Chaotic Evolution in Quantum Mechanics

A quantum system is described, whose wave function has a complexity which increases exponentially with time. Namely, for any fixed orthonormal basis, the number of components required for an accurate representation of the wave function increases exponentially.Comment: 8 pages (LaTeX 16 kB, followed by PostScript 2 kB for figure

The glasma initial state and JIMWLK factorization

We review recent work on understanding the next to leading order corrections to the classical fields that dominate the initial stages of a heavy ion collision. We have recently shown that the leading ln(1/x) divergences of these corrections to gluon multiplicities can be factorized into the JIMWLK evolution of the color charge density distributions.Comment: 4 pages, 2 figures. Talk given by T.L. at Strong and Electroweak Matter 2008 (SEWM08), August 26-29, 2008, Amsterdam, The Netherland

Time-dependent changes in the brain arachidonic acid cascade during cuprizone-induced demyelination and remyelination

Phospholipases A(2) (PLA(2)) are the enzymatic keys for the activation of the arachidonic acid (AA) cascade and the subsequent synthesis of pro-inflammatory prostanoids (prostaglandins and tromboxanes). Prostanoids play critical roles in the initiation and modulation of inflammation and their levels have been reported increased in several neurological and neurodegenerative disorders, including multiple sclerosis (MS). Here, we aimed to determine whether brain expression PLA(2) enzymes and the terminal prostagland in levels are changed during cuprizone-induced demyelination and in the subsequent remyelination phase. Mice were given the neurotoxicant cuprizone through the diet for six weeks to induce brain demyelination. Then, cuprizone was withdrawn and mice were returned to a normal diet for 6 weeks to allow spontaneous remyelination. We found that after 4-6 weeks of cuprizone, sPLA(2)(V) and cPLA(2), but not iPLA(2)(VI), gene expression was upregulated in the cortex, concomitant with an increase in the expression of astrocyte and microglia markers. Cyclooxygenase (COX)-2 gene expression was consistently upregulated during all the demyelination period, whereas COX-1 sporadically increased only at week 5 of cuprizone exposure. However, we found that at the protein level only sPLA(2)(V) and COX-1 were elevated during demyelination, with COX-1 selectively expressed by activated and infiltrated microglia/macrophages and astrocytes. Levels of PGE(2), PGD(2), PGI(2) and TXB(2) were also increased during demyelination. During remyelination, none of the PLA(2) isoforms was significantly changed, whereas COX-1 and -2 were sporadically upregulated only at the gene expression level. PGE(2), PGI(2) and PGD(2) levels returned to normal, whereas TXB(2) was still upregulated after 3 weeks of cuprizone withdrawal. Our study characterizes for the first time time-dependent changes in the AA metabolic pathway during cuprizone-induced demyelination and the subsequent remyelination and suggests that sPLA(2)(V) is the major isoform contributing to AA release

Quantum control and the Strocchi map

Identifying the real and imaginary parts of wave functions with coordinates and momenta, quantum evolution may be mapped onto a classical Hamiltonian system. In addition to the symplectic form, quantum mechanics also has a positive-definite real inner product which provides a geometrical interpretation of the measurement process. Together they endow the quantum Hilbert space with the structure of a K\"{a}ller manifold. Quantum control is discussed in this setting. Quantum time-evolution corresponds to smooth Hamiltonian dynamics and measurements to jumps in the phase space. This adds additional power to quantum control, non unitarily controllable systems becoming controllable by ``measurement plus evolution''. A picture of quantum evolution as Hamiltonian dynamics in a classical-like phase-space is the appropriate setting to carry over techniques from classical to quantum control. This is illustrated by a discussion of optimal control and sliding mode techniques.Comment: 16 pages Late

Topological quenching of the tunnel splitting for a particle in a double-well potential on a planar loop

The motion of a particle along a one-dimensional closed curve in a plane is considered. The only restriction on the shape of the loop is that it must be invariant under a twofold rotation about an axis perpendicular to the plane of motion. Along the curve a symmetric double-well potential is present leading to a twofold degeneracy of the classical ground state. In quantum mechanics, this degeneracy is lifted: the energies of the ground state and the first excited state are separated from each other by a slight difference ¿E, the tunnel splitting. Although a magnetic field perpendicular to the plane of the loop does not influence the classical motion of the charged particle, the quantum-mechanical separation of levels turns out to be a function of its strength B. The dependence of ¿E on the field B is oscillatory: for specific discrete values Bn the splitting drops to zero, indicating a twofold degeneracy of the ground state. This result is obtained within the path-integral formulation of quantum mechanics; in particular, the semiclassical instanton method is used. The origin of the quenched splitting is intuitively obvious: it is due to the fact that the configuration space of the system is not simply connected, thus allowing for destructive interference of quantum-mechanical amplitudes. From an abstract point of view this phenomenon can be traced back to the existence of a topological term in the Lagrangian and a nonsimply connected configuration space. In principle, it should be possible to observe the splitting in appropriately fabricated mesoscopic rings consisting of normally conducting metal

Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field

The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck’s constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution

Solvable three-state model of a driven double-well potential and coherent destruction of tunneling

A simple model for a particle in a double well is derived from discretizing its configuration space. The model contains as many free parameters as the original system and it respects all the existing symmetries. In the presence of an external periodic force both the continuous system and the discrete model are shown to possess a generalized time-reversal symmetry in addition to the known generalized parity. The impact of the driving force on the spectrum of the Floquet operator is studied. In particular, the occurrence of degenerate quasienergies causing coherent destruction of tunneling is discussed—to a large extent analytically—for arbitrary driving frequencies and barrier heights

Maximal couplings in PT-symmetric chain-models with the real spectrum of energies

The domain ${\cal D}$ of all the coupling strengths compatible with the reality of the energies is studied for a family of non-Hermitian $N$ by $N$ matrix Hamiltonians $H^{(N)}$ with tridiagonal and ${\cal PT}-$symmetric structure. At all dimensions $N$, the coordinates are found of the extremal points at which the boundary hypersurface $\partial {\cal D}$ touches the circumscribed sphere (for odd $N=2M+1$) or ellipsoid (for even $N=2K$).Comment: 18 pp., 2 fig

QCD at small x and nucleus-nucleus collisions

At large collision energy sqrt(s) and relatively low momentum transfer Q, one expects a new regime of Quantum Chromo-Dynamics (QCD) known as "saturation". This kinematical range is characterized by a very large occupation number for gluons inside hadrons and nuclei; this is the region where higher twist contributions are as large as the leading twist contributions incorporated in collinear factorization. In this talk, I discuss the onset of and dynamics in the saturation regime, some of its experimental signatures, and its implications for the early stages of Heavy Ion Collisions.Comment: Plenary talk given at QM2006, Shanghai, November 2006. 8 pages, 8 figure