PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

The $PT$ symmetric potential $V_0[\cos(2\pi x/a)+i\lambda\sin(2\pi x/a)]$ has a completely real spectrum for $\lambda\le 1$, and begins to develop complex eigenvalues for $\lambda>1$. At the symmetry-breaking threshold $\lambda=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to result in a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for a broad initial wave packet this growth is suppressed, and instead a saturation leading to a constant maximum amplitude is observed. We revisit this problem by explicitly constructing the Bloch wave-functions and the associated Jordan functions and using the method of stationary states to find the dependence on the longitudinal distance $z$ for a variety of different initial wave packets. This allows us to show in detail how the saturation of the linear growth arises from the close connection between the contributions of the Jordan functions and those of the neighbouring Bloch waves.Comment: 15 pages, 7 figures Minor corrections, additional reference

Use of Equivalent Hermitian Hamiltonian for PTPT-Symmetric Sinusoidal Optical Lattices

We show how the band structure and beam dynamics of non-Hermitian $PT$-symmetric sinusoidal optical lattices can be approached from the point of view of the equivalent Hermitian problem, obtained by an analytic continuation in the transverse spatial variable $x$. In this latter problem the eigenvalue equation reduces to the Mathieu equation, whose eigenfunctions and properties have been well studied. That being the case, the beam propagation, which parallels the time-development of the wave-function in quantum mechanics, can be calculated using the equivalent of the method of stationary states. We also discuss a model potential that interpolates between a sinusoidal and periodic square well potential, showing that some of the striking properties of the sinusoidal potential, in particular birefringence, become much less prominent as one goes away from the sinusoidal case.Comment: 11 pages, 8 figure

On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space

A class of pseudo-hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark(Zeeman) effect with complex electric(magnetic) field, non-hermitian general quadratic form of N boson(fermion) operators, symmetric and asymmetric XXZ spin-chain in complex magnetic field, non-hermitian Haldane-Shastry spin-chain and Lipkin-Meshkov-Glick model.Comment: 29 pages, revtex, minor changes, version to appear in Journal of Physics A(v3

PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras

Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space related J-selfadjoint extensions for PTQM setups with ultra-localized potentials.Comment: 11 page

On complexified mechanics and coquaternions

While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and coquaternionic (split-signature analogue of quaternions) extensions of Hamiltonian mechanics are introduced, and are shown to offer a unifying framework for complexified classical and quantum mechanics. In particular, quantum theories characterised by complex Hamiltonians invariant under space-time reflection are shown to be equivalent to certain coquaternionic extensions of Hermitian quantum theories. One of the interesting consequences is that the space-time dimension of these systems is six, not four, on account of the structures of coquaternionic quantum mechanics.Comment: 11 pages, version to appear in Journal of Physics

Pharmacogenomics of GLP-1 receptor agonists: a genome-wide analysis of observational data and large randomised controlled trials

Background: In the treatment of type 2 diabetes, GLP-1 receptor agonists lower blood glucose concentrations, body weight, and have cardiovascular benefits. The efficacy and side effects of GLP-1 receptor agonists vary between people. Human pharmacogenomic studies of this inter-individual variation can provide both biological insight into drug action and provide biomarkers to inform clinical decision making. We therefore aimed to identify genetic variants associated with glycaemic response to GLP-1 receptor agonist treatment. Methods: In this genome-wide analysis we included adults (aged ≥18 years) with type 2 diabetes treated with GLP-1 receptor agonists with baseline HbA1c of 7% or more (53 mmol/mol) from four prospective observational cohorts (DIRECT, PRIBA, PROMASTER, and GoDARTS) and two randomised clinical trials (HARMONY phase 3 and AWARD). The primary endpoint was HbA1c reduction at 6 months after starting GLP-1 receptor agonists. We evaluated variants in GLP1R, then did a genome-wide association study and gene-based burden tests. Findings: 4571 adults were included in our analysis, of these, 3339 (73%) were White European, 449 (10%) Hispanic, 312 (7%) American Indian or Alaskan Native, and 471 (10%) were other, and around 2140 (47%) of the participants were women. Variation in HbA1c reduction with GLP-1 receptor agonists treatment was associated with rs6923761G→A (Gly168Ser) in the GLP1R (0·08% [95% CI 0·04–0·12] or 0·9 mmol/mol lower reduction in HbA1c per serine, p=6·0 × 10−5) and low frequency variants in ARRB1 (optimal sequence kernel association test p=6·7 × 10−8), largely driven by rs140226575G→A (Thr370Met; 0·25% [SE 0·06] or 2·7 mmol/mol [SE 0·7] greater HbA1c reduction per methionine, p=5·2 × 10−6). A similar effect size for the ARRB1 Thr370Met was seen in Hispanic and American Indian or Alaska Native populations who have a higher frequency of this variant (6–11%) than in White European populations. Combining these two genes identified 4% of the population who had a 30% greater reduction in HbA1c than the 9% of the population with the worse response. Interpretation: This genome-wide pharmacogenomic study of GLP-1 receptor agonists provides novel biological and clinical insights. Clinically, when genotype is routinely available at the point of prescribing, individuals with ARRB1 variants might benefit from earlier initiation of GLP-1 receptor agonists. Funding: Innovative Medicines Initiative and the Wellcome Trus

