Showing papers in "Il Nuovo Cimento B in 1972"

Dimensional renorinalization : The number of dimensions as a regularizing parameter

Abstract: We perform an analytic extension of quantum electrodynamics matrix elements as (analytic) functions of the number of dimensions of space(ν). The usual divergences appear as poles forν integer. The renormalization of those matrix elements (forν arbitrary) leads to expressions which are free of ultraviolet divergences forν equal to 4. This shows thatν can be used as an analytic regularizing parameter with, advantages over the usual analytic regularization method. In particular, gauge invariance is mantained for anyν.

Ground and first excited states of excitons in a magnetic field

Abstract: An exact calculation of the ground state and of the first excited state withm = 0 and even parity of a hydrogenic system in a magnetic field is described, and results are given for the energy and the main features of the wave functions. Tor the excited state the shape of the nodal surface is given, and it is shown that in this case no contradiction exists between the noncrossing rule and the nodal-surface criterion for the connection of the levels in the low- and high-field regions.

Vacuum-field solutions in the brans-dicke theory

Abstract: The field equations of the Brans-Dicke theory are solved for a vacuum with the aid of a space-time metric of Friedmann type. Nonstatic solutions are found showing that, in general, a Birkhoff theorem does not exist for the Brans-Dicke theory. Solutions are also found which may be interpreted as being contrary to Mach’s principle.

A modified Navier-Stokes equation, and its consequences on sound dispersion

Abstract: Taking into account some considerations developed in the past by Cattaneo we suggest a modified Navier-Stokes equation which includes the inertial property of the momentum flux. The linearized system of hydrodynamic equations where the inertial properties of both momentum and heat flux are taken into account leads to a new dispersion equation which does not exhibit the theoretical disadvantages of the Kirchhoff equation. Also the consistency with the experimental results about sound dispersion in rarefied monoatomic gases is definitely improved.

Magneto-plasma surface waves in solids

Abstract: The effect of a d.c. magnetic field on the dispersion relation of surface plasmons in a metal is studied. Numerical results, including the effects of retardation, are presented for both the Faraday and the Azbel-Kaner geometries.

Hooke's symmetries and nonrelativistic cosmological kinematics. i.

Abstract: The structure of the Hooke groups and of their central extensions is studied and compared with that of the Galilei group. The local exponents of the Hooke groups are obtained by integration of the Lie algebra. The invariants of the extended Hooke groups are deduced from the de Sitter’s symmetries. The kinematical observables are identified with elements of their envelopping algebras and the constants of motion are given. Knowing that the main difference between the Hooke and Galilei kinematics can be accounted for by a harmonic potential, we propose some indications for a Hooke cosmological model.

Analysis of indirect two-photon interband transitions and of direct three-photon transitions in semiconductors

Abstract: The indirect two-photon interband transition rate has been studied by analysing the absorption coefficient for the photon Ħϖ1 in the presence of a second photon Ħϖ2 as a function of frequency near the band edge at different temperatures. The structure of the curves is similar to that for one-photon indirect absorption and shows discontinuities when Ħϖ1 + Ħϖ2 ±ĦΩ —Eg ≈ 0, ĦΩ being the energy of the phonon which makes the transition possible, andEg the indirect energy gap. The results are compared with those for three-photon processes, and it is found that indirect transitions are at least one order of magnitude more probable at the available photon densities. Selection rules for indirect two-photon transitions are discussed for Ge and GaP. It is shown that the process which has highest probability in Ge involves only the L.O. phonon, whereas in GaP all phonons are allowed in the transitions except the L.O. phonon. Numerical calculations of the absorption coefficient are performed for the case of GaP near the indirect energy gap to show the dependence on frequency and temperature. The results indicate that presently available optical laser sources should be of sufficient intensity to produce observable two-photon indirect transitions. Preliminary experimental evidence is discussed.

Differential Cross-Section for the inner-shell ionization of medium-heavy atoms by the electrons

Abstract: A theoretical study on the differential cross-section for the inner-shell ionization of medium-heavy atoms has been made. Special attention has been paid to the study of the angular distribution of the scattered electrons, scattered after ionizing a50Sn atom, and to the momentum distribution of the electrons ejected from the atom.

