
Introduction to Clifford Algebra (2006) - beefman
https://www.av8n.com/physics/clifford-intro.htm
======
DreamScatter
Check my clifford algebra implementation

[https://grassmann.crucialflow.com](https://grassmann.crucialflow.com)

The Grassmann.jl package provides tools for doing computations based on multi-
linear algebra, differential geometry, and spin groups using the extended
tensor algebra known as Leibniz-Grassmann-Clifford-Hestenes geometric algebra.
Combinatorial products included are ∧, ∨, ⋅, *, ⋆, ', ~, d, ∂ (which are the
exterior, regressive, inner, and geometric products; along with the Hodge
star, adjoint, reversal, differential and boundary operators). The kernelized
operations are built up from composite sparse tensor products and Hodge
duality, with high dimensional support for up to 62 indices using staged
caching and precompilation. Code generation enables concise yet highly
extensible definitions. The DirectSum.jl multivector parametric type
polymorphism is based on tangent bundle vector spaces and conformal projective
geometry to make the dispatch highly extensible for many applications.
Additionally, the universal interoperability between different sub-algebras is
enabled by AbstractTensors.jl, on which the type system is built.

~~~
elcritch
Wow! I keep finding these really great Julia packages that implement grad
level math / science concepts. It's great! Unfortunately not sure what I'd do
with Clifford Algebra myself.

------
dang
Related from 2016:
[https://news.ycombinator.com/item?id=12938727](https://news.ycombinator.com/item?id=12938727)

2015:
[https://news.ycombinator.com/item?id=9746051](https://news.ycombinator.com/item?id=9746051)

Related a bit more generally:

2017
[https://news.ycombinator.com/item?id=15932739](https://news.ycombinator.com/item?id=15932739)

[https://news.ycombinator.com/item?id=14947065](https://news.ycombinator.com/item?id=14947065)

2016
[https://news.ycombinator.com/item?id=13239632](https://news.ycombinator.com/item?id=13239632)

------
oddthink
Does anyone have a good summary of how this relates to the differential
geometry world with its n-forms and n-vectors? For example, I'm used to
thinking of the wedge product as operating over n-forms and requiring a metric
(or volume element) transformation to work over vectors. Similarly, I don't
see any discussion of behaviors under coordinate transformations.

~~~
pdonis
_> Does anyone have a good summary of how this relates to the differential
geometry world with its n-forms and n-vectors?_

This might be helpful as a start:

[https://en.wikipedia.org/wiki/Exterior_algebra](https://en.wikipedia.org/wiki/Exterior_algebra)

 _> I don't see any discussion of behaviors under coordinate transformations._

That's because, as the article emphasizes, you don't need coordinates (or
basis vectors, which is the term used in the article) to work with geometric
objects. You _can_ use them, but you don't _need_ to.

Similarly, vectors, bivectors, tensors, etc. can all be defined without making
use of their behavior under coordinate transformations. Textbooks that
emphasize coordinates might make it appear otherwise, but that's not the case.

~~~
oddthink
So, like many things, this is both true and false. The object isn't the
coordinates, but you do need two kinds of things to make physics work. (Maybe
you don't for geometry, though, not sure.)

If you have a displacement vector, and a potential gradient, they can't be the
same "kind" of thing (i.e. both vectors), because their dot-product should be
preserved. If the slope gets half as long, it's twice as steep.

~~~
pdonis
_> you do need two kinds of things to make physics work_

I'm not sure what two kinds of things you are referring to, but it doesn't
seem like they are "geometric objects" and "coordinates". All I'm saying is
that you don't need coordinates; they are a convenience, not a necessity. I am
not saying you need only one kind of geometric object.

