Nuclear β-decay. I

In order to clarify the validity of the theory of nuclearβ-transition a new formulation is given. The nuclear properties are contained in form factors, arising from a decomposition of the matrix element of the V-A operator for nuclear states. The connection of these form factors with nuclear physics matrix elements is tabulated. The total transition amplitude can be written as an integral over these form factors folded with off-shell amplitudes for elastic and inelastic electron scattering on nuclei.     

An exact operator solution of the quantum Liouville field theory

Exact, closed form results are given expressing the quantum Liouville field theory in terms of a canonical free pseudoscalar field. The classical conformal transformation properties and a Bäcklund transformation of the Liouville model are briefly reviewed and then developed into explicit operator statements for the quantum theory. This development leads to exact expressions for the basic operator functions of the Liouville field: ?_μΦ, and e^nΦ. An operator product analysis is then used to construct the Liouville energy-momentum tensor operator, which is shown to be equal to that of a free pseudoscalar field. Dynamical consequences of this equivalence are discussed, including the relation between the Liouville and free field energy eigenstates. Liouville correlation functions are partially analyzed, and remaining open questions are discussed.     

Renormalized operator form of quantum action principle

The derivatives of the renormalized field operator with respect to the parameters (coupling constants, masses) are discussed. Two ways of obtaining the finite result in terms of renormalized perturbation expansion are shown. Throughout the paper the operator language is used; in particular the operator formula for renormalized powers of the field operators (normal products) is employed. The λ?⁴ theory is considered as an illustrative example.     

Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ^∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.     

Selfadjoint Extensions of the Neumann Laplacian in Domains with Cylindrical Outlets

Let be a domain with N cylindrical outlets to infinity. The solutions of the Neumann Problem for the Poisson equation are characterized within the theory of self-adjoint extensions of the operator L. Here L is the symmetric operator associated to the problem in , on , in weighted L ² -spaces. The results are applied to examples in the theory of continuum mechanics. Received: 10 June 1996\,/\,Accepted: 16 October 1996     

Generalization of the radiative corrections to β and μ decays in the SU(2)L × U(1) gauge model

Previous results for the radiative corrections to the ratio of decay probabilities of β and μ decays are generalized to the case of arbitrary symmetry breaking in the framework of the SU(2)_L × U(1) gauge model. In this more general case, the results remain very simple: the corrections are given by the corresponding expressions of the local V-A theory with m_W substituted for the cutoff Λ plus a positive definite contribution which depends on θ_W and R = m_W²/m_Z².Thus, the corrections in the general case are still constrained to have large values as required by Cabibbo universality and the present experimental values on β and μ decays. For reasonable values of m_Z and m_W the results in the general case are numerically close to those previously obtained in the simplest version of the theory. To illustrate this point the corrections are compared for two different choices of the parameters of the theory. In an appendix we discuss, rather briefly, the theoretical arguments that allow us to generalize pprevious discussions the case of fractionally charged quarks.     

The influence of ambiguities in the pion production operator on (p, π+) cross sections

Various aspects of the pionic stripping mechanism for (p, π⁺) reactions are studied in a momentum-space DWBA framework. A Foldy-Wouthuysen nonrelativistic reduction of a pseudoscalar field theory leads to several ambiguities in the pion production operator H′. The extent to which these ambiguities are reflected in the (p, π⁺) differential cross section is studied, for the particular case of 185 MeV protons incident on ¹²C. The results indicate that the O(1/m³) terms (m is the nucleon mass) in the production operator have as important an effect on the differential cross section as the O(1/m²) nucleon recoil term. The freedom of carrying out a further unitary transformation on H′ is found to have a substantially larger effect on the cross section than does the uncertainty associated with the relativistic transformation properties in the external nucleonnucleus potential. The size of these several effects suggests that the DWBA pionic stripping theory in its present form cannot serve as a sensitive probe of nuclear structure. Our results indicate the need for a more consistent relativistic theory of (p, π⁺) reactions.     

Self-similarity and asymptotic stability for coupled nonlinear Schrödinger equations in high dimensions

This paper is concerned with systems of coupled Schrödinger equations with polynomial nonlinearities and dimension n≥1. We show the existence of global self-similar solutions and prove that they are asymptotically stable in a framework based on weak-L^p spaces, whose elements have local finite L²-mass. The radial symmetry of the solutions is also addressed.     

Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge (x_uA_u^a(x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is $\text{1}\text{g}$. The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is $\text{1}\text{g}$. (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers $\text{1}\text{g}$ is suggested.     

