Real Energies Research Articles

Chiral phase transition in a covariant nonlocal NJL model

The properties of the chiral phase transition at finite temperature and chemical potential are investigated within a nonlocal covariant extension of the Nambu–Jona-Lasinio model based on a separable quark–quark interaction. We consider both the situation in which the Minkowski quark propagator has poles at real energies and the case where only complex poles appear. In the literature, the latter has been proposed as a realization of confinement. In both cases, the behaviour of the physical quantities as functions of T and μ is found to be quite similar. In particular, for low values of T the chiral transition is always of first order and, for finite quark masses, at certain “end point” the transition turns into a smooth crossover. In the chiral limit, this “end point” becomes a “tricritical” point. Our predictions for the position of these points are similar, although somewhat smaller, than previous estimates. Finally, the relation between the deconfining transition and chiral restoration is also discussed.

Complex Calogero model with real energies

We show that and how PT symmetry (interpreted as a "weakened Hermiticity") can be extended to the exactly solvable two- and three-particle Calogero model.

The field-assisted emission effect on the acceptor energy of GaN:Mg in admittance spectroscopy measurements

A simulation program for admittance calculations in a semiconductor Schottky junction with a trap was developed using physical parameters including the Poole–Frenkel electric field-assisted emission effect. The simulation revealed limitations in the validity of the admittance measurement technique in determining thermal activation energies of semiconductors. The controversy in the observed thermal activation energy of the GaN:Mg structures between the Hall effect and admittance spectroscopy measurements is explained using the simulation results. The admittance measurement technique can only determine the real energies when the overall doping concentration of the shallow (NS) and deep (NT) levels is low and while NT⩽NS. While the condition of NT⩾NS causes a reduction in the apparent measured energies, the electric field built up at the crossover by NS or/and NT further reduces the energies by lowering the barrier potential for the trap emission. Particularly if NT≫NS, the electric field induced by the deep level itself induces the reduction in the barrier potential. This is the case for the rather highly doped GaN:Mg structures.

Supersymmetry without hermiticity within [formula omitted] symmetric quantum mechanics

A new model of supersymmetry between bosons and fermions is proposed. Its representation space is spanned by states with PT symmetry and real energies but the inter-related partner Hamiltonians themselves remain complex and non-Hermitian. The formalism admits vanishing Witten index.

Causality in relativistic many body theory

The stability of the nuclear matter system with respect to density fluctuations is examined exploring in detail the pole structure of the electro-nuclear response functions. Making extensive use of the method of dispersion integrals we calculate the full polarization propagator not only for real energies in the spacelike and timelike regime but also in the whole complex energy plane. The latter proved to be necessary in order to identify unphysical causality violating poles which are the consequence of a neglection of vacuum polarization. On the contrary it is shown that Dirac sea effects stabilize the nuclear matter system shifting the unphysical pole from the upper energy plane back to the real axis. The exchange of strength between these real timelike collective excitations and the spacelike energy regime is shown to lead to a reduction of the quasielastic peak as it is seen in electron scattering experiments. Neglecting vacuum polarization one also obtains a reduction of the quasielastic peak but in this case the strength is partly shifted to the causality violating pole mentioned above which consequently cannot be considered as a physical reliable result. Our investigation of the response function in the energy region above the threshold of nucleon anti-nucleon production leads to another remarkable result. Treating the nucleons as point-like Dirac particles we show that for any isospin independent NN-interaction RPA-correlations provide a reduction of the production amplitude for $p\bar p$-pairs by a factor 2.

Infrared study of H/D isotope exchange reaction of alkanes and Brønsted acidic hydroxyl groups on FER zeolite

H/D isotope exchange between n-alkanes and the acidic OD groups in ferrierite was observed by infrared spectroscopy. The real activation energies were estimated by adding the heats of adsorption to the apparent activation energies, resulting in almost the same value for all n-alkanes. The transition state for the reaction is discussed.