Canonical transformations and quadratic hamiltonians

Abstract: It is shown that there exist symmetry transformations in phase space that preserve Hamilton’s canonical equations of motion for one Hamiltonian, but not for all. Examples of these « canonoid » transformations are given and their relation to canonical transformations is developed. It is demonstrated that a sufficient condition for a canonoid transformation to be canonical is that it preserve Hamilton’s equations for all Hamiltonians quadratic in theq’s andp’s.

Gauge invariance and the quantization of mass (of gravitational charge)

Abstract: In a previous paper (referred to as I in the text) it was shown that the Weyl principle of gauge invariance leads to the relationshipGm2 =ħc for a particle of inertial massm obeying the Dirac equation, whereG is the Newtonian gravitational constant. Instead of interpreting this equation to mean thatG takes on the extremely large valueħc/m2 inside a particle like an electron (as we did in I), we now write it in the formGm2/c = ħ and treat it as a quantization condition on the square of the gravitational charge √Gm. We show that this same quantization condition can be obtained from an angular-momentum component in the general-relativistic two-body problem as well as from the Machian definition of inertial mass in a rotating universe by using the Dirac-Schwinger procedure for quantizing charge. From this quantization condition we now deduce that the fundamental particle in Nature (the uniton) has an inertial mass equal to about 10-5 g. The possibility of using the uniton to shed light on the mystery of the « missing mass » in the Universe is discussed. Other cosmological implications of the uniton are also discussed and it is suggested that unitons can clear up the solar-neutrino discrepancy.

Phonons in a strong static magnetic field

Abstract: The Hamiltonian of an ionic crystal lattice in the harmonic approximation and in the presence of a static magnetic field is diagonalized in terms of creation and annihilation operators. Phonon energies and polarizations are determined in this theory by means of the solutions of an eigenvalue problem of the first kind with twice the dimension of the corresponding problem without magnetic field. The energy eigenvalues of the Hamiltonian are linear combinations of the phonon energies ħωj(k) with integers as coefficients, just as in the case without a magnetic field. For crystals in which every ion is at a center of inversion symmetry the phonon spectrum will conserve its inversion symmetry ink-space. For crystals without inversion symmetry this will no longer be true. In every case the polarizations of the phonons will be elliptical.

Some reflections on the nature of entropy, irreversibility and the second law of thermodynamics

Abstract: We discuss the conceptual problem of understanding irreversibility and the arrow of time We point out that by generalizing the concepts of system and state, and the mechanics which governs them, it is possible to create the conceptual framework in which the occurrence of irreversibility can be understood naturally The destruction of correlations is the principal mechanism which, in the generalized evolution given by a stochastic dynamics, leads to the increase of entropy and all the consequences of the second law In the Appendix, we treat the case of two interacting spin systems and show that the destruction of correlations and the appropriate stochastic dynamics lead to irreversibility

Kinetic derivation of nonstationary general relativistic thermodynamics

Abstract: Chernikov, who generalized Grad’s thirteen-moment method for an approximate formal solution of the Boltzmann equation to the case of the general relativistic Boltzmann equation (applicable also to a photon gas), did not exploit his formalism completely but developed the last eight moment equations for dissipation phenomena no further than a stationary theory. However, this stationary theory results in the possibility of the dissipation processes propagating with infinite speed, which is not acceptable. In this paper the complete Chernikov thirteen-moment approximation is developed to obtain nonstationary transport equations, providing linearized collision integrals, for heat and viscous stresses which lead only to finite dissipation speeds. The flux, density and entropy production and therefore the entropy-balance equation representing the second law of thermodynamics is approximated in terms of moments (and microscopic quantities) at the nonstationary level. An expression for entropy production is also deduced as a linear function of the collision integrals. Furthermore, the Gibbs equation, free energy and chemical potential are given in a nonstationary form. A linearized nonstationary heat-conduction equation is derived for a system in mechanical equilibrium without viscous stresses.

Relativistic classical spinning-particle mechanics

Abstract: A generalized Lagrangian, chosen to be a function of both velocity and acceleration, is found to yield an equation of motion for a relativistic classical spinning particle. Quantization of the corresponding Hamiltonian produces a wave equation which, like the Dirac equation, is of first order in time and manifestly covariant.

