Showing papers in "Il Nuovo Cimento in 1961"

Field Theories with Superconductor Solutions

Abstract: The conditions for the existence of non-perturbative type « superconductor » solutions of field theories are examined. A non-covariant canonical transformation method is used to find such solutions for a theory of a fermion interacting with a pseudoscalar boson. A covariant renormalisable method using Feynman integrals is then given. A « superconductor » solution is found whenever in the normal perturbative-type solution the boson mass squared is negative and the coupling constants satisfy certain inequalities. The symmetry properties of such solutions are examined with the aid of a simple model of self-interacting boson fields. The solutions have lower symmetry than the Lagrangian, and contain mass zero bosons.

Structure of a quantized vortex in boson systems

Abstract: For a system of weakly repelling bosons, a theory of the elementary line vortex excitations is developed. The vortex state is characterised by the presence of a finite fraction of the particles in a single particle state of integer angular momentum. The radial dependence of the highly occupied state follows from a self-consistent field equation. The radial function and the associated particle density are essentially constant everywhere except inside a core, where they drop to zero. The core size is the de Broglie wavelength associated with the mean interaction energy per particle. The expectation value of the velocity has the radial dependence of a classical vortex. In this Hartree approximation the vorticity is zero everywhere except on the vortex line. When the description of the state is refined to include the zero point oscillations of the phonon field, the vorticity is spread out over the core. These results confirm in all essentials the intuitive arguments ofOnsager andFeynman. The phonons moving perpendicular to the vortex line are coherent excitations of equal and opposite angular momentum relative to the substratum of moving particles that constitute the vortex. The vortex motion resolves the degeneracy of the Bogoljubov phonons with respect to the azimuthal quantum number.

General relativity without coordinates

Abstract: In this paper we develop an approach to the theory of Riemannian manifolds which avoids the use of co-ordinates. Curved spaces are approximated by higher-dimensional analogs of polyhedra. Among the advantages of this procedure we may list the possibility of condensing into a simplified model the essential features of topologies like Wheeler’s wormhole and a deeper geometrical insight.

Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern

Abstract: At the beginning (Sect. 2) we consider such field theories, which are characterized by operators defined on spacelike surfaces. In this connection Haag’s theorem is discussed. Then the properties of the field operator algebra are investigated (Sect. 3). It follows that the algebraB, defined in Sect. 3·5, is already cyclic. Furthermore, the algebraB, essentially arrived at by adjoining toB the projection operator onto the vacuum state, is irreducible.

Solution of the equations for the green’s functions of a two dimensional relativistic field theory

Abstract: The explicit solution of the coupled set of equations for the Green’s functions for the self-coupled field theory model of Thirring is given. It is found that the infra-red problem causes no special difficulty. The question of how to define products of singular field operators at coincident space-time points arises and it is shown that the commutation relations for such products are not consistently given by making use of the formal expressions and the canonical commutation relations. The problem of gauge invariance in an external field is studied and it is shown that:a) the current and charge density do not commute at equal times, andb) this is necessary for and consistent with the gauge invariance of the field equations.

Primary cosmic radiation and extensive air showers

Abstract: The hypothesis, that separate sources of cosmic ray particles of widely differing strength predominate in different energy regions, has been examined. In particular the following questions are discussed : Can such separate sources dominate neighbouring energy intervals of the primary particle spectrum without introducing appreciable changes in slope or discontinuities in the size-frequency distribution of air showers? Can such separate contributions, if they exist, nevertheless be distinguished experimentally with existing techniques? Do available data on extensive air showers provide evidence for or against the hypothesis? It appears that, if such separate sources existed and even if they differed in strength by factors as large as thousand, no significant departure from smoothness in the size-frequency relation of air showers would occur, provided only that all chemical components in the primary radiation have identical spectra when expressed in terms of the magnetic rigidity of the particles or energy per nucleon, and provided further that the chemical composition of the accelerated particle beam does not deviate too radically from its composition at lower energies. While under these conditions even an infinitly sharp cut-off in rigidity of the particles supplied by the stronger of the contributing sources will not produce a significant disturbance in the size frequency spectrum of air showers, it will nevertheless produce discontinuities and irregularities in the dependence of certain other shower parameters on shower size.-Irregularities which so far have found no adequate plausible explanation have been reported by various investigators at a shower size corresponding to a primary energy of about 1015 eV. Each of these observations seems to support the hypothesis that a rather sharp rigidity cut-off occurs in the source which supplies most cosmic ray particles below this energy. Additional measurable irregularities can be predicted which have, however, not yet been discovered.-The considerations leading to this hypothesis emphasize the important role which the complexity of the chemical composition of primary cosmic radiation must play in the interpretation of air shower phenomena. In view of the resulting complexity in the origin of showers of a given size, it is suggested that a classification of air showers according to the maximum energy of the nuclear-active particles in the axis may be more relevant to an understanding of the various underlying nuclear and electromagnetic processes than the customary classification on which is based the total number of shower particles.

