Lax Pair Tensors and Integrable Spacetimes

The use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed. Some properties of the tensorial Lax pair formulation are stated. A mechanical system with a well-known Lax representation—the three-particle open Toda lattice—is geometrized by a suitable canonical transformation. In this way the Toda lattice is realized as the geodesic system of a certain Riemannian geometry. By using different canonical transformations we obtain two inequivalent geometries which both represent the original system. Adding a timelike dimension gives four-dimensional spacetimes which admit two Killing vector fields and are completely integrable.     

Invariance under conformal coordinate and point transformations

Coordinate and point transformations are studied in the context of conformal symmetry. When invariance requirements on arbitrary rank tensors are involved in both contexts, the similitudes and differences in transformation laws and invariance conditions are analysed in connection with those on tensor densities of weight W. Physically interesting tensors like the metric tensor, the electromagnetic field and the energy-momentum tensor are specifically examined. Some remarks on scalar fields and densities are added.     

A YANG–MILLS ELECTRODYNAMICS THEORY ON THE HOLOMORPHIC TANGENT BUNDLE

Considering a complex Lagrange space ([24]), in this paper the complex electromagnetic tensor fields are defined as the sum between the differential of the complex Liouville 1-form and the symplectic 2-form of the space relative to the adapted frames of the Chern–Lagrange complex nonlinear connection. In particular, an electrodynamics theory on a complex Finsler space is obtained.

We show that our definition of the complex electrodynamics tensors has physical meaning and these tensors generate an adequate field theory which offers the opportunity of coupling with the gravitation. The generalized complex Maxwell equations are written.

A gauge field theory of electrodynamics on the holomorphic tangent bundle is put over T′M and the gauge invariance to phase transformations is studied. An extension of the Dirac Lagrangian on T′M coupled with the electrodynamics Lagrangian is studied and it offers the framework for a unified gauge theory of fields.     

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.     

On the Noether character of higher local conserved quantities for some two-dimensional models

In the case of two-dimensional conformal-invariant field-theoretical models, it is shown that the higher conserved energy-momentum tensors are of a Noether character. They are generated from generalized nonlinear translations. There are also found nonlinear gauge transformations which give higher local conserved charges. For some of these transformations, the corresponding infinite Lie algebras are investigated.     

Towards a quantization of gauge fields on de Sitter group by functional integral method

A formulation of the de Sitter symmetry as a purely inner symmetry defined on a fixed Minkowski space-time is presented. We define the generators of the de Sitter group and write the structure equations using a constant deformation parameter λ. The conserved gauge currents are calculated, and their physical meaning is given. Local gauge transformations and the corresponding covariant derivative depending on the gauge fields are also obtained. We study the behavior of gauge fields, the torsion and curvature tensors and give a regularization technique in terms of the ζ function.     

Couplings of N=1 chiral spinor multiplets

We derive the action for n_L≥1 chiral spinor multiplets coupled to vector and scalar multiplets. We give the component form of the action, which contains gauge invariant mass terms for the antisymmetric tensors in the spinor superfield and additional Green–Schwarz couplings to vector fields. We observe that supersymmetry provides mass terms for the scalars in the spinor multiplet that do not arise from eliminating an auxiliary field. We construct the dual action by explicitly performing the duality transformations in superspace and give its component form.     

Some D type metric forms for spacetimes with shear,nongeodesic, and nondiverging principal null congruences

In the context of a null tetrad formalism it is assumed that the two real vectors of the tetrad have neither expansion nor rotation and that they are nongeodesic and/or have shear. This assumption is rarely used in search of algebraically special spacetimes. General relations for the connection and for the Ricci and Weyl tensors are presented. An additional assumption that these vectors are principal leads to Petrov type D spacetimes, reducible in some cases to conformally flat ones. Twenty-two explicit metric forms, probably hitherto unknown, are found.     

On the exponential of the 2-forms in relativity

A study of intrinsic properties of proper Lorentz tensors (tensor fields defining proper Lorentz transformations at every point of space-time) is made, giving rise to their covariant decompositions. The exponential series for a generic 2-form is covariantly summed, and the resulting proper Lorentz tensor is expressed as a linear combination of the metric tensor, the 2-form, its dual and its energy tensor. Some covariant expressions for the 2-form corresponding to the logarithmic branches of a proper Lorentz tensor are given. Some properties of the Lorentz group are easily found, concerning the surjectivity, local injectivity and local inversibility of the exponential map.     

