The Schr\"odinger Functional - a Renormalizable Probe for Non-Abelian Gauge Theories

Following Symanzik we argue that the Schr\"odinger functional in lattice gauge theories without matter fields has a well-defined continuum limit. Due to gauge invariance no extra counter terms are required. The Schr\"odinger functional is, moreover, accessible to numerical simulations. It may hence be used to study the scaling properties of the theory and in particular the evolution of the renormalized gauge coupling from low to high energies. A concrete proposition along this line is made and the necessary perturbative analysis of the Schr\"odinger functional is carried through to 1-loop order.Comment: 68 page

COMPLEX HADAMARD MATRICES AND APPLICATIONS

A complex Hadamard matrix is a square matrix H ∈ M N (C) whose entries are on the unit circle, |H ij | = 1, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, F N = (w ij) with w = e 2πi/N. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. CONTENT

Preliminares al estudio de la huella en lingüística

The present paper constitutes a brief advance of much longer and more detailed ongoing work on the concept of “trace” in contemporary linguistic theory, particularly in syntax. It is commonly believed that the idea was coined by Noam Chomsky. However, we already detect its use, with a very accurate value, in the early work of Zellig Harris on mathematical linguistics or, to be more precise, on mathematical structures of language. In its origins, rather than being an index responsible for marking the location occupied by a unit previous to its syntactic movement (which always takes the form of fronting ), the trace was the result of a matrix product between n-adic functions. Thus, in Harris the trace is primarily a concept anchored in matrix calculus, or, put it differently, an algebraic notion. Chomsky’s notion, on its turn, is closely related with the LISP programming language. This text seeks to provide a preliminary analysis of the conceptual complexity implied in the concept of trace, which linguists should become aware of, for otherwise they will be doomed to be entangled in misunderstandings unfruitful to our discipline for decades to come.El presente documento constituye un breve avance de una obra en curso mucho más larga y más detallada sobre el concepto de “huella” en la teoría lingüística contemporánea, particularmente en la sintaxis. Se cree, por lo común, que la idea fue acuñada por Noam Chomsky. Sin embargo, ya detectamos su uso, con un valor muy preciso, en los primeros trabajos de Zellig Harris sobre lingüística matemática o, para ser más exactos, sobre estructuras 2matemáticas del lenguaje. En sus orígenes, en lugar de ser un índice responsable de marcar la ubicación de una unidad antes de su movimiento sintáctico (que siempre toma la forma de fronting), la traza o huella era el resultado de un producto matricial entre funciones n-ádicas. Por lo tanto, en Harris la huella es principalmente un concepto anclado en el cálculo matricial o, dicho de otro modo, una noción algebraica. La noción de Chomsky, por su parte, está estrechamente relacionada con el lenguaje de programación LISP. EL presente texto busca proporcionar un análisis preliminar de la complejidad conceptual implícita en el concepto de huella, del cual los lingüistas deben tomar conciencia, porque de lo contrario estarán condenados a enredarse en malentendidos infructuosos para nuestra disciplina durante las próximas décadas

Advanced Concepts in Particle and Field Theory

Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the elementary particles that are the basic building blocks of nature. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and conceptual connections across these fields. This title, first published in 2015, has been reissued as an Open Access publication

Finiteness conditions for monoids and small categories

Abstract available page vi-vii