$N$ -Body Quantum Scattering Theory in Two Hilbert Spaces: $N$ -Body Integral Equations

Scattering for a nonrelativistic system of \(\) distinguishable and spinless particles interacting via short-range pair potentials is considered. Half-on-shell integral equations (the CG equations) are proposed, the solutions of which determine approximate scattering amplitudes that converge to the exact scattering amplitude. It is proved, under mild Holder integrability assumptions, that these apparently singular equations actually have a compact kernel for real energies and, consequently, a unique solution. The CG equations have a structure that is much simpler than the Yakubovskii equations and similar to that of coupled-reaction-channel equations. The driving terms look like distorted-wave Born integrals and nonorthogonality integrals. However, there is no restriction to channels with only two asymptotic bound clusters and for all channels, no matter how many bound clusters, appropriate boundary conditions are exactly satisfied. This work completes the establishment of a rigorous mathematical link between the solutions of the half-on-shell CG equations and the on-shell transition operators defined in time-dependent multichannel scattering theory, and it provides for the first time a rigorous theoretical basis for practical calculations of scattering amplitudes for certain problems with \(\).

Meson properties in an extended non-local NJL model

We consider a non-local version of the NJL model, based on a separable quark-quark interaction. The interaction is extended to include terms that bind vector and axial-vector mesons. The non-locality means that no further regulator is required. Moreover the model is able to confine the quarks by generating a quark propagator without poles at real energies. Working in the ladder approximation, we calculate amplitudes in Euclidean space and discuss features of their continuation to Minkowski energies. Conserved currents are constructed and we demonstrate their consistency with various Ward identities. Various meson masses are calculated, along with their strong and electromagnetic decay amplitudes. We also calculate the electromagnetic form factor of the pion, as well as form factors associated with the processes γγ* → π0 and ω → π0γ*. The results are found to lead to a satisfactory phenomenology and lend some dynamical support to the idea of vector-meson dominance.

The behaviour of the propagating and evanescent modes of a 2D quantum wire in a perpendicular magnetic field

The modes of a 2D quantum wire in a perpendicular magnetic field are studied in detail for the real electron energies which are appropriate in nanostructure scattering problems. Propagating and evanescent modes are considered for both asymmetric and symmetric confining potentials. The symmetry of the mode functions and the real and imaginary parts ofkare investigated in both cases. For asymmetric potentials the evanescentkvalues usually have non-zero real parts. For symmetric potentials purely imaginarykvalues also appear when ϵ is just below a propagation threshold. Calculated wavefunctions and dispersion curves for both Re(k) and Im(k) are presented for simple examples of both cases and the transformation from one to the other is described. The total current carried by an arbitrary superposition of modes (both propagating and evanescent) is calculated. The part of the current carried by interfering evanescent modes involves a complex analogue of the group velocity.

Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states.

A formalism of degenerate perturbation theory is presented, in which the initial basis set of functions is fully adapted to the perturbation. This full adaptation is obtained by imposing a simple condition which leads to an iteration procedure free of singularities. The formalism includes as a special case nondegenerate perturbation theory. By using it we calculate the dispersion coefficients of alkali-metal dimers for molecular states which dissociate into one atom in the ground state and the other in one of the first two S, P, or D excited states. The dispersion forces are extracted from the first- and second-order energy corrections. Model potentials are used in order to describe the motion of the valence electron in the field of the alkali-metal positive-ion core. Using the first-order wave-function correction, we investigate the leading terms of the long-range expansion of the electronic transition dipole moments. An extension of the Dalgarno-Lewis method is developed in order to handle the radial matrix elements which involves a reduced Green's function for real and complex energies. The results are compared with previous computations.

Covariant off-shell model for dynamical quark confinement

A model for quark propagation is developed in which quarks cannot go on-shell and thus are confined dynamically. The approach is based on a recently proposed time-ordered formulation of the relativistic many-body problem made manifestly covariant by an off-shell modification of Lorentz transformations. It is argued that the present off-shell formalism is better suited for studies of dynamical confinement since it provides a straightforward off-shell extension of the Dirac equation. In the usual Lorentz-covariant formulation, by contrast, spinor solutions do not exist for quarks which cannot go on-shell. Defining the quark propagator by a nonlinear time-ordered one-gluon-loop expansion, similar to a Dyson-Schwinger equation, an energy-dependent quark mass is determined by a nonlinear self-consistency condition. The corresponding time-ordered propagator is found to possess no poles for real energies, thus ensuring confinement. Evidence is given that it is analytic in the entire upper half of the complex energy plane as required by causality. In the model, the self-energy integral exists without regularization, with the convergence being provided by the chiral limit of the quark mass itself. The self-consistency condition exhibits the scaling behavior of the renormalization group and provides an eigenvalue condition for the low-energy value of the quark-gluon coupling constant. Its numerically determined value of g 2/4π= 4.712 can actually be derived in the soft-gluon limit as g 2/4π=3/2π. Implications of these findings and possible further applications are discussed.

