Superluminal Lorentz Transformations Research Articles

Tachyonic Dirac Equation Revisited

In this paper, we revisit the two theoretical approaches for the formulation of the tachyonic Dirac equation. The first approach works within the theory of restricted relativity, starting from a Lorentz invariant Lagrangian consistent with a spacelike four-momentum. The second approach uses the theory of relativity extended to superluminal motions and works directly on the ordinary Dirac equation through superluminal Lorentz transformations. The equations resulting from the two approaches show mostly different, if not opposite, properties. In particular, the first equation violates the invariance under the action of the parity and charge conjugation operations. Although it is a good mathematical tool to describe the dynamics of a space-like particle, it also shows that the mean particle velocity is subluminal. In contrast, the second equation is invariant under the action of parity and charge conjugation symmetries, but the particle it describes is consistent with the classical dynamics of a tachyon. This study shows that it is not possible with the currently available theories to formulate a covariant equation that coherently describes the neutrino in the framework of the physics of tachyons, and depending on the experiment, one equation rather than the other should be used.

Hyperbolic Superluminal Scator Algebra

An alternate time-space framework is proposed by means of a hyperbolic scator algebraic formalism where the deformed Lorentz transformations are an immediate consequence of the formal properties of the algebra. We survey the group properties of the hyperbolic product in the restricted, super-restricted and zero magnitude subsets as well as in the complete scator set. Some properties of the magnitude are derived together with a decomposition of 1 + n dimensional scators in n products of scator elements with scalar component and only one non-vanishing director component. These results are used to describe the scator properties in 1+ n dimensions. The super-restricted subset is associated with superluminal frames or tachyons. The various compositions of subluminal and superluminal events in the same direction follow the canonical superluminal Lorentz transformations interpretation. Composition of orthogonal components produce novel results. Products of scators with components within and outside the light pyramid yield sub or super luminal scators as well as scators belonging to a mixed state.

String inspired model and superluminal Lorentz transformations

String inspired model and superluminal Lorentz transformations

Two Fundamental Representations of Localized Pulse Solutions to the Scalar Wave Equation

A study is undertaken of two fundamental representations suitable for the derivation of localized pulse (LW) solutions to the scalar wave equation. The first one uses superpositions over products of plane waves moving in opposite directions along the characteristic variables z - ct and Z + ct. This bidirectional representation, introduced in an earlier publication, has proved instrumental in advancing our understanding of Focus Wave Mode (FWM)-like pulses. The second representation, based on the Lorentz invariance of the scalar wave equation, uses products of plane waves propagating along the subluminal and superluminal boost variables. This representation is suitable for the derivation of X-wave-type solutions. Subluminal and superluminal Lorentz transformations are used to derive closed-form LW solutions to the scalar wave equation by boosting known solutions of other equations, e.g., the 2-D scalar wave equation, the Helmholtz equation and Laplace's equation. Several of these LW solutions are deduced in this manner and their properties are discussed. Of particular interest is the derivation of a novel finite energy LW solution, named the Modified Focus X-Wave pulse. It is characterized by low sidelobe levels, a desirable property for applications, e.g., in pulse echo techniques used in medical imaging.

Two Fundamental Representations of Localized Pulse Solutions To the Scalar Wave Equation - Abstract

A study is undertaken of two fundamental representations suitable for the derivation of localized pulse (LW) solutions to the scalar wave equation. The first one uses superpositions over products of plane waves moving in opposite directions along the characteristic variables z - ct and z + ct. This bidirectional representation, introduced in an earlier publication, has proved instrumental in advancing our understanding of Focus Wave Mode (FWM)-like pulses. The second representation, based on the Lorentz invariance of the scalar wave equation, uses products of plane waves propagating along the subluminal and superluminal boost variables. This representation is suitable for the derivation of X-wave-type solutions. Subluminal and superluminal Lorentz transformations are used to derive closed-form LW solutions to the scalar wave equation by boosting known solutions of other equations, e.g., the 2-D scalar wave equation, the Helmholtz equation and Laplace's equation. Several of these LW solutions are deduced in this manner and their properties are discussed. Of particular interest is the derivation of a novel finite energy LW solution, named the Modified Focus X-Wave pulse. It is characterized by low sidelobe levels, a desirable property for applications, e.g., in pulse echo techniques used in medical imaging.

