Scattering of Two Spin-Half Particles in a Three-Dimensional Approach

Scattering of two spin- ${\frac{1}{2}}$ particles is formulated in a three-dimensional approach based on a simple three-dimensional momentum-spin basis. Both cases of identical and nonidentical particles are considered. The azimuthal behaviour of the potential and of the T-matrix elements leads to a set of integral equations for the T-matrix elements in two variables, i.e. the momentum magnitude and the scattering angle. Observables can be directly calculated from these T-matrix elements. Some symmetry relations for the T-matrix elements reduce the number of equations to be solved.     

Self-Dual Noncommutative {\phi^4} -Theory in Four Dimensions is a Non-Perturbatively Solvable and Non-Trivial Quantum Field Theory

We study quartic matrix models with partition function \({\mathcal{Z}[E, J] = \int dM}\) exp(trace \({(JM - EM^{2} - \frac{\lambda}{4} M^4)}\) ). The integral is over the space of Hermitean \({\mathcal{N} \times \mathcal{N}}\) -matrices, the external matrix E encodes the dynamics, \({\lambda > 0}\) is a scalar coupling constant and the matrix J is used to generate correlation functions. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula which gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation which must be solved case by case for given E. These results imply that if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing β-function. As the main application we prove that Euclidean \({\phi^4}\) -quantum field theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the infinite volume limit, is exactly solvable and non-trivial. This model is a quartic matrix model, where E has for \({\mathcal{N} \to \infty}\) the same spectrum as the Laplace operator in four dimensions. Using the theory of singular integral equations of Carleman type we compute (for \({\mathcal{N} \to \infty}\) and after renormalisation of \({E, \lambda}\) ) the free energy density (1/volume) log \({(\mathcal{Z}[E, J]/\mathcal{Z}[E, 0])}\) exactly in terms of the solution of a non-linear integral equation. Existence of a solution is proved via the Schauder fixed point theorem. The derivation of the non-linear integral equation relies on an assumption which in subsequent work is verified for coupling constants \({\lambda \leq 0}\) .     

Suppressed Dispersion for a Randomly Kicked Quantum Particle in a Dirac Comb

I study a model for a massive one-dimensional particle in a singular periodic potential that is receiving kicks from a gas. The model is described by a Lindblad equation in which the Hamiltonian is a Schrödinger operator with a periodic δ-potential and the noise has a frictionless form arising in a Brownian limit. I prove that an emergent Markov process in an adiabatic limit governs the momentum distribution in the extended-zone scheme. The main result is a central limit theorem for a time integral of the momentum process, which is closely related to the particle’s position. When normalized by $t^{\frac{5}{4}}$ the integral process converges to a time-changed Brownian motion whose diffusion rate depends on the momentum process. The scaling $t^{\frac{5}{4}}$ contrasts with $t^{\frac{3}{2}}$ , which would be expected for the case of a smooth periodic potential or for a comparable classical process. The difference is a wave effect driven by momentum reflections that occur when the particle’s momentum is kicked near the half-spaced reciprocal lattice of the potential.     

The structure of para-hydrogen clusters

The path integral Monte Carlo calculated radial distributions of para-hydrogen clusters $({\rm p}\text{-}{\rm H}_2)_N$ consisting of N = 4-40 molecules interacting via a Lennard-Jones potential at $T=1.5~{\rm K}$ show evidence for additional peaks compared to radial distributions calculated by diffusion Monte Carlo ( $T=0~{\rm K}$ ) and path integral Monte Carlo at $T \leq 0.5~{\rm K}$ . The difference in structures is attributed to quantum delocalization at the lowest temperature. The new structures at finite temperatures appear to be consistent with classical structures calculated for an effective Morse potential, which in order to account for the large zero point energy, is substantially softer than the Lennard-Jones potential.     

Spin-1 particle in a homogeneous magnetic field

The Corben-Schwinger theory gives imaginary values of the energy, forS ₃ ² =1 states, in very intensive magnetic fields. The theory proposed by the author, which is most satisfactory in the nonrelativistic approximation, does not have this defect forS ₃ ² =1 states, but it appears forS ₃ ² =0 states.     

