Semiclassical quantum mechanics

We consider the 0 limit of the quantum dynamics generated by the HamiltonianH()=–(²/2m)+V. We prove that the evolution of certain Gaussian states is determined asymptotically as 0 by classical mechanics. For suitable potentialsV inn3 dimensions, our estimates are uniform in time and our results hold for scattering theory.Supported in part by the National Science Foundation under Grant PHY 78-08066     

The evans-vigier field,B (3): Derivation of the de Broglie matter-wave equation from the Hamilton-Jacobi equation

The emergence of the Evans-Vigier fieldB ⁽³⁾ of vacuum electromagnetism has been accompanied by a novel charge quantization condition inferred from 0(3) gauge theory. This finding is used to derive the de Broglie matter-wave equation from the classical Hamilton-Jacobi (HJ) equation of one electron in the electromagnetic field. The HJ equation is used with the charge quantization condition to show that, in a perfectly elastic photon-electron interaction, complete transfer of angular momentum occurs self-consistently, and the electron acquires the angular momentum of the photon. In this limit the electron travels infinitesimally near the speed of light, and its concomitant electromagnetic fields become indistinguishable from those of the uncharged photon. This result independently proves the validity of the charge quantization condition and demonstrates unequivocally the existence of the vacuum fieldB ⁽³⁾.     

Topological charges in monopole theories

Let us consider a monopole theory with a compact, simply connected gauge group and the Higgs field in the adjoint representation. Using root theory we show that.(i) The homotopy class of the Higgs field is ap-tuple of integers wherep is the dimension of the centre of the residual symmetry group. These Higgs charges can be expressed as surface integrals of differential forms.(ii) To any invariant polynomial on the Lie algebra is associated a topological invariant which turns out to be a combination of the Higgs charges.(iii) Electric charge is quantized. The monopole's magnetic charge is a combination — with the Higgs charges as coefficients — ofp basic magnetic charges which satisfy generalized Dirac conditions.The example ofG=SU(N) is worked out in detail.     

Topological electric charge

By treating magnetic charge as a gauge symmetry through the introduction of a magnetic pseudo four-vector potential, we show that it is possible to construct a topological electric charge from a theory which originally contains gauge magnetic charge. This is an explicit realization of the Montonen-Olive conjecture that there should exist a dual theory to the usual 't Hooft-Polyakov monopole theory in which the roles of the gauge and topological charges are reversed. The physical distinction between 't Hooft-Polyakov monopoles and the dual theory with electric charge is that the strong and weak coupling regimes are reversed. Physically this leads to the mass of the electrically charged soliton being on the order of (1/137)M _W as opposed to the much larger mass (on the order of 137M _W) of the magnetically charged soliton. Thus even forM _W in the TeV range such an electrically charged particle could be observed at some future accelerator.     

The shape resonance

For a class of Schrödinger operatorsH:=–(²/2m)+V onL ²( ⁿ ), with potentials having minima embedded in the continuum of the spectrum and non-trapping tails, we show the existence of shape resonances exponentially close to the real axis as 0. The resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling.     

One-Loop Quantum Corrections to Thermodynamics of Black Holes with Global Monopoles

Quantum corrections are studied for a black holewith a global monopole charge in a 2D model obtained byspherisymmetric reduction of the 4D action. Thebackreaction of the Hawking radiation on the geometry is studied perturbatively for conformal matter.It is shown that the metric and the position of horizonchange by an amount of order . Within theoff-shell approach the one-loop thermodynamicquantities, energy, and entropy are found. They are shownto contain two parts, one due to the hole itself and oneto the hot gas surrounding it. The deviation of thequantum-corrected entropy from the classical one is given.     

Large deviations from classical paths. Hamiltonian flows as classical limits of quantum flows

We prove that in the limit0, the probability for the paths of the stochastic jump process associated to the quantum time evolution to be in a tublet around the classical trajectory is of order 1–exp{–A/}. We give some applications of this result to the study of the classical limit of Wigner functions.     

