First quantization of braided Majorana fermions

Abstract

A Z 2 -graded qubit represents an even (bosonic) "vacuum state" and an odd, excited, Majorana fermion state. The multiparticle sectors of N , braided, indistinguishable Majorana fermions are constructed via first quantization. The framework is that of a graded Hopf algebra endowed with a braided tensor product. The Hopf algebra is Upglp1|1qq, the Universal Enveloping Algebra of the glp1|1q superalgebra. A 4 ˆ4 braiding matrix B t defines the braided tensor product. B t , which is related to the R-matrix of the Alexander-Conway polynomial, depends on the braiding parameter t belonging to the punctured plane (t P C ˚); the ordinary antisymmetry property of fermions is recovered for t " 1. For each N , the graded dimension m|n of the graded multiparticle Hilbert space is computed. Besides the generic case, truncations occur when t coincides with certain roots of unity which appear as solutions of an ordered set of polynomial equations. The roots of unity are organized into levels which specify the maximal number of allowed braided Majorana fermions in a multiparticle sector. By taking into account that the even/odd sectors in a Z 2 -graded Hilbert space are superselected, a nontrivial braiding with t ‰ 1 is essential to produce a nontrivial Hilbert space described by qubits, qutrits, etc., since at t " 1 the N -particle vacuum and the antisymmetrized excited state encode the same information carried by a classical 1-bit.

First quantization of braided Majorana fermions

“…It was shown in [5] that Z 2 -graded Majorana qubits can be braided within a First Quantization formalism. We revisit in this Section the main ingredients of this construction which makes use of a graded Hopf algebra endowed with a braided tensor product.…”

“…The starting point of [5] is the glp1|1q superalgebra. A single Majorana fermion describes a Z 2 -graded qubit which defines a bosonic vacuum state |0y and a fermionic excited state |1y:…”

“…It was shown in [5] that the braiding of Z 2 -graded Majorana qubits can be realized within a graded Hopf algebra endowed with a braided tensor product (this approach was based on the framework proposed in [6]). In the present paper the First Quantization of Z 2 -graded braided Majorana qubits is shown to be connected and expressed by several different constructions.…”

“…At first this paper solves a question left unanswered in [5,7]: how to directly relate the observed roots-of-unity truncations of the multi-particle braided spectra to quantum group reps data; it is shown that the truncations are also recovered from (superselected) representations of the quantum group U q pospp1|2qq.…”

“…-In Section 2 we revisit the [5] construction of the multi-particle Z 2 -graded braided Majorana qubits obtained from a graded Hopf algebra endowed with a compatible braided tensor product; -In Section 3 we present different suitable parametrizations of the truncations of the spectra at roots of unity; -In Section 4 we recover the roots-of-unity truncations from a quantum group perspective, pointing out that it is given by (superselected) reps of the quantum supergroup U q pospp1|2qq; -In Section 5 the braiding of Majorana qubits is realized via intertwining operators acting on ordinary (i.e., not braided) tensor products; -In Section 6 the indistiguishability of identical particles is recovered as a superselection; -Section 7 presents, following Leites-Serganova's papers, a sketchy introduction to the notion of the Volichenko algebras and their related metasymmetries; -Section 8 presents the generalized "mixed-bracket" Heisenberg-Lie algebras which interpolate bosons/fermions and reproduce the multi-particle sectors of the graded Majorana qubits; -In Section 9 it is shown that the s Ñ 8 limit of the mixed-bracket Heisenberg-Lie algebras reduce to (anti)commutators describing a set of parafermionic oscillators; -Section 10 presents the mixed-bracket Heisenberg-Lie algebras as dynamical metasymmetries of an ordinary differential Matrix Schrödinger equation in 0 `1 dimensions; -In the Conclusions we give comments about the link of the presented constructions with parastatistics and discuss future perspectives; -Appendix A clarifies the difference between generalized mixed-bracket Heisenberg-Lie algebras and the parastatistics induced by quons; -In Appendix B a nonminimal realization of the intertwining operators for the third root of unity is related to ternary algebras.…”