Reflection positivity from a vacuum Markov process

Four different positivity statements

A positive Boltzmann weight defines a probability measure. A kernel with nonnegative entries preserves nonnegative functions. A nonnegative self-adjoint transfer operator has nonnegative spectral values. Reflection positivity is a statement about a sesquilinear form obtained by reflecting observables. These properties are related by the source construction, but they are not interchangeable definitions.

For example, K = [[0.1,0.9],[0.9,0.1]] is a strictly positive, reversible Markov matrix with stationary distribution (1/2,1/2). Its eigenvalues are 1 and −0.8. It preserves probabilities but cannot equal exp(−H) for a self-adjoint H, since such an exponential has positive spectral values. A positive kernel alone therefore does not justify taking a real self-adjoint logarithm.

The source supplies stronger input: a symmetrized heat-kernel transfer operator, normalized by its top eigenvalue. Its nonnegativity is an operator statement, and its positivity-improving kernel supplies the unique positive vacuum. Both facts enter the reconstruction. Keeping them separate also clarifies why the abstract three-state puncture model, written in an excitation-adapted basis, is not itself presented as a coordinate Markov kernel.

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

The vacuum supplies the reversible probability measure

Let TΩ = Ω with Ω normalized and strictly positive relative to the configuration measure μ. Define dν = Ω²dμ and map a function f to Ωf. This is a unitary map U_Ω from L²(ν) to the original Hilbert space. The transformed transfer operator is T̂ = U_Ω⁻¹TU_Ω.

If t(X,Y) is the original kernel, its kernel relative to ν is t̂(X,Y) = t(X,Y)/[Ω(X)Ω(Y)]. The vacuum equation implies ∫t̂(X,Y)dν(Y) = 1. Symmetry of t gives reversibility, while operator nonnegativity and the spectrum are preserved by the unitary transformation. The vacuum becomes the constant function one.

The stationary one-time distribution is therefore ν, not an arbitrary Haar slice distribution. Kolmogorov consistency constructs a two-sided stationary chain indexed by integer transfer times. Conditioned on its value X₀, the past and future are independent. Reversibility identifies the reflected past with another copy of the future conditioned on the same boundary value. This conditional structure is the engine of the finite reflection-positivity proof.

Source: Vacuum obstructions and form Feshbach reduction, §4.1, equations (35)–(37) and Proposition 4.1, printed/PDF p.10.

Why complex conjugation is essential

Let θ act on a path by (θX)_k = X_−k. Reflection on observables is antilinear: ΘF = conjugate(F ∘ θ). The conjugation matters. Without it, even a constant observable F = i would give a negative product. The original PDF includes this overbar, although a plain-text extraction can lose it.

For a bounded cylinder F depending on times 0 through m, conditional independence gives

For two cylinders, polarization yields the inner product of their conditional expectations. Thus the reflection form is positive semidefinite rather than necessarily positive definite. A nonzero random variable can have zero conditional expectation at the reflection boundary, and consequently zero reconstructed norm.

This proof concerns reflection through a time slice. A reversible chain with a negative transfer eigenvalue can still satisfy this particular conditional-square identity. The additional nonnegative transfer operator is what permits the source’s one-step translation to have a self-adjoint Hamiltonian logarithm. The full construction uses both pieces.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 4.2 and equation (38), printed/PDF p.10; the conjugation was checked in the rendered original.

An exact complex-valued cylinder calculation

Let X take values ±1 with uniform stationary law and transition matrix K_t = ½[[1+t,1−t],[1−t,1+t]], where 0 < t < 1. Its constant eigenfunction has eigenvalue one; the function X has eigenvalue t. Consequently E[X_j|X₀] = t^jX₀ and E[X_jX_k] = t^|j−k|.

Choose the genuinely complex future cylinder F = X₁ + iX₂. Its conditional expectation is (t + it²)X₀, so its reflection norm is t² + t⁴. Direct multiplication confirms the cancellation of the imaginary cross terms:

The random variable N = X₁ − tX₀ instead has zero reflection norm, since E[N|X₀] = 0, despite ordinary variance 1 − t². Quotienting by such null vectors leaves the two-dimensional space of functions of X₀. The reconstructed transfer eigenvalues are 1 and t; its Hamiltonian energies are 0 and −log t per declared step. The calculation illustrates reconstruction, rather than assuming a Hilbert-space norm from a Euclidean probability variance.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 4.2, printed/PDF p.10, specialized here to a declared two-state chain.

