Velocity-dependent Potential Research Articles

LINKED ${\mathcal{PT}}$-SYMMETRY TO SUPERSYMMETRY IN A CLASS OF NON-HERMITIAN HAMILTONIANS

We introduce and study a class of non-Hermitian Hamiltonians which have velocity dependent potentials. Since stability cannot be advocated directly from the classical potential, we show that the energy spectra are real and bounded from below which proves the stability of the spectra of all members in the class. We find that the introduced class of non-Hermitian Hamiltonians do have a corresponding superpartner class of non-Hermitian Hamiltonians. We were able to introduce supercharges which in conjunction with the corresponding super Hamiltonians constitute a closed super algebra. Among the introduced Hamiltonians, we show that non-[Formula: see text]-symmetric Hamiltonians can be transformed into their corresponding superpartner Hamiltonians via a specific canonical transformation while the [Formula: see text]-symmetric ones failed to be mapped to their corresponding superpartner Hamiltonians via the same canonical transformation. Since canonical transformations preserve the spectrum, we conclude that non-[Formula: see text]-symmetric Hamiltonians out of the introduced class of Hamiltonians have the same spectrum as the corresponding superpartner Hamiltonians and thus supersymmetry (Susy) is broken for such Hamiltonians. This kind of intertwining of [Formula: see text]-symmetry and Susy is new as all the so far discussed cases concentrate on Hamiltonians of broken [Formula: see text]-symmetry that have broken Susy too while we showed that Susy can be also broken for non-[Formula: see text]-symmetric and non-Hermitian Hamiltonians.

Perturbation Theory for Proton-Neutron Scattering: Perturbing the Energy

Using the time-independent Schrodinger equation with a velocity-dependent potential the energy dependence of the corresponding scattering phase shifts, when the energy is changed by a small amount ΔE from an arbitrary unperturbed value E0, is studied. We expand kcot δ as a power series in ΔE and obtain analytic formulas for the effective range expansion parameters to all orders in the perturbing energy. Formulas for the corresponding wave function changes are also developed. At low energies the Bethe formula for the effective range is reproduced and an expression for the shape-dependent term is obtained. The derived formalism is relevant to fields like nuclear and atomic physics. Examples that demonstrate the effectiveness of the derived formalism are presented.

Effect of the Velocity-Dependent Potentials on the Bound State Energy Eigenvalues

We investigate the effect of isotropic velocity-dependent potentials on the bound state energy eigenvalues for the first time for any quantum states of the Coulomb and harmonic oscillator potentials within the framework of the asymptotic iteration method. When the velocity-dependent term is selected as a constant parameter ρ0, we present that the energy eigenvalues can be obtained analytically for both Coulomb and harmonic oscillator potentials. However, when the velocity-dependent term is considered as a harmonic oscillator type ρ0r2, taking the velocity-dependent term as a perturbation, we present how to obtain the energy eigenvalues of the Coulomb and harmonic oscillator potentials for any n and ℓ quantum states by using perturbation expansion and numerical calculations in the asymptotic iteration method procedure.

Effective-string-theory constraints on the long-distance behavior of the subleading potentials

The dynamics of heavy quarkonium systems in the strong coupling regime reduces to a quantum mechanical problem with a number of potentials which may be organized in powers of $1/m$, $m$ being the heavy quark mass. The potentials must be calculated nonperturbatively, for instance, in lattice QCD. It is well known that the long-distance behavior of the static ($1/{m}^{0}$) potential is well reproduced by an effective string theory. We show that this effective string theory, if correct, should also reproduce the long-distance behavior of all $1/m$ suppressed potentials. We demonstrate the practical usefulness of this result by finding a suitable parameterization of the recently calculated $1/m$ potential. We also calculate the $1/{m}^{2}$ velocity-dependent and spin-dependent potentials. Once Poincar\'e invariance is implemented, the shapes of most of the spin-independent potentials are fully predicted in terms of the string tension and the shapes of the spin-dependent ones in terms of a single parameter.