 _> If you have a displacement vector, and a potential gradient, they can't be
the same "kind" of thing_

Agreed. You need both vectors and covectors, or more generally "things with
upper indexes" and "things with lower indexes". But you don't need coordinates
to work with those things. The indexes do not have to represent components.
They can represent "slots" (at least that's what Misner, Thorne, and Wheeler
call them in their classic GR textbook), in which you can insert vectors (for
lower index slots) or covectors (for upper index slots) in order to obtain
other geometric objects (and ultimately numbers, which are what you compare
with actual measurements).

~~~
oddthink
Yeah, I have a physics / GR background, so MTW is the lens through which I see
all of this. That and Schutz's Geometrical Methods of Mathematical Physics.

I agree that we don't need coordinates. Things are things.

But what always got me about the geometric algebra stuff was that they used
bivectors for areas, which seems like the wrong thing. If you're integrating a
vector field over it, you want a 2-form, not a bivector. I suppose the
distinction doesn't matter as long as you're just in Euclidean space, but even
then, if you need to drop down to coordinates and do the actual integral,
you're still going to want to have changes of variables that work. That leaves
me with a kernel of doubt that they're doing the right thing.

------
peter_d_sherman
Excerpt:

 _" It is traditional to write down four Maxwell equations.

However, by using Clifford algebra, we can express the same meaning in just
one very compact, elegant equation."_

[https://www.av8n.com/physics/clifford-intro.htm#eq-max-
ga](https://www.av8n.com/physics/clifford-intro.htm#eq-max-ga)

------
aesthesia
I've read a number of introductions to Clifford algebras, and I'm always left
with the question of what the geometric product is supposed to mean. The wedge
product and dot product are easy to understand and have obvious
interpretations. But other than being a gadget from which you can extract
these other products, I don't see what the geometric product is for, or why it
should be the primary object of consideration.

~~~
edflsafoiewq
Despite the name the motivation for the geometric product is principally
algebraic, ie. it's useful for doing algebraic manipulation. It does not,
AFAIK, possess any geometric meaning outside of special cases.

(It's "geometric" in the sense it doesn't depend on a choice of basis I
guess.)

~~~
DreamScatter
Actually, it has a lot to do with geometry, in fact all of geometric algebra
can be constructed from the geometric product, which serves as a foundation
for not just algebra but also geometry. It encodes various geometric
properties.

~~~
edflsafoiewq
Yes, it "encodes" them and they can be "constructed" from it, but it does not
have a direct geometric interpretation in the way, say, the cross product
does.

------
vtomole
Clifford algebra is a big part of quantum computation. The Clifford gates
([https://en.wikipedia.org/wiki/Clifford_gates](https://en.wikipedia.org/wiki/Clifford_gates))
along with Magic state distillation
([https://en.wikipedia.org/wiki/Magic_state_distillation](https://en.wikipedia.org/wiki/Magic_state_distillation))
can be used to perform fault-tolerant quantum computation.

Edit: Clifford groups are not the same as Clifford algebras. I was wrong!

~~~
knzhou
That’s the Clifford _group_ , though. Is it actually related to the Clifford
algebra, beyond being named after the same guy?

~~~
vtomole
A group is an algebraic structure. Please reference
[https://en.wikipedia.org/wiki/Clifford_algebra#Clifford_grou...](https://en.wikipedia.org/wiki/Clifford_algebra#Clifford_group)

~~~
knzhou
Yes, I’m aware of that. But _an algebra_ is a very different thing from “an
algebraic structure”.

~~~
lisper
I'm not sure "very different" is a fair characterization. The two are closely
related:

[https://en.wikipedia.org/wiki/Algebraic_structure](https://en.wikipedia.org/wiki/Algebraic_structure)

An algebraic structure on a set A (called the underlying set, carrier set or
domain) is a collection of operations on A of finite arity, together with a
finite set of identities, called axioms of the structure that these operations
must satisfy. In the context of universal algebra, the set A with this
structure is called an algebra,[1] while, in other contexts, it is (somewhat
ambiguously) called an algebraic structure, the term algebra being reserved
for specific algebraic structures that are vector spaces over a field or
modules over a commutative ring.

