Assumptions, results and research status
Exact symmetry decomposition; decoupling conditional
Assumptions
- The Hamiltonian commutes with the finite cyclic auxiliary-translation symmetry.
- Auxiliary charge means a character of that symmetry and is kept distinct from electric or nonabelian gauge charge.
- The physical vacuum is unique, positive and auxiliary neutral.
- Statements about charged thresholds, complete excitation gaps and continuum decoupling specify their sector and regulator dependence.
Results
- Cyclic symmetry gives exact orthogonal charge-sector projections.
- A diverging threshold on the nonneutral reducing subspace removes those states from every fixed low-energy interval.
- Neutrality alone leaves invariant auxiliary observables and physical gauge dynamics.
- The programme requires holonomy concentration, scalarization, geometric compatibility and route independence in addition to charged divergence.
Open questions
- Can a uniform threshold be proved for all nontrivial auxiliary characters simultaneously?
- Can neutral auxiliary observables become scalar on the physical observable core?
- Can the remaining neutral four-dimensional gauge sector be shown to have a uniform mass gap?
Specify which symmetry supplies the charge
The word “charge” can describe different representation labels. Gauge charge concerns the action of local internal frame transformations and Gauss constraints. Electric charge is a particular physical generator after gauge-group and matter identification. Auxiliary translation charge labels the representation of the cyclic shift τ↦τ+1 in the regulator.
These labels should not be identified by notation alone. A nontrivial Fourier mode around the auxiliary circle is not automatically an electrically charged particle, and a winding number is not automatically a measured gauge charge. Each proposed identification needs a representation map and a coupling to the physical observables.
For a cycle of N sites, let R be the simultaneous shift of all auxiliary link variables. It satisfies RN=I. The frozen Hamiltonian commutes with R, so the Hilbert space decomposes into its character sectors.
The auxiliary circle is a modelling device for recurrent completion. The six nested inner levels and seventh completion describe a different organizational statement; they do not supply N independent time dimensions or N physical charges.
Source: Vacuum obstructions and form Feshbach reduction, §§1 and 4.2, pp.2–3,11.
Construct the charge projections exactly
Let ωN=e2πi/N. The projection onto the sector with R eigenvalue ωNq is
PqPq′=δqq′Pq, ΣqPq=I.
The identities follow from summing finite geometric series. Since R is unitary, the projections are orthogonal. Since H commutes with R, every sector reduces H and its spectral calculus.
The neutral projection is P₀=N−1ΣRj. Vacuum uniqueness and positivity imply RΩ=Ω, so Ω belongs to that sector. The other sectors contain no vacuum component.
If an observable O transforms as ROR−1=ωNqO, then O maps charge k to charge k+q modulo N. In particular OΩ belongs to charge q. For q≠0 its vacuum expectation is zero, provided the vector and observable are in the stated domain.
This exact decomposition is stronger than a heuristic Fourier interpretation because it concerns the symmetry acting on the actual quantum Hilbert space. It remains distinct from the finite-difference eigenproblem on matrix-valued link sequences.
Source: Vacuum obstructions and form Feshbach reduction, §4.2, p.11.
What a charged threshold controls
Suppose H restricted to a nonneutral sector obeys H≥Γq. For a charged observable with ψ=OΩ in that sector, the spectral theorem gives
This is a sector-specific correlation bound. It controls the states that O can create from the vacuum. An observable in a different sector can have a different threshold, and a neutral observable can create excitations not seen by this inequality.
For a changing finite cyclic group, a pointwise claim for each labelled character should be strengthened to a uniform bound on the entire nonneutral complement when that complement is to be eliminated. The minimum over all nontrivial characters can change as N increases.
The most useful decoupling assumption is (I−P₀)H(I−P₀)≥Γτ(I−P₀), with Γτ→∞. This concerns physical energy units, including the outer-time normalization. A large eigenvalue of an auxiliary difference matrix is not the same hypothesis.
Such a threshold can help explain why an observer with fixed energy resolution fails to excite the nonneutral auxiliary sector. It does not by itself establish that the entire observable theory is four dimensional.
Source: Auxiliary-time lattice gauge framework, §30.1 and Program Requirement 30.2, printed p.60.
A diverging charged gap leaves the neutral gap free
An exact four-state model demonstrates the logical separation. Take an orthonormal basis consisting of a vacuum, a neutral excitation and two oppositely charged states. For N≥3 set
Rn=diag(1,1,e2πi/N,e−2πi/N).
The operators commute. Choose εn=1/n and Γn=n. Every nonneutral energy diverges, while the full excitation gap is 1/n and tends to zero. The vacuum remains unique for every finite n.
The example is not a lattice gauge theory. It proves that charged divergence alone cannot imply a uniform neutral gap. Any argument making that inference requires an additional relation between the sectors, and that relation must be proved in the actual model.
