The vacuum obstruction to a sharp low-energy window

Mohamed Haj Yousef

Assumptions, results and research status

Exact obstruction and conditional repair

Assumptions

T is bounded, positive and self-adjoint, with normalized vacuum TΩ=Ω.
The proposed gap threshold has r=e^(−aₜδ) in (0,1).
The orthogonal window P has nonzero vacuum weight on both sides, PΩ≠0 and QΩ≠0.
The high block D=QGQ is strictly positive before its inverse is used.

Results

The Feshbach operator annihilates the transported low vacuum PΩ exactly.
Replacing the correct G off-diagonal blocks with T blocks makes the mixing quadratic form strictly too large on that vector.
Puncturing the low window removes the rank-one obstruction.
Uniform low/high contractions and punctured relative mixing would yield an explicit margin proportional to the physical time step.

Open questions

Can the corrected contractions and mixing estimates be proved uniformly for the specified lattice family?
Can a genuine spectral completion band replace the sharp multiplication window while containing the vacuum exactly?
Can the resulting effective four-dimensional theory satisfy a uniform nonabelian gap estimate?

The sharp window is a configuration cut

Let K(X)≥0 be the total mixed-plaquette energy at a fixed regulator and let K denote multiplication by this function on gauge-invariant wavefunctions. For ε>0, the projection P=1_[0,ε](K) multiplies a wavefunction by the indicator of the region A_ε={X:K(X)≤ε}. Its complement is Q=I−P.

This cut is useful for locating configurations with a small covariant interruption defect. It is not generally an energy spectral projection of the physical Hamiltonian. In particular, it need not preserve the excited space Ω^⊥.

The finite-regulator vacuum Ω is strictly positive. If both A_ε and its complement contain open sets, then

∥PΩ∥²=∫_Aε|Ω|²dμ>0,
∥QΩ∥²=∫_X∖Aε|Ω|²dμ>0.

The source proves that K is continuous, nonconstant and has minimum zero. Thus every genuine window with 0<ε<∥K∥ splits the vacuum. Making ε=0 collapses the multiplication low space to zero when the exact zero set has Haar measure zero; making ε≥∥K∥ removes the high sector entirely.

Source: Vacuum obstructions and form Feshbach reduction, §§4.3 and 5.4–5.5, pp.11,14–15.

Keep the vacuum correction explicitly

The physical target is H≥δ on Ω^⊥, with H=−a_t⁻¹log T and δ>0. Put r=e^−a_tδ. On the excited space this is equivalent to T≤rI. On the full space, rI−T has the negative eigenvalue r−1 on the vacuum.

The correct full-space operator adds precisely the missing rank-one term:

G=rI−T+(1−r)|Ω⟩⟨Ω|,
GΩ=0,
G≥0 ⇔ H≥δ on Ω^⊥.

This correction has off-diagonal P/Q components whenever the window divides Ω. Therefore PGQ is not simply −PTQ. Dropping the rank-one term in a mixing estimate changes the operator being tested.

The safe order of work is to identify the vacuum, remove it on a reducing excited space where possible, or transport it through the chosen split. Only then should strict coercivity or a relative-mixing inequality be imposed on the remaining low vectors. The obstruction below occurs even when the full Hamiltonian already has the desired gap.

Source: Vacuum obstructions and form Feshbach reduction, Proposition 6.1, p.15.

Transport the vacuum through the high block

Write ω=PΩ and ζ=QΩ, and suppose D=QGQ≥dQ with d>0. The Q component of G(ω+ζ)=0 gives

ζ=−D⁻¹QGω.

The low vacuum component therefore carries exactly the high component required to rebuild Ω. Eliminating the high block yields F=PGP−PGQD⁻¹QGP, and the P component of the same equation gives Fω=0.

Since F is self-adjoint, its kernel line is orthogonal to all cross terms. Define the normalized line projection P_ω=|ω⟩⟨ω|/∥ω∥² and the punctured low projection P°=P−P_ω. The space Ran P° reduces F.

