How do you know that you have succeeded-Constructive Quantum Field Theory and Lagrangian

It's not necessarily possible to uniquely identify a quantum field theory from its Wightman data.

If you've done the construction yourself, you'll know which theory it is, because you chose the coordinates on the space of fields, constructed the measure and the observables, and showed they obey Osterwalder-Schrader. But the Wightman data you construct this way could have been reached starting with some other set of coordinates on a different space of fields. There might be a dual QFT which gives the same Wightman data, starting with a different set of generators.

The canonical example here is the sigma model to a torus $mathbb{R}/Rmathbb{Z}$ of radius $R$. (I'm not sure if this thing has been built via constructive methods, but it's easy to build by canonical quantization.) This QFT gives the same Wightman data as the sigma model to the torus of radius $1/R$. The only difference is the choice of generators and the underlying classical limits.

The axioms don't tell you what theory you constructed. For that you need to go beyond the construction of correlation functions of the elementary field $phi$ (the basic chapter on renormalization in QFT textbooks) and produce, e.g., by a point-splitting procedure, correlations with insertion of composite fields like $phi^3$. You should then identify your theory via the equation of motion, e.g., $-Deltaphi+m^2phi=-lambdaphi^3$ holding inside correlations. To see how this is done rigorously, see the article by Feldman and Rączka in Ann. Phys. 1977 or the recent article by Gubinelli and Hofmanová.

Also, an interesting example is the following. Let ${rm Conf}_n(mathbb{R}^2)$ denote the configuration space of $n$ points in $mathbb{R}^2$, i.e., the set of tuples $(x_1,ldots,x_n)$ made of $n$ distinct points in $mathbb{R}^2$. Consider the functions $S_n:{rm Conf}_n(mathbb{R}^2)rightarrowmathbb{R}$ given by $$ S_n(x_1,ldots,x_n)=sqrt{sum_q prod_{1le i<jle n}|x_i-x_j|^{frac{q_iq_j}{2}}} $$ where the sum is over "neutral configurations of charges" $q=(q_1,ldots,q_n)in{-1,1}^n$ such that $sum_i q_i=0$.

One can show that $forall n,exists K_n>0,forall (x_1,ldots,x_n)in{rm Conf}_n(mathbb{R}^2)$, $$ S_n(x_1,ldots,x_n)le K_nprod_{i=1}^{n}left(min_{jneq i}|x_j-x_i|right)^{-frac{1}{8}} . $$ The quickest proof I know for this inequality is following the optimal matching argument in Appendix A of "Complex Gaussian multiplicative chaos" by Lacoin, Rhodes and Vargas. Then it is not hard to show that for every Schwartz function $finmathscr{S}(mathbb{R}^{2n})$, the integral $$ int_{{rm Conf}_n(mathbb{R}^2)}S_n(x_1,ldots,x_n) f(x_1,ldots,x_n) d^2x_1cdots d^2x_n $$ converges and defines a temperate distribution in $mathscr{S}'(mathbb{R}^{2n})$. This is explained in Section 2 of my CMP article "A Second-Quantized Kolmogorov-Chentsov Theorem via the Operator Product Expansion".

Now it is a fact that the resulting distributions $S_nin mathscr{S}'(mathbb{R}^{2n})$ satisfy the Osterwalder-Schrader Axioms, and therefore can be analytically continued into Wightman distributions satisfying the Wightman Axioms and thus via a GNS type construction, give in the end a quadruple $(mathcal{H},U,Omega,phi)$ obeying the Gårding-Wightman Axioms. The $S_n$ are also the moments of a probability measure on $mathscr{S}'(mathbb{R}^{2})$.

Quiz: What is the Lagrangian of this theory?

I will come back later with an answer, but regarding the construction via probability measures, I explained this already so I will not repeat myself and refer to

Reformulation - Construction of thermodynamic limit for GFF

A set of questions on continuous Gaussian Free Fields (GFF)

A roadmap to Hairer's theory for taming infinities

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Quiz answer: It is the Ising CFT. Note that I tried to see if one has an equation of motion of $phi^4$ type but my computations got out of hand rather quickly when looking for an explicit $phi^3$.