First quantization of braided Majorana fermions

Abstract: A ${\mathbb Z}_2$-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 ${U}({\mathfrak {gl}}(1|1))$, the Universal Enveloping Algebra of the ${\mathfrak{gl}}(1|1)$ superalgebra. A $4\times 4$ 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 ($t\in {\mathbb C}^\ast$); 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 ${\mathbb Z}_2$-graded Hilbert space are superselected, a nontrivial braiding with $t\neq 1$ 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.