相关论文: Quantum group symmetry of integrable systems with …
We discuss some families of integrable and superintegrable systems in $n$-dimensional Euclidean space which are invariant to $m\geq n-2$ rotations. The integrable invariant Hamiltonian $H=\sum p_i^2+V(q)$ commutes with $n-2$ integrals of…
In a series of papers we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra ${\cal A}_{\Gamma}$ defined on a transformation…
We argue that the description of Feynman loop integrals as integrable systems is intimately connected with their motivic properties and the action of the Cosmic Galois Group. We show how in the case of a family of fishnet graphs, coaction…
Group field theory is a background-independent approach to quantum gravity whose starting point is the definition of a quantum field theory on an auxiliary group manifold (not interpreted as spacetime, but rather as the finite-dimensional…
It is well known that the unitary evolution of a closed $M-$level quantum system can be generated by a non-Hermitian Hamiltonian $H$ with real spectrum. Its Hermiticity can be restored via an amended inner-product metric $\Theta$. In…
Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference…
{We point out some obstacles raised by the lost of symmetry against the extension to the case of an interacting particle of the approach that {\sl deductively} establishes the Quantum Theory of a free particle according to the group…
We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT) -- a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M…
I describe, in the simplified context of finite groups and their representations, a mathematical model for a physical system that contains both its quantum and classical aspects. The physically observable system is associated with the space…
A group $\Gamma$ is defined to be cofinitely Hopfian if every homomorphism $\Gamma\to\Gamma$ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic…
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…
The states of the physical algebra, namely the algebra generated by the operators involved in encoding and processing qubits, are considered instead of those of the whole system-algebra. If the physical algebra commutes with the interaction…
Given a dynamical twist for a finite dimensional Hopf algebra we construct two weak Hopf algebras, using methods of Xu and Etingof-Varchenko, and show that they are dual to each other. We generalize the theory of dynamical quantum groups to…
Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of…
We introduce the framework of Hopf algebra field theory (HAFT) which generalizes the notion of group field theory to the quantum group (Hopf algebra) case. We focus in particular on the 3d case and show how the HAFT we considered is…
We prove that all eight KO groups for a real C*-algebra can be constructed from homotopy classes of unitary matrices that respect a variety of symmetries. In this manifestation of the KO groups, all eight boundary maps in the 24-term exact…
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S-matrix, Feynman…
We discuss the canonical structure of a class of integrable quantum mappings, i.e. iterative canonical transformations that can be interpreted as a discrete dynamical system. As particular examples we consider quantum mappings associated…
An operator deformed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its $q \to 1$ limit appear to be the most general Yang-Baxter algebra underlying…