Relativistic kinetic theory of quantum systems: I. Wigner functions for a relativistic spin-12 system

For a dilute and nondegenerate relativistic spin-$\text{1}\text{2}$ system two kinds of Wigner functions are defined: one has sixteen spinor components and the other four spin components. Their relationship is established. Statistical expressions for the current density, the energy-momentum density and the spin density are obtained in terms of both kinds of Wigner functions. The transformation properties of the latter under Lorentz transformations are discussed.     

Relativistic kinetic theory of quantum systems: V. Transport equations for the massive components of an eeveve-mixture

Covariant transport equations are derived for the Wigner function matrices describing the massive components of a non-degenerate e$\text{e}$v_e$\text{v}$_e-mixture. The collision terms pertaining to the various collision processes involve spin-dependent transition amplitudes which are specified in terms of an effective Hamiltonian density obtained from the Weinberg-Salam theory.     

Quantum effects in the Bertotti-Robinson metric

Scalar and spin $\text{1}\text{2}$ fields have been studied in the Bertotti-Robinson space-time and analytical solutions have been obtained for the Klein-Gordon equation and the Dirac equation. For large e particle creation takes place as in the pseudoeuclidean metric under the influence of a constant external field.     

Quantum maps

We quantize area-preserving maps M of the phase plane q, p by devising a unitary operator $\text{U}$ transforming states | φ_n〉 into | φ_n+1〉. The result is a system with one degree of freedom q on which to study the quantum implications of generic classical motion, including stochasticity. We derive exact expressions for the equation iterating wavefunctions ψ_n(q), the propagator for Wigner functions W_n(q,p), the eigenstates of the discrete analog of the quantum harmonic oscillator, and general complex Gaussian wave packets iterated by a $\text{U}$ derived from a linear M. For | ψ_n〉 associated with curves $\text{L}$_n in q, p, we derive a semiclassical theory for evolving states and stationary states, analogous to the familiar WKB method. This theory breaks down when $\text{L}$_n gets so complicated as to develop convolutions of area ? or smaller. Such complication is generic; its principal morphotologies are“whorls” and “tendrils,” associated respectively with elliptic and hyperbolic fixed points of M. Under $\text{U}$, ψ_n(q) eventually transforms into a new sort of wave that no longer perceives the details of $\text{L}$_n. For all regimes, however, the smoothed | ψ_n(q)|² appears semiclassically appears to be given accurately by the smoothed projection of $\text{L}$_n onto the q axis, both smoothings being over a de Broglie wavelength. The classical, quantum, and semiclassical theory is illustrated by computations on the discrete quartic oscillator map. We display for the first time stochastic wavefunctions, dominated by dense clusters of caustics and characterized by multiple scales of oscillation.     

Wigner crystal model for quasi one-dimensional systems based on TCNQ

The value of the electron transfer, ρ, from donors (Rb, NMP) to acceptors (TCNQ) is studied by calculating the electrostatic energy of the Rb-TCNQ and NMP-TCNQ crystals as a function of ρ and using the spin susceptibility data for the NMS-TCNQ crystal. It is shown that for the Rb-TCNQ crystal ρ = 1 and for the NMP-TCNQ crystal $\text{ρ =}\text{1}\text{8}$. The Wigner crystal model for highly conductive TCNQ salts is discussed.     

On the dynamics of the Ising model in a transverse field

Including mode coupling terms we give the equation of motion for the classical version of the transverse Ising model. The critical behaviour for d ? 2 is dominated by the mode coupling terms. The corresponding kinetic coefficient diverges with $\text{z =}\text{1}\text{2}\text{(d ? 2)}$. For 2 < d ? 4 the model behaves like a relaxator model.     

