Quantum gravitational effects suggest a minimal length, or spacetime interval, of order of the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. One implication is that quantum states with norm below some very small threshold do not exist. The exclusion of what Everett referred to as maverick branches is necessary for the emergence of the Born Rule in no collapse quantum mechanics. We discuss this in the context of quantum gravity, showing that discrete models (such as simplicial or lattice quantum gravity) indeed suggest a discrete Hilbert space with minimum norm. These considerations are related to the ultimate level of fine-graining found in decoherent histories (of spacetime geometry plus matter fields) produced by quantum gravity.
Quantum gravitational effects suggest a minimal length, or spacetime interval, of order of the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. One implication is that quantum states with norm below some very small threshold do not exist. The exclusion of what Everett referred to as maverick branches is necessary for the emergence of the Born Rule in no collapse quantum mechanics. We discuss this in the context of quantum gravity, showing that discrete models (such as simplicial or lattice quantum gravity) indeed suggest a discrete Hilbert space with minimum norm. These considerations are related to the ultimate level of fine-graining found in decoherent histories (of spacetime geometry plus matter fields) produced by quantum 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.
We explain the measure problem (cf. origin of the Born probability rule) in no-collapse quantum mechanics. Everett defined maverick branches of the state vector as those on which the usual Born probability rule fails to hold — these branches exhibit highly improbable behaviors, including possibly the breakdown of decoherence or even the absence of an emergent semi-classical reality. Derivations of the Born rule which originate in decision theory or subjective probability (i.e. the reasoning of individual observers) do not resolve this problem, because they are circular: they assume, a priori, that the observer occupies a non-maverick branch. An ab initio probability measure is sometimes assumed to explain why we do not occupy a maverick branch. This measure is constrained by, e.g. Gleason’s theorem or envariance to be the usual Hilbert measure. However, this ab initio measure ultimately governs the allocation of a self or a consciousness to a particular branch of the wave function, and hence invokes primitives which lie beyond the Everett wave function and beyond what we usually think of as physics. The significance of this leap has been largely overlooked, but requires serious scrutiny.
In this thesis, we lay out the goal, and a broad outline, for a program that takes quantum mechanics in its minimal form to be the fundamental ontology of the universe. Everything else, including features like space-time, matter and gravity associated with classical reality, are emergent from these minimal quantum elements. We argue that the Hilbert space of quantum gravity is locally finite-dimensional, in sharp contrast to that of conventional field theory, which could have observable consequences for gravity. We also treat time and space on an equal footing in Hilbert space in a reparametrization invariant setting and show how symmetry transformations, both global and local, can be treated as unitary basis changes. Motivated by the finite-dimensional context, we use Generalized Pauli Operators as finite-dimensional conjugate variables and define a purely Hilbert space notion of locality based on the spread induced by conjugate operators which we call "Operator Collimation." We study deviations in the spectrum of physical theories, particularly the quantum harmonic oscillator, induced by finite-dimensional effects, and show that by including a black hole-based bound in a lattice field theory, the quantum contribution to the vacuum energy can be suppressed by multiple orders of magnitude. We then show how one can recover subsystem structure in Hilbert space which exhibits emergent quasi-classical dynamics. We explicitly connect classical features (such as pointer states of the system being relatively robust to entanglement production under environmental monitoring and the existence of approximately classical trajectories) with features of the Hamiltonian. We develop an in-principle algorithm based on extremization of an entropic quantity that can sift through different factorizations of Hilbert space to pick out the one with manifest classical dynamics. This discussion is then extended to include direct sum decompositions and their compatibility with Hamiltonian evolution. Following this, we study quantum coarse-graining and state-reduction maps in a broad context. In addition to developing a first-principle quantum coarse-graining algorithm based on principle component analysis, we construct more general state-reduction maps specified by a restricted set of observables which do not span the full algebra (as could be the case of limited access in a laboratory or in various situations in quantum gravity). We also present a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. Throughout the thesis, we discuss implications of our work in the broader goal of understanding quantum gravity from minimal elements in quantum mechanics.
