On geometrodynamics and null fields

The possibility of describing null electromagnetic fields by purely metric concepts has recently been subject to some doubt. Following a method devised by Hlavatý, we here investigate the relations that a Riemannian manifold must satisfy in order to correspond to a null electromagnetic field. It is shown that in most cases the fulfilment of five geometrical relations is a necessary and sufficient condition for the existence of a null electromagnetic field. The latter is unique, except for an arbitrary constant phase factor (as in the case of non-null fields). However, in some exceptional cases, there is a larger degree of arbitrariness in the null electromagnetic field that corresponds to a given metric. Such fields (which always possess wave fronts) are not reducible to metric concepts. We then turn to examine how it can occur that null electromagnetic fields require the fulfilment of five relations, rather than three, as non-null ones. In order to settle this question, we make an attempt to consider null fields as a limiting case of non-null ones, by superimposing an arbitrary infinitesimal non-null field on a finite null one. It is then shown that the Rainich vector of such a field does not have a well defined limit, when the perturbing non-null field tends to zero. It is thereby inferred that null electromagnetic fields really have a special status within the frame of geometrodynamics.     

The preferred null symmetries of a null electromagnetic field interacting with a null free gravitational field

The real null vector 1 ^a of the Newman-Penrose formalism is preferred to correspond to a geometrical symmetry as well as a dynamical symmetry. The 16 types of geometrical symmetries expressed through the vanishing of the Lie derivatives of certain tensor fields with respect to 1 ^a are examined separately. Two types of dynamical symmetries are imposed simultaneously on 1 ^a : A null electromagnetic field and a null gravitational field are both chosen to have the same propagation vector 1 ^a . By adopting freedom conditions on 1 ^a , it is shown that the symmetries of the null electromagnetic field are shared neither by the free gravitational field nor by the gravitational potentials. In fact the following five preferred null symmmetries are found to be proper: motion, affine collineation, special curvature collineation, curvature collineation, and Ricci collineation. The scalars characterizing the coupled fields are found to be constant with respect to 1 ^a .     

Einstein-Maxwell null fields with non-zero twist

It is shown that iff ^a is the propagation vector of a null Maxwell field in a space-time with metric⁰ g _ab , then it is also the propagation vector in the space-timeg _ab =⁰ g _ab +2fi _a i _b . This result, together with the Robinson metric for vacuum gravitational fields and Hughston's generalization to radiating fields, is used to set up equations for combined gravitational and electromagnetic null-fields with special reference to fields with non-zero twist.     

Principal null directions of the neutrino and derived Maxwell fields

Bineutrino fields are constructed from neutrinos described by the two-component Weyl equation. If we demand that the bineutrino fields fulfil the set of Maxwell equations, we have a classical version of the “neutrino theory of light”. With the help of the spin coefficient formalism, it is established that the energy tensor of the neutrino fields cannot be positively definite, and that the principal null direction of the bineutrino Maxwell fields does not coincide with the principal null direction of the underlying neutrino fields.     

Collapse of a null fluid

An exact cylindrically symmetric solution of Einstein's gravitational field equations is given for a null fluid imploding radially along an infinite axis. This solution plays an important rôle in the late stages of collapse of a long cylinder of matter. One might expect that self-gravitational effects due to the increasingly relativistic mass of the collapsing matter would create arbitrarily large gravitational fields. It is shown, however, that in the null-fluid approximation the metric is everywhere regular.     

Symmetry generators for null geodesic motion

The concept of conformai dynamical symmetry (CDS) is introduced as a basic tool for the analysis of null geodesic motion. CDSs are characterized as generators of Jacobi fields along null geodesies. Methods for the construction of CDS's based on the connections between CDSs and conformai Killing tensors or on the behavior of CDSs under conformai transformations are then discussed. Finally, a comprehensive analysis of conserved quantities canonically related to CDSs is given.     