A periodic trace can retain excited weight

For the same two-state spectrum, a periodic chain of M transfer steps gives the normalized trace state T^M/Tr(T^M). Its expectation of the excited-state projector is t^M/(1+t^M). The pure vacuum expectation is zero. They agree only in the large-M limit at fixed t < 1, not at finite periodic length.

The order of limits is visible without any field theory. Take t_n = exp(−1/n) and M_n = n. Then the excited weight is exactly 1/(e+1) at every n. Each fixed regulator separately approaches the vacuum as M tends to infinity, yet this simultaneous sequence does not.

If t_n = exp(−a_nΔ_n), vacuum preparation in this two-level example requires M_na_nΔ_n → ∞. With many excited states, their multiplicities and trace tails must also be controlled; the two-level condition alone is not a general uniform trace bound. Constructing the infinite two-sided vacuum process directly avoids confusing a finite-temperature periodic trace with the desired vacuum state.

Source: Vacuum obstructions and form Feshbach reduction, Remark 4.3, printed/PDF p.11.

How lattice boundary weights become squares

The projected-lattice source gives another finite reflection-positivity route. Split the action into positive-time, reflected negative-time and boundary plaquette terms. Boundary weights must admit a suitable positive character expansion, so that the factors crossing the reflection plane can be sewn into a sum of reflected squares. Positivity of each Boltzmann value by itself does not prove this factorization.

For the stated compact-group trace weight with nonnegative coupling, the expansion is natural: Re χ_ρ = (χ_ρ + χ_ρ̄)/2. Expanding the exponential produces nonnegative coefficients multiplying tensor-product characters; decomposition into irreducibles has nonnegative multiplicities. Compactness controls the uniformly convergent exponential series. The geometric boundary decomposition must still place these characters on the correct half-lattice variables.

After Haar integration, the resulting terms are absolute squares of half-lattice integrals. This realizes the same general principle as the conditional-expectation proof: reflected histories are paired through common boundary data. The auxiliary circle is part of the regulated configuration; reflection acts on the declared transfer-time coordinate. Exchanging those roles would require a different symmetry theorem.

Source: Projected lattice gauge synthesis, §2.2, printed/PDF p.12, and Appendix A.1, p.53.

What survives a pointwise limit

For finitely many positive-time observables A₁,…,A_m, form the matrix M⁽ⁿ⁾_ij = S_n((ΘA_i)A_j). Reflection positivity says c* M⁽ⁿ⁾c ≥ 0 for every complex vector c. If every entry converges on one common algebra, the limiting quadratic form is nonnegative because it is the limit of these nonnegative real numbers.

This argument needs convergence of the reflected products that appear in the form. Convergence of a few individual Wilson-loop means is insufficient. Renormalized insertions, products, time translates and continuity in the chosen completion must be available before the form can be extended beyond a finite list.

The closedness argument supplies no strict lower bound. In the two-state example, t tends to one while every reflection form remains positive. The limiting transfer is the identity and the nontrivial excitation energy tends to zero. Reflection positivity can therefore persist when a gap closes or the vacuum becomes degenerate. Uniform gap transfer needs additional spectral and comparison-space assumptions.

Source: Auxiliary-time lattice framework, Lemma 28.1, printed p.57 / PDF p.64, and Theorem 29.2, printed p.59 / PDF p.66.

The remaining reconstruction obligations

The exact finite construction is valuable because it specifies a vacuum, a Hilbert space, a positive transfer and its generator without relying on a formal path integral. A continuum construction must additionally control the renormalized observable algebra and prove covariance, regularity, locality, clustering and nontriviality for its limiting functional.

The source isolates restoration of four-dimensional Euclidean covariance because its Hamiltonian regulator begins with a spatial lattice and a distinguished transfer direction. Equality of temporal and spatial normalizations, admissible counterterms and continuity of rotations must be established along the chosen trajectory. A finite positive transfer does not determine that isotropic limit or its physical unit calibration.

The resulting research task is therefore concrete: construct a common reflected core, prove convergence of all required finite forms, establish the remaining OS properties, and connect the reconstructed generator to the same uniformly gapped forms used in the spectral argument. Auxiliary invisibility then remains a further identification condition. Reflection positivity is a rigorous part of this chain, with a precisely delimited contribution.

Source: Auxiliary-time lattice framework, Requirements 28.3–28.4 and Proof Audit 28.5, printed p.58 / PDF p.65.

Continue reading

• The exact vacuum construction
• Comparison spaces and gap transfer
• Neutrality and character sectors
• The continuum limit programme