Projective Partition Functions and the Emergence of Particle Masses: From QCD Path Integrals to Diagram Hilbert Space

We investigate a formal correspondence between QCD path integrals and projective partition functions in a Diagram Hilbert Space framework. In the real-space formulation of QCD, oscillatory path integrals over quark and gluon fields are divergent, rendering direct computation intractable. By complexifying the integration domain and performing an imaginary-space projection, these integrals become convergent and mathematically controllable, while retaining full physical content. In parallel, we consider a partition function defined as a product over projective operators acting in a Diagram Hilbert Space, which naturally selects stable eigenstates corresponding to particle masses. We demonstrate that, despite the different formal appearances, both structures are isomorphic at their core: mass generation emerges universally as a consequence of projective summation over fundamental excitations. This equivalence provides a unified perspective on particle mass emergence and offers a framework bridging QCD semiclassical expansions with topological Hilbert-space constructions.