2026-10-01 · chapter Yang–Mills and the mass gapPart 5 Lattice gauge theoryreflection positivity and the transfer matrix

Reflection positivity and the transfer matrix — if the integral reflected in a mirror is positive, a Hamiltonian stands up on the lattice

The theory on a lattice (5-01) is written as the following probability measure.

$dU$ is the Haar measure of total weight 1, and $\beta$ the coefficient of the action (for example, $\beta=4$ for $SU(2)$ with $g^2=1$). The last coordinate $t$ is called imaginary time. Here there are only integrals and expectation values, with no Hilbert space and no Hamiltonian. The mass gap is a gap in the spectrum of the Hamiltonian (4-11), so we first recover the operators. The tool is the reflection positivity of 4-10, which on the lattice can be proved from the action. §6.5 of the problem statement writes 'Reflection positivity holds for the Wilson approximation [36], a major advantage' ([36] is Osterwalder–Seiler 1978) Known.

For the lattice reflected in the mirror to coincide with the original, the $2t_0$ of the plane $t=t_0$ must be an integer. If $t_0$ is an integer, the plane passes through lattice points (site reflection); if it is a half-integer, it cuts the midpoints of the vertical (time-direction) links (link reflection).

A link variable traversed backwards is the inverse ($U_{y\to x}=U_{x\to y}^{-1}$). Let $\Lambda_+$ be the set of links whose two ends both have $t\ge t_0$ (for link reflection, $t\gt t_0$), and $\mathcal A_+$ the set of all functions determined by those variables alone. Reflection positivity is the condition of the same form as (4) of 4-10,

$\theta$ maps plaquettes to plaquettes (reversing the orientation of those containing the time direction), and $\mathrm{Re\,tr}$ is unchanged by inversion, so $S_W(\theta U)=S_W(U)$.

Example (Figure 1). At $t_0=\tfrac12$ a vertical link crossing the plane is mapped to itself with orientation reversed ($(\theta U)_b=U_b^{-1}$), belongs to neither half, and straddles the plane. The plaquettes having it as an edge join the upper and lower halves. At $t_0=0$ the horizontal links at $t=0$ are mapped to themselves, and nothing straddles.

Put the weight $\prod_te^{Ks_ts_{t+1}}$ on a chain of spins $s_t=\pm1$ (the one-dimensional Ising model of 2-09 with zero external field; $K\lt0$ is also allowed). The transfer matrix is

For site reflection ($t_0=0$), fixing $s_0$, the sum $\varphi(s_0)$ over the upper half and the sum over the lower half are complex conjugates of each other, and whatever the sign of $K$, $\langle\overline{\Theta F}F\rangle\propto\sum_{s_0}|\varphi(s_0)|^2\ge0$. For link reflection ($t_0=\tfrac12$) one straddling coupling remains, and

is a quadratic form in $T$. It is non-negative for every $\varphi$ only when $T\ge0$, that is, $K\ge0$. Example. For $F=s_1$, with link reflection $\langle s_0s_1\rangle=\tanh K$ becomes $-0.4621$ at $K=-\tfrac12$ and (3) is violated, while with site reflection $\langle s_{-1}s_1\rangle=\tanh^2K=0.2136\ge0$ Computed.

There are two kinds of "positive". One is that the entries are positive. For a matrix with all entries positive, the largest eigenvalue is not repeated and is larger than the absolute values of the other eigenvalues (the Perron–Frobenius theorem) Known. The entries of the $T$ in (4) are positive whatever the sign of $K$, so $\lambda_0\gt|\lambda_1|$, and the correlation $(\tanh K)^t$ decreases exponentially. The other is being positive as an operator. $T\ge0$ only when $K\ge0$, and if $K\gt0$ one can write $T=e^{-H}$ with a real $H$. For $K\lt0$ the correlation decreases while changing sign, and $-\log\lambda_1$ is not real. Site-reflection positivity corresponds to $T^2\ge0$, and link-reflection positivity to $T\ge0$ Known.

