33 research outputs found
Foundations for a theory of emergent quantum mechanics and emergent classical gravity
Quantum systems are viewed as emergent systems from the fundamental degrees
of freedom. The laws and rules of quantum mechanics are understood as an
effective description, valid for the emergent systems and specially useful to
handle probabilistic predictions of observables. After introducing the
geometric theory of Hamilton-Randers spaces and reformulating it using Hilbert
space theory, a Hilbert space structure is constructed from the Hilbert space
formulation of the underlying Hamilton-Randers model and associated with the
space of wave functions of quantum mechanical systems. We can prove the
emergence of the Born rule from ergodic considerations. A geometric mechanism
for a natural spontaneous collapse of the quantum states based on the
concentration of measure phenomena as it appears in metric geometry is
discussed.We show the existence of stable vacua states for the quantized matter
Hamiltonian. Another consequence of the concentration of measure is the
emergence of a weak equivalence principle for one of the dynamics of the
fundamental degrees of freedom. We suggest that the reduction of the quantum
state is driven by a gravitational type interaction.
Such interaction appears only in the dynamical domain when localization of
quantum observables happens, it must be a classical interaction. We discuss the
double slit experiment in the context of the framework proposed, the
interference phenomena associated with a quantum system in an external
gravitational potential, a mechanism explaining non-quantum locality and also
provide an argument in favour of an emergent interpretation of every
macroscopic time parameter. Entanglement is partially described in the context
of Hamilton-Randers theory and how naturally Bell's inequalities should be
violated.Comment: Extensive changes in chapter 1 and chapter 2; minor changes in other
chapters; several refereces added and others update; 192 pages including
index of contents and reference
On -jet field approximations to geodesic deviation equations
Let be a smooth manifold and a semi-spray defined on a
sub-bundle of the tangent bundle . In this work it is proved
that the only non-trivial -jet approximation to the exact geodesic deviation
equation of , linear on the deviation functions and invariant
under an specific class of local coordinate transformations is the Jacobi
equation. However, if the linearity property on the dependence in the deviation
functions is not imposed, then there are differential equations whose solutions
admit -jet approximations and are invariant under arbitrary coordinate
transformations. As an example of higher order geodesic deviation equations we
study the first and second order geodesic deviation equations for a Finsler
spray.Comment: Accepted version in International Journal of Geometric Methods in
Modern Physics; 21 page