A velocity-dependent potential of a rigid body in a rotating frame

We derive a velocity-dependent potential for describing the dynamics of a rigid body in a rotating frame. We show that, as for one-particle systems, the different components of this potential can be associated with electromagnetic analogs. We provide some examples to demonstrate the feasibility of using the potential as an alternative description of rigid body problems.

Preferred-frame andCP-violation tests with polarized electrons

We used a torsion pendulum containing {approx_equal}10{sup 23} polarized electrons to search new interactions that couple to electron spin. We limit CP-violating interactions between the pendulum's electrons and unpolarized matter in the Earth or the Sun, test for rotation and boost-dependent preferred-frame effects using the Earth's rotation and velocity with respect to the entire cosmos, and search for exotic velocity-dependent potentials between polarized electrons and unpolarized matter in the Sun and Moon. We find CP-violating parameters |g{sub P}{sup e}g{sub S}{sup N}|/(({Dirac_h}/2{pi})c) 1 AU. We test for preferred-frame interactions of the form V=-{sigma}{sup e}{center_dot}A, V=-B{sigma}{sup e}{center_dot}v/c, or , where v is the velocity of the Earth with respect to the cosmic microwave background restframe and i, j represent the equatorial inertial coordinates X, Y, and Z. We constrain all 3 components of A, obtaining 1{sigma} upper limits |A{sub X,Y}|{<=}1.5x10{sup -22} eV and |A{sub Z}|{<=}4.4x10{sup -21} eV that may be compared to the benchmark value m{sub e}{sup 2}/M{sub Planck}=2x10{sup -17} eV. Interpreting our constraint on A in terms of noncommutative geometry, we obtain an upper bound of (355l{sub GUT}){sup 2} on the minimum observable area, where l{sub GUT}=({Dirac_h}/2{pi})c/(10{sup 16} GeV) is the grandmore » unification length. We find that |B|{<=}1.2x10{sup -19} eV. All 9 components of C are constrained at the 10{sup -17} to 10{sup -18} eV level. We determine 9 linear combinations of parameters of the standard model extension; rotational-noninvariant and boost-noninvariant terms are limited at roughly the 10{sup -31} GeV and 10{sup -27} GeV levels, respectively. Finally, we find that the gravitational mass of an electron spinning toward the galactic center differs by less than about 1 part in 10{sup 21} from an electron spinning in the opposite direction. As a byproduct of this work, the density of polarized electrons in Sm Co{sub 5} was measured to be (4.19{+-}0.19)x10{sup 22} cm{sup -3} at a field of 9.6 kG.« less

Is cosmic parity violation responsible for the anomalies in the WMAP data?

In this Letter, I argue that a parity violating extension to general relativity can simultaneously explain the observed loss in power and provides a first step for explaining the alignment at a preferred axis (‘Axis of Evil’) in the low multipole moments of the WMAP data. This observational possibility also provides an experimental window for an inflationary leptogenesis mechanism arising from large-scale parity violation. Similar to the arguments of Contaldi, Peloso, Kofman and Linde, large scale power is suppressed from the backreaction of the parity violating term by inducing a velocity dependent potential. We also argue that this modification can supress power of odd parity multipoles on large scales, a pattern that is observed in the WMAP alignment anomaly.

On the origin of the anomalous precession of Mercury’s perihelion

Firstly, we recover an ancient work of Gerber in 1898 as a precursor of the retarded theories, see [Gerber P. Die räumliche und zeitliche Ausbreitung der Gravitation (Space and temporary propagation of gravitation). Z Math Phys 1898;43:93–104]. In that paper Gerber gave an explanation of the anomalous precession of Mercury’s perihelion in terms of a velocity-dependent potential. In the present paper an explanation of the anomalous precession of Mercury’s perihelion is given in terms of a simple retarded potential, which, at first orders, coincides with Gerber’s potential, and which agrees with the author’s previous works [Giné J. On the origin of quantum mechanics. Chaos, Solitons & Fractals 2006;30(3):532–41; Giné J. On the origin of gravitational quantization: the Titius–Bode law. Chaos, Solitons & Fractals 2007;32(2):363–69].

