Feshbach reduction with the vacuum and domains intact

Remove the vacuum before testing a positive energy

For a nonnegative Hamiltonian with HΩ=0, the desired excitation statement is H≥δ on Ω^⊥. Testing H−δ on the full space would fail immediately on Ω. Begin with the excited space if the projections preserve it.

If a proposed low projection P splits the vacuum, first use the vacuum-corrected transfer operator G=rI−T+(1−r)|Ω⟩⟨Ω| and transport ω=PΩ through the high block. Its Schur complement satisfies Fω=0. Strict estimates then belong on P°=P−|ω⟩⟨ω|/∥ω∥².

A genuine completion-band projection is designed to contain Ω exactly. Its low space still contains the vacuum, so the strict low-energy estimate must remove that line. Containing the vacuum simplifies the geometry of the reduction; it does not make the ground energy positive.

The calculations below can be read on an already punctured excitation space. The parameter z is an energy threshold, and the question is whether all remaining energies lie at or above z.

Source: Vacuum obstructions and form Feshbach reduction, §6, pp.15–18.

Derive the block square identity

Write H−z in blocks as A−zP, B, B† and D−zQ. Assume D≥ΓQ and z<Γ. For p in the low space and q in the high space, add and subtract the auxiliary response (D−z)⁻¹B†p:

The second term is nonnegative and can be made zero by choosing the high response q=−(D−z)⁻¹B†p. Therefore H−z≥0 if and only if F(z)≥0. This is stronger than a variational upper bound: it is an exact equivalence under its domain and invertibility hypotheses.

For an eigenvalue problem, F(E)p=0 determines the low component of an eigenvector, and the eliminated formula reconstructs its high component. The energy dependence is essential. Replacing E by a fixed reference value gives an approximation whose error needs a separate bound.

The minus sign is fixed by minimization over the high variable. Coupling to a positive high sector reduces the low quadratic form; a positive diagonal high block does not guarantee positive full dynamics.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 10.1, pp.23–24.

Why unbounded Hamiltonians need forms

A gauge Schrödinger Hamiltonian contains differential operators on a compact product of groups. It is unbounded despite the finite regulator. A projection depending on the slow field can generate derivatives when commuted through the kinetic term. Its off-diagonal block PHQ may therefore fail to extend to a bounded operator.

Even the expression PHP needs a domain. A legitimate reduction must show that the relevant projection preserves Dom H, or more naturally the closed form domain Dom q. A discontinuous multiplication indicator is especially problematic because multiplying a Sobolev function by it can create a jump.

The form theorem asks for Dom q=P Dom q⊕Q Dom q. This asserts both that the projected pieces belong to the form domain and that they reconstruct every form vector. The high form must then define a complete Hilbert norm strong enough to represent the cross term.

There is also a distinction between compressing and applying a function of an operator. Generally Qe^−aHQ≠e^−aQHQ. A large expectation of H in a Q vector can coexist with substantial low-energy spectral weight. For example, an equal superposition of energies 0 and 100 has expectation 50 but e^−H expectation approximately 1/2. Direct transfer bounds require direct proof unless Q reduces H.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 10.3 and Remark 10.4, pp.24–25.

Represent the cross term in the high-form norm

Let q be a closed semibounded form and q_z=q−z∥·∥². On its projected domain, write q_z[p+r]=a_z[p]+d_z[r]+2Re b[p,r]. Assume d_z is closed and strictly positive and that r↦b[p,r] is continuous in the norm d_z[r]^1/2.

The Riesz representation theorem supplies a unique K_zp in the high form space satisfying d_z[K_zp,r]=b[p,r]. Define the effective form by

Expanding the final square proves the identity. Choosing r=−K_zp proves q_z≥0 exactly when f_z≥0. No bounded PHQ is needed.

If the correction obeys d_z[K_zp]≤ηa_z[p] with 0≤η<1 and a_z≥0 on the punctured space, then f_z≥(1−η)a_z. This is the natural relative estimate for the exact auxiliary regulator. It is a theorem about what such a cross-form bound would imply; obtaining the bound remains model specific.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 10.3, pp.24–25.

An exact two-by-two positivity test

For H with diagonal entries a,d and real coupling b, the eigenvalues are

When d>z, F(z)=a−z−b²/(d−z). Thus λ₋≥z is equivalent to F(z)≥0. The ordinary determinant condition (a−z)(d−z)≥b² is the same statement after multiplication by the positive denominator.

Take a=1,d=9,b=2. The lower eigenvalue is about 0.527864. At z=0.5, the reduced margin is 0.5−4/8.5≈0.0294118, so the threshold is certified. At z=0.6 it is 0.4−4/8.4≈−0.0761905, so it fails, exactly as the eigenvalue says.

