Conventions and units

These conventions apply to the teaching course unless a page explicitly introduces a source’s notation. A different convention in a paper is not a disagreement: translate the action, observables and units before comparing results.

Prerequisites: elementary gauge theory. Outcome: track the sign, normalization, representation and physical units of each expression. The source notation crosswalk derives conversions for selected papers; groups and representations explains global-form restrictions.

Scope and gauge groups

The default teaching setting is pure Yang–Mills theory on four-dimensional spacetime, at zero temperature and $\theta=0$. There are no dynamical quarks, Higgs fields or other matter fields. External test charges may be introduced as probes. Finite lattices and finite boxes are regulators or intermediate settings, with their boundary conditions stated. The official problem statement, rather than this choice of teaching conventions, defines the required theorem.

The Clay formulation asks for a nontrivial theory for every compact simple gauge group $G$. Explicit matrix formulas in the course usually specialize to $SU(N)$ with $N\geq2$. A result for $SU(3)$, for large $N$, or for one chosen global form of a group retains that scope.

For $SU(N)$, $\operatorname{tr}$ is the ordinary trace in the fundamental representation. Generators are Hermitian and satisfy

$$\operatorname{tr}(T^aT^b)=\frac12\delta^{ab}, \qquad [T^a,T^b]=if^{abc}T^c, \qquad A_\mu=A_\mu^aT^a.$$

A normalized loop trace is written explicitly as $\operatorname{tr}/N$. For another compact simple group, choose a positive invariant quadratic form $\langle\cdot,\cdot\rangle_G$ on the Lie algebra. Our $SU(N)$ choice is $\langle X,Y\rangle_G=2\operatorname{tr}(XY)$. Changing that form changes the normalization of the coupling; fundamental-trace formulas are not automatically universal group formulas.

Curvature and the coupling

The coupling sits outside the curvature:

$$D_\mu=\partial_\mu-iA_\mu, \qquad [D_\mu,D_\nu]=-iF_{\mu\nu},$$ $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-i[A_\mu,A_\nu].$$

For a gauge transformation $\Omega(x)$,

$$A_\mu'=\Omega A_\mu\Omega^{-1} +i\Omega\partial_\mu\Omega^{-1}, \qquad F_{\mu\nu}'=\Omega F_{\mu\nu}\Omega^{-1}.$$

The Euclidean action is

$$S_E=\frac{1}{2g^2}\int d^4x\, \operatorname{tr}(F_{\mu\nu}F_{\mu\nu}) =\frac{1}{4g^2}\int d^4x\,F_{\mu\nu}^aF_{\mu\nu}^a.$$

Repeated spacetime indices sum over all ordered pairs. Equivalently,

$$S_E=\frac{1}{g^2}\int d^4x \sum_{\mu<\nu}\operatorname{tr}(F_{\mu\nu}^2).$$

The factor of two is the double counting of the antisymmetric index pair. For general $G$, replace the component expression by $(4g^2)^{-1}\int\langle F_{\mu\nu},F_{\mu\nu}\rangle_G$.

To translate to the common convention with the coupling inside the curvature, set $A_\mu=g\mathcal A_\mu$ for a fixed coupling parameter. Then $D_\mu=\partial_\mu-ig\mathcal A_\mu$, $F_{\mu\nu}=g\mathcal F_{\mu\nu}$ and

$$\mathcal F_{\mu\nu} =\partial_\mu\mathcal A_\nu-\partial_\nu\mathcal A_\mu -ig[\mathcal A_\mu,\mathcal A_\nu], \qquad S_E=\frac14\int d^4x\, \mathcal F_{\mu\nu}^a\mathcal F_{\mu\nu}^a.$$

This is a normalization dictionary, not an assertion that bare and renormalized fields can be identified without renormalization factors. Lesson 1 introduces the action; Tong’s notes supply the broader field-theory context.

Minkowski and Euclidean notation

Minkowski signature is $\eta_{\mu\nu}=\operatorname{diag}(+,-,-,-)$ with indices $\mu=0,1,2,3$. The Minkowski action is

$$S_M=-\frac{1}{2g^2}\int d^4x\, \operatorname{tr}(F_{\mu\nu}F^{\mu\nu}).$$

Euclidean indices run over $1,2,3,4$, with $x_4=\tau$ and contractions using $\delta_{\mu\nu}$. The formal continuation $t=-i\tau$ turns physical evolution $e^{-iHt}$ into $e^{-\tau H}$ and the path-integral weight into $e^{-S_E}$ with the corresponding field continuation. Writing this formal substitution is not a construction of the quantum theory. The positivity lesson explains what connects Euclidean data to physical states.

Physical units

We set $\hbar=c=1$. Energy, mass and inverse length have the same dimension. In four spacetime dimensions,

Quantity	Mass dimension
Coordinate $x$, lattice spacing $a$	$-1$
Gauge potential $A_\mu$	$1$
Curvature $F_{\mu\nu}$	$2$
Coupling $g$, action $S$	$0$
Energy, mass, Hamiltonian $H$	$1$
String tension $\sigma$	$2$

These are engineering dimensions; renormalized composite operators can have anomalous scaling dimensions.

