Applications of the conformal group in physics

The conformal group C(3,1) of Minkowski space-time M is a fifteen parameter Lie group of transformations leaving the causal structure invariant. It contains the Lorentz group. The causal structure is the definition of precedence of events deduced from the forward light-cones of space-time. The light-cone structure is equivalent to a class of general scaling related metrics called the conformal structure. The causal invariances of a manifold are discussed in terms of the conformal structure. Metric scalings, which are transformations within the conformal structure, are examined along with the conformally flat case of a conformal structure. Simple scale invariant fields on a manifold are developed, and the two types of dimension, canonical and metric, are contrasted. C(3,1) acts on M nonlinearly. It is locally equivalent to the group SO0(4,2) acting linearly on a six dimensional space C called conformal space. SU(2,2), the spin covering of SO0(anti 4,anti 2), acts on twistor space. Properties and relations of the three spaces M, C, and T are investigated. Using the results for the conformally flat metrics, the chronogeometric redshift theory of I. E. Segal is discussed. This model introduces a new mechanism to describe the astrophysical redshift of galaxies, the isotropic microwave background, and the unusual qualities of quasars. The result is derived classically and contrasted with his quantum derivation. It is shown that broadening his hypotheses permits a family of similar results with consequential lower predictive power. Difficulties in generalizing to an arbitrary metric gravitational background arise. Only the conformally flat geometries are tractable. The Robertson--Walker--Friedmann case is worked out showing mixed expansion and chronogeometric redshift contributions