Analytic continuation of thermal N-point functions from imaginary to real energies.

We consider thermal $n$-point Green functions in the framework of quantum field theory at finite temperature. We show how analytic continuations from imaginary to real energies relate these functions originally defined in the imaginary-time formalism to retarded and advanced real-time ones. The described method is valid to all orders of perturbation theory. It has the further advantage that it is independent of approximations often applied in actual finite-order calculations.

On parasupersymmetric Hamiltonians and vector mesons in magnetic fields

A new Sakata-Taketani equation describing vector mesons in interaction with a constant magnetic field is proposed. It leads to real energies as opposed to relativistic equations, which have already been pointed out, associated with this particular interacting context. In particular, the connection between this new proposal and a specific model coming from parasupersymmetric quantum mechanics is studied.

Measurement of real free energies of transfer of individual ions from water to other solvents with the jet cell

Resolution of the activities of solutions of electrolytes into the individual ionic contributions cannot be carried out rigorously and requires the introduction of extrathermodynamic assumptions which have inherent uncertainties. The most commonly used approaches are basically similar in that they are based on the assumed solvent independence of the difference in the enthalpy or Gibbs energy of transfer of pairs of model solutes, e.g., tetraphenylarsonium and tetraphenylborate ions, or ferricinium ion and ferrocene. In this work we follow an alternative approach pioneered by Parsons involving measurement in the jet (Kenrick) cell of outer-potential differences between solutions of the same electrolyte in two solvents. These potential differences provide the ‘real’ free energies of transfer of individual ions which, in turn, differ from the usual Gibbs energies of transfer by the work required to transfer the ion through the dipolar layers at the two solvent-gas interfaces. One objective of this work was to improve the reliability of real free energy of transfer measurements, which are experimentally demanding, to within ca. ±0.5 kJ-mol−1 in order to match typical uncertainties in Gibbs transfer energies of electrolytes. This goal was met, in most instances, by careful evaluation of experimental parameters (particularly jet pressure). A major improvement over previous measurements was made by adding a supporting electrolyte which allowed stable potentials to be obtained at test electrolyte concentrations as low as 10−4M. Real free energy changes are reported for the transfer of silver ion from water to methanol, ethanol, acetonitrile, propylene carbonate and dimethyl sulfoxide, as well as for the transfer of chloride ion from water to methanol and ethanol. Reliable data of this kind may lead to improved understanding of either the properties of the surfaces of solvents or the interactions of model solutes with solvents, depending on which of the two fields develops most.

Length scaling of corner-to-corner propagation on fractal lattices

Closed formulas for Schro\ifmmode\ddot\else\textasciidieresis\fi{}dinger Green functions connecting opposite corners of two fractal lattices are presented. The lattices are the 3-simplex S and another, R, not previously analyzed. While the corner-to-corner propagation amplitude goes to zero as a function of the lattice size L with stretched-exponential behavior at most values of energy, power-law scaling of the form ${\mathit{L}}^{\mathrm{\ensuremath{\beta}}}$ is found for a certain infinite set of real energies. Values of the exponent \ensuremath{\beta} are distributed in a multifractal way over the spectrum of the infinite-lattice limit.

Green's function and a generalized Lloyd formula for the density of states in disordered muffin-tin alloys.

