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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 2pi.
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.
List of contents
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.
About the author
Franco Strocchi, after graduating with maximum cum laude at the University of Pisa and at Scuola Normale Superiore in 1961, was lecturer at both Institutions, research associate and visiting professor at Princeton University, visiting Schroedinger professor at Wien University, visiting scientist at CERN theory division, full professor of theoretical physics at the International School of Advanced Studies (SISSA, Trieste) and professor at Scuola Normale Superiore. He gave invited lectures at various meetings and International Schools and he is the author of several books on the subjects of his research works (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 2008, 2010, An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press, International Series of Monographs in Physics, 2013, 2016, Gauge Invariance and Weyl-Polymer Quantization, Springer 2016, A primer of Analytical Mechanics, Springer 2018) and of more than one hundred scientific papers.
Summary
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 2π.
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.