The role of electronic density of states in metallic Lanthanum

Observations of X-ray photoabsorption in metallic Lanthanum in the vicinity of the 3d _5/2-edge are given. Two separated absorption peaks are detected at the quantum energies =830.6eV and =834.0eV. The results are interpreted according to the model given in communication I–IV [1].     

p-Form electrodynamics

A generalization of gauge theory in which the gauge potential1-form is replaced by a p-form is studied. Charged particles are then replaced by elementary extended objects of dimension p–1. It is shown that this extension is compatible with space-time locality only if the gauge group is U(1). A source which is a closed p–1 surface has zero total charge and corresponds to a particle-antiparticle pair. Its quantum rate of production in an external uniform field is evaluated semiclassically. The analog of the Dirac magnetic pole is constructed. It is another extended object, of dimension n–p–3, where n is the dimension of space-time. The electric and magnetic charges obey the Dirac quantization condition. This condition is derived in two different ways. One method makes use of local gauge patches and the other brings in singular gauge transformations. A topological mass term is introduced and it is shown that it can coexist with a magnetic pole when n=2p+1, provided the topological mass is quantized.     

Precision measurement of ℏ/m Cs based on photon recoil using laser-cooled atoms and atomic interferometry

The recoil of an atom due to the absorption of up to 64 photons is measured, using laser-cooled cesium atoms which are made to interfere in an atomic fountain. Measurement of the photon recoil allows a determination of /m _Cs, and hence the fine-structure constant. The measurement is described and a detailed theoretical and experimental study of potential systematic errors is presented. A relative precision in the photon recoil measurement of 0.1 ppm is obtained in two hours of data collection. The measurement is currently 0.85 ppm below the accepted value of /m _Cs. We cannot now formally ascribe a systematic error, but suspect that the bulk of the discrepancy is due to imperfections of the interferometer beams used to induce the Raman transitions.     

Wigner Kirkwood ℏ-expansion of the density matrix in inhomogeneous superfluid Fermi systems

The generalized density matrix of superfluid inhomogeneous Fermi systems is expanded in powers of up to order ². This constitutes the generalisation of the Wigner Kirkwood -expansion of the density matrix of normal fluid systems to the pairing case.One of the authors (P.S.) is very grateful to D. Gogny for contributions in an early stage of this work several years back. He also acknowledges fruitful discussions with M. Centelles and X. Vinas.     

Dependence ofℏω on the mass number of nuclei under the assumption of a trapezoidal average nuclear density

The variation of the harmonic oscillator energy level spacing with the mass number of nuclei is investigated under the assumption of a trapezoidal distribution for the average nuclear density and the results are compared with those obtained with a Fermi distribution. The dependence of the parameters on the mass number is also discussed.     

Effects of the blackbody radiation on the harmonically bound electron-energy level shifts and consistency conditions

The influence of the vacuum radiation field on the harmonically bound electron (frequency ₀) is considered. The electron is minimally coupled to the blackbody radiation field. The dynamics of the system is exactly solvable. The high (k _B T₀) and low (k _B T₀) temperature expansions of the kinetic and potential energy are given. In the high temperature regime theT ²-dependent dynamic Stark shift is found whereas in the low temperature regime there is no temperature dependent shift. The position correlation function of the electron shows in the low temperature regime a unclassical algebraic decay (t ^–4,t/k _B T).     

On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential

In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrödinger equation for sufficiently small >0, in which sense we call them semiclassical bound states. We assume that 1p< forn=1,2 and 1p<1+4/(n–2) forn3, and thatV is in the class(V) _a in the sense of Kato for somea. For any finite collection {x ₁,...,x _N} of nondegenerate critical points ofV, we construct a solution of the forme ^–iEt/v(x) forE<a, wherev is real and it is a small perturbation of a sum of one-lump solutions concentrated nearx ₁,...,x _N respectively. The concentration gets stronger as 0. And we also prove these solutions are positive, and unstable with respect to perturbations of initial conditions for possibly smaller >0. Indeed, for each such collection of critical points we construct 2 ^N–1 distinct unstable bound states which may have nodes in general, and the above positive bound state is just one of them.     

