Runge-Lenz vector, accidental SU(2) symmetry, and unusual multiplets for motion on a cone

We consider a particle moving on a cone and bound to its tip by 1/r or harmonic oscillator potentials. When the deficit angle of the cone divided by 2π is a rational number, all bound classical orbits are closed. Correspondingly, the quantum system has accidental degeneracies in the discrete energy spectrum. An accidental SU(2) symmetry is generated by the rotations around the tip of the cone as well as by a Runge-Lenz vector. Remarkably, some of the corresponding multiplets have fractional “spin” and unusual degeneracies.     

CPT violation does not lead to violation of Lorentz invariance and vice versa

We present a class of interacting nonlocal quantum field theories, in which the CPT invariance is violated while the Lorentz invariance is present. This result rules out a previous claim in the literature that the CPT violation implies the violation of Lorentz invariance. Furthermore, there exists the reciprocal of this theorem, namely that the violation of Lorentz invariance does not lead to the CPT violation, provided that the residual symmetry of Lorentz invariance admits the proper representation theory for the particles. The latter occurs in the case of quantum field theories on a noncommutative space–time, which in place of the broken Lorentz symmetry possesses the twisted Poincaré invariance. With such a CPT-violating interaction and the addition of a C-violating (e.g., electroweak) interaction, the quantum corrections due to the combined interactions could lead to different properties for the particle and antiparticle, including their masses.     

关于空间平移不变性的一些评注

推广了量子力学中关于对称性的分析,与对称性相关的守恒量相应地推广为含时守恒量;证明了单粒子(包括相对论情况)在静态均匀磁场和含时线性势中运动具有空间平移不变性;求出了波函数的变换关系和相应的含时守恒量.     

Phenomenology and conformal field theory or can string theory predict the weak mixing angle?

We show that the weak mixing angle θ_w is the same for continuously connected classical vacua of the heterotic string which have chiral fermions in their massless spectra. We also show that the world-sheet quantum field theory for any classical vacuum with spacetime supersymmetry possesses an N = 2 superconformal invariance.     

Invariance operator groups for a charged particle in an external electromagnetic field

In this paper a general group theoretical approach is given for the problem of a charged particle moving in an external electromagnetic field F. From a knowledge of the symmetry transformations of the field (Galilean or Poincaré), it is possible to explicitly construct groups of operators which commute with the operators of the equations of motion (classical, quantum mechanical, Klein-Gordon or Dirac) using the concept of compensating gauge transformations together with a uniquely chosen map π: F → A fixing the gauge of the potential A. Other choices of gauges give rise to isomorphic operator groups. The general structure of the possible symmetry groups of the fields is discussed and the corresponding invariance operator groups are explicitly given for (almost) arbitrary fields. The structure of these groups is then investigated and it is shown in particular that a large class of fields give rise to non-Type I groups, i.e. to groups which have (unitary continuous) representations whose corresponding von Neumann algebras have non-discrete factors. A general criterion for these pathological cases is given. As an application, we study the problem of a Bloch electron in arbitrary constant uniform electric and magnetic fields.     

Supersymmetry-Based Approach to Quantum Particle Dynamics on a Curved Surface with Non-zero Magnetic Field

We present the N=2 supersymmetric formulation for the classical and quantum dynamics of a nonrelativistic charged particle on a curved surface in the presence of a perpendicular magnetic field. For a particle moving on a constant-curvature surface in a constant magnetic field, our Hamiltonian possesses the shape-invariance property in addition. On the surface of a sphere and also on the hyperbolic plane, we exploit the supersymmetry and shape-invariance properties to obtain complete solutions to the corresponding quantum mechanical problems.     

Quantum cyclotron resonance in a degenerate system. Quasienergetic states

We study the quantum dynamics of a charged particle rotating in a constant magnetic field and in the field of a monochromatic wave propagating perpendicular to the magnetic field under the condition of cyclotron resonance. Quasienergies and quasienergy functions are numerically calculated in the resonance approximation. The passage to the limit from the quantum model to the classical one is discussed.Lobachevskii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 3-4, pp. 232–240, March–April, 1995.     

