Clocks and gravity

We consider the assumption that clocks measure proper time-that is, in a gravitational field ideal clocks are governed by the equationds ²=g _ij dxⁱ dx^j-and give some theoretical and experimental constraints on clock measurements. In particular, we find that if we assume that clocks are governed by an equation of the formds ⁴=c _ijkl dxⁱ dx^j dx^k dx^l, then this equation must reduce to the quadratic equation in a weak, spherically symmetric, static gravitational field (at least to first order in the Newtonian gravitational potentialU), otherwise additional contributions to the time-delay effect of radar propagation (that are not observed) are predicted.     

Coordinate-dependent 3+1 formulation of the general relativity equations for a perfect fluid

This paper defines mass, momentum, and energy densities for a perfect fluid, and derives a coordinate-dependent 3+1 decomposition of the equation of motion in terms of a scalar potential c ² [(–g _g44) 1/2 –1] and a vector potentialA _i cg _4i /(–g ₄₄)^1/2. The momentum equation has the form of the Euler equation except there is an additional force proportional to the vector potential and the rate of change of kinetic energy per unit volume. The momentum and energy equations are integrated to obtain the equations previously derived for a particle. The momentum equation is solved for the total acceleration of a fluid element. The equations are exact and do not depend on the choice of coordinate system.     

Interaction of charged particle flows with plasmons in two-dimensional electron gas

The theory of interaction of charged particle beams with 2D electron gas, located in the field of the delta-like potential wallU(y) =U _o(y) has been developed. The kinetic equation for plasmons in 2D electron gas was obtained; the conditions of arising of their instability are given and the corresponding expressions for the increment of plasmons are found.     

A bound on the number of bound states in the Schrödinger equation

A boundS _l is given for the number of bound statesn _i in thelth partial wave corresponding to a spherically symmetric potential in non-relativistic quantum mechanics. This bound is given by whereV _a(l, r) is the attractive part of the effective potentialV(r)+l(l+1)/r ². Extensive comparative study ofS _i and the Bargmann inequality is made.     

The loop expansion for the effective potential in theP(φ)2 quantum field theory

We study the loop expansion for the effective potential, defined as the Fenchel transform (convex conjugate) of the pressure in an external field, in theP()₂ quantum field theory. For values of the classical fielda for which the classical potentialU ₀(a)=P(a)+1/2m ² a ² equals its convex hull and has nonvanishing curvature we prove that the 1-PI loop expansion is asymptotic as 0. We also give an example of a double well classical potential for which the 1-PI loop expansion fails to be asymptotic, and find the true asymptotics.This paper is a condensed version of the author's Ph.D. thesis for the Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Y4     

Fermi-liquid quasi-particles and -superconductivity in the two-dimensional Hubbard model

A frequency- and momentum-renormalization-group acceleration together with an analytical approach is used to obtain the retarded Green's function in the self-consistent and conserving fluctuation-exchange (FLEX) approximation for the two-dimensional Hubbard model in the normal state and in the superconducting state. The analytical expressions for this approach are given. For band-fillings near half filling the self-energy in the normal state exhibits Fermi-liquid behaviour for, low temperatures and frequencies near the chemical potential, if the momentum is chosen near the Fermi-surface. However due to the presence of large many body effects the observed Fermi-liquid region near the chemical potential and near the Fermi-surface is very small. Results for the single particle density of states, the occupation number and the spectral function are presented. The superconducting state with symmetry is obtained for U = 2 to U = 6 and a (U, n)-phase diagram for the transition temperature T_c is presented. A maximum T_c/t of 0.0275 is obtained for U = 6 near half filling.     

Letter: Normal Forces in Stationary Spacetimes

For geodesic motion of a particle in a stationary spacetime the U ₀ component of particle 4-velocity is constant and is considered to be a conserved mechanical energy. We show that this concept of a conserved mechanical energy can be extended to particles that move under the influence of a normal force, a force that, in the stationary frame, is orthogonal to the motion of the particle. We illustrate the potential usefulness of the concept with a simple example.     

