Related papers: Computations in Finite Groups and Quantum Physics
This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the…
A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its…
This paper is dedicated to a detailed analysis and computation of quantum states of causal fermion systems. The mathematical core is to compute integrals over the unitary group asymptotically for a large dimension of the group, for various…
We study quantum mechanics in the stochastic formulation, using the functional integral approach. The noise term enters the classical action as a local contribution of anticommuting fields. The partition function is not invariant under…
We show that non-relativistic Quantum Mechanics can be faithfully represented in terms of a classical diffusion process endowed with a gauge symmetry of group Z_4. The representation is based on a quantization condition for the realized…
We define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient…
Compact-group representations on Banach spaces are known to be norm-continuous precisely when they have finite spectra. For a quantum group with continuous-function algebra $\mathcal{C}(\mathbb{G})$ norm continuity can be cast analogously…
New status in quantum mechanics is connected with recent achievements in the inverse problem. With its help instead of about ten exactly solvable models which serve as a basis of the contemporary education there are infinite (!) number,…
We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. We want to know which vector-valued finite element spaces have bases invariant under permutation of vertex indices. The…
The conceptual setting of quantum mechanics is subject to an ongoing debate from its beginnings until now. The consequences of the apparent differences between quantum statistics and classical statistics range from the philosophical…
Quantum mechanics started out as a theory to describe the smallest scales of energy in Nature. After a hundred years of development it is now routinely employed to describe, among others, quantum computers with thousands of qubits. This…
In this research notebook in the four-part, quantum computation and applications, quantum computation and algorithms, quantum communication protocol, and universal quantum computation for quantum engineers, researchers, and scientists, we…
Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. These consist of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose…
A quantum version of the action principle in a simple covariant dynamical theory of two relativistic particles is formulated. The central object of this new formulation of quantum theory is a stationary eigenvalue of the quantum action.…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It has recently been shown that computers that exploit…
How can quantum mechanics be (i) the fundamental theoretical framework of contemporary physics and (ii) a probability calculus that presupposes the events to which, and on the basis of which, it assigns probabilities? The question is…
This paper proposes a method of unifying quantum mechanics and gravity based on quantum computation. In this theory, fundamental processes are described in terms of pairwise interactions between quantum degrees of freedom. The geometry of…
A finite quantum hypergroup is a finite-dimensional unital algebra $A$ over the field of complex numbers. There is a coproduct on $A$, a coassociative map from $A$ to $A\otimes A$ assumed to be unital, but it is not required to be an…
We review the geometrical formulation of Quantum Mechanics to identify, according to Klein's programme, the corresponding group of transformations. For closed systems, it is the unitary group. For open quantum systems, the semigroup of…