Geometric quantization in the presence of an electromagnetic field

Some aspects of the formalism of geometric quantization are described emphasizing the role played by the symmetry group of the quantum system which, for the free particle, turns out to be a central extensionG(_m) of the Galilei groupG. The resulting formalism is then applied to the case of a particle interacting with the electromagnetic field, which appears as a necessary modification of the connection 1-form of the quantum bundle when its invariance group is generalized to alocal extension ofG. Finally, the quantization of the electric charge in the presence of a Dirac monopole is also briefly considered.     

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.     

Magnetic catalysis and magnetic oscillations in the Nambu-Jona-Lasinio model

The phase structure of the four-dimensional Nambu-Jona-Lasinio model in the presence of a chemical potential μ and an external magnetic field H is investigated at comparatively small values of the bare coupling constant (G<G _c). It is shown that only for magnetic-field strengths in excess of some critical value H _c(μ) does the magnetic field induce a spontaneous breakdown of chiral symmetry. On the phase portrait of the model, there are infinitely many massless chiral-invariant phases; in addition, there is one massive phase characterized by spontaneously broken chiral invariance. It is because of this phase structure of the system that some physical features of its ground state, including magnetization, pressure, and particle density, oscillate as H → 0. Changes in the vacuum properties of the model are accompanied by first-or second-order phase transitions.     

Dobrushin states for classical spin systems with complex interactions

We consider a classical spin system on the hypercubic lattice with a general interaction of the form $$ H = \frac{\beta } {4}\sum\limits_{\begin{array}{*{20}c} {x,y:} \\ {|x - y| = 1} \\ \end{array} } {|s_x - s_y | - h} \sum\limits_x {x{}_x + } \sum\limits_A {\lambda _A \prod\limits_{y \in A} {S_y } } $$ are the spin variables, Β is the inverse temperature,h is the magnetic field, andλ _A are translation-invariant coupling constants satisfyingλ _A = 0 if diamA > l. No symmetry relating the configurationss ={sinx} and-s=-s _x is assumed. In dimension d-3, we construct low-temperature States which break the translation invariance of the system by introducing so-called Dobrushin boundary conditions which force a horizontal interface into the system. In contrast to previous constructions, our methods work equally well for complex interactions, and should therefore be generalizable to quantum spin systems.     

非中心力场中经典粒子的轨道参数方程与对称性

把非中心力场中经典粒子运动微分方程写成Ermakov方程的形式，得到Ermakov不变量.用改变时间坐标标度的方法得到用能量H和Ermakov不变量表示的轨道参数方程，并研究两守恒量（能量和Ermakov不变量）相应的无限小变换的Noether对称性、Lie对称性和形式不变性.研究结果表明：与两守恒量相应的无限小变换既具有Noether对称性，也具有Lie对称性和形式不变性. 关键词： 非中心力场 轨道参数方程 守恒量 对称性     

Long-range potentials and the electromagnetic polarizabilities

The long-range spin and velocity independent forces of electromagnetic origin which act between any two systems are studied for those cases in which no forces of this type exist to order e². It is shown that they are uniquely determined by the charge, magnetic moment, and polarizabilities of both systems, not only to the dominant order r^?n, but also to the next one r^?(n+1). These potentials provide the link between Compton scattering polarizabilities (response to real photons) and classically defined polarizabilities (response to static electromagnetic field). The two definitions are shown to be equivalent for neutral spinless systems; the problems arising for a neutral particle with magnetic moment are studied in detail. The r^?(n+1) terms have no classical counterpart, since they are due to the relativistic quantum propagation of the system which carries charge or magnetic moment. The results are of general validity with analyticity, crossing, unitarity, and gauge invariance as only inputs. The most general conclusion is that the polarizabilities represent electromagnetic properties of a system at order e², as the charge and magnetic moment do at order e. Thus they give the strength of the response to electric and magnetic fields, independently of the specific characteristics of the electromagnetic agent.     