Schrödinger equation as an operator acting on test functions describing observing systems and the EPR correlation

A reformulation of quantum mechanics is introduced by regarding the Schrödinger equationE(f ⁺) = 0 for the retarded particle wavef ⁺ as an operator (functional) acting on the test functionf ^– satisfying the boundary conditions of the observing system: E(f ⁺),f ^– = 0. The variational expression for the transition amplitude of a particle between the particle source and the detector naturally arises in the dual space of the particle field and the test function. In the two-slit electron interference experiment, the test function plays the role of the quantum potential which carries the information of the detector and the slit locations backwards in time, while in the Einstein-Podolsky-Rosen process the test function describes the time reversed process of a pair of spatially separated fermions with arbitrarily chosen spin orientations progressing backwards in time to form a spherically symmetric compound state. The separation of the kinematics (spin correlation and the dynamics (spacetime aspect) of the EPR process is pointed out.     

Nonstationary quantum mechanics. III. Quantum mechanics does not incorporate classical physics

It is shown that disagreement between the prediction of classical and conventional quantum mechanics about momentum probabilities exists in the case of a quasiclassical motion. The discussion is based on the detailed consideration of two specific potentials:U(x)=x and the oscillatory potentialU(x)=m ² x ²/2. The results of the present Part III represent a further development of the idea in Todorov (1980) about the possible inefficiency of conventional theory in the case of potentials swiftly varying with time.     

Twistfree and Papapetrou vacuum spacetimes with a spacelike killing vector

We show that, for the case of vacuum solutions of the Einstein equations with a spacelike hypersurface orthogonal Killing vector /³ and associated metricds ² =e ^2U (dx ³)² +e ^–2U _ab dx ^a dx ^b whereU is not a constant, there exists at every point of the quotient 3-space a plane of vectorsK ^a such that £_KR_ab=0 andK ^a R_ab=0 whereR{inab} is the Ricci tensor formed from _ab . Then in the case whereU{in,a} is a timelike or spacelike vector in the quotient 3-space, Petrov type I solutions of the vacuum field equations are obtained. In the simpler case whereU{in,a} is a null vector in the quotient 3-space, the complete solution of the vacuum field equations is obtained. It is shown that this solution is Petrov type III of Kundt's class. For the case of Papapetrou solutions where there is a twist potential which is a function ofU, solutions corresponding to the twistfree solutions are given.     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

Representations of Lie algebras from representations of quantum groups

In paper [*] (P. Moylan: Czech. J. Phys., Vol. 47 (1997), p. 1251) we gave an explicit embedding of the three dimensional Euclidean algebra (2) into a quantum structure associated with U _q(so(2, 1)). We used this embedding to construct skew symmetric representations of (2) out of skew symmetric representations of U _q(so(2, 1)). Here we consider generalizations of the results in [*] to a more complicated quantum group, which is of importance to physics. We consider U _q(so(3, 2)), and we show that, for a particular representation, namely the Rac representation, many of the results in [*] carry over to this case. In particular, we construct representations of so(3, 2), P(2, 2), the Poincaré algebra in 2+2 dimensions, and the Poincaré algebra out of the Rac representation of U _q(so(3, 2)). These results may be of interest to those working on exploiting representations of U _q(so(3, 2)), like the Rac, as an example of kinematical confinement for particle constituents such as the quarks.     

Nonclassical type representations of the q-deformed algebra U′q(son)

The nonstandard q-deformation U_q(so_n) of the universal enveloping algebra U(so _n ) has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of U(so _n ). But U_q(so_n) also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of U_q(so_n).     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

Algebraic structures and eigenstates for integrable collective field theories

Conditions for the construction of polynomial eigen-operators for the Hamiltonian of collective string field theories are explored. Such eigen-operators arise for only one monomial potentialv(x)=x ² in the collective field theory. They form aw -algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non-zero-energy polynomial eigen-operators. This analysis leads us to consider a particular potentialv(x)=x ²+g/x ². A Lie algebra of polynomial eigen-operators is then constructed for this potential. It is a symmetric 2-index Lie algebra, also represented as a subalgebra ofU(sl(2)).Work supported in part by the Department of Energy under contract DE-AC02-76ER03130-Task AWork supported by Brown University Exchange Program P.I. 135     

