论文标题
美元
${\mathbb Z}_2\times {\mathbb Z}_2$-graded mechanics: the quantization
论文作者
论文摘要
在上一篇论文ARXIV：2003.06470中，我们介绍了$ {\ Mathbb Z} _2 _2 \ times {\ Mathbb Z} _2 _2 $ - 级经典的机制，并提出了一个通用框架，并在Lagrangian设置中构建了一个构建的一般框架，在Lagrangian环境中，在Lorldline Sigma Models dy Mathiant the $ n MATHBB BB BB BBB BB BB { Z} _2 $ - 级超级级。在这项工作中，我们首先讨论了其中一些模型的古典汉密尔顿表述，然后介绍其规范量化。作为结构的最简单应用，我们恢复了$ {\ Mathbb Z} _2 \ times {\ Mathbb Z} _2 $ - graded Quantum Hamiltonian由Bruce和Doplij引入了Arxiv：1904.06975。我们证明，这是大型$ {\ mathbb z} _2 \ times {\ Mathbb z} _2 $ - 加法的量子模型的第一个示例。我们尤其得出了由Hermitian，矩阵，差分运算符给出的相互作用的多粒子量子哈密顿量。相互作用的术语显示为矩阵中的非对角线条目。提出了纳赛电荷的构建，包括古典和量子。对不同$ {\ Mathbb Z} _2 \ times {\ Mathbb z} _2 $ graded symmetries的全面讨论。
In the previous paper arXiv:2003.06470 we introduced the notion of ${\mathbb Z}_2\times{\mathbb Z}_2$-graded classical mechanics and presented a general framework to construct, in the Lagrangian setting, the worldline sigma models invariant under a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded superalgebra. In this work we discuss at first the classical Hamiltonian formulation of some of these models and later present their canonical quantization. As the simplest application of the construction we recover the ${\mathbb Z}_2\times{\mathbb Z}_2$-graded quantum Hamiltonian introduced by Bruce and Duplij in arXiv:1904.06975. We prove that this is the first example of a large class of ${\mathbb Z}_2\times{\mathbb Z}_2$-graded quantum models. We derive in particular interacting multiparticle quantum Hamiltonians given by Hermitian, matrix, differential operators. The interacting terms appear as non-diagonal entries in the matrices. The construction of the Noether charges, both classical and quantum, is presented. A comprehensive discussion of the different ${\mathbb Z}_2\times{\mathbb Z}_2$-graded symmetries possessed by the quantum Hamiltonians is given.
