arXiv subjects

Amir H. Fatollahi

Publications and source records attributed to Amir H. Fatollahi.

At least 19 recordsLinked to original sources

arXiv2022

Weak Coupling Limit of U(1) Lattice Model in Fourier Basis

The transfer-matrix of the U(1) lattice model is considered in the Fourier basis and in the weak coupling limit. The issues of Gauss law constraint and gauge invariant states are addressed in the Fourier basis. In particular, it is shown that in the strong coupling limit the gauge invariant Fourier states are effectively the finite size closed loop currents. In the weak coupling limit, however, the link-currents along periodic or infinite spatial directions find comparable roles as gauge invariant states. The subtleties related to the extreme weak coupling of the transfer-matrix in the Fourier basis are discussed. A careful analysis of the zero eigenvalues of the matrix in the quadratic action leads to a safe extraction of the diverging group volume in the limit $g\to 0$. By means of the very basic notions and tools of the lattice model, the spectrum at the weak coupling limit for any dimension and size of lattice is obtained analytically. The spectrum at the weak coupling limit corresponds to the expected one by the continuum model in the large lattice limit.

hep-lat↗

arXiv2021

Diagrammatic Strong Coupling Expansion of U(1) Lattice Model in Fourier Basis

The transfer-matrix of U(1) lattice gauge theory is investigated in the field Fourier space, the basis of which consists of the quantized currents on lattice links. Based on a lattice version of the current conservation, the transfer-matrix elements are shown to be non-zero only between current-states that differ in circulating currents inside plaquettes. In the strong coupling limit, a series expansion is developed for the elements of the transfer-matrix, to which a diagrammatic representation based on the occurrence of virtual link and loop currents can be associated. With $g$ as the coupling, the weight of each virtual current in the expansion is $1/g^2$, by which at any given order the relevant diagrams are determined. Either by interpretation or through their role in fixing the relevant terms, the diagrams are reminiscent of the Feynman ones of the perturbative small coupling expansions.

hep-lat↗

arXiv2021

A Lattice Inspired Model for Monopole Dynamics

The site-reduction of U(1) lattice gauge theory along the spatial directions is used to model the monopole dynamics. The reduced theory is that of the angle-valued coordinates on the discrete worldline. Below the critical coupling $g_{c}=1.125$ and temperature $T_c=0.335$ the model exhibits a first order phase transition. It is argued that the phase structure matches with the proposed role for magnetic monopoles in the confinement mechanism based on the dual Meissner effect.

hep-th↗

arXiv2020

Identities for Droplets with Circular Footprint on Tilted Surfaces

Exact mathematical identities are presented between the relevant parameters of droplets displaying circular contact boundary based on flat tilted surfaces. Two of the identities are derived from the force balance, and one from the torque balance. The tilt surfaces cover the full range of inclinations for sessile or pendant drops, including the intermediate case of droplets on a wall (vertical surface). The identities are put under test both by the available solutions of a linear response approximation at small Bond numbers as well as the ones obtained from numerical solutions, making use of the Surface Evolver software. The subtleties to obtain certain angle-averages appearing in identities by the numerical solutions are discussed in detail. It is argued how the identities are useful in two respects. First is to replace some unknown values in the Young-Laplace equation by their expressions obtained from the identities. Second is to use the identities to estimate the error for approximate analytical or numerical solutions without any reference to an exact solution.

physics.flu-dyn↗

arXiv2020

Lost in Normalization

The consequences of the gauge-coupling dependent normalization-factor of $1/g^α$ in the transfer-matrix of 2d U(1) lattice gauge theory are explored. It is seen by the $α=1$ choice that the lowest energy develops a minimum at coupling $g_*=1.125$, leading to a \textit{multi-valued} Gibbs energy similar to the systems with the first-order phase transition. It is argued how the $1/g$ normalization may be regarded as a lost normalization in the commonly used change of variable to the dimensionless angle-variables. Based on the continuum limit at the next-leading order and the Ostrogradsky formulation of higher-order time-derivatives theories, it is argued that the spectrum at continuum is compatible only with the $α=1$ choice.

