Hyperbolic symmetries, inflaton–phantom cosmology, and inflation

Hyperbolic ring based formulation for thermo field dynamics, quantum dissipation, entanglement, and holography

“…Previous treatments using hyperbolic complex plane include for example the unification of diverse scalar fields of cosmological and particle physics interest [22]; in particular an inflaton-phantom cosmology has been established without invoking concepts such as field theory Euclideanization, and the use of noncanonical forms for the Lagrangian terms. As opposed to traditional treatments on the phantom fields dynamics, bounded potentials are constructed along the universal prescriptions, and the spontaneous symmetry breaking physics can be studied in a conventional way in cosmological scenarios.…”

“…Considering that the real exponentials in Eq. 22are solutions to the equations of motion, the analytically continued solutions can be obtained through the transformations ω k → iω k , and k → ik, within the standard scheme, and through ω k → jω k , and k → jk within the case at hand, which lead to the purely hyperbolic expression (22). In both cases the complexification does not change the relative sign between ω 2 k , and k 2 in the dispersion relations, but the relative sign of the dissipative term γ 2 is different in each case, since in the former i 2 = −1, and in the later j 2 = 1.…”

“…If the spectral decomposition of the field operator is constructed from the solution (22), considering the full range k ∈ (−∞, +∞), taking into account the usual Dirac delta functions resulting from the commutators between the annihilaton and creation operators, then the field operator commutators will undergo UV divergences; therefore we have to re-consider the decomposition range and the distribution representing the Dirac delta on the hyperbolic ring.…”

Hyperbolic ring based formulation for thermo field dynamics, quantum dissipation, entanglement, and holography

“…Thus, for the same equations of motion, the corresponding dispersion relation depends on which complex plane the solutions are analytically continued. If one attempts to solve the hyperbolic ring based differential equation (20), by using the usual complex phase, then the product ij will appear, which has not been defined within the ring H; at this point one may extend the ring for including the usual complex unit i, and then to proceed with the quantization prescription (see [31]). In fact, when the solution for equations of motion considered previously is continued into such an extended ring that includes both complex units (i, j), then the corresponding dispersion relation turns out to be that in Eq.…”

“…Previous treatments using hyperbolic complex plane include for example the unification of diverse scalar fields of cosmological and particle physics interest [20]; in particular an inflaton-phantom cosmology has been established without invoking concepts such as field theory Euclideanization, and the use of noncanonical forms for the Lagrangian terms. As opposed to traditional treatments on the phantom fields dynamics, bounded potentials are constructed along the universal prescriptions, and the spontaneous symmetry breaking physics can be studied in a conventional way in cosmological scenarios.…”