Measurement of the W mass in e+e−e^+e^- collisions at production threshold

In June 1996, the LEP centre-of-mass energy was raised to 161 GeV. Pair production of W bosons in e+e- collisions was observed for the first time by the LEP experiments. An integrated luminosity of 11 pb-1 was recorded in the ALEPH detector, in which WW candidate events were observed. In 6 events both Ws decay leptonically. In 16 events, one W decays leptonically, the other into hadrons. In the channel where both Ws decay into hadrons, a signal was separated from the large background by means of several multi-variate analyses. The W pair cross-section is measured to be sigma_WW = 4.23 +-0.73 (stat.) +- 0.19 (syst.) pb From this cross-section, the W mass is derived within the framework of the Standard Model: MW = 80.14 +- 0.34 (stat.) +- 0.09 (syst.) +- 0.03 (LEP~energy) GeV/c2

Study of muon-pair production at centre-of-mass energies from 20 to 136 GeV with the Aleph detector

The total cross section and the forward-backward asymmetry for the process $e^+ e^- \rightarrow \mu^+ \mu^- (n \gamma)$ are measured in the energy range 20-136 GeV by reconstructing the effective centre-of-mass energy after initial state radiation. The analysis is based on the data recorded with the ALEPH detector at LEP between 1990 and 1995, corresponding to a total integrated luminosity of 143.5 $\mathrm{pb}^{-1}$. Two different approaches are used: in the first one an exclusive selection of events with hard initial state radiation in the energy range 20-88 GeV is directly compared with the Standard Model predictions showing good agreement. In the second one, all events are used to obtain a precise measurement of the energy dependence of $\sigma^0$ and $A_{\mathrm{FB}}^0$ from a model independent fit, enabling constraints to be placed on models with extra Z bosons

Search for Bs0B^{0}_{s} oscillations using inclusive lepton events

A search for Bs oscillations is performed using a sample of semileptonic b-hadron decays collected by the ALEPH experiment during 1991-1995. Compared to previous inclusive lepton analyses, the prop er time resolution and b-flavour mistag rate are significantly improved. Additional sensitivity to Bs mixing is obtained by identifying subsamples of events having a Bs purity which is higher than the average for the whole data sample. Unbinned maximum likelihood amplitude fits are performed to derive a lower limit of Dms>9.5 ps-1 at 95% CL. Combining with the ALEPH Ds based analyses yields Dms>9.6 ps-1 at 95% CL.A search for B0s oscillations is performed using a sample of semileptonic b-hadron decays collected by the ALEPH experiment during 1991-1995. Compared to previous inclusive lepton analyses, the proper time resolution and b-flavour mistag rate are significantly improved. Additional sensitivity to B0s mixing is obtained by identifying subsamples of events having a B0s purity which is higher than the average for the whole data sample. Unbinned maximum likelihood amplitude fits are performed to derive a lower limit of Deltam_s>9.5ps^-1 at 95% CL. Combining with the ALEPH D-s based analyses yields Deltam_s>9.6ps^-1 at 95% CL

Measurement of the W mass by direct reconstruction in e+e−e^+ e^- collisions at 172 GeV

The mass of the W boson is obtained from reconstructed invariant mass distributions in W-pair events. The sample of W pairs is selected from 10.65~pb$^{-1}$ collected with the ALEPH detector at a mean centre-of-mass energy of 172.09 \GEV. The invariant mass distribution of simulated events are fitted to the experimental distributions and the following W masses are obtained: $WW \to q\overline{q}q\overline{q } m_W = 81.30 +- 0.47(stat.) +- 0.11(syst.) GeV/c^2$, $WW \to l\nu q\overline{q}(l=e,\mu) m_W = 80.54 +- 0.47(stat.) +- 0.11(syst.) GeV/c^2$, $WW \to \tau\nu q\overline{q} m_W = 79.56 +- 1.08(stat.) +- 0.23(syst.) GeV/C62$. The statistical errors are the expected errors for Monte Carlo samples of the same integrated luminosity as the data. The combination of these measurements gives: $m_W = 80.80 +- 0.11(syst.) +- 0.03(LEP energy) GeV/^2$