Physical Vacuum – Part V Spectral Vacuum Mechanism: A Constructive Approach to Confinement-Like Spectral Properties

Note on this version (December 16, 2025): Revised version incorporating referee suggestions: clarified scope as a multichannel (0+1)D proof-of-principle toy model, refined terminology (“spectral confinement class” for confinement-like spectral properties), expanded proofs for Lemmas 1–3, strengthened references to deterministic localization, and updated abstract/disclaimers.

Abstract

This work develops a constructive operator-level framework for a quantum field theory exhibiting confinement-like spectral properties based on the Spectral Vacuum Mechanism (SVM). Instead of starting from a classical gauge-field Lagrangian, the theory is defined directly as the infinite-channel limit of a sequence of finite-dimensional multichannel Hamiltonian systems with compact internal symmetry. Physical excitations arise as localized positive eigenmodes of the spectral Hamiltonian, interpreted as the second variation around a nontrivial vacuum configuration.

Under explicit structural assumptions — bounded self-adjoint local terms, short-range positive-type interactions, and a gate threshold in the connectivity structure preventing zero-mode delocalization — we establish reflection positivity at finite channel number, uniform lower bounds on the energy density, and existence of a well-defined infinite-channel limit. The resulting Euclidean correlation functions are compatible with the Osterwalder–Schrader axioms in the channel-time direction.

A strict mass gap Δ > 0 is obtained via a variational lower bound tied to the universal gate scale, leading to exponential clustering of correlation functions and the absence of massless non-singlet asymptotic states. These properties define a spectral confinement class sharing necessary features with the confining phase of pure non-Abelian gauge theories.

The construction is rigorously established in a multichannel (0+1)D framework as a proof-of-principle toy model; extension to full (3+1)D, including spatial gauge transformations and renormalization, remains an open problem requiring substantial additional development. The results offer a complementary perspective on mechanisms of confinement and mass generation.

Keywords: confinement; mass gap; constructive quantum field theory; spectral vacuum; localization; toy model; Osterwalder–Schrader axioms

Other works by the author on similar topics:

[1] Physical Vacuum – Part I Spectral vacuum mechanism (SVM) (lepton masses), Published December 13, 2025 | Version v1, https://zenodo.org/records/17931024

[2] Physical Vacuum – Part II Spectral vacuum mechanism (SVM) (quark sector), Published December 13, 2025 | Version v1, https://zenodo.org/records/17923554

[3] Physical Vacuum – Part III Composite (hadronic) excitations in the Spectral Vacuum Mechanism (SVM), Published December 13, 2025 | Version v1, https://zenodo.org/records/17923690

[4] Physical Vacuum – Part IV Spectral vacuum mechanism (SVM) Hadronic sector in the Spectral Vacuum Mechanism: baryons and mesons from the universal gate phase, Published December 15, 2025 | Version v1, https://zenodo.org/records/17942901