We consider the Korringa-Kohn-Rostoker coherent-potential-approximation Green's function for a disordered system of muffin-tin potentials and evaluate its integral over the Wigner-Seitz cell rigorously throughout the complex energy plane. The resulting expression is shown to possess a perfect derivative form, provided the angular-momentum summations are extended to infinity; for a finite angular-momentum cutoff this result is valid to a good approximation. For real energies our expression constitutes a generalization of the Lloyd formula for the density of states. The conventional Lloyd formula contains unphysical singularities that are not only unsatisfactory from a formal viewpoint, but also make its use increasingly cumbersome in multicomponent alloys as the number of atoms in the unit cell becomes larger. Our expression, on the other hand, is free from such singularities, and thus provides a useful basis for evaluating density of states and its energy integral in muffin-tin systems.

A new method for the direct calculation of resonance parameters with application to the quasibound states of the H2X1? g + system

A new method for the direct calculation of resonance parameters is presented. It is based upon searching for poles of the scattering matrix at complex energies. This search is expedited by the use of analytic derivatives of the scattering matrix with respect to the total energy. This procedure is applied initially to a single channel problem, but is generalizable to more complicated systems. Using the most accurate available potential energy data, we calculate resonance parameters for all of the physically important quasibound states of the ground electronic state of the hydrogen molecule. Corrections to the Born-Oppenheimer potential are included and assessed. The new method has no difficulty locating resonances with widths greater than about 1×10−7 cm−1. It is easier to find narrow resonances by monitoring the dependence of the imaginary part of the reactance matrix on the real part of a complex energy than to monitor the dependence of the eigenphase sum on energy at real energies.

GW approach to the calculation of electron self-energies in semiconductors.

Various approximation schemes concerning the calculation of the electron self-energy M for a semiconductor in the (bubble) GW scheme of Hedin [Phys. Rev. 139, A796 (1965)] are discussed. It is shown by using a contour-deformation procedure in the complex energy plane that M, as obtained in the first iteration cycle of the GW scheme, is Hermitian for real energies \ensuremath{\Vert}\ensuremath{\varepsilon}\ensuremath{\Vert}3${\ensuremath{\varepsilon}}_{g}$/2, where ${\ensuremath{\varepsilon}}_{g}$ is the unperturbed energy gap, and non-Hermitian for \ensuremath{\Vert}\ensuremath{\varepsilon}\ensuremath{\Vert}>3${\ensuremath{\varepsilon}}_{g}$/2. The Taylor expansion for M around the midgap energy value \ensuremath{\varepsilon}=0 has a convergence radius of 3${\ensuremath{\varepsilon}}_{g}$/2. Extended use of a (truncated) Taylor series at \ensuremath{\Vert}\ensuremath{\varepsilon}\ensuremath{\Vert}>3${\ensuremath{\varepsilon}}_{g}$/2 is not capable of giving the non-Hermitian part of M, while there is also no guarantee that the Hermitian part is correctly obtained in this way.

A novel method of calculating the RPA dielectric function for a semiconductor at real energies

We demonstrate a novel method for direct calculation of the RPA dielectric function for a semiconductor at real energies based on a special point integration procedure. The method is relevant in the context of model-free ab initio calculations of dielectric properties and self-energies in semiconductors.

Density of states for the gravitational field in black-hole topologies.

Using a previously developed formulation of black-hole thermodynamics for a system of finite size, we show that the partition function of the related canonical ensemble can be used to obtain the density of states in the microcanonical ensemble. We work with the partition function in a zero-loop approximation based on classical solutions with fixed boundary data and obtain the corresponding density of states by using an inverse Laplace transform. The computation requires the introduction of a uniformizing variable so that a path can be defined along which the classical action of a stable black hole is single valued in the integration over imaginary inverse temperatures. Although the zero-loop partition function is not a Laplace transform, its inversion integral yielding the density of states is nevertheless well defined, and we show that it is the exact inverse of a Fourier-Laplace integration over all real energies, positive and negative. We argue that (1) this need for all positive and negative energies and (2) a constraint on the boundary data for obtaining the partition function from classical solutions are both consequences of the zero-loop approximation that should be absent in a complete quantum theory. We also find that the single-valued action enables a discussion of negative temperatures for positive mean energies that appear to be essential for a satisfactory relation between the partition function and the density of states in the zero-loop approximation. The full significance of this observation must await further study.