Dear Christian,

As you say, the spinor definition is that of a mathematician. What I’ve long found problematic (as does Morris) is the profusion of definitions.

I’ve read the introduction and conclusion of your Rethinking paper – and while I get the drift of what seems an intriguing approach to the  comparatively modern problem “to derive and explain space-time from the principles of interaction” – it would take me a long time to work properly through everything you’ve presented in between.

As for the spinor as a point in an abstract phase space, I think I see a connection to the earlier discussions we shared on the interpretation of the complex i .

Regards - Paul

Since replying above (and thinking rather more intuitively than analytically) it seems that it may be possible to justify Morris’s claim via the sense in which David Hestenes regards both quaternions as well as complex numbers as spinors:

In relation to the division algebra H:

“A quaternion is a spinor. The identification of quaternions with spinors is fully justified not only because they have equivalent algebraic properties, but more important, because they have the same geometric significance.” (p. 14 of [1] below)

In relation to the division algebra C:

“By multiplying two vectors in the i-plane we get a quantity called a spinor of the i-plane” But Hestenes continues “our i is a bivector so it has geometric and algebraic properties beyond those traditionally accorded to “imaginary numbers”. ” (p. 51 of [2] below)

In relation to the division algebra O:

Octonionic spinors are discussed e.g. in [3] below.

So we can plausibly take as spinors a unit element of each of these algebras:

• a unit octonion                                             1+ ai+bj+ck+dl+em+fn+gp
• a unit quaternion                                         1+ ai+bj+ck
• a unit complex number (as a bivector)    1 + ai

And if the division algebras C, H and O all admit spinors, it remains to find a convincing reason to locate spinors within the division algebra R of the Reals.

As all three structures listed above combine the arithmetic unit 1 (1^2 = +1) with seven, three or a single basis element obeying  (i^2 = -1), in one sense the termination of this latter “series” would be the arithmetic unit alone, having no basis element squaring to -1 at all.

• a unit real                                                    1

But whether or not that is sufficient to consider a unit Real as a spinor under this definition leaves me no alternative but to track through Morris’s book.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.412.2276&rep=rep1&type=pdf

https://www.amazon.co.uk/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148

https://arxiv.org/pdf/hep-th/0302113.pdf

Further on this question: in particular, Morris’s claim, shown in the Question, that:

“Each division algebra has its own set of spinors (unit length elements)....a spinor is an element of the rotation matrix of a division algebra.”

 1.       The non-associativity of octonions means that they do not have matrix representations, according to [1]

2.      And, more suggestively than proof: While John Baez points out in his well-known octonions article [2] [my bold]:

• “the octonions are not a Clifford algebra, since they are non-associative. Nonetheless, there is a profound relation between Clifford algebras and normed division algebras."

...according to the Wikipedia article on Grassmann Numbers [3]:

• "When the number of fermions is fixed and finite, an explicit relationship between anticommutation relations and spinors is given by means of the spin group. This group can be defined as the subset of unit-length vectors in the Clifford algebra, and naturally factorizes into anti-commuting Weyl spinors. "

I therefore read [2] and [3] as indicating that while octonions are recognised as a normed division algebra, and can be realised as spinors [e.g. 4] they do not, at least according to Baez, form a Clifford Algebra [but 5 does describe "Octonionic representations of Clifford algebras"] and so may not form unit-length vectors in a Clifford Algebra, which if true would seem to invalidate Morris's definition of spinors - the basis of the original Question (?)

[NB Late edits, as a result of digging deeper, [4] and [5] are now raising my doubts about this approach - anyone?]

I would add that I’ve recently received a private communication from a real expert in the field [FF], who also argues for a general view of spinors, but from a very different angle, which he summarises as follows:

• “What can be modelled by spinors? Anything whose properties depend on the properties of the universal covering group of the transformations that one admits into the theory. This too is a matter of choice, and the choice is determined by what is being modelled. ”

 … but I do not see a way to measure Morris’s claim against this definition.

https://en.wikipedia.org/wiki/Octonion

http://math.ucr.edu/home/baez/octonions/node6.html

https://en.wikipedia.org/wiki/Grassmann_number

https://arxiv.org/abs/hep-th/0503210

https://arxiv.org/abs/hep-th/9407179