The exact finite-regulator Hamiltonians

Start with a configuration manifold and a sector

A fixed outer-time slice has finitely many spatial and auxiliary links, each taking values in a compact group G. Their product X=G^E is a compact manifold with product Haar measure μ. The unreduced Hilbert space is L²(X,μ). Gauge transformations act at vertices and transform each link at both endpoints.

Averaging this action over the compact vertex gauge group defines an orthogonal projection Π_G. The physical slice space is its range. Gauge invariance is imposed on wavefunctions, so a state is a function of the entire compatible link configuration rather than an arbitrary tensor product of independently physical links.

Boundary conditions and global sectors are part of the model. On a torus, center or magnetic labels can constrain the allowed configurations and observables. A theorem in a trivial sector should retain that qualification when it is used in a larger argument.

The finite number of links does not make the Hilbert space finite dimensional. Each continuous compact group contributes infinitely many representation modes. Compact configuration space instead gives the elliptic kinetic operator a discrete spectrum with finite multiplicities.

Source: Vacuum obstructions and form Feshbach reduction, §3.1–3.3, pp.6–8.

A faithful representation fixes the defect

Let ρ:G→U(d_ρ) be faithful. Choose an invariant inner product on the Lie algebra and define c_ρ>0 by −Tr[dρ(X)dρ(Y)]=c_ρ(X,Y). The normalized plaquette defect is

Its eigenvalue expression is a sum of terms 1−cos θ_j, so it is nonnegative. It vanishes only when ρ(g)=I, and faithfulness then gives g=e. Conjugation invariance makes the defect gauge invariant on a closed plaquette.

Near the identity, ℓ_ρ(exp X)=½∥X∥²+O(∥X∥⁴). This identifies the quadratic stiffness in the chosen convention. Changing representation without changing c_ρ changes that normalization and therefore changes the numerical meaning of a kinetic or magnetic coefficient.

The gauge group is prescribed in this part of the programme. These operator results work after G has been selected; they do not derive the Standard Model group from the existence of the auxiliary cycle.

Source: Vacuum obstructions and form Feshbach reduction, §2.1, pp.3–4.

Freeze the kinetic and potential coefficients

The kinetic operator L_n is a positive sum of bi-invariant link Laplacians. For the frozen normalization in the source, the coefficient of a spatial-link Laplacian is g_n²/(2a_nN_τ,n), and the coefficient of an auxiliary-link Laplacian is g_n²a_n/(2a_τ,n²N_τ,n). Both are strictly positive at a fixed regulator.

The potential is a nonnegative sum of spatial and mixed plaquette defects. Define the mixed-energy multiplication function by

The spatial part has coefficient 1/(g_n²a_nN_τ,n). All these functions are bounded at fixed n because X is compact. Their bounds can nevertheless grow along the regulator sequence.

Strict ellipticity gives essential self-adjointness of L on smooth functions, a nonnegative closure and compact resolvent. Bi-invariance makes it commute with gauge transformations. The uniform coefficients also preserve translations, lattice rotations and cyclic auxiliary shifts.

Freezing the coefficients prevents circular reasoning in which an auxiliary parameter is retuned after observing the desired gap. The source’s two-loop-inspired numerical sequence is a declared reference sequence; its perturbative form is not a proof of a nonperturbative continuum trajectory.

Source: Vacuum obstructions and form Feshbach reduction, §§2.2–3.2, pp.4–7.

Build a positive transfer operator

For outer-time step a_t>0, form the heat operator P=e^−a_tL on the gauge-invariant slice space and the multiplication operator M=e^−a_tV/2. Then define

Self-adjointness follows from the symmetric placement of M. Operator positivity follows from ⟨f,T⁽⁰⁾f⟩=⟨Mf,P Mf⟩≥0. The heat operator is trace class, and bounded multiplication preserves that property. Since V is bounded, M has a bounded inverse; injectivity of the heat operator then gives injectivity of T.

Compact-group heat kernels are strictly positive at positive time. Their product, followed by gauge averaging and multiplication by the two positive potential factors, gives a strictly positive kernel. This proves positivity improvement: every nonzero nonnegative wavefunction is sent to a strictly positive wavefunction.

The compact positive-kernel theorem then gives a simple top eigenvalue and a normalized strictly positive eigenfunction Ω. The normalization of T makes the top eigenvalue one. Positivity as an operator and positivity improvement are different properties, and both have been checked here: the first controls spectral signs, while the second supplies vacuum uniqueness.

Source: Vacuum obstructions and form Feshbach reduction, §§3.3–3.4, pp.7–9.

Take the logarithm by spectral calculus

Because T is injective, its spectral measure has no atom at zero, even though zero can be a limit point of eigenvalues. Define

This is an unbounded operator in general. Its domain consists of vectors for which ∫a_t⁻²|log t|²d⟨ψ,E_T(t)ψ⟩ is finite; its quadratic-form domain uses a_t⁻¹|log t| instead. Spectral calculus supplies a densely defined nonnegative self-adjoint Hamiltonian, and HΩ=0.

