A “No-Go” theorem for the existence of a discrete action principle

In this paper, we study the problem of the existence of a least-action principle for invertible, second-order dynamical systems, discrete in time and space. We show that, when the configuration space is finite and arbitrary state transitions are allowed, a least-action principle does not exist for such systems. We dichotomize discrete dynamical systems with infinite configuration spaces into those of finite type for which this theorem continues to hold, and those not of finite type for which it is possible to construct a least-action principle. We also show how to recover an action, by restriction of the phase space of certain second-order discrete dynamical systems. We provide numerous examples to illustrate each of these results.     

On finite action solutions of the nonlinear σ-model

The instanton and anti-instanton solutions of the two-dimensional O(3) σ-model are special examples of harmonic maps, which have been studied extensively in the mathematical literature. We give an elementary and self-contained proof that these solutions are the only continuous maps for which the action is finite and stationary under variations, without assuming any additional boundary conditions at infinity. An element of the proof is the vanishing of the stress tensor for a finite action solution, which actually holds true for the general O(N) σ-model. For the two-dimensional O(2l + 1) σ-model we exhibit explicit finite action solutions that do not lie in any lower dimensional sphere; the existence of such solutions has been pointed out in the mathematical literature. We also present a rigorous proof, based on Derrick's scaling argument, that there are no nonconstant finite action solutions in more than two dimensions.     

<Emphasis Type="Italic">D</Emphasis> instanton on the linear-dilaton background

It is shown that, in addition to a standard asymptotically flat D instanton, IIB superstring theory contains an instanton-type excitation on the linear-dilaton background. The new solution is asymptotically flat, but the dilaton grows linearly at infinity. The total action of this configuration diverges, but the instanton action proper becomes finite upon subtracting an infinite background contribution.     

Exact analysis of Coulomb gauge vacuum degeneracies in (2 + 1) dimensions

The solutions of the two-dimensional euclidean σ-model provide an infinite number of pure gauge field configurations satisfying the Coulomb gauge condition, in (2 + 1) dimensions. For vacuum gauge fields associated with finite action instanton solutions of the σ-model, we find that the winding number n configuration leads to n negative eigenvalues for the ghost operator, up to a finite calculable degeneracy.     

General transformation theory of Lagrangian mechanics and the Lagrange group

The general transformation theory of Lagrangian mechanics is revisited from a group-theoretic point of view. After considering the transformation of the Lagrangian function under local coordinate transformations in configuration spacetime, the general covariance of the formalism of Lagrange is discussed. Next, the group of Lagrange (for alln-dimensional Lagrangian systems) is introduced, and some important features of this group, as well as of its action on the set of Lagrangians, are briefly examined. Only finite local transformations of coordinates are considered here, and no variational transformation of the action is required in this study. Some miscellaneous examples of the formalism are included.     

Complexification of gauge theories

For the case of a first-class constrained system with equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations G is equivalent to the single process of dividing out the initial phase space by the complexification G_C of G. For the particular case of a phase space action that is the lift of a configuration space action, conditions are found under which, in finite dimensions, the physical phase space of a gauge system with first-class constraints is diffeomorphic to a manifold imbedded in the physical configuration space of the complexified gauge system. Similar conditions are shown to hold for the infinite-dimensional example of Yang-Mills theories. As a physical application we discuss the adequateness of using holomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory.     

Thermodynamics of the hot BIon

We investigate the thermodynamics of the recently obtained finite temperature BIon solution of [G. Grignani, T. Harmark, A. Marini, N.A. Obers, M. Orselli, Heating up the BIon, arXiv:1012.1494 [hep-th]], focusing on two aspects. The first concerns comparison of the free energy of the three available phases for the finite-temperature brane-antibrane-wormhole configuration. Based on this we propose a heuristic picture for the dynamics of the phases that involves a critical temperature below which a stable phase exists. This stable phase is the finite temperature analogue of the thin throat branch of the extremal brane-antibrane-wormhole configuration. The second aspect that we consider is the possibility of constructing a finite temperature generalization of the infinite spike configuration of the extremal BIon. To this end we identify a correspondence point at the end of the throat where the thermodynamics of the D3-F1 blackfold configuration can be matched to that of k non-extremal black fundamental strings.     

Topological properties of superconducting junctions

Motivated by recent developments in the field of one-dimensional topological superconductors, we investigate the topological properties of s-matrix of generic superconducting junctions where dimension should not play any role. We argue that for a finite junction the s-matrix is always topologically trivial. We resolve an apparent contradiction with the previous results by taking into account the low-energy resonant poles of s-matrix. Thus no common topological transition occurs in a finite junction. We reveal a transition of a different kind that concerns the configuration of the resonant poles.     

