By Ivan T. Todorov
Those notes arose from a chain of lectures first awarded on the Scuola Interna zionale Superiore di Studi Avanzati and the overseas Centre for Theoretical Physics in Trieste in July 1980 after which, in a longer shape, on the Universities of Sofia (1980-81) and Bielefeld (1981). Their goal has been two-fold. First, to introduce theorists with a few history in crew representations to the concept of twistors with an emphasis on their conformal houses; a quick advisor to the literature at the topic is designed to compensate partly for the imcompleteness and the one-sidedness of our evaluate. Secondly, we current a scientific research of po sitive strength conformal orbits when it comes to twistor flag manifolds. they're interpre ted as cl assi ca 1 part areas of "conformal parti cl es"; a characteri sti c estate of such debris is the dilation invariance in their mass spectrum which, there fore, is composed both of the purpose 0 or of the countless period 222 o
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Extra resources for Conformal Description of Spinning Particles
35)]. As a typical point on such an orbit we can take Z = (i,O;O,I) , . (i) , ". 8. 37) 30 3. 1 admit a unique (up to a factor) SU(2,2)-invariant symplectic form 2s dl;- II dl; Ws = de s = ---. 2) ~1; ± is a reat conformal invariant 1-form on Fl' The 2-form Ws is the imaginary part of the SU(2,2)-invariant Kahler metric (2s/~1;)d~a ®1I~d1;8 on F~, which is called the Fubini-Study metric. The standard evaluation of Poisson brackets requires the introduction of independent local coordinates on F~.
40) . 41) valid for p2 =0. 31,33)] in terms of a complex (left-handed) symmetric sPintenso~vfield ~AB with 2-point function <~AB(xH*AB(O) >0 = f eiPXeAAeBB(dP)o . 45c) A. 13. 46) One advantage of introducing the potentials 2 and 2* is the possibility they offer to write down a local Lagrangian for the electromagnetic field. 45). 47b) we used the identity !. ] The standard Lagrangian formalism for the electromagnetic field requires the A usual sacrifices. Since ts belongs to the representation (1/2, 1/2) of SL(2,C) [and not to a representation of the type (j, j +1) which appears in the above-mentioned Weinberg's theorem] its covariant quantum field theoretic treatment requires an indefinite metric space H.
SU(2,2) Orbits in the Grassmann Manifold The introduction of twistors in the preceding sections and the display of their relation to the 0(4,2) Clifford algebra in Appendix A is designed to allude to physicists' experience with Dirac spinors (and representation theory). There is a more direct way of displaying the correspondence F2 ~CR which should be more appealing to mathematicians (cf. [M5]). 26 We start with the familiar representation of points z ECM in terms of 2 x2 (complex) matrices -z zD+z3 zl_iZ2) = z~a ~ = ( Z1.
Conformal Description of Spinning Particles by Ivan T. Todorov