Perturbation theory of Wightman functions

A perturbative expansion of the Wightman functions, and more generally of vacuum expectation values of products of time-ordered and anti-time-ordered products, is derived for ₄ ⁴ field theory. The result is expressed as a sum over generalized Feynman graphs. The derivation is based exclusively on the equation of motion and the Wightman axioms. Neither canonical commutation relations nor asymptotic conditions are needed at any point. In the zero-mass case the individual graphs are infrared divergent, but the sum over all graphs of a given order is convergent.     

Nets ofC*-algebras and classification of states

The concept of locality in quantum physics leads to mathematical structures in which the basic object is an operator algebra with a net of distinguished subalgebras (the local subalgebras). Such nets provide a classification of the states of this algebra in equivalence classes determined by local or asymptotic properties. The corresponding equivalence relations are natural generalizations of the (more stringent) standard quasiequivalence relation (they are also useful for classifying states by their properties with respect to automorphism groups). After discussing general nets from this point of view we investigate in the last section more specialized nets (funnels of von Neumann algebras) with special emphasis on their locally normal states.The research in this paper was supported in part by the N.S.F. and the Ministère de l'Education Nationale.     

Spectra of states,and asymptotically abelianC*-algebras

For each state of aC*-algebra its spectrum is defined and shown to coincide with the spectrum of the naturally associated modular operator. For strongly clustering states of asymptotically abelianC*-algebras the spectrum is minimal among the states in the same quasi-equivalence class, hence is a *-isomorphic invariant for the weak closure of the G.N.S.-representation. Furthermore, the non-zero elements in the spectrum of strongly clustering states form a multiplicative group.     

Reconstruction theorem and cluster properties of Wightman functions in noncommutative quantum field theory

The rigorous definition of quantum field operator is done in any theory where usual product between corresponding test functions is substituted by the star product. The important example of such a theory is noncommutative quantum field theory. Cluster properties of Wightman functions are proved in these theories.     

Spectral functions and hole nuclear states

An explicit expression for the spectral function of hole nuclear states is given in the framework of the coherent fluctuation model. It is shown that the spectral function and single-particle width, centroid energy, quasiparticle effective mass, etc., are functionals of the ground-state nucleon density distribution. Calculation of the spectral functions for⁵⁸Ni,⁴⁰Ca,²⁸Si is carried out and the results are compared with the experimental data.     

Group actions on C*-algebras, 3-cocycles and quantum field theory

We study group extensions , where acts on a C^*-algebraA. Given a twisted covariant representation ,V of the pairA, we construct 3-cocycles on with values in the centre of the group generated byV(). These 3-cocycles are obstructions to the existence of an extension of byV() which acts onA compatibly with . The main theorems of the paper introduce a subsidiary invariant which classifies actions of onV() and in terms of which a necessary and sufficient condition for the the cohomology class of the 3-cocycle to be non-trivial may be formulated. Examples are provided which show how non-trivial 3-cocycles may be realised. The framework we choose to exhibit these essentially mathematical results is influenced by anomalous gauge field theories. We show how to interpret our results in that setting in two ways, one motivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge groups. In order to make comparisons with the usual approach to cohomology in gauge theory we conclude with a Lie algebra version of the invariant and the 3-cocycle.     

Hartree-Fock theory of nuclear deformations and high spin states

A brief review of the Hartree-Fock approximation to nuclear shapes and nuclear spectra is presented. Various aspects of rotation-alignment effect and high spin states are illustrated in the Hartree-Fock model.     

The renormalization of single-particle states in nuclear field theory

The single-particle states are renormalized by taking into account the emission and subsequent absorption of a phonon to all orders of perturbation theory. The Ward identity provides a useful normalization condition to the new dressed excitations. Other orthonormalization properties are interpreted in terms of anticommutation and energy weighted sum rules. A two-level model with a monopole phonon is treated in detail as an example.     

On the equivalence of the Euclidean and Wightman formulation of field theory

A mistake in the paper [1] on the “Axioms for Euclidean Green's Functions” is corrected in the following sense: thanks to these axioms the Euclidean Schwinger functionsS _n can be analytically continued to the corresponding Wightman functionsW _n possessing all the correct analyticity properties and satisfying a generalized positivity condition in the complex domain. It is however suggested by the proof that their tempered behaviour near the Minkowski points cannot be guaranteed without additional assumptions.     

A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. II

We discuss some properties of a non-commutative generalization of the classical moment problem (them-problem) previously introduced. It is shown that there is a connexion between the determination of the problem and the self-adjointness properties in the corresponding Hilbert space. This generalizes the well-known connexion between the determination of the measure in the classical moment problem and the self-adjointness properties of the polynomials as operators in the correspondingL ²-space. The dependence of them-problem on the choice ofC*-semi-norms and on the action of *-homomorphisms is also investigated. As an application, it is shown that if a quantum field (in a very general sense) is essentially self-adjoint then them-problem for the Wightman functional is determined on the quasi-localizableC*-algebra and that the corresponding representation of the localizable algebra generates the bounded observables of the field. It is pointed out that (ultraviolet and spatially) cut-off fields fall in this class and, therefore, are in one to one correspondance with states on the quasi-localizableC*-algebra.Laboratoire associé au Centre National de la Recherche Scientifique.     

A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. I.

A (non-commutative) generalization of the classical moment problem is formulated on arbitrary *-algebras with units. This is used to produce aC*-algebra associated with the space of test functions for quantum fields. ThisC*-algebra plays a role in theories of bounded localized observables in Hilbert space which is similar to that of the space of test functions in quantum field theories (namely it is represented in Hilbert space). The case of local quantum fields which satisfy a slight generalization of the growth condition is investigated. Laboratorie associé au Centre National de la Recherche Scientifique.     

Outer automorphisms and reduced crossed products of simpleC*-algebras

Every outer automorphism of a separable simpleC*-Algebra is shown to have a pure state which is mapped into an inequivalent state under this automorphism. The reduced crossed product of a simpleC*-algebra by a discrete group of outer automorphisms is shown to be simple.     

Channel-coupling contribution to the widths of decay nuclear states and to their wave functions

By using the formalism of the quantum theory of fission, the amplitudes of partial decay widths and the asymptotic behavior of the wave function for a decaying nucleus are found with allowance for open-decay-channel coupling not only for fission, but also for the binary decays of nuclei through protonic, alphaparticle, cluster, and other channels.     

A note on the decrease of truncated Wightman functions for large space-like separation of the arguments

The truncated Wightman functions cannot decrease arbitrarily fast for large space-like separation of the arguments. For certain configurations they can fall off at most exponentially.     

Dynamical Entropy of Generalized Quantum Markov Chains on Gauge-invariant C *-algebras

We prove that the mean entropy and the dynamical entropy are equal for generalized quantum Markov chains on gauge-invariant C ^*-algebras.     

Excitation of collective nuclear states by fast particles

The elastic and inelastic scattering of a fast particle by a vibrating nucleus is discussed in the semicalssical approximation. For elastic scattering it is shown that the effect of the vibrations can be described by an effective deformed optical potential which is axially symmetric about the incident direction. Explicit results are obtained for inelastic scattering. In an appendix, the validity of the semiclassical approximation for potential scattering is discussed and numerical tests made.