Boutet de Monvel / Buchholz / Iagolnitzer Rigorous Quantum Field Theory

A Festschrift for Jacques Bros
1. Auflage 2006
ISBN: 978-3-7643-7434-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 251, 326 Seiten, eBook

Reihe: Progress in Mathematics

Jacques Bros has greatly advanced our present understanding of rigorous quantum field theory through numerous fundamental contributions. The impact of his work is also visible in several articles in this book. Quantum fields are considered as genuine mathematical objects, whose various properties and relevant physical interpretations have to be studied in a well-defined mathematical framework. Key topics: Analytic structures of QFT, renormalization group methods, gauge QFT, stability properties and extension of the axiomatic framework, QFT on models of curved spacetimes, QFT on noncommutative Minkowski spacetime.

Zielgruppe

Research

Autoren/Hrsg.

Buchholz, Anne

Iagolnitzer, Detlev

Moschella, Daniel

Boutet de Monvel, Anne

Buchholz, Detlev

Iagolnitzer, Daniel

Moschella, Ugo

Inhaltsverzeichnis

Local Counterterms on the Noncommutative Minkowski Space.- Massless Scalar Field in a Two-dimensional de Sitter Universe.- Locally Covariant Quantum Field Theories.- Asymptotic Abelianness and Braided Tensor C*-Categories.- Yang-Mills and Some Related Algebras.- Remarks on the Anti-de Sitter Space-Time.- Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory.- Action Ward Identity and the Stückelberg-Petermann Renormalization Group.- On the Relativistic KMS Condition for the P(ø)2 Model.- The Analyticity Program in Axiomatic Quantum Field Theory.- Renormalization Theory Based on Flow Equations.- Towards the Construction of Quantum Field Theories from a Factorizing S-Matrix.- String-Localized Covariant Quantum Fields.- Quantum Anosov Systems.- DFR Perturbative Quantum Field Theory on Quantum Space Time, and Wick Reduction.- On Local Boundary CFT and Non-Local CFT on the Boundary.- Algebraic Holography in Asymptotically Simple, Asymptotically AdS Space-times.- Non-Commutative Renormalization.- New Constructions in Local Quantum Physics.- Physical Fields in QED.- Complex Angular Momentum Analysis and Diagonalization of the Bethe-Salpeter Structure in Axiomatic Quantum Field Theory.

Leseproben

1 Introduction (P. 1)

After the great scienti.c revolutions of Special Relativity and of Quantum Mechanics in the first half of the twentieth century, Relativistic Quantum Field Theory (QFT) was introduced to provide a synthesis of these two new paradigms (in Kuhn’s words). As such, QFT is one of the major advances for theoretical physics in the second half of the last century.

The main reason for emphasizing the importance of QFT lies in the impressive effort that it represents towards a unified understanding of the structure of matter at subatomic scales, as it emerges in the collision phenomena of particle physics in the range of energies presently explored in particle accelerators. But another reason is the extraordinary mathematical wealth of this theoretical framework, which makes it a fascinating domain of research for mathematical physics and may also stimulate the interest of mathematicians.

Foreseen by Dirac in 1933 and finaly established discovered by Feynman around 1949 for the treatment of quantum electrodynamics, the so-called "path-integral formalism" of quantum field theory is also now considered as a powerful tool for providing perturbative methods of computation in many problems of theoretical physics, namely in statistical mechanics and in string theory. However, the use of relativistic quantum fields as a basic concept of mathematical physics underlying all the phenomena of particle physics in a very large range of energies represents a much more ambitious program.

This program was indeed stimulated by the success of the quantum electrodynamics (QED) formalism for computing the electron-photon, electronelectron and electron-positron scattering amplitudes. Even today, it is by no means understood why the perturbative expansion of QED in powers of the coupling parameter, that is the electric charge of the electron, provides such a spectacular agreement with experimental data, although in practice it is reduced to the computation of the very first terms of the series (the only ones which are computable).

This is even more surprising since we now know that the series cannot be convergent, so that, paraphrasing Wigner’s words, one may wonder about the "unreasonable effectiveness" of perturbation theory in four-dimensional QED. In the 1950s, however, the success of these computations was credited to the smallness of the coupling parameter of QED, and this situation stimulated research for other methods of investigation of QFT, which could apply to quantum field models with large coupling: it was in fact the very concept of a quantum field which appeared as the most powerful and promising one for explaining the phenomena of strong nuclear interaction of particle physics.

During all that period the study of phenomena of weak nuclear interaction also benefitted, like QED, from the success of a perturbative QFT formalism reduced to very few terms. The demand for a more general nonperturbative treatment of QFT motivated an important community of mathematical physicists to work out a model-independent axiomatic approach to relativistic quantum field theory [2,8,9,11,14].

Their first task was to provide a mathematically meaningful concept of a relativistic quantum field in terms of "operator-valued distribution in the Hilbert space of quantum physical states", in such a way that the notion of ingoing and outgoing particle states could be introduced in terms of appropriate asymptotic forms of the field operators.