Every vector in Ran P° is orthogonal to Ω, because it already lies in Ran P and is orthogonal to PΩ. The reduced positivity question is therefore F≥0 on Ran P°, together with the exact zero eigenvector ω. It is not a demand that the entire low window have a strictly positive floor.

The bounded Schur square identity then makes F≥0 equivalent to G≥0 when D>0. Transporting the vacuum repairs the decomposition without changing the physical target.

Source: Vacuum obstructions and form Feshbach reduction, §6.1, pp.15–16.

The exact inequality that rules out the shortcut

Set α=1−r>0 and p=∥ω∥². From QGω=−Dζ and the definition of G, obtain QTω=(D+αpI)ζ. Meanwhile Fω=0 implies ⟨ω,PGPω⟩=⟨ζ,Dζ⟩>0.

If one substitutes the transfer off-diagonal block into the mixing term, its value becomes

⟨ω,PTQD⁻¹QTPω⟩
=⟨ζ,Dζ⟩+2αp∥ζ∥²+α²p²⟨ζ,D⁻¹ζ⟩
>⟨ω,PGPω⟩.

Both extra terms are strictly positive because p>0, ζ≠0 and D is positive. Therefore no η≤1 can satisfy PTQD⁻¹QTP≤ηPGP on the entire low space.

This is a mathematical no-go statement about that proposed estimate. It does not prove that the physical gap is absent. It proves that a particular route to certifying the gap is invalid when its window splits the positive vacuum.

The obstruction is independent of gauge fixing, volume or numerical precision. Stronger estimates on unrelated finite blocks cannot repair an inequality that already fails on the exact transported vacuum vector.

The two additional terms have a useful interpretation. The uncorrected transfer block transports not only the response required by GΩ=0 but also the piece produced by the missing vacuum projector. The erroneous estimate then squares both contributions in the positive high inverse. A cancellation that is exact for G becomes a strictly positive surplus when expressed with T alone. This is why treating the correction as a harmless change only to the diagonal is insufficient.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 1.1 and §6.2, pp.3,16–17.

See the failure in two dimensions

Choose p∈(0,1), Ω=(√p,√(1−p)) and v=(√(1−p),−√p). Let T=|Ω⟩⟨Ω|+t|v⟩⟨v|, with 0<t<r<1. Then T is positive with a strictly positive matrix kernel, and G=(r−t)|v⟩⟨v| is already nonnegative.

Take P to project onto the first coordinate. The scalar low and high blocks and the incorrect mixing term are

A=(r−t)(1−p), D=(r−t)p,
M_wrong=(1−t)²(1−p)/(r−t),
M_wrong/A=[(1−t)/(r−t)]²>1.

For p=½, t=0.2 and r=0.8, the low diagonal is A=0.3, but the incorrect mixing subtraction is 0.533333…. It would report a negative reduced value. The correct G coupling is −0.3, giving A−0.3²/0.3=0 exactly.

The actual excited energy is −log(0.2)/a_t, which exceeds the requested threshold −log(0.8)/a_t. The failed estimate therefore produces a false obstruction even in a model where the gap is known explicitly.

Here the low space has only the transported vacuum line, so puncturing leaves no low excitation. A larger model can have a nontrivial punctured low space; the algebraic cause and its repair are the same.

The relative violation is independent of p, even though its absolute size changes with the window weight. As p approaches one, the high vacuum tail becomes small and both low quadratic forms become small, but their ratio stays [(1−t)/(r−t)]². A floating-point check with only an absolute tolerance could therefore conceal the failure near that endpoint. At p=1 exactly the high vacuum tail vanishes and the theorem’s split-vacuum assumption no longer holds. A numerical explorer should distinguish that exceptional case from every nearby genuine split and report the relative ratio as well as the cancellation residual.

