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Renormalization group in algebraic quantum field theory: past, present and future

Roberto Conti, Claudio D’Antoni, Gerardo Morsella

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

ABSTRACT. We present an overview of some recent results on the renormalization group analysis in the framework of local quantumphysics. In particular, we emphasize the role of the theory of superselection sectors, through themagnifying glass provided by the scaling limit, that leads to intrinsic notions of preserved and confined sectors. In the final section we discuss some further results and open problems. 1.

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ABSTRACT. We present an overview of some recent results on the renormalization group analysis in the framework of local quantumphysics. In particular, we emphasize the role of the theory of superselection sectors, through themagnifying glass provided by the scaling limit, that leads to intrinsic notions of preserved and confined sectors. In the final section we discuss some further results and open problems. 1.
For any given algebra of local observables in Minkowski space an associated scaling algebra is constructed on which renormalization group (scaling) transformations act in a canonical manner. The method can be carried over to arbitrary space-time manifolds and provides a framework for the systematic analysis of the short distance properties of local quantum field theories. It is shown that every theory has a (possibly non-unique) scaling limit which can be classified according to its classical or quantum nature. Dilation invariant theories are stable under the action of the renormalization group. Within this framework the problem of wedge (Bisognano-Wichmann) duality in the scaling limit is discussed and some of its physical implications are outlined.
We start with a simple introduction into the renormalization group (RG) in quantum field theory and give an overview of the renormalization group method. The third section is devoted to essential topics of the renorm-group use in the QFT. Here, some fresh results are included. Then we turn to the remarkable proliferation of the RG ideas into various fields of physics. The last section summarizes an impressive recent progress of the "QFT renormalization group" application in mathematical physics.
Abstract Introduced as a quantum extension of Maxwell's classical theory, quantum electrodynamic (QED) has been the first example of a quantum field theory (QFT). Eventually, QFT has become the framework for the discussion of all fundamental interactions at the microscopic scale except, possibly, gravity. More surprisingly, it has also provided a framework for the understanding of second order phase transitions in statistical mechanics. In fact, as hopefully this work illustrates, QFT is the natural framework for the discussion of most systems involving an infinite number of degrees of freedom with local couplings. These systems range from cold Bose gases at the condensation temperature (about ten nanokelvin) to conventional phase transitions (from a few degrees to several hundred) and high energy particle physics up to a TeV, altogether more than twenty orders of magnitude in the energy scale. Therefore, although excellent textbooks about QFT had already been published, I thought, many years ago, that it might not be completely worthless to present a work in which the strong formal relations between particle physics and the theory of critical phenomena are systematically emphasized. This option explains some of the choices made in the presentation. A formulation in terms of field integrals has been adopted to study the properties of QFT. The language of partition and correlation functions has been used throughout, even in applications of QFT to particle physics. Renormalization and renormalization group (RG) properties are systematically discussed. The notion of effective field theory (EFT) and the emergence of renormalizable theories are described. The consequences for fine-tuning and triviality issue are emphasized. This fifth edition has been updated and fully revised.
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.
We review recent developments in tensor network approaches, focusing on renormalization group methods. Since they are free from the negative sign and complex action problems, there is growing interest in their application to lattice field theories, particularly with a view toward future studies of quantum chromodynamics (QCD) at finite temperature and density. They are also of broad interest in quantum field theory, with recent advances in approaches that allow one to directly investigate universal aspects of critical behavior by making use of theoretical insights from conformal field theory. We highlight several recently explored topics that are expected to play important roles in forthcoming tensor-network studies of QCD.