Realisation of a Lorentz algebra in Lorentz violating theory

A Lorentz non-invariant higher derivative effective action in flat spacetime, characterised by a constant vector, can be made invariant under infinitesimal Lorentz transformations by restricting the allowed field configurations. These restricted fields are defined as functions of the background vector in such a way that background dependence of the dynamics of the physical system is no longer manifest. We show here that they also provide a field basis for the realisation of a Lorentz algebra and allow the construction of a Poincaré invariant symplectic two-form on the covariant phase space of the theory.     

Reduction of wave function associated with electromagnetic fields for imaginary mass system to standard helicity representation of Lorentz group

Reduced expansions for electromagnetic fields associated with spin-1 particles of imaginary mass (tachyons) have been derived in terms of standard helicity representations of inhomogeneous Lorentz group. The effects of wave equation and reality condition on these reduced expansions to satisfy Maxwell’s field equations have been derived. The reduced expansions of charge and current source densities associated with these fields have also been derived and it has been shown that these fields cannot satisfy the Maxwell’s equations in the absence of both the current and charge source densities.     

Grassmann-Cartan algebra and Klein-Gordon equations

Given a Riemannian structure (M, g), a hypothesis is investigated that if= _p=0 ⁿ _p (M) is submitted to the differential condition (g++)=0, =mc/—which implies that each component of fulfills the Klein-Gordon equation (- ²) _p =0, ought to be interpreted as a natural complex of the bosonic fields. Then it is found that the complex admits the interpretation in the sense of first quantization with (M) being a convex set of states, with the structure of a Hilbert space over . The definite spin states of bosons are then pure states which are not conserved by the temporal evolution.     

Successive Lorentz transformations of the electromagnetic field

A velocity-orientation formalism to deal with compositions of successive Lorentz transformations, emphasizing analogies shared by Lorentz and Galilean transformations, has recently been developed. The emphasis in the present article is on the convenience of using the velocity-orientation formalism by resolving a paradox in the study of successive Lorentz transformations of the electromagnetic field that was recently raised by Mocanu. The paradox encountered by Mocanu results from the omission of the Thomas rotation (or, precession) which is involved in the composition of two Lorentz transformations with corresponding noncollinear velocity parameters. By resolving this paradox, we expose (i) the central role that the Thomas rotation plays in special relativity, (ii) the need to consider in special relativity orientations in addition to velocities between inertial frames, and (iii) the power and elegance of the novel velocity-orientation formalism. A similar paradox in STR that Mocanu pointed out, also resulting from the omission of the Thomas rotation, has been resolved in a previous communication.     

Analytic solution of gauge field equations for Lorentz gravity

The exterior analytic solution for a static, spherically symmetric system is given by means of a set of gauge field equations from Lorentz gravity in the curvature coordinate. The correction contributed by the gravitational gauge field in the exterior of a static sphere is obtained for the gravity.     

Lorentz invariance of wave equations and galilean invariance of diffusion equations

The Lorentz invariance of a certain class of wave equations and on the other hand the galilean invariance of a certain class of diffusion equations is discussed.     

Modified gravitational equations on braneworld with Lorentz invariant violation

The modified gravitational equations to describe a four-dimensional braneworld in the case with the Lorentz invariant violation in a bulk spacetime is presented. It contains a trace part of the brane energy-momentum tensor and the coefficients of all terms describe the Lorentz violation effects from the bulk spacetime. As an application, we apply this formalism to study cosmology. In respect to standard effective Friedmann equations on the brane, Lorentz invariance violation in the bulk causes a modification of this equations that can lead to significant physical consequences. In particular, the effective Friedmann equation on the brane explicitly depends on the equation of state of the brane matter and the Lorentz violating parameters. We show that the components of five-dimensional Weyl curvature are related to the matter on brane even at low energies. We also find that the constraints on the theory parameters are depend on the equation of state of the energy components of the brane matter. Finally, the stability of the model depend on the specific choices of initial conditions and the parameters β _i .     

