Q&A summary

Questions here accompanied by answers as they are settled in discussion.

Topics

(I'm putting a few starters up, but feel free to place more urgent discussions above them.)

Spinors

Incidentally, I find [Dorst's] point in Q11 quite interesting [1] – essentially that some portion of a CA (obtained by removing the + operation, and only allowing products of scalars and vectors: the "factorizable part") is geometrically useful, and the remainder is not. I suspect that under "products" one must include more than only the geometric product, though...

I'm not so sure about his answer to Q11. It's all very well to say that multivectors that aren't factorisable (or writeable as exponentials) aren't identifiable with a physical meaning, so one should only consider products and exponentials and ban use of the + sign. But there is at least one object this doesn't apply to that on the face of it (unless one can work around it) appears very significant: namely the idempotent projector proportional to (1 + e_i), that is an essential factor of spinors, at least as they are understood traditionally and in group theory.

((Some GA proponents redefine -- in my view misuse -- the word "spinor" to mean something different, viz. an even multivector. In my view, this is highly confusing. The proponents (I think) seem to justify this by saying if you've got an equation where both sides are multiplied by the projector -- ie a spinor equation -- you can cross out the projector either side, to give a new equation which is a multivector equation (possibly imposing some alternate restrictive choice on the multivector). That's all very well, but it's no longer a spinor equation. It may be possible to show that it is a 1:1 isomorphic equation (I'm not sure), but one should be honest that it is now an equation in a different sort of quantity. This is something I haven't fully worked through, but I don't think it's possible to systematically 1:1 replace conventional spinors as used in the group theory analysis of rotations with even multivectors -- I don't think the degree-of-freedom count matches up. It may be that the GA physicists have in mind only those spinors that turn up in particular equations representing particles (or spinor-fields of particles), and want to claim that instead of spinor fields those particle fields can be represented as even multivector fields, so "spinor" being associated with a particle should be associated with an even multivector; but it seems to me that (at least in group theory) the meaning of "spinor" is wider than that, and the word has a particular meaning, so should be used more directly corresponding with that meaning)).

Sorry, rant over. But the word "spinor" is one I feel I was very misled on by the GA literature; and as a result I added misleading content to the spinor article which is still there; and one day I will properly get my head round what should have been there instead. But this is enough for one post. Jheald (talk) 23:20, 2 February 2012 (UTC)[reply]

I have seen various attempts to answer the question "what is a spinor?" – enough to realize that this is a non-trivial concept. It is not an area I have any real familiarity with, but I'm sure physicists will happily use any definition that matches the observations (and hijack any convenient mathematical term for it in the process).

As to Dorst's contention in Q11, my immediate reaction was "in the classical geometric setting, maybe, but in the quantum-mechanical setting in which weighted sums of spinors are inherent in the math (as members of a family of solutions), you're out of your tree". I nevertheless find the suggestion that we may be using a definable subset of the CA in the classical geometric setting of interest. — Quondum ^☏✎ 07:56, 3 February 2012 (UTC)[reply]

As far as I can comprehend - they are vectors with the special characteristic that they become negative when rotated by 2π rads, and positive again after another 2π rads, i.e. 4πn rads (where n is any integer) preserves the spinor.

What exactly rotates? I guess an object, which has a vector (the spinor) attached to it and fixed in some orientation. When the object is rotated about some axis, the vector (spinor) remains in the same direction for an even number of 2π rotations (2πn rad, for n even), in the opposite for an odd number of 2π rotations (2πn, for n odd).

One way to think of a spinor, according to Prof. R. Penrose and his book Road to Reality, is a book attached to a belt, which is fixed flat at one end. Rotating the book about some axis makes the belt behave like a spinor: for 2πn (n odd) the belt is twisted through an angle of 2π rads, but for 2πn (n even), the belt can always be untwisted to the "flat state". The belt keeps track of the parity of the full number of rotations.

There is also this youtube clip: [2].

As Penrose continues, the way to think of spinorial objects (e.g. Spin-½ fermions) is by imagining an imaginary belt attached to the object, the object can rotate about some axis like the book, the imaginary belt will twist around into either a flat state, or a twisted state, i.e. exactly two states correspond to an object which is apparently at the same angle.

They are used in QM and QFT because wavefunctions which incorporate spin are spinors, i.e. spinors describe particles with spin.