Ion cyclotron drift instability driven by cross-field current

An ion cyclotron drift instability is excited by a cross field current due to E_r × B drift of electrons and ion heating is observed in a plasma column in a magnetic field when a dc radial electric field E_r is applied.     

Relationship between σxx and σyx - integral quantum hall effect

The possibility of a Kramers-Kronig relation between the longitudinal and transverse conductivities of a 2-D electron gas in the presence of a magnetic field is investigated. The longitudinal (σ_xx) and the transverse (σ_yx) conductivities are expressed as the real and imaginary part of a complex conductivity (σ). A formal expression is derived for σ within the independent particle approximation. In the classical limit, σ is analytic in the upper complex ω_c plane, where ω_c is the cyclotron frequency. Hence the usual Kramers-Kronig relation holds for σ^c. We modeled a quantum correction (σ^q) to σ and found that σ^q is non-analytic in the upper half plane, hence the total σ does not satisfy a Kramers-Kronig relation.     

Time reversal and symmetries of time correlation functions

ABSTRACT

The time reversal invariance of classical dynamics is reconsidered in this paper with specific focus on its consequences for time correlation functions and associated properties such as transport coefficients. We show that, under fairly common assumptions on the interparticle potential, an isolated Hamiltonian system obeys more than one time reversal symmetry and that this entails non trivial consequences. Under an isotropic and homogeneous potential, in particular, eight valid time reversal operations exist. The presence of external fields that reduce the symmetry of space decreases this number, but does not necessarily impair all time reversal symmetries. Thus, analytic predictions of symmetry properties of time correlation functions and, in some cases, even of their null value are still possible. The noteworthy case of a constant external magnetic field, usually assumed to destroy time reversal symmetry, is considered in some detail. We show that, in this case too, some of the new time reversal operations hold, and that this makes it possible to derive relevant properties of correlation functions without the uninteresting inversion of the direction of the magnetic field commonly enforced in the literature.     

Quantum regime of a plasma‐wave‐pumped free‐electron laser in the presence of an axial magnetic field

The quantum regime of a plasma‐whistler‐wave‐pumped free‐electron laser (FEL) in the presence of an axial‐guide magnetic field is presented. By quantizing both the plasma whistler field and axial magnetic field, an N‐particle three‐dimensional Hamiltonian of quantum‐FEL (QFEL) has been derived. Employing Heisenberg evolution equations and introducing a new collective operator which controls the vertical motion of electrons, a quantum dispersion relation of the plasma whistler wiggler has been obtained analytically. Numerical results indicate that, by increasing the intrinsic quantum momentum spread and/or increasing the axial magnetic field strength, the bunching and the radiation fields grow exponentially. In addition, a spiking behavior of the spectrum was observed with increasing cyclotron frequency which provides an enormous improvement in the coherence of QFEL radiation even in a limit close‐to‐classical regime, where an overlapping of these spikes is observed. Also, an upper limit of the intrinsic quantum momentum spread which depends on the value of the cyclotron frequency was found.     

Microwave photoresistance of a double quantum well at high filling factors

The effect of millimeter wave radiation on the electronic transport in a GaAs double quantum well at a temperature of 4.2 K in a magnetic field of up to 2 T has been studied. Resistance (conductance) oscillations have been shown to appear in the two-dimensional electronic system under investigation at high filling factors. The magnetic field positions of the oscillation maxima are determined by the condition Δ_SAS/? = lω_c, where Δ_SAS = (E ₂ ? E ₁) is the size quantization sublevel splitting in the quantum well, ω_c is the cyclotron frequency, and l is a positive integer. It has been found that the microwave field substantially modifies the oscillations in the double quantum well, which results in alternating two-frequency oscillations of photoresistance with the inverse magnetic field.     

A refined Mermin argument for the two-dimensional jellium

We consider the two-dimensional classical jellium with periodic boundary conditions and with translation invariance broken by an external periodic localizing one-body potential. Using a refinement of Mermin's argument, we show that a symmetry breakdown of translation invariance (appearance of a crystalline phase with positional long-range order) does not occur, if the correlation functionh(r) decays faster than 1/r ² asr.