author={Andrzej Trautman and Darwin},
This review article begins with a short history of the notions associated with spinors; it describes several distinct ‘square root’ ideas occurring in connection with Clifford algebras, spin groups and pure spinors. Applications of pure spinors to geometry and physics are briefly presented. An appendix contains a simple derivation of the Vahlen-Ahlfors, fractional-linear form of Möbius transformations. 
Introductory and historical remarks Clifford (1878) introduced his ‘geometric algebras’ as a generalization of Grassmann algebras, complex numbers and quaternions. Lipschitz (1886) was the first to
This paper is dedicated to the memory of Matey Mateev , a missing friend Clifford Algebras and Spinors
Expository notes on Clifford algebras and spinors with a detailed discussion of Majorana, Weyl, and Dirac spinors. The paper is meant as a review of background material, needed, in particular, in now
Institute for Mathematical Physics on Complex Structures in Physics on Complex Structures in Physics
Complex numbers enter fundamental physics in at least two rather distinct ways. They are needed in quantum theories to make linear diier-ential operators into Hermitian observables. Complex
Tensor- and spinor-valued random fields with applications to continuum physics and cosmology
In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random
Quantum fluctuations of geometry in a hot Universe
The fluctuations of spacetime geometries at finite temperature are evaluated within the linearized theory of gravity. These fluctuations are described by the probability distribution of various


An introduction to the spinorial chessboard
Simple spinors and real structures
A concept of a real index associated with any maximal totally null subspace in a complexified vector space endowed with a scalar product, and also with any complex simple (pure) spinor, is introduced
Heat Kernels and Dirac Operators
The past few years have seen the emergence of new insights into the Atiyah-Singer Index Theorem for Dirac operators. In this book, elementary proofs of this theorem, and some of its more recent
Self-duality in four-dimensional Riemannian geometry
We present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis. In particular we apply this to the self-dual
Pauli-Kofink Identities and Pure Spinors
A machinery producing identities between the bilinear covariants of spinors, devised by Pauli and Kofink, is extended to the n-dimensional case and applied to pure spinors.
The index of elliptic operators on compact manifolds
1. A. H. Clifford, Naturally totally ordered commutative semigroups, Amer. J. Math. 76(1954), 631-646. 2. , Totally ordered commutative semigroups, Bull. Amer. Math. Soc. 64 (1958), 305-316. 3. O.
Physical space-time and nonrealizable ${\text{CR}}$-structures
Space-time views leading up to Einstein's general relativity are described in relation to some of Poincare's early ideas on the subject. The basic geometry of twistor theory is introduced as it
The conformal geometry of complex quadrics and the fractional‐linear form of Möbius transformations
A new derivation is given of the Vahlen (1902) form of the local conformal transformations of Cn, v ■ (av+b)(cv+d)−1, where v∈Cn and a, b, c, d are suitable elements of the complex Clifford algebra