Topological Spin Excitations Induced by an External Magnetic Field Coupled to a Surface with Rotational Symmetry

We study the Heisenberg model in an external magnetic field on curved surfaces with rotational symmetry. The Euler–Lagrange static equations, derived from the Hamiltonian, lead to the inhomogeneous double sine-Gordon equation. Nonetheless, if the magnetic field is coupled to the metric elements of the surface, and consequently to its curvature, the homogeneous double sine-Gordon equation emerges and a $2\pi $ -soliton solution is obtained. In order to satisfy the self-dual equations, surface deformations are predicted to appear at the sector where the spin direction is opposite to the magnetic field. On the basis of the model, we find the characteristic length of the $2\pi $ -soliton for three specific rotationally symmetric surfaces: the cylinder, the catenoid, and the hyperboloid. On finite surfaces, such as the sphere, torus, and barrels, fractional $2\pi $ -solitons are predicted to appear.     

Electroweak radiative corrections to $$W^+W^-\gamma $$ production at the ILC

We consider holographic superconductors in a rotating black string spacetime. In view of the mandatory introduction of the \(A_\varphi \) component of the vector potential we are left with three equations to be solved. Their solutions show that the rotation parameter \(a\) influences the critical temperature \(T_\mathrm{c}\) and the conductivity \(\sigma \) in a simple but non-trivial way.     

Fractional Klein–Gordon Equations and Related Stochastic Processes

This paper presents finite-velocity random motions driven by fractional Klein–Gordon equations of order $\alpha \in (0,1]$ . A key tool in the analysis is played by the McBride’s theory which converts fractional hyper-Bessel operators into Erdélyi–Kober integral operators. Special attention is payed to the fractional telegraph process whose space-dependent distribution solves a non-homogeneous fractional Klein–Gordon equation. The distribution of the fractional telegraph process for $\alpha = 1$ coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Klein–Gordon equation. Fractional planar random motions at finite velocity are also investigated, the corresponding distributions obtained as well as the explicit form of the governing equations. Fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations $B(n,\alpha )$ (with uniformly-distributed orientation) are considered. The parameter $n$ of $B(n,\alpha )$ is itself a random variable with fractional Poisson distribution, so that fractionality acts as a subsampling of the changes of direction. Finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained. Extensions to $N$ -dimensional fractional random flights are envisaged as well as the fractional counterpart of the Euler–Poisson–Darboux equation to which our theory applies.     

Simplicial Ricci Flow

We construct a discrete form of Hamilton’s Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, ${{\mathcal S}}$ . These new algebraic equations are derived using the discrete formulation of Einstein’s theory of general relativity known as Regge calculus. A Regge–Ricci flow (RRF) equation can be associated to each edge, ?, of a simplicial lattice. In defining this equation, we find it convenient to utilize both the simplicial lattice ${{\mathcal S}}$ and its circumcentric dual lattice, ${{\mathcal S}^*}$ . In particular, the RRF equation associated to ? is naturally defined on a d-dimensional hybrid block connecting ? with its (d?1)-dimensional circumcentric dual cell, ? ^*. We show that this equation is expressed as the proportionality between (1) the simplicial Ricci tensor, Rc _? , associated with the edge ${\ell\in{\mathcal S}}$ , and (2) a certain volume weighted average of the fractional rate of change of the edges, ${\lambda\in \ell^*}$ , of the circumcentric dual lattice, ${{\mathcal S}^*}$ , that are in the dual of ?. The inherent orthogonality between elements of ${\mathcal S}$ and their duals in ${{\mathcal S}^*}$ provide a simple geometric representation of Hamilton’s RF equations. In this paper we utilize the well established theories of Regge calculus, or equivalently discrete exterior calculus, to construct these equations. We solve these equations for a few illustrative examples.     

Equivalence of Blocks for the General Linear Lie Superalgebra

We develop a reduction procedure which provides an equivalence (as highest weight categories) from an arbitrary block (defined in terms of the central character and the integral Weyl group) of the BGG category ${\mathcal{O}}$ for a general linear Lie superalgebra to an integral block of ${\mathcal{O}}$ for (possibly a direct sum of) general linear Lie superalgebras. We also establish indecomposability of blocks of ${\mathcal{O}}$ .     

Classical {\mathcal{W}}-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

We derive explicit formulas for λ-brackets of the affine classical \({\mathcal{W}}\) -algebras attached to the minimal and short nilpotent elements of any simple Lie algebra \({\mathfrak{g}}\) . This is used to compute explicitly the first non-trivial PDE of the corresponding integrable generalized Drinfeld–Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov’s equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h ˇ?3 functions, where h ˇ is the dual Coxeter number of \(\mathfrak{g}\) . In the case when \(\mathfrak{g}\) is \({\mathfrak{sl}_2}\) both these equations coincide with the KdV equation. In the case when \(\mathfrak{g}\) is not of type \({C_n}\) , we associate to the minimal nilpotent element of \(\mathfrak{g}\) yet another generalized Drinfeld–Sokolov hierarchy.     

