Glossary

Use this page while reading a lesson or paper. Each entry links to a fuller explanation; the conventions fix signs, units and normalizations. An unfamiliar mathematical word should be a short detour, not a barrier to the physics.

Prerequisites: the QFT concepts in the entry diagnostic. Outcome: identify the precise meaning of a term in its model, then follow the linked calculation or source. For paper-specific symbols use the notation crosswalk.

Area law

Decay of a Wilson-loop expectation at least exponentially with the area enclosed by a large loop. A typical bound is $|\langle W(C)\rangle|\leq C_0e^{-\sigma A(C)}$, with the meaning of area, allowed loops and constants specified. Under additional assumptions, rectangular loops relate this behavior to the energy of external static sources. An area law is not itself a construction of continuum Yang–Mills theory or a proof of its physical mass gap. Read a precise theorem.

Asymptotic freedom

The decrease of the renormalized coupling at increasing energy scales in a specified renormalization scheme. In pure four-dimensional Yang–Mills theory, the leading beta-function coefficient has the sign that produces this behavior. It explains short-distance perturbative control; it does not establish the long-distance spectrum. Derive the running coupling.

Bare coupling

A parameter in the regulated action, denoted $g_0$ here. Its value depends on the regulator and the physical trajectory being followed. A renormalized coupling $g(\mu)$ is defined through a prescription at scale $\mu$; it is not interchangeable with $g_0$. Follow the continuum trajectory.

Compact simple gauge group

The nonabelian group specifying the gauge symmetry in the Clay problem. “Compact” makes normalized invariant group integration available; “simple” here refers to the absence of proper nonzero Lie-algebra ideals. $SU(N)$, $N\geq2$, is the main example used in this course. A compact group with simple Lie algebra need not have a nontrivial center, and groups sharing a Lie algebra can differ globally. Groups and representations explains why a result for one group or probe need not establish the statement for every compact simple group. See also the target and Haar measure.

Confinement

A family of statements about the long-distance behavior of color and external sources. For suitable representations and theories, a linearly increasing static-source potential or a Wilson-loop area law provides a useful diagnostic. The diagnostic must be specified: screening, the gauge group and dynamical matter affect its applicability. Confinement and a gap in the gauge-invariant physical spectrum are distinct claims. Compare the observables.

Connected correlator

For two operators, the expectation with its factorized vacuum contribution removed: $\langle AB\rangle_c=\langle AB\rangle-\langle A\rangle\langle B\rangle$. For a suitable observable $\mathcal O$, vacuum subtraction removes the constant vacuum term that would otherwise conceal decay. Quantum local products must also be renormalized and smeared appropriately. Derive its spectral representation.

Continuum limit

Removal of a short-distance regulator while keeping specified physical quantities fixed. On a lattice this means $a\to0$ along a suitable choice of bare parameters, with renormalized observables under control. A finite physical mass then has $a m_{\mathrm{phys}}\to0$. Taking a larger box at fixed $a$ is a different limit. Work through the scales.

Correlation length

The distance scale governing asymptotic exponential decay of a specified connected correlation function. For $C_{\mathcal O}(t)=\int e^{-Et}\,d\rho_{\mathcal O}(E)$ with positive, nonzero spectral measure, the temporal decay threshold is $m_{\mathcal O}=\inf\operatorname{supp}\rho_{\mathcal O}$ and $\xi_{\mathcal O}=1/m_{\mathcal O}$ when $m_{\mathcal O}>0$. Power-law prefactors can accompany this exponential. A channel may miss lower states, so its correlation length need not equal the inverse full gap. Study decay and operator overlap.

Cutoff

A restriction that makes some otherwise uncontrolled part of a theory manageable. A lattice spacing supplies an ultraviolet cutoff of order $1/a$; a finite box supplies an infrared restriction. A claim proved with a cutoff in place needs a further argument before it becomes a claim about the unregulated theory. Distinguish the two limits.

Dimensional transmutation

The replacement of a dimensionless renormalized coupling specified at a reference scale by a dimensionful scale such as $\Lambda$. At leading order, $\Lambda=\mu\exp[-1/(2b_0g^2(\mu))]$. This relation introduces a scale into the quantum description; proving a nonzero physical mass in units of that scale is additional work. Derivation and exercise.

Euclidean field theory

A formulation using Euclidean time and statistical weights such as $e^{-S_E}$. Physical time evolution is related to the semigroup $e^{-tH}$. A suitable Euclidean theory can yield a Hilbert-space quantum theory through reconstruction, but this requires more than writing a formal integral or having a positive probability density. Start with Euclidean correlators and reconstruction.

Finite volume

A theory restricted to a box with stated boundary conditions. Discrete spatial momenta can create a positive finite-box excitation energy even when the infinite-volume theory is gapless. The course uses a massless scalar with Dirichlet boundary conditions to demonstrate this without silently deleting a zero mode. See the worked example.

Gauge invariance

Independence from a redundant choice of local gauge frame. Curvature transforms by conjugation, so traces of suitable curvature products and closed holonomies give classical gauge-invariant observables. Their quantum definitions can require renormalization. Gauge-fixed fields are useful calculation variables; their correlation functions do not automatically have the physical positive spectral interpretation. Identify physical observables.

Glueball

A color-singlet excitation in a pure gauge theory, usually discussed through channels labelled by spin, parity and, where applicable, charge conjugation. A “glueball operator” creates a superposition of spectral components; its name does not guarantee an isolated one-particle state. Learn how masses are extracted.