Claim: on a lattice with the Wilson action, (3) holds for all real $\beta$ with site reflection, and for $\beta\ge0$ with link reflection. Reflection positivity of lattice gauge theory and the existence of a positive self-adjoint transfer matrix are due to Osterwalder–Seiler 1978. Their abstract states that physical positivity was verified, and continues 'This implies the existence of a positive self-adjoint transfer matrix' Known. The conditions on $\beta$ for each kind of reflection rest on the proofs (6) to (8) below Paper. An example where link reflection actually fails at $\beta\lt0$ is in section 08 Computed. Below are outlines of the proofs (with periodic time there are two planes, but it is the same).

Site reflection — fixing the variables on the plane

No plaquette straddles. Distributing the action of the plaquettes on the plane half up and half down gives $S_W=S_++S_-$ and $S_-(U)=S_+(\theta U)$. Fixing the variables $U_0$ on the plane, the lower integral becomes the complex conjugate of the upper one by the substitution $U_\lt=\theta U_\gt$ (Haar measure is invariant under inversion), so

The sign of $\beta$ is used nowhere.

Link reflection — opening the straddling plaquettes into sums of squares

The straddling vertical links have distinct upper endpoints, so gauge transformations at those points can set them to 1 (6-02 §02). For this, a general $F$ is first replaced by $\widetilde F$, averaged over gauge transformations at the points of the layer just above the plane ($t=1$). $\Theta F$ is determined by the variables with $t\le0$ only, and does not touch the points of that layer. So changing the integration variables $U\to U^g$ by a gauge transformation $g$ of that layer (neither the action nor Haar measure changes), $\langle\overline{\Theta F}F\rangle=\langle\overline{\Theta F}F^g\rangle$, and averaging over $g$ gives $\langle\overline{\Theta F}\widetilde F\rangle$. Doing the same at the mirror layer ($t=0$), the left-hand side of (3) equals $\langle\overline{\Theta\widetilde F}\,\widetilde F\rangle$. A straddling plaquette becomes $\mathrm{Re\,tr}(U_{\theta\ell}U_\ell^\dagger)$, involving only the upper horizontal link $\ell$ and its image, and with the entries $h_{ij}=(U_\ell)_{ij}\in\mathcal A_+$ ($\Theta h_{ij}=(U_{\theta\ell})_{ij}$) it can be written

Expanding the weight of the straddling plaquettes $\exp\bigl(\tfrac{\beta}{2N}\sum_k\overline{\Theta h_k}h_k\bigr)$ ($h_k$ ranges over all $h_{ij}$ and $\overline{h_{ij}}$; constant factors omitted) as a series, it takes the form $\sum_\alpha c_\alpha\overline{\Theta H_\alpha}H_\alpha$ with products $H_\alpha$ of the $h_k$, and if $\beta\ge0$ then $c_\alpha\ge0$. The upper and lower halves no longer share variables, so

Just as $T\ge0$ was needed in (5) for the chain, $\beta\ge0$ is needed here. An example where it actually fails is in section 08.

Given reflection positivity, the reconstruction of 4-10 makes a Hilbert space and an operator that advances time by one lattice unit; on the lattice this operator (the transfer matrix) can be written down directly. Make time periodic (length $L_t$) and set as many vertical links as possible to 1 by gauge transformations (temporal gauge); then one link $V_x$ remains on the time loop at each spatial point $x$. Integrating over the remaining $V_x$ is the same as applying the gauge transformation $g_x=V_x$ to all links at one time and averaging, and that average is the projection $P_0$ onto gauge-invariant states. The state space is the space $\mathcal H=L^2(G^{E_s},dU)$ of functions of the spatial links $E_s$ at one time, and $T$ is the integral operator with kernel

($S_s$ is the action of the spatial plaquettes at one time, and the product in the middle comes from the vertical plaquettes). Then

($O$ is a gauge-invariant real observable, $\lambda_n$ and $|n\rangle$ the eigenvalues and eigenvectors of $P_0T$, $\Omega=|0\rangle$; the same calculation as (8) of 2-09).

