The book gives an introduction to Weyl non-regular quantization suitable for the description of physically interesting quantum systems, where the traditional Dirac-Heisenberg quantization is not applicable. The latter implicitly assumes that the canonical variables describe observables, entailing necessarily the regularity of their exponentials (Weyl operators). However, in physically interesting cases -- typically in the presence of a gauge symmetry -- non-observable canonical variables are introduced for the description of the states, namely of the relevant representations of the observable algebra.

In general, a gauge invariant ground state defines a non-regular representation of the gauge dependent Weyl operators, providing a mathematically consistent treatment of familiar quantum systems -- such as the electron in a periodic potential (Bloch electron), the Quantum Hall electron, or the quantum particle on a circle -- where the gauge transformations are, respectively, the lattice translations, the magnetic translations and the rotations of 2.

Relevant examples are also provided by quantum gauge field theory models, in particular by the temporal gauge of Quantum Electrodynamics, avoiding the conflict between the Gauss law constraint and the Dirac-Heisenberg canonical quantization. The same applies to Quantum Chromodynamics, where the non-regular quantization of the temporal gauge provides a simple solution of the U(1) problem and a simple link between the vacuum structure and the topology of the gauge group.

Last but not least, Weyl non-regular quantization is briefly discussed from the perspective of the so-called polymer representations proposed for Loop Quantum Gravity in connection with diffeomorphism invariant vacuum states.

First primer on this emerging topic Written by a leading expert in the field Suitable as advanced textbook and as self-study guide Includes supplementary material: sn.pub/extras

Autorentext

Franco Strocchi graduated at Pisa University and at Scuola Normale Superiore, Pisa, and then has been Research Fellow of the National Institute of Nuclear Physics (INFN) and Lecturer at Pisa University and at Scuola Normale Superiore. Later, he has been Full Professor of Theoretical Physics at the International School for Advanced Studies in Trieste (SISSA) and Professor of Theoretical Physics at Scuola Normale Superiore, Pisa. He is the author of more than one hundreds of papers on theoretical and mathematical physics and of the following books: Elements of Quantum Mechanics of Infinite Systems (World Scientific, 1985); Selected Topics on the General Properties of Quantum Field theory, (World Scientific, 1993); Symmetry Breaking (Springer, 2005, 2008); An Introduction to the Mathematical Structure of Quantum Mechanics (World Scientific, 2005, 2008, 2010); An Introduction to Non-Perturbative Foundations of Quantum Field Theory (Oxford University Press, 2013, 2016); Gauge Invariance and Weyl-polymer Quantization (Springer, 2016).He has been Research Associate and Visiting Professor at Princeton University (USA), Visting Scientist at Cern, Visiting Schroedinger Professor at the University of Vienna.

Inhalt

Introduction.- Heisenberg quantization and Weyl quantization.- Delocalization, gauge invariance and non-regular representations.- Quantum mechanical gauge models.- Non-regular representations in quantum field theory.- Diffeomorphism invariance and Weyl polymer quantization.- A generalization of Stone-von Neumann theorem.- Bibliography.- Index.