On the degenerate dynamics of branched hamiltonians

Abstract

Branched Hamiltonians and the corresponding singularity are present in several inter esting physical systems: Lovelock extension of General Relativity in higher dimensions, classical time crystals, k-essence fields, Horndeski theories, compressible fluids, and nonlinear electrodynamics. The emergent ill defined sympletic structure and tricky dynamical evolution poses challenges to a consistent interpretation. In this thesis, multi-valued Hamiltonians are investigated in the framework of degenerate dynamical system, whose sympletic form does not have a constant rank, allowing novel features and interpretations not present in previous investigations. In particular, it is shown how the multi-valuedness is associated with a dynamical mechanism of dimensional reduction, as some degrees of freedom turn into gauge symmetries when the system degenerates. In the case of classical time crystal, there is no “moving” ground state nor brick wall solution, as described previously. Moreover, the degenerate dynamics of a k–essence model enables it to be responsible for both primordial inflation and the present observed acceleration of the cosmological background geometry, while also admitting a non-singular de Sitter beginning of the Universe (it arises from de Sitter and ends in de Sitter). Furthermore, the model is free of pathologies such as propagating superluminal perturbations, negative energies, and perturbation instabilities. Henceforth, in thesis is demonstrated that the degenerate dynamics offer a consistent interpretation, under which the degeneracy and consequent branching is not a problem but a dynamical feature.