I suppose this is not much help, but as far as I understand spinors qualitatively this is what they appear to be. -- F = q(E + v × B) 16:56, 8 March 2012 (UTC)[reply]

Spinors are tricky. I think the idea of a spinor as "pretty much anything with 4π periodicity" may be an oversimplification. I like Penrose's picture (Figure 11.4 in The Road to Reality) in which he gives a geometric interpretation for quantitative "adding" of rotations in 3D. In 3D they have three degrees of freedom, which may make them tempting to map onto vectors (just like bivectors and the cross product), but in any other number of dimensions the number of degrees of freedom do not match (e.g. 1 degree of freedom in 2D, 6 –or is it 7?– degrees of freedom in 4D). Thinking of spinors as rotations (or more correctly rotors) seems to me to be a far closer match, but I'm not sure it's perfect. I think there is even some disagreement about the definition of a spinor. — Quondum ^☏✎ 20:28, 8 March 2012 (UTC)[reply]

Hi- thanks for your response. I thought as much this would be an oversimplification, since then I don't properly understand them, this is just the accumulated knowledge I have of them so far. You'll notice I trailed off as to what the spinor is, and even Penrose seems to do the same (first a statement that spinors are some forms of vector - then trail off by not explaining a spinor but a vague description of spinorial objects). Of course he doesn't like to scare readers away - making the exposure of the content more approachable etc. If its vector - how exactly can we draw the spinor? If the book (attached to belt) as described Penrose is a spinorial object ... how can we draw the spinor as a vector on the book diagram?

Perhaps the Möbius strip interpretation above is the most simplest and correct one of a vector as a spinor? -- F = q(E + v × B) 07:49, 10 March 2012 (UTC)[reply]

One of my longstanding ambitions is to really feel I understand what spinors are. I haven't got there yet.

Be aware, though, that a number of authors in the GA community use the word to mean something that is not the same mathematical object as generally understood by the mathematics and physics community.

These members of the GA community redefine the word to be a slight generalisation of the GA concept of a rotor. A rotor R is a way to represent rotations, such that a (multi)vector quantity A is mapped to R A R^-1. These GA folk define a spinor to be "a (normalised) element that preserves grade under a sandwiching product in a Clifford algebra" [i.e. S x S⁻¹] (Dorst et al (2007), p. 195 (Google books); so something that maps a bivector to a bivector, a vector to a vector, etc. in such a sandwiching product.

But this is, at least on the face of it, not straightforwardly what the rest of the world means by spinor. Before GA, the rotors which act like R above have traditionally been thought of very closely in terms of their representation as matrices (typically, though not necessarily, matrices of complex numbers). These matrices, called spin matrices, fall out as one of the possible representations in the group representation theory of rotation groups; in a way which encodes more information about the rotation than just the rotation-matrix form. Conventionally, a spinor is a column-thing that such a matrix can act on -- it's the mathematical thing that gets transformed by the representation of the rotation; I suppose, a way of keeping score of the result after multiplying by one matrix and then another and another etc. From there, one can (in the usual way with matrices) see the spin matrices as concatening column-by-column the effect they have on a number n of base spinors, where n is the dimensionality of the matrices, so one can form in turn the idea of the matrices as operating on a spinor space, spanned by the n base spinors.

If one tries to translate these notions almost literally into a GA language, one finds the spinor space is a sub-algebra of the whole GA, formed by multiplying the algebra by a quantity like ½(1+e₁), where e₁^2 = +1. This is an idempotent (it squares to give itself), and acts as a projector, projecting the whole algebra into a smaller sub-algebra. But it's quite hard, on the face of it, to make sense in GA terms of what the objects in this smaller projected space mean. It's quite normal, in GA, to come across objects which combine scalars and bivectors: this is the form rotations typically have. But objects which combine a scalar with a vector are not part of the normal routine, and it's not obvious what kind of physical thing they might represent.

However, it turns out that, at the very least in the physically important Pauli and Dirac cases, it is possible to make an isomorphism between spinor-space and the space of rotation rotors (or rotation-boosts in the Dirac case), so something like the Dirac equation can be interpreted as an equation constraining a rotation-boost, giving it some real tangible physical meaning.

But I'm not sure whether such an isomorphism can generally be made. (I'm not sure, for example, that the count of degrees of freedom would match up). So until I am more sure about that, I am very wary of assuming such an isomorphism can be found generally, and even more so of redefinitions of the word spinor, at least until I can see that the two definitions really can be inter-related. Until then, I'd tend to go with eg Charles Gunn ([3], page 21) writing "An element of the spin group is called a rotor. Some authors (Perwass 2009) refer to an element of the spin group as a spinor but this conflicts with the accepted definition of spinor in the mathematics community, so we avoid using the term here."

