Symmetric Tensor Fields Research Articles

MASSIVE FIELDS WITH ARBITRARY INTEGER SPIN IN HOMOGENEOUS ELECTROMAGNETIC FIELD

We study the interaction of gauge fields of arbitrary integer spins with the constant electromagnetic field. We reduce the problem of obtaining the gauge-invariant Lagrangian of integer spin fields in the external field to purely algebraic problem of finding a set of operators with certain features using the representation of the high-spin fields in the form of vectors in a pseudo-Hilbert space. We consider such a construction up to the second order in the electromagnetic field strength and also present an explicit form of interaction Lagrangian for a massive particle of spin s in terms of symmetrical tensor fields in linear approximation. The result obtained does not depend on dimensionality of space–time.

On the cohomology of sℓ( m + 1, [formula omitted]) acting on differential operators and sℓ( m + 1, [formula omitted])-equivariant symbol

Let D k λμ ( R m) denote the space of differential operators of order ≤ k from the space of λ-densities of R m into that of μ-densities and let S δ k ( R m) be the space of k- contravariant, symmetric tensor fields on R m valued in the δ-densities, δ = μ − λ. Denote by sℓ m + 1 the projective embedding of sℓ( m + 1, R ) as a subalgebra of the Lie algebra of vector fields of R m. One computes the cohomology of sℓ m + 1 with coefficients in the space of differential operators from S δ p ( R m) into S δ q ( R m). It is non vanishing only for some critical values of δ. For m = 1, these are the values pointed out by H. Gargoubi in a completely different context [6]. This allows to determine the condition under which the short exact sequence of sℓ m + 1 - modules0 → D λμ k − 1( R m) → D λμ k( R m) σ → S δ k → 0 is split (σ is the symbol map). From this one recovers and generalizes useful results about the structure of the sℓ m + 1 - moduleD λμ( R m) = ∪ kD λμ k( R m) [3, 5, 6, 8]. The cohomology of the latter is also computed.

DUALITY INVARIANCE OF COSMOLOGICAL SOLUTIONS WITH TORSION

We show that for a string moving in a background consisting of maximally symmetric gravity, dilaton field and second rank antisymmetric tensor field, the O(d) ⊗ O(d) transformation on the vacuum solutions gives inequivalent solutions that are not maximally symmetric. We then show that the usual physical meaning of maximal symmetry can be made to remain unaltered even if torsion is present and illustrate this through two toy models by determining the torsion fields, the metric and Killing vectors. Finally we show that under the O(d) ⊗ O(d) transformation this generalized maximal symmetry can be preserved under certain conditions. This is interesting in the context of string related cosmological backgrounds.

Massive fields with arbitrary integer spin in symmetrical Einstein space

We study the propagation of gauge fields with arbitrary integer spins in the symmetrical Einstein space of any dimensionality. We reduce the problem of obtaining a gauge-invariant Lagrangian of integer spin fields in such a background to the algebraic problem of the search for a set of operators with certain features by means of the representation of higher-spin fields in the form of vectors in a pseudo-Hilbert space. We consider such a construction at linear order in the Riemann tensor and scalar curvature and also present an explicit form of interaction Lagrangians and gauge transformations for massive particles of spins 1 and 2 in terms of symmetrical tensor fields.

Massive symmetric tensor field on AdS

The two-point Green function of a local operator in CFT corresponding to a massive symmetric tensor field on the AdS background is computed in the framework of the AdS/CFT correspondence. The obtained two-point function is shown to coincide with the two-point function of the graviton in the limit when the mass vanishes.

Homogeneous Fedosov star products on cotangent bundles II: GNS representations, the WKB expansion, traces, and applications

This paper is part II of a series of papers on the deformation quantization on the cotangent bundle of an arbitrary manifold Q. For certain homogeneous star products of Weyl ordered type (which we have obtained from a Fedosov type procedure in part I, see [M. Bordemann, N. Neumaier, S. Waldmann, Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation, Comm. Math. Phys. 198 (1998) 363–396]) we construct differential operator representations via the formal GNS construction (see [M. Bordemann, S. Waldmann, Formal GNS construction and states in deformation quantization, Comm. Math. Phys. 195 (1998) 549–583]). The positive linear functional is integration over Q with respect to some fixed density and is shown to yield a reasonable version of the Schrödinger representation where a Weyl ordering prescription is incorporated. Furthermore we discuss simple examples like free particle Hamiltonians (defined by a Riemannian metric on Q) and the implementation of certain diffeomorphisms of Q to unitary transformations in the GNS (pre-)Hilbert space and of time reversal maps (involutive anti-symplectic diffeomorphisms of T ∗ Q ) to anti-unitary transformations. We show that the fixed-point set of any involutive time reversal map is either empty or a Lagrangian submanifold. Moreover, we compare our approach to concepts using integral formulas of generalized Moyal-Weyl type. Furthermore we show that the usual WKB expansion with respect to a projectable Lagrangian submanifold can be formulated by a GNS construction. Finally we prove that any homogeneous star product on any cotangent bundle is strongly closed, i.e. the integral over T ∗ Q w.r.t. the symplectic volume vanishes on star-commutators. An alternative Fedosov type deduction of the star product of standard ordered type using a deformation of the algebra of symmetric contravariant tensor fields is given.

