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(Dear mods, I have edited the question. The previous version was probably too confused.)

I can convince myself that spinors exist in 2D using complex numbers. The elementary example is a (complex number) $f (\theta) = \exp(i \theta /2)$. When the argument is increased by $2\pi$, it flips sign ($\exp(i (\theta + 2\pi) /2) $ = $-\exp(i \theta /2)$), and requires another rotation of $2\pi$ to bring it back to the original value, and it is therefore a spinor.

I want to understand why we can deduce, hopefully in a similarly easy way, that spinors can exist even in a three (odd-dimensional) space.

Thanks

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  • $\begingroup$ What did you mean by $\exp(i\theta/2)\exp(2\pi)=-\exp(i\theta/2)$? (It's obviously not correct as stated.) Perhaps you meant $f(\theta)=\exp(i\theta/2)$ satisfies $f(\theta+2\pi)=-f(\theta)$. $\endgroup$ – arctic tern Jan 13 '17 at 16:24
  • $\begingroup$ The edited equality is also false. :-) $\quad$ $\endgroup$ – arctic tern Jan 15 '17 at 7:52
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    $\begingroup$ Also, another comment. If by "spinor" you mean a vector in the irrep of Spin(n) coming from restricting the irrep of the even subalgebra Cliff(n), which is isomorphic to Cliff(n-1), then the space of spinors cannot be odd-dimensional. $\endgroup$ – arctic tern Jan 15 '17 at 8:02
  • $\begingroup$ @arctic tern Of course. Guilty as charged and that'll teach me to be so careless. $\endgroup$ – user_of_math Jan 15 '17 at 14:38
  • $\begingroup$ @arctic tern Could you perhaps expand your comment into an answer? $\endgroup$ – user_of_math Jan 15 '17 at 14:54
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This answer is not an elementary explanation for the existence of spinors, but pointers to references which might allow an elementary understanding of spinors, hopefully sufficient enough to understand their construction, and thus why they exist.

According to Wikipedia,

...Spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product.

Thus I propose attempting to first understand rotors, and then leveraging that knowledge to try to understand spinors. This may or may not work for you, but is one approach I feel might be worth keeping in the back of your pocket in case other approaches do not work.

  • Probably the simplest introduction to rotors I have been able to find is this tutorial introduction by Gull, Lasenby, and Doran (available for free online here). The first section is an introduction, the third section discusses rotors.

  • Perhaps the second-simplest introduction to rotors I have been able to find is MacDonald's survey (available for free online here) -- rotors are already explained in the second section.

  • Chisholm gives a very readable introduction to rotors (available for free online here), which however does not mention spinors.

  • Perhaps more abstract (although on the other hand also with more pictures) is the treatment given by Lundholm and Svensson (available for free online here), which does at the end define and discuss spinors.

A readable introduction to both rotors and spinors can be found in Alan Bromborksy's book (available for free online here) introduces rotors in the first chapter and spinors in the second chapter. Keep in mind though that it seems somewhat fast-paced, so it probably is not an elementary introduction. I am not sure though since I have not read all of it myself.

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  • $\begingroup$ Thanks - I downloaded the Gull, Lasenby and Doran writeup and will read through it for starters. Anyway, it seems that one needs to understand geometric algebra and the geometric product to understand spinors. $\endgroup$ – user_of_math Jan 17 '17 at 13:14
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UPDATE3 In reply to:

"why we can deduce, ... that spinors can exist even in a three (odd-dimensional) space"

Do we ever deduce that? Or do we define it?

Respectfully: I think you're still thinking about these objects as being "in" the space they're associated with.

Spinors live in a spinor space, even when they refer to something in another space.

Try looking at it this way: I would agree that we can describe e.g. vector fields in which there is a 3D vector associated with each point in, for example, physical (3D) space. But the vectors are still elements of a vector space, they are not part of the real space even though we assign their position and coordinates using physical space measures.

However, I see no reason why one couldn't define vectors of any number of dimensions associated with a space of any other number of dimensions, without the coordinates necessarily measuring aspects of the space with which they are associated.

