Take an Euclidean vectorspace (V, g). Then V is embedded in the Clifford algebra Cl(V,g). In fact V = F^1Cl(V,g) \cap Cl(V,g)_1  where F^pCl(V,g) is the set of elements of the Clifford algebra that can be written as a sum of products of at most p elements of V and Cl(V,g)_1 is the ideal of elements that can be written as a sum of an odd number of elements.  

Now take v \in V with g(v,v) = 1. Then for v \in Cl(V,g) we have v^2 = -1.  The map  R: V --> Cl(V,g),  w --> vwv  takes values in  V\subset Cl(V,g) because  clearly vwv is odd and

         vwv = -v^2w + v(vw + wv) = w - 2g(v,w)v     (*)

can be written as a sum of elements of V, so  vwv is in F^1Cl(V,g).  Moreover the explicit formula (*) shows that R is just a reflection in the hyperplane orthogonal to v.

Roger,

Thank you for opening the question up in terms of Clifford Algebras, and for showing how the (initially counter-intuitive) spinorial character of reflections (v^2 = -1) emerges.

I would still be interested if anyone can point me to where Penrose got (or himself expanded on) what appears to be an algebra of 'pure' (?) reflections. 

Paul

Paul

I am afraid that I am not familiar with Clifford algebra! 

Ann

Dear Dimiter

I get that Penrose’s reflection gammas are operators, but being myself more physics-oriented than a mathematician, prefer a geometric representation to achieve a comfortable understanding of the algebra (such as Rogier’s earlier reply, which I follow algebraically, but still need to decode/translate a bit further for my complete satisfaction).  

When you write "double reflection gives the same object", that reads like “gamma-squared = +1”, rather than -1 (?)

Initially, I was puzzled by the idea that a squared reflection could demonstrate spinorial character (γ^2 = -1). I thought I’d found a good geometric representation of this, until I re-read the Penrose text a moment ago, and saw that his construction of quaternions from base reflections involves pairs of distinct γs so that his quaternion base units i, j, k, equal, respectively, γ2γ3,  γ3γ1 and  γ1γ2.

So I’ll have to play with it a bit more…

Thanks for the stimulus - Paul

Further, Dimiter

I can see now that a pair of distinct γs will act in the same way as a quaternion, but I’m still wondering if the following makes sense of γ^2 = -1 for (as Penrose says) a spinorial object:

In Ch. 11 of The Road to Reality, Penrose demonstrates with a book & belt that quaternionic base units i, j, k, (even sub algebra of Cl0,3(R), and others) each represent a π rotation (180°) in Euclidean space. So  e.g. j^2 = -1 represents a rotation by 2π (two rotations by π) resulting in the expected spinorial inversion.

If I sketch a circle with perpendicular axes crossing the centre and imagine j pointing towards me, then a point on the circle is returned to itself via the double rotation operation represented by j^2 ,  and results in the (essentially topological) factor of -1 for an odd number of 2π rotations.

As for reflections: If I consider γ to be the reflection from the same original point as mentioned above, across a perpendicular axis, to its polar opposite on the circle – and then repeat the reflection γ [as Penrose says “in the same direction, by which I assume he means ”along the same axis” (i.e. reversing the reflection)] then I reach the same final state as the 2π rotation; so I can agree that γ^2 = -1 also.

However: consider the well-known image of rotations in terms of a ball* of radius π, representing all possible rotations in Euclidean 3-space:

• centre = no rotation (other definitions are possible),
• the orientation of any vector in the ball represents the axis of the rotation in 3-space,
• the length of the vector represents the angle of rotation, out to a maximum length representing π
• further rotation jumps to the polar point of the ball as π-rotation in one direction is identical to (-π) rotation, so all points at the surface are to be identified with their polar opposites.

But, while the spinorial character of rotations involves a continuous path though the ball of rotations, for reflections the path is discontinuous.   

So does γ^2 = -1 make sense for arbitrary reflections? Or can we compose an arbitrary rotation by a series of infinitesimal reflections?

Paul

*  Incidentally, using this image one can also see (separately from Penrose’s demonstration) how rotations can be spinorial, via the shrinkage/non-shrinkage of paths through the ball representing even/odd numbers of 2π rotations.)

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