Green Functions for the Wrong-Sign Quartic

It has been shown that the Schwinger-Dyson equations for non-Hermitian theories implicitly include the Hilbert-space metric. Approximate Green functions for such theories may thus be obtained, without having to evaluate the metric explicitly, by truncation of the equations. Such a calculation has recently been carried out for various $PT$-symmetric theories, in both quantum mechanics and quantum field theory, including the wrong-sign quartic oscillator. For this particular theory the metric is known in closed form, making possible an independent check of these approximate results. We do so by numerically evaluating the ground-state wave-function for the equivalent Hermitian Hamiltonian and using this wave-function, in conjunction with the metric operator, to calculate the one- and two-point Green functions. We find that the Green functions evaluated by lowest-order truncation of the Schwinger-Dyson equations are already accurate at the (6-8)% level. This provides a strong justification for the method and a motivation for its extension to higher order and to higher dimensions, where the calculation of the metric is extremely difficult

Extending PT symmetry from Heisenberg algebra to E2 algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the Hamiltonian H=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where again g is real. As in the case of PT-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry in which all the eigenvalues are real and a region of broken PT symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.Comment: 8 pages, 7 figure

The quantum anharmonic oscillator in the Heisenberg picture and multiple scale techniques

Multiple scale techniques are well-known in classical mechanics to give perturbation series free from resonant terms. When applied to the quantum anharmonic oscillator, these techniques lead to interesting features concerning the solution of the Heisenberg equations of motion and the Hamiltonian spectrum.Comment: 18 page

Symmetry Considerations for the Detection of Second-Harmonic Generation in Cuprates in the Pseudogap phase

A proposal to test the proposed time-reversal and inversion breaking phase in the Pseudogap region of the Cuprate compounds through the variation of Second-harmonic generation intensity with temperature and polarization and angle of incidence is presented.Comment: 7 pages, 2 figures include

Entanglement Efficiencies in PT-Symmetric Quantum Mechanics

The degree of entanglement is determined for an arbitrary state of a broad class of PT-symmetric bipartite composite systems. Subsequently we quantify the rate with which entangled states are generated and show that this rate can be characterized by a small set of parameters. These relations allow one in principle to improve the ability of these systems to entangle states. It is also noticed that many relations resemble corresponding ones in conventional quantum mechanics.Comment: Published version with improved figures, 5 pages, 2 figure

Continuous canonical correlation analysis

Given a bivariate distribution, the set of canonical correlations and functions is in general finite or countable. By using an inner product between two functions via an extension of the covariance, we find all the canonical correlations and functions for the so-called Cuadras-Aug´e copula and prove the continuous dimensionality of this distribution

Nuclear weapons and European interests : Discussions in the Assembly of the Western European Union, 1955–1975

In the 1950s and 1960s the Defence Committee of the Western European Union (WEU) Assembly submitted a series of conspicuous reports on nuclear control issues, culminating in a plea for an Atlantic nuclear force. The article analyses the reports and discussions in the Assembly in the field of defence as well as its attempts to have their recommendations adopted by national parliaments. Internal political and national differences notwithstanding, the representatives in the WEU Assembly developed a clear understanding of European interests.

Our shifting perspectives on the oceans

In the last 15 years new research findings have radically reshaped our understanding of human effects on ocean ecosystems. Here I describe five major shifts in perspective that reveal our impacts to be more severe and persistent than previously appreciated. Firstly, scientists have delved deep into the past and found that the global expansion of European nations across the planet caused the large-scale loss of marine megafauna. In the past century, expansion of industrial scale fishing has continued the process, massively reducing the biomass of exploited species. Secondly, once depleted we are finding that populations rarely rebound rapidly, contrary to a widespread belief in greater resilience of marine compared to terrestrial species. Thirdly, marine ecosystems are being shifted into alternative states that are less desirable from the human perspective and may be stable. It could be difficult, or impossible in some cases, to reverse impacts once inflicted. Fourthly, marine species are at risk of extinction. Loss of shallow water marine habitats is proceeding as rapidly as on land, many species have small geographic ranges, and many possess life history characteristics that leave them highly susceptible to overexploitation. Finally, the deep sea is not beyond harm. Depletion of shallow water fisheries and technological advances are opening up the deep to exploitation and its collateral impacts. If we are to reverse these negative trends we must establish large-scale networks of marine reserves that are off limits to damaging activities and fishing. Such reserves would protect biodiversity, and recover and sustain the world's fisheries productivity

Keeping a reflective journal : reflections of a mature student

Reflective journals are used increasingly in Higher Education. Examples from an unstructured journal kept by the author in her role as a new Mechanical Engineering student raise issue of what 'ought' to be in journals, how emotional effects should be dealt with and how the subjective nature of reflection may cause problems for assessment

Uniqueness of Ground States for Short-Range Spin Glasses in the Half-Plane

We consider the Edwards-Anderson Ising spin glass model on the half-plane $Z \times Z^+$ with zero external field and a wide range of choices, including mean zero Gaussian, for the common distribution of the collection J of i.i.d. nearest neighbor couplings. The infinite-volume joint distribution $K(J,\alpha)$ of couplings J and ground state pairs $\alpha$ with periodic (respectively, free) boundary conditions in the horizontal (respectively, vertical) coordinate is shown to exist without need for subsequence limits. Our main result is that for almost every J, the conditional distribution $K(\alpha|J)$ is supported on a single ground state pair.Comment: 20 pages, 3 figure