It's probably not good to think of a spinor as a vector. Spinors are indeed elements of a linear space, but it's not the same linear space as physical vectors. Jheald (talk) 09:27, 10 March 2012 (UTC)[reply]

I seem to remember that precisely in the 4D (Dirac spinor) case the isomorphism does not hold. A normalized Dirac spinor has (AFAIK) 7 degrees of freedom, where as a nomalized rotor has 6. So this would be worth looking into. ('tis also possible that I have it wrong.) — Quondum ^☏✎ 13:39, 10 March 2012 (UTC)[reply]

This discussion is pretty surreal from my point of view :) As a mathematician, I would wait until I have concrete definitions to examine before philosophizing about what they mean, but from above I can see that others prefer the reverse approach.

I've read two reasonably lucid mathematical descriptions of spinors which we should all probably see. They are: Elie Cartan's preface in his book The theory of spinors, and secondly a "fundamental" paper by Weyl and Brauer Spinors in n-dimensions. They all seem to say that spinors are elements of representations of rotation groups, which is a pretty concrete description. Philosophically, it sounds like that they carry more information about the action of a rotation than the rotation itself does. If anyone is interested in a pdf of the Brauer-Weyl paper, please email me. Rschwieb (talk) 15:39, 10 March 2012 (UTC)[reply]

Here is something which illustrates that spinors are elements of a linear space and form a rotation group in QM (refs: QM (2nd edn) Schuam's outlines, p.147, and QM, E.Abers):

Given a state vector $|\alpha \rangle \,\!$ in one coordinate system, which is an eigenstate of the spin operator:

{\hat {S}}^{2}|\alpha \rangle =s(s+1)\hbar ^{2}|\alpha \rangle \,\!

[where the eigenvalues are of course s(s+1)], it can be rotated by an angle θ about a unit vector u defining an axis of rotation (its direction along the axis is not important) into a new coordinate system $|\alpha '\rangle \,\!$ :

|\alpha '\rangle =e^{-\theta \mathbf {u} \cdot \mathbf {\hat {S}} /\hbar }|\alpha \rangle \,\!

where

\mathbf {\hat {S}} ={\frac {\hbar }{2}}{\boldsymbol {\hat {\sigma }}},\quad {\boldsymbol {\hat {\sigma }}}=({\hat {\sigma }}_{1},{\hat {\sigma }}_{2},{\hat {\sigma }}_{3})\,\!

is the Pauli spin vector and

${\hat {\sigma }}_{1}={\begin{pmatrix}0&1\\1&0\\\end{pmatrix}},\quad {\hat {\sigma }}_{1}={\begin{pmatrix}0&-i\\i&0\\\end{pmatrix}},\quad {\hat {\sigma }}_{1}={\begin{pmatrix}1&0\\0&-1\\\end{pmatrix}},\,\!$

are the Pauli matricies.

The rotation matrix corresponding to the transformation is:

{\hat {U}}_{R}=e^{-\theta \mathbf {u} \cdot \mathbf {\hat {S}} /\hbar }={\begin{pmatrix}\cos(\theta /2)&-\sin(\theta /2)e^{-i\varphi }\\\sin(\theta /2)e^{i\varphi }&\cos(\theta /2)\\\end{pmatrix}}\,\!

The rotation of a spin vector (aka spinor) transforms differently to a spatial vector, so rotations are used to define spin vectors.

This source actually gives the interpretation of a spin vector as a type of vector...

Although I don't understand group theory in detail, these rotation matrices certainly form a rotation group (they satisfy the group axioms). Also the vectors are of course elements of an abstract linear space, as said Jheald before also.

You will not find this at the main article on spinors, but the Pauli matrix article should be helpful in understanding how spinors behave.