Variation of the local field through the liquid-vapour interface

We derive an integral equation for the self-consistent local field E^loc(z) within an inhomogeneous non-polar fluid, with particular application to the liquid-vapour interface. Approximate solutions are given for the cases of induced atomic dipoles oriented perpendicular and parallel to the interface. For the perpendicular case we relate the average field to the local field and thus obtain an equation for the static dielectric constant ?(z) in terms of the density profile n(z). The departures of the local field from Lorentz form $\text{E}{}^{\mathrm{<mtext>ext</mtext>}}\text{/(1 + (}\text{8}\text{3}\text{)παn(z))}$ and of the dielectric constant from the Clausius-Mossotti form ($\text{1 + (}\text{8}\text{3}\text{)πan(z))/(1 ? (}\text{4}\text{3}\text{)παn(z))}$ are shown to be small. For the parallel case we discuss fringing of the external field and show that the dipoles align themselves with the average field, not the external field. The departure of the local field from $\text{E}{}^{\mathrm{<mtext>ave</mtext>}}\text{/(1 ? (}\text{4}\text{3}\text{παn(z))}$ is shown to be small.     

Solution of the Schrödinger equation in terms of classical paths

An expression in terms of classical paths is derived for the Laplace transform $\text{Ω(s)}$ of the Green function G of the Schrödinger equation with respect to $\text{1}\text{h}\text{?}$. For an analytic potential V(r), the function Ω is also analytic in the plane of the complex action variable s; its singularities lie at the values $\text{S}$ of the action along each possible (complex) classical path, including the paths which reflect from singularities of the potential. Accordingly, G may be written as a sum of terms, each of which is associated with such a classical path, and contains the factor $\text{exp}\text{(}\text{iS}\text{h}\text{?}\text{)}$. This expansion formally solves the problem of constructing waves out of the corresponding (complex) classical paths. A similar expression, in terms of closed paths, is derived for the density ? of eigenvalues of the Schrödinger equation. In situations when the eigenvalues are dense, ? is well approximated by the contributions of the shortest closed paths: while the path of vanishing length corresponds to the Thomas-Fermi approximation and its smooth corrections, the other paths yield contributions which oscillate and are damped as $\text{exp}\text{(}\text{iS}\text{h}\text{?}\text{)}$. This result also holds for nonanalytic potentials V(r). If the spectrum is continuous, closed classical paths yield oscillations in the scattering phase-shift. The analysis is also extended to multicomponent wave functions (describing, e.g., motion of particles with spin, or coupled channel scattering); along a classical path, the internal degree of freedom varies adiabatically, except through points at which it is not coupled to the potential, where it may undergo discrete changes.     

Application of finite strain theory to non-cubic crystals

The application of finite strain theory to a crystal of orthorhombic or higher elastic symmetry, which is subjected to a purely extensional deformation parallel to the crystallographic axes, yields stress-strain relations and effective elastic constants for hydrostatic stresses. Alternative formulations of the theory are obtained when the free energy is written as Taylor series in E, the invariant analogue of the Eulerian strain tensor, or η, the Lagrangian strain tensor. In most cases, the formulation in terms of E provides a better approximation when the Taylor series are truncated at the second or third order. In the case of cubic crystals, the stress-strain relations reduce to the Birch equation. For non-cubic crystals, the P-V relations calculated using the non-cubic and Birch equations will differ due to the effects of elastic anisotropy. A comparison between the second-order approximations of the non-cubic stress-strain relations and the cubic Birch equation suggests that the difference in volume will be less than 1% for most materials. The difference in volume is reduced when the third-order approximations are used. When the third-order terms are retained, the stress-strain relations calculated using the Eulerian formulation agree with measured linear compression data for quartz to 150 kbar $\text{(}\text{P}\text{K}{}_0\text{= 0.4).}$ For zinc, the calculated pressure-volume relation agrees with the shock-wave Hugoniot up to 750 kbar $\text{(}\text{P}\text{K}{}_0\text{= 1.2),}$ although both calculated and Hugoniot P-V relations disagree with X-ray compression data. At pressures greater than 300 kbar, the calculated axial ratio of zinc approaches that for other hexagonal metals $\text{(}\text{c}\text{a}\text{? 1.63).}$     

On the correlation between diffusion and self-diffusion processes of adsorbed molecules in a simple microkinetic model

Combining the formalism of transition state theory with an appropriate collisional term, a general relation between diffusion and self-diffusion coefficients (D and D¹ respectively) in zeolites is derived. For low pore filling factors, this relation reduces to Darken's equation, which is shown to coincide in this case with the simple equation $\text{D = (}\text{d}\text{dc}\text{)(D}{}^1\text{c)}$, where c denotes the concentration of adsorbate. Concerning the intracrystalline self-diffusion of ethane in NaCaA zeolites, it is shown, that up to pore filling factors θ ≈ 0.8, Darken's equation should be valid, whereas for higher concentrations considerable deviations from this simple relation may occur and the general formula, as derived in this paper, should be applied.     

