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Two-dimensional massless fermion field theories with quartic interactions are analyzed. These models 10 3235

Elementary particle physics has made remarkable progress in the past ten years. We now have, for the first time, a comprehensive theory of particle interactions. One can argue that it gives a complete and correct description of all non-gravitational physics. This theory is based on the principle of gauge symmetry. Strong, weak, and electromagnetic interactions are all gauge interactions. The importance of a knowledge of gauge theory to anyone interested in modern high energy physics can scarcely be overstated. Regardless of the ultimate correctness of every detail of this theory, it is the framework within which new theoretical and experimental advances will be interpreted in the foreseeable future.The aim of this book is to provide student and researcher with a practical introduction to some of the principal ideas in gauge theories and their applications to elementary particle physics. Wherever, possible we avoid intricate mathematical proofs and rely on heuristic arguments and illustrative examples. We have also taken particular care to include in the derivations intermediate steps which are usually omitted in more specialized communications. Some well-known results are derived anew, in a way more accessible to a non-expert.The book is not intended as an exhaustive survey. However, it should adequately provide the general background necessary for a serious student who wishes to specialize in the field of elementary particle theory. We also hope that experimental physicists with interest in some general aspects of gauge theory will find parts of the book useful.The material is based primarily on a set of notes for the graduate courses taught by one of us (L.F.L.) over the past six years at the Carnegie-Mellon University and on lectures delivered at the 1981 Hefei (China) Summer School on Particle Physics (Li 198 l ). It is augmented by material covered in seminars given by the other author (T.P.C.) at the University of Minnesota and elsewhere. These notes have been considerably amplified, reorganized, and their scope expanded. In this text we shall assume that the reader has had some exposure to quantum field theory. She or he should also be moderately familiar with the phenomenology of high energy physics. In practical terms we have in mind as a typical reader an advanced graduate student in theoretical physics; it is also our hope that some researchers will use the book as a convenient guide to topics that they wish to look up.Modern gauge theory may be described as being a 'radically conservative theory' in the sense used by J. A. Wheeler (see Wilczek 1982b). Thus, one extrapolates a few fundamental principles as far as one can, accepting some 'paradoxes' that fall short of contradiction. Here we take as axioms the principles of locality, causality, and renormalizability. We discover that a VI Preface Preface Vil gramme. A certain amount of repetition is deliberately built into the book so that the reader can pick and choose different sections without any serious problems. An experimentally inclined reader, w...

A.symptotically free gauge theories of the strong interactions are constructed and analyzed. The reasons for doing this are recounted, including a review of renormalization-group techniques and their application to scaling phenomena. The renormalization-group equations are derived for Yang-Mills theories. The parameters that enter into the equations are calculated to lowest order and it is shown that these theories are asymptotically free. More specifically the effective coupling constant, which determines the ultraviolet behavior of the theory, vanishes for large spacelike momenta. Fermions are incorporated and the construction of realistic models is discussed. %'e propose that the strong interactions be mediated by a "color" gauge group which commutes with SU(3) &(SU(3). The problem of symmetry breaking is discussed. It appears likely that this would have a dynamical origin. It is suggested that the gauge symmetry might not be broken and that the severe infrared singularities prevent the occurrence of noncolor singlet physical states. The deep-inelastic structure functions, as well as the electron-positron total annihilation cross section are analyzed. Scaling obtains up to calculable logarithmic corrections, and the naive light-cone or parton-model results follow. The problems of incorporating scalar mesons and breaking the symmetry by the Higgs mechanism are explained in detail.

We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rd log 2 d) measurement settings, compared to standard methods that require d 2 settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low-rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed. We present both theoretical bounds and numerical simulations.The tasks of reconstructing the quantum states and processes produced by physical systems -known respectively as quantum state and process tomography [1] -are of increasing importance in physics and especially in quantum information science. Tomography has been used to characterize the quantum state of trapped ions [2] and an optical entangling gate [3] among many other implementations. But a fundamental difficulty in performing tomography on many-body systems is the exponential growth in the state space dimension. For example, to get a maximum-likelihood estimate of a quantum state of 8 ions, Ref.[2] required hundreds of thousands of measurements and weeks of post-processing.Still, one might hope to overcome this obstacle, because the vast majority of quantum states are not of physical interest. Rather, one is often interested in states with special properties: pure states, states with particular symmetries, ground states of local Hamiltonians, etc., and tomography might be more efficient in such special cases [4].In particular, consider pure or nearly pure quantum states, i.e., states with low entropy. More precisely, consider a quantum state that is essentially supported on an r-dimensional space, meaning the density matrix is close (in a given norm) to a matrix of rank r, where r is small. Such states arise in very common physical settings, e.g. a pure state subject to a local noise process [20].A standard implementation of tomography [5,6] would use d 2 or more measurement settings, where d = 2 n for an nqubit system. But a simple parameter counting argument suggests that O(rd) settings could possibly suffice -a significant improvement. However, it is not clear how to achieve this performance in practice, i.e., how to choose these measurements, or how to efficiently reconstruct the density matrix. For instance, the problem of finding a minimum-rank matrix subject to linear constraints is NP-hard in general [7].In addition to a reduction in experimental complexity, one might hope that a post-processing algorithm which takes as input only O(rd) ≪ d 2 numbers could be tuned to run considerably faster than standard methods. Since the output of the procedure is a low...

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