Group structure of superluminal Lorentz transformations and their representation

Constructing the generalized superluminal Lorentz transformations for inertial frames with nonpreferred orientation, we have investigated their group theoretical and commutation properties. The extended homogeneous Lorentz group in terms of these transformations has been constructed and its generators have been derived; these have been shown to generate the ray representation of the resulting Poincaré group.

Generalized Lorentz transformations in six-dimensional space-time

We have completely solved for the first time Cole's conditions for subluminal as well as superluminal Lorentz transformations by considering the more general dependence of the transformation submatrices on the velocity matrix. The apparent imaginary factor in the superluminal Lorentz transformations (SLT) is interpreted through congruence transformation as real coordinate SLT.

Erratum to Formal and physical properties of the generalized (subluminal and Superluminal) Lorentz transformations

We investigate the mathematical and physical properties of the generalized Lorentz transformations (both subluminal and Superluminal). The form here adopted for the Superluminal Lorentz transformations is the one—recently introduced by us—which satisfies the requested group-theoretical properties. We clarify the role of the interpretation procedure of the imaginary quantities also from the formal point of view, for both the «longitudinal» and the «transverse» co-ordinates. Careful attention is devoted to define four-momentum and three-velocity for tachyons. At last, the shape of a tachyon—obtained by applying to an ordinary particle ageneric Superluminal Lorentz transformation (without rotations)—is studied. As a simplifying tool, we make recourse also to the «light-cone co-ordinates» and to «dilation-invariant» co-ordinates.

Symmetries of fields associated with dyons

Investigating the dynamical symmetries of the system of dyons under subluminal and imaginary superluminal Lorentz transformations, it has been demonstrated that, though the parity and time reversal are not symmetric operations for this system, they possess strong symmetry under charge conjugation and space-time inversions in superluminal frame (i.e. for tachyonic dyons). It has also been shown thatCPT is not conserved for tachyonic dyons and that the rotational symmetry in charge space for the system of dyons is responsible for the conservation of residual angular momentum carried by dyons in generalized fields.

Behaviour of dyons in superrelativity. - II

By undertaking the study of the behaviour of dyons under real superluminal Lorentz transformations, it has been demonstrated that their field equation, equation of motion and the Lorentz force acting on them in their interaction with a superluminal electromagnetic field do not remain invariant under these transformation except in the direction of the tachyon corridor. It has also been shown that for undertaking the study of behaviour of dyons under superrelativity the best choice is that of imaginary Lorentz transformations.

Perspectives of superluminal Lorentz transformations

The difficulties connected with the introduction of Superluminal Lorentz transformations both as maps between frames of a single manifold and as maps between two manifolds are briefly discussed. It is suggested that fibre bundles with more than one canonical projection should be used. An example is given in which the bundle manifold is a Lie group.

The introduction of Superluminal Lorentz transformations: A revisitation

We revisit the introduction of the Superluminal Lorentz transformations which carry from “bradyonic” inertial frames to “tachyonic” inertial frames, i.e., which transform time-like objects into space-like objects, andvice versa. It has long been known that special relativity can be extended to Superluminal observers only by increasing the number of dimensions of the space-time or—which is in a sense equivalent—by releasing the reality condition (i.e., introducing also imaginary quantities). In the past we always adopted the latter procedure. Here we show the connection between that procedure and the former one. In other words, in order to clarify the physical meaning of the imaginary units entering the classical theory of tachyons, we have temporarily to call into play anauxiliary six-dimensional space-time M(3, 3); however, we are eventually able to go back to the four-dimensional Minkowski space-time. We revisit the introduction of the Superluminal Lorentz transformations also under another aspect. In fact, the generalized Lorentz transformations had been previously written down in a form suited only for the simple case of collinear boosts (e.g., they formed a group just for collinear boosts). We express now the Superluminal Lorentz transformations in a more general form, so that they constitute a group together with the ordinary—orthochronousand antichronous—Lorentz transformations, and reduce to the previous form in the case of collinear boosts. Our approach introduces either real or imaginary quantities, with exclusion of (generic) complex quantities. In the present context, a procedure—in two steps—for interpreting the imaginary quantities is put forth and discussed. In the case of a chain of generalized Lorentz transformations, such a procedure (when necessary) is to be applied only at the end of the chain. Finally, we justify why we call “transformations” also the Superluminal ones.