A vector product formulation of special relativity and electromagnetism

The vector product method developed in previous articles for space rotations and Lorentz transformations is extended to the cases of four-vectors, anti-symmetric tensors, and their transformations in Minkowski space. The electromagnetic fields are expressed in six-vector form using the notationH +iE, and this vector form is shown to be relativistically invariant. The wave equations of electromagnetism are derived using these vector products. The following three equations are deduced, which summarize electrodynamics in a compact form: (1) Maxwell's four equations expressed as one, (2) the scalar and vector potential wave equations combined into one relation, and (3) the wave equations for the electric and magnetic fields and the continuity equation combined together. Space inversion, time reversal, and magnetic monopoles are also treated.On leave of absence from the University of Tel Aviv.     

Bäcklund transformations and the construction of the Atiyah-Ward ansätze for self-dual SU(2) gauge fields

Simple expressions for the ansätze of Atiyah and Ward are given in Yang's R gauge. Some non-linear transformations connecting solutions of the field equations are exhibited. These transformations are analogous to the Bäcklund transformations of the Sine-Gordon equation, etc., and provide an essentially algebraic inductive proof of the ansätze. Neither the sources for the anti-self-dual Maxwell fields used in the ansätze nor the reality conditions are discussed.     

Quantum fluctuations of vacuum stress tensors and spacetime curvatures

We analyze the quantum fluctuations of vacuum stress tensors and spacetime curvatures, using the framework of linear response theory which connects these fluctuations to dissipation mechanisms arising when stress tensors and spacetime metric are coupled. Vacuum fluctuations of spacetime curvatures are shown to be a sum of two contributions at lowest orders; the first one corresponds to vacuum gravitational waves and is restricted to light-like wavevectors and vanishing Einstein curvature, while the second one arises from gravity of vacuum stress tensors. From these fluctuations, we deduce noise spectra for geodesic deviations registered by probe fields which determine ultimate limits in length or time measurements. In particular, a relation between noise spectra characterizing spacetime fluctuations and the number of massless neutrino fields is obtained.     

Gauge Fields in General Parametrical Approach and Their Finslerian Reduction

The covariance principle of general relativity is extended to internal space. Associated gauge fields and tensors are systematically described, whereupon the variational principle is set up for all gauge fields by applying a Palatini-type method, thereby giving rise to an attractive self-contained theory in which the Einstein equations are intrinsically synthesized with the generalized Yang-Mills equations. General gauge-covariant physical field equations are formulated, showing that currents, external + internal spin tensors, and energy-momentum tensors can be introduced unambiguously under these general conditions and that the associated conservation laws can be derived. The electromagnetic field finds its gauge-geometric origin as the gauge field related to internal densities. To be operative with the tensor indices of external and internal types, this general theory must be bimetric. The assumptions that the gauge-covariant derivatives of metric tensors should vanish simplify the theory to the level of a Finslerian gauge approach.     

Amplitude ambiguities in second-harmonic generation frequency-resolved optical gating

We construct field shapes with distinct amplitude profiles that have nearly identical second-harmonic generation frequency-resolved optical gating (SHG FROG) traces. Although such fields are not true mathematical ambiguities, they result in experimentally indistinguishable FROG traces. These fields are neither time-reversed copies nor pulselets with a mere relative phase difference, which are well known nontrivial ambiguities for SHG FROG. We also show that for certain example fields, second-order interferometric autocorrelation is more sensitive to the pulse shape than is SHG FROG.     

Schwarzschild geometry,once more

Schwarzschild geometry exhibits interesting features when the field equations are decomposed with respect to a system of freely falling observers. We only use quantities behaving like tensors under a restricted group of transformations of the reference system. Moreover Lorentz transformations with non-constant velocities drastically change the physical picture of the theory.     

Note on Dirac–Kähler massless fields

We obtain the canonical and symmetrical Belinfante energy-momentum tensors of Dirac–Kähler’s fields. It is shown that the traces of the energy-momentum tensors are not equal to zero. We find the canonical and Belinfante dilatation currents which are not conserved, but a new conserved dilatation current is obtained. It is pointed out that the conformal symmetry is broken. The canonical quantization is performed and the propagator of the massless fields in the first-order formalism is found.     

A comment on the symmetries of Kerr black holes

Conditions are given for the linear dependence of the two Killing vectors, found by Hughston and Sommers to exist in a class of Einstein-Maxwell fields of Petrov typeD. The Killing tensors associated with these fields are shown to be contracted products of Killing Yano tensors.     

Linear and angular coherence momenta in the classical second-order coherence theory of vector electromagnetic fields

A new concept of vector and tensor densities is introduced into the general coherence theory of vector electromagnetic fields that is based on energy and energy-flow coherence tensors. Related coherence conservation laws are presented in the form of continuity equations that provide new insights into the propagation of second-order correlation tensors associated with stationary random classical electromagnetic fields.     

Dirac Operators on Taub-NUT Space: Relationship and Discrete Transformations

It is shown that the N = 4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant constant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N = 4 supersymmetry.