Unfortunately I have caused deviation from GR and EM into QM and rotations, and should have started this discussion on spinors at your rotations subpage, Rschwieb, so apologies for that, feel free to move and re-section... --F=q(E+v×B) ⇄ ∑_ic_i 08:39, 22 March 2012 (UTC)[reply]

It's all very well producing formulae, but until we are all on the same page, we're not going to get very far (I for one can't interpret the formulae, and a "rotation axis" is an ill-defined concept in GA in general). And a good place to start is understanding the maths at an abstract level, before relating it to physics concepts such as spin. (And working in 3D as here also does not help: there are so many shortcuts in the maths that my geometric intuition doesn't keep up. In 3D there is too much room for confusion of vectors and bivectors etc., and the accurate modelling of the physics uses 4D math.) So for example, it would be good for us to understand the mathematical distinction between the two types of spinor Jheald mentioned: the rotation group, and the general grade-preserving (and orientation-preserving) sandwiching group, how one can be a larger group than the other, and how these relate to the (special) Clifford group, the spin group etc. — Quondum ^☏✎ 11:11, 22 March 2012 (UTC)[reply]

AFAIK, you pick out the invertible elements of a CA to get the Clifford group then Pin is all products of non-null normed vectors and Spin is restricted to even numbers of vectors. Rotors (R_barR = 1) form rotor subgroup (+ Clifford even = SL(2,C)).

So Pin is cover (2:1) of O, Spin is cover of SO, rotor subgroup is cover of Lorentzian case (special).Selfstudier (talk) 11:55, 10 July 2012 (UTC)[reply]

When I get the time, I intend to tackle spinors using Cartan's book The theory of spinors as the starting point. The preface gave me hope that there would be information on how the abstract description was used in physics. Rschwieb (talk) 13:38, 22 March 2012 (UTC)[reply]

The extent that GA is "coordinate free"

One of the selling points touted by authors in GA is that it is "coordinate free", a claim which is a bit hard to interpret. Of course, the Clifford algebra construction does not depend on a particular basis. I think a lot of people interpret the claim as "You can do linear algebra without picking a basis" or at least "you don't have to refer to a basis as much". See what Dorst has to say (question 10). He seems to indicate that the conformal model is the most coordinate free, but I'm unfamiliar with it at this moment. Rschwieb (talk) 13:06, 2 February 2012 (UTC)[reply]

I find the claim in Q10 that "... only in the conformal model ... does GA allow fully coordinate-free computations" a little strange. It seems to refer to the greater abstraction and hiding of extraneous representation information, though this depends on what you are representing. So for example, one might represent a line in G(3) as two vectors to points on the line; the position of the points on the line is irrelevant, but even in coordinate-free GA, extraneous (i.e. irrelevant) information about the position of these points on the line is explicit. CGA hides this this extraneous information behind the abstraction of the line being a single object. To call only this "fully coordinate free" is to my mind indulging in hyperbole. CGA(3)=G(5) is strictly richer than G(3), but what is being abstracted is not coordinates.

Incidentally, I find the point in Q11 quite interesting – essentially that some portion of a CA (obtained by removing the + operation, and only allowing products of scalars and vectors: the "factorizable part") is geometrically useful, and the remainder is not. I suspect that under "products" one must include more than only the geometric product, though... — Quondum ^☏✎ 14:35, 2 February 2012 (UTC)[reply]

The "co-ordinate free" or "component free" emphasis goes back a very long way in the literature. It may even go back to some of Hestenes's earliest papers. I haven't done a literature search, but I suspect it may originally have come about as a counterpoint to the way the Dirac spinor in the Dirac equation is broken up into components -- the contrast being that GA tends to try to treat objects as a whole, that transform as a whole, rather than breaking them into components. There contrast might also have been being drawn with the Gamma matrices, which may often have been taught in a way that emphasised matrix computations, there to make possible those calculations, rather than as algebraic entities with a meaning in their own right. So it is, I think, staking a claim that there are theorems you can prove, manipulations you can perform, and understanding you can obtain, by treating the various objects as entities in their own right, without necessarily ever representing them numerically.

Though pedagogically, I think there is a case for what Snygg does in his new book (2012), namely to bring in a set of 3 matrices at the top of page 4, write down some multiplication properties, then declare within ten lines that

"In the formalism of Clifford algebra, one never deals with the components of any specific matrix representation. We have introduced the matrices of (2.3) only to demonstrate that there exist entities that satisfy (2.4) and (2.5). "

Which is nice; though for a more practical hands-on let's-draw-some-pictures perspective Ian Bell cites Irving Kaplansky writing about Paul Halmos:

"We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury." [4]

(As Bell discusses, and eg Richard Wareham takes further (thesis, ch 4), in fact we generally do rather better than naive matrix multiplications).