Study of fields associated with spin-1 massive particles carrying magnetic charge

The reduction and second quantization of electromagnetic fields associated with spin-1 massive particles carrying magnetic charge only have been carried out and the expansions for Hamiltonian and momentum operators have also been derived. It has been demonstrated that a magnetic monopole at rest behaves as an electrically charged tachyon as the consequence of complex superluminal Lorentz transformations.

Wave-particle duality and extended special relativity

It is shown that the wave-particle duality implies the extension of special relativity to superluminal frames. The real one-dimensional superluminal Lorentz transformations are obtained as a special case of the Parker-Antippa transformations. Some interrelations between extended special relativity and the many-wave hypothesis are discussed.

Critical comments on the discussion about tachyonic causal paradoxes and on the concept of superluminal reference frame

The discussions of the tachyonic causal paradoxes and the concept of superluminal reference frame are criticized. The essence of the construction of the known paradoxes is revealed. Some possibilities of eliminating these paradoxes without contradicting the theory of relativity, are discussed. The tachyonic causal loop in an arbitrarily dimensional flat space-time is formally defined. The logical relations between assumptions on existence (or nonexistence) of the tachyonic causal loops and of inertial reference frames preferred in the tachyon kinematics are given. Such frames are not preferred in relation to bradyons and luxons, and maybe are not preferred in the dynamics of the tachyons. The theorem is proved which shows that the discussion on the tachyonic causal loops concerns also the preferred frames. The operational definitions of spacelike, timelike, and null vectors are given. It is shown that superluminal transformations and reference frames do not exist inside the theory of relativity. It is also shown that the so-called superluminal Lorentz transformations are not in fact transformations but mappings. It is concluded that the existence of tachyonic phenomena is not contradictory to the theory of relativity, while the concept of usual superluminal reference frame is contradictory to that theory.

Quaternionic formulation for electromagnetic-field equations

The quaternionic formalism for subluminal field equations (Maxwell’s equations) and its interrelationship with complex superluminal Lorentz transformations have been given and it has been shown that the quaternionic forms of relativistic equations describe tachyons.

О форме тахионов

We study some aspects of the experimental behaviour of tachyons, in particular by finding out their « apparent » shape. A Superluminal particle, which in its own rest frame is spherical or ellipsoidal (and with an infinite lifetime), would « appear » to a laboratory frame as occupying the whole region of space bound by a double cone and a twosheeted hyperboloid. Such a structure (the tachyon « shape ») rigidly travels with the speed of the tachyon. However, if the Superluminal particle has a finite lifetimein its rest frame, then in the laboratory frame it gets afinite space extension. As a by-product, we are able to interpret physically the imaginary units entering—as is well known—the transverse co-ordinates in the Superluminal Lorentz transformations. The various particular or limiting cases of the tachyon shape are thoroughly considered. Finally, some brief considerations concerning possible experiments to look for tachyons are added.

Study of superluminal electromagnetic fields

Behavior of Maxwell’s equations under imaginary and real superluminal Lorentz transformations respectively has been reexamined and the expression of a Lorentz force acting on an electric charge interacting with superluminal electromagnetic fields has been derived. It has been shown that in four-dimensional space-time as well as in six-dimensional space-time the electrically charged tachyon interacting with electromagnetic fields behaves neither as a purely electric charge nor as a purely magnetic monopole.

Comment on “the behavior of maxwell's equations under real superluminal lorentz transformations”

Negi et al. have recently obtained field equations for the superluminal electromagnetic field, in theories based on real superluminal transformations along a “tachyon corridor”. Their results differ from equations obtained some time ago by the present authors. We trace the source of the discrepancy to the failure of Negi et al. to consistently transform all relevant quantities from the old to the new Lorentz frame.

Revisiting the superluminal Lorentz transformations and their group-theoretical properties

We briefly revisit the introduction of generalized Lorentz transformations (GLT) in four dimensions, by writing them in a new form which satisfies the requested group-theoretical properties. The new GLTs, in the particular case of boosts, reduce to the ones by Mignani and Recami (the latter, in fact, forming a group only in the case of collinear boosts). Some other aspects are briefly clarified.