The example is also a warning about judging blocks separately. Both a and d are positive, yet b²>ad makes the full lower eigenvalue negative. The Schur correction measures precisely the effect that a diagonal inspection misses.

In a finite dangerous cluster, the same test becomes a matrix positivity calculation with an independently bounded model error. A positive model eigenvalue is useful only to the extent that it exceeds that error.

The boundary d=z deserves separate treatment. The displayed Schur inverse is then undefined, but the exact matrix still exists. If b=0 and a≥z, the shifted matrix is positive semidefinite with a zero high direction. If b≠0, its determinant is −b²<0 and it has a negative eigenvalue. A singular denominator therefore signals failure of the stated positive-invertible-high-block method, not the disappearance of the original eigenvalue problem. One may enlarge the retained space to include the zero high direction or use a separately justified singular-block theorem; silently dividing by a very small number supplies neither.

Near a boundary, an exact sign certificate should also retain its numerical margin. If the smallest computed value differs from zero by less than the independently controlled error, the calculation does not establish strict positivity or strict negativity. The finite example has explicit eigenvalues and determinant formulas, making it a useful consistency test for both the algebra and the presentation of numerical thresholds.

When a fast gap actually suppresses mixing

If ∥B∥≤b and D≥ΓQ, then 0≤B(D−z)⁻¹B†≤b²/(Γ−z)P. Suppose A≥ΔP on the excitation space. A sufficient condition for H≥z is

This inequality displays the competing quantities. Γ may diverge while b also diverges. If b remains bounded, the correction vanishes on fixed energy windows. If b=c√Γ, the correction tends to c²; if b grows faster, a large high energy may fail to suppress its effect.

In the form version, the analogous requirement is a relative cross-form error that tends to zero after the high coercivity is inserted. Bounds on derivatives of a fast projection are part of that estimate because slow kinetic energy acts on the changing auxiliary eigenfunctions.

A local band gap alone does not bound the global mixing. Overlapping blocks and gauge constraints can correlate the local bands. The global projection must be constructed with those correlations included and its exact relationship to the full form proved.

The source therefore demands a simultaneous scaling window: a diverging nonlinear fast gap, a vanishing adiabatic error and vanishing induced four-dimensional interactions for the same frozen parameter sequence.

Source: Auxiliary-time lattice gauge framework, §§17–19, printed pp.44–49.

Keep every term in the resolvent estimate

When F(z) is invertible, solving the two block equations gives the low block of (H−z)⁻¹ as F(z)⁻¹. Its off-diagonal block is −F(z)⁻¹B(D−z)⁻¹, and the high diagonal contains both the bare high resolvent and a feedback term.

Consequently a useful bound for the full high row is the sum of 1/(Γ−z), b∥F(z)⁻¹∥/(Γ−z), and b²∥F(z)⁻¹∥/(Γ−z)². Omitting the feedback term can invalidate a claimed resolvent estimate near a low spectral value.

A resolvent bound also requires a spectral separation from the low eigenvalues. If F(z) approaches singularity, its inverse can grow even while Γ is very large. Uniform convergence should therefore be stated on a declared contour or energy region with its resolvent control.

These bookkeeping details matter when a local effective calculation is transported across scales. The small parameter must control the actual quantity being compared, including all blocks and the chosen energy dependence.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 10.1, p.24.

Where the exact identity enters the lattice programme

The redesigned route first derives local fast forms from the exact global Friedrichs form. It then seeks uniform nonlinear spectral bands and gauge-covariant derivative estimates, patches them into a global projection containing Ω, and proves the form-domain and cross-form conditions above.

The effective form can then be carried by an isometry to a four-dimensional gauge Hilbert space. Its comparison with an ordinary gauge Hamiltonian must account for kinetic renormalization, geometric band corrections and all induced gauge-invariant interactions. Gauge invariance restricts these terms but does not establish their smallness.

After auxiliary elimination, a separate analysis must establish the four-dimensional nonabelian infrared gap, construct the continuum vacuum theory and identify its observable net intrinsically. Schur–Feshbach reduction is a rigorous transport mechanism between estimates; it is not a substitute for the estimates it consumes.

This makes the proof obligations reviewable. One can check the vacuum first, the form domain second, the high coercivity third, and the cross-form margin fourth. A failure at any stage identifies a specific mathematical gap in the reduction rather than an ambiguity about the word “effective.”

Source: Auxiliary-time lattice gauge framework, Program Requirements 14.6, 17.3, 18.5 and 19.5, printed pp.38,44–49.

Continue reading

• The exact vacuum-window counterexample
• Interactions generated by elimination
• Dangerous clusters and finite model margins
• Constructing a nonlinear completion band
• A nonempty low space after vacuum puncture
• Counterexamples that locate a gap proof’s missing premise
• A nonzero vacuum-punctured low window