40 research outputs found
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics
After recalling episodes from Pascual Jordan's biography including his
pivotal role in the shaping of quantum field theory and his much criticized
conduct during the NS regime, I draw attention to his presentation of the first
phase of development of quantum field theory in a talk presented at the 1929
Kharkov conference. He starts by giving a comprehensive account of the
beginnings of quantum theory, emphasising that particle-like properties arise
as a consequence of treating wave-motions quantum-mechanically. He then goes on
to his recent discovery of quantization of ``wave fields'' and problems of
gauge invariance. The most surprising aspect of Jordan's presentation is
however his strong belief that his field quantization is a transitory not yet
optimal formulation of the principles underlying causal, local quantum physics.
The expectation of a future more radical change coming from the main architect
of field quantization already shortly after his discovery is certainly quite
startling. I try to answer the question to what extent Jordan's 1929
expectations have been vindicated. The larger part of the present essay
consists in arguing that Jordan's plea for a formulation without ``classical
correspondence crutches'', i.e. for an intrinsic approach (which avoids
classical fields altogether), is successfully addressed in past and recent
publications on local quantum physics.Comment: More biographical detail, expansion of the part referring to Jordan's
legacy in quantum field theory, 37 pages late
TOWARDS A HAWAIʻI SOIL HEALTH INDEX: IDENTIFYING SENSITIVE AND PRACTICAL INDICATORS OF CHANGE ACROSS LAND USE AND SOIL DIVERSITY
M.S.M.S. Thesis. University of Hawaiʻi at Mānoa 201
Algebraic Topology for Data Scientists
This book gives a thorough introduction to topological data analysis (TDA),
the application of algebraic topology to data science. Algebraic topology is
traditionally a very specialized field of math, and most mathematicians have
never been exposed to it, let alone data scientists, computer scientists, and
analysts. I have three goals in writing this book. The first is to bring people
up to speed who are missing a lot of the necessary background. I will describe
the topics in point-set topology, abstract algebra, and homology theory needed
for a good understanding of TDA. The second is to explain TDA and some current
applications and techniques. Finally, I would like to answer some questions
about more advanced topics such as cohomology, homotopy, obstruction theory,
and Steenrod squares, and what they can tell us about data. It is hoped that
readers will acquire the tools to start to think about these topics and where
they might fit in.Comment: 322 pages, 69 figures, 5 table
Soziologie - Sociology in the German-Speaking World
This book provides the first systematic overview of German sociology today. Thirty-four chapters review current trends, relate them to international discussions and discuss perspectives for future research. The contributions span the whole range of sociological research topics, from social inequality to the sociology of body and space, addressing pressing questions in sociological theory and innovative research methods
A walk in the noncommutative garden
This text is written for the volume of the school/conference "Noncommutative
Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in
noncommutative geometry, based on a discussion of significant examples of
noncommutative spaces in geometry, number theory, and physics. The paper also
contains an outline (the ``Tehran program'') of ongoing joint work with Consani
on the noncommutative geometry of the adeles class space and its relation to
number theoretic questions.Comment: 106 pages, LaTeX, 23 figure
Soziologie - Sociology in the German-Speaking World
This book provides the first systematic overview of German sociology today. Thirty-four chapters review current trends, relate them to international discussions and discuss perspectives for future research. The contributions span the whole range of sociological research topics, from social inequality to the sociology of body and space, addressing pressing questions in sociological theory and innovative research methods