hep-lat↗

arXiv2019

First-Order Phase Transition by XY Model of Particle Dynamics

A gas-liquid type of phase transition is found based on the particle dynamics on radius-$R$ circle in which the coordinate appears as the angle-variable of 1D XY-model. Due to the specific appearance of compact-space radius (volume) in the present interpretation of XY-model, the ground-state develops a minimum at some critical radius, leading to the multi-valued Gibbs energy similar to systems with first-order phase transition.

cond-mat.stat-mech↗

arXiv2019

On U(1) Gauge Theory Transfer-Matrix in Fourier Basis

The properties of the transfer-matrix of U(1) lattice gauge theory in the Fourier basis are explored. Among other statements it is shown: 1) the transfer-matrix is block-diagonal, 2) all consisting vectors of a block are known based on an arbitrary block vector, 3) the ground-state belongs to the zero-mode's block. The emergence of maximum-points in matrix-elements as functions of the gauge coupling is clarified. Based on explicit expressions for the matrix-elements we present numerical results as tests of our statements.

hep-lat↗

arXiv2017

Worldline as a Spin Chain

The general theoretical ground for the models based on the compact angle coordinates is presented. It is observed that the proper dependence on compact coordinates has to be through the group elements and is achieved most naturally in a discrete-time formulation of the theory. By the construction, the discrete worldline inlaid by compact coordinates resembles the spin chains of magnetic systems. As examples, the models based on the groups U(1), $\mathbb{Z}_N$ and SU(2) are explicitly constructed and their exact energy spectra are obtained. As the consequence of minima in the spectra, the models exhibit a phase transition of first-order. The dynamics by U(1) group is attempted to be fitted to the proposed role for monopoles in the dual Meissner effect of confinement mechanism.

hep-th↗

arXiv2016

Phase Transition by 0-Branes of U(1) Lattice Gauge Theory

The site reduction of U(1) lattice gauge theory is used to model the 0-branes in the dual theory. The reduced theory is the 1D plane-rotator model of the angle-valued coordinates on discrete world-line. The energy spectrum is obtained exactly via the transfer-matrix method, with a minimum in the lowest energy as a direct consequence of compact nature of coordinates. Below the critical coupling $g_c=1.125$ and temperature $T_c=0.335$ the system undergoes a first order phase transition between coexistent phases with lower and higher gauge couplings. The possible relation between the model and the proposed role for magnetic monopoles in confinement mechanism based on dual Meissner effect is pointed.

hep-th↗

arXiv2016

Regge Trajectories by 0-Brane Matrix Dynamics

The energy spectrum of two 0-branes for fixed angular momentum in 2+1 dimensions is calculated by the Rayleigh-Ritz method. The basis function used for each angular momentum consists of 80 eigenstates of the harmonic oscillator problem on the corresponding space. It is seen that the spectrum exhibits a definite linear Regge trajectory behavior. It is argued how this behavior supports the picture by which the bound-states of quarks and QCD-strings are governed by the quantum mechanics of matrix coordinates.

hep-th↗

arXiv2015

Coordinate/Field Duality in Gauge Theories: Emergence of Matrix Coordinates

The proposed coordinate/field duality [Phys. Rev. Lett. 78 (1997) 163] is applied to the gauge and matter sectors of gauge theories. In the non-Abelian case, due to indices originated from the internal space, the dual coordinates appear to be matrices. The dimensions and the transformations of the matrix coordinates of gauge and matter sectors are different and are consistent to expectations from lattice gauge theory and the theory of open strings equipped with the Chan-Paton factors. It is argued that in the unbroken symmetry phase, where only proper collections of field components as colorless states are detected, it is logical to assume that the same happens for the dual coordinates, making matrix coordinates the natural candidates to capture the internal dynamics of baryonic confined states. The proposed matrix coordinates happen to be the same appearing in the bound-state of D0-branes of string theory.

