# Is there any Classical Mechanics system which needs to be described by a spinor?

We need an ordinary number (scalar) to describe a harmonic oscillator, and a vector to describe, for example, a pendulum.

Is there any similarly simple system which we need to describe using a (two-component Weyl) spinor?

• Thanks for your comment! Unfortunately, I wasn't able to find classical models mentioned in the Wiki article. (But I learned that "8 Euclidean dimensions" counts as "low dimensions" on Wikipedia) – JakobH Feb 9 at 10:16
• Related: physics.stackexchange.com/q/444730/50583, physics.stackexchange.com/q/261215/50583, mathoverflow.net/q/66681 | Also related: physics.stackexchange.com/a/33217/50583, in which David Bar Moshe exhibits a classical phase space $S^2$ with the spinorial $\mathrm{SU}(2)$ rather than $\mathrm{SO}(3)$, but it does not obviously correspond to any "real" classical system. So your question boils down to the analogue of this question with "torus" replaced by "sphere". – ACuriousMind Feb 9 at 10:47
• Goldstein's Classical Mechanics (Second ed., not third) discusses spinors in classical systems, though it amounted to just rotations using SU(2) as the double covering of SO(3) IIRC. – Alex Nelson Feb 11 at 14:26
• @AlexNelson. Hestenes' paper (op. cit.) says (referencing 1950 edition) "...missed opportunities to exploit spinors in classical mechanics by Goldstein ... Altogether, the chapter is a jumble of three different descriptions of rotations only loosely tied together. Moreover, it is unnecessarily restricted in generality." Thanks, I should look out 2nd edn. – iSeeker Feb 11 at 15:28

Yes, there is a simple system which is efficiently described using 2-component spinors: the dynamics of a classical spinning top.

As Andrew Steane says, in his excellent undergrad introduction to spinors https://arxiv.org/abs/1312.3824

It appears that [Felix] Klein originally designed the spinor to simplify the treatment of the classical spinning top in 1897. The more thorough understanding of spinors as mathematical objects is credited to Elie Cartan in 1913. They are closely related to Hamilton’s quaternions (about 1845).

Since one definition of a spinor is its 720-degree identity under rotation, Hamilton’s quaternions, used to describe e.g. rotations in computer imagery and aeronautics, can also be considered as spinors of a relatively simple, but classical kind.

For a more detailed discussion, see Chapter 5 of Laszlo Tisza’s MIT lectures on Applied Geometric Algebra ( https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009/lecture-notes-contents/Ch5.pdf )

The two-component complex vectors are traditionally called spinors. We wish to show that they give rise to a wide range of applications. In fact we shall introduce the spinor concept as a natural answer to a problem that arises in the context of rotational motion.

Tisza’s chapter works through from triads to Cayley-Klein parameters and thence to 2-complex component spinors. Technically, his spinors would be non-relativistic Pauli spinors rather than Weyl spinors as they transform under SU(2) rather than SL(2,C).

The same topic (no pun intended) is touched on by David Hestenes in his New Foundations for Classical Mechanics (2nd ed. pp. 466+) and, in terms of Cayley-Klein parameters only, by Springborn in https://arxiv.org/pdf/math/0007206.pdf (Goldstein, as advised by Alex Nelson above, mentions spinors in his 2nd edn (1980); but not in his 3rd edn (2002), nor, according to Hestenes in the 1st edn 1950 or 1951).

Late addition: David Hestenes paper Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics addresses the OP's question more directly.

"A wider use of spinors in classical mechanics ought to dispel the pervasive mistaken impression that spinors are a special feature of quantum mechanics."

But it should be borne in mind that Hestenes' definition of spinor rests on an interpretation of the i in the usual complex plane as a bivector, enabling him to label the complex plane as the "spinor i-plane". This interpretation is elaborated in 'Section 2-2. The Algebra of a Euclidean Plane' in his New Foundations for Classical Mechanics (op. cit.).

• Thanks! I remember reading about Hestenes' geometrical algebra approach a while ago but remember very little. Is there a short way to understand how the defining property of spinors (rotation by 360° yields a minus sign, rotation by 720° the original state) can be understood using a classical spinning top or Hestenes' bivectors? – JakobH Feb 12 at 8:11
• @JakobH It won't satisfy purist mathematicians, but I recommend, initially, a look at Section 11.3 of Roger Penrose's Road to Reality in which he shows how to illustrate this property, most convincingly, using a book and a long belt (a variant of the well known Tangloids/Dirac Scissors/Balinese Candle trick). It's also illustrated in en.wikipedia.org/wiki/Spinor and much discussed on its Talk pages.The full explanation involves the topology of shrinking paths through the solid-sphere representation of all rotations in 3D, SO(3) & SU(2) etc., but I can't find an easy reference just now. – iSeeker Feb 12 at 17:15
• @JakobH (Cont...) I don't think the character of bivectors alone can clarify the 360°/720° property, nor a top's spinning motion per se. You might get an easier handle on it via the 'orientation entanglement relation'; see: en.wikipedia.org/wiki/Orientation_entanglement – iSeeker Feb 12 at 19:43