[Paper Review] On the spinor formalism for even n

K. V. Andreev|arXiv (Cornell University)|Oct 21, 2011

Algebraic and Geometric Analysis5 references3 citations

TL;DR

This paper develops a comprehensive spinor formalism for even-dimensional spaces, focusing on n=6 and n=8, using Clifford algebras, connecting operators, and geometric constructions involving quadrics and twistors. It establishes explicit isomorphisms, constructs curvature spinors, and generalizes the Cartan triality principle via the Klein correspondence, revealing deep connections between spinor structures, the Weyl tensor, and octonionic algebra for n=8.

ABSTRACT

Spinor formalism is the formalism induced by solutions of the Clifford equation (the connecting operators). For the space-time manifold (n = 4), these operators, connecting the tangent and spinor bundle, are operators that are represented by the Dirac matrices in the special basis. Reduced connecting operators are represented by the Pauli matrices. In order to uniquely prolong the Killing equation from the tangent bundle onto the spinor bundle over the space-time manifold, it is necessary to pass to the complexification of the manifold and the corresponding bundles, and then to pass to the real representation. Returning the reverse motion, one can already obtain two copies of the spinor bundle. Their set (the pair-spinor) allows to construct the Lie operator analogues for the spinors (and the pair-spinors). Similar procedure is feasible for any even n. For n=6, the specified formalism is closely connected with the Bogolyubov-Valatin transformations. For n mod 8=0, being based on the Bott periodicity, the reduced connecting operators generate the structural constants of an hypercomplex algebra (without division for n> 8) with the alternative-elastic, flexible (Jordan), and "norm" identities. For n = 8, such the algebra is the octonion algebra. In addition, in the article the various options of the prolonging of the connection to the spinor bundle with even-dimensional base are considered, and the corresponding curvature spinors are constructed.

Motivation & Objective

• To develop a systematic spinor formalism for even-dimensional spaces, particularly n=6 and n=8, based on Clifford algebra and real/complex representations.
• To construct explicit isomorphisms and double coverings using connecting operators for orthogonal groups in even dimensions.
• To generalize the Cartan triality principle to the Klein correspondence and explore its geometric realization in n=8.
• To establish a geometric representation of twistors in R⁶(2,4) and relate them to quadrics and Grassmannian geometry.
• To derive curvature spinors and link them to the Weyl tensor and Bianchi identities via spinor analogs of the Riemann curvature tensor.

Proposed method

• Constructs real and complex representations of Clifford algebras via involutions and real inclusions, using matrix forms of the tensor S for dimensional reductions.
• Employs connecting operators ηΛKL to realize double coverings of orthogonal groups and to relate spinor and vector representations.
• Applies the Clifford equation and infinitesimal transformation techniques to derive Lie operator analogues and curvature spinor structures.
• Uses the Rosenfeld null-pair construction and geometric transitions between quadrics CQ6 and C̃Q6 to model the inductive step from n=6 to n=8.
• Establishes identities for the octonion algebra (e.g., Moufang and alternative identities) and proves their role in the n=8 spinor formalism.
• Derives the twistor equation and relates it to the conformal Killing equation and normalized Grassmannian geometry via the Klein correspondence.

Experimental results

Research questions

• RQ1 How can the spinor formalism be systematically constructed for even n ≥ 4 using Clifford algebras and connecting operators?
• RQ2 What is the geometric and algebraic structure of curvature spinors in n=6 and n=8, and how do they relate to the Weyl tensor and Bianchi identities?
• RQ3 How does the Klein correspondence generalize the Cartan triality principle in the context of n=8 spinor geometry?
• RQ4 What is the role of the octonion algebra and its structure constants in the spinor formalism for n=8?
• RQ5 How are twistors in R⁶(2,4) geometrically represented, and how do they relate to quadrics and Grassmannian subspaces?

Key findings

• The paper constructs explicit isomorphisms between spinor and vector representations for n=6 and n=8 using connecting operators ηΛKL.
• For n=6, it classifies tensors with Riemann curvature symmetries and defines curvature spinors via spinor analogs of the Riemann tensor.
• A geometric representation of twistors in R⁶(2,4) is established through the correspondence CP³ ⊃ CP¹ ↔ CP¹ ⊂ K₆.
• For n=8, the spinor formalism is built on the octonion algebra, with structure constants derived from the A-operator identities and the central Moufang identity.
• The inductive step from n=6 to n=8 is realized via the Rosenfeld null-pair and the construction of dual quadrics CQ6 and C̃Q6, linked by connecting operators.
• The paper proves a theorem on two quadrics, showing that the geometry of CQ6 and C̃Q6 is equivalent under a spinor duality, with implications for Majorana spinors and invariant structures.