The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space $\mathcal{X}$ as a model for the space. However, we consider the time as a real variable. In this framework, the formalism of Dirac-von Neumann can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})^{3}$ as a model of a space-time. For simplicity, we work out our models using $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})$ as the configuration space. Quantum mechanics (QM), in the sense of Dirac-von Neumann, on the Hilbert space $L^{2}(\mathbb{R}\times\mathcal{X})$ is a non-local theory: the Hamiltonians are non-local operators, and thus, spooky action at a distance is allowed. The paradigm asserting that the universe is non-locally real implies that the proposed version of QM admits realism. This version of QM can be specialized to standard QM by using Hamiltonians acting on wavefunctions supported on the region $\mathbb{R}\times\mathbb{R}$. We apply the developed formalism to the measurement problem. We propose a new mechanism for the collapse of the wavefunction. The mechanism resembles the one proposed by Ghirardi, Ramini, and Weber, but there are significant differences. The most important feature is that the Schrödinger equation describes the dynamics at all times, even at the moment of measurement. We also discuss a model for the two-slit experiment, where bright and dark states of light (proposed recently) naturally occur.
The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space X as a model for the space. However, we consider the time as a real variable. In this framework, the formalism of Dirac-von Neumann can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose R×(R×X)3 as a model of a space-time. For simplicity, we work out our models using R×(R×X) as the configuration space. Quantum mechanics (QM), in the sense of Dirac-von Neumann, on the Hilbert space L2(R×X) is a non-local theory: the Hamiltonians are non-local operators, and thus, spooky action at a distance is allowed. The paradigm asserting that the universe is non-locally real implies that the proposed version of QM admits realism. This version of QM can be specialized to standard QM by using Hamiltonians acting on wavefunctions supported on the region R×R. We apply the developed formalism to the measurement problem. We propose a new mechanism for the collapse of the wavefunction. The mechanism resembles the one proposed by Ghirardi, Ramini, and Weber, but there are significant differences. The most important feature is that the Schrödinger equation describes the dynamics at all times, even at the moment of measurement. We also discuss a model for the two-slit experiment, where bright and dark states of light (proposed recently) naturally occur.
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.
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.
We present a resolution of the Wigner's Friend paradox within a framework of quantum mechanics (QM) on the hybrid space RxQ_{p}, where Q_{p} denotes the field of p-adic numbers, regarded as a model of discrete microscopic space at the Planck-Bronstein scale. In this framework, wavefunction collapse is not an independent postulate but a dynamical consequence of the Schrödinger equation with non-local Hamiltonians: wavefunctions localize onto compact supports during measurement interactions, producing definite pointer readings without the intervention of observers or the exchange of information between subsystems. We model both Wigner and his Friend as classical apparatuses and show that each produces a definite reading through independent applications of the collapse mechanism, thereby eliminating the conflict between their descriptions of reality. The framework is consistent with the principal no-go theorems in finite- and infinite-dimensional Hilbert spaces associated with extended Wigner's Friend scenarios -- including those of Frauchiger-Renner, Brukner, Bong et al., and Guérin et al. -- since it requires no agents capable of recording or reasoning about outcomes, thereby vacating the observer-dependent assumptions that drive those theorems. We illustrate the collapse mechanism explicitly through a toy model of a particle in a box, comparing the standard description with the new one. The non-locality intrinsic to QM on L2(RxQ_{p}) permits realism at the cost of locality, and the Absoluteness of Observed Events holds in our framework without requiring observer independence.
The central idea of this work is the concept of prespace, a hypothetical structure that is postulated to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the implications of the existence of this structure for quantum theory. I compare space and space-time to other spaces used in physics, such as configuration space, phase space, and Hilbert space. I support the 'property view' of space, opposing both the traditional views of space and space-time, substantivalism and relationism. I argue that configuration space has a special status in the microscopic world similar to the status of position space in the macroscopic world. The prespace structure is compared with a computational system, in particular to a cellular automaton, in which space and all physical quantities are broken into discrete units. One way open for a prespace metaphysics can be found if physics is made fully discrete. I suggest as a heuristic principle that the physical laws of our world are such that the computational cost of implementing those laws on an arbitrary computational system is minimized. I discuss the 'measurement problem' of quantum mechanics. Considering how quantum theory could be made fully discrete leads naturally to a suggestion of how standard linear quantum mechanics could be modified to give a solution to the measurement problem, calling this model Critical Complexity Quantum Mechanics (CCQM). I compare CCQM with some other proposed solutions to the measurement problem, in particular, the spontaneous localization model of Ghirardi, Rimini, and Weber. Finally, I argue that the measure of the complexity of quantum mechanical states introduced in CCQM also provides a new definition of entropy for quantum mechanics, and suggests an objective foundation for statistical mechanics, thermodynamics, and the arrow of time.
