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Compiled to illustrate the recent history of Quantum Field Theory
and its trends, this collection of selected reprints by Jurg
Froehlich, a leading theoretician in the field, is a comprehensive
guide of the more mathematical aspects of the subject. Results and
methods of the past fifteen years are reviewed. The analytical
methods employed are non-perturbative and, for the larger part,
mathematically rigorous. Most articles are review articles
surveying certain important developments in quantum field theory
and guiding the reader towards the original literature.The volume
begins with a comprehensive introduction by Jurg Froehlich.The
theory of phase transitions and continuous symmetry breaking is
reviewed in the first section. The second section discusses the
non-perturbative quantization of topological solitons. The third
section is devoted to the study of gauge fields. A paper on the
triviality of 4 - theory in four and more dimensions is found in
the fourth section, while the fifth contains two articles on
"random geometry". The sixth and final part addresses topics in
low-dimensional quantum field theory, including braid statistics,
two-dimensional conformal field theory and an application to
condensed matter theory.
Compiled to illustrate the recent history of quantum field theory
and its trends, this collection of selected reprints by Frohlich
aims to be a comprehensive guide of the more mathematical aspects
of the subject. Results and methods of the past 15 years are
reviewed. The analytical methods employed are non-perturbative and,
for the larger part, mathematically rigorous. Most articles are
review articles surveying certain important developments in quantum
field theory and guiding the reader towards the original
literature. The theory of phase transitions and spontaneous
symmetry breaking is reviewed in the first section. The second
section discusses the non-perturbative quantization of topological
solitons. The third section is devoted to the study of gauge
fields. A paper on the triviality of (labda phi) to the power of 4
- theory in four and more dimensions is found in the fourth
section, while the fifth contains two articles on "random
geometry". The sixth and final part addresses topics in
low-dimensional quantum field theory, including braid statistics,
two-dimensional conformal field theory and an application to
condensed matter theory. The volume begins with a comprehensive
introduction by Jurg Frohlich.
This book reviews recent results on low-dimensional quantum field
theories and their connection with quantum group theory and the
theory of braided, balanced tensor categories. It presents
detailed, mathematically precise introductions to these subjects
and then continues with new results. Among the main results are a
detailed analysis of the representation theory of U (sl ), for q a
primitive root of unity, and a semi-simple quotient thereof, a
classfication of braided tensor categories generated by an object
of q-dimension less than two, and an application of these results
to the theory of sectors in algebraic quantum field theory. This
clarifies the notion of "quantized symmetries" in quantum
fieldtheory. The reader is expected to be familiar with basic
notions and resultsin algebra. The book is intended for research
mathematicians, mathematical physicists and graduate students.
This book collects lecture courses and seminars given at the Les
Houches Summer School 2010 on "Quantum Theory: From Small to Large
Scales." Fundamental quantum phenomena appear on all scales, from
microscopic to macroscopic. Some of the pertinent questions include
the onset of decoherence, the dynamics of collective modes, the
influence of external randomness and the emergence of dissipative
behaviour. Our understanding of such phenomena has been advanced by
the study of model systems and by the derivation and analysis of
effective dynamics for large systems and over long times. In this
field, research in mathematical physics has regularly contributed
results that were recognized as essential in the physics community.
During the last few years, the key questions have been sharpened
and progress on answering them has been particularly strong. This
book reviews the state-of-the-art developments in this field and
provides the necessary background for future studies. All chapters
are written from a pedagogical perspective, making the book
accessible to master and PhD students and researchers willing to
enter this field.
English summary: The mechanical-deterministic world-view plunged
into crisis at the end of the 19th century and for the next 85
years quantum mechanics succeeded the naive realism of the old
world view. Jorg Frohlich considers this paradigm shift and
reflects on the physics of the 20th century from a
historical-philosophic perspective: What stood behind these
concepts, who essentially contributed to the new forms of
knowledge? After considering these questions, he comes to the
conclusion: theoretical physics does not explain natural events, it
only describes them. German description: Das
mechanistisch-deterministische Weltbild geriet am Ende des 19.
Jahrhunderts in eine Krise - und vor nunmehr 85 Jahren folgte der
Abschied vom naiven Realismus des alten Weltbildes mit der
Quantenmechanik. Juerg Frohlich nimmt diese Paradigmenwechsel auf
und reflektiert die Physik des 20. Jahrhunderts aus
historisch-philosophischer Perspektive: was steht hinter diesen
Begriffen, wer trug wesentlich zu neuen Erkenntnissen bei? Nach
Erorterung dieser Fragen kommt er zu dem Schluss: aDie theoretische
Physik erklart das Naturgeschehen nicht, sie beschreibt es nur.
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