(a) The kernel is real symmetric, so $T$ is self-adjoint (the content of site reflection). (b) $T=VKV$ ($V$ the operator multiplying by $e^{-S_s/2}$, $K$ the product in the middle), and if $\beta\ge0$ then $K\ge0$ (section 06), so $T\ge0$ (the content of link reflection). (c) For $\beta\gt0$, $V$ multiplies by a positive function and so has an inverse, and the $\lambda_r$ of section 06 are all positive, so $K$ does not have 0 as an eigenvalue either, and $T$ does not have 0 as an eigenvalue Paper. Lüscher 1977 calls this property 'strictly positive' in its title. (d) On a finite box the kernel of $P_0T$ is continuous and positive and the space of gauge orbits is compact, so by a generalisation of Jentzsch's theorem on integral operators with continuous positive kernels on compact spaces (of Krein–Rutman type; the integral-operator version of the Perron–Frobenius theorem), $\lambda_0$ is not repeated Known (the generalisation is Schaefer 1974; see the table of sources).

From here on we take $\beta\gt0$. By (b)(c) all eigenvalues are positive, so since $\lambda_0$ is not repeated, $\lambda_1\lt\lambda_0$. Thus on $P_0\mathcal H$

stands up ($a$ is the lattice spacing). Example. The gap in physical units is $\Delta=m_L/a$. That $m_L\gt0$ for each box ($\beta\gt0$) is automatic from (d), and what is asked is whether $\Delta$ remains positive as $L\to\infty$ and $a\to0$ (4-11 §08, 5-09).

$K$ is a product of operators $k$, one per link. For a single link $(kf)(U)=\int_Ge^{\frac\beta N\mathrm{Re\,tr}\,UU'^{-1}-\beta}f(U')\,dU'$, and the kernel is a function unchanged by conjugation $W\to hWh^{-1}$ of $W=UU'^{-1}$ (a class function). Diagonalising it needs two words from representation theory.

• We write the character of a representation $r$ (of dimension $d_r$) as $\chi_r(W)=\mathrm{tr}\,r(W)$. It is a class function, and between irreducible representations $\int_G\chi_r(W)\overline{\chi_s(W)}\,dW=\delta_{rs}$ (orthogonality) Known.
• Convolution with a class function acts as a scalar multiple on the space spanned by the matrix elements $r(U)_{ij}$ of each irreducible representation (from Schur's lemma in 1-04). $L^2(G)$ is spanned by these matrix elements (the Peter–Weyl theorem), so $k$ is determined by its eigenvalue on each irreducible representation Known.

For details, (3.1) in 6-02 §03. The eigenvalue on the representation $r$ is

Expanding the exponential in powers of $\mathrm{tr}\,W$ and $\overline{\mathrm{tr}\,W}$, each power is a character of a tensor-product representation, and the multiplicities of irreducible representations are non-negative, so if $\beta\ge0$ then $\lambda_r\ge0$, that is, $K\ge0$. If $\beta\gt0$ all the coefficients of the expansion are positive, and every irreducible representation appears in some tensor product of the fundamental representation and its complex conjugate Known, so $\lambda_r\gt0$. For $SU(2)$, with the eigenvalues of $W$ written $e^{\pm i\theta}$ ($0\le\theta\le\pi$), $\mathrm{tr}\,W=2\cos\theta$ and $\chi_j=\sin((2j+1)\theta)/\sin\theta$. The Haar measure for integrating class functions is $\tfrac2\pi\sin^2\theta\,d\theta$, which is the density of the uniform distribution on $S^3$ of 5-01 seen through the angle $\theta$ from the north pole (the simplest case of Weyl's integration formula). From these Known

($I_n(\beta)=\frac1\pi\int_0^\pi e^{\beta\cos\theta}\cos n\theta\,d\theta$ is the modified Bessel function; agreeing with numerical integration to relative error below $10^{-10}$ Computed; the ratio $\lambda_j/\lambda_0$ is the $u_j$ of (4) in 5-02). For $\beta\lt0$ it is negative for half-integer $j$. From the asymptotic form at large $\beta$, $I_\nu(\beta)\approx\frac{e^\beta}{\sqrt{2\pi\beta}}\bigl(1-\frac{4\nu^2-1}{8\beta}\bigr)$ (DLMF 10.40.1), the energy per step is

$j(j+1)$ is the value of the Casimir $E^aE^a$ (the $J^2$ of 1-04). Speaking in metaphor, a single link is a top spinning on $SU(2)$ (a three-dimensional sphere), and $E^a$ is its angular momentum. Precisely, at large $\beta$, $k$ approaches the heat kernel $e^{-\frac{g^2}2E^aE^a}$ on the group (6-02 §04).