Statistical Emission in Nuclear Reactions and nuclear level density

Abstract: The statistical model in nuclear reactions has been developed extensively in the past decades, mainly by V. Weisskopf. However a clear understanding of the experimental situation regarding low and medium energy nuclear reaction is not yet settled. The interpretation is complicated by the fact that often the reactions proceed via other mechanisms, for instance direct effects. In the present work an attempt is made to show the possibility of explaining a certain group of nuclear reactions with a statistical model. The experiments discussed refer to :a) resonance measurements for slow neutrons;b) energy spectra of particles emitted in the reactions (n, p) and (n, n′) and (p, n) for incident energies ranging between 4 and 16 MeV;c) cross-sections of (n, p) and (n, n′) at 14 MeV. From the set of experiments discussed it is possible to obtain a consistent table of thea parameter of nuclear level density formula.

Notes on the static parity non-conserving internucleon potential

Abstract: If the weak Lagrangian density ℒ consists of self-interacting charged currents, the inclusion into these of a term linear in the pion field leads to a static parity non-conserving internucleon potentialV, the general order of magnitude of whose effects no longer depends on the quadratic pion terms needed for the conservation of the vector current. If neutral currents also contribute to ℒ in such a way as to make it an isotopic scalar, the inclusion of linear pion terms has no effect onV, which then remains inappreciable unless the vector current is conserved. The origins of such pion terms are discussed; it emerges incidentally that as a consequence of time reversal invariance,V cannot contain contributions arising from the exchange of a single pion.

Connection between wightman functions and green functions inp-space

Abstract: Dans le present travail, apres avoir repris l’etude des proprietes d’analyticite de la fonctionW de Wightman dans le cas ou le temps seul est variable complexe, nous en deduisons la fonction de GreenG, etendant ainsi par une nouvelle methode les resultats de O. Steinmann relatifs a la fonction a 4 points. La fonctionG a pour valeur frontiere la transformee de Fourier de la valeur moyenne du vide du produitT des champs et prolonge analytiquement la fonction retardee de L.S.Z. dans l’espace des impulsions. Finalement on etablit un ensemble de proprietes qui caracterisentG en ce sens que siG possede ces proprietes, il existe une et une seule fonction Open image in new window possedant les proprietes habituelles et telle queG en derive.

Closed formulae for the statistical weights

Abstract: The branching ratios between the different charge distributions of mesons emerging with a given multiplicity from a high-energy collision are computed from isospin conservation and a statistical hypothesis on the probability of the possible end states. The method uses the Wigner-type of projection operator in group space and yields a closed formula, which is entirely general, for the weights of the different charge states. The formula is especially simple in the case ofn isospin 1 particles coupled to a low total isospin. Cases withn isospin 1 particles and one or two isospin 1/2 particles are simply related to the former result.

Consequences of unitary symmetry for weak and electromagnetic transitions

Abstract: Under the condition of invariance under the 3-dimensional unitary group, the strong-interaction Lagrangian is assumed to consist of a part invariant under the unitary group, plus a correction' breaking the invariance. Properties that follow for weak and electromagnetic amplitudes and for the assumption that symmetry-breaking corrections.