Tunneling conductance in a gapped graphene-based superconducting structure: Case of massive Dirac electrons

The tunneling conductance in a NG/SG graphene junction in which the graphene was grown on a SiC substrate is simulated. The carriers in the normal graphene (NG) and the superconducting graphene (SG) are treated as massive relativistic particles. It is assumed that the Fermi energy in the NG and SG are E_FN400 meV and E_FS400 meV+U, respectively. Here U is the electrostatic potential from the superconducting gate electrode. It is seen that the Klein tunneling disappears in the case where a gap exist in the energy spectrum. As U→∞, the zero bias normalized conductance becomes persistent at a minimal value of G/G₀1.2. The normalized conductance G/G₀ is found to depend linearly on U with constant slope of , where is the size of the gap Δ opening up in the energy spectrum of the graphene grown on the SiC substrate. It is found that G/G₀2+αU for potentials in the range −270 meV<U<0 meV and G=0 for potentials U<−270 meV. As α→∞, the conductance for eV=Δ (V is the bias voltage placed across the NG/SG junction) can be approximated by a unit step function G(eV=Δ,U)/G₀2Θ(U). This last behavior indicates that a NG/SG junction made with gapped graphene could be used as a nano switch having excellent characteristics.     

The CWKB approach to non-reflecting potential and cosmological implications

We discuss the method of calculating the reflection coefficient using complex trajectory WKB (CWKB) approximation to understand the non-reflecting nature of the potentialU(x) = -U ₀/cosh²(x/a). We show that the repeated reflections between the turning points whose paths are in conformity with Bogolubov transformation technique are essential in obtaining the non-reflecting condition. We also discuss the implications of the results when applied to the particle production scenario. We use the CWKB technique developed by one of the authors (SB) to obtain the results which agree very well with those obtained by exact quantum mechanical calculations.     

Solution of the Bethe-Goldstone equation for the reaction matrix in finite nuclei

A method for solving the BG equation for the reaction matrixt in finite nuclei is presented. The application of this method is demonstrated for a one-dimensional case, which is similar to the problem where the internucleon potential acts only in the relatives-state. The single particle potential has a harmonic oscillator form and the phenomenological internucleon potentialv(r) contains a hard core and an attractive part of the Yukawa type. By taking the exclusion principle into account exactly an infinite system of integral equations is obtained. It is proved that the solution of the corresponding finite system converges to the exact solution. An iteration method for solving such a finite system with an arbitrary number of equations is developed. Its main feature consists in the exclusion of the dependence on the hard core part ofv(r) (which is treated as the limit case of a rectangular repulsive potential with a variable heightv ₀). This exclusion transforms the original system to a system of integral equations depending only on the attractive part ofv(r) and to a linear algebraic system. Both these systems can be solved by iteration for all values ofv ₀ as well as for v₀= +. The numerical results confirm the rapid convergence of the proposed iteration method and demonstrate that the solution of the finite system with a sufficiently large number of equations approximates the exact solution very precisely.     

Spin physics with internal targets and the blast detector

The aim of this paper is to give a set of central elements of the algebras U_q(so_m) and U _q(iso _m ) when q is a root of unity. They are surprisingly arise from a single polynomial Casimir element of the algebra U_q(so₃). It is conjectured that the Casimir elements of these algebras under any values of q (not only for q a root of unity) and the central elements for q a root of unity derived in this paper generate the centers of U_q(so_m) and U _q(iso _m ) when q is a root of unity.     

The inverse scattering matrix for the Schrödinger equation when the potentialq(x) ∈L 1 1 with a singular term

When the potentialq(x) L ₁ ¹ with a singular term, the continuities of the scattering matrix of the Schrödinger equation are investigated. By means of the transformation approach, we arrive at the conclusion that the scattering matrix S(k) of such a potential is continuous for the wholek,- <k < .