{F} $ are for antenna theory applications, and can be dropped for conventional electromagnetism. This is a very compact representation of the fields, but can be unpacked to yield the usual: $$\begin{aligned}\mathbf{E} &= - \boldsymbol{\nabla} \phi - \frac{\partial {\mathbf{A}}}{\partial {t}} - \frac{1}{{\epsilon}} \boldsymbol{\nabla} \times \mathbf{F} \\ \mathbf{H} &= - \boldsymbol{\nabla} \phi_\textrm{m} - \frac{\partial {\mathbf{F}}}{\partial {t}} + \frac{1}{{\mu}} \boldsymbol{\nabla} \times \mathbf{A}.\end{aligned}$$
7. Using the potential representation above, you can find various interesting (and compact) multivector field representations. For example, given a spherical potential $$ \mathbf{A} = \frac{e^{-j k r}}{r} \vec{A}( \theta, \phi ),$$ you can show that the far field ($r \gg 1 $) electromagnetic field has the form $$F = -j \omega \left( { 1 + \hat{\mathbf{r}} } \right) \left( { \hat{\mathbf{r}} \wedge \mathbf{A}} \right).$$

It's a bit of a cheat to give physics examples for a question that asked for geometrical examples. However, in many cases, there is geometry behind these examples, such as the directly encoding of the propagation direction and the transverse plane in various field solutions above.

Answer 2

In the physics of electromagnetic fields, Maxwell's equations, when expressed in Geometric Algebraic form yield a multivector field that has scalar, vector, bivector, and trivector components thus populating a 3-spatial dimensional multivector fully.

See this short but excellent exposition by author @alan-macdonald

https://www.youtube.com/watch?v=iv5G956UGfs