(Forgive me if this isn't maths so much as pedagogy): Imagine, if you will, a fine 2-dimensional lattice (not quite a continuous space) representing the surface of an agricultural zone on planet Earth. At various points we can identify farms (by, say, the lattice-location closest to their farm office) with different combinations of produce, and if there are, say, 57 varieties of produce across all the farms, then we can associate a 57D vector, one per farm, with some points in the 2D lattice...

UPDATE2 (in reply to modified question) Your identification of the usual complex numbers as being themselves spinors, while not widely recognised as such AFAIK, is consistent with the views of the estimable David Hestenes who has done much to bring Geometric Algebra to the attention of physicists.You might do well to start with http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf in which is demonstrated more clearly the spinorial character of your exp(iθ/2), using a bilinear transformation of vectors.

UPDATE in reply to Comment below from OP (and written prior to OP's revision of original question): Are you thinking of complex numbers or spinors as being "in" the manifold? it might be better to think of them as simply associated to it, but not "living inside" it. For example, a quaternion (which some recognise as a form of spinor) describing a rotation in R3 is (in my limited understanding) an element of a quaternion-space that happens to be able to describe a particular rotation, but is not itself "in" R3.

There was an earlier comment suggesting that mathematicians had moved on from concern with spinors, and I guess the commenter was pointing in the direction of Clifford Algebras and beyond.

So, if you haven't seen them already, you might want to look at the ever-expanding Wiki pages on Clifford Algebras and the one on Spinors. Though they refer to Reals, it's still not clear to me that spinors can be described without at least implicitly involving complex numbers. For example: the Spinor wiki page says:

"Thus the (real) spinors in three-dimensions are quaternions_" 

...yet the quaternion unit vectors i, j, k obey relations like i^2 = -1 (referring now to a pi-rotation).

The Reals, R(1), considered as a primordial Clifford Algebra, sit at the top of a pyramidal table of Clifford Algebras (below which sit the Complex Numbers C(1), together with R(2), and another level down the quaternions, H(1)), so it seems clear that not all Clifford Algebras admit spinors, but I leave it to someone more expert to say just which ones definitely don't admit spinors.

If you find the Wiki pages hard-going, another route in is via the late Pertti Lounesto's wonderful compendium, "Clifford Algebras and Spinors" (London Mathematical Society/Cambridge U. Press), which starts at an elementary level, though it's now quite expensive and may be hard to get.

ORIGINAL REPLY (To initial version of question since edited by OP)

How have you reached the conclusion that "spinor geometry is independent of the existence of complex numbers"?

I am likely at the same level as yourself in pursuing such insights (though trivially elementary to mathematicians - except, famously, to Sir Michael Atiyah).

However, in my limited physicist's understanding, spinors are indissolubly linked to some sense of rotation/angle; and complex numbers are, in geometric terms, about manipulating changes of angle (rotation), so I would be surprised - and also interested to learn - if spinors can be encoded without complex/hypercomplex numbers, either explicitly or implicitly.

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  • $\begingroup$ Alas, my understanding is very limited. I understand that spinors arise from having a spin-structure on a manifold, which is something that (apparently) can be defined on many 3 manifolds. On the other hand, complex geometry requires even dimensions to make sense, so they cannot be the same thing. $\endgroup$ – user_of_math Jan 3 '17 at 16:20
  • $\begingroup$ user_of_math Would you like to check your formula in the revised question? NB: exp(2π) is a real number; and it's not -1. Do you mean something more like (exp(i(θ+2π)/2)? Also, exp(2πi) = +1. $\endgroup$ – iSeeker Jan 3 '17 at 18:45
  • $\begingroup$ User_of_maths : Have you lost interest in your question? Is there a way I can re-engage you? $\endgroup$ – iSeeker Jan 7 '17 at 11:03
  • $\begingroup$ Thanks for your replies and sorry for my belated response. Of course, I am aware that vectors dwell in the tangent space to a manifold, and do not live on the manifold itself. $\endgroup$ – user_of_math Jan 7 '17 at 19:24
  • $\begingroup$ I am looking for something similar for spinors, I guess. Although a tangent space is far more easy and intuitive to understand than whatever the spinor-related structure is. $\endgroup$ – user_of_math Jan 7 '17 at 19:26

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