At this fixed regulator the simple eigenvalue one of the compact operator T is isolated. If the remaining spectrum lies below r<1, then H≥−a_t⁻¹log r on Ω^⊥. This is a strictly positive finite-regulator gap.

Nothing in this compactness argument bounds r uniformly as the number of links grows, the volume increases or a_t tends to zero. For a fixed physical gap Δ, the desired transfer bound is r≤e^−a_tΔ, which approaches one as a_t shrinks. A spectral plot without its time-step normalization can therefore misrepresent the physical conclusion.

Source: Vacuum obstructions and form Feshbach reduction, Definition 3.4 and Proposition 3.5, p.9.

Define the completion Hamiltonian directly

The redesigned regulator adds a gauge-invariant holonomy-closure potential. At each spatial site, H_x is the product around the auxiliary circle. Put 𝓗(V)=Σ_xℓ_ρ(H_x) and V^cb=V+κ_H𝓗. The superscript labels the completion-band regulator.

On the gauge-invariant Sobolev domain, define the closed form

The potential is a bounded real form perturbation at fixed regulator. The representation theorem therefore defines a unique self-adjoint Friedrichs operator H₀^cb. Compact resolvent and positivity improvement give a simple positive ground state. Subtract its ground energy E₀^cb to obtain H^cb≥0, and define T^cb=e^−a_tHcb exactly.

This direct definition removes an ambiguity. The logarithm of one symmetric heat-potential step is generally not equal to L+V. The first construction and the Friedrichs construction are related regulators, with their own exact generators. A local fast Hamiltonian introduced later must be derived from the selected global form rather than assumed to coincide with it.

Source: Vacuum obstructions and form Feshbach reduction, §7.1, pp.19–21.

What fixed-regulator product convergence permits

For fixed regulator and fixed t, the symmetric product formula gives

Every approximating kernel remains strictly positive and gauge invariant. This is useful both conceptually and numerically: increasingly small symmetric steps approximate the exact unshifted Schrödinger semigroup at the chosen regulator.

The conclusion does not provide a convergence rate uniform in the regulator. Commutators, potential norms and derivative bounds can grow with N and with the physical lattice size. A simultaneous limit m=m(n) therefore needs estimates beyond the fixed-n product theorem.

A finite computation should state which object it diagonalizes: one normalized transfer step, a finite representation approximation to the Friedrichs operator, or an auxiliary tangent matrix. Their low eigenvalues are not interchangeable. If representation truncation is used, boundary errors and preservation of gauge constraints should be checked against the exact model.

An exact benchmark is a constant potential V=cI. Then the one-step operator is e^−a_tce^−a_tL. Its norm is e^−a_tc, since L has a constant zero mode, so normalization removes the constant and gives T=e^−a_tL. The logarithmic Hamiltonian is exactly L. The Friedrichs operator is L+cI, whose ground-energy subtraction also gives L. Both constructions agree in this commuting case. For a nonconstant potential, L differentiates V and the factors need not commute, which is why this benchmark does not justify their general identification.

This check also separates a vacuum-energy convention from a coupling prediction. Adding the same constant to every state changes neither normalized transfer probabilities nor excitation gaps. Changing a plaquette or kinetic coefficient generally changes relative energies and physical response. It cannot be removed by normalizing the top transfer eigenvalue. A numerical implementation can use the constant-potential case to check its normalization while retaining independent convergence checks for the interacting case.

Source: Vacuum obstructions and form Feshbach reduction, Proposition 7.4 and Remark 7.5, pp.20–21.

Check the surviving physical normalization

Set all auxiliary links to the identity and let the spatial links be independent of τ. Every mixed plaquette is then the identity, and summing the spatial potential over N auxiliary slices cancels the factor 1/N:

The surviving classical magnetic term has the declared ordinary lattice normalization. This is a useful consistency check: the regulator has not accidentally retained N copies of the magnetic energy. It is a pointwise identity on a configuration stratum, not a theorem that the quantum Hilbert space has already reduced to ordinary Yang–Mills.

The next step is a gauge-covariant slow–fast disintegration of the exact global form. It must identify physical base directions, auxiliary normal directions, cross terms, boundary constraints and chart overlaps without double counting. The source explicitly leaves this step as a programme requirement.

Only after that identification can local completion spectra feed a global effective Hamiltonian. The existence of both exact Hamiltonians is a firm starting point; uniform reduction, continuum existence and physical universality are separate achievements still required by the programme.

Source: Vacuum obstructions and form Feshbach reduction, §3.5 and Remark 7.5, pp.9,21.

Continue reading

• Vacuum and reconstruction
• Exact auxiliary-cycle geometry
• The sharp-window vacuum obstruction
• What a continuum limit must preserve