Example Computed. $E_{1/2}$ is 0.0967 at $\beta=16$ ($g^2=\tfrac14$) ((14) gives 0.0938), and 0.0236 at $\beta=64$ (0.0234). The deviation of the ratio $E_{1/2}/(14)$ (the relative error) shrinks from 1.032 to 1.008, roughly as $1/\beta$ (the difference itself shrinks from 0.00295 to 0.00018, as $1/\beta^2$). At strong coupling, $\beta=1$, $E_1=3.24$ departs from the approximation 4. For the ratio $E_1/E_{1/2}\to8/3$, see 6-02 §07.

The $H_a$ of (11) is the logarithm of an integral operator, and its form is not visible. A local form becomes visible if only the spacing $a_t$ in the time direction is made finer than the spatial spacing $a$. In four dimensions, we put separate coefficients on the plaquettes containing the time direction and those that are purely spatial,

From the number of plaquettes per volume $1/(a^3a_t)$ and (7) of 5-01, the classical continuum limit is the action (8) of 5-01 in either direction. As $a_t\to0$, $\beta_t\to\infty$, so for a step in the time direction the general form of (14), $-\log(\lambda_r/\lambda_0)\approx\frac N{\beta_t}C_2(r)=a_t\frac{g^2}{2a}C_2(r)$, can be used ($C_2$ is the value of the Casimir, normalised by $\mathrm{tr}\,T^aT^b=\tfrac12\delta^{ab}$). This is because near $W=e^{iX^aT^a}$, $\frac\beta N\mathrm{Re\,tr}\,W\approx\beta-\frac\beta{4N}X^aX^a$, and $k$ is close to the heat kernel at time $N/\beta$ (for the fundamental representation of $SU(3)$ at $\beta=640$, the ratio is 1.003 Computed). The spatial part $V^2=e^{-S_s}$ has $\beta_s\propto a_t$, so it has the form $e^{-a_t(\cdots)}$ from the start. Together $T=e^{-a_tH_{\mathrm{KS}}+O(a_t^2)}$, with

The first term is the energy of the electric field (the spinning of the top on each link), the second that of the magnetic field, and the states are restricted to those satisfying the Gauss law ($P_0\mathcal H$). This form was given by Kogut–Susskind 1975, and the abstract of Creutz 1977 writes 'Taking the timelike lattice spacing to zero, we extract the Kogut-Susskind Hamiltonian from the transfer matrix in the gauge A0 = 0' Known. The coefficients of (15)(16) are calculations in the conventions of this chapter, and the limit is an approximation eigenvalue by eigenvalue. From canonical quantisation, see 4-01.

Example. When $g^2$ is large the first term dominates, and the first excitation is a state in which an electric field in the fundamental representation goes once round the boundary of a single plaquette, with energy $\frac{g^2}{2a}\cdot4\cdot C_2(\text{fund})$. For $SU(N)$, $C_2(\text{fund})=\frac{N^2-1}{2N}$, and for $SU(2)$ ($j=\tfrac12$, $C_2=\tfrac34$) it is $\frac{3g^2}{2a}$. The energy of a line of electric field is proportional to its length (5-02).

Take the group to be $\mathbb Z_2=\{\pm1\}$ (the detour of 2-09, 5-10), spacetime three-dimensional (space a periodic $L\times L$ lattice), and the action $\beta\sum_P(1-U_P)$. The factor for a single link is $e^{-\beta}$ times the matrix (4) with $K$ replaced by $\beta$, and the ratio of its eigenvalues is $\tanh\beta$. For $\beta\lt0$ it is negative.

In the eigenvectors of the factor of each link (the electric-field basis), it is decided for each link whether a line of electric field passes through it. The Gauss law says that an even number of lines meet at each point, gauge-invariant states are labelled by collections $e$ of closed lines, and $K$ is a diagonal matrix determined only by the total length $|e|$ of the lines.

