Related papers: A Relationship Between Spin and Geometry
We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in…
Twisted geometry is a piecewise-flat geometry less rigid than Regge geometry. In Loop Gravity, it provides the classical limit for each step of the truncation utilized in the definition of the quantum theory. We define the torsionless…
This article is based on a talk given by the author at MSRI in the workshop "Connections for Women" in January 2013, while being a part of the program "Noncommutative Algebraic Geometry and Representation Theory" at MSRI. One purpose of the…
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…
Massless higher-spin fields show no preference for any value of the cosmological constant in $3d$. All matter-free higher-spin gravities in 3d are equivalent to Chern-Simons theories with an appropriate choice of gauge algebra. For various…
We present an introduction to the geometry of higher order vector and co--vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor…
In this short pedagogical presentation, we introduce the spin groups and the spinors from the point of view of group theory. We also present, independently, the construction of the low dimensional Clifford algebras. And we establish the…
Possible geometric frameworks for a unified theory of gravity and electromagnetism are investigated: General relativity is enlarged by allowing for an arbitrary complex linear connection and by constructing an extended spinor derivative…
We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra $\mathbb{R}_{2,1}$, whose minimal version may be…
The real number system is geometrically extended to include three new anticommuting square roots of plus one, each such root representing the direction of a unit vector along the orthonormal coordinate axes of Euclidean 3-space. The…
By virtue of the Noether theorems, the vast gauge redundancy of general relativity provides us with a rich algebra of boundary charges that generate physical symmetries. These charges are located at codimension-2 entangling surfaces called…
The concept of superintegrability in quantum mechanics is extended to the case of a particle with spin s=1/2 interacting with one of spin s=0. Non-trivial superintegrable systems with 8- and 9-dimensional Lie algebras of first-order…
The algebras of non-relativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic…
The Einstein-Hilbert action in three dimensions and the transformation rules for the dreibein and spin connection can be naturally described in terms of gauge theory. In this spirit, we use covariant coordinates in noncommutative gauge…
In these continuation papers (VI and VII) we are interested in approach the problem of spin from a classical point of view. In this first paper we will show that the spin is neither basically relativistic nor quantum but reflects just a…
We present an introduction to the geometry of higher order vector and co-vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor…
Within a geometric and algebraic framework, the structures which are related to the spin-statistics connection are discussed. A comparison with the Berry-Robbins approach is made. The underlying geometric structure constitutes an additional…
I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random…
We study the problem of interacting theories with (partially)-massless and conformal higher spin fields without matter in three dimensions. A new class of theories that have partially-massless fields is found, which significantly extends…
This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter -A. There is an…