Renormalization Group And Fixed Points PDF Download

In this dissertation, the distinction between conformal invariance and two similar symmetries, scale invariance and Weyl invariance, is clarified. For the distinction between scale and conformal invariance, we focus on d = 4. Using the operator product expansion in momentum space as well as the form of the scale anomaly in d = 4, we are able to complement other recent work in order to rule out all but a very small class of scale-but-not-conformally invariant theories. The techniques used are similar to those used in another attempted proof of ours that scale invariance implies conformal invariance in d = 4. We comment on where exactly the proof failed as well as the usefulness of the techniques involved. We also show that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions d ≤ 10. We include possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify anomalies. We also present an example of the usefulness of conformal symmetry when combined with energy positivity constraints. We argue that all consistent quantum field theories obey a spacetime-averaged weak energy inequality. If this condition is violated, the theory has states that are indistinguishable from states of negative total energy by any local measurement, and we expect instabilities or other inconsistencies. We apply this condition to 3D and 4D conformal field theories, and find the constraints it places on the OPE coefficients of the theory.