But yes, I guess the point is you can do a lot without introducing any explicit component-type representation at all.

Dorst is taking this a step further. Normally GA is also co-ordinate free in the sense that it is manifestly covariant under 3D rotations, or Lorentz transformations -- i.e. you can change your basis vectors, and know that your equations in terms of vector a or multivector A are completely unchanged (even though that a might now be represented by completely different numerical component-by-component values, if one really felt that was something necessary to present). However, normally GA is not quite completely co-ordinate agnostic, because it singles out a particular preferred point as the Origin, around which all rotations are performed. On the other hand, in CGA, because one can now represent translations as an integral part of the algebra, there is no longer any particular privileged Origin; instead one can translate anywhere to become the Origin and rotate about there, all within the sandwich formalism. This is what Dorst is getting at with Q10. Jheald (talk) 23:20, 2 February 2012 (UTC)[reply]

I concur, though I'd add that the desireability of a coordinate-independent formalism must have been pioneered with tensors, and merely adopted by the GA crowd. So I wouldn't give Hestenes any credit for that: it has been a cornerstone of special and general relativity (in the concept of covariance), even if a manifestly coordinate-free version such as Penrose's abstract index notation may not have been around at the start.

While I agree that a formalism that manifestly removes the origin for dealing with an affine space is desireable, I object to Dorst referring to it as "coordinate-free". It is simply the wrong term. — Quondum ^☏✎ 07:56, 3 February 2012 (UTC)[reply]

GA and general metrics

It is said that GA is compatible with metric spaces, but I have not seen any examples other than metrics of +1 or -1. For a general metric (a distance function of (x,y,z) and (x', y', z')) what does GA look like? Rwgray34 2012-02-02T02:52:01

In general any symmetric bilinear metric is comfortably accommodated by GA. In particular, it is straightforward to choose a basis so that the metric has only +1, 0, and −1 on the diagonal and zero elsewhere. The occurrence of such standard bases is merely a matter of convenience and any basis would suffice; it is merely a linear transformation involved. Or to answer your question more directly: In the case of a GA with a general metric, we have basis vectors

{e i}

such that

e i e j + e j e i = 2 g ij

. — Quondum ^☏✎ 06:22, 2 February 2012 (UTC)[reply]

I think there might be a misunderstanding in terminology here. A "differential geometry" metric space and a "topological metric space" are different. You mentioned a topological metric, but I believe that authors are talking about the differential geometry type of metric, that is, a bilinear form on a vector space. For real spaces all of these bilinear forms are equivalent to forms with 0's 1's and -1's, so it suffices to study those.

Of course, any positive definite bilinear form gives rise to a topological metric. I don't know off the top of my head if the converse is true. In any case, the interpretation of "positive definite distance" breaks down for signatures with -1's, since you can have negative and zero lengths.

If I saw "compatible with metric spaces" when talking about GA, I'd bet they're talking about the fact given a (real) metric space V with bilinear form < , >, the associated Clifford algebra's product encodes the bilinear form <a,b>=(ab+ba)/2. Rschwieb (talk) 12:58, 2 February 2012 (UTC)[reply]

Uhhh... you're at risk of losing me. I suspect that I only think in terms of differential-type metric (in any event, one associated with a quadratic form on a vector space). Topological spaces appear to deal with potentially highly nonlinear, but always positive-definite, metrics. So I'd say that many positive-definite metrics will not give rise to bilinear forms. So I do not know what made you think anyone may have been referring to a topological metric. — Quondum ^☏✎ 15:11, 2 February 2012 (UTC)[reply]

Sorry, I meant that I thought Bob might be confusing the two notions of metric. Clearly, Q, we are not used to having more than two people involved in a discussion :) When Bob mentioned a distance function it sounded an awful lot like the topological notion of metric space. This is the notion I was used to for four years before I met this strange (to me) usage of "metric" in differential geometry. Rschwieb (talk) 18:25, 2 February 2012 (UTC)[reply]

A representation of conics in a projective GA

How to represent non-sphere conics (ellipsoid, paraboloid, hyperboloid) in GA or conformal GA? Is it then easier to find the intersection of two conics, or a conic and a line, or a conic and a plane? Rwgray34 2012-02-02T02:52:01

