Conformal Description of Spinning Particles

ISBN-13: 9783540168904 / Angielski / Miękka / 1986 / 76 str.

Ivan T. Todorov

Opis

These notes arose from a series of lectures first presented at the Scuola Interna- zionale Superiore di Studi Avanzati and the International Centre for Theoretical Physics in Trieste in July 1980 and then, in an extended form, at the Universities of Sofia (1980-81) and Bielefeld (1981). Their objective has been two-fold. First, to introduce theorists with some background in group representations to the notion of twistors with an emphasis on their conformal properties; a short guide to the literature on the subject is designed to compensate in part for the imcompleteness and the one-sidedness of our review. Secondly, we present a systematic study of po- sitive energy conformal orbits in terms of twistor flag manifolds. They are interpre- ted as cl assi ca 1 phase spaces of conformal parti cl es; a characteri sti c property of such particles is the dilation invariance of their mass spectrum which, there fore, consists either of the point zero or of the infinite interval 222 o

Spis treści

A Guide to the List of References.- 1. The Conformal Group of a Conformally Flat Space Time and Its Twistor Representations.- 1.1 Conformal Classes of Pseudo-Riemannian Metrics.- 1.2 Connection and Curvature Forms — a Recapitulation. The Weyl Curvature Tensor.- 1.3 Global Conformal Transformations in Compactified Minkowski Space. Conformal Invariant Local Causal Order on $$ \overline {\text} $$.- 1.4 The Lie Algebra of the Conformal Group and Its Twistor Representations.- 2. Twistor Flag Manifolds and SU(2,2) Orbits.- 2.1 Seven Flag Manifolds in Twistor Space. Conformal Orbits in F1 =PT.- 2.2 Points of Compactified Space-Time as 2-Planes in Twistor Space.- 2.3 An Alternative Realization of the Isomorphism $$ \overline {{\text}} \Leftrightarrow {{\text}_2} $$ SU(2,2) Orbits in the Grassmann Manifold.- 2.4 Higher Flag Manifolds.- 3. Classical Phase Space of Conformal Spinning Particles.- 3.1 The Conformal Orbits F1+ and F1? as Phase Spaces of Negative and Positive Helicity O-Mass Particles.- 3.2 Canonical Symplectic Structure on Twistor Space; a Unified Phase Space Picture for Free O-Mass Particles.- 3.3 The Phase Space of Spinless Positive Mass “Conformal Particles”.- 3.4 The 10-Dimensional Phase Space of a Timelike Spinning Particle.- 3.5 The 12-Dimensional Phase Space F1,2,3?.- 4. Twistor Description of Classical Zero Mass Fields.- 4.1 Quantization of a Zero Mass Particle System: The Ladder Representations of U(2,2).- 4.2 Local Zero Mass Fields. Second Quantization.- 4.3 The Neutrino and the Photon Fields in the Twistor Picture.- 4.4 Remark on the Quantization of Higher-Dimensional Conformal Orbits.- Appendix A.Clifford Algebra Approach to Twistors. Relation to Dirac Spinors.- A.1 Clifford Algebra of O(6,?) and Bitwistor Representation of the Lie Algebra SO(6,?).- A.2 The Homomorphism SL(4,?) ? SO(6,?). Inequivalent 4-Dimensional Analytic Representations of SL(4,?).- A.3 Conformal Dirac Spinors.- References.