Preface Acknowledgements 1. Introduction 1.1. Motivation 1.2. Structure of the work 2. Assumptions for deterministic dynamical models at the hypothesized fundamental scale 2.1. On the problem of reversibility versus non-reversibility at the fundamental level 2.2. A general notion of dynamical system 2.3. Pseudometric structures 2.4. Causal structure 2.5. Characteristics of the fundamental dynamical systems 2.6. Remarks on the assumptions on the fundamental dynamical systems 2.7. The approximation from discrete to continuous dynamics 2.8. The need of a maximal proper acceleration 3. Hamilton-Randers dynamical systems 3.1. Geometric framework 3.2. Diffeomorphism invariance of the theory as a consistence requirement for the interpretation 3.3. Measure and metric structures 3.4. Deterministic dynamics for the fundamental degrees of freedom 3.5. Notion of Hamilton-Randers dynamical systems 3.6. Postulate on the existence of a geometric flow Ut 3.7. Properties of the Ut flow 3.8. Interpretation of the t-time parameter and the semi-period T 3.9. Periods for composite Hamilton-Randers systems 3.10. Conservation of the mass parameter 3.11. Emergence of the quantum energy-time uncertainty relation 3.12. t-time inversion operation 3.13. Notion of external time parameter τ 3.14. The classical Hamiltonian function of the fundamental degrees of freedom 3.15. General properties of the bare Uτ flow Kinematical properties of the Ut dynamics Heritage kinematical properties of the Uτ dynamics 3.16. Re-definition of the t-time parameter and Ut flow 3.17. The deterministic, local dynamics of the sub-quantum degrees of freedom 3.18. A fundamental result in Hamilton-Randers theory 3.19. Geometric character of the Hamiltonian dynamics 3.20. Macroscopic observers and metric structures 4. Hilbert space formulation of Hamilton-Randers systems 4.1. Hilbert space formulation of classical systems and its application to Hamilton-Randers systems Naturalness of the Heisenberg representation EMERGENT QUANTUM MECHANICS AND EMERGENT GRAVITY Decomposition of the identity operator 4.2. Momentum basis 4.3. The pre-Hilbert space generated by the collection of ontological states Complex structure of the vector space HF unThe scalar product in the vector space HF un 4.4. Quantum Hamiltonian and Heisenberg dynamics associated to a Hamilton-Randers dynamical system 4.5. Sub-quantum mechanical and quantum mechanical operators 4.6. On the non-commutativity of the spacetime suggested by Hamilton-Randers theory 4.7. On the operator interpretation of the external time coordinate 4.8. Geometric issues in the quantization of classical in Hamilton-Randers systems 5. Emergence of the quantum mechanics Hilbert space for Hamilton-Randers systems 5.1. Quantum states from Hamilton-Randers systems 5.2. Quantum states as equivalent classes 5.3. Highly oscillating relative phases states 5.4. Scalar product in the vector space H ′ and the definition of the quantum pre-Hilbert space H Definition of the Hilbert space H 5.5. Representations of the sub-quantum degrees of freed...

General relativity is incomplete because it cannot describe quantum effects of space-time. The complete theory of quantum gravity is not yet known and to date no observational evidence exists that space-time is quantized. However, in most approaches to quantum gravity the space-time manifold of general relativity is only an effective limit that, among other things like higher curvature terms, should receive corrections stemming from space-time defects. We here develop a modification of general relativity that describes local space-time defects and solve the Friedmann equations. From this, we obtain the time-dependence of the average density of defects. It turns out that the defects' average density dilutes quickly, somewhat faster even than radiation.

Caianiello's derivation of Quantum Geometry through an isometric embedding of the spacetime (M,g) in the pseudo-Riemannian structure (T * M, g * AB ) is reconsidered. In the new derivation, a non-linear connection and the bundle formalism induce a Lorentzian-type structure in the 4-dimensional manifold M that is covariant under arbitrary local coordinate transformations in M. If models with maximal acceleration are required to be non-trivial, gravity should be supplied with other interactions in a unification framework.

and INFN Sezione di TriesteIn a recent paper [1] a general mechanism for emergence of cosmological space-time geometry from a quantum gravity setting was devised and departure from standard dispersion relations for elementary particle were predicted. We elaborate here on this approach extending the results obtained in that paper and showing that generically such a framework will not lead to higher order modified dispersion relations in the matter sector. Furthermore, we shall discuss possible phenomenological constraints to this scenarios showing that space-time will have to be by nowadays classical to a very high degree in order to be consistent with current observations.

Using the Finsler structure living in the phase space associated to the tangent bundle of the configuration manifold, deterministic models at the Planck scale are obtained. The Hamiltonian function are constructed directly from the geometric data and some assumptions concerning time inversion symmetry. The existence of a maximal acceleration and speed is proved for Finslerian deterministic models. We investigate the spontaneous symmetry breaking of the orthogonal symmetry SO(6N ) of the Hamiltonian of a deterministic system. This symmetry break implies the non-validity of the argument used to obtain Bell's inequalities for spin states. It is introduced and motivated in the context of Randers spaces an example of simple 't Hooft model with interactions.