Gravity and duality between coordinates and matter fields

We use the duality between the local Cartezian coordinates and the solutions of the Klein-Gordon equation to parametrize locally the spacetime in terms of wave functions and prepotentials. The components of metric, metric connection, curvature as well as the Einstein equation are given in this parametrization. We also discuss the local duality between coordinates and quantum fields and the metric in this later reparametrization.     

Polaron transport in a long phonon waves approximation

One seeks to understand directly the smooth electron distribution functions long ago observed in semiconductors by Haynes and Westphal, and found to satisfy the Einstein relation for diffusion.A zero-order approximation to the distribution function analogous to Chandrashekhar's solution of the Brownian motion problem is established. This is done on the basis of the expression of Feynman et al., for the density matrix of the electron with phonon coordinates eliminated. Justification of the underlying approximation of expansion in long phonon waves, as well as a discussion of the relation of the scheme to the Boltzmann or rate equation approach, is given.     

Folding model analysis of the nucleus–nucleus scattering based on Jacobi coordinates

This paper presents the results of scattering of ¹⁶O+²⁰⁹Bi interaction near the Coulomb barrier. The interaction potential between two nuclei is calculated using the double folding model with the effective nucleon–nucleon (NN) interaction. The calculations of the exchange part of the interaction were assumed to be of finite-range and the density dependence of the NN interaction is accounted for. Also the results are compared with the zero-range approximation. All these calculations are done using the wave functions of the two colliding nuclei in place of their density distributions. The wave functions are obtained by the D-dimensional wave equation using the hyperspherical calculations on the basis of Jacobi coordinates. The numerical results for the interaction potential and the differential scattering are in good agreement with the previous works.     

Global structure of the superstring partition function and resolution of the supermoduli measure ambiguity

We investigate the global structure of the fermionic string partition function on the supermoduli space M_g^sup. In particular we show how the recently discovered moduli total-derivative ambiguity is due to a non-trivial cocycle on M_g^sup, present if an atlas of coordinates of M_g^sup can be found, whose transition functions contain even, nilpotent components. We find a correction to the usual Berezin measure on M_g^sup, given by the so-called Rothstein volume form, that eliminates the above boundary ambiguities of the supermoduli measure at any genus. We discuss also the so-called theorem of holomorphic factorization in this particular case and its relation to the physical requirement of modular invariance.     

Prepotentials in superstring world sheet supergravity

The vielbein of (p, 0) supergravity is solved for in terms of prepotentials. The properties of the prepotentials under supercoordinate, Lorentz, and super-Weyl transformations are discussed, and a proof given that (p, 0) superspace is superconformally flat. The superdeterminant of the (p, 0) supergravity vielbein is shown to transform as a density superfield. Using this density, the Einstein and heterotic string actions are constructed in an arbitrary gauge, and proven to be invariant under super-Weyl transformations.     

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

In the classical theory of Brownian motion we can consider the Langevin equation as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, we can reinterpret the Langevin equation as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta we can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. We may associate, according to Weyl functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions we can express the streaming as a continuity equation of the Wigner function. We find that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation.     

Ideal hydrodynamics outside and inside a black hole: Hamiltonian description in Painlevé-Gullstrand coordinates

We show that when the Painlevé-Gullstrand coordinates are used in their Cartesian version, the Hamiltonian of relativistic ideal hydrodynamics in the vicinity of a nonrotating black hole differs by only one simple term from the corresponding Hamiltonian in a flat spacetime. The interior region of the black hole is also described in a unified way, because there is no singularity on the event horizon in Painlevé-Gullstrand coordinates. We present the exact solution describing the steady accretion of extremely hard matter (? ∝ n ²) onto a moving black hole up to the central singularity. In the local induction approximation, we derive the equation of motion for a thin vortex filament against the background of such an accretion flow. We explicitly calculate the Hamiltonian for a fluid with an ultrarelativistic equation of state, ? ∝ n ^4/3, and solve the problem of a centrally symmetric steady flow of such matter.     

Heavy ion collisions in three dimensions by the relaxation-time method

A quantum transport equation with two-body collisions included via a relaxation-time method, earlier applied to two-dimensional (slab) collisions, is now extended to three-dimensional calculations A density matrix is constructed from self-consistent field wave functions and is time-evolved in cartesian coordinates. This dynamical model of the nucleus is applicable at all nonrelativistic energies. The semiclassical limit is discussed. Results are shown for ¹⁶O-¹⁶O collisions between 40 and 200 MeV/A lab energies. Hot spots and conditions for fragmentation are discussed. The threshold for breakup of the compound system formed in a head-on collision lies between 40 and 60 MeV/A lab energies. At these energies, the maximum density-averaged thermal excitation energy is 7 and 10 MeV/A (average temperatures 8 and 11 MeV), respectively, compared with a binding energy of 8 MeV/A. The system does not thermalize completely, and the distribution in momentum space is not quite isotropic when breaking up.     

The Condition of Invariance of the Energy of an Atomic-Molecular System with Respect to Rotations of the Jacobi Coordinates

An additional condition for the wave functions of the ground and excited states of atomic-molecular systems, which follows from the energy invariance with respect to rotations of the Jacobi coordinates, is studied. This condition makes a substantial contribution to the conventional criterion of the quality of wave functions, based on the variational principle for the average energy value. The explicit form of this condition is found for an arbitrary interaction potential dependent on the particle-particle distances and the requirements for realization of this condition are ascertained. Variational calculations of ³He²⁺ μ^? e ^? and ⁴He²⁺ μ^? e ^? mesoatoms in the basis of correlated exponential functions dependent on the particle-particle distances are performed with a detailed optimization of all nonlinear parameters for each individual energy level. A high sensitivity of the condition under study to the quality of approximate wave functions and the efficiency of the detailed optimization of the nonlinear variational parameters in calculation of each individual energy level are demonstrated.     

