Space-times carrying a quasirecurrent pairing of vector fields

A quasirecurrent pairing of vector fields {X₁, X₂} (defined in [1]) is investigated on a space-time in two cases: (1)X ₁ is spacelike andX ₂ is timelike; (2) X₁ is null andX ₂ space-like. The physical interpretation of these vector fields is given.     

Bicovariant differential geometry of the quantum group SL h (2)

There are only two quantum group structures on the space of two by two unimodular matrices, these are the SL _q (2) and the SL _h (2) quantum groups. The differential geometry of SL _q (2) is well known. In this Letter, we develop the differential geometry of SL _h (2), and show that the space of left invariant vector fields is three-dimensional.     

Vacuum spacetimes with a two-dimensional orthogonally transitive group of homothetic motions

Vacuum spacetimes with a two-dimensional orthogonally transitive groupH ₂ of proper homothetic motions acting on nonnull orbits are investigated with the aid of the Geroch-Held-Penrose formalism. It is found that these spacetimes admit in general anH ₃ of homothetic motions containing two commuting and hypersurface orthogonal Killing vector fields. The metric equations are integrated, and the line elements of the spacetimes in question are explicitly given in a diagonal form.     

Lie-Backlund vector fields for the nonlinear system,Q t=AQxx+F(Qx,Q)

We have analyzed the class of nonlinear second-order equations written asQ _t=AQ_xx +F(Qx, Q) withQ =( _v ^u ) andA, F are, respectively, matrix and vector functions depending onQ, Q _x, from the point of view of Lie-Backlund vector fields. When the vector functionF does not depend onQ _x, these equation set reduces to the coupled diffusion equations discussed by Steeb. But our generalized system encompasses a large class of physically meaning full nonlinear equations, such as (i) dispersive water waves and (ii) a completely anisotropic Heisenberg spin chain. We also exhibit a new nonlinear coupled system which do have nontrivial Lie-Backlund vector fields. Also our approach yields more information about the symmetry generators for a wider class of nonlinear equations than the function space approach of Fuchsteiner in a much simpler way.     

Current helicity pattern of large-scale magnetic field on the photosphere

Following Pevtsov and Latushko, we study the current helicity pattern of the large-scale magnetic field on the photosphere. We use the same technique as theirs to derive the vector magnetic field (B _r , B _θ , B _ϕ ) from full-disk longitudinal magnetograms based on the assumption that large-scale magnetic fields evolve rather slowly and the variations of the longitudinal magnetic fields within certain time duration are caused by the changing position angles only. Different from their study, we have calculated the current helicity maps directly from the derived vector magnetograms, rather than from obtaining the latitudinal variation of h _c by ignoring the role of B _θ component and averaging B _r and B _ϕ over all solar longitudes. This approach significantly strengthens the evidence of the hemispheric rule presented in the reconstructed vector magnetic field. Our study shows that the established hemispheric sign rule, that is, positive helicity sign in the southern hemisphere and negative helicity sign in the northern hemisphere, is applicable everywhere in the global magnetic field, namely, also evident in weak fields outside active regions, and that the obtained sign pattern is independent of the longitudinal magnetograms and the parameters that we have used.     

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.     

T1 and T2 effects during radio-frequency pulses in spoiled gradient echo sequences

Finite pulse durations in diverse pulse schemes lead to the reduction of the magnitude of the magnetization vector due to T₁ and T₂ effects during the radio-frequency pulses. This paper presents an analysis of the steady state signal in the presence of relaxation effects during radio-frequency pulses in MRI spoiled gradient echo sequences. It is shown that minor attenuations of the magnetization vector can have dramatic consequences on the measured signal, and may thus entail a loss in SNR benefits at high static magnetic fields if a careful analysis is not performed. It is emphasized that it is the time-integrated magnetization vector trajectory that matters for these effects and not only the pulse duration. Some experimental results obtained on a phantom at 3 T verify this analysis.     

Linearized interactions of scalar and vector fields with the higher spin field in <Emphasis Type="Italic">AdS</Emphasis><Subscript><Emphasis Type="Italic">D</Emphasis></Subscript>

The explicit form of linearized gauge arid generalized “Weyl invariant” interactions of scalar and general higher even spin fields in the AdS _D space constructed in [1] is reviewed. Also a linearized interaction of vector field with general higher even spin, gauge field is obtained. It is shown that the gauge invariant action of linearized vector field interacting with the higher spin field also includes the whole tower of invariant actions for couplings of the same vector field with the gauge fields of smaller even spin.     

Vector-spinor space and field equations

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists a vector-spinor space with N_v vector dimensions and N_s spinor dimensions, where N_v=2k and N_s=2 ^k, k3. This space is decomposed into a tangent space with4 vector and4 spinor dimensions and an internal space with N_v–4 vector and N_s–4 spinor dimension. A variational principle leads to field equations for geometric quantities which can be identified with physical fields such as the electromagnetic field, Yang-Mills gauge fields, and wave functions of bosons and fermions.     

