The 7d cross product is almost certainly not useful for physics. It's not uniquely defined! e_1 x e_2 can equal any of the other basis vectors.
The cross product in physics is the wedge product. (which is also featured in 'geometric algebra', as another commenter mentioned, though I would contend that GA is the wrong approach). It produces area vectors from two vectors, and happily extends to any dimensions. For instance the electromagnetic field tensor is a bivector (F = d_u A_v - d_v A_u) which is basically a 4d curl of the 4-potential A, and has bivectorial components for each (uv) plane in spacetime.
I just said GA gives the most elegant formulation in another comment. What alternative do you suggest? Seeing that you mention the wedge product and bivectors, may you be thinking in something based in differential forms?
I'm fascinated by GA, although I have not had the chance to use it heavily yet. I am very interested if you know something better.
I think that the wedge product is fantastic and the geometric product is basically useless -- it has no geometric interpretation, complicates things massively, and conceals intuition wherever it is used. So I'd argue that 'exterior algebra' (which is the language used by differential forms, plus the exterior derivative) is basically the right framework for physics.
(The reason the geometric product is used in GA is that it's the extension of complex/quaternion multiplication to higher dimensions, and allows for inverting (non-zero-norm) multivectors . But I don't think that makes it worth including. It's not true a priori that inverting vectors is useful to physics, and it only really makes sense when they're being used as linear transformations -- in which case you can just use linear transformations explicitly.
The exterior algebra by itself is not enough for physics, since it doesn’t “know” about the inner product, and so it cannot tell when vectors or wedges should be perpendicular.
Given the inner product, you can define the “hodge star”, which restores this information. Another way of restoring the information is to use a modification of the exterior algebra, where rather than having the product uv = u wedge v, we deform the product by setting uv = u wedge v + <u, v>. This is called a Clifford algebra [1], and when the input vector space is R^3 and the input inner product is the Euclidean dot product, this Clifford algebras is exactly the “geometric algebra”.
I prefer to read about Clifford algebras, because the language is much less sensationalistic than many of the articles I’ve seen on geometric algebra. It also generalises to not just inner products, but any symmetric bilinear form, such as the Minkowski form on R^4 which is used in the study of relativity.
Not using the geometric product is sort of like working as a chef but not allowing yourself any knives with blades between 3 inches – 15 inches long. Only a paring knife, a hatchet, and a machete. Yes, it is possible to accomplish everything you wanted using a confusing mishmash of other abstractions (the raw numerical data turns out basically the same), but it’s really really inconvenient in comparison.
> it has no geometric interpretation
This is ridiculous. The geometric product is thoroughly “geometrical”.
A more accurate translation of your sentence is “I ajkjk have not personally thought enough about it to visualize a multivector”.
I have no problem visualizing multivectors, and have studied them extensively. What I have failed to find any intuition for is the general geometric product of two multivectors (without restricting to a particular grade). If you have one, I would love to hear it.
How would you define rotations using only the wedge product? It is also terribly useful being able to invert these rotations. You could make it work in some way, but it is not clear to me that the result will be simpler than a GA (or quaternion) based approach.
The GA approach is in there; I just think the 'geometric product' itself is not that fundamental.
A bivector x^y can be used as a generator of rotations, by contracting vectors with it: x.(x^y) = y, y.(x^y) = -x. If you define R(v) = v.(x^y), then general rotation is via the exponential map: e^(Rt)x = (cos t) x + (sin t) Rx.
(To be clear: exterior algebra as the algebra of the wedge product operation alone is not enough to do physics. The contraction operation is also necessary (and Hodge star, which is contraction with a particular volume element). But I don't think the geometric product as a standalone operation is useful compared to these components.)
I am having problems to follow your argument. I mean, I understand what you are saying, but not how you arrive to your conclusions.
You say the geometric product complicates things, but while you can derive any geometric algebra result simply defining this product (which does not have a difficult definition at all), in the approach you are suggesting you need to define not only the wedge product (which I do not find conceptually simpler than the geometric product) but also contractions. Both approaches are valid, and in certain contexts the differential forms one may be better, but I do not think you can claim it is simpler (when you need to define more concepts) or more fundamental (if you can define wedge product and contraction as a special case of the geometric product, I would say the geometric product is more fundamental).
Nevertheless, as I said I would be glad of learning some simpler tool than geometric algebra, and I am not an expert, so I will give you more details about my use case to see if you can come up with a good suggestion (it may happen that although differential forms are a better tool for physics in general, they are not for my particular problems).
I work in the field of crystallography. We have to constantly deal with crystal orientations (calculate misorientations between them, distributions of orientations in polycrystals, define properties and behavior in basis to these distributions, and so on). The field has traditionally used Euler angles (ugh!), but when new PhD students start developing computer models, obviously they run into problems, and then someone (usually me) suggests them to learn about quaternions.
The students gladly start looking into the subject, but reading definitions that talk about generalizations of complex numbers to four dimensions of stories about mathematical vandalism in bridges do not really help them to build an intuitive idea of how quaternions work, so I usually give them a brief and very informal lecture on geometric algebra (in a very simplified way, I avoid introducing the wedge product for example). After learning the geometric product, they see quaternions as a quite simple concept. We then do a few exercises, like rotating some vectors, calculating misorientations, defining symmetry operators,...
Now, you claim the geometric product complicates things, so I am intrigued. However, I do not see how the life of the students could be made any easier without it. I would have to explain what is the wedge product and contractions, rotations expressed as an exponential have the problem of trigonometric functions (and consequent rounding problems), and some simple operations like calculating the rotation between two vectors become problematic. The connection with quaternions is also much less obvious.
I have a very basic understanding of differential forms. I found fascinating the Discrete Differential Geometry introduction by Keenan Crane and have read some other introductory material, but that's about it. If you can point me in the right direction to learn how to best use it for my problems, I would appreciate it.
My stance is: the wedge product is the fantastic mathematical tool you are looking for, to make sense of rotations and generalizations of complex numbers to higher dimensions, and the "geometric product" (of Hestenes) is not that useful, and almost every time it is useful it's useful because it's got the wedge product as a special case. I think that the vector calculus that should be used in physics should consist of "wedge products and interior products" as the fundamental operations, and the 'geometric product' should probably not be mentioned.
That said, this is only my current judgment of the situation, which may change someday. I'm waiting to be convinced that the geometric product is so useful. I just find it so very cludgy when, for instance, GA texts are filled with use of 'grade projection" : take a geometric product with all its terms and then project off the grade you're interested in. It seems like the wrong tool is being used, and almost every time there's a multivector of 'mixed grade', it's regrettable.
(Indeed, wedge products are used everywhere already, in hiding -- integration, determinants, matrix minors, commutators, cross products, rotation matrices, projective geometry, grassmann algebra, ...)
Isn't the three-dimensional vector cross product also not uniquely defined, since the choice between the right-hand rule and left-hand rule is arbitrary?
It is certainly much more uniquely defined, since the direction of the result is determined. But yes, the cross product is defined relative to an orientation and changes coordinates accordingly.
That's one of many reasons why the wedge product is more natural. If I could have my way there would not be a single cross product in the entirety of physics.
The cross product in physics is the wedge product. (which is also featured in 'geometric algebra', as another commenter mentioned, though I would contend that GA is the wrong approach). It produces area vectors from two vectors, and happily extends to any dimensions. For instance the electromagnetic field tensor is a bivector (F = d_u A_v - d_v A_u) which is basically a 4d curl of the 4-potential A, and has bivectorial components for each (uv) plane in spacetime.