Finite reflection positivity and rooted bounds

Finite reflection positivity and rooted bounds

Cut the periodic lattice perpendicular to a coordinate axis. A leaf is the set of links tangent to one slice; threads are the links joining successive slices. The supplied finite argument starts with the positive kernel), the real inner product of real and imaginary matrix entries. Finite sums, pullbacks and nonnegative multiples preserve positivity. Products preserve it by tensoring Gram vectors; the exponential preserves it by its nonnegative power series and finite-matrix limits. At identity thread the joining-leaf kernel is

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"
β
m
#
X
Wβ(X, Y ; 1) = e−βqEβ(X)Eβ(Y ) exp
l
Re tr(XlY ∗
l)
.
(RP1)

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Here q counts crossing plaquettes and Eβ allocates the tangential half-action.
For β ≥0 it is
continuous and positive semidefinite. Endpoint gauge changes and normalized Haar invariance give
Z
Wβ(X, Y ; u) du =
ZZ
Wβ(u · X, v · Y ; 1) du dv.
(RP2)

The identity follows by replacing one thread variable by its left translate and combining the two endpoint gauge actions. The right side is positive semidefinite: apply the positive-kernel quadratic inequality to finite linear combinations of the averaged feature vectors, or approximate the integrals by simple functions in the same Gram identity. Compact boundedness justifies Fubini and the continuity of the averaged kernel. Site reflections have shared seam variables and constant conditional cross kernel. Link reflections have two distinct periodic seam threads, including at the shortest allowed half-length, and the product of their averaged kernels, with one argument order reversed, is positive.

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The seam/open-link assembly changes variables only by left/right multiplication and inversion,
preserving the product Haar prior by iterated invariance. To make the splitting precise, list the seam
variables, the two seam threads when present, and the open variables on the two halves. Assemble
the original link configuration from these coordinates by their assignments and the endpoint gauge
multiplications along the cut direction. Its plaquette reading splits the action into the two halves
and seam factor.
Gauge-invariant probes strictly in one half read that half and shared seam
variables. Conditional on the seam, with half-action factors included in f, the unnormalized pairing
is
ZZ
f(X)f(Y )Kseam(X, Y ) dX dY ≥0.

The integrated Gram inequality proves its sign. Integrating over the seam preserves nonnegativity. Bounded measurable probes are integrable against the compact Wilson density. Dividing by the strictly positive partition function yields

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⟨F F ◦θ⟩β ≥0.
(RP3)

The reflection includes orientation inversions; no crossing link inside the probe is replaced. On an admissible rectangular tiling with cells, define

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*Y
c
F (c)
+
1/v
R(F) =
.
β

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The copied functions retain the prescribed translations, reflections and conjugations. For V (F•) =
|⟨Q
c F (c)
c ⟩β|, reflection Cauchy–Schwarz across each translated cell seam gives

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V (A)2 ≤V (A+)V (A−),
(RP4)

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where each assignment folds the corresponding half into the other. In a finite alphabet the maximum
exists. Both folded values are at most that maximum; if it is positive the inequality forces both
to equal it, and if zero equality is immediate. Thus maximizing assignments remain maximizing
under both folds. Starting with a one-cell-wide identical strip, successive boundary folds extend
it to a half-period and then the full coordinate. A fold in another coordinate preserves equality
along the coordinates already filled, so repeating in each coordinate yields a constant maximizing
assignment, without a power-of-two hypothesis. Normalize each chamber function by R(Fc) + ε.
Every constant assignment has value at most one, hence the mixed maximum does too. Rescaling
and ε ↓0 prove

*Y
+

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≤
Y
c
R(Fc).
(RP5)
c
F (c)
c
β

Copied scalars give): a copied scalar is either or ̄, and the absolute value of their full product is. Expanding the 2 assignments in the copied product of and applying RP5 bounds its expectation by (. Taking the root proves subadditivity. This proves all the rooted homogeneity, subadditivity and mixed-product estimates used below directly for the Wilson law.

For the lift, reflect frames and raw links by their geometric vertex/link assignment, with inverse raw links under orientation reversal. The path identity then gives. Thus has exactly the RP3 pairing by measure recovery. Every copied chamber product likewise has the original expectation, so its rooted seminorm and RP4–RP5 bounds are identical. All seam and support records are retained. This proves preservation of the finite positive form and rooted estimates before the later Hilbert completion.

Positive-form recovery on the retained probe algebra. Let be the supplied bounded gauge-invariant chamber algebra, with its strict half-space support convention, and let Equip its image with the reflected form

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BZ(J+F, J+H) =
Z
(F ◦π) (H ◦π) ◦θZ deμ =
Z
F H ◦θ dμW = B(F, H).

The middle identity uses the geometric intertwining and normalized pushforward together; positivity alone would not identify the form. The section gives (, so this is an injective

identification of the retained probe algebra. For any finite list, the entire reflected Gram matrix is identical on both sides, not only its diagonal. Consequently identifies the zero-form subspaces and; it induces an isometry of onto, and hence of their completions. This statement concerns precisely the retained probe image. For each copied chamber product the same identities give), including zero roots, so normalization by and the limit0 commute with recovery. Strict support is transported with the endpoints of each raw link and its endpoint frames; the original readout support is recovered on the section. Neither a new seam convention nor a positivity assertion on arbitrary raw-frame functions is needed.