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

Variational Ansatz for PT-Symmetric Quantum Mechanics

A variational calculation of the energy levels of a class of PT-invariant quantum mechanical models described by the non-Hermitian Hamiltonian H= p^2 - (ix)^N with N positive and x complex is presented. Excellent agreement is obtained for the ground state and low lying excited state energy levels and wave functions. We use an energy functional with a three parameter class of PT-symmetric trial wave functions in obtaining our results.Comment: 9 pages -- one postscript figur

Complex periodic potentials with real band spectra

This paper demonstrates that complex PT-symmetric periodic potentials possess real band spectra. However, there are significant qualitative differences in the band structure for these potentials when compared with conventional real periodic potentials. For example, while the potentials V(x)=i\sin^{2N+1}(x), (N=0, 1, 2, ...), have infinitely many gaps, at the band edges there are periodic wave functions but no antiperiodic wave functions. Numerical analysis and higher-order WKB techniques are used to establish these results.Comment: 8 pages, 7 figures, LaTe

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

Ten Conversations about Identity Preservation: Implications for Cooperatives

Motivation: While it appears the modern economy demands ever increasing amounts of differentiation, opportunities for grain producers to create and capture significant new sources of value remains elusive. Opportunities appear to loom large to help remove risk and improve quality in the grain supply chain through preservation of product identity, producers, producer groups, and cooperatives are frustrated at the low level of value available to them from IP demand. Why do premiums remain low? And, what is the role of group action in these new differentiated markets? Objectives: This research report helps to explain this apparent paradox underlying the economics of the value proposition for IP grains. Methodology: Needs assessments were conducted on procurement executives using a semi-structured instrument. Results: The study demonstrates that understanding identity preservation business opportunities requires an understanding of the buy-side proposition. Respondents described how they balance the risk mitigation and market uplift features of a supply offering with the risks of narrowing the supply base. A model of the buyer's calculus is constructed. The results show how for producers and producer groups to drive value up the chain they need to shift away from solely a new product focus. Instead attention needs to be directed towards technologies, delivery systems, and organizational models that when bundled with new products make end-users more competitive. A second insight was the limited role of group action in meeting end-user needs. Where value markets existed, internalized groups rather than "arm's length" group transactions were the norm. Plan for Discussion: The cooperative movement was grounded in group action giving individual producers power in the market. The motivation to unite was very clear. In the post-industrial agrifood system though, why do buyers want to purchase from a group? What is the role of the group, from a buy-side perspective, in the modern economy? How should effective groups be structured? Key Words: identity preservation, supply chain management, value creation, group actionidentity preservation, supply chain management, value creation, group action, Agribusiness,

Non-Hermitian extension of gauge theories and implications for neutrino physics

An extension of QED is considered in which the Dirac fermion has both Hermitian and anti-Hermitian mass terms, as well as both vector and axial-vector couplings to the gauge field. Gauge invariance is restored when the Hermitian and anti-Hermitian masses are of equal magnitude, and the theory reduces to that of a single massless Weyl fermion. An analogous non-Hermitian Yukawa theory is considered, and it is shown that this model can explain the smallness of the light-neutrino masses and provide an additional source of leptonic CP violation.Comment: 23 pages, 1 figure, JHEP style; corrections to match published versio

Small Mercury Relativity Orbiter

The accuracy of solar system tests of gravitational theory could be very much improved by range and Doppler measurements to a Small Mercury Relativity Orbiter. A nearly circular orbit at roughly 2400 km altitude is assumed in order to minimize problems with orbit determination and thermal radiation from the surface. The spacecraft is spin-stabilized and has a 30 cm diameter de-spun antenna. With K-band and X-band ranging systems using a 50 MHz offset sidetone at K-band, a range accuracy of 3 cm appears to be realistically achievable. The estimated spacecraft mass is 50 kg. A consider-covariance analysis was performed to determine how well the Earth-Mercury distance as a function of time could be determined with such a Relativity Orbiter. The minimum data set is assumed to be 40 independent 8-hour arcs of tracking data at selected times during a two year period. The gravity field of Mercury up through degree and order 10 is solved for, along with the initial conditions for each arc and the Earth-Mercury distance at the center of each arc. The considered parameters include the gravity field parameters of degree 11 and 12 plus the tracking station coordinates, the tropospheric delay, and two parameters in a crude radiation pressure model. The conclusion is that the Earth-Mercury distance can be determined to 6 cm accuracy or better. From a modified worst-case analysis, this would lead to roughly 2 orders of magnitude improvement in the knowledge of the precession of perihelion, the relativistic time delay, and the possible change in the gravitational constant with time