Global Invariants for Variable-Mass Systems

We investigate the effect of mass loss on the invariants of single particle motion in potentials having translational or rotational symmetry. These systems are non-Hamiltonian in the physical momenta, but for non-velocity-dependent potentials can be made formally Hamiltonian by treating the velocities as canonical momenta. Applying Noether's theorem to the formal Hamiltonian then yields global invariants corresponding to its symmetries. For velocity-dependent potentials, an exact invariant is constructed from the (non-Hamiltonian) vector field. The results are applied to charged particle motion in a magnetic dipole and agree well with numerical calculations.

Perturbation theory for isotropic velocity-dependent potentials: Scattering case

The time-independent Schr\odinger equation with an isotropic velocity-dependent potential is considered. Treating the velocity-dependent interaction as a small perturbation, we develop analytical formulas for the changes in the scattering phase shifts and wave functions. It is shown that only the zeroth-order solution and the perturbing potential are needed to determine the phase-shift and wave-function corrections. No prior knowledge of the unperturbed scattering-states continuum is required. In order to test the validity of our approach, we applied it to an exactly solvable model for nucleon-nucleon scattering. The results of the perturbation formalism compare quite well with those of the exactly solvable model. The developed formalism can be applied in problems concerning pion-nucleon, nucleon-nucleon, and electron-atom scattering. It may also be useful in studying the scattering of electrons in semiconductor heterostructures.

Effect of ordering ambiguity in constructing the Schrödinger equation on perturbation theory

This work explores the application of perturbation formalism, developed for isotropic velocity-dependent potentials, to three-dimensional Schrodinger equations obtained using different orderings of the Hamiltonian. It is found that the formalism is applicable to Schrodinger equations corresponding to three possible ordering ambiguities. The validity of the derived expressions is verified by considering examples admitting exact solutions. The perturbative results agree quite well with the exactly obtained ones.

Perturbation theory for isotropic velocity-dependent potentials: Bound-states case

Starting from the time-independent Schrodinger equation we develop formulae for the changes in the bound-state energies in the presence of an isotropic, velocity-dependent perturbing potential. The corresponding changes in the wave functions are also obtained. Unlike the case of the standard perturbation theory, determination of the changes in the energy and the wave function of a state only requires knowledge of the unperturbed ground-state wave function in addition to the perturbing potential. Evaluations of the energy changes and the corresponding wave functions are given for two examples in the s-wave case.

Effective descriptions of complex quantum systems: path integrals and operator ordering problems

We study certain aspects of the effective, occasionally called collective, description of complex quantum systems within the framework of the path integral formalism, in which the environment is integrated out. Generalising the standard Feynman-Vernon Caldeira-Leggett model to include a non-linear coupling between ``particle'' and environment, and considering a particular spectral density of the coupling, a coordinate-dependent mass (or velocity-dependent potential) is obtained. The related effective quantum theory, which depends on the proper discretisation of the path integral, is derived and discussed. As a result, we find that in general a simple effective low-energy Hamiltonian, in which only the coordinate-dependent mass enters, cannot be formulated. The quantum theory of weakly coupled superconductors and the quantum dynamics of vortices in Josephson junction arrays are physical examples where these considerations, in principle, are of relevance.

Cartesian and Lagrangian Momentum

Historical, physical, and geometrical relations between two different momenta, characterized here as Cartesian and Lagrangian, are explored. Cartesian momentum is determined by the mass tensor, and gives rise to a kinematical geometry. Lagrangian momentum, which is more general, is given by the fiber derivative, and produces a dynamical geometry. This differs from the kinematical in the presence of a velocity-dependent potential. The relation between trajectories and level surfaces in Hamilton-Jacobi theory can also be Cartesian and kinematical or, more generally, Lagrangian and dynamical.

On pseudo-Hermitian Hamiltonians and their Hermitian counterparts

In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT-invariant Hamiltonian, with potential V=igx^3. In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P.