On the electron-muon mass doublet from general relativity

Abstract: This paper examines a consequence of combining separate aspects of the author’s research program toward the development of a self-consistent field theory of a closed system. These were the separate studies that indicated that: 1) the inertial mass of matter is derivable from an explicit dynamical relation in terms of the geometrical field that describes a closed system in general relativity, and 2) the result of a self-consistent approach to electrodynamics that any region of spacetime must be populated with a countable and finite system of particleantiparticle pairs, each in its ground state of null energy and null momentum. The latter followed from an application of Noether’s theorem in a nonlinear field theory of electrodynamics. In this article, a linearized version of the theory that couples the matter and the metric field equations is studied. It is shown that the inertial mass field for spinor particles, being a 2-dimensional matrix field, has two distinct mass eigenvalues. From a study of the space-time curvature as due to the background gas of particle-antiparticle pairs, whenever the mass of an observed particle might be measured, semi-quantitative arguments are presented which imply that the ratio of these two eigenvalues for the electron mass doublet is the order of 3/2α ≅205 (α being the fine-structure constant)—a number that is quite close to the muon-electron mass ratio. The analysis implies that to be consistent with the magnitude of the electron mass, the specific volume of the background gas of pairs is the order of (10−15 cm)3. It is also noted that a further implication of this background gas of pairs should be the appearance of « electron jets » and the accompanying electromagnetic radiation, in those parts of the Universe where the temperatures of the ordinary mass distribution would reach values compatible with the dissociation energy of these pairs (corresponding toT ∼ 109 ‡K)—this provides a possible explanation for some of the observed characteristics of quasars.

Orbital and vortical motion in the Kerr metric

Abstract: The motion is analysed, in general, in the Kerr metric with the use of first integrals. Some of the high-energy particles and photons are found to move in a giant vortex around the axis of symmetry above and below the equatorial plane, dragged by the gravitational field.

Stopping particles underground.

Abstract: An attempt has been made to interpret the experimental results on stopping muons underground, which appear to show an anomalously high stopping rate, in terms of atmospheric muons, muons from the decay of locally produced pions and muons from neutrino interactions. The results of calculations of the various contributions are given and the necessary corrections to the experimental data are discussed. It is shown that the experimental results are in reasonable agreement with the expected contributions from the various sources and that no new process is required.

Equilibrium in special relativity.

Abstract: It is shown that there are no energy currents perpendicular to the forces on an object. Many reasons for the erroneous nature of the von Laue energy current are given. This energy current is of no help in formulating equilibrium conditions in the conventional synchronous formulation, where additionalad hoc momenta due to the nonconservation of simultaneity have to be added. Equilibrium maintained by continuous time-dependent forces cannot be defined by the simultaneous vanishing of the forces. On the other hand, in the asynchronous formulation of Cavalleri and Salgarelli, equilibriumis given in terms of only the forces and torques. The relevant torque tensor ℐΜΝ is defined. This tensor is the sum of quantities evaluated at different times, corresponding to the Lorentz transformation of simultaneous measurements in the proper inertial frameS. Consequently, if ℐΜΝ= 0, then ℐΜΝ1, the torque in another inertial frameS′, also vanishes. Numerous objections are fully discussed. It is shown that there is no paradox in the Trouton-Noble experiment. Finally, it is shown why it is impossible to formulate the usual Hamiltonian for interacting particles in special relativity in terms of the co-ordinate timet; instead, one must use individual proper timesΤ.

The cosmic-ray spectral modulation above 2 GV. IV. — The influence on the attenuation coefficient of the nucleonic component

Abstract: In this Part IV we study the influence of the primary cosmic-ray modulation on the attenuation coefficient of the nucleonic component at different latitudes and altitudes. The direct investigation of the time behaviour of the station attenuation coefficients is based on the largest amount of refined data so far used,i.e. the revised data of the IGY neutron monitors 1957–1965 obtained in Part 1 and the IQSY supermonitor data 1965–1969. The results of the correlation between this time behaviour and the time changes of the nucleonic intensity are in agreement with the estimates based on the solar-cycle modulation of the primary spectrum found in Part III. The results are also presented in a form suitable to their use for an accurate pressure correction of the data of the network of neutron monitors.