On a gauge theory of elementary interactions

Abstract: A theory of strong as well as weak interactions is proposed using the idea of having only such interactions which arise from generalized gauge transformations.

Inelastic form factors

Abstract: Electron inelastic scattering by nucleons or nuclei in the process e/sup -/ + A -- e/sup -/ + B + C is related to the photonuclear reaction gamma + A - B + C by using first-order ealculations in the electromagnetic coupling constant alpha = (e/sup 2//4 pi ) = (1/137). (L.N.N.)

Analytic behavior of the scattering amplitude at zero energy

Abstract: It is shown that the scattering amplitude has a two-sheeted branch point at zero kinetic energy. The analytic continuation into the second sheet is discussed in detail.

Theory of the low-energy pion-pion interaction — II

Abstract: It is shown that whenP-wave pion-pion scattering is large at low energies, the integral equations previously formulated by the authors require a cut-off. Because of the cut-off and the unstable nature of the solution, the numerical integration procedure becomes much more involved. The original equations are therefore replaced by a series of conditions at the symmetry point, and the unphysical cuts of the partial-wave amplitudes are replaced by a corresponding series of poles. Within this framework one need not speak of a cut-off, but one new parameter appears. Self-consistent solutions can be found in which aP-wave resonance is sustained by a « bootstrap » mechanism; that is, a strong attractive force in theI=1 state results from the exchange of a resonating pair ofP-wave pions. The symmetry-point conditions used would be modified by the cut-off and quantitative accuracy is not attempted; however, this and other corrections are not expected to change the qualitative nature of our solutions. Rough estimates of the corrections are made.

On the μ-mesonic atoms

Abstract: In an attempt to get a deeper insight into the problem of the « missing X-rays » in the capture ofμ-mesons in the light elements, a complete cascade calculation of theμ-meson transitions in the mesonic atom was conducted. The results indicate again that the calculatedK X-ray yields belowZ=6 and theL X-ray yields belowZ=14 are much above the experimental values. As was noted before by other authors, the deficiency of X-rays increases linearly with decreasing quantum energy between about 90 keV and 20 keV. The experimental yields agree with the calculations above 90 keV, and are lower than the calculated values, by a factor of about 5, at 20 keV. Good agreement between the calculations and experiment is obtained, however, for the (approximately energy-independent) ratios of the higher to basic X-ray yields, provided that a proper choice of the initial meson distribution is made. A comparison between the calculated Auger electron yields and the experimental results forμ−-capture in nuclear emulsions seems to indicate that the « mixed » Auger-radiative transitions introduced by Ruderman are indeed effective (apparently close to 100%) in the de-excitation of the metastable 2s level in CNO. If this mixed transitions play the same role also in the lightest elements (Li, Be, B), the problem of the « missing » X-rays becomes more serious than before. The expected Auger electron spectrum forμ−-capture in the heavy elements (AgBr) of nuclear emulsion has also been calculated.

The definition of electromagnetic radiation

Abstract: Momentum and energy of electromagnetic radiation emitted from a charge are Lorentz-covariantly defined The non-local measurement of a Lorentz inlariant, the rate of radiation energy emission is proposed as a necessary and sufficient criterion for the emission of radiation at a gilen time This measurement is a field measurement which can be performed atany distance from the charge and need not be made in the wale zone It can be regarded as an operational definition of radiation in terms of rate of energy emission These results permit a simple derilation of the equation of motion of a charged particle which aloids the self-energy difficulties A question concerning the principle of equilalence and radiation in uniformly accelerated motion is answered

Über die Vollständigkeit lorentzinvarianter Felder in einer zeitartigen Röhre

Abstract: It will be shown that a complete field is already complete in a timelike tube. This is a consequence of invariance and spectrum condition alone.

