Physics NotePublished
Volichenko-type metasymmetry of braided Majorana qubits
F. Toppan
Publication details
- Authors
- F. Toppan
- Type
- Physics Note
- Status
- Published
- Publication date
- 12/06/2024
- Description
- This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits and clarifies their role; in particular, ``mixed-bracket" Heisenberg-Lie algebras are introduced. These algebras belong to a more general framework than the {\it Volichenko algebras} defined in 1990 by Leites-Serganova as {\it metasymmetries} which do not respect even/odd gradings and lead to mixed brackets interpolating ordinary commutators and anticommutators.\\ In a previous paper braided ${\mathbb Z}_2$-graded Majorana qubits were first-quantized within a graded Hopf algebra framework endowed with a braided tensor product. The resulting system admits truncations at roots of unity and realizes, for a given integer $s=2,3,4,\ldots$, an interpolation between ordinary Majorana fermions (recovered at $s=2$) and bosons (recovered in the $s\rightarrow \infty$ limit); it implements a parastatistics where at most $s-1$ indistinguishable particles are accommodated in a multi-particle sector.\\ The structures discussed in this work are:\\ - the quantum group interpretation of the roots of unity truncations recovered from a (superselected) set of reps of the quantum superalgebra ${\cal U}_q({\mathfrak{osp}}(1|2))$;\\ - the reconstruction, via suitable intertwining operators, of the braided tensor products as ordinary tensor products (in a minimal representation, the $N$-particle sector of the braided Majorana qubits is described by $2^N\times 2^N$ matrices);\\ - the introduction of mixed brackets for the braided creation/annihilation operators which define generalized Heisenberg-Lie algebras;\\ - the $s\rightarrow \infty$ untruncated limit of the mixed-bracket Heisenberg-Lie algebras producing parafermionic oscillators;\\ - ({\it meta})symmetries of ordinary differential equations given by matrix Schr\"{o}dinger equations in $0+1$ dimension induced by the braided creation/annihilation operators;\\ - in the special case of a third root of unity truncation, a nonminimal realization of the intertwining operators defines the system as a ternary algebra.
- Year
- 2024
- Number
- CBPF-NF-002/24
- Linked to a thesis
- No
- Linked to a grant
- No