Colored null flags and para-Fermi fields

An inverse problem of deriving the concept of quantized fields from a certain observable conserved current is investigated. It is found that a natural framework in which to attack the problem is provided for by what we shall call Green's ansatz of null decomposition of the current. The null decomposition naturally yields a set ofcolored null flags hoisted at each space-time point, a null flag comprizing a real null vector and an associated real null six-vector, and is invariant under all permutations of colors. From the fact that to any null flag there corresponds a two-component spinor it follows that the color permutation group is extended tocolor groups O(p) orU(p), wherep is the number of null flags considered. It is shown that para-Weyl (para-Fermi) fields of orderp2 can be deduced from the (chiral) set ofp colored null flags, and that the color groupU(p) is singled out that functions as the gauge group of para-Fermi theory.     

一个二次多项式混沌系统的均匀化及其熵分析

利用已有理论给出了一个二次多项式混沌系统,证明了该系统与Tent映射拓扑共轭,给出了该混沌系统的概率密度函数;并根据此概率密度函数,得到了轨道均匀分布的反三角函数映射;对均匀化前后的混沌系统在不同参数下产生序列的信息熵、Kolmogorov熵、离散熵的特性进行了分析,结果显示均匀化后产生的混沌序列混沌程度不改变且具有更好的均匀性.     

Einstein-maxwell null fields and extended massless particles with helicity

The concept of extended elementary particles proposed by the author in previous papers is briefly outlined in a slightly modified form. In this framework special Einstein-Maxwell null fields are interpreted as extended massless particles with helicity. A model of extended photons is proposed.     

On the Robinson theorem and shearfree geodesic null congruences

Null electromagnetic fields and shearfree geodesic null congruences in curved and flat spacetimes are studied. We point out some mathematical problems connected with the validity of the Robinson theorem. The problem of finding nonanalytic twisting congruences in the Minkowski space is reduced to the construction of holomorphic functions with specific boundary conditions.Institute of Theoretical Physics, University of Warsaw Poland     

OPE for null Wilson loops and open spin chains

Maximal helicity-violating scattering amplitudes in N=4$N=4$ supersymmetric Yang–Mills theory are dual to Wilson loops on closed null polygons. We perform their operator product expansion analysis in two-dimensional kinematics in the soft-collinear approximation which corresponds to the case when some light-cone distances vanish. We construct the expansion in terms of multi-particle “heavy”–light operators, where the “heavy” fields are identified with the Wilson lines defining the OPE channel and the light fields emerge from the curvature of the contour. The correlation function of these define the remainder function. We study the dilatation operator for these operators at one-loop order and find that it corresponds to a non-compact open spin chain. This provides an alternative view on elementary excitations propagating on the GKP string at weak coupling, which now correspond to particles traveling along an open spin chain. The factorized structure of the Wilson loop in the soft limit allows one to represent the two-loop correction to the octagon Wilson loop as a convolution formula and find the corresponding remainder function.     

Effect of the initial plasma parameters on the structure of the current sheets developing in two-dimensional magnetic fields with a null line

The effect of the initial plasma parameters on the structure of the plasma of the current sheets that form in two-dimensional magnetic fields with a null line is studied by holographic interferometry. The evolution of the plasma sheets that develop in an initial low-density plasma, where a gas is mainly ionized by a pulse current passing through the plasma and initiating the formation of a current sheet, has been comprehensively studied for the first time. At the early stage of evolution, the spatial structure of such a plasma sheet differs substantially from the classic current sheets forming in a dense plasma. Nevertheless, extended plasma sheets with similar parameters form eventually irrespective of the initial plasma density.     

Transient and cyclic behavior of cellular automata with null boundary conditions

One-dimensional cellular automata (CA) over finite fields are studied in which each interior cell is updated to contain the sum of the previous values of its two nearest neighbors. Boundary cells are updated according to null boundary conditions. For a given initial configuration, the CA evolves through transient configurations to an attracting cycle. The dependence of the maximal transient length and maximal cycle length on the number of cells is investigated. Both can be determined from the minimal polynomial of the update matrix, which in this case satisfies a useful recurrence relation. With cell values from a field of characteristic two, the explicit dependence of the maximal transient length on the number of cells is determined. Extensions and directions for future work are presented.Deceased.     

