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Exact finite-lattice SU(2) Wilson area law in two dimensions

Statement

Assume ${A C}_{ω}$ . Let $M, N \geq 1$ be integers and use the finite $M$ -by- $N$ square lattice with free boundary (no periodic identifications and all boundary links retained). Its vertices are $(i, j)$ for $0 \leq i \leq M$ , $0 \leq j \leq N$ ; its positively oriented edges are eastward $h_{i, j} : (i, j) \to (i + 1, j)$ for $0 \leq i < M$ , $0 \leq j \leq N$ , and northward $v_{i, j} : (i, j) \to (i, j + 1)$ for $0 \leq i \leq M$ , $0 \leq j < N$ ; and its faces $p_{i, j}$ for $0 \leq i < M$ , $0 \leq j < N$ have counterclockwise boundary words $(h_{i, j}, v_{i + 1, j}, {\overset{ˉ}{h}}_{i, j + 1}, {\overset{ˉ}{v}}_{i, j})$ . Use the matrix group $S U (2) = {U \in M_{2} (C) : U^{*} U = I_{2}, \det U = 1}$ with matrix multiplication and its identity defining representation. In real coordinates its elements are

U (x) = (\begin{array}{cc} x_{0} + i x_{3} & x_{2} + i x_{1} \\ - x_{2} + i x_{1} & x_{0} - i x_{3} \end{array}), x \in S^{3} : = {x \in R^{4} : \sum_{j = 0}^{3} x_{j}^{2} = 1} .

Let $σ_{3}$ be the induced round surface-area measure on $S^{3}$ , with total area $2 π^{2}$ , and let $μ_{2}$ be the pushforward of $σ_{3} / (2 π^{2})$ under this map. Each positively oriented edge carries one link matrix; reverse links use inverses. For each face $p$ , $U_{p}$ is the product of its boundary links in traversal order; for a loop $C$ , $U_{C}$ is the analogous ordered edge product. Use the Wilson action $S_{β} (U) = β \sum_{p} (1 - \frac{1}{2} {Tr}_{2} U_{p})$ and normalized loop $W (C) = \frac{1}{2} {Tr}_{2} U_{C}$ . Let $β = 4 / g_{0}^{2}$ for dimensionless $g_{0} > 0$ , and restrict to the explicit strong-coupling window $0 < β \leq 1$ (equivalently $g_{0}^{2} \geq 4$ ). For any axis-aligned rectangle contained in the lattice with corners $(a, c)$ and $(b, d)$ , where $0 \leq a < b \leq M$ and $0 \leq c < d \leq N$ , let $C_{a, b; c, d}$ be its boundary traversed counterclockwise from $(a, c)$ . Its area in plaquette units is $A = (b - a) (d - c)$ . Define $Z (S U (2)) : = {z \in S U (2) : z U = U z for every U \in S U (2)}$ and $χ : Z (S U (2)) \to {\pm 1}$ by $z = χ (z) I_{2}$ . Then $Z (S U (2)) = {I_{2}, - I_{2}}$ and the defining representation has the nontrivial central character $χ (I_{2}) = 1$ , $χ (- I_{2}) = - 1$ . Put

λ (β) : = \frac{\int_{S U (2)} \frac{1}{2} {Tr}_{2} (U) e^{β {Tr}_{2} (U) / 2} d μ_{2} (U)}{\int_{S U (2)} e^{β {Tr}_{2} (U) / 2} d μ_{2} (U)},

where $τ (U) = \frac{1}{2} {Tr}_{2} (U)$ . Then $0 < λ (β) < 1$ and the exact finite-lattice expectation is

⟨ W (C_{a, b; c, d}) ⟩_{β} = λ (β)^{A} = \exp ⁣ (- σ_{l a t} (β) A), σ_{l a t} (β) : = - \log λ (β) > 0.

The parameter $β$ and the lattice string tension $σ_{l a t}$ are dimensionless. The identity holds for every finite container and every stated rectangle; no infinite-area, infinite-volume, or continuum limit is taken, and no continuum confinement or mass-gap claim follows.

Statement references

Finite link configurations, gauge action, and product measure

Normalized trace of ordered lattice holonomy

SU(2) Wilson action and one-plaquette probability

Independent plaquette variables on a finite rectangle

One-plaquette matrix average and strong-coupling coefficient

Gauge invariance of the original loop observable under tree gauge fixing

Normalized link and plaquette probabilities

Finite product sigma-algebra

Finite link product measure

Finite product measure existence and rectangle formula

Finite-volume expectation integral

Fubini's theorem for bounded matrix-entry products

Meyer, §1.4.1.1, printed pp. 22–24; the cited 4D leading-order argument is contextual, and the finite 2D identity is proved here

Lohmayer, §§4.1–4.2.2, printed pp. 59–64; the exact 2D factorization mechanism is rederived for the stated finite free-boundary lattice

Facts & Assumptions

Given: ${A C}_{ω}$ , the finite free-boundary square lattice, the defining $S U (2)$ representation, the Wilson action with $0 < β \leq 1$ , and a rectangular loop $C_{a, b; c, d}$ .

[A1]

${A C}_{ω}$ is inherited through the finite non-Abelian lattice datum; the present integration proof uses only finite products and explicit coordinate changes. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Finite non-Abelian lattice gauge data: link configurations are $G$ -valued, the vertex group acts by endpoint multiplication, and normalized product Haar measure is used on the finite link set.