Examples of algebraic structures include groups, rings, fields, and lattices.

~~~
klodolph
I’m going to agree with knzhou here. Unfortunately, the terminology in
mathematics can be misleading.

It is important to note context, and note the part where the article you
quoted uses the works “ambiguously”, because the word “algebra” has more than
one meaning.

In this case, a group is not an algebra (because we are talking in the context
of algebras over a field or ring, not universal algebras).

It is unfortunate that the words are defined this way, but you have to deal
with it. A “universal algebra” is a very different concept from an “algebra”
(over a ring or field) even though one is an example of the other.

It’s like saying that “book” is a very different concept from “The Great
Gatsby”.

~~~
monoideism
Huh? I thought a a group, ring, etc. was precisely an example of a "universal
algebra". You seem to contradict yourself, at times agreeing with this
statement ("even though one is an example of the other"), at times not ("a
group is not an algebra").

Edit: Wolfram Mathworld agrees with me: "Universal algebra studies common
properties of all algebraic structures, including groups, rings, fields,
lattices, etc."
[http://mathworld.wolfram.com/UniversalAlgebra.html](http://mathworld.wolfram.com/UniversalAlgebra.html)

~~~
gjm11
The word "algebra" means a number of different things.

It's the name of a whole field in mathematics, which covers objects like
groups, rings, fields, and so forth. The sort of things defined, very
handwavily, in terms of operations you can do with their elements and the
equations they satisfy.

It's the name of a rather specific kind of mathematical structure: roughly, a
vector space together with a way of multiplying its elements. (Motivating
example: n-by-n matrices.)

It's the name (but usually with some qualifiers to make it clearer what you
mean) for a broad range of mathematical structures, of which the one in the
previous paragraph is a special case. See e.g.
[https://en.wikipedia.org/wiki/F-algebra](https://en.wikipedia.org/wiki/F-algebra).

Groups _are_ among the things studied in the field of algebra. Groups _aren
't_ algebras in the second, specific, sense. They _are_ F-algebras. Most
mathematicians, most of the time, would not call groups "algebras" without
some qualifier like that F- prefix.

The term "Clifford algebra" also means some different things.

 _A Clifford algebra_ is a particular sort of algebra-in-the-second-sense. The
field called _Clifford algebra_ is the study of those things. These days
people more often say "geometric algebra" rather than "Clifford algebra" for
that meaning.

There are mathematical objects called Clifford groups. They are not at all the
same thing as Clifford algebras, and you can study Clifford algebras in some
depth without paying any attention to the Clifford groups. But they _are_
closely related to the Clifford algebras.

Both Clifford groups and Clifford algebras have applications in quantum
physics and, more specifically, in quantum computing. But so far as I know the
ways in which you use them in quantum computing has very little to do with the
ways in which you use Clifford algebras for doing geometry. It is quite common
in mathematics for the same (or equivalent) objects and structures to turn up
in multiple places, in unrelated-looking ways. Sometimes this gives rise to
deep connections between different fields; sometimes it's just a coincidence.
I don't know enough about either quantum computation or geometric algebra to
know which of those is going on here, but my intuition leans toward
"coincidence".

So. vtomole's original comment was kinda-right and kinda-wrong: yes, there is
a connection between Clifford algebras and quantum computation, but it doesn't
have much to do with the stuff discussed, e.g., at the far end of the top-
level link here. knzhou's question was a good one, pointing out that the two
topics are quite separate. vtomole's reply "A group is an algebraic structure
..." didn't make any _untrue statements_ but did miss the point; the fact that
a group is an algebraic structure doesn't mean that something called "the X
group" necessarily has anything to do with something called "the X algebra"
\-- though it happens that in this case there is a connection. (vtomole
clearly got the point soon after, as seen from their subsequent replies.)
knzhou was correct to point out that vtomole's reply missed the point. lisper,
again, didn't say anything untrue but I think he missed the point. klodolph's
comment about algebras versus universal algebras versus algebra was spot-on
and the only reason why I went into more detail above is that it was
apparently too brief to be clearly understood. monoideism is _right_ that
(e.g.) a group is a universal algebra, _wrong_ to say that klodolph is self-
contradictory, and I think _missing the point_ that "algebra" is used with
different meanings on different occasions, and in the phrase "Clifford
algebra" the specific meaning in question is _not_ "universal algebra". (Even
though a Clifford algebra is, also, an example of a universal algebra.)

The fact that a group _is_ a universal algebra doesn't at all licence any sort
of blurring of the distinction between Clifford groups and Clifford algebras.
The meaning of "algebra" in "Clifford algebra" is _not_ "universal algebra" or
"F-algebra", it is "vector space with multiplication", and a group simply _isn
't_ one of those (well, _some_ groups are, but e.g. the Clifford groups are
not).

------
bialpio
Something that inhibits my understanding of a lot of the things beyond section
2.6: according to the article, equation 4b is supposed to also work for scalar
^ vector and scalar ^ scalar, but both those cases seem to contradict defn 20.
- according to 4b, scalar ^ scalar will always be 0 (a^b = (ab - ba)/2 = 0,
since scalar multiplication commutes), same for scalar ^ vector (a^V = (aV -
Va)/2 = 0, since scalar times vector also commutes). Definition 20 on the
other hand makes sense (scalar ^ scalar is a regular multiplication, same for
scalar ^ vector where it's just scaling). Am I missing something? Why does 4b
try to generalize to grades <= 1 in a way that contradicts the other
definition (and is counter-intuitive to me)? It all makes sense if 4b only
applies to grades = 1.

------
platz
> The traditional form of the Maxwell equations is not manifestly invariant
> with respect to special relativity, because it involves a particular
> observer’s time and space coordinates. However, we believe the underlying
> physical laws are relativistically invariant.

so are maxwell's laws actually relativistically invariant or not?

