Related papers: Discrete quantum structures
We further elaborate on a phase-space picture for a system of $N$ qubits and explore the structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves satisfying certain additional properties and…
A particular family of Discrete Time Quantum Walks (DTQWs) simulating fermion propagation in $2$D curved space-time is revisited. Usual continuous covariant derivatives and spin-connections are generalized into discrete covariant…
We introduce quantum monadic and quantum cylindric algebras. These are adaptations to the quantum setting of the monadic algebras of Halmos, and cylindric algebras of Henkin, Monk and Tarski, that are used in algebraic treatments of…
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…
Discrete quantum groups were introduced as duals of compact quantum groups by Podle\'s and Woronowicz in 1990. They have been studied intrinsically by Effros and Ruan (1994) and by the author (1996). In a more recent note (2025), we have…
A general algebraic approach, incorporating both invariance groups and dynamic symmetry algebras, is developed to reveal hidden coherent structures (closed complexes and configurations) in quantum many-body physics models due to symmetries…
We propose the assumption of quantum mechanics on a discrete space and time, which implies the modification of mathematical expressions for some postulates of quantum mechanics. In particular we have a Hilbert space where the vectors are…
It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum…
The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor…
We consider some general aspects of the new noncommutative or quantum geometry coming out of the theory of quantum groups, in connection with Planck scale physics. A generalisation of Fourier or wave-particle duality on curved spaces…
We define Ozawa's notion of bi-exactness to discrete quantum groups, and then prove some structural properties of associated von Neumann algebras. In particular, we prove that any non amenable subfactor of free quantum group von Neumann…
Quantum algorithms are sequences of abstract operations, performed on non-existent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke…
Double coverings of the orthogonal groups of the real and complex spaces are considered. The relation between discrete transformations of these spaces and fundamental automorphisms of Clifford algebras is established, where an isomorphism…
We study actions of compact quantum groups on type I factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz' results concerning Peter-Weyl theory for…
We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group…
Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing…
Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise…
The proposed theory of causally structured discrete fields studies integer values on directed edges of a self-similar graph with a propagation rule, which we define as a set of valid combinations of integer values and edge directions around…
We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for…
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…