The hopf algebra of vector fields on complex quantum groups

We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A _n , B _n , C _n , and D _n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.Humboldt Fellow.     

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field Q on a supermanifold M is called homological, if Q ² = 0. The operator of Lie derivative L _Q makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator L _Q and are represented by Q-invariant tensors made up of the homological vector field and a symmetric connection on M by means of the algebraic tensor operations and covariant differentiation.     

Pauli Operator and Aharonov–Casher Theorem¶ for Measure Valued Magnetic Fields

We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual A∈L ² _loc condition on the vector potential, which does not allow to consider such singular fields. We extend the Aharonov–Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted L ² estimate on a singular integral operator. Received: 14 May 2001 / Accepted: 5 September 2001     

Real complexified quantum groups,their vector fields and quantum enveloping algebras

We construct complex quantum groups associated with the Lie algebras of typeA _n–1 ,B _n ,C _n andD _n which are considered as real algebras. Following the ideas of Faddeev, Reshetikhin and Takhtayan, we obtain the Hopf algebras of regular functionalsU _R , on these real complexified quantum groups. Theq-analogues of the left invariant vector fields of the quantum enveloping algebras are defined. These quantum vector fields are functionals over the corresponding real formA of the complex quantum groupA. The equivalence of the Hopf algebra of regular functionals and the algebra of complex quantum vector fields is shown by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals. In the special exampleA ₁ , we derive theq-deformed real complexified enveloping algebraU _q sl(2, ) with six generators.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.Based on the papers: [i]Drabant B., Schlieker M., Weich W., and Zumino B.: PreprintLMU-TPW 1991-5 (to appear in Commun. Math. Phys.) [ii]Chryssomalakos C., Drabant B., Schlieker M., Weich W., and Zumino B.: Preprint UCB 92/03 (to appear in Commun. Math. Phys.) [iii]Drabant B., Juro B., Schlieker M., Weich W., and Zumino B.: Preprint MPI-Ph/92-39 (submitted to Lett. Math. Phys.)     

Real Kasner and related complex “windmill” vacuum spacetime metrics

The Kasner family of vacuum solutions of Einstein's field equations admits a simply-transitiveH ₄, a four-parameter local homothetic group of motions which has an AbelianG ₃ subgroup. It is shown that a complex transformation of coordinates and constants exists which maps this family from the normal Kasner form into a form of vacuum metrics whose Weyl tensors are each Petrov type I and which were published in 1932 by Lewis. These metrics also admit a similarH ₄; however for one particular metric (for one parameter value) theH ₄ becomes aG ₄ and the resultant metric is one which was rediscovered by Petrov in 1962. These Lewis metrics are thus shown to be Kasner metrics over complex fields. Here they are calledwindmill metrics because of the rotating relationship between the coordinates and the Killing vector fields admitted. The principal null directions of thereal Kasner and the windmill metrics are discussed; the two families then provide illustrations of two degenerate classes of spacetime metrics whose Weyl tensors are of Petrov type I, as discussed elsewhere by Arianrhod and McIntosh. An extension of the windmill-type generation of metrics to some other families of metrics is also discussed.     

Black holes with metric and fields of the same form

We present two rotating black hole solutions with axion ξ, dilaton f{\phi} and two U(1) vector fields. Starting from a non-rotating metric with three arbitrary parameters, which we have found previously, and applying the “Newman–Janis complex coordinate trick” we get a rotating metric g _μν with four arbitrary parameters namely the mass M, the rotation parameter a and the charges electric Q _E and magnetic Q _M . Then we find a solution of the equations of motion having this g _μν as metric. Our solution is asymptotically flat and has angular momentum J = M a, gyromagnetic ratio g = 2, two horizons, the singularities of the solution of Kerr, axion and dilaton singular only when r = a cos θ = 0 etc. By applying to our solution the S-duality transformation we get a new solution, whose axion, dilaton and vector fields have one more parameter. The metrics, the vector fields and the quantity l = x+ie^-2f{\lambda=\xi+ie^{-2\phi}} of our solutions and the solution of: Sen for Q _E , Sen for Q _E and Q _M , Kerr–Newman for Q _E and Q _M , Kerr, Reference Kyriakopoulos [Class. Quantum Grav. 23:7591, 2006, Eqs. (54–57)], Shapere, Trivedi and Wilczek, Gibbons–Maeda–Garfinkle–Horowitz–Strominger, Reissner–Nordstr?m, Schwarzschild are the same function of a, and two functions ρ ² = r(r + b) + a ² cos² θ and Δ = r(r + b) − 2Mr + a ² + c, of a, b and two functions for each vector field, and of a, b and d respectively, where a, b, c and d are constants. From our solutions several known solutions can be obtained for certain values of their parameters. It is shown that our two solutions satisfy the weak the dominant and the strong energy conditions outside and on the outer horizon and that all solutions with a metric of our form, whose parameters satisfy some relations satisfy also these energy conditions outside and on the outer horizon. This happens to all solutions given in the “Appendix”. Mass formulae for our solutions and for all solutions which are mentioned in the paper are given. One mass formula for each solution is of Smarr’s type and another a differential mass formula. Many solutions with metric, vector fields and λ of the same functional form, which include most physically interesting and well known solutions, are listed in an “Appendix”.     

