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Fermions in odd space-time dimensions: Back to basics

机译:时空维度中的费米子:回到基础

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It is a well-known feature of odd space-time dimensions d that there exist two inequivalent fundamental representations A and B of the Dirac gamma matrices. Moreover, the parity transformation swaps the fermion fields living in A and B. As a consequence, a parity-invariant Lagrangian can only be constructed by incorporating both the representations. Based upon these ideas and contrary to long-held belief, we show that in addition to a discrete exchange symmetry for the massless case, we can also define chiral symmetry provided the Lagrangian contains fields corresponding to both the inequivalent representations. We also study the transformation properties of the corresponding chiral currents under parity and charge-conjugation operations. We work explicitly in 2 + 1 dimensions and later show how some of these ideas generalize to an arbitrary number of odd dimensions.
机译:奇异时空维d的一个众所周知的特征是,存在狄拉克伽马矩阵的两个不等价的基本表示形式A和B。此外,奇偶变换转换了生活在A和B中的费米子场。因此,只能通过合并两个表示来构造奇偶不变的拉格朗日矩阵。基于这些想法并且与长期以来的信念相反,我们表明,除了针对无质量情况的离散交换对称性之外,我们还可以定义手性对称性,前提是拉格朗日包含对应于两个不等价表示形式的场。我们还研究了奇偶和电荷共轭操作下相应手性电流的转化性质。我们在2 + 1维度中进行了明确的工作,随后展示了其中的某些想法如何推广到任意数量的奇数维度。

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