Semiclassical Models for Virtual Antiparticle Pairs, the Unit of Charge e, and the QCD Coupling α s

New semiclassical models of virtual antiparticle pairs are used to compute the pair lifetimes, and good agreement with the Heisenberg lifetimes from quantum field theory (QFT) is found. The modeling method applies to both the electromagnetic and color forces. Evaluation of the action integral of potential field fluctuation for each interaction potential yields /2 for both electromagnetic and color fluctuations, in agreement with QFT. Thus each model is a quantized semiclassical representation for such virtual antiparticle pairs, to good approximation. When the results of the new models and QFT are combined, formulae for e and _s (q) are derived in terms of only and c.     

Exact monopole solutions in gauge theories for an arbitrary semisimple compact group

We construct an exact n-parametric monopole and dyon solutions for an arbitrary compact gauge group G of rank n by using the symmetry between cylindrically symmetric instanton equations in Euclidean space R ₄ and monopole equations in Minkowski space R _3,1 (with Higgs scalar field in adjoint representation). The solutions are spherically symmetric with respect to the total momentum operator represents the minimal embedding of SU(2) in G. Explicit expressions for the monopole magnetic charge and mass matrices are obtained. The remarkable aspect of our results is the existence of discrete series of the monopole solutions, which are labelled by n quantum numbers and degenerated in the latter ones at a fixed monopole mass matrix.     

Probability Backflow for a Dirac Particle

The phenomenon of probability backflow, previously quantified for a free nonrelativistic particle, is considered for a free particle obeying Dirac's equation. It is shown that probability backflow can occur in the opposite direction to the momentum; that is to say, there exist positive-energy states in which the particle certainly has a positive momentum in a given direction, but for which the component of the probability flux vector in that direction is negative. It is shown that the maximum possible amount of probability that can flow backwards, over a given time interval of duration T, depends on the dimensionless parameter = (4/mc²T)^1/2, where m is the mass of the particle and c is the speed of light. At = 0, the nonrelativistic value of approximately 0.039 for this maximum is recovered. Numerical studies suggest that the maximum decreases monotonically as increases from 0, and show that it depends on the size of m, , and T, unlike the nonrelativistic case.     

Cosmological Models with Variable “Constants”

The behavior of the constants, G, c, , a, e, m _i, and , considering them asvariable, in the framework of a flat cosmological model with FRW symmetriesdescribed by a bulk viscous fluid and considering mechanisms of adiabatic mattercreation are investigated. Two cases are studied; one with radiationpredominance and another of matter predominance. It is found that with the solution obtainedour model verifies these basic principles: Lorentz invariance and generalcovariance, Mach, Equivalence and causality. Finally, to emphasize that the envisaged modelsare free from the main problem: Planck's, horizon and entropy. With regard tothat model with matter predominance it is seen that mechanisms of creation ofmatter cannot be considered since if these are taken into account thetemperature would increase instead of remaining constant while the universe expands.     

Large spin corrections in SYM : Still a linear integral equation

Anomalous dimension and higher conserved charges in the sl(2) sector of SYM for generic spin s and twist L are described by using a novel kind of non-linear integral equation (NLIE). The latter can be derived under typical situations of the SYM sectors, i.e. when the scattering need not depend on the difference of the rapidities and these, in their turn, may also lie on a bounded range. Here the non-linear (finite range) integral terms, appearing in the NLIE and in the dimension formula, go to zero as s→∞. Therefore they can be neglected at least up to the O(s⁰) order, thus implying a linear integral equation (LIE) and a linear dimension/charge formula respectively, likewise the ‘thermodynamic’ (i.e. infinite spin) case. Importantly, these non-linear terms go faster than any inverse logarithm power (lns)⁻ⁿ, n>0, thus extending the linearity validity.     

Semiclassical Behaviour of Expectation Values in Time Evolved Lagrangian States for Large Times

We study the behaviour of time evolved quantum mechanical expectation values in Lagrangian states in the limit 0 and t. We show that it depends strongly on the dynamical properties of the corresponding classical system. If the classical system is strongly chaotic, i.e. Anosov, then the expectation values tend to a universal limit. This can be viewed as an analogue of mixing in the classical system. If the classical system is integrable, then the expectation values need not converge, and if they converge their limit depends on the initial state. An additional difference occurs in the timescales for which we can prove this behaviour; in the chaotic case we get up to Ehrenfest time, t ln (1/), whereas for integrable system we have a much larger time range.