By Franco Strocchi
The publication supplies an advent to Weyl non-regular quantization compatible for the outline of bodily attention-grabbing quantum platforms, the place the conventional Dirac-Heisenberg quantization isn't acceptable. The latter implicitly assumes that the canonical variables describe observables, entailing inevitably the regularity in their exponentials (Weyl operators). even if, in bodily attention-grabbing situations -- quite often within the presence of a gauge symmetry -- non-observable canonical variables are brought for the outline of the states, specifically of the suitable representations of the observable algebra.
In common, a gauge invariant flooring kingdom defines a non-regular illustration of the gauge based Weyl operators, offering a mathematically constant therapy of established quantum structures -- reminiscent of the electron in a periodic capability (Bloch electron), the Quantum corridor electron, or the quantum particle on a circle -- the place the gauge modifications are, respectively, the lattice translations, the magnetic translations and the rotations of 2π.
Relevant examples also are supplied via quantum gauge box conception types, particularly via the temporal gauge of Quantum Electrodynamics, averting the clash among the Gauss legislation constraint and the Dirac-Heisenberg canonical quantization. an identical applies to Quantum Chromodynamics, the place the non-regular quantization of the temporal gauge offers an easy resolution of the U(1) challenge and a straightforward hyperlink among the vacuum constitution and the topology of the gauge group.
Last yet now not least, Weyl non-regular quantization is in brief mentioned from the point of view of the so-called polymer representations proposed for Loop Quantum Gravity in reference to diffeomorphism invariant vacuum states.