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This paper extends canonical quantization procedures in quantum field theories by introducing flat bundles on infinite-dimensional functional manifolds of local time, with the S-matrix represented as a time-ordered exponential. The authors generalize Hamiltonian evolution and functional renormalization group equations within this framework, construct moduli spaces of flat connections with rational structure, and identify physical states as points in these moduli spaces. A key theoretical outcome is that spacetime concepts, including particle configuration spaces, emerge as spectral sets of functional differential operators rather than being assumed as foundational inputs.
Why it matters
This framework targets longstanding challenges in theoretical physics, including the first-principles treatment of bound states in quantum chromodynamics and precision calculations for hydrogen and muonium, which are relevant to high-stakes tests of the Standard Model. If the approach proves robust, it could offer new mathematical tools for understanding confinement and atomic-scale quantum systems with greater rigor.
Understand the Science
Abstract: In this paper we discuss extensions of the canonical quantization procedure in quantum field theories. We focus specifically on S-matrix representation as a T-exponent. This extension involves flat bundles on certain infinite dimensional functional manifolds of local time. The motivating problem is first principles treatment of bound states in quantum chromodynamics as well as precision physics of hydrogen atom and the muonium. Our main results include systematic treatment of flat bundles in an infinite dimensional setting, generalization of Hamiltonian evolution and functional renormalization group evolution equations in quantum field theories. We discuss several results from finite dimensional theory that have analogies in the functional setting. This includes construction of moduli space of flat connections and isomonodromic deformations. One of the outcomes of our analysis is a construction of a rich family of functional flat bundles with rational connections. This class of connections exhibits a rich set of mathematical properties. In particular, we construct examples of spaces fundamental groups of which have a definable continuum of generators. Physical states correspond to points in the moduli space of bundles on these spaces. On the physics side of things, we conclude that spacetime notions, such as spaces of particle configurations, emerge effectively as spectral sets of functional differential operators.
Source: Flat Bundles on Function Manifolds and Evolution Equations in Quantum Field Theories