Foundations for a theory of emergent quantum mechanics and emergent classical gravity

Abstract

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...

A note on Clifford bundles and certain Finsler type spaces

“…The theory of Finsler spaces of positive signature can be formulated in the context of robust geometric frameworks [2,3,4]. The case of Finsler structures with Lorentzian metrics still lacks of a general framework, but such models have found many applications in mathematical physics, ranging from application in emergent quantum mechanics [5,7], geometries of extended relativistic gravitational models [6], dual dispersion relations [11] and as a spacetime geometry compatible with clock hypothesis of relativity [8]. It is 1 email: rigato39@gmail.com this ample range of applicability of the Finsler concept of space that partially motivates many studies aiming to extend (pseudo)-Riemannian theory to the Finsler setting.…”

On the origin of the weak equivalence principle in a theory of emergent quantum mechanics

“…It is the main purpose of this paper to offer a derivation of the weak equivalence principle in the context of an emergent theory of gravity, a theory that the author is developing and exploring in our theory of Hamilton-Randers dynamical systems [16]. The framework of emergent quantum theory that we pursue is based upon the hypothesis that there is a deeper level of description of physical systems than the current quantum theory is able to offer.…”

“…At the more fundamental level, the level of the dynamics of the fundamental degrees of freedom, the dynamics is non-reversible. In particular, there is a necessary different notion of time (different time parameters) than in the quantum description (see chapters 2 and 3 in [14,16]). Let us remark that this kind of dynamics is different than in slow/fast dynamics [3] and is on the basis of our interpretation of quantum correlations [16,17].…”

“…In particular, there is a necessary different notion of time (different time parameters) than in the quantum description (see chapters 2 and 3 in [14,16]). Let us remark that this kind of dynamics is different than in slow/fast dynamics [3] and is on the basis of our interpretation of quantum correlations [16,17]. This type of explanation differs considerably from the offered by superdeterminism [25].…”

“…In emergent quantum mechanics is essential, since it allows to consider deterministic systems from a quantum mechanical point of view. This is essential in the coarse grained transition from fundamental degrees of freedom to emergent quantum states (see chapters 4 and 5 in [14,16]). Similar techniques have been investigated by Hooft in his theory of quantum cellular automata and in earlier work [22,23,24,25].…”