Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

AdS3/AdS2 degression of massless particles

We study a 3d/2d dimensional degression which is a Kaluza-Klein type mechanism in AdS3 space foliated into AdS2 hypersurfaces. It is shown that an AdS3 massless particle of spin s = 1, 2, …, ∞ degresses into a couple of AdS2 particles of equal energies E = s. Note that the Kaluza-Klein spectra in higher dimensions are always infinite. To formulate the AdS3/AdS2 degression we consider branching rules for AdS3 isometry algebra o(2,2) representations decomposed with respect to AdS2 isometry algebra o(1,2). We find that a given o(2,2) higher-spin representation lying on the unitary bound (i.e. massless) decomposes into two equal o(1,2) modules. In the field-theoretical terms, this phenomenon is demonstrated for spin-2 and spin-3 free massless fields. The truncation to a finite spectrum can be seen by using particular mode expansions, (partial) diagonalizations, and identities specific to two dimensions.

Magnetic Ordering in Systems of Identical Particles with an Arbitrary Spin

The Wigner–Eckart theorem is used for considering the collective effects related to ordering spins in systems of identical particles in ferro- and antiferromagnetic electronic systems, as well as magnetic effects that occur in high spin systems. The Hamiltonian, written in the spin representation in the form obtained by Heisenberg, Dirac, and van Vleck used to describe spin ordering in systems of particles with spin ½, is not appropriate for a description of particle systems with a spin different from ½. “High” spin particles in the spin representation need other forms of the Hamiltonian of the exchange interaction in the spin representation. The Hamiltonian for high-spin particles has been developed from the first principles, and the effects of magnetic ordering in systems of identical particles with arbitrary spin are considered in this review. An effect of giant negative magnetoresistance in the Indium antimonide has been interpreted from the exchange contribution of a high spin heavy holes point of view.

Double spinor Calabi-Yau varieties

Consider the ten-dimensional spinor variety in the projectivization of a half-spin representation of dimension sixteen. The intersection X of two general translates of this variety is a smooth Calabi-Yau fivefold, as well as the intersection Y of their projective duals. We prove that although X and Y are not birationally equivalent, they are derived equivalent and L-equivalent in the sense of Kuznetsov and Shinder.

The simulation of non-Abelian statistics of Majorana fermions in Ising chain with Z2 symmetry

In this paper, we numerically study the non-Abelian statistics of the zero-energy Majorana fermions at the end of Majorana chain and show its application to quantum computing by mapping it to a spin model with special symmetry. In particular, by using transverse-field Ising model with Z2 symmetry, we verify the nontrivial non-Abelian statistics of Majorana fermions. Numerical evidence and comparison in both Majorana representation and spin representation are presented. The degenerate ground states of a symmetry protected spin chain therefore provide a promising platform for topological quantum computation.

Spin representations with negative indices

We consider the spin representation of Artin's braid group, which has a negative index of one and was originally given by D. D. Long and explicitly computed by J.P.Tian. In our work, we find sufficient conditions under which the complex specialization of that representation, namely $\alpha :B_{n}\to GL_{n^{2}}(\mathbb C)$, is unitary relative to a nonsingular hermitian matrix.

Entanglement capabilities of the spin representation of (3+1)D-conformal transformations

Relying on a mathematical analogy of the pure states of the two-qubit system of quantum information theory with four-component spinors we introduce the concept of the intrinsic entanglement of spinors. To explore its physical sense we study the entanglement capabilities of the spin representation of (pseudo-) conformal transformations in (3+1)-dimensional Minkowski space-time. We find that only those tensor product structures can sensibly be introduced in spinor space for which a given spinor is not entangled.