Haar measure

The translation-invariant measure on a group. For a compact group it can be normalized so that $\int_G dU=1$. Invariance under left and right multiplication makes it the natural measure for lattice link variables and is central to exact lattice gauge invariance. Mathematics bridge.

Lattice gauge theory

A regulated formulation with group elements on links and gauge-invariant interactions built from loops. On a finite lattice with compact gauge group, the Wilson measure is a genuine finite-dimensional integral. The continuum quantum theory still requires a controlled limiting argument. Construct the measure.

Mass gap

A positive lower threshold for the nonzero energy spectrum of the physical Hamiltonian above its vacuum sector. With vacuum energy zero, the target is $\operatorname{spec}(H)\cap(0,\Delta)=\varnothing$ for some $\Delta>0$, in a nontrivial continuum, infinite-volume theory. The threshold need not be an isolated eigenvalue. A gauge-fixed gluon parameter, a finite-box level or an auxiliary stochastic gap is a different statement. Exact target and spectral theorem.

Operator overlap

The component of $\mathcal O\Omega$ in a particular state or spectral range. In discrete notation, the squared amplitude $|\langle n|\mathcal O|\Omega\rangle|^2$ weights that state’s contribution to a correlator. A symmetry can force this weight to vanish. Even a perfectly measured correlator can therefore miss the lowest physical excitation. See a solvable example.

Osterwalder–Schrader reconstruction

The passage from Euclidean correlation functions satisfying appropriate axioms to a positive Hilbert space, a vacuum, observables and physical time evolution. Reflection positivity supplies the crucial nonnegative inner-product form. The theorem also requires other hypotheses, including appropriate covariance and regularity; checking one positivity inequality is not a complete reconstruction. The 1973 formulation must be read with the 1975 extension and correction. Learn the construction and inspect the deeper reconstruction guide.

Physical Hilbert space

The positive-inner-product state space containing physical states after gauge redundancy and constraints have been handled. The mass-gap Hamiltonian acts here. Unphysical polarizations and ghosts introduced in a gauge-fixed calculation are not additional physical excitations to be included in its spectrum. Gauss’s law and states.

Reflection positivity

Nonnegativity of $\langle(\Theta F)F\rangle$ for suitable functionals $F$ supported at positive Euclidean times, where $\Theta$ combines time reflection with complex conjugation and any required tensor transformation. This condition links Euclidean data to a positive physical inner product. A positive probability measure alone need not satisfy it; neither condition alone proves a mass gap. Work through a counterexample.

Renormalization

The adjustment of parameters and the definitions of observables as a regulator changes, so that specified physical quantities remain meaningful. Composite operators can require subtractions and mixing with other operators. Constructing counterterms is an important step, but does not by itself prove convergence, physical reconstruction or a spectral gap. See what a continuum limit preserves.

Spectral measure

A measure encoding how a vector is distributed across the spectrum of a self-adjoint operator. For $\psi$, the Hamiltonian gives $d\rho_\psi(E)=d\langle\psi,P_H(E)\psi\rangle$, a positive measure that can have discrete and continuous parts. Its Laplace transform is $\langle\psi,e^{-tH}\psi\rangle$. Mathematics bridge and correlator lesson.

String tension

The coefficient $\sigma$ of a linear term $V(R)\sim\sigma R$ in a suitable static-source potential, or the corresponding asymptotic area coefficient when the Wilson-loop interpretation applies. It has dimensions of mass squared. The dimensionless lattice quantity is $a^2\sigma$. Connect loops and static sources and compare mass ratios.

Transfer matrix

An operator that advances a lattice system by one Euclidean time step. When the required positivity properties hold and a suitable normalization is chosen, it is related to a Hamiltonian by $\mathsf T=e^{-aH}$. A gap for this regulated physical operator still needs control through volume and continuum limits. Semigroups bridge and positivity lesson.

Uniform estimate

A bound whose relevant constants remain controlled while a specified parameter varies. To preserve a positive physical gap through a limit, positivity at every cutoff is insufficient: the bound must survive in physical units and pass to the limiting theory. Always ask “uniform in what?” Mathematics bridge.

Vacuum

A lowest-energy physical state. We denote a normalized chosen vacuum by $\Omega$ and shift its energy to zero, so $H\Omega=0$. Simple connected-correlator formulas assume that the chosen sector has a one-dimensional zero-energy space; otherwise the whole zero-energy projection must be accounted for. Vacuum subtraction and precise conventions.

Wilson loop

The normalized trace of parallel transport around a closed curve. On the lattice it is a trace of an ordered product of links. Closing the curve makes the result gauge invariant. Its large-loop behavior probes external-source physics; continuum loops may also require renormalization. Physical interpretation and lattice definition.

Yang–Mills theory

A gauge field theory whose classical action is quadratic in its curvature. The nonabelian commutator in the curvature supplies self-interactions. This site’s default teaching setting is pure four-dimensional quantum Yang–Mills theory, at zero temperature and $\theta=0$. The official problem quantifies over compact simple gauge groups and governs the required theorem. “Existence” means constructing the intended quantum theory with appropriate properties; “mass gap” is a further spectral property of that construction. Read the first lesson.

Primary starting points

• Jaffe and Witten, Quantum Yang–Mills Theory: the official target and its mathematical context.
• David Tong, Gauge Theory: graduate-level physical background.
• Wilson, Confinement of quarks: lattice gauge theory and Wilson loops.
• Osterwalder and Schrader, Axioms for Euclidean Green’s Functions: the Euclidean reconstruction framework.
• Osterwalder and Schrader, Axioms for Euclidean Green’s Functions II: the 1975 extension and correction, to be read with the first paper.

Conventions · Course overview · Mathematics bridges