physics.gen-ph↗

arXiv2014

Closedness of orbits in a space with SU(2) Poisson structure

The closedness of orbits of central forces is addressed in a three dimensional space in which the Poisson bracket among the coordinates is that of the SU(2) Lie algebra. In particular it is shown that among problems with spherically symmetric potential energies, it is only the Kepler problem for which all of the bounded orbits are closed. In analogy with the case of the ordinary space, a conserved vector (apart from the angular momentum) is explicitly constructed, which is responsible for the orbits being closed. This is the analog of the Laplace-Runge-Lenz vector. The algebra of the constants of the motion is also worked out.

physics.class-ph↗

arXiv2014

The similarity of attractive and repulsive forces on a lattice

On a lattice, as the momentum space is compact, the kinetic energy is bounded not only from below but also from above. It is shown that this, somehow removes the distinction between repulsive and attractive forces. In particular, it is seen that a region with attractive force would appear forbidden for states with energies higher than a certain value, while repulsive forces could develop bound-states. An explicit transformation is introduced which transforms the spectrum of a system corresponding to a repulsive force, to that of a similar system corresponding to an attractive force. Explicit numerical examples are presented for discrete energies of bound-states of a particle experiencing repulsive force by a piecewise constant potential. Finally, the parameters of a specific one dimensional translationally invariant system on continuum are tuned so that the energy of the system resembles the kinetic energy of a system on a one dimensional lattice. In particular, it is shown that the parameters could be tuned so that while the width of the first energy band and its position are kept finite, the energy gap between the first energy band and the next energy band go to infinity, so that effectively only the first energy band is relevant.

quant-ph↗

arXiv2014

On the relation between the spin and the magnetic moment of the proton

In the context of the quark model of hadrons the spin and the magnetic moment of proton can not be taken proportional. This is in contradiction with the widely used relation between these two properties of the proton. This apparent difficulty is addressed by the most elementary notions of the relevant physics. In particular it is emphasized that the widely used relation is only valid in the lowest orders of perturbation, in which transitions between different baryons do not occur. For other processes where such transitions do occur, such as inelastic scattering off the protons, the quark model relation for the magnetic moment is used to give an estimation for the amplitude of transition between states with different total spins.

hep-ph↗

arXiv2014

New identities for sessile drops

A new set of mathematical identities is presented for axi-symmetric sessile drops on flat and curved substrates. The geometrical parameters, including the apex curvature and height, and the contact radius, are related by the identities. The validity of the identities are checked by various numerical solutions both for flat and curved substrates.

physics.flu-dyn↗

arXiv2014

Eigenvalue problem for radial potentials in space with SU(2) fuzziness

The eigenvalue problem for radial potentials is considered in a space whose spatial coordinates satisfy the SU(2) Lie algebra. As the consequence, the space has a lattice nature and the maximum value of momentum is bounded from above. The model shows interesting features due to the bound, namely, a repulsive potential can develop bound-states, or an attractive region may be forbidden for particles to propagate with higher energies. The exact radial eigen-functions in momentum space are given by means of the associated Chebyshev functions. For the radial stepwise potentials the exact energy condition and the eigen-functions are presented. For a general radial potential it is shown that the discrete energy spectrum can be obtained in desired accuracy by means of given forms of continued fractions.

physics.gen-ph↗

arXiv2013

Central force problem in space with SU(2) Poisson structure

The central force problem is considered in a three dimensional space in which the Poisson bracket among the spatial coordinates is the one by the SU(2) Lie algebra. It is shown that among attractive power-law potentials it is only the Kepler one that all of its bound-states make closed orbits. The analytic solution of the path equation under the Kepler potential is presented. It is shown that except the Kepler potential, in contrast to ordinary space, all of the potentials for which all of the almost circular orbits are closed are non-power-law ones. For the non-power-law potentials examples of the numerical solutions of the path equations are presented.

hep-th↗

arXiv2013

Entropy as a measure of diffusion

The time variation of entropy, as an alternative to the variance, is proposed as a measure of the diffusion rate. It is shown that for linear and time-translationally invariant systems having a large-time limit for the density, at large times the entropy tends exponentially to a constant. For systems with no stationary density, at large times the entropy is logarithmic with a coefficient specifying the speed of the diffusion. As an example, the large time behaviors of the entropy and the variance are compared for various types of fractional-derivative diffusions.

cond-mat.stat-mech↗