Lattice orientation and normalization

The link assigned to $x\to x+a\hat\mu$ is an inverse transporter: as a matrix on column vectors it maps the fiber at $x+a\hat\mu$ back to the fiber at $x$. Thus

$$U_{x,\mu}'=\Omega_xU_{x,\mu}\Omega_{x+a\hat\mu}^{-1}, \qquad U_{x,\mu}\simeq e^{-iaA_\mu(x)}$$

for a smooth connection. Reversing a link replaces its matrix by its inverse. The elementary product is

$$U_{x,\mu\nu}=U_{x,\mu}U_{x+a\hat\mu,\nu} U_{x+a\hat\nu,\mu}^{-1}U_{x,\nu}^{-1}.$$

It transforms by conjugation at $x$ and has leading curvature term $-ia^2F_{\mu\nu}$. By comparison, the continuum column-vector transporter along an oriented curve is $\mathcal P\exp(i\int A)$; it transforms with the destination matrix on the left. The two descriptions agree after reversing the transporter orientation. The real traced plaquette is unchanged by this inversion.

For $SU(N)$ fundamental links, the Wilson action and measure are

$$S_W=\beta_W\sum_p \left(1-\frac1N\operatorname{Re}\operatorname{tr}U_p\right), \qquad \beta_W=\frac{2N}{g_0^2}, \qquad d\mu(U)=Z^{-1}e^{-S_W}\prod_\ell dU_\ell.$$

Each plaquette is counted once; each Haar measure is normalized to one. The course’s lattice lesson writes $\beta$ for this $\beta_W$. Its smooth-field expansion gives precisely the continuum action above, including the factor $1/2$ in front of the full trace contraction. See the derivation and Wilson’s original paper.

The strong-coupling paper in lesson 11 instead has a weight proportional to $\exp[Nb\sum_p\operatorname{Re}\operatorname{tr}U_p]$. In our notation,

$$\beta_W=N^2b, \qquad b=\frac{2}{Ng_0^2}.$$

Neither $\beta_W$ nor $b$ is the renormalization-group beta function. We write that function as $\beta_{\mathrm{RG}}(g)=\mu\,dg/d\mu$. With the generator normalization above, its leading pure-$SU(N)$ term is $-11N g^3/[3(4\pi)^2]$. Lesson 3 explains its use.

Converting lattice quantities

Quantity	Physical units	Lattice units
Spatial side length	$L=N_sa$	$N_s$
Euclidean time extent	$T=N_ta$	$N_t$
Energy or mass	$m_{\mathrm{phys}}$	$m_{\mathrm{latt}}=a m_{\mathrm{phys}}$
Correlation length	$\xi_{\mathrm{phys}}$	$\xi_{\mathrm{latt}}=\xi_{\mathrm{phys}}/a$
String tension	$\sigma_{\mathrm{phys}}$	$\sigma_{\mathrm{latt}}=a^2\sigma_{\mathrm{phys}}$
Loop area	$A_{\mathrm{phys}}$	$A_{\mathrm{latt}}=A_{\mathrm{phys}}/a^2$

For periodic Euclidean time, temperature is $1/T$; zero temperature requires $T\to\infty$. Taking $N_s\to\infty$ at fixed $a$ removes the spatial volume cutoff. Taking $a\to0$ at fixed physical $L$ requires increasing $N_s$. A finite nonzero physical mass corresponds to a vanishing lattice mass and a diverging correlation length in sites. Lessons 7 and 8 explain why these distinctions matter.

Vacuum, spectrum and correlation functions

The Hamiltonian $H$ acts on the physical, positive-inner-product Hilbert space. We subtract the vacuum energy so that $H\geq0$ and normalize $\langle\Omega,\Omega\rangle=1$. Let $P_0$ project onto $\ker H$. A positive gap means

$$H\geq\Delta(1-P_0),\qquad \Delta>0,$$

as a quadratic-form inequality, with nontrivial excited spectrum. The full spectral threshold is $\inf(\operatorname{spec}(H)\setminus\{0\})$; it need not be an eigenvalue. This is the physical Hamiltonian, not a gauge-fixed propagator or a simulation’s auxiliary Markov generator.

For the course’s simple connected-correlator formulas, work in a chosen vacuum sector with $\ker H=\mathbb C\Omega$. For an appropriately renormalized and smeared operator, set

$$\widetilde{\mathcal O}=\mathcal O-\langle\Omega,\mathcal O\Omega\rangle, \qquad \psi=\widetilde{\mathcal O}\Omega, \qquad C_{\mathcal O}(t)=\langle\psi,e^{-tH}\psi\rangle =\int_0^\infty e^{-tE}\,d\rho_\psi(E).$$

Smearing and domain assumptions ensure that $\psi$ has finite norm. If the zero-energy space is larger, use $(1-P_0)\mathcal O\Omega$ when removing all zero-energy contributions. The positive measure $\rho_\psi$ contains only spectral components with which this vector overlaps.

Its lower support edge $m_{\mathcal O}$ governs the asymptotic temporal decay rate. If $m_{\mathcal O}>0$, define the corresponding exponential correlation length as $\xi_{\mathcal O}=1/m_{\mathcal O}$. A continuum threshold can add power-law factors, and $m_{\mathcal O}$ can exceed the full gap. The spectral lesson and the overlap lesson make these qualifications explicit.

Reflection positivity is the separate Euclidean condition $\langle(\Theta F)F\rangle\geq0$ for positive-time functionals, with the required conjugation and reflection of field components. Together with the other reconstruction hypotheses, it supports the physical Hilbert-space interpretation. It supplies neither a gap nor the remaining axioms by itself. Read Osterwalder and Schrader’s 1973 paper together with their 1975 extension and correction, and use lesson 10 for the first worked example.

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