Narrow Results

In this paper, we investigate the kinetic energy and angular momentum of free particles in a class of rotating cylindrical gravitational wave spacetimes, using the Noether symmetry approach. For that purpose, we determine the conserved quantities and subsequently analyze them for the velocities of free particles in the spacetimes, thereby simplifying the relationship between velocities and coordinates. Our analysis demonstrates that arbitrary choices of metric coefficients lead to fluctuations in the velocity of free particles, thereby causing variations in their kinetic energy under the influence of rotating cylindrical gravitational waves. Furthermore, we derive expressions for the angular momentum imparted to free particles by these waves, considering various scenarios for metric coefficients. Interestingly, we observe that the angular momentum of free particles initially experiences an increase over time but eventually starts to decrease after a certain duration.

We develop the theory of the momentum map for the Maxwell–Lorentz equations for a spinning extended charged particle. The development relies on the Hamilton–Poisson structure of the system. This theory is indispensable for the study of long-time behavior and radiation of the solitons of this system, in particular for the proof of orbital stability of the corresponding solitons moving with constant speed and rotating with constant angular velocity.

We apply the theory to the translation and rotation symmetry groups of the system. The general theory of the momentum map results in formal equations for the corresponding invariants. We solve these equations obtaining expressions for the momentum and the angular momentum.

We check the conservation of these expressions and their coincidence with known classical invariants. For the first time, we specify a general class of finite energy initial states which provide the conservation of angular momentum.

The experimental data on the capture and evaporation residue cross-sections obtained in the ⁴⁸Ca+²⁰⁸Pb reaction were analyzed in the framework of the dynamical model based on the dinuclear system concept and advanced statistical method to clarify the reaction mechanism. The experimental excitation function of the capture reactions was decomposed into contributions of the fusion–fission, quasifission and fast-fission processes. Total evaporation residues and ones after neutron emission were only calculated and compared with the available experimental data.

In Randall–Sundrum models, by the use of general Noether theorem, the covariant angular momentum conservation law is obtained with respect to the local Lorentz transformations. The angular momentum current also has superpotential and is therefore identically conserved. The space-like components J_ij of the angular momentum for Randall–Sundrum models are zero. But the component J₀₄ is infinite.

The entropy and area spectra for D-dimensional Gauss–Bonnet black holes are investigated from two different approaches, which are the quasinormal modes approach developed by Hod, Kunstatter and Maggiore, and the very recent method proposed by Ropotenko. The two methods give the same results, which show that the entropy spectrum of Gauss–Bonnet black holes is equidistantly quantized with spacing ΔS = 2πℏ, while the area spectrum is not equally spaced.

We study the non-uniqueness problem of the gauge-invariant angular momentum separation for the case of QED, which stems from the recent controversy concerning the proper definitions of the orbital angular momentum and spin operator of the individual parts of a gauge field system. For the free quantum electrodynamics without matter, we show that the basic requirement of Euclidean symmetry selects a unique physical angular momentum separation scheme from the multitude of the possible angular momentum separation schemes constructed using the various gauge-invariant extensions (GIEs). Based on these results, we propose a set of natural angular momentum separation schemes for the case of interacting QED by invoking the formalism of asymptotic fields. Some perspectives on such a problem for the case of QCD are briefly discussed.

It is well known that an electron has either spin-up or spin-down state and a photon has two possible polarizations called spin $+1$ or spin $-1$. But when two particles are created, the two particles can have 50% of one state and 50% in the other. This is called the two particles in quantum entanglement. The spooky thing is that an event at one point in the universe can instantaneously affect the event that is arbitrarily far away between these two particles.^a The entanglement of the two particles can be electron or photon. We believe, in order to study this phenomenon we have to study further than the previously established principles of quantum mechanics, that is, to study how an electron creates a photon and how it interacts with the photon emitted.

Quantum entanglement simplified-video results — Quantum entanglement and spooky action at a distance, youtube.com, two years ago.

For a charge-monopole pair, we have another definition of the orbital angular momentum, and the transverse part of the momentum including the vector potential turns out to be the so-called geometric momentum that is under intensive study recently. For the charge on the spherical surface with the monopole at the origin, the commutation relations between all components of both the geometric momentum and the orbital angular momentum satisfy the so(3,1) algebra. With construction of the geometrically infinitesimal displacement operator based on the geometric momentum, the so(3,1) algebra implies the Aharonov–Bohm phase shift. The related problems such as charge and flux quantization are also addressed.

I discuss the momentum and angular momentum decomposition problem in the Abelian Higgs model. The usual gauge-invariant extension (GIE) construction is incorporated naturally into the framework of quantum gauge transformation à la Strocchi and Wightman and with this, I investigate the momentum and angular momentum separation in a class of GIE conditions which correspond to the so-called “static gauges”. Using this language, I find that the so-called “generator criterion” does not generally hold even for the dressed physical field. In the case of U(1) symmetry breaking, I generalize the standard GIE construction to include the matter field degrees of freedom so that the usual separation pattern of momentum and angular momentum in the unitarity gauge can be incorporated into the same universal framework. When the static gauge condition could not uniquely fix the gauge, I show that this GIE construction should be expanded to take into account the residual gauge symmetry. In some cases, I reveal that the usual momentum or angular momentum separation pattern in terms of the physical dressed field variables needs some type of modification due to the nontrivial commutator structure of the underlying quantum gauge choice. Finally, I give some remarks on the general GIE constructions in connection with the possible commutator issues and modification of momentum and angular momentum separation patterns.

Using noncommutative infinitesimal rotations, we work out a formulation for angular momentum which leads to an expression for nuclear magnetic resonance (NMR). We also indicate a cosmic footprint of these considerations, much like the 21 cm radio signal of cosmic Hydrogen. Here, the motivation is that there are a number of “second-order effects” which provide results over and above the usual well-known results.

