Giovanni Morchio

Quantum Delocalization of the Electric Charge

The classical Maxwell–Dirac and Maxwell–Klein–Gordon theories admit solutions of the field equations where the corresponding electric current vanishes in the causal complement of some bounded region of Minkowski space. This poses the interesting question of whether states with a similarly well localized charge density also exist in quantum electrodynamics. For a large family of charged states, the dominant quantum corrections at spacelike infinity to the expectation values of local observables are computed. It turns out that certain moments of the charge density decrease no faster than the Coulomb field in spacelike directions. In contrast to the classical theory, it is therefore impossible to define the electric charge support of these states in a meaningful way.

Charge density and electric charge in quantum electrodynamics

The convergence of integrals over charge densities is discussed in relation with the problem of electric charge and (nonlocal) charged states in quantum electrodynamics. Delicate points like the domain dependence of local charges as quadratic forms and the class of time smearing ensuring strong convergence of integrals of charge densities are analyzed and shown to be crucial in QED, also for the control of vacuum polarization effects leading to time dependence of the charge (Swieca phenomenon). The possibility of constructing physical charged states in the Feynman–Gupta–Bleuler gauge as limits of local state vectors is discussed, compatibly with the vanishing of the Gauss charge on local states. A modification of the Dirac exponential factor which yields the physical Coulomb fields from the Feynman–Gupta–Bleuler fields is shown to remove the infrared divergence of scalar products of local and physical charged states, allowing for a construction of physical charged fields with well-defined correlation func...

Quantum Mechanics and Stochastic Mechanics for Compatible Observables at Different Times

Abstract Bohm mechanics and Nelson stochastic mechanics are confronted with quantum mechanics in the presence of noninteracting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with quantum mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of noninteracting systems at different times.

The extension problem for partial Boolean structures in quantum mechanics

Alternative partial Boolean structures, implicit in the discussion of classical representability of sets of quantum mechanical predictions, are characterized, with definite general conclusions on the equivalence of the approaches going back to Bell and Kochen–Specker. An algebraic approach is presented, allowing for a discussion of partial classical extension, amounting to reduction of the “number of contexts,” classical representability arising as a special case. As a result, known techniques are generalized and some of the associated computational difficulties overcome. The implications on the discussion of Boole–Bell inequalities are indicated.

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.

Spectral stochastic processes arising in quantum mechanical models with a non-L2 ground state

A functional integral representation is given for a large class of quantum mechanical models with a non-L2 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.A functional integral representation is given for a large class of quantum mechanical models with a non--L2 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.

Bell Inequalities as Constraints on Unmeasurable Correlations

The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases, in particular for observables with compatibility relations described by tree graphs. This implies classical representability of any set of correlations 〈Ai〉, 〈B〉, 〈AiB〉, and the equivalence of the Bell-Clauser-Horne inequalities to a non void intersection between the ranges of values for the unmeasurable correlation 〈A1A2〉 associated to different choices for B. The same analysis applies to the Hardy model and to the “perfect correlations” discussed by Greenberger, Horne, Shimony and Zeilinger. In all the cases, the dependence of an unmeasurable correlation on a set of variables allowing for a classical representation is the only basis for arguments about violations of locality and causality.

Chiral symmetry breaking and θ vacuum structure in QCD

Abstract 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.

Mathematical structure of the temporal gauge in quantum electrodynamics

The conflict between Gauss’ law constraint and the existence of the propagator of the gauge fields, at the basis of contradictory proposals in the literature, is shown to lead to only two alternatives, both with peculiar features with respect to standard quantum field theory. In the positive (interacting) case, the Gauss’ law holds in operator form, but only the correlations of exponentials of gauge fields exist (nonregularity) and the space translations are not strongly continuous, so that their generators do not exist. Alternatively, a Kallen–Lehmann representation of the two point function of Ai satisfying locality and invariance under space–time translations, rotations and parity is derived in terms of the two point function of Fμν; positivity is violated, the Gauss’ law does not hold, the energy spectrum is positive, but the relativistic spectral condition does not hold. In the free case, θ-vacua exist on the observable fields, but they do not have time translationally invariant extensions to the gau...

Effective non-symmetric Hamiltonians and Goldstone boson spectrum☆

Abstract A generalization of the Goldstone theorem is derived which includes the case of non-symmetric Hamiltonians. In particular, this provides the modification of the Goldstone boson spectrum due to a non-symmetric perturbation. Such a generalized Goldstone theorem is particularly relevant in the case of long range interactions, where the removal of the infrared cutoff leads to a non-symmetric effective Hamiltonian, starting from a symmetric infrared cutoff dynamics. This phenomenon is shown to be related to the infrared counterterms which are necessary to guarantee the stability under time evolution of the algebra of essentially localized observables. Explicit applications of the above structures are discussed, like the Heisenberg model, the chiral symmetry sum rules, the f-sum rule, and the long-wavelength perfect screening sum rule for Coulomb systems in uniform background, the Kibble model, etc.