Finite-lattice positivity

Finite-lattice positivity

Reflection positivity starts at the finite Wilson law. Cutting a torus into two reflected halves rewrites each seam contribution as a positive kernel, so that a future observable paired with its reflected conjugate has nonnegative expectation. The finite-dimensional statement gives the null-space quotient used later to build a Hilbert space. It is also the input to the chessboard bounds: positivity converts a product of separated chamber observables into a product of individual rooted bounds. The details below retain the seam and orientation conventions needed for both uses.

First the finite Wilson law is an actual probability measure. Every unitary matrix has Re tr: each diagonal entry has modulus at most one. Thus every plaquette price lies 2]. If there are plaquettes and0, its continuous density relative to the product Haar probability measure obeys

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e−2βF ≤e−βS(U) ≤1,
e−2βF ≤Z ≤1.

In particular normalization is legitimate and bounded measurable cylinders are integrable. A site gauge change multiplies a link on its two ends; by Haar invariance its change of variables preserves the product prior. Its plaquette holonomy is conjugated at the base site, so the trace and the action are unchanged. The normalized law is consequently gauge invariant. A signed lattice permutation merely reindexes plaquettes, possibly reversing their holonomies; inversion preserves their real traces. It too preserves both the prior and the law. The local representatives in (PF12), and their finite products, therefore have precisely the finite-lattice expectations used below.

Reflection positivity is the nonnegativity of the expectation of a future observable times its reflected complex conjugate. It is a property of this finite law, before any quotient or completion. The Wilson measures used throughout have that property at every nonnegative coupling. We give the finite-dimensional argument, since it also supplies the rooted bounds used in extraction.

First, the kernel

on is positive semidefinite: it is the real Euclidean inner product of the real and imaginary matrix entries. Finite sums, pullbacks, nonnegative multiples and pointwise products of positive

semidefinite kernels are positive semidefinite. For the product assertion, express each finite positive semidefinite matrix as a Gram matrix and tensor its Gram vectors. The exponential of such a kernel is positive semidefinite as well, by its power series with nonnegative coefficients and entrywise limits of the resulting finite matrices.

Cut the periodic lattice perpendicular to a coordinate axis. A leaf consists of the links tangent to one slice, and a thread consists of the links joining successive slices. With the thread fixed at the identity, the factor joining two leaves has the form

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

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where q counts the crossing plaquettes and Eβ is the positive factor from the allocated half of the
tangential action. Thus (RP1) is a continuous positive semidefinite kernel for β ≥0.
For general threads, use their endpoint gauge action on the leaf links. Haar invariance gives the
identity
Z
Wβ(X, Y ; u) du =
ZZ
Wβ(u · X, v · Y ; 1) du dv.
(RP2)

Both integrations use the product normalized Haar measure on the thread variables. 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. All integrands are continuous and bounded on compact spaces, so this passage and Fubini are justified. The averaged kernel in (RP2) is continuous by dominated convergence.

Now split the lattice into the two halves of the reflection. For a reflection through sites, the two halves share the seam-site variables, and after fixing them the cross kernel is the constant kernel 1. For a reflection through links, there are two seam threads on the periodic lattice. They are distinct even at the shortest allowed half-length. Their joint Haar law is the product thread law. After integration, the cross kernel is the product of the two kernels (RP2), one with its leaf arguments reversed. It is again positive semidefinite.

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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 by their coordinate
assignments and the endpoint gauge multiplications along the cut direction. Each change of coor-
dinates multiplies a Haar link variable on the left or right by fixed already listed group elements,
or inverts that variable. Iterated Haar invariance shows that this assembly preserves the product
prior. Reading each plaquette shows that the action splits into the two half factors and the seam
factor just described. A probe F supported strictly in one half reads only that half and the shared
seam variables; its gauge invariance permits the endpoint gauge changes. Its reflected conjugate
reads the other half.

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Consequently, conditional on the seam variables, the unnormalized reflected pairing has the
form
ZZ
f(X)f(Y ) Kseam(X, Y ) dX dY,

with positive semidefinite. This integral is nonnegative by the integrated Gram inequality. The half-action factors have been included in. Bounded measurable probes are allowed: the Wilson density is bounded on the compact finite configuration space, and their reflected products are bounded and measurable, so all integrations are absolutely integrable. Integrating the conditional nonnegative quantities and dividing by the strictly positive Wilson partition function proves

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⟨F F ◦θ⟩β ≥0.
(RP3)

Here is the actual geometric link reflection, including inverse links. In particular no crossing link has been replaced inside the probe. We next derive the chessboard inequality on an admissible rectangular tiling. Let its number of cells. For a function in its strict reference chamber, write for its translated, reflected and, where prescribed, conjugated copy in cell

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*Y
c
F (c)
+
1/v
R(F) =
.
β

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For different chamber functions Fc, let V (F•) = |⟨Q
c F (c)
c ⟩β|. The Cauchy–Schwarz inequality from
(RP3), applied across any translated cell seam, gives

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V (A)2 ≤V (A+)V (A−),
(RP4)

where andare the assignments obtained by reflecting the corresponding half-assignment into the other half.

For a finite alphabet of functions, choose a cell assignment maximizing . Both folded values in (RP4) are at most this maximum. If the maximum is positive, (RP4) forces both folded values to equal it; if it is zero this is immediate. Hence the property of being a maximizing assignment is preserved by every fold. Such a fold-closed property has a constant assignment: in one coordinate start with a one-cell-wide strip of identical slices and fold at its preceding boundary. This extends the strip by one slice while preserving the property. Continue until a half-period is filled, then reflect it to fill that coordinate. A fold in another coordinate preserves equality along the coordinates already filled. Repeat in all four coordinates. This is a finite construction and needs no power-of-two restriction on the number of slices.

For arbitrary, apply this maximum principle to the finite alphabet

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eFc = Fc/(R(Fc) + ε),
ε > 0.

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Every constant assignment of a normalized function has value at most 1, because its value is
[R(Fc)/(R(Fc) + ε)]v. The maximum principle bounds their mixed assignment by 1. Rescale each
cell and then let ε ↓0 to obtain

*Y
+

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≤
Y
c
R(Fc).
(RP5)
c
F (c)
c
β

The same root is a seminorm on the chamber functions. Complex homogeneity follows because a copied scalar is either or ̄, and the absolute value of their full product is To prove subadditivity, expand the full copied product of into its 2 assignments of The triangle inequality and (RP5) bound its expectation by (. Take the positive-th root. This proves all the rooted homogeneity, subadditivity and mixed-product estimates used below directly for the Wilson law.