Approximation of the semigroup generated by the Hamiltonian of Reggeon field theory in Bargmann space

The Reggeon field theory is governed by a non-self adjoint operator constructed as a polynomial in A, A * , the standard Bose annihilation and creation operators. In zero transverse dimension, this Hamiltonian acting in Bargmann space is defined by H λ ′ , μ = λ ′ A * 2 A 2 + μ A * A + i λ A * ( A * + A ) A , where i 2 = − 1 , λ ′ , μ and λ are real numbers and the operators A , A * satisfy the commutation relation [ A , A * ] = I . As the quantum mechanical system described by H λ ′ , μ has a velocity-dependent potential containing powers of momentum up to the fourth, the problem of existence of Hamiltonian path integral for the evolution operator e − t H λ ′ , μ of this theory is of interest on its own. In particular, can we express e − t H λ ′ , μ as a limit of “integral” operators? In this article one considerably reduces the difficulty by studying the Trotter product formula of H λ ′ , μ to reach two objectives: • The first objective is to prove a very specific error estimate for the error in a Trotter product formula in trace-norm for H viewed as the sum of the operators λ ′ A * 2 A 2 and μ A * A + i λ A * ( A * + A ) A . • The second objective of this work is to give a approximation of the semigroup generated by H λ ′ , μ when H λ ′ , μ is split in the sum of λ ′ A * 2 A 2 + μ A * A and i λ A * ( A * + A ) A . We note that this case is entirely different. In fact, the usual Trotter product formula is not defined, because the interaction operator A * ( A * + A ) A is not the infinitesimal generator of a semigroup on Bargmann space. For λ ′ > 0 and ɛ > 0 , we choose an approximation operator θ ɛ = [ I − ɛ i λ A * ( A * + A ) A ] e − ɛ ( λ ′ A * 2 A 2 + μ A * A ) and we give a connection between θ ɛ and e − ɛ H λ ′ , μ . This choice allows us to give in [A. Intissar, Note on the path integral formulation of Reggeon field theory, preprint] a “generalized Trotter product formula” for T μ = μ A * A + i λ A * ( A + A * ) A , i.e., for limit case as λ ′ = 0 and answers to the above question.

On Integrable Hamiltonians with Velocity Dependent Potentials

We study strongly and weakly integrable 2-dimensional Hamiltonian systems with velocity dependent potentials. We determine the set of conditions which must be satisfied in order to allow the existence of an independent second invariant polynomial in the momenta. We then investigate the linear case for which a complete solution of the problem can be obtained. We recover the classical set of linear strongly integrable systems and provide several new examples of weakly integrable systems whose equations of motion can be explicitly solved at a fixed value of the energy.

What can WMAP tell us about the very early Universe? New physics as an explanation of the suppressed large scale power and running spectral index

The Wilkinson Microwave Anisotropy Probe microwave background data may be giving us clues about new physics at the transition from a ``stringy'' epoch of the Universe to the standard Friedmann-Robertson-Walker description. Deviations on large angular scales of the data, as compared to theoretical expectations, as well as running of the spectral index of density perturbations, can be explained by new physics whose scale is set by the height of an inflationary potential. As examples of possible signatures for this new physics, we study the cosmic microwave background spectrum for two string inspired models: (1) modifications to the Friedmann equations and (2) velocity dependent potentials. The suppression of low $``l''$ modes in the microwave background data arises due to the new physics. In addition, the spectral index is red $(n&lt;1)$ on small scales and blue $(n&gt;1)$ on large scales, in agreement with data.

String theory in electromagnetic fields

A review of various aspects of superstrings in background electromagnetic fields is presented. Topics covered include the Born-Infeld action, spectrum of open strings in background gauge fields, the Schwinger mechanism, finite-temperature formalism and Hagedorn behaviour in external fields, Debye screening, D-brane scattering, thermodynamics of D-branes, and noncommutative field and string theories on D-branes. The electric field instabilities are emphasized throughout and contrasted with the case of magnetic fields. A new derivation of the velocity-dependent potential between moving D-branes is presented, as is a new result for the velocity corrections to the one-loop thermal effective potential.

Classical Lagrangian and quantum phase of the dipole

The non-relativistic Lagrangians of the electric and magnetic dipole are derived consistently with classical electrodynamics. The corresponding velocity-dependent potential leads to a generalized quantum phase of the magnetic dipole and to a related field-free quantum effect.