Closed lines fall into four classes by the parities of their winding numbers in the $x$ and $y$ directions, and $V$ (which adds or removes the line round a plaquette's boundary) preserves the class. Adding a boundary does not change the parity of the length, so lines in class $(1,0)$ have length of the same parity as $L$. Hence for $\beta\lt0$ with $L$ odd, $K\le0$ in classes $(1,0)$ and $(0,1)$, and $T=VKV\le0$. If $L$ is even, the ratios $\lambda/\lambda_0$ are the same for $\beta$ and $-\beta$. Flipping the signs of the $x$-direction links on every other row gives $U_p\to-U_p$, and $V(-\beta)$ becomes similar to a constant multiple of $V(\beta)$. If $L$ is even, all closed lines have even length, so the $K$ of (17) also changes only by a constant factor Paper (agreeing for all eigenvalues at $L=2$ and the top ones at $L=4$ Computed).

Each class has dimension $2^{L^2-1}$; $L\le3$ is diagonalised directly and $L=4$ with the Lanczos method (a method that finds only the extreme eigenvalues of a large symmetric matrix by iteration) (at $L=3$ the two agree to the 8 to 11 digits printed Computed). Let $F=W(1)$ be the Wilson line going once round in the $x$ direction at time 1. The left-hand side of (3) for link reflection ($t_0=\tfrac12$) is $\langle W(0)W(1)\rangle=\sum_n|\langle0|W|n\rangle|^2\lambda_n/\lambda_0$ ($n$ in class $(1,0)$), and at $L=3$, $\beta=-0.5$ it is $-0.1306$, negative. For site reflection ($t_0=1$, $F=W(2)$), $\langle W(0)W(2)\rangle=+0.0183$. Monte Carlo without the transfer matrix ($3\times3\times18$) also gives $-0.1305(18)$ and $+0.0189(11)$, agreeing within errors Computed.

ComputedThe numbers in this figure were computed on this machine.

ComputedTable 1 The gap $m=\log(\lambda_0/\lambda_1)$ of the zero class (states with no winding lines), and the lowest energy $E_w=\log(\lambda_0/\lambda_0^{(1,0)})$ of class $(1,0)$ (lattice units).

How to read it. (i) For $\beta\gt0$ all eigenvalues are positive and $m\gt0$ ((d) of section 05). (ii) At small $\beta$, $m\approx-4\log\tanh\beta$ (a line of length 4; the $m_0$ of 5-02), and $E_w\approx-L\log\tanh\beta$ (proportional to the length of the winding line: the confining form). (iii) At large $\beta$, $E_w$ becomes rapidly smaller with $L$. (iv) Near $\beta=0.75$, $m$ moves a lot with $L$. Three-dimensional $\mathbb Z_2$ gauge theory is dual to the three-dimensional Ising model and is said to have a phase transition (Wegner 1971) Known, and the $m$ of finite boxes alone does not determine the infinite-volume gap. (v) For $\beta\lt0$ with $L=3$, all 512 eigenvalues of classes $(1,0)$ and $(0,1)$ are at most 0 (the smallest is $-0.788\lambda_0$ at $\beta=-1$), and $\langle W(0)W(t)\rangle$ changes sign with the parity of $t$ ($-0.0027$ at $\beta=-0.5$, $t=3$) Computed.

Kept in the limit. (3) is an inequality, so if expectation values converge along a sequence of lattices and the mirror plane lies on a lattice plane at every stage, the limiting measure is also reflection positive (6-01 §07) Known. In the four-dimensional non-abelian case, however, the existence of that continuum limit has not been shown (6-10).

Can be lost when the action is changed. For improvements adding dimension-6 terms (the Symanzik improvement of 5-05), the abstract of Lüscher–Weisz 1984 writes 'physical positivity is lost in general', and states that the transfer matrix can have complex eigenvalues (correlations with damped oscillation) and that negative weights necessarily appear in the spectral decomposition of the two-point function Known. The $\beta\lt0$ of section 08 is the simplest example of a transfer matrix that is no longer positive (the eigenvalues are real and negative). For continuum regularisations, see 4-10 §08.

What is not said. The tools of this chapter do not answer whether $m_L\gt0$ survives as $L\to\infty$ (the strong-coupling side is 5-04) and $a\to0$. The recovery of rotational symmetry is also a task on the side of the limit.

What this chapter can say

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