Publications

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Quantum cylindrical integrability in magnetic fields

Ondřej Kubů, Libor Šnobl

SciPost Phys. Proc. 14, 032 (2023) · published 24 November 2023 | · pdf

We present the classification of quadratically integrable systems of the cylindrical type with magnetic fields in quantum mechanics. Following the direct method used in classical mechanics by [F Fournier et al 2020 J. Phys. A: Math. Theor. 53 085203] to facilitate the comparison, the cases which may a priori differ yield 2 systems without any correction and 2 with it. In all of them the magnetic field B coincides with the classical one, only the scalar potential W may contain a $\hbar^2$-dependent correction. Two of the systems have both cylindrical integrals quadratic in momenta and are therefore not separable. These results form a basis for a prospective study of superintegrability.
Group invariants for Feynman diagrams

Idrish Huet, Michel Rausch de Traubenberg, Christian Schubert

SciPost Phys. Proc. 14, 031 (2023) · published 24 November 2023 | · pdf

It is well-known that the symmetry group of a Feynman diagram can give important information on possible strategies for its evaluation, and the mathematical objects that will be involved. Motivated by ongoing work on multi-loop multi-photon amplitudes in quantum electrodynamics, here I will discuss the usefulness of introducing a polynomial basis of invariants of the symmetry group of a diagram in Feynman-Schwinger parameter space.
Hyperquaternions and physics

Patrick R. Girard, Romaric Pujol, Patrick Clarysse, Philippe Delachartre

SciPost Phys. Proc. 14, 030 (2023) · published 24 November 2023 | · pdf

The paper develops, within a new representation of Clifford algebras in terms of tensor products of quaternions called hyperquaternions, several applications. The first application is a quaternion 2D representation in contradistinction to the frequently used 3D one. The second one is a new representation of the conformal group in (1+2) space (signature $+--$) within the Dirac algebra $C_(2,3) \simeq $$\mathbb{C\otimes H\otimes H}$ subalgebra of $\mathbb{H\otimes H\otimes H}$. A numerical example and a canonical decomposition into simple planes are given. The third application is a classification of all hyperquaternion algebras into four types, providing the general formulas of the signatures and relating them to the symmetry groups of physics.

1 citation
Study of entanglement in symmetric multi-quDits systems through information diagrams

Julio Guerrero, Antonio Sojo, Alberto Mayorgas, Manuel Calixto

SciPost Phys. Proc. 14, 029 (2023) · published 24 November 2023 | · pdf

Along this paper, we analyze the entanglement properties of symmetric multi-quDits systems in a special type of states created from the generalization to U(D) of the usual spin coherent states. By means of parity operators, we define what we call multicomponent Schrödinger cat states as parity adapted coherent states. Introducing the tool of information diagrams, i.e. representations of pairs of entropy measures, we analyze the correlation structure of this type of states and their M-wise reduced density matrices.
On old relations of Lie theory, classical geometry and gauge theory

Rolf Dahm

SciPost Phys. Proc. 14, 025 (2023) · published 24 November 2023 | · pdf

Having been led by hadron interactions and low-energy photoproduction to SU(4) and non-compact SU*(4) symmetry, the general background turned out to be projective geometry (PG) of $P^3$, or when considering line and Complex geometry to include gauge theory, aspects of $P^5$. Point calculus and its dual completion by planes introduced quaternary (quadratic) "invariants" $x_{\mu}x^{\mu}=0$ and $p_{\mu}p^{\mu}=0$, and put focus on the intermediary form $(xu)$ and its treatment. Here, the major result is the identification of the symmetric 20 of SU(4) comprising nucleon and Delta states as related to the quaternary cubic forms discussed by Hilbert in his work on full invariant systems. So PG determines geometrically the scene by representations (reps) and invariant theory without having to force affine restrictions and additional (spinorial or gauge) rep theory.
Newton mechanics, Galilean relativity, and special relativity in $\alpha$-deformed binary operation setting

Won S. Chung, Mahouton N. Hounkonnou

SciPost Phys. Proc. 14, 024 (2023) · published 23 November 2023 | · pdf

We define new velocity and acceleration having dimension of $(Length)^{\alpha}/(Time)$ and $(Length)^{\alpha}/(Time)^2,$ respectively, based on the fractional addition rule. We discuss the formulation of fractional Newton mechanics, Galilean relativity and special relativity in the same setting. We show the conservation of the fractional energy, characterize the Lorentz transformation and group, and derive the expressions of the energy and momentum. The two body decay is discussed as a concrete illustration.
Generalized Heisenberg-Weyl groups and Hermite functions

Enrico Celeghini, Manuel Gadella, Mariano A. del Olmo

SciPost Phys. Proc. 14, 023 (2023) · published 23 November 2023 | · pdf

A generalisation of Euclidean and pseudo-Euclidean groups is presented, where the Weyl-Heisenberg groups, well known in quantum mechanics, are involved. A new family of groups is obtained including all the above-mentioned groups as subgroups. Symmetries, like self-similarity and invariance with respect to the orientation of the axes, are properly included in the structure of this new family of groups. Generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of these new groups, are introduced.
Generalisation of affine Lie algebras on compact real manifolds

Rutwig Campoamor-Stursberg, Marc de Montigny, Michel Rausch de Traubenberg

SciPost Phys. Proc. 14, 022 (2023) · published 23 November 2023 | · pdf

We report on recent work concerning a new type of generalised Kac-Moody algebras based on the spaces of differentiable mappings from compact manifolds or homogeneous spaces onto compact Lie groups.
Hilbert space structure and classical limit of the low energy sector of U(N) quantum Hall ferromagnets

Manuel Calixto, Alberto Mayorgas, Julio Guerrero

SciPost Phys. Proc. 14, 021 (2023) · published 23 November 2023 | · pdf

Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify and construct the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites. The carrier Hilbert space of irreducible representations of U(N) is described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M)×U(N-M). Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.
The tower of Kontsevich deformations for Nambu-Poisson structures on $\mathbb{R}^d$: Dimension-specific micro-graph calculus

Ricardo Buring, Arthemy V. Kiselev

SciPost Phys. Proc. 14, 020 (2023) · published 23 November 2023 | · pdf

In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multi - vectors, e.g., Poisson bi - vectors; the Nambu - determinant Poisson brackets are differential - polynomial in the Casimir(s) and density $\varrho$ times Levi - Civita symbol. We resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Levi - Civita symbol×$\varrho$. Using this micro - graph calculus, we show that Kontsevich's tetrahedral $\gamma_3$-flow on the space of Nambu - determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we realize the trivializing vector field $\vec{X}$ over $\mathbb{R}^3$ using micro - graphs. This $\vec{X}$ projects to the known trivializing vector field for the $\gamma_3$-flow over $\mathbb{R}^2$.

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