Pion production and the higher resonances in pion-nucleon scattering

The reaction π + N → 2π + N has been studied in the vicinity of the higher resonances in the pion-nucleon cross section. The Low equation for the production amplitude is transformed into an integral equation by isolating the true one-meson intermediate states and discarding higher order contributions. The only part kept in the inhomogeneous term corresponds to the collision of a pion in the nucleon cloud with the incident pion in the resonant T = J = 1 state, which is simulated by an unstable vector Boson. Crossed terms are neglected and the 2π-N state is described by the static model. The terms kept in the sum over states describe the rescattering (off the nucleon) of one of the outgoing pions. The required off-the-energy-shell elastic scattering amplitude is approximated by the 3-3 resonance formula of Chew and Low. With these simplifications the Low equation for the production amplitude reduces to an easily soluble linear integral equation. The rescattering amplitude, which dominates the inhomogeneous term in the resonance region, is proportional to the 3-3 scattering amplitude of one of the outgoing pions. Although the result provides some support for the conventional isobar model, it is important to note that the largeness of the rescattering term arises from scattering far off the energy shell, rather than by “real” excitation as in the phenomenological isobar model. Quantitative calculations for the $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ channel leading to a p-wave ($\text{J =}\text{3}\text{2}$) and an s-wave pion produce a maximum in the cross section near 600 Mev incident pion lab energy. For a π-π resonance energy squared S = 10, agreement with experiment is obtained with a width about one third that suggested by nucleon electromagnetic structure. In our approximation, the well known 600 Mev $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ isospin $\text{1}\text{2}$ resonance occurs at the same energy as the 800 Mev $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ isospin $\text{3}\text{2}$ resonance. It is assumed, but not proved, that the neglected terms are responsible for the splitting of the resonance energies. When this splitting is taken into account, the predicted charge state ratios near the second resonance agree well with existing data. The “third” resonance occurs for the state having two p-wave pions, according to the present theory, although no numerical calculations were made for this case. This point of view suggests that the $\text{F}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, $\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, and $\text{P}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ incident channels contribute to the third resonance.     

Elementary excitations in quadrupolar systems

We consider $\text{S ?}\text{3}\text{2}$ isotropic quadrupolar ordered systems and derive elementary excitations at low temperature. A Holstein-Primakoff type transformation and a linear approximation are used. For $\text{S =}\text{3}\text{2}$, the spectrum is made of four degenerate acoustic branches. For S ? 2, only two degenerate branches satisfy the Goldstone theorem: they describe Δm = ± 1 excitations similar to librons in molecular crystals. The two degenerate branches describing Δm = ± 2 excitations have a gap at k = 0 although the hamiltonian is isotropic. For a special $\text{S =}\text{3}\text{2}$ cubic hamiltonian, a Goldstone mode is found in the spectrum and related to a continuous degeneracy of the ground state. A comparison between $\text{S =}\text{1}\text{2}$ dipolar and $\text{S =}\text{3}\text{2}$ quadrupolar systems is presented.     

Dirac equation with cylinder-symmetrical potential

For a potential V(?), $\text{?=(x}{}^2\text{+y}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, the Dirac equation is reduced to a pair of first-order differential equations in ?. The fine structure of relativistic “channeling” electrons bound to a crystal axis is calculated.     

