By Gerardo F. Torres del Castillo
This systematic and self-contained therapy of the speculation of 3-dimensional spinors and their purposes fills an incredible hole within the literature. with no utilizing the known Clifford algebras often studied in reference to the representations of orthogonal teams, spinors are built during this paintings for third-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 therapy of spinors, requiring no earlier wisdom of spinors or complex wisdom of Lie groups
* 3 chapters dedicated to the definition, homes and functions of spin-weighted features, with all history given.
* certain remedy of spin-weighted round harmonics, homes and lots of purposes, with examples from electrodynamics, quantum mechanics, and relativity
* wide selection of issues, 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 really is compatible as a path or seminar textual content, as a reference textual content, and will even be used for self-study.
Read Online or Download 3-D Spinors, Spin-Weighted Functions and their Applications PDF
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Additional info for 3-D Spinors, Spin-Weighted Functions and their Applications
33) Cjm sYjm. j=lslm=-j In effect, if f is a function with spin weight s > 0, the product f (jA(jl ... -" 2s ° 7 has spin weight (when f has spin weight s < 0, we consider in place of f). 34) j=Om=-j where b AB ... c (j, m) are some constants totally symmetric in the 2s indices A, B, ... , C. •. 29)] 00 f = j L L bAB ... c(j, m)oAoB ... 35) Each product s Y sm ' Yjm can be expressed as a linear combination of spinweighted spherical harmonics of spin weight s and orders j + s, j + s - 1, ... 37) 2.
2 2/+1 . d h A ... BC ... D A ... BC ... D ~ 1 l+m (2/)! (/-m)! (l-m)! ~~ ~~. (/-m)! I'd . 10) l-ml+m are components of the spherical harmonics dA ... BC ... DO A •.. oB(jC ... (jD with respect to an orthonormal basis. Thus, expressing the spherical harmonic dA ... BC ... 10) we have dA ... BC ... DO A ... oB(jC ... )/-m)! ~~ ' " ( l)m ~ - l-ml+m m=~ (/-m) l's, (l+m) 2's ~ where the factors (-I)m have been introduced in order to get agreement with the convention employed in quantum mechanics.
L - m')! 50) which means thatthe representation ofSO(3) given by the matrices D~'m is unitary. 49) we have D~'m(¢' e, X) = mm (2l)! (l - (l+m') m')! e, X» _ e imx 2's ~(101 ... 0 1 ~(jo . 29) we find that DIm'm('I', A. 31), e) m' I (A. 52) 2. Spin- Weighted Spherical Harmonics 54 (cf Goldberg et al. 1967, Torres del Castillo and Hernandez-Guevara 1995). This last equation shows that D~'m(d~'m(8)e-imX. (l-m ' )!