Superluminal coordinate transformations: Four-dimensional case

Abstract

We investigate, from a group-theoretical point of view, the possibility of implementing the so-called extended principle of relativity. This consists in postulating that the set of all equivalent reference frames contains frames whose relative velocities are larger than c, in addition to those whose coordinates are related by proper orthochronous Lorentz transformations. We show that implementing the extended principle of relativity by means of either real or complex linear transformations results in strong conflicts with experiment and/or intractable problems of interpretation. We then briefly analyze alternative approaches to four-dimensional superluminal transformations, in which the extended principle of relativity is either weakened or completely abandoned.

Quantum principle of relativity

“…Second, both branches of solutions form a symmetry group only in the considered 1+1 dimensional scenario. This is not the case in the 1+3 dimensional case [3], therefore we will carefully discuss this case separately in section 5. For now we stick to the 1+1 scenario, in which equations (9) describe a hyperbolic rotation by the angle Î ( ) .…”

“…. This cannot be a symmetry group, because it involves transformations such as direction-dependent time dilation, which are not observed [3]. Therefore superluminal transformations in 1+3 dimensional spacetime should not be symmetries.…”

Quantum Principle of Relativity

“…Second, both branches of solutions form a symmetry group only in the considered 1+1 dimensional scenario. This is not the case in the 1+3 dimensional case [3], therefore we will carefully discuss this case separately in section 5. For now we stick to the 1+1 scenario, in which equations (9) describe a hyperbolic rotation by the angle…”

Majorana spinors and extended Lorentz symmetry in four-dimensional theory

“…Consider spinor transformations (14) and (15) in which the matrix S depends on the point of the space-time manifold. Let D µ = ∂ µ + Ω µ be a covariant derivative of the spinor field ϕ where Ω µ is the spin connection.…”