The Spinor Enigma

While studying some intriguing papers by Vadim Varlamov (refer to my penultimate post, "The Quantum Leap: A Journey Beyond Reality"), I visited his Russian blog. I couldn't resist translating one of his posts about spinors into English. Here it is:

"What is a Spinor ???"

V.V. Varlamov: What is a spinor?

No one fully understands spinors. Their algebra is formally understood, but their geometric significance remains enigmatic.

– Michael Atiyah [1, p. 430]

Michael Atiyah (1929-2019)

The quotation from Atiyah about the "mystery" of spinors suggests that in the search for the fundamental element of matter (prima materia), the concept of the spinor plays a crucial role. Let us explore the origins of this notion, tracing back to Hamilton's quaternions, and delve into the geometric (Cartan) and algebraic (Brauer-Weil) definitions of spinors. The theory of Clifford algebras has its roots in Hamilton's quaternions [2] and Grassmann's theory of extension [3].

William Rowan Hamilton (1805-1865)

Hermann Günther Grassmann (1809-1877)

William Kingdon Clifford (1845-1879)

The algebra introduced by Clifford [4] generalizes quaternion algebra to multidimensional spaces. Consequently, a quaternion structure emerges within Clifford algebra as a tensor product of quaternion algebras, i.e., a tensor product of four-dimensional algebras (the dimension of quaternion algebra is 4). While studying the rotations of n-dimensional Euclidean space R(n), Lipschitz [5] discovered that the rotation group of space R(n) with determinant +1 is doubly represented by the spinor group.

Rudolph Lipschitz (1832-1903)

It is known [6] that the motion groups of n-dimensional non-Euclidean spaces S(n) are isomorphic to the rotation groups of spaces R(n+1). Since Clifford algebras are isomorphic to matrix algebras, spinor representations of the motions of spaces S(n) can be viewed as representations of these motions by linear transformations of vectors in the corresponding spaces. The vectors in these spaces are called spinors of spaces S(n). The concept of the spinor was introduced by Cartan [7].

Élie Joseph Cartan (1869-1951)

Van der Waerden [8] notes that the term "spinor" was coined by Ehrenfest after the publication of the famous paper by Goudsmit and Uhlenbeck [9] on the spinning electron. More precisely, the geometric meaning of spinor representations of the motions of non-Euclidean spaces S(n) is that the coordinates of spinors can be considered as the coordinates of flat generators of the maximal dimension of the absolutes of these spaces. The spinor representations of the motions of these spaces coincide with those transformations of spinors that correspond to transformations of the absolutes under motions. The absolute is a set of infinitely distant points in non-Euclidean space. For example, in the case significant to physics, it is known that the connected motion group of three-dimensional non-Euclidean space S(1,2) (Lobachevsky space) is isomorphic to the connected rotation group of four-dimensional pseudo-Euclidean space R(1,3) (Minkowski space-time), which coincides with the Lorentz transformation group of special relativity. Thus, the spinor representation of the connected motion group of space S(1,2) (by complex second-order matrices with determinant +1) is also a spinor representation of the Lorentz group. Therefore, each spinor of space S(1,2) corresponds to a certain point of the absolute of space S(1,2), and each point of the absolute of space S(1,2) corresponds to an isotropic line of space R(1,3) passing through a certain point of this space. The absolute of Lobachevsky space S(1,2) is homeomorphic to the extended complex plane C ∪ {∞}.

Nikolai Ivanovich Lobachevsky (1792-1856)

The geometric interpretation of spinors and spinor representations outlined above was proposed by Cartan [10] (see also [6]).

In his book [11], Chevalley notes that the concept of the spinor given by Cartan is quite complex, and a simpler representation of the theory, based on the use of Clifford algebras, was proposed in the paper by Brauer and Weyl [12].

Luitzen Egbertus Jan Brouwer (1881-1966)

Hermann Klaus Hugo Weyl (1885-1955)

Claude Chevalley (1909-1984)

In his exposition of the algebraic theory of spinors, Chevalley follows the paper [12]. Thus, alongside Cartan's geometric interpretation, an algebraic approach to describing spinors and spinor representations emerged, further developed in works [13-16]. The productivity and maturity of the algebraic approach demonstrate that a spinor is primarily an object of algebraic nature. According to the algebraic definition, a spinor is defined by an element of the minimal left ideal of the Clifford algebra Cl(V, Q), where V is a vector space equipped with a non-degenerate quadratic form Q. For even n, the minimal left ideal of the algebra Cl(V, Q) corresponds to the maximal totally isotropic subspace U ⊂ V of dimension n/2, i.e., it is isomorphic to the spin space S of dimension 2^(n/2) [17].

In the pursuit of understanding the primordial essence of matter, spinors remain elusive, their true nature cloaked in layers of complexity and profound significance. From the elegant formulations of Hamilton’s quaternions to the enigmatic representations in Clifford algebras, spinors encapsulate a mystery that beckons further exploration, a tantalizing glimpse into the very fabric of the universe.

REFERENCES

1. Farmelo G. The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genious.New York: Basic Books, 2009

2. Hamilton W.R. Lectures on Quaternions. Dublin, 1853.

3. Grassmann H. Die Ausdehnungslehre. Berlin, 1862.

4. Clifford W.K. Applications of Grassmann’s extensive algebra. Amer. J. Math. 1878.V. 1. P. 350.

5. Lipschitz R. Untersuchungen ̈uber die Summen von Quadraten. Max Cohen und Sohn,Bonn, 1886.

6. Rosenfeld B.A. Non-Euclidean geometries (in Russian). М. : ГИТТЛ, 1955. 744 p.

7. Cartan E. Les groupes projectifs qui ne laissent invariante aucune multiplicit ́e plane// Bull. Soc. math. France. 1913. V. 41. P. 53–96.

8. van der Waerden B.L. “Exclusion Principle and Spin,” in Theoretical Physics in the Twentieth Century, Interscience, New York 1960.

9. Uhlenbeck G.E., Goudsmit S. Spinning Electrons and the Structure of Spectra  Nature. 1926. V. 117. P. 264–265.

10. Cartan, E. Theory of Spinors: Dover, 1981.

11. Chevalley C. The Algebraic Theory of Spinors. New York: Columbia University Press,1954.

12. Brauer R., Weyl H. Spinors in 𝑛 dimensions  Amer. J. Math. 1935. V. 57. P. 425–449.

13. Porteous I.R. Topological Geometry. van Nostrand, London, 1969.

14. Crumeyrolle A. Orthogonal and Symplectic Clifford Algebras, Spinor Structures.Kluwer Acad. Publ., Dordrecht, 1991.

15. Lounesto P. Clifford Algebras and Spinors. Cambridge Univ. Press, Cambridge, 2001.

16. Riesz M. Sur certain notions fondamentales en th ́eorie quantique relativiste / C.R.10𝑒 Congr ́es Math. Scandinaves (Copenhagen, 1946). Jnl. Cjellerups. Forlag, Copenhagen,1947, pp. 123–148.

17. Ablamowicz R. Construction of Spinors via Witt Decomposition and Primitive Idempotents: A Review / Clifford Algebras and Spinor Structures. Kluwer Academic Publishers, 1995. P. 113–124.