The corrected conditional estimate

On Ran P°, the vacuum projection vanishes. Thus QGP°=−QTP° and P°GP°=rP°−P°TP°. The transfer blocks can now be used in the punctured relative-mixing inequality without dropping a rank-one contribution.

Assume numbers m₀>δ, m_Q>δ and 0≤η<1 satisfy

P°TP°≤e^−a_tm₀P°,
QTQ≤e^−a_tm_QQ,
P°TQD⁻¹QTP°≤ηP°GP°.

The second inequality makes D strictly positive. The first and third yield F≥(1−η)(e^−a_tδ−e^−a_tm₀)P°. The vacuum line remains an exact zero mode, so the bounded Schur criterion then proves the desired gap on the declared neutral excitation space.

These inequalities are assumptions of the corrected theorem. The source proves their implication but does not prove them uniformly for the lattice family. The puncture repairs the algebra; it does not manufacture the missing nonabelian estimates.

In particular, the high transfer contraction is an estimate on QTQ itself. A lower bound for the compressed Hamiltonian QHQ cannot simply be exponentiated to infer it when Q does not reduce H.

Source: Vacuum obstructions and form Feshbach reduction, Lemma 6.7 and Proposition 6.8, p.18.

Keep the physical-time-step margin

The difference of exponentials has an exact integral form:

e^−a_tδ−e^−a_tm₀=a_t∫_δ^m₀e^−a_tsds.

If 0<a_t≤a_t,max, then the punctured margin is at least ca_t, with c=(1−η)(m₀−δ)e^{−a_t,maxm₀}>0. This is the scaling appropriate to a fixed energy gap as the transfer step tends to zero.

A dimensionless one-step estimate that vanishes with a_t is therefore not automatically weak. Its meaning depends on whether the quotient by a_t stays positive. Conversely, a positive number at every regulator can vanish too quickly to produce a uniform physical gap.

The source also gives a necessary vacuum-tail test. If QTQ≤ρQ with ρ<1, then p=∥PΩ∥² must obey p≥(1−ρ)/(2−ρ). It follows by expanding ⟨Ω,TΩ⟩ and using T≥0. This condition can rule out proposed windows with too little vacuum weight, but satisfying it does not establish the required high contraction.

Source: Vacuum obstructions and form Feshbach reduction, Proposition 6.6 and Proposition 6.8, pp.17–18.

Why the programme moves to quantum bands

A hard multiplication cut has a second limitation: multiplication by a discontinuous indicator need not preserve a Schrödinger form domain. A sharp cutoff of an H¹ wavefunction can create a jump and an uncontrolled derivative. The bounded transfer-gap calculation is legitimate on its Hilbert space, while an unbounded Hamiltonian reduction needs a separate domain argument.

The redesigned programme therefore seeks spectral projections of exact local completion Hamiltonians. A kinetic term gives genuine normalizable quantum bands. A holonomy closure term addresses continuous auxiliary flat moduli. The global band must contain Ω exactly, so the physical vacuum is compatible with the split from the beginning.

Neither local compact resolvent nor a tangent stiffness estimate supplies that global band automatically. Gauge constraints, overlapping blocks, boundary sectors and geometric derivative terms must be controlled. The later form-Feshbach theorem is the correct interface once those tasks have been completed.

The obstruction is useful because it removes a false target and specifies the corrected one. It leaves a concrete sequence of mathematical tasks rather than an apparent proof resting on an impossible inequality.

The replacement must also respect physical normalization. Moving the cutoff, changing a local Hamiltonian or flattening a spectrum creates a new model unless a comparison theorem controls the change. A finite representation truncation can produce genuine local ground projections, but it may change exact unitary link identities and introduce boundary contributions. These alternatives are legitimate research choices only when their errors are carried into the final spectral margin. They cannot be inserted silently to rescue the failed unpunctured bound.

Source: Vacuum obstructions and form Feshbach reduction, §§7 and 10, pp.19–21,23–25.

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