Defining quantum Yang–Mills theory

Writing a Lagrangian tells us the intended local dynamics. It does not automatically produce a nonperturbative quantum theory with well-defined observables, a physical Hilbert space, and controlled behavior at every scale. The existence part of the Yang–Mills problem asks us to close that gap.

This volume follows one important route: define regulated Euclidean observables, control their limits, and recover a physical quantum theory. A direct Hamiltonian construction may use a different route to the required outcomes.

What must actually be constructed?

Prerequisites: the problem and the idea of a regulated path integral. The distribution bridge introduces smearing without assuming prior constructive field theory.

Outcome: state which observable, limiting argument and physical structure a proposed construction supplies, and which remain to be proved.

The formal expression

$$\langle O\rangle= \frac{\int O[A]e^{-S[A]}\,\mathcal DA} {\int e^{-S[A]}\,\mathcal DA}$$

is a guide until its measure and observables have been defined. A finite lattice turns it into ordinary integrals over compact groups. The challenge then becomes understanding what remains as the ultraviolet and volume cutoffs are removed.

The resulting object must be the intended Yang–Mills theory. Constructing an arbitrary positive measure or an unrelated gapped model would not identify it. The official statement, sections 3–4, explains the role of local gauge-invariant observables and short-distance behavior.

Read the lattice definition for the finite starting point and distributions for why continuum fields require smearing.

Correlation functions carry the information

Euclidean correlation functions can encode a quantum theory, provided they obey sufficiently strong structural conditions. These include suitable symmetry, regularity, positivity, and compatibility properties. Positivity is not one interchangeable condition: an ordinary nonnegative probability weight does not by itself guarantee a physical Hilbert-space interpretation.

Reflection positivity tests expressions built from observables supported at positive Euclidean times and their reflected counterparts. Schematically,

$$\langle(\Theta F)F\rangle\geq0.$$

Here $\Theta$ combines time reflection with the appropriate conjugation. This expression supplies a candidate squared norm. Null vectors are identified before completing the resulting inner-product space.

The reflection-positivity lesson explains this mechanism through a free-field example. It also distinguishes a motivation or proof sketch from a full reconstruction theorem. The 1975 Osterwalder–Schrader extension and correction belongs alongside the 1973 formulation; positivity alone does not replace its additional hypotheses.

The limits must preserve the structure

At finite regulator, many expressions are harmless because only finitely many degrees of freedom occur. Continuum limits can create singularities. Infinite-volume limits can change the spectrum or vacuum structure.

A typical argument therefore needs more than a convergent partition function. It must control the relevant renormalized observables and establish the required properties of their limiting correlations. A correlation-decay estimate is useful only with its dependencies and units specified.

The bridge on uniform estimates gives a simple example: every member of a sequence can have a positive gap even though the limiting theory is gapless.

Construction and dynamics meet

Once an appropriate physical Hamiltonian has been constructed, a mass-gap argument must exclude arbitrarily low positive-energy excitations. Exponential decay can help, but the class of observables and the relation to the physical state space matter.

Bałaban’s conditional ultraviolet-stability bounds illustrate a major component of constructive control. In Large field renormalization II, published version, CMP 122 (1989), Theorem 1, p. 355, the entire sequence of effective couplings is assumed to remain sufficiently small. The conclusion controls effective densities with the inherited representation hypotheses; it does not itself construct all limiting correlations or a physical gap. The dossier records the finite-torus lattice setting, group and action, and the article’s historical qualification about the coupling argument. Three-dimensional stochastic renormalization addresses a separate construction question with its own dimension, matter and lifetime restrictions.

A reading route

Construction question	Core concept page
Which objects and properties identify the intended quantum theory?	Axioms and the target
Why quotient null vectors, and what does a reflected norm mean?	Reflection positivity
Why is a gauge-invariant product not automatically a defined quantum observable?	Renormalized observables
Which bounds survive the cutoff and volume limits?	Limits and uniformity

Begin with the continuum limit, then read reflection positivity. Use the bridges on Gaussian measures and spectral theory as needed. Finish with the Proof Map, which shows how construction, identification, and the physical gap fit into the larger target.