Differential cohomology and locally covariant quantum field theory
Differential cohomology and locally covariant quantum field theory
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DOI：
10.1142/s0129055x17500039
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发表时间：
2014-06
期刊：
影响因子：
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通讯作者：
C. Becker;Alexander Schenkel;R. Szabo
中科院分区：
文献类型：
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作者：
C. Becker;Alexander Schenkel;R. Szabo
我们研究全局双曲洛伦兹流形范畴上的微分上同调。洛伦兹度规允许我们定义一个自然的变换，它的核推广了麦克斯韦方程，并适合微分上同调的基本精确序列的限制。我们考虑微分上同调群的光滑Pontryagin群，它是特征标群的子群。我们证明了这些群适合于微分上同调的基本精确序列的光滑映射，并使它们具有由广义麦克斯韦拉格朗日量导出的自然前辛结构。由此产生的前辛阿贝尔群量化使用CCR-函子，这产生了一个协变函子从我们的范畴的全局双曲洛伦兹流形的C*-代数的范畴。我们证明了这个函子满足局部协变量子场论的因果性和时间片公理，但它违反了局部性公理。我们表明，这种违反恰恰是由于这样一个事实，即我们的函子的拓扑子函子描述的庞特里亚金的某些奇异上同调群。作为副产品，我们开发了微分上同调群上的Fr\'echet-Lie群结构。
We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the CCR-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of C*-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fr\'echet-Lie group structure on differential cohomology groups.