[F2]

Wilson loop on a finite lattice: $W (C) = \frac{1}{2} {Tr}_{2} (U_{C})$ for the defining representation, with edge matrices multiplied in traversal order.

[F3]

Finite SU(2) Wilson plaquette measure: $S_{β} = β \sum_{p} (1 - τ (U_{p}))$ , $τ (U) = \frac{1}{2} {Tr}_{2} U$ , and the one-plaquette law is $d ν_{β} = z_{β}^{- 1} e^{β τ} d μ_{2}$ .

[F4]

Independent plaquette variables on a finite planar rectangle: in a chosen tree gauge, the rectangle's plaquette holonomies are independent with law $ν_{β}$ when all face weights are the same Wilson weight.

[F5]

SU(2) one-plaquette matrix average at strong coupling: $E_{ν_{β}} [U] = λ (β) I_{2}$ and $0 < λ (β) < 1$ .

[F6]

Wilson loop is invariant under vertex gauge transformations: the normalized trace of the original closed-loop holonomy is unchanged by every vertex gauge transformation, including the configuration-dependent transformation to the tree-gauge representative.

[F7]

The finite link and plaquette measures are product probabilities on their finite product sigma-algebras; the product formula supplies their rectangle values. Probability measures and probability spaces The product sigma-algebra and its finite iterates The product measure of two sigma-finite measure spaces For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

[F8]

An integrand bounded by $1$ on the finite product probability space is in $L^{1}$ , and Fubini applies to its product-coordinate sections. Measure space and integral Integrable real and complex functions, and their integrals Fubini's theorem for $L^{1}$ functions on a sigma-finite product

Proof

1.1givenA1F1F3F4algebra

The Wilson action weight is $e^{- β N_{p}} \prod_{p} e^{β τ (U_{p})}$ , where $N_{p} = M N$ ; the constant $e^{- β N_{p}}$ cancels in normalized expectations. By the coordinates in [F3], $τ (U) = x_{0} \in R$ ; since $U^{- 1} = U^{*}$ , $τ (U^{- 1}) = \overline{τ (U)} = τ (U)$ , and cyclicity of trace makes $τ$ conjugation-invariant. Therefore $w_{p} (U) = e^{β τ (U)}$ is positive, continuous, conjugation-invariant and inversion-invariant, so it meets [F4]'s hypotheses. Thus [F4] gives the plaquette law $ν_{β}$ after the tree-gauge change of variables. The inherited axiom base is ${A C}_{ω}$ [A1].

2.1givenF1F2F4F6step 1.1algebra

Let $U^{t r e e}$ be the unique tree-gauge representative of the original link configuration supplied by [F4]. By [F6], $W (C; U^{t r e e}) = W (C; U)$ pointwise, so evaluating the original expectation after this configuration-dependent gauge fixing preserves the loop observable; [F4] supplies the corresponding law of its plaquette variables. Choose the vertical tree column equal to the left side $x = a$ of $C_{a, b; c, d}$ . All horizontal links and the left-side vertical links are then the identity. For each row $j$ , $P_{i, j} = V_{i + 1, j} V_{i, j}^{- 1}$ and $V_{a, j} = I_{2}$ , so the recursion $V_{i + 1, j} = P_{i, j} V_{i, j}$ yields $V_{b, j} = P_{b - 1, j} \dots P_{a, j}$ . The counterclockwise boundary from $(a, c)$ therefore has ordered holonomy $U_{C} = (P_{b - 1, c} \dots P_{a, c}) (P_{b - 1, c + 1} \dots P_{a, c + 1}) \dots (P_{b - 1, d - 1} \dots P_{a, d - 1})$ , a product of exactly $A = (b - a) (d - c)$ plaquette matrices.

3.1F4F5F8step 2.1algebraF7

By [F4] the plaquette variables in this ordered product have the product probability law $ν_{β}^{\otimes A}$ . Each defining-representation matrix entry has modulus at most $1$ , so every entry of the finite matrix-product integrand is bounded and belongs to $L^{1}$ by [F8]. Fubini applied entry by entry therefore replaces each independent factor by its matrix expectation without changing its order. By [F5] each expectation is $λ I_{2}$ , so the expected loop holonomy is $(λ I_{2})^{A} = λ^{A} I_{2}$ .

4.1F2F5step 3.1algebra

Taking half the trace gives $⟨ W (C) ⟩_{β} = λ^{A}$ . Since $0 < λ < 1$ by [F5], $σ_{l a t} = - \log λ$ is positive and $λ^{A} = e^{- σ_{l a t} A}$ for every positive integer area $A$ . The conclusion is the stated finite-lattice model deduction only.

5.1givenalgebra∎

To identify the center, write $D = diag (i, - i)$ and $J = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ ; both belong to $S U (2)$ . If $A = (\begin{array}{cc} a & b \\ c & d \end{array}) \in Z (S U (2))$ , then $A D = D A$ forces $b = c = 0$ . The conditions $A \in S U (2)$ then give $A = diag (z, z^{- 1})$ with $∣ z ∣ = 1$ , and $A J = J A$ forces $z = z^{- 1}$ , hence $z = \pm 1$ . Conversely, both scalar matrices $\pm I_{2}$ commute with every element of $S U (2)$ . Since the defining representation is the identity matrix representation, its character on this center is $χ (\pm I_{2}) = \pm 1$ , as stated.

Exact finite-lattice SU(2) Wilson area law in two dimensions

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources