Abstract:
This contribution is the third of a series of articles devoted to the
physics of discrete spaces. After the building of space-time [1] and the foundation of quantum theory [2] one studies here how the three fundamental
interactions could emerge from the model of discrete space-time that we have
put forward in previous contributions. The gauge interactions are recovered. We
also propose an original interpretation of gravitational interactions.

Abstract:
This paper presents a new modified quantum mechanics, Critical Complexity Quantum Mechanics, which includes a new account of wavefunction collapse. This modified quantum mechanics is shown to arise naturally from a fully discrete physics, where all physical quantities are discrete rather than continuous. I compare this theory with the spontaneous collapse theories of Ghirardi, Rimini, Weber and Pearle and discuss some implications of these theories and CCQM for a realist view of the quantum realm.

Abstract:
't Hooft has recently developed a discretisation of (2+1) gravity which has a multiple-valued Hamiltonian and which therefore admits quantum time evolution only in discrete steps. In this paper, we describe several models in the continuum with single-valued equations of motion in classical physics, but with multiple-valued Hamiltonians. Their time displacements in quantum theory are therefore obliged to be discrete. Classical models on smooth spatial manifolds are also constructed with the property that spatial displacements can be implemented only in discrete steps in quantum theory. All these models show that quantization can profoundly affect classical topology.

Abstract:
Gibbs measure is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random field is defined is itself random. Moreover, this randomness is not given apriori and independently of the configuration, but rather they depend on each other, and both are given by Gibbs procedure; We call the resulting object a Gibbs family because it parametrizes Gibbs fields on different graphs in the support of the distribution. We study also quantum (KMS) analog of Gibbs families. Various applications to discrete quantum gravity are given.

Abstract:
In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a `fine enough' rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.

Abstract:
This article is an application of the theory of discrete spaces to cosmology. Its conclusions are necessarily speculative. An interesting aspect is that it gives possible solutions to many pending problems within a unique framework. Let us cite a scenario for the Big-Bang that avoids any initial mathematical singularity, an interpretation of dark matter that does not involve any hadronic matter, a description of the formation of stellar and galactic black holes and, for the later, a description of quasars, their characteristics and their source of energy. Finally dark energy is also given an interpretation through modifications of the laws of gravity.

Abstract:
In this contribution we use the model of discrete spaces that we have put forward in former articles to give an interpretation to the phenomena of quantum entanglement and quantum states reduction that rests upon a new way of considering space and time.

Abstract:
We show that the model of discrete spaces that we have
proposed in previous contributions gives a comprehensive and detailed interpretation
of the properties of the standard model of particles. Moreover the model also suggests
the possible existence of a non-standard family of particles.

Abstract:
The purpose of this paper is to present an introduction to a point of view for discrete foundations of physics. In taking a discrete stance, we find that the initial expression of physical theory must occur in a context of noncommutative algebra and noncommutative vector analysis. In this way the formalism of quantum mechanics occurs first, but not necessarily with the usual interpretations. The basis for this work is a non-commutative discrete calculus and the observation that it takes one tick of the discrete clock to measure momentum.