The source’s mixed-flat trivial-holonomy stratum gives a concrete reason this issue appears in the auxiliary regulator. It contains ordinary τ-independent spatial gauge configurations. Their dynamics survive even when auxiliary interruption is zero. Their nonabelian infrared spectrum is therefore a separate problem.
Neutral composite operators can also carry dynamics. A product OqO−q is neutral under the cyclic symmetry, but symmetry alone says nothing about its spectral weight, binding energy or overlap with independent neutral modes.
Source: Vacuum obstructions and form Feshbach reduction, §5.2, pp.12–13.
What the neutral projection leaves behind
Cyclic averaging removes observables with a nontrivial character, but preserves every invariant function. The trace of auxiliary holonomy is a basic example: changing the starting point of the cyclic product conjugates the holonomy, and the trace is unchanged.
A sum over τ of a local relative-configuration observable is also invariant under simultaneous shift. Neutrality therefore permits information about relative auxiliary patterns, even when a single labelled auxiliary slice is unobservable.
Gauge invariance does not remove this issue. Traced holonomy is also invariant under local frame changes. It can survive both the gauge projection and auxiliary neutrality. Two symmetry filters can leave a common invariant sector with nontrivial dynamics.
This is why the source calls the neutral space a physical candidate. A genuine four-dimensional identification must prove that these residual observables become irrelevant, scalar, infinitely costly or identified with already declared physical data.
A scalar limit means that the observable no longer distinguishes physical states beyond its expectation value. It is much stronger than the statement that the observable commutes with a symmetry. An operator can commute with every symmetry and still have a rich spectrum.
Source: Vacuum obstructions and form Feshbach reduction, §4.2, p.11.
A precise low-energy decoupling lemma
Let Pn commute with Hn and suppose the complementary spectrum starts at Γn→∞. Fix any finite observation energy E. Once Γn>E, the complementary spectral projection below E vanishes:
The proof is simply the spectral theorem on reducing subspaces. The restriction of H to the complement has no spectrum in [0,E], so its spectral projection on that interval is zero.
The commutation assumption is essential to this exact conclusion. A merely approximate low-band projection can have small high-sector leakage into low-energy states, and that leakage needs an error estimate. The form-Feshbach method supplies one route to such estimates when the band is not reducing.
Applied to P₀, the lemma shows eventual neutrality of every bounded-energy state. It does not show that all neutral auxiliary operators act as scalars there. That is a distinct statement about operator matrix elements on the retained sector.
The result also illustrates why the order of limits matters. E is held fixed in physical units while Γ diverges. Allowing the observation energy to grow alongside Γ can retain states that a fixed-energy limit removes.
Source: Auxiliary-time lattice gauge framework, Lemma 30.1, printed p.60.
State auxiliary invisibility on observable matrix elements
The auxiliary framework asks for several compatible limits. Nonneutral thresholds must diverge. Auxiliary holonomies must concentrate near the declared trivial or admissible topological data. Most importantly, centered neutral auxiliary observables must lose their physical matrix elements.
Here Bn is a bounded local neutral auxiliary observable, and A₁,A₂ belong to the declared physical observable core. The expression tests whether B can distinguish or connect physical states after its scalar expectation is removed.
Holonomy concentration alone is useful but narrower. It constrains one family of auxiliary observables; it does not automatically control all neutral relative configurations, all bounded operator insertions or all geometric band corrections.
The completion-band Kato curvature must also obey the no-extra-gauge condition. A positive rank-one fast state has no Abelian Berry phase in its positive gauge, but its Born–Huang correction can survive and requires a quantitative relative-form bound.
Finally, physical correlations must be independent of admissible choices of N, aτ, closure potential, auxiliary gauge fixing and equivalent fast-band frame. This route independence is part of removing a regulator rather than turning its arbitrary details into new physical predictions.
Source: Auxiliary-time lattice gauge framework, Program Requirement 30.2, printed pp.60–61.
The neutral four-dimensional theory still needs dynamics
After genuine auxiliary invisibility, the retained observables should be generated by four-dimensional Wilson loops and invariant curvature data. Their Hamiltonian can still have nontrivial neutral excitations. A physical mass gap asks for a uniform positive lower bound above its unique vacuum, not merely for the removal of auxiliary characters.
The source locates the central remaining Yang–Mills problem in nonabelian coercivity and renormalization from the ultraviolet regime to a massive infrared domain. A fast auxiliary band may improve locality or suppress unwanted variables; it does not determine the infrared spectral margin of the surviving gauge field.
This yields a useful reporting rule. State the symmetry that defines the charge, the exact subspace on which the threshold holds, the units of the energy, and whether the constants are uniform in volume and cutoff. Then state separately what has been proved for neutral modes and for the full observable theory.
With those qualifications, a charged-threshold result is substantial and interpretable. It becomes one component of a complete construction, rather than an ambiguous substitute for all the different meanings of a gap.
Source: Auxiliary-time lattice gauge framework, §§24–26 and 30, printed pp.54–56,59–61.
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Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.