Quantum mechanics in a discrete space-time

Pavel Šťovíček, J. Tolar

An-Wei Zhang

Miguel Lorente

An-Wei Zhang

In this letter we briefly investigate the mathematical structure of space-time in the framework of discretization. It is shown that the discreteness of space-time may result in a new mechanical system which differ from the usual quantum mechanics (QM).

Martin Leckey

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.

Wilson A. Zúñiga-Galindo

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.