Classical Hamiltonian perturbation theory without secular terms or small denominators

We consider perturbation theory in ? for the classical Hamiltonian H = H₀ + ?H₁, where H₀ gives rise to a known motion and ? is small. First we demonstrate how the usual secular terms and small denominators arise from a straightforward expansion in ? and argue that they are artifacts of the method. Then we present an alternative perturbation theory based on an analysis of the operator (s ? L)^?1, where s is a complex number and L is the Liouville operator corresponding to H. This perturbation series contains neither secular terms nor small denominators. In the case of almost multiply periodic systems we show, to lowest non-trivial order in ?, how our series reproduces the standard results both in the resonant and nonresonant regions — all in one analytic formula. As a final exercise we demonstrate that energy is conserved at order ?ⁿ⁺¹ when the accuracy of the theory is order ?ⁿ.     

Isovector Q balls

Finite-energy solutions rotating at a constant angular velocity in isovector space are considered within theories that possess global SO(3) symmetry (isovector theories). It is shown that, for a nonlinear O(3) model (n field), such solutions exist only in the one-dimensional case. For the isovector theory proposed by S. Weinberg in order to describe low-energy properties of the π mesons, such solutions exist at some values of parameters that appear in this theory. Some properties of these solutions are studied.     

On Vlasov–Manev Equations. I: Foundations, Properties, and Nonglobal Existence

We consider the classical stellar dynamic (Vlasov) equation with a so-called Manev correction (based on a pair potential γ/r + ε/r ²). For the pure Manev potential γ = 0 we discuss both the continuous case and the N-body problem and show that global solutions will not exist if the initial energy is negative. Certain global solutions can be constructed from local ones by a transformation which is peculiar for the ε/r ² law. Moreover, scaling arguments are used to show that Boltzmann collision terms are meaningful in conjunction with Manev force terms. In an appendix, a formal justification of the Manev correction based on the quasirelativistic Lagrangian formalism for the motion of a particle in a central force field is given.     

Indirect interaction of Mössbauer nuclei

An expression is deduced for the operator of the indirect interaction of nuclei via the electromagnetic field. The properties of Mössbauer nuclei are described within the pseudospin formalism, which is usually used in the theory of optical two-level systems. The indirect interaction of pseudospins is derived by a method adopted from the theory of superconductivity. It is found that the potentials of this interaction involve terms decreasing as r ^?3, r ^?2, and r ^?1. The estimates demonstrate that the two-particle interaction can contribute significantly to the width of the resonance line, for example, in crystals whose cells contain thulium nuclei.     

Antigravitating black hole solitons with scalar hair in N=4 supergravity

We present some new solutions of the equations of the N=4 supergravity theory which represent black holes with scalar, electric and magnetic charges. The solutions are parameterized by the mass and 6 electric and 6 magnetic charges which can be assembled into a complex 6-vector, Z^N. One can act on the solutions with SO(6)×U(1) to obtain new solutions with the same mass M but charges Z^N related by SO(6)×U(1) transformations, the U(1) factor corresponding to the duality subgroup of the hidden SU(1, 1) ssymetry of the N=4 model. In a certain limiting case the black holes have zero temperature and behave like solitons. In this case multisoliton solutions are exhibited which antigravitate, i.e. are in static equilibrium. We also present some solutions of the Kaluza-Klein theory which were anticipated by Scherk which also antigravitate. However, these latter solutions contain naked singularities. A discussion is also given of the relation of these solutions to dimensional reduction which has relevance for the black holes in the N=8 supergravity theory.     

Nonlocal gravitational models and exact solutions

A nonlocal gravity model with a function f (□^?1 R), where □ is the d’Alembert operator, is considered. The algorithm, allowing to reconstruct f(□^?1 R), corresponding to the given Hubble parameter and the state parameter of the matter, is proposed. Using this algorithm, we find the functions f(□^?1 R), corresponding to de Sitter solutions.     

On the finiteness of scaling sum rules

The convergence of sum rules relating the matrix elements of local operators to integrals over deep-inelastic structure functions is studied critically. It is found that the matrix elements may always be written as the q² → ? ∞ limit of finite expressions, regardless of the (Regge) asymptotic behavior of the structure functions or the possible occurrence of J = 0 fixed singularities. The correct form of the sum rule for the operator Schwinger term is taken as a paradigm case. It is derived from the Bjorken-Johnson-Low theorem and agrees with the results of parton model and light-cone analyses. It readily encompases the results of second order φ³ theory (where the Schwinger term diverges logarithmically) and second order vector gluon theory (where it vanishes). Sufficient conditions for the finiteness of the operator Schwinger term are the scaling of the longitudinal structure function and the absence of J = 0 fixed singularities with nonpolynomial residues. The treatment is readily applicable to other scaling and fixed q² sum rules needing regulation. A compendium of these is given.     

On torsion-free vacuum solutions of the model of de Sitter gauge theory of gravity (II)

It is shown that all torsion-free vacuum solutions of the model of de Sitter (dS) gauge theory of gravity are the vacuum solutions of Einstein field equations with the same positive cosmological constant. Furthermore, for the gravitational theories with more general quadratic gravitational Lagrangian (F ² + T ²), the torsion-free vacuum solutions are also the vacuum solutions of Einstein field equations.