Over these past few years several quantum-gravity research groups have been exploring the possibility that in some Planck-scale nonclassical descriptions of spacetime one or another form of nonclassical spacetime symmetries might arise. One of the most studied scenarios is based on the use of Hopf algebras, but previous attempts were not successful in deriving constructively the properties of the conserved charges one would like to obtain from the Hopf structure, and this in turn did not allow a crisp physical characterization of the new concept of spacetime symmetry. Working within the example of κ-Minkowski noncommutative spacetime, known to be particularly troublesome from this perspective, we observe that these past failures in the search of the charges originated from not recognizing the crucial role that the noncommutative differential calculus plays in the symmetry analysis. We show that, if the properties of the κ-Minkowski differential calculus are correctly taken into account, one can easily perform all the steps of the Noether analysis and obtain an explicit formula relating fields and energy-momentum charges. Our derivation also exposes the fact that an apparent source of physical ambiguity in the description of the Hopf-algebra rules of action, which was much emphasized in the literature, actually only amounts to a choice of conventions and in particular does not affect the formulas for the charges.1 The space indices j, l take values in {1, 2, 3} while 0 is the time index. We shall later also use the spacetime indices µ, ν, α, which take values in {0, 1, 2, 3}.2 Rather than our length scale λ a majority of authors use the energy scale κ, which is the inverse of λ (λ → 1/κ). 3 One notices that κ-Minkowski and the κ-Poincaré Hopf algebra form a "Heisenberg double" [8,9], i.e. κ-Minkowski and κ-Poincaré are linked, as algebras, in a way that is rather similar to the relationship between classical Minkowski spacetime and the classical Poincaré Lie algebra.
We study the quantization of a linear scalar field, whose symmetries are described by the κ-Poincaré Hopf-algebra, via deformed Fock space construction. The one-particle sector of the theory exhibits a natural (planckian) cut-off for the field modes. At the multi-particle level the non-trivial co-algebra structure of κ-Poincaré leads to a deformed bosonization in the construction of Fock space states. These physical states carry energy-momentum charges which are divergenceless and obey a deformed dispersion relation. * Electronic address: marzano@perimeterinstitute.ca † Electronic address: antonino.marciano@roma1.infn.it
We investigate a (super-)renormalizable and ghost-free theory of gravity, showing that under a natural (exponential) ansatz of the form factor and a suitable truncation it can give rise to the Starobinsky inflationary theory in cosmological frameworks, and thus offering a theoretical justification of its origin. We study the corresponding inflationary evolution and we examine the generation of curvature perturbations, adapting the f (R)-like equations in a symmetry-reduced FLRW metric. Furthermore, we analyze how the ultraviolet regime of a simply renormalizable and unitary theory of gravity is also compatible with the Starobinsky action, and hence we show that such a theory could account for an inflationary phase of the Universe in the ultraviolet regime.
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