Quantization via Linear homotopy types

Abstract

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.

89 pages; these are expanded talk notes for three talks that I gave in February 2014 at Paris Diderot and at ESI in Vienna, one together with Joost Nuiten

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Cited by in corpus (8)

• Iterated spans and classical topological field theories
• Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams
• Ambidexterity and the universality of finite spans
• Syntax and Semantics of Linear Dependent Types
• On Constructive Axiomatic Method
• Algebraic K-theory, K-regularity, and T-duality of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Indexed type theories
• A Categorical Semantics for Linear Logical Frameworks