Experimental results on the proton-nucleus collisions at 27 GeV in emulsion

Abstract: We are examining the caracteristics of 3 226 stars produced by collisions of 27 GeV protons against emulsions nuclei. We obtain the mean multiplicities ns = 6.6 + 0.1,n h= 7.2±0.2. The mean energy of the secondary shower particles is 2.3 ±0.2 GeV. The energy transfer to the secondary mesons in the l.s. isK ∼ 0.6. From the study of the angular distribution we obtain the mean number of collisions inside the nucleus to be 1.1 and 27 for light anf heavy nuclei respectively. The mean free path for absorption is λ =38.0±1.0 which according to an optical model corresponds to a. nuclear mean free path λn = (4.3±0.3) fermi.

On radiative corrections due to soft photons

Abstract: It is shown how the infra-red divergent part, due to soft virtual photons, of any scattering amplitude can be factorized in a Lorentz invariant manner. It is further shown that if the energy resolution for a process is negligible compared to the electron rest mass then the whole soft photon contribution to the transition probability density can be factorized leaving a Lorentz-invariant remainder. This factorization is then extended to the more realistic case of an energy resolution which is of the same order of magnitude as the electron rest mass, or larger. Finally, the magnitudes of radiative corrections are discussed.

A theoretical approach to high-energy pion phenomena

Abstract: A theoretical analysis of the main features of high-energy strong interactions is performed. First of all, by applying dispersion theory to the elastic scattering amplitude for high-energy and low-momentum transfer we have obtained expressions for the shape of diffraction scattering and for the total cross-sections. The form of those results has suggested a definite model for the most important inelastic processes. In this model a well-defined group of Feynman graphs, the « most peripheral » ones, gives the dominating effect in high-energy collisions.

Off-shell pion-nucleon scattering and dispersion relations

Abstract: The problem of pion-nucleon scattering when one of the pions is virtual is discussed. By assuming that one-dimensional dispersion relations in the energy variable can be written for suitably defined invariant amplitudes, and that also in the off-shell case the dispersive integrals are dominated by the,T = J = 3/2 amplitude, an Omnes-type integral equation is deduced for this amplitude. The rigorous solution of this equation is discussed and reduced to an approximate simpler form which is more suitable for numerical calculations. This form is particularly simple for squared four-momenta of the virtual pion less than ∼ 5 squared pion masses.

A self-consistent field theory of quantum electrodynamics

Abstract: A theoretical approach based on the postulate that the laws of nature must be described in terms of field variables associated only with elementary interactions is presented in the development of a self-consistent field theory of quantum electrodynamics. Maxwell's equations are presented in spinor form. A self-consistent field theory is discussed in terms of the principle of least action, field equations, and the continuity condition. The Pauli exclusion principle is considered. Positronium and pair annihilationcreation process are presented in terms of field equation solutions and the energy and momentum of positronium in the annihilation state. The Lamb shift and the field equation linearizations of the hydrogen spectrum are discussed.

An introduction to the general theory of relativity

Abstract: Fundamental aspects of general relativity are considered, with special emphasis on cosmological problems. Points of disagreement between special and general relativity are discussed. Subjects investigated include rotating coordinate systems, the equivalence principle, tensor analysis, Riemannian geometry, the Schwarzschild and Einstein fields, and Mach's principle. (T.F.H.)

Minimal electromagnetic coupling for spin two particles

Abstract: It is noted that if the same Lagrangian that describes the linearized theory of general relativity is extended to charged massed spin 2 particles the coupling cannot be minimal electromagnetic.

Vector field associated with the unitary theory of the Sakata model

Abstract: For the Sakata elementary particles (p,n,A), a traceless Hermitian 3 x 3 matrix is found from which the existence . of eight vector mesons may be inferred. These mesons include four that resemble K mesons, three that resemble pi mesons, and an isoscalar p/sup 0/ particle. It is shown that psuedo-vector mesons may be produced by gauge transformations on the Hermitian matrix. Similarly, a set of 3 x 3 matrices is found that affords an approximation to a gauge theory for wesk vector bosons. (T.F.H.)