I take non-locality to be the Michelson–Morley experiment of the early 21st century, assume its universal validity, and try to derive its consequences. Spacetime, with its locality, cannot be fundamental, but must somehow be emergent from entangled coherent quantum variables and their behaviors. There are, then, two immediate consequences: (i). if we start with non-locality, we need not explain non-locality. We must instead explain an emergence of locality and spacetime. (ii). There can be no emergence of spacetime without matter. These propositions flatly contradict General Relativity, which is foundationally local, can be formulated without matter, and in which there is no “emergence” of spacetime. If these be true, then quantum gravity cannot be a minor alteration of General Relativity but must demand its deep reformulation. This will almost inevitably lead to: matter not only curves spacetime, but “creates” spacetime. We will see independent grounds for the assertion that matter both curves and creates spacetime that may invite a new union of quantum gravity and General Relativity. This quantum creation of spacetime consists of: (i) fully non-local entangled coherent quantum variables. (ii) The onset of locality via decoherence. (iii) A metric in Hilbert space among entangled quantum variables by the sub-additive von Neumann entropy between pairs of variables. (iv) Mapping from metric distances in Hilbert space to metric distances in classical spacetime by episodic actualization events. (v) Discrete spacetime is the relations among these discrete actualization events. (vi) “Now” is the shared moment of actualization of one among the entangled variables when the amplitudes of the remaining entangled variables change instantaneously. (vii) The discrete, successive, episodic, irreversible actualization events constitute a quantum arrow of time. (viii) The arrow of time history of these events is recorded in the very structure of the spacetime constructed. (ix) Actual Time is a succession of two or more actual events. The theory inevitably yields a UV cutoff of a new type. The cutoff is a phase transition between continuous spacetime before the transition and discontinuous spacetime beyond the phase transition. This quantum creation of spacetime modifies General Relativity and may account for Dark Energy, Dark Matter, and the possible elimination of the singularities of General Relativity. Relations to Causal Set Theory, faithful Lorentzian manifolds, and past and future light cones joined at “Actual Now” are discussed. Possible observational and experimental tests based on: (i). the existence of Sub- Planckian photons, (ii). knee and ankle discontinuities in the high-energy gamma ray spectrum, and (iii). possible experiments to detect a creation of spacetime in the Casimir system are discussed. A quantum actualization enhancement of repulsive Casimir effect would be anti-gravitational and of possible practical use. The ideas and concepts discussed here are not yet a theory, but at most the start of a framework that may be useful.
I take non-locality to be the Michaelson Morley experiment of the early 21st Century, assume its universal validity, and try to derive its consequences. Spacetime, with its locality, cannot be fundamental, but must somehow be emergent from entangled coherent quantum variables and their behaviors. There are, then, two immediate consequences: i. If we start with non-locality, we need not explain non-locality. We must instead explain an emergence of locality and spacetime. ii. There can be no emergence of spacetime without matter. These propositions flatly contradict General Relativity, which is foundationally local, can be formulated without matter, and in which there is no “emergence” of spacetime.If these be true, then quantum gravity cannot be a minor alteration of General Relativity but must demand its deep reformulation. This will almost inevitably lead to: Matter not only curves spacetime, but “creates” spacetime.We will see independent grounds for the assertion that matter both curves and creates spacetime that may invite a new union of quantum gravity and General Relativity.This quantum creation of spacetime consists in: i. Fully non-local entangled coherent quantum variables.ii. The onset of locality via decoherence. iii. A metric in Hilbert Space among entangled quantum variables by the sub-additive von Neumann Entropy between pairs of variables. iv. Mapping from metric distances in Hilbert Space to metric distances in classical spacetime by episodic actualization events.v. Discrete spacetime is the relations among these discrete actualization events. vi. “Now” is the shared moment of actualization of one among the entangled variables when the amplitudes of the remaining entangled variables change instantaneously.vii. The discrete, successive, episodic, irreversible actualization events constitute a quantum arrow of time. viii. The arrow of time history of these events is recorded in the very structure of the spacetime constructed. ix. Actual Time is a succession of two or more actual events.The theory inevitably yields a UV cut-off of a new type. The cut-off is a phase transition between continuous spacetime before the transition and discontinuous spacetime beyond the phase transition.This quantum creation of spacetime modifies general relativity and may account for Dark Energy, Dark Matter, and the possible elimination of the singularities of General Relativity.Relations to Causal Set Theory, faithful Lorentzian manifolds, and past and future light cones joined at “Actual Now” are discussed.Possible observational and experimental tests based on: i. the existence of Sub- Planckian photons, ii. knee and ankle discontinuities in the high energy gamma ray spectrum, and iii. possible experiments to detect a creation of spacetime in the Casimir system are discussed. A quantum actualization enhancement of repulsive Casimir effect would be anti-gravitational and of possible practical use.The ideas and concepts discussed here are not yet a theory, but at most the start of a framework that may be useful.