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Introduction to Clifford Algebra (2006)
- vtomole 7y agoClifford 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 7y agoThat’s the Clifford group, though. Is it actually related to the Clifford algebra, beyond being named after the same guy?
- vtomole 7y agoA group is an algebraic structure. Please reference https://en.wikipedia.org/wiki/Clifford_algebra#Clifford_group https://en.wikipedia.org/wiki/Clifford_algebra#Clifford_grou...
- knzhou 7y agoYes, I’m aware of that. But an algebra is a very different thing from “an algebraic structure”.
- vtomole 7y agooops yeah my apologies!
- lisper 7y agoI'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 7y agoI’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 7y agoHuh? 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 7y agoThe 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).
- Koshkin 7y agoNo, it's not related.
- vtomole 7y agoYou are right. My confusion. Sorry.
- dktoao 7y agoKinda 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)
- Random_ernest 7y agoI was in this very situation, thinking I want more than "just" EE. Applied for a PhD position in a branch where mostly mathematicians work, could not be happier with the decision. Do it. Scratch that itch while you still can.
- hackernewsname 7y agoWhat steps did you take to go from industry back into academia?
- Random_ernest 7y agoSorry if my comment implied that I had left academia. I simply decided to not go into industry as an electrical engineer but pursue a PhD in a non-related, more math-heavy subject.
- Koshkin 7y agoYou might enjoy http://peeterjoot.com/writing/geometric-algebra-for-electrical-engineers/ http://peeterjoot.com/writing/geometric-algebra-for-electric... https://www.academia.edu/21674685/Geometric_Algebra_for_Electrical_and_Electronic_Engineers https://www.academia.edu/21674685/Geometric_Algebra_for_Elec...
- DreamScatter 7y agogoing 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
- msla 7y agoHere'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.
- m4r35n357 7y agoLooks really impressive. For those interested in this sort of thing, another huge and varied collection of mathematical/physics articles is located at https://www.mathpages.com/ https://www.mathpages.com/
- OldGuyInTheClub 7y agoHe'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 7y ago> 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 7y agoDenker 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=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 https://books.google.com/books?id=8tJMDwAAQBAJ&pg=PP16&lpg=P...
- playing_colours 7y agoIs interest in GA just a local fashion or there are objective reasons in recent revival of interest?
- m4r35n357 7y agoI never got the hang of direction of cross products, still don't know which hand rules to use for eg motors & generators (well I think I knew once but as never at ease). This looks like what I wished I had learned instead!
- dktoao 7y agoRead a little bit of the introduction. If you are familiar with using complex numbers and vector cross products, you will see the advantage pretty quickly
- Koshkin 7y agoI 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 7y agoWhy does it worry you?
- Koshkin 7y agoThis 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.
- monoideism 7y ago
- Squithrilve 7y agoDoes anyone know any other good resources (esp. books) on this subject? (Clifford/geometric algebra)
- gibsonf1 7y agoI highly recommend https://bivector.net/ https://bivector.net/ In the docs section, the Geometric Algebra Primer by Jaap Suter is excellent. http://www.jaapsuter.com/geometric-algebra.pdf http://www.jaapsuter.com/geometric-algebra.pdf
- lazycrazyowl 7y agoClifford Algebras: An Introduction Author(s): D. J. H. Garling Series: London Mathematical Society Student Texts 78 Publisher: Cambridge University Press, Year: 2011 ISBN: 1107096383
- SAI_Peregrinus 7y agoIn 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_Classical_Mechanics.html?id=AlvTCEzSI5wC https://books.google.com/books/about/New_Foundations_for_Cla... Geometric Algebra For Computer Science by Dorst, Fontijne, and Mann: http://www.geometricalgebra.net/index.html http://www.geometricalgebra.net/index.html
- dangirsh 7y agoI'm currently going through Geometric Algebra for Physicists, and find it exceedingly clear: https://g.co/kgs/ciKXVS https://g.co/kgs/ciKXVS
- chombier 7y agoA survey of geometric algebra and geometric calculus, A Macdonald - Advances in Applied Clifford Algebras, 2017 - Springer edit: link http://www.astro.umd.edu/~jph/GAandGC.pdf http://www.astro.umd.edu/~jph/GAandGC.pdf
- dang 7y agoRelated 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
- adamnemecek 7y agoIf 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.
- vmchale 7y agoNeat, thank you!
- DreamScatter 7y agoCheck 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 7y agoWow! 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.
- K0SM0S 7y agoOh wow, this is nothing short of awesome. Your docs¹ are great at first glance, too. Incredible work! Thanks so much, this might prove invaluable for my (personal) research. I guess I have to learn Julia too now, eh. :) [1]: https://grassmann.crucialflow.com/dev/ https://grassmann.crucialflow.com/dev/
- oddthink 7y agoDoes 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.
- DreamScatter 7y agoMy website is based on differential geometric algebra: https://grassmann.crucialflow.com/dev/algebra https://grassmann.crucialflow.com/dev/algebra
- pdonis 7y ago> 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 7y agoSo, 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.
- ErotemeObelus 7y agoSaved
- aesthesia 7y agoI'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 7y agoDespite 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 7y agoActually, 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 7y agoYes, 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.
- orbots 7y agoI've come to see it more as the inverse of a division operator. Like a quaternion can be defined as the ratio of two vectors q = a/b. then qb = a and qc will rotate c the same amount as needed to take b into a. dual quaternions have a similar motivating derivation.
- ajkjk 7y agoI agree with this. Most of the benefits of GA are from embracing the wedge product, not the geometric product. However, I do think there is something there. There is a concept of vector inversion that it makes possible. I have been trying to crystallize this for a long time, though.
- platz 7y ago> 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 7y agoThere 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.
- kragen 7y ago> 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 є₀) Holy. Shit. Is this for real?
- senderista 7y agoYawn. The differential forms version is almost as compact (2 equations) and far more interpretable (if you understand connections and curvature).
- kragen 7y agoThanks! I don't know anything about the differential-forms version.
- peter_d_sherman 7y agoExcerpt: "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
- bialpio 7y agoSomething 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.