A phase cell approach to Yang-Mills theory

For the abelian Yang-Mills theory, a one-to-one correspondence is established between continuum gauge potentials and compatible lattice configurations on an infinite sequence of finer and finer lattices. The compatibility is given by a block spin transformation determining the configuration on a lattice in terms of the configuration on any finer lattice. Thus the configuration on any single lattice is not an approximation to the continuum field, but rather a subset of the variables describing the field.It is proven that the Wilson actions on the lattices monotonically increase to the continuum action as one passes to finer and finer lattices. Configurations that minimize the continuum action, subject to having the variables fixed on some lattice, are studied.This work was supported in part by the National Science Foundation under Grant No. PHY-85-02074     

Ground States of the 2D Sticky Disc Model: Fine Properties and N 3/4 Law for the Deviation from the Asymptotic Wulff Shape

We investigate ground state configurations for a general finite number N of particles of the Heitmann-Radin sticky disc pair potential model in two dimensions. Exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic Wulff shape which is a regular hexagon: There are arbitrarily large N with ground state configurations deviating from the nearest regular hexagon by a number of ～N ^3/4 particles. We also prove that for any N and any ground state configuration this deviation is bounded above by ～N ^3/4. As a consequence we obtain an exact scaling law for the fluctuations about the asymptotic Wulff shape. In particular, our results give a sharp rate of convergence to the limiting Wulff shape.     

Line vortices in a three-dimensional ordered Josephson medium

Two equilibrium configurations of a line vortex in a three-dimensional ordered Josephson medium are considered: (i) the vortex core is at the center of a cell and (ii) the vortex core is on a contact. Infinite systems of equations describing these configurations are derived. In going to a finite system, the currents far away from the center are neglected. A new technique for solving the finite system of equations is suggested. It does not require smallness of phase discontinuities at all vortex cells and, therefore, can be applied for any values of pinning parameter I down to zero. The structures and energies of both equilibrium states for isolated line vortices are calculated for any I from the range considered. For I >0.3, a vortex can be thought of as fitting a square of 5×5 cells. For lower I, the vortex energy can be expressed as a sum of the energies of the small discrete core and the quasi-continuous outside. The core energy is comparable to the energy of the outside and is a major contributor to the vortex energy when I is not too small. For any I, the energy of the vortex centered on the contact is higher than the energy of the configuration centered at the center of the cell.     

Zero temperature dynamics of Ising model on a densely connected small world network

The zero temperature quenching dynamics of the ferromagnetic Ising model on a densely connected small world network is studied where long range bonds are added randomly with a finite probability p. We find that in contrast to the sparsely connected networks and random graph, there is no freezing and an initial random configuration of the spins reaches the equilibrium configuration within a very few Monte Carlo time steps in the thermodynamic limit for any p ≠0. The residual energy and the number of spins flipped at any time shows an exponential relaxation to equilibrium. The persistence probability is also studied and it shows a saturation within a few time steps, the saturation value being 0.5 in the thermodynamic limit. These results are explained in the light of the topological properties of the network which is highly clustered and has a novel small world behaviour.     

Optimal lattice domain-wall fermions

I show that the conventional formulations of lattice domain-wall fermion with any finite N(s) (in the fifth dimension) do not preserve the chiral symmetry optimally and propose a new action which preserves the chiral symmetry optimally for any finite N(s).     

Lenses as an atom-photon interface: A semiclassical model

Strong interaction between the light field and an atom is often achieved with cavities. Recent experiments have used a different configuration: a propagating light field is strongly focused using a system of lenses, the atom being supposed to sit at the focal position. In reality, this last condition holds only up to some approximation; in particular, at any finite temperature, the atom position fluctuates. We present a formalism that describes the focalized field and the atom sitting at an arbitrary position. As a first application, we show that thermal fluctuations do account for the extinction data reported in M. K. Tey et al., Nature Physics 4, 924 (2008).     

Noncolliding Squared Bessel Processes

We consider a particle system of the squared Bessel processes with index ν>−1 conditioned never to collide with each other, in which if −1<ν<0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function J _ν is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.     

Radiative decay of mesons and anomaly in 2+1 dimensions

In 2+1 space-time, we study the radiative decay of mesons with Yukawa couplings to two and four component fermions. In a four component theory, the electromagnetic current is free of parity odd radiative corrections; however an axialU(1) current is shown to develop a finite vacuum expectation value. Unlike even dimensions this anomaly is temperature dependent and does not play any role in the radiative decay. We calculate the effective action for the mesons and show that a finite kinetic term for these fields is induced at the one-loop level and vanishes as the temperature goes to infinity.     

New non-Abelian effects on D-branes

We extend the Myers dielectric effect to configurations with angular momentum. The resulting time-dependent N D0 brane bound states can be interpreted as describing rotating fuzzy ellipsoids. A similar solution exists also in the presence of a RR magnetic field, that we study in detail. We show that, for any finite N, above a certain critical angular momentum is energetically more favorable for the bound state system to dissociate into an Abelian configuration of N D0 branes moving independently. We have investigated this problem in the low-energy expansion of the non-Abelian D brane action for generic N. In the case N=2 we find explicit solutions of the full non-Abelian Born–Infeld D brane dynamics, which remarkably have the same structure and confirm the features of the low-energy approximation. We further study D string configurations representing fuzzy funnels deformed by the magnetic field and by the rotational motion.