论文标题
编织的Majorana Fermions的首次量化
First quantization of braided Majorana fermions
论文作者
论文摘要
$ {\ mathbb z} _2 $ graded Qubit代表一个均匀的（玻感）“真空状态”和一个奇怪的，兴奋的Majorana fermion状态。 $ n $，编织，难以区分的Majorana fermions的多片扇区是通过首次量化构建的。该框架是赋予编织张量产品的分级HOPF代数的框架。 Hopf代数为$ {u}（{\ Mathfrak {gl}}（1 | 1））$，$ {\ Mathfrak {gl}}}（1 | 1）$ superalgebra的通用包络代数。 $ 4 \ times 4 $编织矩阵$ b_t $定义了编织的张量产品。 $ b_t $，与亚历山大 - 连线多项式的$ r $ -matrix有关，取决于属于刺穿平面的编织参数$ t $（$ t \ in {\ mathbb c}^\ ast $）;费米子的普通反对称性能以$ t = 1 $的形式回收。对于每个$ n $，计算分级多片的分级尺寸$ m | n $。除了通用情况外，当$ t $与某些统一根相一致时，会发生截断，这些统一起源是有序的一组多项式方程的解决方案。统一的根部分为水平，这些水平指定了多片扇区中允许编织的Majorana fermions的最大数量。通过考虑到$ {\ Mathbb z} _2 $ raded Hilbert空间中的偶数/奇数部门是超级选择的，这是一个非平凡的编织，$ t \ neq 1 $是生产Qubits，Qutrits所描述的非繁琐的Hilbert Space的必不可少的必不可少的。经典$ 1 $ - 携带的信息。
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.
