Gauge Invariance and Symmetry Breaking by Topology and Energy Gap

For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra $\mathcal A$, with a non-trivial center $\mathcal Z$, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of $\mathcal A$. In particular, this gives rise to a gauge symmetry described by the action of $\mathcal Z$. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries which do not commute with the topological invariants represented by elements of $\mathcal Z$ are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem.Comment: 23 page

Local Gauss law and local gauge symmetries in QFT

Local gauge symmetries reduce to the identity on the observables, as well as on the physical states (apart from reflexes of the local gauge group topology) and therefore their use in Quantum Field Theory (QFT) asks for a justification of their strategic role. They play an intermediate role in deriving the validity of Local Gauss Laws on the physical states (for the currents which generate the related global gauge group); conversely, we show that local gauge symmetries arise whenever a vacuum representation of a local field algebra ${\cal{F}}$ is used for the description/construction of physical states satisfying Local Gauss Laws, just as global compact gauge groups arise for the description of localizable states labeled by superselected quantum numbers. The above relation suggests that the Gauss operator, which by locality cannot vanish in ${\cal{F}}$, provides an intrinsic characterization of the realizations of a gauge QFT in terms of a local field algebra and of the related local gauge symmetries.Comment: 14 pages Corrected typo

Asymptotic Abelianness and Braided Tensor C*-Categories

By introducing the concepts of asymptopia and bi-asymptopia, we show how braided tensor C*-categories arise in a natural way. This generalizes constructions in algebraic quantum field theory by replacing local commutativity by suitable forms of asymptotic Abelianness.Comment: 20 pages, no figures. Final version, as to appear in "Rigorous Quantum Field Theory", Progress in Mathematics, Volume 25

A non-perturbative argument for the non-abelian Higgs mechanism

The evasion of massless Goldstone bosons by the non-abelian Higgs mechanism is proved by a non-perturbative argument in the local BRST gauge.Comment: The new version differs from version 1 by the correction of a few misprint

Quantum mechanics on manifolds and topological effects

A unique classification of the topological effects associated to quantum mechanics on manifolds is obtained on the basis of the invariance under diffeomorphisms and the realization of the Lie-Rinehart relations between the generators of the diffeomorphism group and the algebra of infinitely differentiable functions on the manifold. This leads to a unique ("Lie-Rinehart") C* algebra as observable algebra; its regular representations are shown to be locally Schroedinger and in one to one correspondence with the unitary representations of the fundamental group of the manifold. Therefore, in the absence of spin degrees of freedom and external fields, the first homotopy group of the manifold appears as the only source of topological effects.Comment: A few comments have been added to the Introduction, together with related references; a few words have been changed in the Abstract and a Note added to the Titl