Clifford algebra structure and Einstein's equations

We point out that a Clifford algebra representation for the Riemannian curvature leads to an equation for gravity similar to the Yang-Mills equation, in a gauge model for gravity with the Lorentz group. Einstein's equation of general relativity emerges as a natural solution in this approach.     

Additional symmetries for integrable equations and conformal algebra representation

We present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our method, additional symmetry equations arise together with their L-A pairs, so that they are integrable themselves. This procedure is based on a modified dressing method. For INEEs in 1+1 dimensions, some appropriate symmetry equations are shown to form the vector fields on a circle S ¹ algebra representation. In contrast to the so-called isospectral deformations, these symmetries result from conformal transformations of the associated linear problem spectrum. For INEEs in 2+1 dimensions, the commutation relations for symmetry equations are shown to coincide with operators , with integer m, p. Some additional results about Kac-Moody algebra applications are presented.     

Lorentz transformations for electromagnetic field vectors and the motion of charges in constant,homogeneous electromagnetic fields

A surface in velocity space which corresponds to Lorentz transformations minimizing the magnitudes of the vectors of a given electromagnetic field is formed. The relation with Lorentz transformations which do not change the magnitudes of the given electromagnetic field is indicated. The behavior of field vectors in a system of reference bound with a charge moving in a constant and homogeneous electromagnetic field is considered. It is shown that an electric charge in a constant and homogeneous electromagnetic field with inclusion of the effect of radiation reaction moves in such a manner that in the reference frame moving with it the magnitudes of the electromagnetic field vectors decrease and their directions approach one another. The time of transition to the state in which the field vectors in the system bound with the charge are mutually parallel is determined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 11, pp. 63–67, November, 1971.     

Causal signal transmission by quantum fields. VI: The Lorentz condition and Maxwell’s equations for fluctuations of the electromagnetic field

The general structure of electromagnetic interactions in the so-called response representation of quantum electrodynamics (QED) is analysed. A formal solution to the general quantum problem of the electromagnetic field interacting with matter is found. Independently, a formal solution to the corresponding problem in classical stochastic electrodynamics (CSED) is constructed. CSED and QED differ only in the replacement of stochastic averages of c-number fields and currents by time-normal averages of the corresponding Heisenberg operators. All relations of QED connecting quantum field to quantum current lack Planck’s constant, and thus coincide with their counterparts in CSED. In Feynman’s terms, one encounters complete disentanglement of the potential and current operators in response picture.     

Space-time algebra for the generalization of gravitational field equations

The Maxwell–Proca-like field equations of gravitolectromagnetism are formulated using space-time algebra (STA). The gravitational wave equation with massive gravitons and gravitomagnetic monopoles has been derived in terms of this algebra. Using space-time algebra, the most generalized form of gravitoelectromagnetic Klein–Gordon equation has been obtained. Finally, the analogy in formulation between massive gravitational theory and electromagnetism has been discussed.     

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU _q (g). They have the same FRT generatorsl ^± but a matrix braided-coproductL=LL, whereL=l ⁺ Sl ^–, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM _q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U _q (sl ₂)) (also known as the quantum Lorentz group) is the semidirect product as an algebra of two copies ofU _q (sl ₂), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.     

A generalized variational algebra and conserved densities for linear evolution equations

The symbolic algebra of Gel'fand and Dikii is generalized to the case of n variables. Using this algebraic approach we give a rigorous characterization of the polynomial kernel of the variational derivative. This is applied to classify all the conservation laws for linear polynomial evolution equations of arbitrary order.Partially supported by the Junta de Energía Nuclear, Madrid.     

Quantum Einstein’s equations and constraints algebra

In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein’s equations. We shall investigate this problem in the de-Broglie-Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein’s equations are derived.     

Discrete and continuous Bogomolny equations through the deformed algebra

We show that there exists one-parameter algebra connecting the discrete and continuous Bogomolny equations. The algebra is the deformation of the extended conformal algebra. This shows that the deformed algebra plays a role of the link between the matrix valued model and the model with one more space dimension higher.