Noether's theorem on a quasistochastic half-group of relativistic endomorphisms (QHGRE)

The quantum theory of free fields is used with Noether's theorem to discuss the polarization features of vector and spinor fields in the QHGRE (generalized de Broglie group) from the viewpoint of local isomorphism of the QHGRE for the product of two ordinary groups (the groups of x , t deformations and rotations). Estimates are made of the contributions to the polarization tensors for the corresponding fields. In both cases the spin tensors are induced by space and time generators of the generalized de Broglie group, and the rotational component of the QHGRE is responsible for the spin properties.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, Vol. 12, No. 1, pp. 32–36, January, 1969.We are indebted to D. D. Ivanenko for discussion of these topics.     

Vector fields on complex quantum groups

Using previous results we construct theq-analogues of the left invariant vector fields of the quantum enveloping algebra corresponding to the complex Lie algebras of typeA _n–1 ,B _n ,C _n , andD _n . These quantum vector fields are functionals over the complex quantum groupA. In the special caseA ₁ it is shown that this Hopf algebra coincides withU _q sl(2, ).     

Critical current density of a sample of melt grown Y1.2Ba1.8Cu2.4O x

Melt grown samples of Y_1.2Ba_1.8Cu_2.4O _x have been prepared and studied for their current carrying capacity. The composition was chosen to include Y₂BaCuO₅ (211) particles in the YBa₂Cu₃O _x (123) phase. The critical current density (J _c) of these samples was studied as a function of magnetic field using magnetization technique. The micrographic investigation shows well aligned grains in this material. The magnetic hysteresis measurements were done using a MPMS SQUID magnetometer up to the fields of 5.5 T. TheJ _c was estimated from the remanent magnetization using Bean model. Isothermal magnetization hysteresis loops at low fields reveal the presence of only one kind of hysteresis loops (corresponding to intragrain magnetizations). This is a valid proof that the weak links are greatly eliminated in these samples prepared by MG process. TheJ _c behaviour as a function of magnetic field has two components, a rapidly decaying exponential function of field and the other component that predominates at higher fields. This could be explained if we assume that the sample contains two phases of superconductors, one having a lowH _c2 becoming normal at fairly medium fields of the order of a few kilogauss will act as pinning centres for the other phase having higherH _c2 and hence higherJ _c at high fields.     

Radiation from a Circular Waveguide Flush-mounted to a Ground Plane covered by a Plasma or a Dielectric Slab

A theory of the radiation from the open end of a circular waveguide fed in the dominant TE₁₁ mode and flush-mounted to an infinite, perfectly conducting flat ground plane covered by a plasma or a dielectric sheath is developed. In this analysis, the contributions to the aperture admittance and the excited fields from the higher order TE_1n (n ≥ 1) and TM_1l(l ≥ 1) modes are included. The aperture admittance for which the higher order modes are considered, has been calculated by two methods: 1. variational technique and 2. successive iteration method. It is shown that for any desired order of accuracy the results of these approaches are identical. Expressions for radiated fields, surface waves for all radial values of r, and the fields which represent glancing incident fields inside the slab are derived. The contribution of the surface waves to the aperture admittance is also presented.     

Spontaneous Symmetry Breaking in Presence of Electric and Magnetic Charges

Starting with the definition of quaternion gauge theory, we have undertaken the study of SU(2) _e ×SU(2) _m ×U(1) _e ×U(1) _m in terms of the simultaneous existence of electric and magnetic charges along with their Yang-Mills counterparts. As such, we have developed the gauge theory in terms of four coupling constants associated with four-gauge symmetry SU(2) _e ×SU(2) _m ×U(1) _e ×U(1) _m . Accordingly, we have made an attempt to obtain the abelian and non-Abelian gauge structures for the particles carrying simultaneously the electric and magnetic charges (namely dyons). Starting from the Lagrangian density of two SU(2)×U(1) gauge theories responsible for the existence of electric and magnetic charges, we have discussed the consistent theory of spontaneous symmetry breaking and Higgs mechanism in order to generate the masses. From the symmetry breaking, we have generated the two electromagnetic fields, the two massive vector W ^± and Z ⁰ bosons fields and the Higgs scalar fields.