On classical hydrodynamics and collective motion as approximation to the nuclear many-body problem

Using time-dependent unitary transformations, one can cast a one-body equation of the time-dependent Hartree-Fock type into a form which is closely related to equations for a classical irrotational fluid. The hydrodynamic equation of state finds its counterpart in a stationary constrained field equation. The hydrodynamic equations in turn can be translated into a classical Hamiltonian formalism with an infinite number of generalised coordinates, which are given as all possible spatial moments of the density. The reduction to a few ones, the “natural collective coordinates” is possible by the choice of appropriate initial conditions. The lowest of the hydrodynamical frequencies can be calculated in closed form by harmonic approximations. For the quadrupole frequency a value ofω=31 A^?1/3 MeV/h is obtained. As expected, the value does not agree with the experiment, but rather is in between the characteristic frequency for theβ-vibration and the isoscalar giant quadrupole vibration.     

Parabolic sturmians approach to the three-body continuum Coulomb problem

The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger-type equation, where the Green’s function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the (e ^?, e ^?, He⁺⁺) system has been solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution has been obtained as the basis set used to describe the potential is enlarged.     

A new parametrized quantum distribution function and its time development

We discuss a new general class of quantum distribution functions characterized by an arbitrary parameter b. The values b = -1, 0, 1 correspond to the anti-standard (Kirkwood), the Wigner, and the standard distribution functions, respectively. An analytic form of the equation of motion is derived. We conclude that, for time-dependent problems involving a potential which is a function of coordinates only, the Wigner distribution function is the optimum one to use, from a simplicity standpoint.     

Thermal relaxation of systems with quadratic heat bath coupling: I. Classical systems

We consider the evolution of systems whose coupling to the heat bath is quadratic in the bath coordinates. Performing an explicit elimination of the bath variables we arrive at an equation of evolution for the system variables alone. In the weak coupling limit we show that the equation is of the generalized Langevin form, with fluctuations that are Gaussian and that obey a fluctuation-dissipation relation. If the system-bath coupling is linear in the system coordinates the resulting fluctuations are additive and the dissipation is linear. If the coupling is nonlinear in the system coordinates, the resulting fluctuations are multiplicative and the dissipation is nonlinear.     

Many-body atomic potentials in Thomas-Fermi theory

Many-body atomic potentials, ?, are functions of the nuclear coordinates, and are defined by differences of ground state energies, E, e.g., ?(1, 2) ≡ E(1, 2) ? E(1) ? E(2). We prove that in Thomas-Fermi theory the n-body potential always has the sign (?1)ⁿ for all coordinates. We also prove that the remainder in the expansion of the total energy E in terms of the ?'s, when truncated at the n-body terms, has the sign (?1)ⁿ⁺¹.     

Form factors,threshold relations,and large angle elastic scattering of pions

The form factor F(q²) of the pion in a simple core model is investigated, together with the deep inelastic electroproduction structure function, F₂(ω). A relation between these two, analagous to the Drell-Yan-West relation for nucleons, is derived and it is found that F₂ is related to either the form factor or the square of the form factor depending on how rapidly F(q²) ultimately falls with momentum transfer. The unitarity equation and its implications for this kind of threshold relation are discussed. The simple core model is also applied to elastic large angle ππ scattering.     

On the connection between the macroscopical and microscopical evolution in an exactly soluble hopping model: I. Neutral particles

The hopping rate equation for neutral particles, on an arbitrary periodical lattice, can be solved exactly. It is shown that if one scales the time t and the distances x(t→λ²t, x→λx) then, in the λ→∞ limit, the particle density tends to the solution of the diffusion equation faster than λ^?3. The diffusion coefficient is the same as obtained from both Kubo and Miller-Abrahams theory via the Einstein relation.     

On multivortex solutions of the weierstrass representation

The connection between the complex sine-Gordon equation on the plane associated with a Weierstrass-type system and the possibility of constructing several classes of multivortex solutions is discussed in detail. It is shown that the amplitudes of these vortex solutions represented in polar coordinates satisfy the fifth Painlevé equation. We perform the analysis using the known relations for the Painlevé equations and construct explicit formulas in terms of the Umemura polynomials, which are τ functions for rational solutions to the third Painlevé equation. New classes of multivortex solutions to the Weierstrass system are obtained through the use of this proposed procedure.     

On the degeneracy of the 2H and 2P states from the atomic configuration d3 in the non-relativistic Hartree-Fock approximation

A new relation between the wave functions of the atomic states ²H and ²P arising from the configuration d³ and the wave functions of two molecular states arising from the configuration h³ in icosahedral symmetry molecules is presented. This relation gives an insight into the unknown origin of the non-relativistic Hartree-Fock degeneracy of the atomic states ²H and ²P.     

QCD predictions for the Q2 dependence of quark and gluon fragmentation functions using Altarelli-Parisi type equations

In the leading logarithmic approximation, the fragmentation functions of quarks and gluons are investigated using Altarelli-Parisi type equations. Using a new method to make the Mellin transformation, the equation is solved. Analytic expressions for the fragmentation functions near z = 0 and z = 1 are also given. Finally, numerical results for the fragmentation functions D_q^π, D_q^K are presented for different Q².