Lagrangian dynamics of polarized media with internal spin in general relativity

The complete system of field equations for a polarized medium with internal spin in interaction with an arbitrary set of fields is derived from a four-dimensional action integral. The general results are then specialized to the interaction with an Einstein-Maxwell field. The application to ideal spinning fluids is discussed.     

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.     

An algorithm for the calculation of the net induction energy and induced multipole tensors in a set of charge distributions with non-linear contributions of all potentials of external fields and both permanent and induced multipoles

This algorithm for the calculation of the induced multipole tensors in a set of charge distributions includes the contributions of partial derivatives of arbitrary order of the potentials defined by both induced and permanent multipoles as well as non-uniform external fields of arbitrary strength. Specific equations are given for both systems with and without translational order. The algorithm, which is based on the Maxwell invariant form, uses direct extensions of algorithms previously developed and tested for the calculation of permanent multipole energies and induced dipole vectors (when the non-uniformity of the field is neglected). The induced tensor components are calculated iteratively. The first approximation, which gives the components as the solution of a set of simultaneous linear equations, includes all nonlinear, non-uniform contributions of permanent multipoles and external fields as well as all contributions linear in derivatives of the induced potentials. The induced tensor components are then used to calculate the net induction energy. The general relations between the polarizability tensors with respect to a centre and the moments of the polarizability densities about the centre are derived.     

On the energy-momentum and spin tensors in the Riemann–Cartan space

General classical theories of material fields in an arbitrary Riemann–Cartan space are considered. For these theories, with the help of equations of balance, new non-trivially generalized, manifestly generally covariant expressions for canonical energy-momentum and spin tensors are constructed in the cases when a Lagrangian contains (a) an arbitrary set of tensorial material fields and their covariant derivatives up to the second order, as well as (b) the curvature tensor and (c) the torsion tensor with its covariant derivatives up to the second order. A non-trivial manifestly generally covariant generalization of the Belinfante symmetrization procedure, suitable for an arbitrary Riemann–Cartan space, is carried out. A covariant symmetrized energy-momentum tensor is constructed in a general form.     

Constraint equations for general hypersurfaces and applications to shells

Hypersurfaces of arbitrary causal character embedded in a spacetime are studied with the aim of extracting necessary and sufficient free data on the submanifold suitable for reconstructing the spacetime metric and its first derivative along the hypersurface. The constraint equations for hypersurfaces of arbitrary causal character are then computed explicitly in terms of this hypersurface data, thus providing a framework capable of unifying, and extending, the standard constraint equations in the spacelike and in the characteristic cases to the general situation. This may have interesting applications in well-posedness problems more general than those already treated in the literature. As a simple application of the constraint equations for general hypersurfaces, we derive the field equations for shells of matter when no restriction whatsoever on the causal character of the shell is imposed.     

Dirac particle spin in strong gravitational fields

Dynamics of the Dirac particle spin in general strong gravitational fields is discussed. The Hermitian Dirac Hamiltonian is derived and transformed to the Foldy-Wouthuysen (FW) representation for an arbitrary metric. The quantum mechanical equations of spin motion are found. These equations agree with corresponding classical ones. The new restriction on the anomalous gravitomagnetic moment (AGM) by the reinterpretation of Lorentz invariance tests is obtained.     

用耦合波理论求解渐变折射率光纤的传播常数和模式场

本文应用耦合波理论来计算任意渐变折射李光纤的传播常数和模场分布.将无限伸展的抛物型折射率分布光纤作为一个理想波导,任意径向不均匀折射率分布光纤便可以视作该理想波导的微扰,其模场可以展开成一组完备的理想波导模的叠加.然后,模场和传播常数可以通过耦合波方程求得.进一步将一阶微分方程组形式的耦合波方程变换成线性方程组,使计算过程大大简化.文中给出了数值计算结果,并将其与准确值进行了比较.     

A note on the equations of motion for the transverse vibration of a Timoshenko beam subjected to an axial force

The equations of motion are derived for a Timoshenko beam of arbitrary section in transverse vibration with an applied axial loading. Some discrepancies are noted by comparison with earlier work and these are examined and accounted for. The application of these more general equations is then considered for certain special cases, including a rotating blade.     

Keller magnetic fields in space of constant holomorphic sectional curvature

Keller magnetic fields on Keller manifolds of constant holomorphic curvature with an arbitrary signature are considered. The metric of the Keller space of constant holomorphic curvature is obtained in general form. Simulation of the Keller magnetic fields of by means of H-projective curves of a flat space is used to reduce the study of the trajectory equations to a study of one ordinary second-order differential equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 13–19, January, 1996.     

