Related papers: The octonionic phase space Hopf map
We interpret an open orbit in a 32-dimensional representation space of Spin(9,1) x SL(2,R) as a substitute for the non-existent group of invertible 2x2 matrices over the octonions and study various natural homogeneous subspaces. The…
We study the topological configurations of the two-component condensates of bosons with the $3$D $\vec{\sigma}\cdot \vec{p}$ Weyl-type spin-orbit coupling subject to a harmonic trapping potential. The topology of the condensate…
We deal with Riemannian properties of the octonionic Hopf fibration S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the…
In the work some relations between three techniques, Hopf's bundle, Kustaanheimo-Stiefel's bundle, 3-space with spinor structure have been examined. The spinor space is viewed as a real space that is minimally (twice as much) extended in…
Rotations in 3 dimensional space are equally described by the SU(2) and SO(3) groups. These isomorphic groups generate the same 3D kinematics using different algebraic structures of the unit quaternion. The Hopf Fibration is a projection…
The octonionic flag manifold $Fl(\mathbb{O})$ is the space of all pairs in $\mathbb{O}P^2\times \mathbb{O}P^2$ (where $\mathbb{O}P^2$ denotes the octonionic projective plane) which satisfy a certain "incidence" relation. It comes equipped…
The spaces of harmonic maps of the projective plane to the four-dimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree…
We show that a two twistor phase space {\`a} priori describing two non localized massless and spinning particles may be decomposed into a product of three independent phase spaces: the (forward) cotangent bundle of the Minkowski space, the…
Given a CMC surface in $R^3$, its traceless second fundamental form can be viewed as a holomorphic section called the Hopf differential. By analogy, we show that for an associative submanifold of a 7-manifold $M^7$ with $G_2$-structure, its…
The $sp(2M)$ invariant unfolded system is considered in the periodic twistor-like spinor space. Complete set of non-trivial charges corresponding to the global symmetry compatible with the periodicity conditions is constructed. Residual…
In this paper we represent the generalization of relativistic quantum mechanics on the base of eght-component values "octons", generating associative noncommutative spatial algebra. It is shown that the octonic second-order equation for the…
Possible quantum mechanical corollaries of changing the vectorial geometrical model of the physical space, extending it twice, in order to describe its spinor structure (in other terminology and emphasis it is known as the Hopf's bundle)…
We study a quantum ladder of interacting fermions with coupled s and p orbitals. Such a model describes dipolar molecules or atoms loaded into a double-well optical lattice, dipole moments being aligned by an external field. The two orbital…
We rewrite the standard 4-dimensional Dirac equation in terms of quaternionic 2-component spinors, leading to a formalism which treats both massive and massless particles on an equal footing. The resulting unified description has the…
The concept of phase space amplitudes for systems with continuous degrees of freedom is generalized to finite-dimensional spin systems. Complex amplitudes are obtained on both a sphere and a finite lattice, in each case enabling a more…
We develop a noncompact version of the Hopf maps based on the split algebras. The split algebras consist of three species: split-complex numbers, split quaternions, and split octonions. They correspond to three noncompact Hopf maps that…
The geometrical application of split octonions is considered. The modified Fano graphic, which represents products of the basis units of split octonionic, having David's Star shape, is presented. It is shown that active and passive…
We study the evolution of spin-orbital correlations in an inhomogeneous quantum system with an impurity replacing a doublon by a holon orbital degree of freedom. Spin-orbital entanglement is large when spin correlations are…
We demonstrate that it is conceptually and computationally favorable to regard spin-weighted spherical harmonics as vector valued functions on the total space $SO(3)$ of the Hopf bundle, satisfying a covariance condition with respect to the…
Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form…