基于线性同伦类型的量子化
Quantization via Linear homotopy types
摘要：
在同伦型理论的基本逻辑框架中，我们讨论了稳定几何同伦理论中二次积分变换的自然形式化。我们观察到，这产生了一个非微扰的局部前量子场论的上同调量子化的过程，并表明量子异常抵消相当于实现这一点的边界的场论，是由真正的（初级）积分变换，因此由线性多项式函子。回顾传统的线性逻辑在对称monoidal范畴中具有语义，并用于形式化量子力学，我们考虑的是它对线性同伦型理论的改进，其语义在稳定同伦型束的稳定无限大范畴（广义上同调理论）中，形式化拉格朗日量子场论，遵循Nuiten，与Haugseng和Hopkins-Lurie最近的工作密切相关。对于对量子化的技术问题感兴趣的读者，我们提供了泊松流形和超弦的非微扰量子化;并深入了解量子反常抵消、全息原理和量子化中的动机结构。对于倾向于解释量子力学的读者，我们将量子叠加和干涉展示为线性同伦型理论中的存在量化。对于倾向于基础的读者，我们提供了一个由Lawvere提出的物理学正式基础的改进，从经典连续介质力学提升到局部拉格朗日量子规范场。
英文摘要：
In the foundational logical framework of homotopy-type theory we discuss a natural formalization of secondary integral transforms in stable geometric homotopy theory. We observe that this yields a process of non-perturbative cohomological quantization of local pre-quantum field theory; and show that quantum anomaly cancellation amounts to realizing this as the boundary of a field theory that is given by genuine (primary) integral transforms, hence by linear polynomial functors. Recalling that traditional linear logic has semantics in symmetric monoidal categories and serves to formalize quantum mechanics, what we consider is its refinement to linear homotopy-type theory with semantics in stable infinity-categories of bundles of stable homotopy types (generalized cohomology theories) formalizing Lagrangian quantum field theory, following Nuiten and closely related to recent work by Haugseng and Hopkins-Lurie. For the reader interested in technical problems of quantization we provide non-perturbative quantization of Poisson manifolds and of the superstring; and find insight into quantum anomaly cancellation, the holographic principle and motivic structures in quantization. For the reader inclined to the interpretation of quantum mechanics we exhibit quantum superposition and interference as existential quantification in linear homotopy-type theory. For the reader inclined to foundations we provide a refinement of the proposal by Lawvere for a formal foundation of physics, lifted from classical continuum mechanics to local Lagrangian quantum gauge field theory.
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