A quasilocal framework of stationary and dynamical untrapped hypersurfaces is introduced to generalize the notions of energy and angular momentum of isolated and dynamical trapping horizons to general strong gravitating systems.

There is an interesting but not-so-popular quantity called pseudo orbital angular momentum (OAM) in the Landau-level system, besides the well-known canonical and mechanical OAMs. The pseudo OAM can be regarded as a gauge-invariant extension of the canonical OAM, which is formally gauge invariant and reduces to the canonical OAM in a certain gauge. Since both of the pseudo OAM and the mechanical OAM are gauge invariant, it is impossible to judge which of those is superior to the other solely from the gauge principle. However, these two OAMs have totally different physical meanings. The mechanical OAM shows manifest observability and clear correspondence with the classical OAM of the cyclotron motion. On the other hand, we demonstrate that the standard canonical OAM as well as the pseudo OAM in the Landau problem is the concept which crucially depend on the choice of the origin of the coordinate system. We try to reveal the relation between the pseudo OAM and the mechanical OAM as well as their observability by paying special attention to the role of guiding-center operator in the Landau problem.

In this paper, the contributions of gravitational instantons to the temperature of the cosmic background radiation are evaluated. The comoving number density in the radiation-dominated and matter-dominated eras is derived. In addition to the temperature resulting from expansion of the universe, the mass of the microscopic black holes required to create the thermal energy of the cosmic microwave background radiation is derived.

The Lie algebra of the group SU₂ is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the {J², U_r} scheme, an alternative to the familiar {J², J_z} quantization scheme corresponding to common eigenvectors of the Casimir operator J² and the Cartan operator J_z. A connection is established between the eigenvectors of the complete set of commuting operators {J², U_r} and mutually unbiased bases in spaces of constant angular momentum.

A simple recipe for generating a complete set of mutually unbiased bases in dimension 2j + 1, with 2j integer and 2j + 1 prime, is developed from a single matrix V_a acting on a space of constant angular momentum j and defined in terms of the irreducible characters of the cyclic group C_{2j + 1}. This recipe yields an (apparently new) compact formula for the vectors spanning the various mutually unbiased bases. In dimension (2j + 1)^e, with 2j integer, 2j + 1 prime and e positive integer, the use of direct products of matrices of type V_a makes it possible to generate mutually unbiased bases. As two pending results, the matrix V_a is used in the derivation of a polar decomposition of SU(2) and of a FFZ algebra.

The existing methods for calculating the energy of stationary states relate it to the energy of the electron, considering it negative in the atom. Formally, choosing a point that corresponds to zero potential energy you can assign a negative value to the electron energy. However, this approach does not answer many other questions, for example, the actual value of the energy of stationary states is unknown, but only the difference in energies between stationary states is known; the concept of “minimum energy of the system” loses its meaning (choosing the origin of the energy reference, we replace the minimum with the maximum, or vice versa); the physical reason for the stability of stationary states is not clear; it is impossible to reveal the physical reason for the introduction of selection rules, since the Heisenberg uncertainty relations exclude the analysis of the transition mechanism, replacing it with the concept of a “quantum leap”. Let us show that the energy of stationary states is the energy of a spherical capacitor, the covers of which are spheres whose radii are equal to the radius of the nuclear and corresponding stationary state. The energy of the ground state in the hydrogen atom is 0.8563997 MeV. The presence of charges and a magnetic field presupposes the circulation of energy in the volume of the atom (the Poynting vector is not zero). Revealed quantization of the angular momentum of the electromagnetic field in stationary states is $Ln=n\hslash$. The change in the angular momentum of the electromagnetic field during transitions between stationary states in atoms removes the physical grounds for introducing selection rules. The analysis shows that the Heisenberg uncertainty relations are not universal, and their application in each specific case must be justified.

I propose that the phase of an electron's wavefunction changes by π when the electron goes around a loop maintaining phase coherence. Equivalently, that the minimum orbital angular momentum of an electron in a ring is ℏ/2 rather than zero as generally assumed, hence, that the electron in a ring has azimuthal zero point motion. This proposal implies that a spin current exists in the ground state of aromatic ring molecules and suggests an explanation for the ubiquitousness of persistent currents observed in mesoscopic rings.

Recently, Ahn et al. created dumbbell-shaped silica nanoparticles, then optically levitated in vacuum, capable of spinning at 300 billion revolutions per minute, driven only by the force and torque of laser. This is by far the fastest spinning object in the world. We show that in this system, there is an additional change in momentum (or extra force) that is proportional to the change in the angular momentum of the particle (or torque). We suggest that this effect also applies to microscopic particles. In the experiment of Ahn et al., although the angular momentum of the particle itself L is large, the rate of change of the angular momentum $\dot{\mathit{L}}$ may not be large. Therefore, experimentally, the physical effects associated with changes in angular momentum can be verified either by applying very strong force and torque to the particle, or by allowing the axis of rotation of the particle to find a way to change very quickly.

We discuss the equations describing the motion of ferrofluids subject to an external magnetic field. We concentrate on the model proposed by Rosensweig, provide an appropriate definition for the effective magnetizing field, and explain the simplifications behind this definition. We show that this system is formally energy stable, and devise a numerical scheme that mimics the same stability estimate. We prove that solutions of the numerical scheme always exist and, under further simplifying assumptions, that the discrete solutions converge. We also discuss alternative formulations proposed in pre-existing work, primarily involving a regularization of the magnetization equation and supply boundary conditions which lead to an energy stable system. We present a series of numerical experiments which illustrate the potential of the scheme in the context of real applications.

We prove an upper bound for the evaluation of all classical SU₂ spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.