The axioms and the target theory

Prerequisites: the problem and fields as distributions.

You will learn to identify what a proposed construction actually defines, separate reconstruction from dynamics, and read an axiom claim without replacing it by a slogan.

A formal action is a specification of intent

A finite-dimensional integral is defined once its measure and integrand are specified. The expression $\int e^{-S[A]}\mathcal DA$ does not come with a translation-invariant infinite-dimensional Lebesgue measure. A construction must say what replaces that notation and which observables it defines.

The official Jaffe–Witten statement, §4, printed p. 6 asks for nontrivial quantum Yang–Mills on $\mathbb R^4$ for every compact simple gauge group, with a positive finite mass gap and sufficiently strong axiomatic properties. It also specifies the role of local gauge-invariant operators and short-distance identification. These requirements prevent an arbitrary gapped model from being relabelled Yang–Mills.

The site’s seven proof obligations organize these issues. They are an editorial map, not seven interchangeable certificates.

Two ways to present the quantum object

In a Hamiltonian description, one expects a physical Hilbert space, a vacuum, spacetime transformations and local observable operators on suitable domains. The Hamiltonian is self-adjoint and its positive-energy spectrum has a physical meaning. Locality relates observables at spacelike separation; it is not merely a short-range interaction in a chosen regulator.

In a Euclidean description, one may instead construct correlation distributions

$$S_n(f_1,\ldots,f_n),$$

where the $f_i$ are smooth test functions and the observable species have been specified. These quantities must be compatible with one another. Positivity, covariance, regularity and the appropriate symmetry properties must hold for the full family, in the precise form required by a reconstruction theorem.

The 1973 Osterwalder–Schrader paper must be read together with its 1975 extension and correction. The second paper’s abstract explicitly states that relationship. A five-name mnemonic for axioms is a reading aid, not the exact corrected theorem. This page supplies a map of the work; it does not reproduce all technical hypotheses of either paper.

What each layer answers

Layer	A concrete question	What it does not supply by itself
Observables	Are renormalized correlation distributions defined on a stated test-function space?	Their Yang–Mills identification or positivity
Physical structure	Can one construct a positive state space and spacetime evolution with the required locality and covariance?	A positive gap
Short-distance identification	Do the observables have the required ultraviolet behavior and conventions?	Infrared control
Dynamics	Is there a common positive lower spectral threshold above the vacuum?	Existence of the underlying theory if it was only formal
Scope	Are cutoff, volume, dimension and gauge group handled?	Extensions to unproved groups or geometries

For example, reflection positivity starts a Hilbert-space construction, while translation and regularity properties are needed to turn time shifts into suitable operators. A nonnegative Hamiltonian may still have spectrum arbitrarily close to zero. Each successful step needs its own conclusion.

Worked audit: a sequence of regulated theories

Suppose someone supplies finite-lattice expectations $\langle O_1\cdots O_n\rangle_{a,L}$, a positive transfer operator and a measured mass $a m(a,L)>0$. What has been given?

First, $a$ is a spacing and $L$ a physical box size; the finite expectation is a regulated object. Second, the transfer operator may define a regulated Hamiltonian on a gauge-invariant subspace. Third, a fitted mass in one channel need not be a bound on that Hamiltonian’s full spectrum.

To approach the target one still needs an observable normalization, convergence in a stated sense, preservation of the reconstruction hypotheses, identification with Yang–Mills, the unbounded-volume limit and a physical gap bound that survives. A positive number at every $a,L$ is insufficient: $m(a,L)=1/L$ is positive for each box and vanishes as $L\to\infty$.

Exercise: a useful wrong model

A free massive scalar field has a well-understood physical Hilbert space and gap. Why is constructing it not a solution to the Yang–Mills problem? What part of the exercise is still useful?

Solution. It is the wrong field content and observable theory; its correlations do not identify the required non-Abelian gauge dynamics or ultraviolet behavior. It is nevertheless a useful controlled model for smearing, positivity, spectral measures and the distinction between existence and a gap. Testing a construction on that model checks a method, not the target.

Read the construction in manageable pieces

Continue to reflection positivity, renormalized observables and limits and uniformity. Together they explain the missing steps without requiring a measure-theory course before entering the main twelve-lesson sequence.