Read-back: YangMills.lattice_reflection_positivity

Data and assumptions

The statement is universally quantified over all of the following.

• A natural number $d$ together with the assumption $d \neq 0$ (so the coordinate index $0 \in \{0,\dots,d-1\}$ exists and is the distinguished "time" coordinate below).
• A type $G$ carrying a group structure, a topology for which multiplication and inversion are continuous, and which is compact; $G$ carries a $\sigma$-algebra which is assumed to be the Borel $\sigma$-algebra of that topology. (No connectedness, non-abelianness, Hausdorff-ness, or simplicity is assumed; $G$ may be trivial or finite.)
• A natural number $N$ (possibly $N = 0$).
• A group homomorphism $\rho : G \to \mathrm{U}(N)$, where $\mathrm{U}(N)$ is the group of $N\times N$ complex matrices $M$ with $M^*M = MM^* = I$. Each $\rho(g)$ is regarded as an $N\times N$ complex matrix when a trace is taken.
• A hypothesis $h_\rho$: the map $g \mapsto \rho(g)$ is continuous (with respect to the topology on $\mathrm{U}(N)$ inherited from entrywise convergence of matrices). This hypothesis is a premise of the theorem; it does not enter into any of the definitions that follow.
• Natural numbers $k$ and $L$ (both possibly $0$).
• A real number $\beta$ (any sign, possibly $0$).
• A natural number $n$ (possibly $0$).
• A family of complex coefficients $c_0,\dots,c_{n-1} \in \mathbb{C}$.
• A family $A_0,\dots,A_{n-1}$, each $A_i$ a finite list (ordered, repetitions allowed, possibly empty) of "loops" in dimension $d$ (defined below).
• A hypothesis $h_A$: for every index $i$ and every loop $\gamma$ occurring in the list $A_i$, $\gamma$ has "positive time" (defined below). This hypothesis is vacuous when $n = 0$ or when every $A_i$ is empty. No hypothesis of simplicity, disjointness, non-emptiness of steps, or containment in any box is imposed on the loops.

The conclusion

Writing $\overline{z}$ for complex conjugation, $\gamma^{\dagger}$ for the reflection of a loop (defined below), $A_i^{\dagger}$ for the list obtained by reflecting each loop of $A_i$ in order, $\mathbin{+\!\!+}$ for list concatenation, and $\langle\,\cdot\,\rangle_{k,L,\beta}$ for the loop correlation (defined below), set

$$S \;:=\; \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} \overline{c_i}\, c_j\, \big\langle A_i^{\dagger} \mathbin{+\!\!+} A_j \big\rangle_{k,L,\beta} \;\in\; \mathbb{C}.$$

The theorem asserts the conjunction

$$\operatorname{Im} S = 0 \quad\text{and}\quad 0 \le \operatorname{Re} S ,$$

i.e. $S$ is a real number that is greater than or equal to $0$. (When $n = 0$ the double sum is empty, $S = 0$, and both clauses hold trivially.)

Unfolding the custom notions

Lattice sites, the box, its edges and plaquettes

A site is a function $x : \{0,\dots,d-1\} \to \mathbb{Z}$, i.e. a point of $\mathbb{Z}^d$; $x_\mu$ denotes its $\mu$-th coordinate. For $\mu \in \{0,\dots,d-1\}$, $e_\mu$ is the site with $(e_\mu)_\mu = 1$ and all other coordinates $0$.

For a natural number $R$, the box is

$$B_R \;=\; \{\, x \in \mathbb{Z}^d \;:\; -R \le x_\mu \le R \text{ for every } \mu \,\}.$$

For $R = 0$, $B_0 = \{0\}$.

An edge is a pair $(x,\mu)$ with $x$ a site and $\mu$ a direction. The box edges are

$$E_R \;=\; \{\,(x,\mu) \;:\; x \in B_R \text{ and } x + e_\mu \in B_R\,\}.$$

For $R = 0$, $E_0 = \varnothing$.

