PT-Symmetric Quantum Mechanics

This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition $H^\dagger=H$ on the Hamiltonian, where $\dagger$ represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian $H$ has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement $H^\ddag=H$, where $\ddag$ represents combined parity reflection and time reversal ${\cal PT}$, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation $H=p^2+x^2(ix)^\epsilon$ of the harmonic oscillator Hamiltonian, where $\epsilon$ is a real parameter. The system exhibits two phases: When $\epsilon\geq0$, the energy spectrum of $H$ is real and positive as a consequence of ${\cal PT}$ symmetry. However, when $-1<\epsilon<0$, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because ${\cal PT}$ symmetry is spontaneously broken. The phase transition that occurs at $\epsilon=0$ manifests itself in both the quantum-mechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians $H=p^2+x^{2N}(ix)^\epsilon$ with $N$ integer and $\epsilon>-N$; each of these complex Hamiltonians exhibits a phase transition at $\epsilon=0$. These ${\cal PT}$-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.Comment: 20 pages RevTex, 23 ps-figure

Universality in Random Walk Models with Birth and Death

Models of random walks are considered in which walkers are born at one location and die at all other locations with uniform death rate. Steady-state distributions of random walkers exhibit dimensionally dependent critical behavior as a function of the birth rate. Exact analytical results for a hyperspherical lattice yield a second-order phase transition with a nontrivial critical exponent for all positive dimensions $D\neq 2,~4$. Numerical studies of hypercubic and fractal lattices indicate that these exact results are universal. Implications for the adsorption transition of polymers at curved interfaces are discussed.Comment: 11 pages, revtex, 2 postscript figure

Broadband Records of Earthquakes in Deep Gold Mines and a Comparison with Results from SAFOD, California

For one week during September 2007, we deployed a temporary network of field recorders and accelerometers at four sites within two deep, seismically active mines. The ground-motion data, recorded at 200 samples/sec, are well suited to determining source and ground-motion parameters for the mining-induced earthquakes within and adjacent to our network. Four earthquakes with magnitudes close to 2 were recorded with high signal/noise at all four sites. Analysis of seismic moments and peak velocities, in conjunction with the results of laboratory stick-slip friction experiments, were used to estimate source processes that are key to understanding source physics and to assessing underground seismic hazard. The maximum displacements on the rupture surfaces can be estimated from the parameter Rv, where v is the peak ground velocity at a given recording site, and R is the hypocentral distance. For each earthquake, the maximum slip and seismic moment can be combined with results from laboratory friction experiments to estimate the maximum slip rate within the rupture zone. Analysis of the four M 2 earthquakes recorded during our deployment and one of special interest recorded by the in-mine seismic network in 2004 revealed maximum slips ranging from 4 to 27 mm and maximum slip rates from 1.1 to 6:3 m=sec. Applying the same analyses to an M 2.1 earthquake within a cluster of repeating earthquakes near the San Andreas Fault Observatory at Depth site, California, yielded similar results for maximum slip and slip rate, 14 mm and 4:0 m=sec

Effect of bevelled silo outlet in the flow rate during discharge

We investigate the effect of a bevelled (or slanted) outlet on the discharge rate of mono-sized spheres from a quasi-two-dimensional silo, using the discrete element method. In contrast to hopper discharges, where the bevelling is across the entire base of the container, we study a bevelled opening that is significantly smaller than the silo width and in which the slanting is limited to half a sphere diameter at the boundary of the outlet. We show that the bevelling increases the flow rate comparably to the inclination in hopper walls. Using Beverloo's model, we relate this increase in rate to what we define as the ‘effective opening’ of the silo and analyse the velocity profiles associated with the discharges. We show that different openings, having effectively the same discharge rates, give rise to distinctly different internal dynamics in the silo. These results have the potential to aid industrial processes by fine-tuning and improving control of silo discharges, with a minimal impact on silo design, thus significantly reducing production and handling costs.Fil: Gago, Paula Alejandra. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Imperial College London; Reino UnidoFil: Madrid, Marcos Andres. Universidad Tecnológica Nacional; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Boettcher, Stefan. University of Emory; Estados UnidosFil: Blumenfeld, Raphael. Imperial College London; Reino UnidoFil: King, Peter. Imperial College London; Reino Unid

Bound States of Non-Hermitian Quantum Field Theories

The spectrum of the Hermitian Hamiltonian ${1\over2}p^2+{1\over2}m^2x^2+gx^4$ ($g>0$), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian $H={1\over2}p^2+{1 \over2}m^2x^2-gx^4$, where the coupling constant $g$ is real and positive, is ${\cal PT}$-symmetric. As a consequence, the spectrum of $H$ is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When $g$ is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian ${\cal PT}$-symmetric $-g\phi^4$ quantum field theory for all dimensions $0\leq D<3$ but is not present in the conventional Hermitian $g\phi^4$ field theory.Comment: 14 pages, 3figure