Quantum counterpart of spontaneously broken classical PT symmetry

The classical trajectories of a particle governed by the PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) have been studied in depth. It is known that almost all trajectories that begin at a classical turning point oscillate periodically between this turning point and the corresponding PT-symmetric turning point. It is also known that there are regions in $\epsilon$ for which the periods of these orbits vary rapidly as functions of $\epsilon$ and that in these regions there are isolated values of $\epsilon$ for which the classical trajectories exhibit spontaneously broken PT symmetry. The current paper examines the corresponding quantum-mechanical systems. The eigenvalues of these quantum systems exhibit characteristic behaviors that are correlated with those of the associated classical system.Comment: 11 pages, 7 figure

Engineering America's Future in Space: Systems Engineering Innovations for Sustainable Exploration

The National Aeronautics and Space Administration (NASA) delivers space transportation solutions for America's complex missions, ranging from scientific payloads that expand knowledge, such as the Hubble Space Telescope, to astronauts and lunar rovers destined for voyages to the Moon. Currently, the venerable Space Shuttle, which has been in service since 1981, provides U.S. capability for both crew and cargo to low-Earth orbit to construct the International Space Station, before the Shuttle is retired in 2010, as outlined in the 2006 NASA Strategic Plan. I In the next decade, NASA will replace this system with a duo of launch vehicles: the Ares I Crew Launch Vehicle/Orion Crew Exploration Vehicle and the Ares V Cargo Launch Vehicle/Altair Lunar Lander. The goals for this new system include increased safety and reliability, coupled with lower operations costs that promote sustainable space exploration over a multi-decade schedule. This paper will provide details of the in-house systems engineering and vehicle integration work now being performed for the Ares I and planned for the Ares V. It will give an overview of the Ares I system-level test activities, such as the ground vibration testing that will be conducted in the Marshall Center's Dynamic Test Stand to verify the integrated vehicle stack's structural integrity against predictions made by modern modeling and simulation analysis. It also will give information about the work in progress for the Ares I-X developmental test flight planned in 2009 to provide key data before the Ares I Critical Design Review. Activities such as these will help prove and refine mission concepts of operation, while supporting the spectrum of design and development tasks being performed by Marshall's Engineering Directorate, ranging from launch vehicles and lunar rovers to scientific spacecraft and associated experiments. Ultimately, the work performed will lead to the fielding of a robust space transportation solution that will carry international explorers and essential payloads for sustainable scientific discovery beyond planet Earth

Outnumb3r3d : intrinsically motivating mathematics for the PlayStation 4

This paper and accompanying poster describes the design of an intrinsically integrated educational game to improve children’s competencies in mental mathematics. A number of researchers have suggested that educational games are more effective when they are closely integrated with their learning content. Specifically work by the lead author has showed that a closer integration between an educational game's core-mechanics and its learning content can be both more appealing (in terms of time spent on-task) and more educationally effective (in terms of learning outcomes) than a less integrated "edutainment" approach. However, cursory approaches to integrating learning content remain common in contemporary educational software, and the literature lacks an exemplar of what can be achieved using an integrated approach. The Outnumb3r3d game was conceived to provide a commercial and theoretical exemplar of intrinsic integration for the Nintendo Wii, but was never completed. This project is now porting the original Wii prototype onto the PlayStation 4 in order to revive Outnumb3r3d as a research project. This paper details the design of Outnumb3r3d with reference to the key theoretical constructs that underlie its pedagogical design. In doing so it provides an example of a game design created to integrate mathematical learning content seamlessly into the game's core mechanics, ensuring that the mathematics is what makes the game intrinsically motivating to play rather than trying to hide or "sugar coat" its learning content. At the time of writing the game’s implementation is still a “work in progress”, but is expected to be the subject of future empirical evaluations into its effectiveness as a teaching tool

Semiclassical analysis of a complex quartic Hamiltonian

It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\half p^2+\half\mu^2x^2-\lambda x^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half \mu^2x^2+igx^3-\lambda x^4, whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter \lambda.Comment: 4 page

PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics

A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian $ix^3$ quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green's function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian PT-symmetric Hamiltonian. The $\mathcal{O}(g^4)$ contribution to the ground-state energy of the Hermitian version of the theory involves graphs with overlapping divergences, and these graphs are extremely difficult to regulate. In contrast, the graphs for the non-Hermitian version of the theory are finite to all orders and they are very easy to evaluate.Comment: 13 pages, REVTeX, 10 eps figure

Complex Extension of Quantum Mechanics

It is shown that the standard formulation of quantum mechanics in terms of Hermitian Hamiltonians is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct to explain experimental data. One might expect that a quantum theory based on a non-Hermitian Hamiltonian would violate unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed physical symmetry C of the Hamiltonian. Using C, an inner product is constructed whose associated norm is positive definite. This construction is completely general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalisation of it.Comment: 4 Pages, Version to appear in PR

Additive Manufacturing a Liquid Hydrogen Rocket Engine

Space Propulsion is a 5 day event being held from 2nd May to the 6th May 2016 at the Rome Marriott Park Hotel in Rome, Italy. This event showcases products like Propulsion sub-systems and components, Production and manufacturing issues, Liquid, Solid, Hybrid and Air-breathing Propulsion Systems for Launcher and Upper Stages, Overview of current programmes, AIV issues and tools, Flight testing and experience, Technology building blocks for Future Space Transportation Propulsion Systems : Launchers, Exploration platforms & Space Tourism, Green Propulsion for Space Transportation, New propellants, Rocket propulsion & global environment, Cost related aspects of Space Transportation propulsion, Modelling, Pressure-Thrust oscillations issues, Impact of new requirements and regulations on design etc. in the Automotive, Manufacturing, Fabrication, Repair & Maintenance industries

Interactions of Hermitian and non-Hermitian Hamiltonians

The coupling of non-Hermitian PT-symmetric Hamiltonians to standard Hermitian Hamiltonians, each of which individually has a real energy spectrum, is explored by means of a number of soluble models. It is found that in all cases the energy remains real for small values of the coupling constant, but becomes complex if the coupling becomes stronger than some critical value. For a quadratic non-Hermitian PT-symmetric Hamiltonian coupled to an arbitrary real Hermitian PT-symmetric Hamiltonian, the reality of the ground-state energy for small enough coupling constant is established up to second order in perturbation theory.Comment: 9 pages, 0 figure

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

Dissociable roles of the inferior longitudinal fasciculus and fornix in face and place perception

We tested a novel hypothesis, generated from representational accounts of medial temporal lobe (MTL) function, that the major white matter tracts converging on perirhinal cortex (PrC) and hippocampus (HC) would be differentially involved in face and scene perception, respectively. Diffusion tensor imaging was applied in healthy participants alongside an odd-one-out paradigm sensitive to PrC and HC lesions in animals and humans. Microstructure of inferior longitudinal fasciculus (ILF, connecting occipital and ventro-anterior temporal lobe, including PrC) and fornix (the main HC input/output pathway) correlated with accuracy on odd-one-out judgements involving faces and scenes, respectively. Similarly, blood oxygen level-dependent (BOLD) response in PrC and HC, elicited during oddity judgements, was correlated with face and scene oddity performance, respectively. We also observed associations between ILF and fornix microstructure and category-selective BOLD response in PrC and HC, respectively. These striking three-way associations highlight functionally dissociable, structurally instantiated MTL neurocognitive networks for complex face and scene perception