Hertz Potentials and Asymptotic Properties of Massless Fields

In this paper we analyze Hertz potentials for free massless spin-s fields on the Minkowski spacetime, with data in weighted Sobolev spaces. We prove existence and pointwise estimates for the Hertz potentials using a weighted estimate for the wave equation. This is then applied to give weighted estimates for the solutions of the spin-s field equations, for arbitrary half-integer s. In particular, the peeling properties of the free massless spin-s fields are analyzed for initial data in weighted Sobolev spaces with arbitrary, non-integer weights.     

Bargmann–Wigner方程与高自旋场方程

在坐标表象中由Bargmann-Wigner方程导出了便于求解的高自旋场方程,并给出了相应的拉氏函数.     

The teleparallel equivalent of general relativity

A review of the teleparallel equivalent of general relativity is presented. It is emphasized that general relativity may be formulated in terms of the tetrad fields and of the torsion tensor, and that this geometrical formulation leads to alternative insights into the theory. The equivalence with the standard formulation in terms of the metric and curvature tensors takes place at the level of field equations. The review starts with a brief account of the history of teleparallel theories of gravity. Then the ordinary interpretation of the tetrad fields as reference frames adapted to arbitrary observers in space–time is discussed, and the tensor of inertial accelerations on frames is obtained. It is shown that the Lagrangian and Hamiltonian field equations allow us to define the energy, momentum and angular momentum of the gravitational field, as surface integrals of the field quantities. In the phase space of the theory, these quantities satisfy the algebra of the Poincaré group.     

Nonlinear dynamics in a Fabry-Perot resonator

We develop a general formalism to describe the dynamical behavior of an ensemble of two-level systems in a Fabry-Perot cavity. Our main result is a set of space and time-dependent, integro-differential equations for the slowly varying radiation and atomic variables, which we derive through a precise analysis of the slowly-varying amplitude approximation in the presence of counter-propagating fields. With the help of new and properly chosen variables we recast these equations in a form that makes their boundary conditions formally identical to those of an ideal Fabry-Perot resonator, and introduce in a natural way a modal structure even for systems with arbitrary mirror reflectivity. We derive simplified forms of these equations in the uniform field limit and within the more general single-frequency approximation. Finally, we extend our formulation to include driven systems such as optically bistable devices and the laser with an injected signal.     

A twistorial description of the dynamics of complexified dirac fields

A method that allows the construction of a twistorial least-action principle for complexified Klein-Gordon fields is adapted to the context of the two-component theory of Dirac fields. It appears that the viability of the adaptation procedures rests crucially upon the freedom in the selection of spinor kernels for the integrands of the universal twistor contour integrals for massive fields carrying arbitrary spin. An intrinsic ambiguity borne by the inner structure of the twistor representation of the dynamics of the spinning fields taken into consideration is then brought out. This feature appears to be reflected by the fact that many Poincaré-invariant Lagrangian densities can be regarded as candidates for entering the new variational principle. It is remarked that all the corresponding equations of motion agree with a single twistor translation of the pertinent field equations. The dynamical expressions that result from the implementation of the procedures turn out to be related to one another through a simple interchange scheme. A set of wave equations for the system allowed for is also obtained. This work was supported financially by the Brazilian agency CNPq.     

Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes

The stochastic and quantum dynamics of open quantum systems interacting with stochastic perturbations in considered. The master equations for one time and multi-time correlation functions of such a system are derived to all orders in the interaction with the stochastic perturbations. The importance of the non-markovian character of such equations in the study of various problems in optical resonance is discussed. The simplified form of the non-markovian master equations in Born approximation is also given. It is shown that such non-markovian master equations in Born approximation are exact if there is only one random perturbation, of the telegraphic signal type, acting on the system. The master equations for the linear response functions of an open system interacting with stochastic perturbations are also derived. The non-markovian master equations for multitime correlations are used to study the behaviour of two level atoms interacting with fluctuating laser fields. Both amplitude and phase fluctuations are taken into account. Explicit results are presented for the spectrum of resonance fluorescence, absorption spectrum, photon antibunching effects etc. The calculations are done for arbitrary values of the relaxation parameters and intial conditions. In general the fluorescence spectrum is found to be asymmetric for off resonant fields.     

A connection between the Einstein and Yang-Mills equations

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.Andrew Mellon Postdoctoral fellow and Fulbright ScholarSupported in part by NSF grant no. PHY 80023     

Variational Equations and Symmetries in the Lagrangian Formalism; Arbitrary Vector Fields

We continue the study of symmetries in the Lagrangian formalism of arbitrary order with the help of the generalized Helmholtz equations (sometimes called the Anderson-Duchamp-Krupka equations). For the case of second-order equations and arbitrary vector fields we are able to establish a polynomial structure in the second-order derivatives. This structure is based on the some linear combinations of Olver hyper-Jacobians. We use as the main tools Fock space techniques and induction. This structure can be used to analyze Lagrangian systems with groups of Noetherian symmetries. As an illustration we analyze the case of Lagrangian equations with Abelian gauge invariance.