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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.18225421 |
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- <p dir="ltr"><strong>Abstract</strong></p> <p dir="ltr">A central obstacle in constructive approaches to four-dimensional Yang-Mills theory is the loss of spectral control upon introducing non-Abelian gauge degrees of freedom. Within the Spectral Vacuum Mechanism (SVM) framework, we have previously obtained explicit lower bounds on the spectral gap for multichannel lattice Hamiltonians and elevated these results to a structural level through the Spectral Confinement Class (SCC). The present work addresses the critical next step: demonstrating that SCC survives the introduction of non-Abelian gauge structure.</p> <p dir="ltr">We consider a truncated SU(2) gauge embedding where gauge representations are restricted to finite spin j ≤ j_max. Even at j_max = 1, this setting exhibits genuine non-Abelian features: non-commuting SU(2) generators [J_a, J_b] = iε_{abc}J_c, multiplet structure with (2j+1)-dimensional representations, and plaquette self-interactions coupling four gauge fields. Truncation preserves operator boundedness while incorporating these essential Yang-Mills-like features.</p> <p dir="ltr"><strong>Our main result (<a href="https://docs.google.com/document/d/1zJO3IYyqeZBlbKZF0toZdCZQ44kfxziR/edit#bookmark=id.x3stbe6eb27n">Theorem B</a>) establishes that spectral confinement can be preserved under this gauge embedding, provided coupling parameters remain within an explicitly controlled window. Specifically, for H = H_m + H_mg + H_g, the spectral gap satisfies</strong></p> <p dir="ltr"><strong>Δ ≥ m₀ − 2z(d)(gV_max^m + κV_max^mg) − C_□(d)λW_max</strong></p> <p dir="ltr"><strong>where z(d) = 2d is the coordination number and C_□(d) = 2d(d−1) is the plaquette incidence constant for d-dimensional cubic lattices. </strong></p> <p dir="ltr"><strong>This constitutes the first nontrivial contact between SVM-based spectral confinement and Yang-Mills-like gauge structure. </strong></p> <p dir="ltr">The dimensional scaling C_□(d) ~ d² provides a spectral explanation for why four dimensions are critical: the plaquette contribution imposes the strongest dimensional penalty.</p> <p dir="ltr">We additionally establish three verifiability criteria strengthening physical contact:</p> <p dir="ltr">(i) A penalty method H_μ = H + μ·∑(G_x)² provides minimal gauge-invariant embedding with controlled gap preservation Δ_phys ≥ 0.9Δ at μ = 10Δ (<a href="https://docs.google.com/document/d/1zJO3IYyqeZBlbKZF0toZdCZQ44kfxziR/edit#bookmark=id.uolonj8mbvpo">Lemma D.1</a>), making the construction testable on quantum simulators.</p> <p dir="ltr">(ii) The gap exhibits Lipschitz continuity |Δ(g,κ,λ) − Δ₀| ≤ L_Lip·δ(g,κ,λ) with explicit constant L_Lip = 4d·V_max + 4d·V_max^mg + 2d(d−1)·W_max, establishing spectral rigidity and excluding fine-tuning (Theorem D.2).</p> <p dir="ltr">(iii) The dimensional scaling C_□(d) ~ d² yields a falsifiable prediction for the correlation length ratio R₄₃ := ξ(d=4)/ξ(d=3) ≈ 0.84 ± 0.2 at normalized coupling λ̂ = 0.3, directly testable via numerical simulation or cold-atom quantum simulator experiments. This closes the verification loop from spectral gap bounds to experimentally measurable observables, transforming abstract confinement estimates into concrete physical predictions.</p> <p dir="ltr"><strong>Keywords</strong></p> <p dir="ltr">Yang-Mills theory • mass gap • spectral confinement • lattice gauge theory • SU(2) gauge group • truncated representations • Hamiltonian formulation • spectral gap bounds • variational estimates • gate condition • dimensional analysis • four dimensions • confinement mechanism • quantum field theory • constructive methods • operator bounds • plaquette dynamics • matter-gauge coupling • exponential clustering • Spectral Vacuum Mechanism • penalty method • gauge invariance • spectral rigidity • Lipschitz continuity • stability zones • falsifiable predictions • quantum simulators • correlation length • dimensional crossover</p> <p dir="ltr"><strong>MSC 2020 Classification</strong></p> <p dir="ltr"><strong>Primary:</strong></p> <p dir="ltr">81T13 (Yang-Mills and other gauge theories in quantum field theory)</p> <p dir="ltr">81T25 (Quantum field theory on lattices)</p> <p dir="ltr">81V05 (Strong interaction, including quantum chromodynamics)</p> <p dir="ltr"><strong>Secondary</strong>:</p> <p dir="ltr">81Q10 (Selfadjoint operator theory in quantum theory, including spectral analysis)</p> <p dir="ltr">47A10 (Spectrum, resolvent)</p> <p dir="ltr">82B20 (Lattice systems)</p> <p dir="ltr"><strong>PACS Numbers</strong></p> <p dir="ltr">11.15.-q (Gauge field theories)</p> <p dir="ltr">11.15.Ha (Lattice gauge theory)</p> <p dir="ltr">12.38.-t (Quantum chromodynamics)</p> <p dir="ltr">12.38.Aw (General properties of QCD)</p> <p dir="ltr"><strong>Subject Areas</strong></p> <p dir="ltr">Mathematical Physics • Quantum Field Theory • Lattice Gauge Theory • Spectral Theory • Constructive Quantum Field Theory • Yang-Mills Theory • Confinement Physics</p> <p dir="ltr"> </p> <p><strong>Other related works by the author:</strong></p> <p dir="ltr">[1] Nemirovsky M., From Quantum Theory to Thermodynamic Applications Based on the Nemirovsky Bound, <a href="https://zenodo.org/records/17977864">https://zenodo.org/records/17977864</a> (2025)</p> <p dir="ltr">[2] Nemirovsky M., Spectral Vacuum Mechanism - Part XIV Spectral Confinement as a Necessary Condition for Quantum Field Theory Confinement Gate-Induced Spectral Localization and Dimensional Constraints, <a href="https://zenodo.org/records/18140235">https://zenodo.org/records/18140235</a> (2025) </p> <p>[3] Nemirovsky M., Spectral Vacuum Mechanism - Part XV Unification of the mass formula in SVM particles of the Standard Model, <a href="https://zenodo.org/records/18207487">https://zenodo.org/records/18207487</a> (2026)</p>