Heisenberg representation in classical general relativity

Abstract: The energy of the gravitational field (as of any other system) is not always the numerical value of the Hamiltonian (for example, not in a Hamilton-Jacobi formulation). We define a classical « Heisenberg representation » which excludes Hamilton-Jacobi-like canonical transformations. Ordinarily, within the Heisenberg representation, there remains only the possibility of time-independent canonical transformation among the dynamical variables. However, the freedom of co-ordinate transformations in general relativity allows many extra « canonical » transformations not found in conventional Lorentz covariant theory. This wider class of canonical formalisms possess all the properties usually associated with the Heisenberg picture in that in each formalism the measurable quantities,gμv(t), are obtained from knowledge of the canonical variables at the same time without any explicit co-ordinate dependence. Further, the Hamiltonian is a constant of motion. Only in Heisenberg frames is the Hamiltonian to be associated with the energy of the system. In spite of the additional freedom of canonical transformations (due to the freedom of co-ordinate change mentioned above), it is shown that the Hamiltonian is numerically the same for a fixed state of the gravitational field in any Heisenberg representation. The energy is then a uniquely definable quantity in the theory. In the process, it is established that two Heisenberg frames can differ by co-ordinate transformations that depend only on the canonical variables and not explicitly on the co-ordinates. These transformations must also preserve the property that at spatial infinity the metric become Lorentz so that the physical boundary conditions be unaltered.

Effect of radiofrequency resonance on the natural line form

Abstract: The form of Zeeman lines in the presence of a resonant rotating r.f. magnetic field is determined. The spontaneous emission distributions are obtained from the steady-state solutions, leading to a prediction of the splitting of Zeeman lines at rotation frequencies corresponding to well separated single- and multiple-quantum resonance frequencies, as well as at the resonance frequency in the Majorana case.

Electromagnetic properties of superconductors

Abstract: A method of calculating the response of superconductors to weak electromagnetic fields is -formulated. The response is generated from a single-particle Green’s function, defined in the presence of an electromagnetic perturbation, that satisfies the conditions imposed by gauge invariance and charge conservation. The response function is thus itself explicitly gauge invariant and charge conserving. The singleparticle Green’s function is obtained by making Gorkov’s factorization of the two-particle Green’s function in the presence of the external field. The symmetry properties of the approximation are used to simplify the calculation considerably. This is trivially carried out in the London limit. An economical derivation is thus obtained of Anderson’s result that the Bardeen, Cooper and Schrieffer quasi-particle excitations must be supplemented by a collective excitation in order to satisfy the conservation laws.

Mesic decays of hypernuclei from K−-capture

Abstract: 422 K- and 13 ∑--capture stars emitting mesic hyperfragments (MHF) have been studied in an attempt to understand the mechanism of hyperfragment production. 123 K- stars emitting8Li fragments were also considered. Several criteria were used to separate K- captures in C, N, O and Ag, Br. Lower limits for the proportion of MHF produced in C, N, O are 57%, 73%, 84%, and 94% for HΛ, HeΛ, LiΛ and MHF of Z ≥ 4. For8Li, this lower limit is 75%. An overall frequency of (1.05±0.05)·10-2 MHF/K--capture and (3.8±1.4) · 10-2 HF/∑--capture is obtained. The energy spectrum of the pions emitted in the capture stars indicates that a substantial fraction of the emitted bound Λ’s originates in ∑-conversion processes. Indication of bound-Λ production from multinucleon K--capture processes was obtained, important in the case of HΛ. The momentum spectra for different hypernuclear species are very similar and the spectra of HeΛ and LiΛ are, in turn, comparable with the momentum spectra of He and8Li respectively. Hyperfragments are emitted preferentially opposite to the direction of emission of an associated pion or fast proton. The backwards-forward (B/F) ratio is for π, MHF events, 1.8±0.4. Two models for the production of hyperfragments are proposed. According to the first (« trapped-Λ ») the Λ created in the elementary process is trapped in the nuclear potential well of the parent nucleus. The HF is then emitted in the process of nuclear de-excitation. In the second model (« prompt-HF ») the HF is emitted promptly following the K- or ∑- interaction with a group of nucleons. It is concluded that both mechanisms may contribute. The occurrence of both processes can explain essentially all the salient features of hyperfragment emission.