What question did this study set out to answer?

This research aims to address five core mathematical problems in quantum field theory through the UCM substrate.

April 16, 2026Open Access

The Well-Defined Path Integral: How the UCM Substrate Resolves Five Foundational Problems of Quantum Field Theory (UCM Paper 54)

Key Points

This research aims to address five core mathematical problems in quantum field theory through the UCM substrate.
Introduced the UCM substrate and its fibre-bundle structure.
Analyzed how the bulk field contributes a finite Lebesgue measure.
Evaluated the gauge field's contribution of a finite Haar measure.
Eliminated the matter integral through gauge-matter unity.
Constructed a complete partition function using well-defined measures.
The UCM substrate resolves the issue of measure existence in infinite dimensions.
Convergence of the Lorentzian integrand is achieved in this framework.
Demonstrated that traditional issues with 4D Yang-Mills can be addressed.
Validated Haag's theorem interactions within the UCM context.
The complete partition function is now a finite product of well-defined measures.

Abstract

The Feynman path integral rests on five unresolved mathematical foundations: (1) the measure does not exist in infinite dimensions; (2) the Lorentzian integrand does not converge; (3) rigorous Euclidean constructions exist only in d<4; (4) 4D Yang-Mills existence is an open Millennium Prize problem; (5) Haag's theorem forbids the interaction picture. The UCM substrate combined with the fibre-bundle structure of Papers 52-53 resolves all five: the bulk field contributes a finite Lebesgue measure, the gauge field contributes a finite Haar measure, and gauge-matter unity eliminates the matter integral. The complete partition function is a finite product of well-defined measures.

The Well-Defined Path Integral: How the UCM Substrate Resolves Five Foundational Problems of Quantum Field Theory (UCM Paper 54)

Key Points

Abstract

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