A plaquette is a triple $(x,\mu,\nu)$. The box plaquettes are

$$P_R \;=\; \{\,(x,\mu,\nu) \;:\; x\in B_R,\ \mu < \nu,\ x+e_\mu \in B_R,\ x+e_\nu\in B_R,\ x+e_\mu+e_\nu \in B_R\,\}.$$

(Here $\mu<\nu$ is comparison of the indices as natural numbers. When $d = 1$ there are no plaquettes at all.)

A configuration on the box of radius $R$ is a function $U : E_R \to G$ (one group element per box edge). When $E_R$ is empty there is exactly one configuration.

Links, plaquette variables, Wilson action

For a configuration $U$ on $E_R$ and an arbitrary edge $e = (x,\mu)$ (not necessarily in the box),

$$\operatorname{link}_U(e) \;=\; \begin{cases} U(e) & e \in E_R,\\ 1_G & e \notin E_R.\end{cases}$$

Thus every edge lying outside the box carries the identity element; nothing in the theorem restricts loops to the box.

For a plaquette $p = (x,\mu,\nu)$ (again arbitrary),

$$\operatorname{plaq}_U(p) \;=\; \operatorname{link}_U(x,\mu)\cdot \operatorname{link}_U(x+e_\mu,\nu)\cdot \operatorname{link}_U(x+e_\nu,\mu)^{-1}\cdot \operatorname{link}_U(x,\nu)^{-1}\;\in G.$$

The Wilson action of $U$ (depending on $\rho$) is the real number

$$S_W(U) \;=\; \sum_{p \in P_R} \operatorname{Re}\, \operatorname{tr}\big(\rho(\operatorname{plaq}_U(p))\big),$$

the trace being that of an $N\times N$ complex matrix (so $S_W \equiv 0$ if $N = 0$ or if $P_R = \varnothing$).

The measure on configurations

$\mu_G$ denotes the left-invariant Haar measure on $G$ built from the positive compact set $G$ itself and normalized so that $\mu_G(G) = 1$; it is a probability measure on the Borel sets of $G$.

The configuration measure $\mu_R$ on the space of configurations $E_R \to G$ is the product measure $\bigotimes_{e\in E_R}\mu_G$ (product of $|E_R|$ copies of $\mu_G$, with the product $\sigma$-algebra). It has total mass $1$; when $E_R = \varnothing$ it is the unit mass on the single configuration.

Gibbs weight and expectation

For $\beta \in \mathbb{R}$ and a configuration $U$, the Gibbs weight is

$$w_\beta(U) \;=\; \exp\big(\beta\, S_W(U)\big) \;>\; 0,$$

regarded as a complex number where needed.

For a function $F$ from configurations on $E_R$ to $\mathbb{C}$, the expectation is the complex number

$$\mathbb{E}_{R,\beta}[F] \;=\; \frac{\displaystyle\int F(U)\, w_\beta(U)\, d\mu_R(U)}{\displaystyle\int w_\beta(U)\, d\mu_R(U)} .$$

Both integrals are Bochner integrals of $\mathbb{C}$-valued functions, with the convention that an integrand which is not integrable (or not almost-everywhere strongly measurable) has integral $0$; and division in $\mathbb{C}$ follows the convention $z/0 = 0$. No integrability or measurability is hypothesised in the theorem for the specific integrands that arise.

Steps, paths, and holonomy

A step is a pair $s = (\mu, \varepsilon)$ with $\mu$ a direction and $\varepsilon \in \{\mathsf{true},\mathsf{false}\}$; its displacement is

$$v(s) \;=\; \begin{cases} e_\mu & \varepsilon = \mathsf{true},\\ -e_\mu & \varepsilon = \mathsf{false}.\end{cases}$$

Given a starting site $x$ and a list of steps $[s_1,\dots,s_m]$, the path vertices are the $m$ sites

$$\big[\,x,\; x+v(s_1),\; x+v(s_1)+v(s_2),\;\dots,\; x+v(s_1)+\dots+v(s_{m-1})\,\big]$$

(the empty list for $m = 0$; the endpoint $x + \sum_{r\le m} v(s_r)$ is not listed).