I'm taking a shot in the dark here. In "classic" GA, geometric objects such as lines, planes, etc. (all through the origin) may be represented by single elements of the algebra (and operations being rotations, reflections, spans and intersections). Add one dimension (homogeneous GA), and an element can represent similar objects that have offsets from the origin (with the added operation of translation). Add a second dimension, and conformal objects (lines→circles, planes→spheres etc.) can be represented. I would be unsurprised if conics could be represented by single elements in a projective space by adding a third extra dimension. But I'm not familiar with even CGA etc., so take this with a pinch of salt. It would be fun to see what is possible in this regard. — Quondum ^☏✎ 06:52, 2 February 2012 (UTC)[reply]

Stretches don't play nicely in CGA, so there isn't a particularly natural way to represent an ellipsoid. Perwass's book [5] has a section where he introduces higher dimensions, so that more general conic sections come out as primitives; but it still doesn't allow stretches to be represented in "sandwich" form, so, as he accepts, it's not quite there.

One idea that occurred to me was that one way to get stretches might be to try to engineer a way to have both hyperbolic rotations and normal rotations on the same set of real axes. At the moment, the sign of (e_ae_b)^2 determines whether the corresponding bivector is going to represent a normal rotation or a hyperbolic rotation -- and if it's one, then you can't have the other. This relates to the Clifford structure being defined by the metric, and the metric is physically meaningful, so transformations which preserve metric distances are privileged.

What I think one may want to do is to add an additional new pseudoscalar to the algebra, so that one can have both normal and hyperbolic rotations -- this should allow the stretches that we want in order to boost one shape into another, but such stretches need to be distinguised as not preserving metric distances.

The GA party line to date has very much been to emphasise real CAs as being what are physically meaningful. I suspect what I'm suggesting above may be close to "complexification", ie moving to a complex CA -- but done in such a way that the new algebra is not just Cl_n(C), but keeps hold of a privileged real metric, so that it still makes meaningful sense to talk of it having a real signature (p,q), rather than a complex signature (p+q).

The Dirac equation, of course, is conventionally presented in a complex CA -- something which has been a bone of contention for the GA crowd, who insist that time is not space, so it is the real CA Dirac equation which is meaningful. But perhaps understanding if you can have a complexified CA but still have a privileged real structure might be helpful there. If "complexification" can be used to construct algebras with stretches on Euclidean space (no pseudo-Euclidean directions), that is understood to stay Euclidean despite that structure, that understanding may be a helpful step forward. Jheald (talk) 17:26, 2 February 2012 (UTC)[reply]

A stretch (by which I assume you mean a scaling in one dimension) is not the only way to map a sphere to an ellipsoid, so the lack of a natural representation for a stretch does not imply the lack of a natural representation for an ellipsoid. However, assuming that all sandwich products in CGA produce conformal mappings, the fact that there is no conformal mapping of a sphere onto an ellipsoid would imply the lack of a natural ellipsoid object in CGA. Though if one plays with the signature of the extra dimensions, one might perhaps get a non-conformal projective GA?

Your suggestion of combining two metrics very closely mirrors a thought I've had: Generate a CA from a non-degenerate othogonal vector basis. Repeat the exercise with the square of one of the basis vectors negated. Average the values in the two multiplication tables. The result should be the same as for one basis vector squaring to zero: a degenerate quadratic form. I suspect this is useful as one way of unifying translations and rotations via a quadratic form (reading between the lines of dual quaternion). I wonder whether a degenerate quadratic form might not have uses such as stretches much like you suggest? (I doubt a second pseudoscalar would work: sandwiching would not work). Anyhow, you seem to have answered the original question, that according to Perwass (at least some) conics can be expressed in a suitable (projective) GA, given extra dimensions. — Quondum ^☏✎ 18:52, 2 February 2012 (UTC)[reply]

If I remember correctly, there are certainly treatments of Clifford Algebras that include the possibility of algebras that include dual numbers, e.g. that algebras Cl_p,q,r that include a number of bases ε_i with the property ε_i² = 0. We even present this as part of our introduction to Clifford algebras at Hypercomplex_number#Clifford_algebras before "putting aside" such bases. Nilpotents and degenerate spaces (I think) can also arise quite naturally as subspaces of larger Clifford algebras -- one classic approach to spinors is to present them as elements of an isotropic subspace. I think homogeneous coordinates are sometimes also presented using such dual numbers, or can be manipulated into such a representation -- again if I remember correctly there was some material that investigated this just before CGA came so much to the fore, precisely as a way to allow both translations and reflections to be possible. So (I think) Clifford algebras incorporating dual numbers are certainly a reasonable thing to read up and consider. But the reason CGA so much put this into the shade is that it allows translations and reflections both to be possible as sandwich operators, and that is what opens up the whole of inversive geometry to GA methods.

It's the fact that we already know how to represent hyperbolic transformations using sandwich operators that makes me think that a system that allowed both hyperbolic transformations and normal rotations of the same underlying axes might be a very interesting thing to look at. Such hyperbolic transformations wouldn't be a stretch alone, strictly speaking: the stretch in one direction would be countered by a squeeze in the other, to make them area preserving. (Perhaps ways could later be explored to dispense with this invariance, with dual numbers perhaps; but preserving the constraint seems initially more interesting, to give a slightly more restricted and structured system to explore, to understand the ins and outs of the algebra).

I think you make a very good point when you underline that there are many ways to map a sphere onto an ellipsoid. Perwass has perhaps found one, but the mapping is not easy to describe; and it seems that it may be that it can't be related to a simple stretch or stretch-squeeze. To be able to extend the methods of GA to affine geometry, in particular to be able to represent more general affine transformations in sandwich form, would be a significant prize. Perwass hasn't been able to do that, and so is dissatisfied with his construction. Jheald (talk) 20:35, 2 February 2012 (UTC)[reply]

Clifford algebras over finite fields

The study of Clifford algebras originated as that of geometric algebra, and were subsequently generalized to Clifford algebras over other fields. Over ℂ finds obvious application in physics. The question arises why they were generalized to finite fields, and what applications this has.

The question of 1-3 and 3-1 signature

So far it seems nobody knows if either one is preferable to the other. While they are nonisomorphic algebras, there is a "tilting" map that connects the two.^[1] Rschwieb (talk) 19:14, 1 February 2012 (UTC)[reply]

I would rephrase that: nobody appears to be able to conclusively rule out either of the two cases for use in physics. For example, a clear preference would be suggested by that the Dirac equation cannot be expressed in Cℓ_1,3(ℝ), as it must be embedded in Cℓ_1,3(ℂ)≅Cℓ₄(ℂ) first, or else expressed in a frame-dependent way, thus violating the mathematical principle of basis-independence. I'm pretty sure that Cℓ_3,1(ℝ) does not suffer from this shortcoming in this context. I suspect this situation is a result of inertia and the physicists' acceptance of "any tool that works". — Quondum ^☏✎ 08:51, 2 February 2012 (UTC)[reply]

GA Dirac equation

See also

Dirac equation

GA gravitation

For clarity, I suggest that this topic refers to using GA as a tool to explore a gravitational straw model in flat space: it is not compatible with general relativity, but has the potential to produce nearly indistinguishable results for weak fields, including measurements by Gravity Probe B. I regard this as an interesting exercise to gain insight into and familiarity with GA, as well as providing a fascinating study of gravity. — Quondum ^☏✎ 05:56, 2 February 2012 (UTC)[reply]

I recently ran across the subject in a Doran-et.al paper. I will have to reread what was written, but I got the impression they said "It's possible to do gravity without curvature in GA". I think it was from the 90's, so possibly experiments have already sunk this approach. Can you give me an idea about what you know and how it might pertain to a good model? Would it only be good as a rough approximation? Is curvature not the best way to look at gravity? Rschwieb (talk) 20:16, 24 February 2012 (UTC)[reply]

I'm not an authority in this area. My impression is that Einstein's curvature-of-spacetime model is not the only mathematical model that is consistent with the observations at this point. There also seems to be some work in GA "flat" background spacetimes (which may not be flat as measured by a metric), such as suggested in Spacetime algebra#A new formulation of General Relativity. I suspect Doran et al's approach may not yet have been developed to the point of being tested, but I have the sense they are aiming at an accurate (i.e. "good") mathematical model of gravity. Please do not misinterpret my intent as stated at the start of this section though: I am not expecting to produce a serious or entirely accurate model of gravity, but I do hope to produce a tractable mathematical model based on a field rather than curvature that is accurate in the weak field approximation (i.e. nothing resembling a black hole) and looks a lot like Maxwell's equations, and gives something to cut our GA teeth on.

References

Gravitoelectromagnetism – a mishmash giving the suggested form of the expected results; two approaches: linearizing general relativity, and applying special relativity to Newton's law of universal gravitation in Minkowski space (we'd follow the latter).