Observation of the lower hybrid waves near the three-dimensional null pair

Magnetic reconnection is a fundamental process in plasma, which is thought to play important roles both in laboratory and natural plasmas through affecting magnetic topology, heating and accelerating particles. During an event on Oct. 1st, 2001, the Cluster tetrahedron circled around the magnetic reconnection region several times, and Xiao et al. first identified the null pair and found that the spectrum of the null-point oscillation shows the maximum power near the lower-hybrid frequency. In this paper we report the observation of electromagnetic and electrostatic wave enhancements near lower hybrid frequency associated with the reconnection process near the null pair. The lower hybrid waves (LHWs) with quasi-perpendicular propagation were identified and also confirmed by the power law of the spectrum of electric and magnetic fields. Supported by the National Natural Science Foundation of China (Grant Nos. 40325012, 40390151, 40574073, 40574074, and 40640420563), the Research Foundation from Ministry of Education of China (Grant No. 40174043), and the China Plan on the International Polar Year     

On the permanence of the null character of Maxwell fields

A critical review of known results about the permanence conditions for the null character of the solutions to the (vacuum) Maxwell equations, is presented. Concomitants of the electromagnetic field and the metric tensor are constructed, which give the principal directions of the field in covariant form. The known permanence conditions are generalized in order to includeall the (local) null fields; the above concomitants allow these conditions to be explicitly formulated in terms of the electromagnetic field.Supported in part by Conselleria de Cultura, Educació i Ciència de la Generalitat Valenciana.     

Relativistic dynamics on a null plane

In view of possible applications to the quark model and to hadron spectroscopy, we investigate relativistic Hamiltonian quantum theories of finitely many degrees of freedom. We make use of the fact that if null planes are used as initial surfaces, the structure of the theory closely resembles nonrelativistic quantum mechanics: the inner variables that describe the structure of the system uncouple from the motion of the system as a whole. The dynamical content of such a theory resides in the operators M, $\text{j}$ of mass and spin that act in the space carrying the inner degrees of freedom. Relativistic invariance is equivalent to the requirement that M and $\text{j}$ generate a unitary representation of U(2). In contrast to this requirement, the condition that the wavefunctions of the system transform covariantly strongly restricts the dynamics. It is proven that for systems containing two constituents, covariance is equivalent to an algebraic relation that involves M and $\text{j}$ — the angular condition. A class of solutions of the angular condition is provided by a particular type of local manifestly covariant wave equations. One nontrivial solution of this class, a relativistic oscillator is given in detail. Confinement models of this type represent an interesting alternative to the solutions of the angular condition that result from the perturbation expansion of a local field theory through the three-dimensional quasipotential versions of the Bethe-Salpeter equation.     

Gravitational fields near space-like and null infinity

Near space-like infinity an initial value problem for the conformal Einstein equations is formulated such that: (i) the data and equations are regular, (ii) space-like and null infinity have a finite representation, with their structure and location known a priori, and (iii) the setting relies entirely on general properties of conformal structures.A first analysis of this problem shows that the solutions develop in general a certain type of logarithmic singularity at the set where null infinity touches space-like infinity. These singularities form an intrinsic part of the solutions' conformal structure. Conditions on the free initial data near space-like infinity are derived which ensure that for solutions developing from these data singularities of this type cannot occur.     

Some remarks on the “null plane development” of a relativistic quantum field theory

We give conditions for the existence of field operators on so-called null planes and discuss some consequences of the necessary restriction of the test function space, concerning Haag's theorem and the possibility of unitary mappings intertwining between free fields of different masses. In the last section we discuss conditions under which a unitary representation of the dilatations in the null plane gives rise to a unitary representation of the dilatations in Minkowski space.     

Relativistic particles and the geometry of 4-D null curves

We study actions in (d+1)$(d+1)$-dimensions associated with null curves, mainly when d=3$d=3$, whose Lagrangian is a linear function on the curvature of the particle path, showing that null helices are always possible trajectories of the particles. We find Killing vector fields along critical curves of the action which correspond to the linear and the angular momenta of the particle. They provide two constants of the motion which can be interpreted in terms of the mass and the spin of the system. Moreover, we are able to integrate both the Euler–Lagrange equations and the Cartan equations in cylindrical coordinates around a certain plane.