On objective corotational rates and their defining spin tensors

In this paper, we prove a general result on objective corotational rates and their defining spin tensors: let Ω* be a spin tensor that is associated with the rotation and deformation of a deforming material body in an arbitrary manner indicated by Ω* = Y(B, D, W), where B and D and W are the left Cauchy Green tensor and the stretching tensor and the vorticity tensor, respectively. Then the corotational rate of σ defined by the spin Ω*, i.e., the tensor field σ* = σ+σΩ*—Ω*σ, is objective for every time-differentiable objective Eulerian symmetric tensor field a if and only if the spin tensor Ω* assumes the form Ω* = W + Y(B, D). where Y(B, D) is an antisymmetric tensor-valued isotropic function. Furthermore, by virtue of certain necessary or reasonable requirements, it is found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. Accordingly, a general explicit basis-free expression for the latter is established in terms of the left Cauchy-Green tensor B, the vorticity tensor W and the stretching tensor D as well as the introduced antisymmetric function. By choosing several particular forms of the latter, it is shown that all commonly-used spin tensors are incorporated into this general expression in a natural way.

Conformally covariant differential operators: symmetric tensor fields

We extend previous work on conformally covariant differential operators to consider the case of second-order operators acting on symmetric traceless tensor fields. The corresponding flat space Green function is explicitly constructed and shown to be in accord with the requirements of conformal invariance.

Finite volume discretization with imposed flux continuity for the general tensor pressure equation

We present a new family of flux continuous, locally conservative, finite volume schemes applicable to the diagonal and full tensor pressure equations with generally discontinuous coefficients. For a uniformly constant symmetric elliptic tensor field, the full tensor discretization is second order accurate with a symmetric positive definite matrix. For a full tensor, an M-matrix with diagonal dominance can be obtained subject to a sufficient condition for ellipticity. Positive definiteness of the discrete system is illustrated. Convergence rates for discontinuous coefficients are presented and the importance of modeling the full permeability tensor pressure equation is demonstrated.

Theoretical Ideas of the XX century for describing electromagnetism

Quantum electrodynamics is a well-accepted theory. But, we believe it useful to look at formalisms which provide alternative ways to describe light, because the development of quantum field theories based primarily on the gauge principle have, in recent years, met with considerable difficulties. There are numerous generalized theories and, mainly, they are characterized by introducing some additional parameters and/or longitudinal modes of electromagnetism. The Majorana–Oppenheimer form of electrodynamics, the Sachs' theory of Elementary Matter, the analysis of the action-at-a-distance concept, presented recently by Chubykalo and Smirnov-Rueda, and the analysis of the claimed ‘longitudinality’ of the anti–symmetric tensor field after quantization are examined in this article. We list also recent advances in the Weinberg 2(2J+1) formalism (which is built on first principles) and in the Majorana theory for neutral particles. They can serve as starting points for constructing the quantum theory of light.

Energy and Momentum in the Tetrad Theory of Gravitation

We study the energy and momentum of an isolated system in the tetrad theory of gravitation, starting from the most general Lagrangian quadratic in torsion, which involves four unknown parameters. When applied to the static spherically symmetric case, the parallel vector fields take a diagonal form, and the field equation has an exact solution. We analyze the linearized field equation in vacuum at distances far from the isolated system without assuming any symmetry property of the system. The linearized equation is a set of coupled equations for a symmetric and skew-symmetric tensor fields, but it is possible to solve it up to $O(1/r)$ for the stationary case. It is found that the general solution contains two constants, one being the gravitational mass of the source and the other a constant vector ${\grave B_\alpha}$. The total energy is calculated from this solution and is found to be equal to the gravitational mass of the source. We also calculate the spatial momentum and find that its value coincides with the constant vector ${\grave B_\alpha}$. The linearized field equation in vacuum, which is valid at distances far from the source, does not give any information about whether the constant vector ${\grave B_\alpha}$ is vanishing or not. For a weakly gravitating source for which the field is weak everywhere, we find that the constant vector ${\grave B_\alpha}$ vanishes.

A polynomial first-order action for the Dirichlet 3-brane

A new first order action for type IIB Dirichlet 3-brane is proposed. Its form is inspired by the superfield equations of motion obtained recently from the generalized action principle. The action involves auxiliary symmetric spin tensor fields. It seems promising for a reformulation of the generalized action in a structure most adequate for investigating the extrinsic geometry of the super-3- brane, but also for further studies of string dualities.