Heterogeneous Memorized Continuous Time Random Walks in an External Force Fields

In this paper, we study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated memorized waiting times, which involves Reimann–Liouville fractional derivative or Reimann–Liouville fractional integral. We show that the mean squared displacement of the test particle \(X_{x}\) which is dependent on its location \(x\) of the form (El-Wakil and Zahran, Chaos Solitons Fractals, 12, 1929–1935, 2001) 1 $$\begin{aligned} \langle \mathbb {X}_x^2\rangle (t)=\langle (\Delta X_x(t))^2\rangle _0\sim |x|^{-\theta }t^{\gamma }, \quad 0<\gamma <1, \quad \theta =d_w-2, \end{aligned}$$ where \(d_w>2\) is the anomalous exponent, the diffusion exponent \(\gamma \) is dependent on the model parameters. We obtain the Fokker–Planck-type dynamic equations, and their stationary solutions are of the Boltzmann–Gibbs form. These processes obey a generalized Einstein–Stokes–Smoluchowski relation and the second Einstein relation. We observe that the asymptotic behavior of waiting times and subordinations are of stretched Gaussian distributions. We also discuss the time averaged in the case of an harmonic potential, and show that the process exhibits aging and ergodicity breaking.     

Meson spectra and mass splitting of \eta_{c}^{} - J/ \psi calculated with a regularized quark potential

The complete Breit potential contains the terms of spin-spin, spin-orbit, orbit-orbit, and tensor force interactions which become singular at short distance. Most of previous calculations of the non-relativistic potential quark model considered only the spin-spin interaction and substituted the $ \delta$ (r) -function by the Gaussian or Yukawa potential in coordinate space. Recently, a method to regularize the Breit potential consists of subtracting terms that cancel the singularity at the origin but leave the intermediate- and long-distance behavior unchanged. Motivated by this work we regularize the Breit potential by multiplying the singular terms in momentum space identically by the form factor [ $ \mu^{2}_{}$ /(q ² + $ \mu^{2}_{}$ )]² of the momentum transfer q , where the screened mass μ increases with the reduced mass of the meson. With the regularized Breit potential we calculate the masses of 30 common mesons and the new $ \eta_{b}^{}$ meson. We find that the calculated masses from light to heavy mesons agree well with experimental data. The inclusion of such a dependence of the reduced mass in the potential regularization improves the spin-spin splittings of $ \eta_{c}^{}$ -J/ $ \psi$ and $ \eta_{b}^{}$ - $ \Upsilon$ (1S) . The spin-orbit and tensor force interactions in the Breit potential lead to the splittings of $ \chi_{{c0}}^{}$ , $ \chi_{{c1}}^{}$ , and $ \chi_{{c2}}^{}$ .     

Universality of the Local Regime for the Block Band Matrices with a Finite Number of Blocks

We consider the block band matrices, i.e. the Hermitian matrices $H_N$ , $N=|\Lambda |W$ with elements $H_{jk,\alpha \beta }$ , where $j,k \in \Lambda =[1,m]^d\cap \mathbb {Z}^d$ (they parameterize the lattice sites) and $\alpha , \beta = 1,\ldots , W$ (they parameterize the orbitals on each site). The entries $H_{jk,\alpha \beta }$ are random Gaussian variables with mean zero such that $\langle H_{j_1k_1,\alpha _1\beta _1}H_{j_2k_2,\alpha _2\beta _2}\rangle =\delta _{j_1k_2}\delta _{j_2k_1} \delta _{\alpha _1\beta _2}\delta _{\beta _1\alpha _2} J_{j_1k_1},$ where $J=1/W+\alpha \Delta /W$ , $\alpha < 1/4d$ . This matrices are the special case of Wegner’s $W$ -orbital models. Assuming that the number of sites $|\Lambda |$ is finite, we prove universality of the local eigenvalue statistics of $H_N$ for the energies $|\lambda _0|< \sqrt{2}$ .     

A Strong Central Limit Theorem for a Class of Random Surfaces

This paper is concerned with d = 2 dimensional lattice field models with action ${V(\nabla\phi(\cdot))}$ , where ${V : \mathbf{R}^d \rightarrow \mathbf{R}}$ is a uniformly convex function. The fluctuations of the variable ${\phi(0) - \phi(x)}$ are studied for large |x| via the generating function given by ${g(x, \mu) = \ln \langle e^{\mu(\phi(0) - \phi(x))}\rangle_{A}}$ . In two dimensions ${g'' (x, \mu) = \partial^2g(x, \mu)/\partial\mu^2}$ is proportional to ${\ln\vert x\vert}$ . The main result of this paper is a bound on ${g''' (x, \mu) = \partial^3 g(x, \mu)/\partial \mu^3}$ which is uniform in ${\vert x \vert}$ for a class of convex V. The proof uses integration by parts following Helffer–Sjöstrand and Witten, and relies on estimates of singular integral operators on weighted Hilbert spaces.     

Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order $k\geqslant 1$ . It is known that the ‘splitting Gibbs measures’ of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geqslant 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.     

Feynman's path integral

Feynman's integral is defined with respect to a pseudomeasure on the space of paths: for instance, letC be the space of pathsq:T?? → configuration space of the system, letC be the topological dual ofC; then Feynman's integral for a particle of massm in a potentialV can be written where $$S_{\operatorname{int} } (q) = \mathop \smallint \limits_T V(q(t)) dt$$ and wheredw is a pseudomeasure whose Fourier transform is defined by for μ∈C′. Pseudomeasures are discussed; several integrals with respect to pseudomeasures are computed.     

Two-dimensional exactly and completely integrable dynamical systems

An investigation of two-dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary Lie algebra \(\mathfrak{g}\) whose grading is consistent with an arbitrary integral embedding of 3d-subalgebra in \(\mathfrak{g}\) has been carried out. We have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a Goursat problem. A method for the construction of a wide class of infinite-dimensional Lie algebras of finite growth has been proposed.     

Toward a Mathematical Holographic Principle

In work started in [17] and continued in this paper our objective is to study selectors of multivalued functions which have interesting dynamical properties, such as possessing absolutely continuous invariant measures. We specify the graph of a multivalued function by means of lower and upper boundary maps \(\tau _{1}\) and \(\tau _{2}.\) On these boundary maps we define a position dependent random map \(R_{p}=\{\tau _{1},\tau _{2};p,1-p\},\) which, at each time step, moves the point \(x\) to \(\tau _{1}(x)\) with probability \(p(x)\) and to \(\tau _{2}(x)\) with probability \(1-p(x)\) . Under general conditions, for each choice of \(p\) , \(R_{p}\) possesses an absolutely continuous invariant measure with invariant density \(f_{p}.\) Let \(\varvec{\tau }\) be a selector which has invariant density function \(f.\) One of our objectives is to study conditions under which \(p(x)\) exists such that \(R_{p}\) has \(f\) as its invariant density function. When this is the case, the long term statistical dynamical behavior of a selector can be represented by the long term statistical behavior of a random map on the boundaries of \(G.\) We refer to such a result as a mathematical holographic principle. We present examples and study the relationship between the invariant densities attainable by classes of selectors and the random maps based on the boundaries and show that, under certain conditions, the extreme points of the invariant densities for selectors are achieved by bang-bang random maps, that is, random maps for which \(p(x)\in \{0,1\}.\)      

Frame-dragging fields and spin 1 gravitomagnetic radiation

Experimental results published in 2004 (Ciufolini and Pavlis in Nature 431:958–960, 2004) and 2011 (Everitt et al. in Phys Rev Lett 106:221101, 1–5, 2011) have confirmed the frame-dragging phenomenon for a spinning earth predicted by Einstein’s field equations. Since this is observed as a precession caused by the gravitomagnetic (GM) field of the rotating body, these experiments may be viewed as measurements of a GM field. The effect is encapsulated in the classic steady state solution for the vector potential field $\zeta $ of a spinning sphere–a solution applying to a sphere with angular momentum J and describing a field filling space for all time (Weinberg in Gravitation and Cosmology, Wiley, New York, 1972). In a laboratory setting one may visualise the case of a sphere at rest $(\zeta =0, \text{ t}<0)$ , being spun up by an external torque at $\text{ t}=0$ to the angular momentum J: the $\zeta $ field of the textbook solution cannot establish itself instantaneously over all space at $\text{ t}=0$ , but must propagate with the velocity c, implying the existence of a travelling GM wave field yielding the textbook $\zeta $ field for large enough t (Tolstoy in Int J Theor Phys 40(5):1021–1031, 2001). The linearized GM field equations of the post-Newtonian approximation being isomorphic with Maxwell’s equations (Braginsky et al. in Phys Rev D 15(6):2047–2060, 1977), such GM waves are dipole waves of spin 1. It is well known that in purely gravitating systems conservation of angular momentum forbids the existence of dipole radiation (Misner et al. in Gravitation, Freeman & Co., New York, 1997); but this rule does not prohibit the insertion of angular momentum into the system from an external source–e.g., by applying a torque to our laboratory sphere.