~~~
g82918
There is a form of them which is invariant. You just have to write them
correctly. Mathematics isn't as amazing as it sounds, there is a some
ambiguity in how symbols are defined. Is your gradient with respect to
rectilinear coordinates, or more general? Etc.

------
playing_colours
Is interest in GA just a local fashion or there are objective reasons in
recent revival of interest?

~~~
Koshkin
I think it's both. Still, in the manner it's happening, the surging abundance
of tutorials on Geometric Algebra somehow feels worrisome. This looks all too
similar to the ever growing number of guides on what are monads and how to use
them in programming. For most people - kind of makes sense, sort of
interesting, sometimes inspiring, practically useless...

~~~
virgil_disgr4ce
Why does it worry you?

~~~
Koshkin
This is almost like, for instance, why are there so many popular accounts on
quantum mechanics (and new ones keep popping up every so often). This makes me
think they are all wrong somehow (and to a significant degree they indeed are
- which is quite understandable in this case, as QM is a tricky subject);
looks like that's what the author of the next one should think, too.

------
adamnemecek
If this interests you, you should check out the bivector community
[https://bivector.net/](https://bivector.net/).

Join the discord [https://discord.gg/vGY6pPk](https://discord.gg/vGY6pPk).

Check out a demo [https://observablehq.com/@enkimute/animated-
orbits](https://observablehq.com/@enkimute/animated-orbits)

Also at the end of February, there is geometric algebra event in Belgium.
[https://bivector.net/game2020.html](https://bivector.net/game2020.html) All
the big names in the field will be there.

------
msla
Here's the page one up in the directory structure:

[https://www.av8n.com/physics/](https://www.av8n.com/physics/)

It's got a lot of very interesting math and physics information.

~~~
OldGuyInTheClub
He's the guy that built the shark for 'Jaws' while an undergraduate. Very very
capable, to say the least. Saw him in the halls during my postdoc but never
had the occasion to talk with him.

~~~
msla
> He's the guy that built the shark for 'Jaws' while an undergraduate.

Fascinating. That thing famously never worked well, and the movie was better
for it.

