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.

This Brief presents an introduction to the theory of the renormalization group in the context of quantum field theories of relevance to particle physics. Emphasis is placed on gaining a physical understanding of the running of the couplings. The Wilsonian version of the renormalization group is related to conventional perturbative calculations with dimensional regularization and minimal subtraction. An introduction is given to some of the remarkable renormalization group properties of supersymmetric theories.

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This is the proceedings of the third Nagoya workshop on Strong Coupling Gauge Theories (SCGT), after SCGT 88 and SCGT 90. As a tradition of the Nagoya SCGT workshops, the focus is on dynamical symmetry breaking with particular emphasis on the nontrivial fixed points and/or large anomalous dimension, which was actually the basis of walking technicolor, strong ETC technicolor and top quark condensate, etc. Special attention is also paid to the fixed point structure in supersymmetric gauge theories, which has recently been highlighted through duality arguments.

At a fundamental level, the interaction of quarks with gluon fields lies at the heart of our understanding of the strong nuclear force. Experimentally, however, we only observe physical hadrons such as protons and pions. This book explores the fascinating physics involved in the path between these contrasting pictures of the world. Along the way, the book discusses symmetries, which play a crucial role in understanding the parameters of the theory, and details of the spectrum of physical particles.This would be the first book to elaborate on the detailed connections between confinement and chiral symmetry, with an emphasis on a unified treatment of the non-perturbative nature of these phenomena. As such, it should be a valuable title on any particle theorist's bookshelf, containing extensive pedagogical material for scientists at the graduate level and above.

This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. It begins with a brief review of phase transitions in simple systems, then goes on to introduce the core ideas of the renormalisation group.