By Ivan T. Todorov
Those notes arose from a chain of lectures first offered on the Scuola Interna zionale Superiore di Studi Avanzati and the foreign 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 heritage in team representations to the idea of twistors with an emphasis on their conformal houses; a quick advisor to the literature at the topic is designed to compensate partially for the imcompleteness and the one-sidedness of our overview. Secondly, we current a scientific learn of po sitive power conformal orbits by way of 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 limitless period 222 o
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E. the expression for the SU(2,2) generators Jab in terms of the z-variables. Every antisymmetric tensor Jab on PQ is proportional to i(ZaZb -ZbZa)/ZZ. The coefficient is determined by the requirement that Jab provide a Poisson bracket realization of the conformal Lie algebra. 31)]. 5. 42). 42)] is to express Jab in terms of the independent canonical coordinates x and p. 44b) The SO(4,2) Lie algebra relations now appear as a straightforward consequence of the canonical commutation relations among xjl and pv .
The eigenvalues of these two central elements determine completely the corresponding, irreducible representation of SU(2) xSU(2) in H. ] This ladder structure of the twistor representations of U(2,2) with respect to their U(2) xU(2) content has given rise to the name "ladder representations" for the series under consideration. They can be integrated to global representations of the 4-fold covering SU(2,2) of the conformal group of Minkowski space and form a most degenerate series of unitary irreducible representations of this group.
5. 42). 42)] is to express Jab in terms of the independent canonical coordinates x and p. 44b) The SO(4,2) Lie algebra relations now appear as a straightforward consequence of the canonical commutation relations among xjl and pv . 39) plays the role of particle momentum. Since <0 in both T+ and T_, energy ~ositivity implies pyO <0. We shall choose in what follows p >0, identifying the classical phase space of a positive energy spinless "conformal particle" with the backward tube T_ ( F;-). I~I.