How do units work for multivectors? (Clifford/Geometric algebra)

Question

Suppose we work with $\mathcal{G}(2)$. If the units of $e_{1}$ and $e_{2}$ are of length, it makes sense that the units of $e_{12}$ would be of area. But then what are the units of scalars? And what about multivectors, e.g. $a=5+3e_{1}-e_{2}+10e_{12}$? How can quantities of different units be added together? Are there some kind of "unit correcting" constants hiding somewhere?

Answer 1

"Unit" vectors i.e. vectors of length 1 do not have units. They are unitless quantities, so there is not problem with $e_1 + e_1e_2$ from this perspective. Typically the quantities we're interested in in geometric algebra are: (1) homogeneous, meaning they are $k$-vectors for some $k$, and (2) have definite units in the sense that we can write $X = xX_0$ where $x$ is some unitful scalar and $X_0$ is a dimensionless multivector. Under these constraints, every expression you can cook up has well-defined units.

There are also nonhomogeneous multivectors that are useful like rotors, but these are unitless in a sort of "fundamental" sense since they are exponentials of bivectors $e^B$, where of course $B$ has to be unitless.

That all being said, there's no fundamental reason we can't have e.g. $(1\,\mathrm m)e_1 + (2\,\mathrm m^2)e_1e_2$, but I would argue that $(1\,\mathrm m)e_1e_2 + (2\,\mathrm m^2)e_1e_3$ and by extension $(1\,\mathrm m)e_1e_2 + (2\,\mathrm m^2)e_3e_4$ make no sense.

Answer 2

But then what are the units of scalars?

Who says there are any? Well, let's pretend there is. The common theme here is that a grade $n$ blade could be used to represent an $n$ dimensional volume. So what do you use $0$-dimensional volumes for? I don't know, maybe like a point charge or something? It depends on what you're using the elements of the algebra to represent in whatever calculus you're carrying out.

This brings us to the next question

How can quantities of different units be added together?

I.e. what is the meaning of all this stuff in the algebra? If you can add stuff together it must make physical sense somehow, right? If there's an element in there, there must be some physical interpretation of it, right?

Well, I don't see a problem with the answer "For all elements? no: Why would there have to be?"

As far as I've seen, the utility of geometric algebras is the ability to encode operators and spaces they operate on in subsets of geometric algebras in useful ways. Finding these encodings is indeed where the magic lies.

Nobody said everything has to generalize to the whole algebra, but that doesn't stop people from trying.

My advice would be to get thoroughly acquainted with "What elements are useful for this application" before pondering the backwards problem "what applications will this element be useful for."