Turing jumps through provability

Fixing some computably enumerable theory $T$, the Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary arithmetic, each $\Sigma_1$ formula is equivalent to some formula of the form $\Box_T \varphi$ provided that $T$ is consistent. In this paper we give various generalizations of the FGH theorem. In particular, for $n>1$ we relate $\Sigma_{n}$ formulas to provability statements $[n]_T^{\sf True}\varphi$ which are a formalization of "provable in $T$ together with all true $\Sigma_{n+1}$ sentences". As a corollary we conclude that each $[n]_T^{\sf True}$ is $\Sigma_{n+1}$-complete. This observation yields us to consider a recursively defined hierarchy of provability predicates $[n+1]^\Box_T$ which look a lot like $[n+1]_T^{\sf True}$ except that where $[n+1]_T^{\sf True}$ calls upon the oracle of all true $\Sigma_{n+2}$ sentences, the $[n+1]^\Box_T$ recursively calls upon the oracle of all true sentences of the form $\langle n \rangle_T^\Box\phi$. As such we obtain a `syntax-light' characterization of $\Sigma_{n+1}$ definability whence of Turing jumps which is readily extended beyond the finite. Moreover, we observe that the corresponding provability predicates $[n+1]_T^\Box$ are well behaved in that together they provide a sound interpretation of the polymodal provability logic ${\sf GLP}_\omega$

Q-stars and charged q-stars

We present the formalism of q-stars with local or global U(1) symmetry. The equations we formulate are solved numerically and provide the main features of the soliton star. We study its behavior when the symmetry is local in contrast to the global case. A general result is that the soliton remains stable and does not decay into free particles and the electrostatic repulsion preserves it from gravitational collapse. We also investigate the case of a q-star with non-minimal energy-momentum tensor and find that the soliton is stable even in some cases of collapse when the coupling to gravity is absent.Comment: Latex, 19pg, 12 figures. Accepted in Phys. Rev.

Extended bound states and resonances of two fermions on a periodic lattice

The high-$T_c$ cuprates are possible candidates for d-wave superconductivity, with the Cooper pair wave function belonging to a non-trivial irreducible representation of the lattice point group. We argue that this d-wave symmetry is related to a special form of the fermionic kinetic energy and does not require any novel pairing mechanism. In this context, we present a detailed study of the bound states and resonances formed by two lattice fermions interacting via a non-retarded potential that is attractive for nearest neighbors but repulsive for other relative positions. In the case of strong binding, a pair formed by fermions on adjacent lattice sites can have a small effective mass, thereby implying a high condensation temperature. For a weakly bound state, a pair with non-trivial symmetry tends to be smaller in size than an s-wave pair. These and other findings are discussed in connection with the properties of high-$T_c$ cuprate superconductors.Comment: 21 pages, RevTeX, 4 Postscript figures, arithmetic errors corrected. An abbreviated version (no appendix) appeared in PRB on March 1, 199

Boson-fermion model beyond mean-field approximation

A model of hybridized bosons and fermions is studied beyond the mean field approximation. The divergent boson self-energy at zero temperature makes the Cooper pairing of fermions impossible.The frequency and momentum dependence of the self- energy and the condensation temperature $T_{c}$ of initially localized bosons are calculated analytically. The value of the boson condensation temperature $T_{c}$ is below $1K$ which rules out the boson-fermion model with the initially localized bosons as a phenomenological explanation of high-temperature superconductivity. The intra-cell density-density fermion-boson interaction dominates in the fermion self-energy. The model represents a normal metal with strongly damped bosonic excitations. The latter play the role of normal impurities.Comment: 16 pages, Latex, 5 figures available upon reques

Nuclear and Particle Physics applications of the Bohm Picture of Quantum Mechanics

Approximation methods for calculating individual particle/ field motions in spacetime at the quantum level of accuracy (a key feature of the Bohm Picture of Quantum Mechanics (BP)), are studied. Modern textbook presentations of Quantum Theory are used throughout, but only to provide the necessary, already existing, tested formalisms and calculational techniques. New coherent insights, reinterpretations of old solutions and results, and new (in principle testable) quantitative and qualitative predictions, can be obtained on the basis of the BP that complete the standard type of postdictions and predictions.Comment: 41 page

EYM equations in the presence of q-stars

We study Einstein-Yang-Mills equations in the presence of gravitating non-topological soliton field configurations, of q-ball type. We produce numerical solutions, stable with respect to gravitational collapse and to fission into free particles, and we study the effect of the field strength and the eigen-frequency to the soliton parameters. We also investigate the formation of such soliton stars when the spacetime is asymptotically anti de Sitter.Comment: 11 pages, to appear in Phys. Rev.

Q-stars in extra dimensions

We study q-stars with global and local U(1) symmetry in extra dimensions in asymptotically anti de Sitter or flat spacetime. The behavior of the mass, radius and particle number of the star is quite different in 3 dimensions, but in 5, 6, 8 and 11 dimensions is similar to the behavior in 4.Comment: 18 pages, to appear in Phys. Rev.

Comment on "Boson-fermion model beyond the mean-field approximation"

In a recent paper [A.S.Alexandrov, J.Phys.:Condens.Matter 8, 6923 (1996); cond-mat/9603111], it has been suggested that there is no Cooper pairing in boson-fermion models of superconductivity. We show that this conjecture is based on an inconsistent approximation that violates an exact identity. Quite generally, the divergence of the fermion t-matrix (the Thouless criterion) is accompanied by the condensation of a boson mode.Comment: LaTeX, 5 pages, 2style files included, 4 embedded EPS figures, submitted to J.Phys.:Condens.Matte

A New Approach to Solve the Low-lying States of the Schroedinger Equation

We review a new iterative procedure to solve the low-lying states of the Schroedinger equation, done in collaboration with Richard Friedberg. For the groundstate energy, the $n^{th}$ order iterative energy is bounded by a finite limit, independent of $n$; thereby it avoids some of the inherent difficulties faced by the usual perturbative series expansions. For a fairly large class of problems, this new procedure can be proved to give convergent iterative solutions. These convergent solutions include the long standing difficult problem of a quartic potential with either symmetric or asymmetric minima.Comment: 54 pages, 3 figures given separatel

The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary

We consider the dimer-monomer problem for the rectangular lattice. By mapping the problem into one of close-packed dimers on an extended lattice, we rederive the Tzeng-Wu solution for a single monomer on the boundary by evaluating a Pfaffian. We also clarify the mathematical content of the Tzeng-Wu solution by identifying it as the product of the nonzero eigenvalues of the Kasteleyn matrix.Comment: 4 Pages to appear in the Physical Review E (2006