~~~
OldGuyInTheClub
Denker describes the development as a case study in "Experimental Techniques
in Condensed Matter Physics at Low Temperatures." His chapter is on
electromagnetic shielding and grounding - important for animated sharks and
microKelvin measurements. Spielberg is indirectly referenced as the director
getting impatient with delays caused by all sorts of hidden electrical
problems.

The book is a compendium of tips and techniques from graduate students in
Cornell's famous low temperature physics lab. Although published in 1988, it
is sufficiently general to be valuable today.

[https://books.google.com/books?id=8tJMDwAAQBAJ&pg=PP16&lpg=P...](https://books.google.com/books?id=8tJMDwAAQBAJ&pg=PP16&lpg=PP16&dq=john+denker+shark+jaws&source=bl&ots=NSchIogj3k&sig=ACfU3U0oUPYXG-
oi68UfBfFBoA8shgQR_A&hl=en&sa=X&ved=2ahUKEwjHppSP59HnAhUOIjQIHTdVBq8Q6AEwCnoECAsQAQ#v=onepage&q=john%20denker%20shark%20jaws&f=false)

------
dktoao
Kinda makes me want to go back to college (I'm an EE) just to re-learn all the
stuff I remember being so mind bending with this elegant new framework. Also,
just so I can be THAT guy who always argues with the professor. Anyone know of
any PhD openings that could use a maverick like me? :) (/s kinda)

~~~
DreamScatter
going to college won't really help you, I quit college so I can abandon
traditional math to completely devote myself to geometric algebra based math,
here is my algebra implementation for example:

[https://grassmann.crucialflow.com](https://grassmann.crucialflow.com)

it isn't taught at universities, it is self taught.. at the university level
you are going to be artificially held back more than you would by studying it
independently

~~~
senderista
This cult-like attitude about geometric algebra is something I have never seen
in any other (legitimate) field of math. Not sure what to attribute it to,
maybe Hestenes being a bit of a crackpot?

~~~
DreamScatter
Actually, it's not a cult-like attitude but is more of an individualist
attitude. In fact, universities are the ones with the cult-like attitude.. so
your statement makes no common sense at all. Geometric algebra is just as
legitimate as any branch of mathematics.. Hestenes is not a crackpot, he is
responsible for continuing the legacy of geometric algebra and passing it onto
several other researchers.

------
Squithrilve
Does anyone know any other good resources (esp. books) on this subject?
(Clifford/geometric algebra)

~~~
SAI_Peregrinus
In addition to gibsonf1's recommendations (not books), the following books are
good. The first 2 are more pure introductions to the math, the third is
applying it to physics, the fourth to CS.

Linear and Geometric Algebra, by Alan Macdonald:
[http://www.faculty.luther.edu/~macdonal/laga/](http://www.faculty.luther.edu/~macdonal/laga/)

Vector and Geometric Calculus, by Alan Macdonald:
[http://www.faculty.luther.edu/~macdonal/vagc/index.html](http://www.faculty.luther.edu/~macdonal/vagc/index.html)

Application to physics: New Foundations for Classical Mechanics by David
Hestenes:
[https://books.google.com/books/about/New_Foundations_for_Cla...](https://books.google.com/books/about/New_Foundations_for_Classical_Mechanics.html?id=AlvTCEzSI5wC)

Geometric Algebra For Computer Science by Dorst, Fontijne, and Mann:
[http://www.geometricalgebra.net/index.html](http://www.geometricalgebra.net/index.html)

------
kragen
> _It is traditional to write down four Maxwell equations. However, by using
> Clifford algebra, we can express the same meaning in just one very compact,
> elegant equation:_

> ∇ _F_ = _J_ /( _c є_ ₀)

Holy. Shit. Is this for real?

~~~
senderista
Yawn. The differential forms version is almost as compact (2 equations) and
far more interpretable (if you understand connections and curvature).

~~~
kragen
Thanks! I don't know anything about the differential-forms version.

------
vmchale
Neat, thank you!

