2026-10-01 · chapter Yang–Mills and the mass gapAppendixTable of notation

The whole book　Contents and reading orders

Table of notation — one convention for each letter, two meanings separated by name

What this chapter shows — The symbols of this book can be looked up by "meaning, chapter of definition, note". The three ways of writing the gauge field, the six ways of writing the coupling, and the normalisation of $C_P$ are set out with conversion formulas, tables and moving figures. Different meanings of the same letter ( $\beta$ , $\Delta$ , $\eta$ , $\sigma$ , $\Omega$ and others) are told apart in one table.

Prerequisite chapters — None. As a dictionary, it can be opened from any chapter. The "Chapter" column of each row is the chapter that defined the symbol.

Said first — This chapter is a list of conventions and contains no claims of content. The correctness of formulas is shown in each chapter; here only the arithmetic of conversions and the normalisation of generators were checked on this machine. The names of assumptions are kept separately in A-06 Index of assumptions. This chapter shows nothing about the mass gap.

The order of this chapter

How to use this table — the common conventions, and three choices of convention
Spacetime and indices — the signature $(-,+,+,+)$ and imaginary time (Table 1)
Groups and Lie algebras — normalisation of generators and structure constants (Tables 2 and 3)
Three ways of writing the gauge field — where $g$ and $i$ go (Table 4). Detour: why physics likes to attach $i$
Lattice symbols — links, plaquettes, link angles (Table 5)
Six ways of writing the coupling — conversions and one-loop coefficients (Table 6, Figure 1, Table 7)
Scales, lengths, masses — lattice units and physical units, the symbol $\asymp$ (Table 8)
Probability, spectrum, relaxation — three $C_P$ and two $\eta$ (Figure 2, Tables 9 and 10)
Different meanings of the same letter — a list for telling them apart (Table 11). Detour: can symbol collisions be eliminated?

How to use this table — one convention for each letter

This chapter is a dictionary to look things up in; it need not be read from the front. The tables have four columns, "symbol, meaning, chapter, note", and "chapter" is the chapter that defined the symbol. Check the correctness of formulas in that chapter.

There are four conventions common to all chapters. From Part 3 on, $\hbar=c=1$ . An index appearing twice in the same term is summed. Both $\ln$ and $\log$ are the natural logarithm. When the same letter has two meanings, this is declared at the start of the chapter (the list is in §09). Beyond that, the appearance of the symbols changes with the following three choices.

\underbrace{\text{way of writing the gauge field}}_{\S04\ \text{three ways}}\ \times\ \underbrace{\text{way of writing the coupling}}_{\S06\ \text{six ways}}\ \times\ \underbrace{C_P\ \text{normalisation}}_{\S08\ \text{two ways}} \tag{1}

The choices in (1) do not change the physical content, only coefficients and signs. But the same condition looks like a different number. Example. The strong-coupling condition of Shen–Zhu–Zhu, $|\beta_{\mathrm{SZZ}}|\lt1/(16(d-1))$ , becomes $\beta_{\mathrm{SZZ}}\lt1/48$ in four dimensions ( $d=4$ ), and in this book's Wilson $\beta$ it is $\beta\lt0.0833$ for SU(2) and $\beta\lt0.1875$ for SU(3) (§06) Computed. The names of assumptions (H, H↑, (S1)–(S8) and others) are in A-06 Index of assumptions, and the legend of the labels is in 0-01.

Spacetime and indices — the signature is $(-,+,+,+)$ , and in imaginary time everything becomes $+$

Physical spacetime is Minkowski spacetime, where only the time direction carries a negative sign in the squared distance (2-04, and the existing article One sign is different). Constructive field theory and lattice arguments work in Euclidean space $\mathbb R^4$ , with time rotated to imaginary values (4-02).

\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1),\qquad t=-i\tau,\qquad A_0=iA_4,\qquad F_{\mu\nu}F^{\mu\nu}\ \longmapsto\ \sum_{\mu,\nu=1}^4F_{\mu\nu}F_{\mu\nu} \tag{2}

(2) gives, from the left, the metric, the rotation (Wick rotation), and the result of rotating. In Euclidean signature the metric is the identity matrix, so upper and lower indices are not distinguished. Example. The light path $-t^2+|\mathbf x|^2=0$ becomes $\tau^2+|\mathbf x|^2$ when rotated, positive except at the origin.

Table 1 Spacetime and indices
Symbol	Meaning	Chapter	Note
$x^\mu,\ \mu=0,\dots,3$	Minkowski coordinates ( $x^0=t$ )	2-04	Some textbooks use the opposite signature $(+,-,-,-)$
$x_4=\tau,\ \mu=1,\dots,4$	Imaginary time and Euclidean indices	4-02	$\tau=it$
$i,j,k=1,2,3$	Spatial indices	3-02	Do not mix with the colour indices $a,b,c$
$E_i,\ B_i$	Colour electric and magnetic fields	3-02	$B_i=\tfrac12\varepsilon_{ijk}F_{jk}$ ; in Euclidean signature $E_{E,j}=F_{4j}$
$d$	Dimension of (Euclideanised) spacetime	0-03	The problem statement has $d=4$
$\mathbb Z^d,\ a\mathbb Z^d$	The integer lattice, the lattice with spacing $a$	5-01	§05

Groups and Lie algebras — generators are normalised by $\operatorname{tr}(T^aT^b)=\tfrac12\delta^{ab}$

The gauge group $G$ is a compact simple Lie group, and in almost all examples it is $\mathrm{SU}(N)$ (1-04). For generators we take Hermitian $T^a$ in the physics style. Unless the normalisation is fixed, both the structure constants and the coupling constant are off by constant factors.

[T^a,T^b]=i\,f^{abc}\,T^c,\qquad \operatorname{tr}(T^aT^b)=\tfrac12\,\delta^{ab},\qquad a,b,c=1,\dots,N^2-1 \tag{3}

In the mathematics style one takes anti-Hermitian $X^a=-iT^a$ , giving the form $[X^a,X^b]=f^{abc}X^c$ without $i$ . It is the same $f^{abc}$ . Example. For SU(2), $T^a=\sigma^a/2$ and $f^{abc}=\varepsilon^{abc}$ ; for SU(3), $T^a=\lambda^a/2$ (half the Gell-Mann matrices). The adjoint Casimir $C_A$ ( $f^{acd}f^{bcd}=C_A\delta^{ab}$ ) is $N$ , the fundamental Casimir $C_F$ ( $\sum_aT^aT^a=C_F\mathbf 1$ ) is $(N^2-1)/2N$ , and the index of the representation $T(R)$ ( $\operatorname{tr}_RT^aT^b=T(R)\delta^{ab}$ ) is $\tfrac12$ for the fundamental representation. What enters the one-loop coefficient (4-06) is $C_A$ and $T(R)$ .

Table 2 Numerical check of the normalisation of generators Computed
Group	$\max\|\operatorname{tr}T^aT^b-\tfrac12\delta^{ab}\|$	$C_F$	$C_A$	Independent non-zero $f^{abc}$ ( $a\lt b\lt c$ )
SU(2)	0	0.75	2	$f^{123}=1$
SU(3)	$1.1\times10^{-16}$	1.333333	3	$f^{123}=1$ ; $f^{147}=f^{246}=f^{257}=f^{345}=\tfrac12$ ; $f^{156}=f^{367}=-\tfrac12$ ; $f^{458}=f^{678}=\tfrac{\sqrt3}2$

Table 2 was obtained by writing out the matrices, computing $f^{abc}=-2i\operatorname{tr}([T^a,T^b]T^c)$ , and checking the residual on substituting back into (3) (at most $1.1\times10^{-16}$ ) and that the Casimir is a constant multiple of the identity matrix.

Table 3 Symbols for groups and Lie algebras
Symbol	Meaning	Chapter	Note
$G,\ \mathfrak g,\ N$	Gauge group, Lie algebra, the size of $\mathrm{SU}(N)$	1-03	Large $N$ is in 7-04
$T^a,\ f^{abc}$	Hermitian generators, structure constants	1-04, 3-02	Normalisation is (3)
$\sigma^a,\ \lambda^a$	Pauli matrices, Gell-Mann matrices	1-04	$\sigma$ is also used in other meanings (§09)
$C_A,\ C_F,\ T(R)$	Casimirs and the index of a representation	4-06	Values in the text
$\chi_r,\ d_r$	Character and dimension of the representation $r$	5-02	$\chi_r(U)=\operatorname{tr}_rU$
$\Omega(x)$	Gauge transformation (a group element at each point)	3-01	The letter $g$ is kept for the coupling

Three ways of writing the gauge field — where to put $g$ , and where to attach $i$

The same gauge field is written in three ways. The physics chapters "separate Hermitian $A$ from the coupling $g$ ", the path-integral and dictionary chapters "absorb $g$ into the field and put $1/g^2$ in front of the action", and the geometry chapters use "an anti-Hermitian 1-form $\mathcal A$ ". The translation is one line.

\hat A=g\,A,\qquad \mathcal A=-i\hat A=-ig\,A,\qquad \hat F=g\,F,\qquad \mathcal F=-i\hat F=-ig\,F \tag{4}

"Absorbing" is just multiplying by $g$ , and "geometry" is just multiplying further by $-i$ . But $-i$ changes the sign of $\operatorname{tr}F^2$ . For Hermitian $X$ , $\operatorname{tr}X^2\ge0$ , while for anti-Hermitian $-iX$ it is $\le0$ , so the geometric action carries a minus sign.

Table 4 Three ways of writing the gauge field (Euclidean action)
	Physics ( $g$ placed in $D$ )	Physics ( $g$ absorbed)	Geometry (anti-Hermitian)
Covariant derivative	$\partial_\mu-igA_\mu$	$\partial_\mu-i\hat A_\mu$	$d+\mathcal A$
Field strength	$\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu]$	$\partial_\mu\hat A_\nu-\partial_\nu\hat A_\mu-i[\hat A_\mu,\hat A_\nu]$	$d\mathcal A+\mathcal A\wedge\mathcal A$
Gauge transformation	$\Omega A\Omega^{-1}-\tfrac ig(\partial\Omega)\Omega^{-1}$	$\Omega\hat A\Omega^{-1}-i(\partial\Omega)\Omega^{-1}$	$\Omega\mathcal A\Omega^{-1}-(d\Omega)\Omega^{-1}$
Action $S_E$	$\tfrac12\int\operatorname{tr}F_{\mu\nu}F_{\mu\nu}=\tfrac14\int F^a_{\mu\nu}F^a_{\mu\nu}$	$\tfrac1{2g^2}\int\operatorname{tr}\hat F_{\mu\nu}\hat F_{\mu\nu}$	$-\tfrac1{g^2}\int\operatorname{tr}\mathcal F\wedge*\mathcal F$
Chapters that use it	3-02, 3-03, Part 4 (except 4-02; 4-03 uses both)	0-01, 4-02, 8-04	1-07, 3-04, 3-05

The topological charge $Q$ is related to the second Chern number $c_2=\frac1{8\pi^2}\int\operatorname{tr}\mathcal F\wedge\mathcal F$ by $Q=-c_2$ ((2) of 3-05; the difference in sign is $(-i)^2=-1$ , coming from $\mathcal F=-igF$ ), and the action has the lower bound $S_E\ge8\pi^2|Q|/g^2$ Known. Example. For SU(2) with $g^2=1$ ( $\beta=4$ ), the action of one instanton is $8\pi^2=78.96$ , and the weight $e^{-S_E}$ is about $5\times10^{-35}$ Computed.

Detour: why physics likes to attach $i$

Detour You may skip this. In quantum mechanics observables are Hermitian operators, with real eigenvalues (2-05). Colour charge is an observable, so physics takes Hermitian generators and puts

i

in the exponent,

e^{i\theta^aT^a}

. On the geometric side, the Lie algebra is the space of tangent vectors at the identity of the group, and

\mathfrak{su}(N)

is just the anti-Hermitian matrices. One can write

e^X

, and neither

i

nor

g

appears in the curvature. Precisely, the two styles differ only by changing the basis by a factor

i

, and represent the same connection on the same group. The differences are only the sign of

\operatorname{tr}F^2

and the

i

in front of

f^{abc}

Lattice symbols — links, plaquettes, Wilson loops

Lattice gauge theory replaces spacetime by a lattice of spacing $a$ and places a group element on each edge (link) (5-01). The key symbols are the Wilson action and its coefficient $\beta$ .

S_W(U)=\beta\sum_P\Bigl(1-\frac1N\operatorname{Re}\operatorname{tr}U_P\Bigr),\qquad \beta=\frac{2N}{g^2},\qquad U_P=U_{x,\mu}U_{x+a\hat\mu,\nu}U_{x+a\hat\nu,\mu}^{-1}U_{x,\nu}^{-1} \tag{5}

(5) sums the deviations from the identity of the product $U_P$ around every small square (plaquette $P$ ). $\beta=2N/g^2$ is the value for which it matches the continuum action, coefficient included, for smooth fields as $a\to0$ (5-01 §05) Known. Unless stated otherwise, this book's $\beta$ is this Wilson $\beta$ . $\beta=4/g^2$ in the SU(2) chapters (10-01, 10-02, 8-05) is the same formula with $N=2$ . A link sees the field only in the form $U_{x,\mu}\approx e^{-igaA_\mu}$ . The parallel transport of 1-07 is its inverse, which is why the sign of the exponent of the plaquette is opposite to 1-07 (the trace does not change).

SU(2) links are also written with angles. The record kept on this machine writes $U=e^{i\vartheta\cdot\sigma}$ , while the book's (3) gives $U=e^{i\theta^aT^a}$ , so they are related by $\theta=2\vartheta$ .

U=e^{i\vartheta\cdot\sigma}=e^{i\theta^aT^a}\quad(\theta^a=2\vartheta^a),\qquad \tfrac12\operatorname{tr}U=\cos|\vartheta| \tag{6}

Example. For $\vartheta=(0.3,-0.2,0.5)$ , $\tfrac12\operatorname{tr}U$ computed with the matrix exponential and $\cos|\vartheta|$ are both $0.815941$ Computed. The difference in how the angle is taken acts as a constant factor on $C_P$ in §08.

Table 5 Lattice symbols
Symbol	Meaning	Chapter	Note
$a,\ L$	Lattice spacing, number of points on a side of the periodic lattice	5-01	$4L^4$ links, $6L^4$ plaquettes
$U_\ell=U_{x,\mu}$	Group element on the link $\ell$	5-01	$U_{x+a\hat\mu,-\mu}=U_{x,\mu}^{-1}$
$P,\ U_P$	Plaquette and the product around it	5-01	$1-\tfrac1N\operatorname{Re}\operatorname{tr}U_P\in[0,2]$
$\mu_\beta$	Wilson measure $Z^{-1}e^{-S_W}\prod dU_\ell$	5-01	$dU$ is the Haar measure
$W(C),\ W_{R\times T}$	Wilson loop $\tfrac1N\operatorname{tr}U_C$ , rectangle	5-02	$T$ is the length in time
$\sigma$	String tension: $\langle W_{R\times T}\rangle\approx e^{-\sigma RT}$	5-02, 5-08	Lattice units. In physical units $\sqrt\sigma$
$\vartheta,\ \theta$	SU(2) link angle, $\theta=2\vartheta$	(6), 10-13	Different from $\theta$ of the θ term
$b,\ k,\ \delta(b)$	Factor per step, number of steps, the shortfall of the bookkeeping	10-02	Definitions of the record kept on this machine

Six ways of writing the coupling — $\beta$ , $\beta_{\mathrm{Ch}}$ , $\beta_{\mathrm{SZZ}}$ , $\lambda$ , $g^2$ , $\alpha$

The same coupling $g^2$ is written with different letters and different normalisations in different literature. Comparing term by term with the action (5) fixes the conversions (repeated from (8a) of 5-01).

\beta=\frac{2N}{g^2},\qquad \beta_{\mathrm{Ch}}=\frac{\beta}{N},\qquad \beta_{\mathrm{SZZ}}=\frac{\beta}{N^2},\qquad \lambda=g^2N=\frac{2N^2}{\beta},\qquad \alpha=\frac{g^2}{4\pi} \tag{7}

$\beta_{\mathrm{Ch}}$ is Chatterjee's form, writing the action as $\beta_{\mathrm{Ch}}\sum_P\operatorname{Re}\operatorname{tr}(I-U_P)$ ; $\beta_{\mathrm{SZZ}}$ is 't Hooft's convention, writing the inverse coupling as $N\beta_{\mathrm{SZZ}}$ (Shen–Zhu–Zhu, 5-04); $\lambda$ is the 't Hooft coupling; $\alpha$ is what physics calls the strength of the coupling. The running coefficients also have a way of being written for each variable (Table 1 of 4-06); in this book $b_0$ is the coefficient for $g$ , and $\kappa$ that for the Wilson $\beta$ .

\mu\frac{dg}{d\mu}=-b_0g^3+\cdots,\qquad b_0=\frac{11N}{48\pi^2},\qquad \frac{d\beta}{d\ln\mu}=\kappa=4Nb_0=\frac{11N^2}{12\pi^2} \tag{8}

The right of (8) is the one-loop prediction that the same physics is kept if $\beta$ is raised by $\kappa\ln2$ each time $a$ is halved Known. $\kappa=11/(3\pi^2)$ for SU(2) is the same number as "the slope of the step formula" of the record kept on this machine, and is the known coefficient written in the lattice convention (10-14).

Table 6 Constants for each group (pure $\mathrm{SU}(N)$ ) Computed
$N$	$\dim G$	$b_0$	$2b_0$	$\kappa$	$\kappa\ln2$	$b_0^{(\alpha)}=11N/12\pi$ ( $\mu^2\,d\alpha/d\mu^2=-b_0^{(\alpha)}\alpha^2$ , Table 1 of 4-06)
2	3	0.04644	0.09288	0.3715	0.2575	0.5836
3	8	0.06966	0.13932	0.8359	0.5794	0.8754

Figure 1 Coupling converter. Moving the Wilson

\beta

with the knob moves the values of the six ways of writing (●) together on logarithmic scales. The vertical dashed line is the edge of the Shen–Zhu–Zhu condition in four dimensions (

d=4

\beta_{\mathrm{SZZ}}=1/(16(d-1))=1/48

, and the coloured band is the side where the condition holds. Switching SU(2) / SU(3) moves the position of the edge.

SU(2), β = 2.5: β_Ch = 1.25, β_SZZ = 0.625, λ = 3.2, g² = 1.6, α = 0.1273

ComputedThe numbers in this figure were computed on this machine. The figure computes (7) on the spot (the same formulas as Table 7).

Table 7 Numerical examples of the conversion, and the edge of the strong-coupling condition in four dimensions ( $d=4$ ) Computed
Case	$\beta$	$\beta_{\mathrm{Ch}}$	$\beta_{\mathrm{SZZ}}$	$\lambda$	$g^2$	$\alpha$
SU(2), $\beta=2.5$	2.5	1.25	0.625	3.2	1.6	0.12732
SU(3), $\beta=6.0$	6	2	0.66667	3	1	0.07958
SU(2), edge of the condition	0.08333	0.04167	0.02083	96	48	3.81972
SU(3), edge of the condition	0.1875	0.0625	0.02083	96	32	2.54648

In 't Hooft's convention, $\beta_{\mathrm{SZZ}}$ and $\lambda$ , the edge of the condition does not depend on $N$ . The SU(2) $\beta=2.5$ often used in numerical computation and the edge of the condition are about 30 times apart in $\alpha$ . The conditions are to be read in the normalisation of each paper; the conversion here is this book's arithmetic.

Scales, lengths, masses — separating lattice units from physical units

The symbols for length and mass come double, in lattice units (numbers with the lattice spacing as 1) and in physical units. If they are confused, "there is a gap on the lattice" and "there is a gap in the continuum" look like the same sentence (4-11).

m_{\rm lat}=\frac1\xi,\qquad \Delta=m_{\rm phys}=\frac{m_{\rm lat}}{a},\qquad \xi_{\rm phys}=a\,\xi \tag{9}

$\xi$ is the correlation length in lattice units, fixed by the rate of exponential decay of two-point correlations ((6) of 2-08). In the continuum limit, $a\to0$ and $\xi\to\infty$ are taken together so that the product $a\xi$ stays constant. Example. If $\xi=20$ and $a=0.05$ , then $\xi_{\rm phys}=1$ . Halving $a$ needs $\xi=40$ , which at one loop corresponds to raising $\beta$ by $\kappa\ln2$ . That two quantities are the same up to constant factors is written with the following symbol.

A\asymp B\quad\Longleftrightarrow\quad \exists\,0\lt c\le C:\ \ c\le A/B\le C\quad(\text{throughout the range considered}) \tag{10}

(10) says that the ratio goes neither to 0 nor to infinity; it is stronger than $\ln A/\ln B\to1$ , which compares only power exponents, and tells apart differences in logarithmic factors (Figure 2 of §08).

Table 8 Symbols for scales, lengths and masses
Symbol	Meaning	Chapter	Note
$H,\ \Omega,\ \mathcal H$	Hamiltonian, vacuum, Hilbert space	4-11	$H\Omega=0$
$\Delta,\ m$	$\Delta$ : the constant in the problem statement that bounds the energy of excitations from below. $m$ : the largest value that can be taken as $\Delta$ (the lower end of the spectrum)	0-03, 4-11	$m=\inf\sigma(H\|_{\Omega^\perp})$ . There is a gap $\Longleftrightarrow$ $m\gt0$
$\xi,\ m_{\rm lat},\ m_{\rm phys}$	Correlation length (lattice units), $1/\xi$ , $m_{\rm lat}/a$	2-08, 8-03	(9)
$\mu$	Renormalisation scale (momentum)	4-06	Same letter as indices and measures (§09)
$\Lambda$	The scale of dimensional transmutation ( $\mu\,e^{-1/(2b_0g^2)}$ at one loop)	4-07	Its value depends on the scheme. The ratio $m/\sqrt\sigma$ does not
$\xi_{\rm RG},\ a:=1/\xi_{\rm RG}$	The bookkeeping length $b^{k(\beta)}$ , and the lattice spacing fixed by it	10-01	Definition of the record kept on this machine. Agrees with $a$ of Part 5 under chain (iii)

Probability, spectrum, relaxation — three $C_P$ and two $\eta$

In the second half of the book, the Poincaré constant $C_P$ , which expresses how fast fluctuations are smoothed out, comes to the centre (1-10). It is the supremum of the ratio of the variance to the mean square of the gradient.

C_P=\sup_f\frac{\operatorname{Var}_{\mu}(f)}{\mathcal E(f)},\qquad \mathcal E(f)=\mathbb E_\mu\Bigl[\sum_\ell|\nabla_\ell f|^2\Bigr],\qquad \lambda_1=\frac1{C_P} \tag{11}

$\lambda_1$ is the gap of the Langevin dynamics that preserves $\mu$ . The same letter $C_P$ is used for three quantities. (α) $C_P$ of the vacuum measure at one time ( $\Delta=1/(2C_P^{(\alpha)})$ ; convention with kinetic term $\tfrac12p^2$ , exact for finitely many degrees of freedom; 2-05 §08, A-06 Table 2), (β) $C_P$ of the path measure (Wilson measure) ( $C_P$ of the record kept on this machine), (γ) the difference in normalisation of (β) according to whether the gradient is measured in the link angle $\vartheta$ or in the field $A=\sqrt\beta\,\vartheta$ (10-01 The line of construction and Assumption H).

\mathcal E_A=\frac{\mathcal E_\vartheta}{\beta},\qquad C_P[\mathcal E_A]=\beta\,C_P[\mathcal E_\vartheta],\qquad \text{in the free approximation}\ \ C_P^{(A)}=\xi^2,\ \ C_P^{(\vartheta)}=\frac{\xi^2}{\beta}\quad(\mathrm{SU}(2)) \tag{12}

The free approximation is the approximation that regards the action as a Gaussian free field keeping only the quadratic terms, and gives $C_P^{(A)}=1/m^2=\xi^2$ ( $m$ in lattice units). $A$ in (12) is $A$ of §04 measured in lattice units ( $\sqrt\beta\,\vartheta^a=-aA^a$ ; the same field up to sign).

\n\n

(12) shows that writing " $C_P\asymp\xi^2$ " without naming the normalisation can change whether the sentence is true. A sentence comparing power exponents (Assumption H) moves only by the logarithm of $\beta$ , and is the same sentence in both normalisations. A band statement, that the ratio stays within constant bounds, changes truth value with the normalisation. That is why this book writes band statements with the normalisation, as $\mathrm H_{\rm band}[\mathcal E_\vartheta]$ (10-01 The line of construction and Assumption H).

Figure 2 Why the normalisation is named (SU(2), free approximation). Horizontal axis

\log_{10}\xi

, vertical axis

C_P/\xi^2

(logarithmic). The solid line is the field normalisation

1

, the dashed line the angle normalisation

1/\beta

\beta

is linked by replacing

\xi

with the one-loop formula for

\xi_{\rm RG}

\xi\approx e^{3\pi^2\beta/11}

(up to a constant factor) (asymptotic scaling assumed Physics). Moving

\xi

with the knob shows the local power exponent

\ln C_P/\ln\xi

in the readout.

ξ = 10⁶: β = 5.133, C_P^(ϑ)/ξ² = 0.1948, power exponent 1.8816 (angle) / 2 (field)

ComputedThe numbers in this figure were computed on this machine. The figure computes (12) and the one-loop formula on the spot (the same formulas as Table 9).

Table 9 $C_P/\xi^2$ in the angle normalisation and the local power exponent (SU(2), free approximation, replacing $\xi$ by the one-loop formula for $\xi_{\rm RG}$ , $\beta=\tfrac{11}{3\pi^2}\ln\xi$ . Asymptotic scaling assumed Physics) Computed
$\xi$	$\beta$	$C_P^{(\vartheta)}/\xi^2$	$\ln C_P^{(\vartheta)}/\ln\xi$	$\ln C_P^{(A)}/\ln\xi$
$10^{2}$	1.7109	0.5845	1.8834	2
$10^{6}$	5.1326	0.1948	1.8816	2
$10^{12}$	10.2652	0.0974	1.9157	2
$10^{48}$	41.0609	0.0244	1.9664	2

The ratio in the angle normalisation goes to 0, so $\mathrm H_{\rm band}[\mathcal E_\vartheta]$ fails in this free approximation. The power exponent $2-\ln\beta/\ln\xi$ is still 1.97 at $\xi=10^{48}$ , but it goes to 2. This is an example showing a convention of notation, and shows nothing about $C_P$ of the Yang–Mills measure.

There are also two exponents. The dynamical exponent $z$ is the power in $C_P\asymp\xi^z$ , and $\eta$ without qualification is $\eta_{\rm dyn}=2-z$ (8-05 §02, 2-10). The static $\eta_s=2\Delta_O-(d-2)$ is the power of correlations fixed by the scaling dimension $\Delta_O$ of the operator $O$ , and the two are equal only when the relation $z=2-\eta_s$ holds. Example. For Glauber dynamics of the two-dimensional Ising model, $\eta_s=1/4$ and $\eta_{\rm dyn}=-0.1665$ (8-03) Physics.

Table 10 Symbols for probability, spectrum and relaxation
Symbol	Meaning	Chapter	Note
$\mathbb E_\mu,\ \operatorname{Var}_\mu,\ \langle\cdot\rangle$	Expectation, variance, physical expectation	1-08	$\langle\psi,\phi\rangle$ is the inner product
$\sigma(H)$	Spectrum of an operator	1-10	Different from the string tension $\sigma$
$C_P,\ \lambda_1,\ \mathcal E_\vartheta,\ \mathcal E_A$	(11)(12)	1-10, 10-01	The three $C_P$ : (α)(β)(γ)
$\tau_{\rm exp},\ \tau_{\rm int}$	Exponential and integrated autocorrelation times	2-10, 5-07	For continuous-time Langevin, $\tau_{\rm exp}=C_P$
$z,\ \eta_{\rm dyn},\ \eta_s$	Dynamical exponent, $2-z$ , static power	2-10, 8-05	Different from the metric $\eta_{\mu\nu}$

Different meanings of the same letter — a list for telling them apart by context

The conventions of mathematics and physics have assigned the same letters to different things, so collisions within one book cannot be avoided. This book keeps the strong conventions as they are and declares the meaning at the start of the chapter. The guide for telling them apart is "subscripts and arguments". Example. $\beta(ba)=\beta(a)-\kappa\ln b+\cdots$ in 8-04 is the Wilson inverse coupling taking the lattice spacing as argument, not the β function $\beta(g)$ .

Table 11 Different meanings of the same letter
Letter	Meaning (chapter)	How to tell apart
$\beta$	Wilson inverse coupling (5-01; the converted $\beta_{\mathrm{Ch}}$ and $\beta_{\mathrm{SZZ}}$ are in §06) / inverse temperature $1/k_BT$ (2-08; in 2-07 the length of the loop $\beta\hbar$ ) / the critical exponent $\beta$ (2-08 §06, $1/8$ for the two-dimensional Ising model) / the β function (4-06) / (β) of the three $C_P$ (10-01)	Without qualification, the Wilson inverse coupling. The β function carries the argument $g$ . The exponent is written alongside $\alpha,\gamma,\nu$
$\Delta$	Mass gap (4-11) / Laplacian $\Delta_{\mathrm{lat}}$ (1-08) / scaling dimension $\Delta_O$ (8-05; the scaling dimension of 7-08 is the same) / the decrease per step $\Delta(b)$ and the MCRG shift $\Delta\beta$ (8-04) / coproduct $\Delta(\Gamma)$ (4-05) / the group quantity $\Delta_G$ (Adhikari–Cao in 9-01; $\Delta_G$ in 9-02 and 9-05 is the mass gap) / difference $\Delta x$ (2-04)	The scaling dimension carries an operator subscript, the Laplacian carries lat, the coproduct takes a diagram as argument. For $\Delta_G$ , see the chapter's declaration
$\eta$	Metric $\eta_{\mu\nu}$ (2-04) / $\eta_{\rm dyn},\eta_s$ (8-05)	With two indices, the metric
$\kappa$	One-loop slope (4-06) / decay rate of the Gaussian chain $1/\xi$ (2-08)	Decay rate only within 2-08
$\lambda$	Gell-Mann matrices (1-04) / 't Hooft coupling (7-04) / eigenvalue $\lambda_1$ (1-10)	With a colour index, a matrix; with a numeral subscript, an eigenvalue
$\sigma$	Pauli matrices (1-04) / string tension (5-02) / spin $\sigma_x$ (2-08) / spectrum $\sigma(H)$ (1-10)	If the argument is an operator, the spectrum
$\Omega$	Gauge transformation $\Omega(x)$ (3-01) / vacuum (4-11)	With an argument point $x$ , a gauge transformation
$\theta$	Angle of the θ term (3-05) / link angle $2\vartheta$ (§05) / $gac$ of the plaquette (5-01)	Multiplied by $Q$ , the θ term
$\Lambda$	Lorentz transformation (2-04) / scale of dimensional transmutation (4-07)	If a matrix, the transformation
$\mu$	Spacetime index / renormalisation scale (4-06) / measure $\mu_\beta$	In subscript position, an index
$a$	Lattice spacing (5-01) / spacing fixed by bookkeeping $1/\xi_{\rm RG}$ (10-01) / starting point of a path (sections 01–06 of 2-07; from section 07, the step width) / colour index (3-02)	Superscript, colour. In 2-07 it switches by section
$\delta$	Kronecker $\delta^{ab}$ (1-04) / shortfall of the bookkeeping $\delta(b)$ (10-02)	With two indices, Kronecker; with the factor $b$ as argument, the shortfall
$P$	Plaquette $P$ (5-01) / subscript of the Poincaré constant $C_P$ (1-10) / assumption (P) (A-06)	In brackets, the assumption; as a subscript of $C$ , the Poincaré constant
$H$	Hamiltonian (4-11) / Assumption H (10-01) / Ising energy function (2-08)	The assumption is the name of a sentence
$I$	Identity matrix ( $\mathrm{tr}(I-U_P)$ in 5-01) / modified Bessel function $I_k(J)=\frac1{2\pi}\int_0^{2\pi}e^{J\cos\theta}\cos k\theta\,d\theta$ (sections 06 and 07 of 0-04)	With a subscript $k$ and an argument, the Bessel function
Ghost	The anticommuting Faddeev–Popov fields $c^a,\bar c^a$ (4-08 §03) / states of negative weight in Pauli–Villars regularisation (4-10)	If a field, Faddeev–Popov; if a state of negative inner product, Pauli–Villars
Orientation of links	Parallel transport carries from the start point to the end point (1-07) / the lattice link $U_{x,\mu}$ is its inverse, carrying from $x+a\hat\mu$ to $x$ , $U\approx e^{-igaA}$ (5-01 §02) / numbering of chains: 3-01 uses $U_j=U(j{+}1,j)$ , 3-02 uses $U_n=U(n,n{+}1)$	The value of the trace does not depend on the orientation. When comparing signs of exponents, see each chapter's declaration (the correspondence with 1-07 in 5-01 §05)
$T$	Generators $T^a$ , index $T(R)$ / length in time (5-02) / temperature (2-08) / transfer matrix $T$ (2-08, 4-11; $T=e^{-aH}$ )	Superscript, a generator. Inside a matrix power or a trace, the transfer matrix

Detour: can symbol collisions be eliminated?

Detour You may skip this. Assigning a different letter to every quantity would eliminate collisions, but we do not, for two reasons. One is distance from the literature: writing the string tension with anything other than

\sigma

would make the tables of lattice papers unreadable as they are. The other is the number of letters: the number of quantities is far larger than the number of letters in commonly used typefaces. Precisely, a collision does harm only when two meanings stand in the same formula or the same table. For example, "(β)" in A-06 Table 2 stands in the same table as the Wilson

\beta

, so it is distinguished with brackets.

What this chapter can say

Matter	What is said	What is not said
Conventions of notation (Tables 1, 3, 5, 8, 10, 11)	A list collecting in one place the conventions used by the chapters of this book. The definitions are as in the linked chapters	Properties of the quantities each symbol refers to (existence, positivity and so on). These can be said only within the range of the labels in each chapter
Normalisation of generators (3) and structure constants (Table 2)	Checked numerically for SU(2) and SU(3) Computed. Standard convention Known	General $G$ (the list in 1-04)
Three ways of writing the gauge field (Table 4)	That they translate into one another by the substitutions (4). $Q=-c_2$ ((2) of 3-05) Known	—
Conversion of couplings (7) and constants (Tables 6 and 7, Figure 1)	Conversions as this book's arithmetic Computed. $b_0$ and $\kappa$ rewrite known one-loop coefficients Known	The content of each paper's conditions (conditions are read in the normalisation of the paper)
Normalisation of $C_P$ ((12), Table 9, Figure 2)	Arithmetic under the free approximation and the one-loop formula (up to constants) Computed	The size of $C_P$ for the lattice Yang–Mills measure. Whether Assumption H holds
The mass gap	—	This chapter is only conventions of notation, and shows nothing, on the lattice or in the continuum

Sources and reproduction

Item	Kind	Source / tools
Definition of each symbol (the "Chapter" column of each table)	Chapters of this book	The linked chapters. Primary sources are as in the source tables of those chapters
Definitions of $\Delta$ and $m$ in the problem statement	Primary source (text checked)	A. Jaffe, E. Witten, “Quantum Yang–Mills Theory” (the official Clay problem description) §4. As in the source table of 4-11
The Wilson action and $\beta=2N/g^2$	Known	K. G. Wilson, Phys. Rev. D 10 (1974) 2445–2459. As in the source table of 5-01
The convention for $\beta_{\mathrm{Ch}}$	Primary source (text checked)	S. Chatterjee, “Yang–Mills for probabilists”, arXiv:1803.01950 (2018) §3. As in the source table of 5-01
The convention for $\beta_{\mathrm{SZZ}}$ and the condition $1/(16(d-1))$	(1.2)(1.3) of the text checked (the reading of the normalisation is in 5-04)	H. Shen, R. Zhu, X. Zhu, Comm. Math. Phys. 400 (2023) 805–851, arXiv:2204.12737. As in the source table of 5-04
The one-loop coefficient $b_0$ and its four ways of writing	Known	D. J. Gross, F. Wilczek (1973), H. D. Politzer (1973), D. Tong's lecture notes §2.4, the QCD chapter of the PDG. As in the source table of 4-06
The three $C_P$ , $\mathcal E_\vartheta$ , $\mathcal E_A$ , $\xi_{\rm RG}$	Definitions of the record kept on this machine	The source tables of 10-01 and 10-13
$\eta_s$ , $\eta_{\rm dyn}$ , $z$	Physics, the record	The source tables of 8-05 and 2-10
The numbers in Tables 2, 6, 7, 9, (6) and §04	Computed on this machine	Python 3, NumPy, SciPy: `generators.py` (normalisation, commutation relations, Casimirs), `constants.py` (the arithmetic of (7)(8)), `normalisation.py` ((12) and the one-loop formula). Figures 1 and 2 compute the same formulas in plain JavaScript

Next chapter to read: A-02 Glossary (look up the meaning of a term in one to three lines, and jump to the chapter). The names of assumptions are in A-06 Index of assumptions.

Revised 2026-10-01: first version.