Given a configuration $U$, the path holonomy $\operatorname{hol}_U(x,[s_1,\dots,s_m]) \in G$ is defined recursively by $\operatorname{hol}_U(x,[\,]) = 1_G$ and

$$\operatorname{hol}_U\big(x, s :: \text{rest}\big) \;=\; \ell_U(x,s)\cdot \operatorname{hol}_U\big(x + v(s), \text{rest}\big), \qquad \ell_U\big(x,(\mu,\varepsilon)\big) = \begin{cases}\operatorname{link}_U(x,\mu) & \varepsilon=\mathsf{true},\\ \operatorname{link}_U(x-e_\mu,\mu)^{-1} & \varepsilon=\mathsf{false}.\end{cases}$$

Edges outside the box contribute the identity factor.

Loops

A loop $\gamma$ in dimension $d$ consists of

• a natural number $\gamma.\mathrm{scale}$,
• a base site $\gamma.\mathrm{base} \in \mathbb{Z}^d$,
• a list of steps $\gamma.\mathrm{steps} = [s_1,\dots,s_m]$ (possibly empty, repetitions allowed, back-tracking allowed),
• a proof that $\sum_{r=1}^m v(s_r) = 0$ (the steps return to the base).

The vertices of $\gamma$ are the path vertices of $(\gamma.\mathrm{base}, \gamma.\mathrm{steps})$ as defined above, computed at the loop's own scale data (no rescaling).

$\gamma$ has positive time iff for every site $x$ among the vertices of $\gamma$, $0 \le x_0$ (the time coordinate is non-negative; $x_0 = 0$ is allowed). A loop with empty step list has no vertices and has positive time vacuously.

Refinement. For $j \in \mathbb{N}$, $\gamma.\mathrm{refine}(j)$ is the loop with scale $\gamma.\mathrm{scale}+j$, base $2^j\cdot\gamma.\mathrm{base}$, and step list obtained by replacing each step $s_r$ by $2^j$ consecutive copies of itself:

$$[\underbrace{s_1,\dots,s_1}_{2^j},\ \underbrace{s_2,\dots,s_2}_{2^j},\ \dots,\ \underbrace{s_m,\dots,s_m}_{2^j}].$$

Loop at scale $k$. $\gamma.\mathrm{atScale}(k)$ is the pair (base, steps) of $\gamma.\mathrm{refine}(k \mathbin{\dot-} \gamma.\mathrm{scale})$, where $\mathbin{\dot-}$ is truncated natural-number subtraction: $k \mathbin{\dot-} \gamma.\mathrm{scale} = k - \gamma.\mathrm{scale}$ if $k \ge \gamma.\mathrm{scale}$ and $=0$ otherwise. Consequently, if $k < \gamma.\mathrm{scale}$ the loop is not coarsened: its base and steps are used unchanged (refinement by $2^0 = 1$).

Time flip and reflection. The time flip of a site is $\tau(x)_\mu = -x_\mu$ if $\mu = 0$ and $\tau(x)_\mu = x_\mu$ otherwise. The reflection of a step is

$$r(\mu,\varepsilon) \;=\; \begin{cases}(\mu,\varepsilon) & \mu = 0,\\ (\mu,\neg\varepsilon) & \mu \ne 0,\end{cases}$$

i.e. steps in the time direction are kept as they are and steps in every spatial direction have their orientation bit negated. The reflected loop $\gamma^{\dagger}$ has the same scale as $\gamma$, base $\tau(\gamma.\mathrm{base})$, and step list

$$[\,r(s_m),\ r(s_{m-1}),\ \dots,\ r(s_1)\,]$$

(reflect each step, then reverse the order). Its displacement sum is $-\tau\big(\sum_r v(s_r)\big) = 0$, so it is again a loop.

Wilson loops, loop products, loop correlation

For a configuration $U$ on $E_R$, a natural number $k$ and a loop $\gamma$, the Wilson loop is the complex number

$$W_U^{k}(\gamma) \;=\; \operatorname{tr}\Big(\rho\big(\operatorname{hol}_U\big(\gamma.\mathrm{atScale}(k)\big)\big)\Big),$$

the holonomy being taken from the base and along the steps of the loop at scale $k$ as just defined. (If $N = 0$ every Wilson loop is $0$; if all edges traversed lie outside the box the holonomy is $1_G$ and the Wilson loop is $\operatorname{tr} I_N = N$.)

