Must all correlation functions come from a measure?

Question

In the Wightman setting a quantum field is defined as an operator valued distribution satisfying the Wightman axioms. Using the vacuum state of this theory we can define correlation/Wightman functions that are known to satisfy certain properties.

We can also work in the opposite direction. Given a set of scalar-valued distributions $\{W_n\}$ obeying certain properties there is an associated unique quantum field theory (in other words, a unique set of operator-valued distributions obeying the Wightman axioms for which the $W_n$ are its correlation functions) thanks to the Wightman reconstruction theorem.

If we are given a measure $\mu$ then we can of course define correlation functions by taking moments of $\mu$ and check if these moments satisfy the conditions needed to invoke the Wightman reconstruction theorem. However the reconstruction theorem itself makes no mention of a measure $\mu$, so $\{W_n\}$ doesn't necessarily have to be moments of some measure.

At what point does the measure $\mu$ enter? Do we always consider a measure because of motivation from physics and the path integral? That is, when rigorously constructing quantum field theories do we always study correlation functions arising from a measure because these are the only theories that are physically relevant? I am trying to distinguish the physics and the mathematics behind these constructions.

Answer 1

Just to clarify formalism, a correlation functions of a Wightman field theory cannot arise from a measure. Measures describe classical probability theories, whereas a Wightman field is noncommutative. What is true is that for a bosonic Wightman field, one can use the Bargmann-Hall-Wightman theorem in order to analytically continue the Wightman functions into the non-coincident Euclidean points. One obtains analytic functions $S(x_1,\dots, x_n)$ (at non-coincident points). These are known as Schwinger functions, and they are symmetric under permutations of the coordinates. Because of this symmetry, one can hope that they truly arise as moments of a probability measure $\mu$.

However, this is actually FALSE in general. A necessary condition for Schwinger functions to arise from a probability measure is what is known as Nelson-Symanzik positivity. The condition is not too surprising: If $\mu$ is a probability measure, then we should expect that for any sequence of test functions $f_0, \dots, f_n$ where $f_i \in \mathcal{S}(\mathbb{R}^{di})$, we have $$\sum_{n,m} S_{n+m}(x_1,\dots, x_n, y_1, \dots, y_m)f^\ast_n(x_1,\dots, x_n)f_m(y_1, \dots, y_m)\geq 0$$

This is the obvious consequence of the claim that $\mu$ is a measure. All this comes from is the fact that the above expression is equal to $$\int \bigg|\sum_{n} \int f_n(x_1,\dots, x_n)\phi(x_1)\dots\phi(x_n)dx_1\dots dx_n\bigg|^2d\mu(\phi)\geq 0$$

Finally, it is not expected that every physical quantum field theory arises from a measure. For example, in gauge theories with fermions, one finds the relation for the axial current $\partial_\mu j^\mu = \frac{ie}{\pi}F$. The occurrence of these complex numbers immediately kills your chances for finding a probability measure generating the Schwinger functions for the observable fields $(j^\mu, F_{\mu\nu})$. (See Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics for details)

However for less complicated models, such as purely bosonic field theories with polynomial interactions, one can typically always expect a measure theoretic/probabilistic representation of the model.