Resistivity of dilute AlMg and AlMn between 0° and 100°C a generalized approach

We have measured the resistivity of $\text{Al}$Mg and $\text{Al}$Mn up to 0.5 at. % impurity concentration between 0° and 100°C. The results for the resistivity for both systems can be analysed within one generalized model in which spinfluctuations as well as changes in the electron-phonon interaction are considered. For $\text{Al}$Mn a calculation of the impurity contribution to d?/dT is attempted. For $\text{Al}$Mg there are indications that the superconducting T_c will increase with Mg concentration.     

On the geometrical aspects of electromagnetism,I

The trajectory of a charged test particle under a Lorentz force is obtained as the geodesic of a riemannian four dimensional manifold. Originally, the geodesic equation is nonlinear in some vector field A′_μ. The nonlinearity is traded in for the correct characteristic $\text{e}\text{m}$ of the test particle through a gauge condition, imposed upon A′_μ, which turns the geodesic into the fully covariant linear and gauge invariant Lorentz equation. Fitting the $\text{e}\text{m}$ ratio inside the gauge leaves F′_μν independent of $\text{e}\text{m}$ and allows its identification with the E.-M. tensor F_μν. This four dimensional approach allows the identification of the fifth coordinate used in Kaluza's geometrization |1,2|. The gauge function appears as the sum of Hamilton-Jacobi function plus an additional term, related to the “length” of the trajectory. It is this latter term which guarantees the correct “normalisation” of the $\text{e}\text{m}$ ratio.     

A phenomenological investigation of ππ → KK partial-wave amplitudes from hyperbolic dispersion relations

We calculate the $\text{ππ →}\text{K}\text{K}$ partial-wave amplitude g_J^(+-), J = 0, 1, 2, 3, in the unphysical region below the $\text{K}\text{K}$ threshold using partial-wave relations derived from hyperbolic dispersion relations. For g₁^(?) the results are in agreement with earlier calculations and we confirm the current low-energy predictions. For g_o⁽⁺⁾ we favour a down-type solution with a large positive $\text{K}\text{K}\text{→}\text{K}\text{K}$ s-wave scattering length.     

On the determination of α2F(ω) in metals by measuring I–V characteristics of “wide” (non-ballistic) point-contact junctions

For metals with small electron and phonon mean free paths (alloys, deformed or amorphous materials), there exists a possibility of determining α²F(ω) by measuring the V dependence of $\text{d}{}^2\text{I}\text{d}\text{V}{}^2$ or $\text{d}{}^3\text{I}\text{d}\text{V}{}^3$ of wide (d ? 10³ Å) point contacts (PC) and then inverting the linear equation relating these quantities to α²F(ω). The procedure is elaborated numerically and tested successfully for model electron-phonon interaction spectra.     

Canonical quantization of polyakov's string in arbitrary dimensions

Last year Polyakov discovered the important role the trace anomaly plays in the relativistic string theory. This result means that one has to add counter terms to the string lagrangian

$\text{L}\text{=}\text{L}{}_{\mathrm{<mtext>string</mtext>}}\text{+C}\text{L}{}_1\text{,}$

, where $\text{L}$₁ contains a cosmological term and a term from the trace anomaly. In the conformal gauge we have

$\text{L}{}_1\text{=}\text{L}{}_{\mathrm{<mtext>Liouville</mtext>}}\text{.}$

. We give a conventional GGRT treatment of this modified lagrangian for the bosonic string. Under the assumption that the exact quantization of Liouville's equation does not yield any additional anomalies, we show that relativistic invariance requires the constant C to be $\text{C =}\text{26 ? D}\text{48π}$, in agreement with Polyakov's result. For D < 26 the string acquires longitudinal modes, and our calculations show explicitly how the longitudinal component of the string receives the degrees of freedom from the Liouville variable. Under the boundary conditions yielding the lowest value of the classical Liouville hamiltonian, the mass spectrum starts with a tachyon $\text{m}{}^2\text{= ?}\text{1}\text{α′}$, independent of D. The lowest-lying longitudinal excitation is $\text{m}{}^2\text{=}\text{1}\text{α′}$. These results are semiclassical. It is shown that an exact quantization of Liouville's equation could remove the tachyon state when D < 26.