L-shell photoelectric cross-sections for heavy atoms at 145 and 279 keV

Abstract: L-shell photoelectric cross-sections for mercury, gold, thallium, lead, bismuth, thorium and uranium at 145 and 279 keV energies have been determined via two methods. The technique employed involves measurement of the total intensity of fluorescentL X-rays that follow the photoelectric absorption of a known flux of γ-rays in theL-shell. A proportional counter has been used as a fluorescentL X-ray detector. The results are in good agreement with the theoretical predictions of Schmickley and Pratt.

The absolute vertical cosmic-ray muon intensity at sea level

Abstract: The absolute vertical intensity of muons with momentum above 0.88 GeV/c at sea level has been measured and found to be (8.22 ± 0.40) · 10−3 cm−2 sr−1 s−1. This value is close to what would be expected from recent determinations by Allkoferet al. and is significantly higher than the normalization value used for a number of years. By using other absolute measurements in this region of momentum, the integral and differential muon intensities at the standard momentum of 1 GeV/c are found to be (7.58 ± 0.40) · 10−3 cm−2 sr−1 s−1 and (3.18±0.17) · 10−3 cm−2 · sr−1 s−1 (GeV/c)−1, respectively.

The North-South anisotropy and the cosmic-ray radial gradient in the vicinity of the earth

Abstract: As was shown by Bercovitch, an estimate of the cosmic ray radial gradient in the solar system can be obtained from the North-South (NS) anisotropy in interplanetary space. In the present analysis the neutron monitor data of the polar looking stations Alert (North) and McMurdo (South) in the years 1965–68(i.e. from the minimum to the maximum of solar activity) were used, omitting days of Forbush decreases and days of polarity inversion in the interplanetary magnetic field. The peak-to-peak NS anisotropy which reverses its direction with the field polarity is found to increase from (0.030 ± 0.025) % in solar minimum to (0.30 ± 0.03)% in solar maximum, while the estimated radial gradient increases from (0.4 ± 0.3)%/a.u. up to (4.2 ± 0.4)%/a.u. The implications of this result on the long-term modulation mechanisms are discussed taking into account the observed constancy in the solar-wind velocity throughout the investigated period.

Newtonian gravitational field theory

Abstract: This paper was inspired by Bondi’s formula for gravitational energy transfer, and recent work on this by Cooperstock and Booth. It concerns only Newtonian gravitation and aims at clarifying the concept of energy in this context. Use is made of gravitational energy density, suggested by Maxwell, but not acceptable to him since it is negative.

Static multiparticle systems in general relativity

Abstract: We develop exact static solutions to the field equations of general relativity that represent axially symmetric distributions of isolated bodies. The solutions obtained do not involve negative mass; the gravitational attraction between the bodies is counterbalanced by the Coulomb repulsion of electrical charges placed on the bodies. Since the assignment of physical interpretations to axially symmetric solutions is not straightforward in general relativity, we also present additional arguments showing that our multiparticle interpretation is indeed correct. Thus we analyse, for the case of two isolated particles, the behavior of the differential invariants and the geodesic motion of test particles, and for all the examples considered we find agreement with our multiparticle interpretation. Our solutions also contain nonsingular event horizons, which we relate to an electrovac theorem recently proved by Israel.

Pseudopotential approach to the band structure of alkali halides

Abstract: A pseudopotential approach to the calculation of the conduction band structure of ionic crystals is proposed along the lines suggested by Heine and Abarenkov. An improved model potential originally suggested by Griuliano and Ruggeri is adopted and discussed in connection with other choices of the potential. Detailed calculations are performed in LiCl, NaCl and KC1, and the results are similar to those obtained with the KKR and the OPW methods, but are more free electronlike, as suggested by Phillips. The minimum of the conduction band is F1 in NaCl and KC1 andL1 in LiCl. The lowest conduction band is nondegenerate and overlaps very little with the higher conduction bands. In the case of KC1 a well-defined secondary minimum exists atX3, while in LiCl and NaClX3 is more likely a saddle point. The strong absorbtion peak above ionization in KC1 is attributed to a resonant exciton associated to the minimum atX3, rather than to a peak in the density of states. The electron-hole interaction should be taken into account to interpret the entire optical excitation spectrum.

On the lifetime of the muon state of the electron-muon mass doublet

Abstract: In a previous paper the electron and muon masses were derived by combining the author’s general relativistic derivation of the inertial masses of spinor particles from the geometrical field and his model of the « physical vacuum » (from a self-consistent field theory of electrodynamics) in terms of an ideal gas of electron-positron pairs in a particular state that was derived from the theory. It was predicted earlier that in this field theory spinor particles occur in mass doublets. In the electron-muon case, the observation of the « heavy electron » (muon) is a consequence of quadrupolar excitation of the physical vacuum of pairs—correspondingly altering the geometrical field in the vicinity of the observed electron, thereby making it more inertial. In this paper, the lifetime of the heavy electron in this physical vacuum will be determined, de-excitation occurring by the transferral of electromagnetic energy from this heavy electron to the pairs of the physical vacuum by means of a quadrupolar transition. To facilitate the calculation with time-dependent perturbation theory, use is made of Fermi’s multiple-meson production theory, and the improvisation of his theory for break-up reactions by the author. In this way, the ensuing quantitative analysis yields a lifetime for the heavy electron that is of the order of 2.09·10-6 s, which is within 5% of the observed muon decay rate. It is then shown how the present theory, in which electromagnetic variables are in terms of 2-component spinor fields, explains the reaction Μ → e++Ν in terms of a process that is purely electromagnetic and does not entail neutrinos.

The generalized-lorentz-gauge description of quantum electrodynamics

Abstract: A unified formulation of quantum electrodynamics in the generalized Lorentz gauge (including, for example, the Feynman, Landau and Yennie gauges) is presented. The formulation is shown to be consistent with the appropriate commutation relations and propagators in the various gauges. All the gauge descriptions are shown to lead to Maxwell’s equations in the Lorentz gauge provided that expectation values of the electric and magnetic field intensities are taken over physically admissible states. It is shown that when the subsidiary condition is properly treated for the case of interacting photons and charged particles, the various gauge descriptions have the same physical content and electron wave function renormalization constants as the Coulomb gauge

Chemical shifts in X-ray absorption edges and effective nuclear charge. A correlation

Abstract: The experimentalK andL III edge shifts of some metals (Cu, Co, Ga, As, Pb and Bi) in their binary compounds have been quantitatively related to effective nuclear charge and electronegativity. It is found that these two seem to be linearly related to the edge shifts for a given metal as well as for a set of isoeloctronic metals ions in the different compounds.

On the apparent visual forms of relativistically moving objects

Abstract: The question of the apparent visual shape of an object moving at relativistic speeds, as perceived by a single observer, is analysed afresh. It is shown by qualitative arguments that the apparent shape is related to the shape at rest through a combination of nonuniform shear and extension/contraction parallel to the direction of motion, which does not reduce to a rotation even in the case of distant objects subtending a small angle at the observer. The two-dimensional projection (as in a photograph) of this distorted shape may coincide with that of the object (suitably rotated) at rest; but we emphasize that it would be grossly misleading to conclude from this, as is generally done in the literature, that distant relativistically moving objects appear as if simply lotated. The « train paradox » is discussed in illustration of this point. Analytical formulae relating the apparent visual shape to the shape at rest are given. Also the striking fact that the apparent speed of the object as seen by visual observation may well exceed the speed of light is brought out. Finally it is pointed out that the phenomenon is closely analogous to the relativistic Doppler effect.

A modified quantization of the spinor field

Abstract: We present a quantization procedure for the spinor field in an external electromagnetic field that leads to a theory with a fixed number of « particles ». As in relativistic quantum mechanics, a « particle » can be found in a particle state, propagating forward in time, or in an antiparticle state, propagating backward. This way, pair creation and annihilation appear as changes in the mode of propagation. We interchange the roles of the creation and annihilation operators for the antiparticles, and we further assume that the Hamiltonian operator displaces the antiparticle backward in time. This can be carried out systematically if we use Dirac’s many-times formalism and numbered particle operators. Although the dynamical problems are very different from those of the conventional formulation, transition amplitudes for finite times are quite similar. The main difference in the perturbation expansion is the absence of terms with closed fermion loops. We also find that the vacuum is a stationary state, and we obtain a closed single-particle theory for the amplitudes with one « particle » which is a modified Dirac equation.