A theory of the S-Matrix, starting from physically plausible assumptions and looking at the mathematical consequences.

Presents the basic mathematical ideas and algorithms of the matrix analytic theory in a readable, up-to-date, and comprehensive manner.

Co-published with and In this era of “Big Data,” institutions of higher education are challenged to make the most of the information they have to improve student learning outcomes, close equity gaps, keep costs down, and address the economic needs of the communities they serve at the local, regional, and national levels. This book helps readers understand and respond to this “analytics revolution,” examining the evolving dynamics of the institutional research (IR) function, and the many audiences that institutional researchers need to serve.Internally, there is a growing need among senior leaders, administrators, faculty, advisors, and staff for decision analytics that help craft better resource strategies and bring greater efficiencies and return-on-investment for students and families. Externally, state legislators, the federal government, and philanthropies demand more forecasting and more evidence than ever before. These demands require new and creative responses, as they are added to previous demands, rather than replacing them, nor do they come with additional resources to produce the analysis to make data into actionable improvements. Thus the IR function must become that of teacher, ensuring that data and analyses are accurate, timely, accessible, and compelling, whether produced by an IR office or some other source. Despite formidable challenges, IR functions have begun to leverage big data and unlock the power of predictive tools and techniques, contributing to improved student outcomes.

This book provides a modern perspective on the analytic structure of scattering amplitudes in quantum field theory, with the goal of understanding and exploiting consequences of unitarity, causality, and locality. It focuses on the question: Can the S-matrix be complexified in a way consistent with causality? The affirmative answer has been well understood since the 1960s, in the case of 2→2 scattering of the lightest particle in theories with a mass gap at low momentum transfer, where the S-matrix is analytic everywhere except at normal-threshold branch cuts. We ask whether an analogous picture extends to realistic theories, such as the Standard Model, that include massless fields, UV/IR divergences, and unstable particles. Especially in the presence of light states running in the loops, the traditional iε prescription for approaching physical regions might break down, because causality requirements for the individual Feynman diagrams can be mutually incompatible. We demonstrate that such analyticity problems are not in contradiction with unitarity. Instead, they should be thought of as finite-width effects that disappear in the idealized 2→2 scattering amplitudes with no unstable particles, but might persist at higher multiplicity. To fix these issues, we propose an iε-like prescription for deforming branch cuts in the space of Mandelstam invariants without modifying the analytic properties of the physical amplitude. This procedure results in a complex strip around the real part of the kinematic space, where the S-matrix remains causal. We illustrate all the points on explicit examples, both symbolically and numerically, in addition to giving a pedagogical introduction to the analytic properties of the perturbative S-matrix from a modern point of view. To help with the investigation of related questions, we introduce a number of tools, including holomorphic cutting rules, new approaches to dispersion relations, as well as formulae for local behavior of Feynman integrals near branch points. This book is well suited for anyone with knowledge of quantum field theory at a graduate level who wants to become familiar with the complex-analytic structure of Feynman integrals.

A Pick function is a function that is analytic in the upper half-plane with positive imaginary part. In the first part of this book we try to give a readable account of this class of functions as well as one of the standard proofs of the spectral theorem based on properties of this class. In the remainder of the book we treat a closely related topic: Loewner's theory of monotone matrix functions and his analytic continuation theorem which guarantees that a real function on an interval of the real axis which is a monotone matrix function of arbitrarily high order is the restriction to that interval of a Pick function. In recent years this theorem has been complemented by the Loewner-FitzGerald theorem, giving necessary and sufficient conditions that the continuation provided by Loewner's theorem be univalent. In order that our presentation should be as complete and trans parent as possible, we have adjoined short chapters containing the in formation about reproducing kernels, almost positive matrices and certain classes of conformal mappings needed for our proofs.

The Maxwell, Einstein, Schr”dinger and Dirac equations are considered the most important equations in all of physics. This volume aims to provide new eight- and twelve-dimensional complex solutions to these equations for the first time in order to reveal their richness and continued importance for advancing fundamental Physics. If M-Theory is to keep its promise of defining the ultimate structure of matter and spacetime, it is only through the topological configurations of additional dimensionality (or degrees of freedom) that this will be possible. Stretching the exploration of complex space through all of the main equations of Physics should help tighten the noose on ?the? fundamental theory. This kind of exploration of higher dimensional spacetime has for the most part been neglected by M-theorists and physicists in general and is taken to its penultimate form here.