For a list of loops $A = [\gamma_1,\dots,\gamma_p]$, the loop product is

$$\Pi_U^{k}(A) \;=\; \prod_{r=1}^{p} W_U^{k}(\gamma_r) \;\in\;\mathbb{C},$$

with $\Pi_U^k([\,]) = 1$.

Finally, the loop correlation appearing in the theorem is

$$\big\langle A \big\rangle_{k,L,\beta} \;=\; \mathbb{E}_{R,\beta}\big[\,U \mapsto \Pi_U^{k}(A)\,\big] \qquad\text{with}\qquad R \;=\; L\cdot 2^{k},$$

that is,

$$\big\langle A \big\rangle_{k,L,\beta} \;=\; \frac{\displaystyle\int \Pi_U^{k}(A)\,\exp\big(\beta\,S_W(U)\big)\, d\mu_{L2^k}(U)} {\displaystyle\int \exp\big(\beta\,S_W(U)\big)\, d\mu_{L2^k}(U)},$$

subject to the "non-integrable integral is $0$" and "$z/0 = 0$" conventions above.

Assembled statement

For every $d \ne 0$; every compact topological group $G$ with its Borel $\sigma$-algebra; every $N$ and every continuous homomorphism $\rho : G \to \mathrm{U}(N)$; every $k, L \in \mathbb{N}$ and $\beta \in \mathbb{R}$; every $n \in \mathbb{N}$, coefficients $c \in \mathbb{C}^n$ and lists of loops $A_0,\dots,A_{n-1}$ such that every vertex of every loop in every $A_i$ has time coordinate $\ge 0$:

$$\sum_{i,j=0}^{n-1} \overline{c_i}\,c_j\; \frac{\displaystyle\int \Pi_U^{k}\big(A_i^{\dagger}\mathbin{+\!\!+}A_j\big)\,e^{\beta S_W(U)}\, d\mu_{L2^k}(U)} {\displaystyle\int e^{\beta S_W(U)}\, d\mu_{L2^k}(U)}$$

has imaginary part exactly $0$ and real part $\ge 0$, where $\Pi_U^k(A_i^{\dagger}\mathbin{+\!\!+}A_j)$ is the product of the Wilson loops at scale $k$ of the reflected loops of $A_i$ (in order) followed by the Wilson loops at scale $k$ of the loops of $A_j$ (in order).

Degenerate cases silently included

• $n = 0$: $S = 0$ and the claim is trivially true. Any $A_i$ may be the empty list, contributing the factor $\Pi = 1$ for its part of the concatenation.
• $N = 0$: all traces are $0$, so $S_W \equiv 0$, $w_\beta \equiv 1$, and $\Pi_U^k(A) = 0$ whenever $A$ is non-empty (and $=1$ when $A$ is empty).
• $L = 0$: $R = 0$, $B_0 = \{0\}$, $E_0 = P_0 = \varnothing$; there is a single configuration, $\mu_0$ is the unit point mass, every link is $1_G$, every Wilson loop equals $N$, and $\langle A\rangle_{k,0,\beta} = N^{|A|}$.
• $d = 1$: no plaquettes exist, so $S_W \equiv 0$ and the Gibbs weight is identically $1$ for every $\beta$.
• $k < \gamma.\mathrm{scale}$ for some loop: truncated subtraction yields refinement by $2^0$, so the loop's own base and steps are used at scale $k$ unchanged.
• Loops (or parts of loops) leaving the box $B_{L2^k}$: the corresponding link variables are the identity; no hypothesis excludes this.
• Loops with empty step lists, with repeated or back-tracking steps, or with intersecting/coincident images are all permitted; positive time only constrains the vertices listed above (at the loop's own scale, not at scale $k$), and permits vertices with $x_0 = 0$.
• $\beta$ may be zero or negative.
• If either integral in the expectation fails to be Bochner integrable it is $0$ by convention; if the denominator is $0$ the quotient is $0$ by convention. The theorem states its conclusion for the values produced by these conventions.
• The continuity hypothesis $h_\rho$ and the compactness/Borel assumptions on $G$ are premises of the theorem; none of the unfolded definitions refer to them.