Text covering the algebra of reflections? (From Penrose's R2R)

[pellis]

Summary:: Where can I find further discussion of the algebra(s) of “basic reflections” (e.g. γ^2 = -1 ), mentioned in Sec. 11.5 of Penrose’s "Road to Reality"?

In Roger Penrose’s Chapter 11 of Road to Reality, titled ‘Hypercomplex Numbers’, he discusses Clifford Algebra elements being constructed out of “basic reflections” γ (gamma) or “first order (‘primary’) entities” in terms of “quaternion-like relations” among these basic reflections, such as γ(i)^2 = -1 etc.

While I’m familiar with the construction of rotations out of reflections, none of my other references on Clifford Algebras or reflection groups (e.g. Pertti Lounesto's Clifford Algebras and Spinors, or works by Coxeter), seem to refer specifically to an algebra of reflections, or identify them as directly constituting any of the simplest Clifford Algebras.

I'm particularly interested to find a visualisable geometric/diagrammatic view of how reflections can act spinorially, such that γ(i)^2 = -1.

Where can I find more about this algebra of basic reflections, if indeed there is anything more to them than Penrose states (relevant pages, 208-211, attached)?

 

[fresh_42]

Your question isn't especially precise. Reflections are usually considered to generate groups, so you can look for such groups. Your example of a Clifford algebra is one representation of one Clifford algebra. so you can look for their representations in general. Reflections as geometric transformations also pave the way to root systems and buildings.

So which way do you want to go and why? It is a bit like asking: "I have read that $e^{i \pi}=-1$. Where can I find more about $i$?"

 

[pellis]

Many thanks for your reply. I'm a chemist rather than a mathematician, and think visually (as also did many of the theoretical physicists and mathematicians surveyed for Jacques Hadamard’s famous study, On the Psychology of Invention in the Mathematical Field, 1944).

I'm fine seeing complex i as a π/2 rotation operation in the complex plane showing, perfectly clearly, i ^2 = -1. Ideally I would like to 'see' something comparable for reflections.

If it involves numerous alternatives (different Clifford algebras?) then I’d prefer to relate it to the quaternions as one example of a Clifford algebra that displays spinoriality (to use the word Penrose uses).

Initially, I was puzzled by the idea that a squared reflection (as discussed by Penrose in the extract attached to my original question) could demonstrate spinorial character (γ^2 = -1).

Penrose then constructs quaternions from pairs of distinct γs so that his quaternion base units i, j, k, equal, respectively, γ2γ3, γ3γ1 and γ1γ2.

I can see how a pair of orthogonal reflections effect a rotation by π to build the base quaternions with their own spinoriality (as illustrated in MTW's well-known textbook: Gravitation).

Incidentally, not only can reflection across a plane be seen as equivalent to a rotation w.r.t. a point not on the line of reflection (as in MTW); but also, as I see it, as equivalent to the result of a rotation by π around the reflection plane itself.

But I still don’t see how a repeated reflection operator acting in the same plane results in some sort of inversion, unless it’s forced by a definition like Clifford’s original 1878 definition for unit vectors in which their Clifford product squares to -1 (or was that just another way of specifying rotational axes like Hamilton's i, j and k?)

One way out of this dilemma could be if successive reflections by the same operator proceed around the coordinate frame, in the sense that e.g. γ1 initially reflects across the x-y plane and a repeated γ1 reflects the result of the first operation across the y-z plane, resulting in an overall inversion; but is it allowable to sequence the same reflection in this manner – shouldn’t one of the other reflection operators, such as γ2, come into effect for the 2nd reflection?

In the case of rotations represented by quaternions, I understand that spinoriality corresponds to one rotation by 2π resulting from a repeated π rotation by e.g. a unit quaternion j yielding the characteristic inversion e.g. j^2 = -1, and this is easily visualised via the Belt Trick, Dirac’s Scissors or in a number of other ways.

However, I’ve just checked https://en.wikipedia.org/wiki/Clifford_algebra#Quaternions where Hamilton’s real quaternion algebra is constructed as the even subalgebra Cl[0]0,3(R) using a set of orthogonal unit vectors of R3 as e1, e2, and e3 , again using Clifford’s original 1878 rules (units square to -1).

On the face of it, these unit vectors seem to play the same role as Penrose’s reflections.

I’ve never really been comfortable with Clifford’s 1878 rules, and my only other way out is to accept that in some abstract sense one can ‘always’ define certain squared operators to yield an inversion, without it necessarily making any visualisable geometric sense…

Hence my desire to try to chase down more of what exactly Penrose was referring to with his reflections.

 

[fresh_42]

I have found two papers related to this, both from famous mathematicians. Coxeter has even "his own groups" which is why I mentioned buildings.
http://www.math.utah.edu/~ptrapa/ma...eter-Quaternions-and-reflections-AMM-1946.pdf
https://reader.elsevier.com/reader/...64E48EC4CAB445271D080D8981C0D3779E38AC7132116

 

[pellis]

Thanks for the refs - I'll find the Coxeter easier going than Cohen, I expect.