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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?

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  • $\begingroup$ A sum of blades of all grades could be used to represent a flag. But I'm not sure if this is meaningful; it might as well be a list of blades instead of a single multivector. The algebra's multiplication ought to be used in some essential way. $\endgroup$
    – mr_e_man
    Feb 1, 2023 at 6:00
  • $\begingroup$ We usually don't care if the metric is scaled by any non-zero number. For example, Euclidean space could have a positive-definite metric, or negative-definite, and the geometry is the same. Likewise, spacetime can be modelled as either $\mathbb R^{3,1}$ or $\mathbb R^{1,3}$. But it does change the algebra; e.g. that of $\mathbb R^{1,0}$ has zero-divisors ($(1+e_1)(1-e_1)=0$), while that of $\mathbb R^{0,1}$ has no zero-divisors (it's isomorphic to $\mathbb C$ which is a field). Yet the even subalgebra is not changed when the metric is scaled. This suggests that we shouldn't mix grades freely. $\endgroup$
    – mr_e_man
    Feb 1, 2023 at 17:00
  • $\begingroup$ Apparently even and odd grades are mixed in quantum mechanics. See Gauge Gravity and Electroweak Theory (around equation 46). $\endgroup$
    – mr_e_man
    Apr 11, 2023 at 5:04

3 Answers 3

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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.

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    $\begingroup$ But when this is expressed in terms of $\mathbb R^{3,1}$ or $\mathbb R^{1,3}$, everything is either purely even or purely odd, as far as I can see. $\endgroup$
    – mr_e_man
    Feb 1, 2023 at 5:39
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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

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Eek... not a fan of the other two responses, both of which work from a starting point of viewing the E field as a one-vector, rather than as as the timelike bivectors of the maxwell 2 form F. I am utterly baffled where that convention has found its popularity; but it does 'work' so you might still regard it as an example of a mixed grade object. Though I think its a poor example when those mixed grades are not essential but everything about the maxwell equation is looking much simpler when viewed as a single grade.

To answer your question; the even subalgebra of STA contains quadvector components as well; and you will encounter those when modelling leptons as geometric_vector_derivative(phi)=0, with phi from the even grade subalgebra.

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