Boundary identifiability of residual stress via the Dirichlet to Neumann map

An interesting problem in nondestructive evaluation is the determination of the residual stress of an elastic body. Residual stress is the stress in a body in the absence of any external forces. In the linear theory of elasticity, the residual stress in a body is represented by a divergence-free, second-order symmetric tensor field with vanishing boundary traction. If the elastic properties of the body are described by Lamé functions and and residual stress T, it is shown in this paper (for dimensions ) that T and are determined at any point on the boundary of the body by the Dirichlet to Neumann map.

Intrinsically Biaxial Systems: A Variational Theory for Elastomers

Elastomeric networks of polymer liquid crystals are a classical example of “solid liquid crystals”. They are organized in aggregates of ellipsoidal shape, which can be represented by a symmetric tensor field L , which is closely related to the nematic order parameter Q . We describe a continuum theory, which takes into account both the mechanical forces and the nematic properties of the system.

Integral geometry of a tensor field on a surface of revolution

from the known integrals of f over the geodesics of metric (0.1) which join the points of the circle ρ = ρ1. His method, exposed in detail in § 3.4 of the book [4], also bases on the above-mentioned geometric fact. Due to this fact, some Volterra integral equation of second kind appears for each Fourier coefficient of the function f(ρ cos θ, ρ sin θ). This equation implies uniqueness of a solution to the problem under assumption (0.2). It was shown in § 3.7 of [4] that the linearization of the inverse kinematic problem for anisotropic media leads to the integral geometry problem for a symmetric tensor field of second degree. As was mentioned there, a solution to the latter is not unique. Theorem 3.6 of [4] states only that some part of components of a sought tensor field can be recovered under the assumption that the other components are known. The integral geometry problem for tensor fields of higher degree is of interest in its own right. For instance, it was shown in Chapter 7 of [5] that, in the case of quasi-isotropic elastic media, the problem of determining the anisotropic part of the elasticity tensor from the results of measuring the phase of a compression wave is equivalent to the integral geometry problem for a symmetric tensor field of fourth degree. The present article treats the integral geometry problem of symmetric tensor fields along the geodesics of a spherically symmetric metric. We completely describe the extent of nonuniqueness for a solution to the problem. The main result states

Schouten–Nijenhuis brackets

The Poisson and graded Poisson Schouten–Nijenhuis algebras of symmetric and antisymmetric contravariant tensor fields, respectively, on an n-dimensional manifold M are shown to be n-symplectic. This is accomplished by showing that both brackets may be defined in a unified way using the n-symplectic structure on the bundle of linear frames LM of M. New results in n-symplectic geometry are presented and then used to give globally defined representations of the Hamiltonian operators defined by the Schouten–Nijenhuis brackets.

A Yang–Mills Theory in Loop Space and Chapline–Manton Coupling

We consider a Yang–Mills theory in loop space with the affine gauge group. From this theory, we derive a local field theory with Yang–Mills fields and Abelian antisymmetric and symmetric tensor fields of the second rank. The Chapline–Manton coupling, i.e. coupling of Yang–Mills fields and a second-rank antisymmetric tensor field via the Chern–Simons three-form is obtained systematically.

The topology of symmetric, second-order 3D tensor fields

The authors study the topology of symmetric, second-order tensor fields. The results of the study can be readily extended to include general tensor fields through linear combination of symmetric tensor fields and vector fields. The goal is to represent their complex structure by a simple set of carefully chosen points, lines, and surfaces analogous to approaches in vector field topology. They extract topological skeletons of the eigenvector fields and use them for a compact, comprehensive description of the tensor field. Their approach is based on the premise: analyze, then visualize. The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. Degenerate points play a similar role as critical points in vector fields. In tensor fields they identify two kinds of elementary degenerate points, which they call wedge points and trisector points. They can combine to form more familiar singularities-such as saddles, nodes, centers, or foci. However, these are generally unstable structures in tensor fields. Based on the notions developed for 2D tensor fields, they extend the theory to include 3D degenerate points. Examples are given on the use of tensor field topology for the interpretation of physical systems.

Gauge Theory of Massive Tensor Field. II: Covariant Expressions

Covariant forms are given to a gauge theory of massive tensor field. This is accomplished by introducing another auxiliary field of scalar type to the system composed of a symmetric tensor field and an auxiliary field of vector type. The situation is compared to the case of the theory in which a tensor field describes a scalar ghost as well as an ordinary massive tensor. In this case only an auxiliary vector field is needed to give covariant expressions for the gauge theory.

Consistent spin-two coupling and quadratic gravitation

A discussion of the field content of quadratic higher-derivative gravitation is presented, together with a new example of a massless spin-two field consistently coupled to gravity. The full quadratic gravity theory is shown to be equivalent to a canonical second-order theory of a massive scalar field, a massive spin-two symmetric tensor field and gravity. The conditions showing that the tensor field describes only spin-two degrees of freedom are derived. A limit of the second-order theory provides a new example of massless spin-two field consistently coupled to gravity. A restricted set of vacua of the second-order theory is also discussed. It is shown that flat-space is the only stable vacuum of this type, and that the spin-two field around flat space is unfortunately always ghost-like.