An Introduction to κ-Deformed Symmetries, Phase Spaces and Field Theory

In this review, we give a basic introduction to the κ-deformed relativistic phase space and free quantum fields. After a review of the κ-Poincaré algebra, we illustrate the construction of the κ-deformed phase space of a classical relativistic particle using the tools of Lie bi-algebras and Poisson–Lie groups. We then discuss how to construct a free scalar field theory on the non-commutative κ-Minkowski space associated to the κ-Poincaré and illustrate how the group valued nature of momenta affects the field propagation.

From the Early Universe to the Modern Universe

The birth of the Universe, its dark components, and the next fundamental level of matter are briefly discussed. The classical cosmological solution for our Universe with a Λ-term has two branches divided by a gap. The quantum process of tunneling between branches took place. A model of a slowly swelling Universe in the result of the multiple reproductions of cosmological cycles arises naturally. The occurrence of baryon asymmetry is briefly discussed. The problem of the cosmological constant is solved and, thus, the crisis of physics connected with this constant is overcome. But we note that dark energy is evolving. Dark matter (part or all) consists of familon-type pseudo-Goldstone bosons with a mass of 10−5–10−3 eV. It follows the composite model of particles. This model reproduces three relativistic phase transitions in the medium of familons at different red shifts, forming a large scale structure of the Universe dark matter that was “repeated” by baryons. Here three generations of elementary particles are absolutely necessary.

Tools for Calculation

This chapter discusses the calculation of quantities observable in elementary particle reactions — cross sections and partial decay widths. It introduces relativistic phase space. It introduces Feynman diagrams and describes their role in visualizing elementary particle reactions.

Geometric relativistic phase from Lorentz symmetry breaking effects in the cosmic string spacetime

In this paper, we have investigated the arising of geometric quantum phases in a relativistic quantum dynamics of a Dirac neutral particle from the spontaneous Lorentz symmetry violation effects in the cosmic string spacetime. We started by the Dirac equation in an effective metric, and we have observed a relativistic geometric phase which stems from the topology of the cosmic string spacetime and an intrinsic Lorentz symmetry breaking effects. It is shown that both Lorentz symmetry breaking effects and the topology of the defect yields a phase shift in the wave function of the nonrelativistic spin-[Formula: see text] particle.

Cosmology of spin-2 fields

The Cauchy-Kowalevski theorem is applied to the solutions of Einstein’s equations and to cosmology. Three fundamental requirements of the theorem: the use of analytic series; the existence of the boundary surfaces; and the setting of the independent initial data are revised, using methods of geometric analysis. It is found that during its relativistic phase the standard model of the universe is completely governed by Einstein’s gravitation, in which the source includes a massive spin-2 field, compatible with the observations of the dark sector of the universe. On the other hand, the exponential growth of the volume of the universe at the inflationary phase, is shown to be predominantly thermodynamical, as opposed to a relativistic process. These two phases are joined by the last inflationary surface endowed with a mirror symmetry, which eventually suggests a possible bouncing universe scenario.