The central motivating idea behind the development of this work is the concept of prespace, a hypothetical structure that is postulated by some physicists to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the rest of reality as we know it, and the implications of the existence of this structure for quantum theory. Understanding how this structure could relate to space and to the rest of reality requires, I believe, that we consider how space itself relates to reality, and how other so-called “spaces” used in physics relate to reality. In the first chapter I compare space and space-time to other spaces used in physics, such as configuration space, phase space and Hilbert space. I support what is known as the “property view” of space, opposing both the traditional views of space and space-time, substantivalism and relationism. I argue that all these spaces are property spaces. After examining the relationships of these spaces to causality, I argue that configuration space has, due to its role in quantum mechanics, a special status in the microscopic world similar to the status of position space in the macroscopic world. In chapter two prespace itself is considered. One way of approaching this structure is through the comparison of the prespace structure with a computational system, in particular to a cellular automaton, in which space or space-time and all other physical quantities are broken down into discrete units. I suggest that one way open for a prespace metaphysics can be found if physics is made fully discrete in this way. I suggest as a heuristic principle that the physical laws of our world are such that the computational cost of implementing those laws on an arbitrary computational system is minimized, adapting a heuristic principle of this type proposed by Feynman (1982). In the final chapter some of the ideas of the previous chapters are applied in an examination of the physics and metaphysics of quantum theory. I first discuss the “measurement problem” of quantum mechanics: this problem and its proposed solution are the primary subjects of the final chapter. It turns out that considering how quantum theory could be made fully discrete leads naturally to a suggestion of how standard linear quantum mechanics could be modified to give rise to a solution to the measurement problem. The computational heuristic principle reinforces the same solution. I call the modified quantum mechanics Critical Complexity Quantum Mechanics (CCQM). I compare CCQM with some of the other proposed solutions to the measurement problem, in particular the spontaneous localization model of Ghirardi, Rimini and Weber (1986). Finally I argue that the measure of complexity of quantum mechanical states I introduce in CCQM also provides a new definition of entropy for quantum mechanics, and suggests a solution to the problem of providing an objective foundation for statistical mechanics, thermodynamics, and the arrow of time.
Introduction Our concern in this chapter is locality in quantum mechanics (QM). Locality is a heuristic physics principle based on the following propositions. No Action-at-a-Distance All the evidence points to the principle that physical actions, taken within restricted (localized) regions of space and time by observers or other agencies such as systems under observation (SUOs), do not cause instantly observable effects on other SUOs at large distances. This does not apply to mathematical/ metaphysical concepts such as quantum wave functions or correlations, as these are conceptual objects (Scarani et al., 2000). Statements about instantaneous wave function collapse are vacuous (have no empirical significance) and are therefore not an issue of significance in physics. Such statements are an issue to theorists who objectivize wave functions, as in Hidden Variables (HV) theory. Action-at-a-distance is generally regarded as anathema by most physicists. For example, Newton's law of universal gravitation is well known for mathematically encoding action-at-a-distance. There is direct evidence, however, in the form of a letter written by Newton to Bentley, that Newton believed that gravity acting “at a distance through a vacuum without the mediation of anything else” was an absurdity (Newton, 2006). The no action-at-a-distance principle is encoded in quantized detector networks (QDN) by the requirement that labstate preparation and consequent signal detection never occur at the same stage. Causal Transmission All physically observable consequences of local actions taken by an observer or SUO are transmitted by identifiable physical processes, such as electromagnetic waves or neutrinos. There is no such thing as magic or action-at-a-distance . In QDN this proposition is taken into account implicitly in the labstate outcome amplitudes at each stage, as these model how information is propagated from stage to stage. When necessary, the information void can be modeled as if there were fields and/or particles propagating through it, giving scope for different mathematical models, such as Euclidean space, curved spacetime, noncommuting spacetimes, and so on. The structure of the information void is essentially a discussion of whatever modules have to be taken into account between labstate preparation and signal detection.