3-D Spinors, Spin-Weighted Functions and their Applications by Gerardo F. Torres del Castillo

By Gerardo F. Torres del Castillo

This systematic and self-contained therapy of the speculation of three-d spinors and their purposes fills a big hole within the literature. with out utilizing the well-known Clifford algebras usually studied in reference to the representations of orthogonal teams, spinors are built during this paintings for 3-dimensional areas in a language analogous to the spinor formalism utilized in relativistic spacetime.

Unique positive factors of this work:

* Systematic, coherent exposition throughout

* Introductory remedy of spinors, requiring no past wisdom of spinors or complicated wisdom of Lie groups

* 3 chapters dedicated to the definition, homes and purposes of spin-weighted services, with all historical past given.

* particular therapy of spin-weighted round harmonics, homes and lots of purposes, with examples from electrodynamics, quantum mechanics, and relativity

* wide selection of subject matters, together with the algebraic type of spinors, conformal rescalings, connections with torsion and Cartan's structural equations in spinor shape, spin weight, spin-weighted operators and the geometrical that means of the Ricci rotation coefficients

* Bibliography and index

This paintings will serve graduate scholars and researchers in arithmetic and mathematical and theoretical physics; it's appropriate as a path or seminar textual content, as a reference textual content, and will even be used for self-study.

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Extra resources for 3-D Spinors, Spin-Weighted Functions and their Applications

Sample text

Taking into account that (u, v, Pu, Pv) and (-u, -v, -Pu, -Pv) correspond to the same point (x, y, Px, Py), it follows that 1. Rotations and Spinors 32 SOo(2,1) acts on the phase space as a dynamical symmetry group of the twodimensional Kepler problem with positive energy (note that k may be positive or negative). Alternative definition Another procedure for defining spinors in a three-dimensional space with indefinite metric, which shows the existence of a homomorphism of SOo(2,1) with SL(2,]R), is obtained by considering the stereographic projection of the circle onto the extended real line.

128) one obtains ~ = (i + ~)/(i - ~) and it follows that 4d~df/(1 - ~f)2 = d~df/(lm~)2. 128) is an isometry and, according to the preceding results, the linear fractional transformations l:: 5 H- a~ +b e~ +d' with a, b, e, d E R such that ad - be = 1, are isometries of d~df/(lm~)2. , Stillwell 1992) or the Poincare half-plane, which models the Lobachevsky geometry. 1 Spherical harmonics The spherical harmonics can be defined in various ways; they are eigenfunctions of the Laplace-Beltrami operator of the sphere and they are the angular part of the separable solutions in spherical coordinates of the Laplace equation in Euclidean space.

O ) ''''/2 e1'l" 1 cos 20 . e il/> sinO) = dll0 1(]1 ~ dn (-~Yl'-l) +dl2 (~Yl'O) +dn (-~Yl'} where we have made use of the standard notation for the spherical harmonics. Since each component 0 1 or ot contains a factor e- il/>j2 and each component 2 0 or (J2 contains a factor eil/>/2, the spherical harmonic of order I, o(AoB ... oC(;DQE ... OF), is an eigenfunction of the operator L z == -i8j84> with 2. (n2 - nl), where nl [resp. n2] is the number of indices AB ... F taking the value 1 [resp. 2].

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