Algebraic fermion bosonization

By exploiting the construction of charged field algebras as canonical extensions of CCR current algebras in 1+1 dimensions and nonregular representations of extended algebras, we provide an algebraic construction of local Fermi fields as ultrastrong limits of bosonic variables in all representations which are locally Fock with respect to the ground-state representation of the massless scalar field.     

Chiral symmetry breaking and vacuum structure in QCD

The solution of the axial U(1) problem, the role of the topology of the gauge group in forcing the breaking of axial symmetry in any irreducible representation of the observable algebra and the θ vacua structure are revisited in the temporal gauge with attention to the mathematical consistency of the derivations. Both realizations with strong and weak Gauss law are discussed; the control of the general mechanisms and structures is obtained on the basis of the localization of the (large) gauge transformations and the local generation of the chiral symmetry. The Schwinger model in the temporal gauge exactly reproduces the general results.     

A Generalization of the Stone–Von Neumann Theorem to Nonregular Representations of the CCR-Algebra

We give a classification, up to unitary equivalence, of the representations of the C*-algebra of the Canonical Commutation Relations which generalizes the classical Stone–von Neumann Theorem to the case of representations which are strongly measurable, but not necessarily strongly continuous. The classification includes all the (nonregular) representations which have been considered in physical models.     

Spectral stochastic processes arising in quantum mechanical models with a non-L 2 ground state

A functional integral representation is given for a large class of quantum mechanical models with a non-L ² ground state. As a prototype, the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal Abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite-volume functional integrals for such models is discussed, and the superselection sectors associated to an observable subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite-volume measures.Supported by DFG, Nr. Al 374/1-2     

Spontaneous symmetry breaking and energy gap generated by variables at infinity

Spontaneous symmetry breaking in the presence of long range instantaneous interactions is studied and the general mechanism underlying it is clarified. A characteristic feature is that the algebraic dynamics does not leave any essentially local algebra stable, i.e. variables at infinity get involved in the time evolution of local variables, so that in each irreducible representation the time evolution fails to be symmetric. For continuous symmetries, the Fourier transform of the vacuum expectation value of charge commutators is related to the energy spectrum at low momenta and a generalized Goldstone theorem is proved which explains the generation of energy gap. This energy gap is further shown to be governed by a classical dynamics at infinity, equivalently by the group generated by the effective Hamiltonian and the charge. Explicit examples are discussed.Work supported in part by INFN, Sezione di Pisa     

A Quantum Mechanical Gauge Model and a Possible Dynamical Solution of the StrongCPProblem

A model describing a quantum mechanical particle on a circle with minimal electromagnetic interaction with a space independent vector potential, and with a potential −M cos(?−θ_M) so that it mimics the massive Schwinger model, is discussed as a prototype of mechanisms and infrared structures which characterize gauge quantum field theories in positive gauges and QCD in particular. The functional integral representation in terms of the field variables which enter in the Lagrangean displays non-standard features, like a complex functional measure (failure of Nelson positivity), a crucial rôle of the boundary conditions, and the decomposition intoθsectors already in finite volume. In the infinite volume limit, one essentially recovers the standard picture whenM=0 (“massless fermions”), but one meets substantial differences forM≠0: for generic boundary conditions, independently of the Lagrangean angle of the topological term, the infinite volume limit selects the sector withθ=θ_Mand provides a natural “dynamical” solution of the strong CP problem. In comparison with previous approaches, the strategy discussed here allows us to exploit the consequences of theθdependence of the free energy density, with a unique minimum atθ=θ_M.     

The Infrared Problem in QED: A Lesson from a Model with Coulomb Interaction and Realistic Photon Emission

The scattering of photons and heavy classical Coulomb interacting particles, with realistic particle–photon interaction (without particle recoil) is studied adopting the Koopman formulation for the particles. The model is translation invariant and allows for a complete control of the Dollard strategy devised by Kulish–Faddeev and Rohrlich (KFR) for QED: in the adiabatic formulation, the Møller operators exist as strong limits and interpolate between the dynamics and a non-free asymptotic dynamics, which is a unitary group; the S-matrix is non-trivial and exhibits the factorization of all the infrared divergences. The implications of the KFR strategy on the open questions of the LSZ asymptotic limits in QED are derived in the field theory version of the model, with the charged particles described by second quantized fields: i) asymptotic limits of the charged fields, \({\Psi_{{\rm out}/{\rm in}}(x)}\), are obtained as strong limits of modified LSZ formulas, with corrections given by a Coulomb phase operator and an exponential of the photon field; ii) free asymptotic electromagnetic fields, \({B_{{\rm out}/{\rm in}}(x)}\), are given by the massless LSZ formula, as in Buchholz approach;   iii) the asymptotic field algebras are a semidirect product of the canonical algebras generated by \({B_{{\rm out}/{\rm in}}}\), \({\Psi_{{\rm out}/{\rm in}}}\);   iv) on the asymptotic spaces, the Hamiltonian is the sum of the free (commuting) Hamiltonians of \({B_{{\rm out}/{\rm in}}}\), \({\Psi_{{\rm out}/{\rm in}}}\) and the same holds for the generators of the space translations.     

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 C ^∞ 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, $ \pi_1{(\mathcal M)}$ appears as the only source of topological effects.     

The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold ${\mathcal {M}}$ , defined by the Lie structure of the vector fields, their action and their module structure over ${C^\infty({\mathcal {M}})}$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ${\Lambda_{R}({\mathcal {M}})}$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ${{\mathcal {M}}}$ ) ${Z}$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z  =  i z, z  =  0 and ${z = \hbar}$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For ${z =\hbar \neq 0}$ , the regular factorial Hilbert space representations of ${\Lambda_{R}({\mathcal{M}})}$ describe quantum mechanics on ${{\mathcal {M}}}$ . For z  =  0, if Diff( ${{\mathcal {M}}}$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on ${{\mathcal {M}}}$ .     

Natural flavour conservation in the neutral currents and the determination of the Cabibbo angle

We point out a conflict between the requirement of natural flavour conservation in the neutral currents and the possibility of determining the generalized Cabibbo angles as a function of the quark masses in the SU(2) × U(1) and in the SU(2)_L × SU (2)_R × U (1) models. The general implications of natural flavour conservation in the neutral currents on the fermion mass matrices are also discussed.