GA electromagnetism

One point of interest, I found out from a physics friend, is that the Kaluza–Klein theory was developed with the idea of adding another dimension to provide a electromagnetic-force-causing-curvature. This is theoretically interesting but experiments at the LHC indicate that this theory is not consistent with the physical world, and the quantum field theory version is still the best.Rschwieb (talk) 19:14, 1 February 2012 (UTC)[reply]

I (R.W. Gray) too have thought about a GA and Kaluza-Klein theory connection. In particular, the 5 dimensional conformal GA model of 3D space.

Books about GA and Electromagnetism: John W. Author, Understanding Geometric Algebra for Electromagnetic Theory, John Wiley & Sons, Inc. 2011. — Preceding unsigned comment added by Rwgray34 (talk • contribs) 00:35, 2 February 2012 (UTC)[reply]

I would suggest renaming this section if Kaluza-Klein theory is the intended topic, as it relates to the geometry of a manifold and not to GA. These are orthogonal concepts. — Quondum ^☏✎ 05:28, 2 February 2012 (UTC)[reply]

Honestly I just need to get to know how Maxwell's equations work in both GA and the "standard" presentation. The KK-theory is not a priority for me since it appears that it will be discarded as a physical theory. I don't dictate the section contents, though :) Rschwieb (talk) 13:14, 2 February 2012 (UTC)[reply]

Well, then, this section is correctly named. Maxwell's EM theory in both formulations (GA and Gibbs's vector "algebra") is a lovely way to get the feel of it. Any investigation into KK or general relativity should have a section to itself, if we get to curved manifolds. — Quondum ^☏✎ 14:52, 2 February 2012 (UTC)[reply]

Quick links to related discussion pages

Quondum's GA Relativistic fields page
Quondum's Trial ideas page
Talk page oriented about the algebraic definition of geometric algebras.

Misc announcements and requests

Hi, I bet you guys can really help me fill in my rotations subpage. It would be really great if each section had a procedure along with a worked out example. Even better, the same example worked out in all methods! I think I can imagine how, with orthogonal matrices, one vector could be rotated to be in the same direction as another vector. I would appreciate a clear description of how this works from someone who has been thinking about it, though. I have no idea how Euler angles or rotations with quaternions work. The point of that page, and hopefully a few more like it, is to have a clear cut comparison of how you can do that task in several ways. Thanks for whatever time you spend on it, I'm sure I will learn something. Rschwieb (talk) 20:49, 3 February 2012 (UTC)[reply]

There is a page Rotation representation (mathematics). It's a fair while since I've looked at it, and it looks like it could do with some tidying up; your critique of it would be useful, as somebody coming to it fresh, asking exactly the question it supposed to be there to answer. Jheald (talk) 21:31, 3 February 2012 (UTC)[reply]

OK cool, thanks for the link. I didn't know there was that specific of an article around. Maybe after reading it I can fill in my own page. For personal use I'm interested in something more terse and uniform. Rschwieb (talk) 02:36, 4 February 2012 (UTC)[reply]

The material at the end of Gimbal lock ("Gimbal lock in applied mathematics") is a bit more terse. Jheald (talk) 02:57, 4 February 2012 (UTC)[reply]

I've added something to the GA case. You'll want to trim/shape it and settle on conventions (symbol for reversion, order of sandwich etc.). Where derivation is desired, just say. I've deliberately omitted some detail as an exercise; I can fill in detail.

Quarternions are most easily understood in terms of GA; it is a matter of defining the mapping of the representations: rotors↔quaternions, 1-vectors↔imaginary quaternions. The latter is a bit like the mapping between the cross product and the exterior product: it requires the Hodge dual (and I sure behind this hides the double cover matter). The problem of vectors vs. pseudovectors will also occur. — Quondum ^☏✎ 07:01, 4 February 2012 (UTC)[reply]

I've been learning a lot from the contributions, thanks. I also wanted to let everyone know that the non-talk portion of this page User:Rschwieb/GA_Discussion has a list of online GA resources I've found to be pretty informative. Rschwieb (talk) 14:09, 7 February 2012 (UTC)[reply]

^ Pertti Lounesto (2001) Clifford Algebras and Spinors, Cambridge University Press, §13 Tilt to the Opposite Metric p174–

[1] Pertti Lounesto (2001) Clifford Algebras and Spinors, Cambridge University Press, §13 Tilt to the Opposite Metric p174–

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