洛伦兹型凯勒-狄拉克费米子
Lorentzian Kähler-Dirac fermions
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中文总结 AI 辅助
本文研究洛伦兹符号时空上的凯勒费米子理论，发现其作为洛伦兹不变的形式或反对称张量场理论非幺正，通过引入修正内积与算符J可恢复幺正性，平直空间中等价于4个狄拉克费米子。
中文摘要 AI 辅助
我们研究具有洛伦兹符号的时空上凯勒费米子的构建，实际聚焦于闵可夫斯基时空，因为即便时空平直，遇到的大部分困难依然存在。我们证明，当将该理论解释为形式或反对称张量场的洛伦兹不变理论时，它是非幺正的。我们表明，只要在希尔伯特空间上采用修正内积，就能恢复幺正性。该修正内积需要插入一个算符J，它与凯勒场的某些模式反交换，以保证所有态具有正模。我们给出J的显式（且局域）形式，并证明它虽与哈密顿量对易，但与张量场的洛伦兹变换性质不兼容。在平直空间中，该幺正化构建等价于4个狄拉克费米子。
英文摘要
We examine the formulation of Kähler fermions on spacetimes with Lorentz signature. In practice we focus on Minkowski spacetime since most of the difficulties that are encountered are visible even when the spacetime is flat. We show that the theory, when interpreted as a Lorentz invariant theory of forms or antisymmetric tensor fields, is non-unitary. We show that unitarity can be restored provided one adopts a modified inner product on the Hilbert space. This modified inner product requires the insertion of an operator J that anti-commutes with certain modes of the Kähler field in such a way as to guarantee all states have positive norm. We give an explicit (and local) form for J and show that while it commutes with the Hamiltonian, it is incompatible with the Lorentz transformation properties of the tensor fields. In flat space, the unitarized formulation is equivalent to 4 Dirac fermions.
