Abstract
We derive finite boost transformations based on the Lorentz sector of the bicross-product-basis $\kappa$-Poincare' Hopf albegra. We emphasize the role of these boost transformations in a recently-proposed new relativistic theory. We find that when the (dimensionful) deformation parameter is identified with the Planck length, which together with the speed-of-light constant has the status of observer-independent scale in the new relativistic theory, the deformed boosts saturate at the value of momentum that corresponds to the inverse of the Planck length.
Highlights
We find that when the deformation parameter is identified with the Planck length, which together with the speed-of-light constant has the status of observer-independent scale in the new relativistic theory, the deformed boosts saturate at the value of momentum that corresponds to the inverse of the Planck length
Several approaches to the unification of General Relativity and Quantum Mechanics have led to arguments in favour of the emergence of a minimum length and/or a maximum momentum, possibly connected with the Planck length (Lp ∼ 10−33cm) and this has motivated some authors to argue that somehow boosts should saturate at the Planck scale
One example of this new type of Relativity postulates, an example based on the introduction of a deformed dispersion relation, was analyzed in detail [3] obtaining various results in leading order in the second (Planck-scale related) observer-independent scale, including the nature of the transformation rules between different inertial observers and kinematic rules1 for particleproduction processes, emphasizing those aspects that lead to the emergence of a minimum length, a minimum length uncertainty and a maximum momentum
Summary
Several approaches to the unification of General Relativity and Quantum Mechanics have led to arguments in favour of the emergence of a minimum length and/or a maximum momentum, possibly connected with the Planck length (Lp ∼ 10−33cm) and this has motivated some authors (see, e.g., Refs. [1,2]) to argue that somehow boosts should saturate at the Planck scale. [3], where, following a close analogy with our present description of fundamental physics (in which Galileo’s Relativity Principle coexists with the observer-independent scale c, the speed-of-light fundamental constant), it was shown that logically consistent relativistic theories can host both c and a second scale, possibly connected with the Planck length, as observer-independent scales One example of this new type of Relativity postulates, an example based on the introduction of a deformed dispersion relation, was analyzed in detail [3] obtaining various results in leading order in the second (Planck-scale related) observer-independent scale, including the nature of the transformation rules between different inertial observers and kinematic rules for particleproduction processes, emphasizing those aspects that lead to the emergence of a minimum length, a minimum length uncertainty and a maximum momentum (minimum wavelength). Some cosmological scenarios based on κ-Poincare were analyzed, leading to encouraging preliminary results [16,17]
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