Is it known whether Wightman's axiomatic QFT is logically equivalent to Osterwalder–Schrader's axiomatic QFT?

Question

Constructive QFT has provided some interesting models for dimension $d < 4$ of space-time, satisfying specific axiomatic versions of QFT. On the other hand, it is a well known fact that an axiomatic theory can admit more than one model, the question here would be: is it known whether there are models that satisfy Wightman's set of axioms but not those of Osterwalder and Schrader, or vice versa?

Answer 1

It depends on what you mean by the Osterwalder-Schrader axioms. If you begin with a Wightman field, the Bargmann-Wightman-Hall lemma tells you that the Wightman functions are (distributional) boundary values of a holomorphic function on the permuted extended forward tube. This domain in particular contains the non-coincident Euclidean points. That is, tuples $(\zeta_1, \dots, \zeta_n)$ where $\zeta_i = (i\tau, \mathbf{x})$, and $\xi_i\neq \xi_j$ for any $i\neq j$. Thus in particular the Wightman functions imply that the Schwinger functions are actually analytic functions away from the non-coincident points. These just so happen to be non-coincident tempered distributions, but obviously analyticity is far more rigid data.

The Osterwalder-Schrader axioms in their most practical form do not assume this analyticity. Instead they only stipulate that you have, a priori, only some tempered distributions that satisfy Euclidean invariance, reflection-positivity, and so on. The idea is that if one can provide sufficiently strong axioms, then you can directly recover the analyticity as a theorem rather than having to prove it constructively.

Now the reason I mention this is that one can define a system of axioms which in the original Osterwalder-Schrader paper are notated $\check{E}$. One of the conditions amounts to saying that the Schwinger functions are Fourier-Laplace transforms, and so are in particular holomorphic. $\check{E}$ is equivalent to $W$. The problem however is that the key analyticity axiom in $\check{E}$ is extremely impractical to check in a real constructive setting. Thus one must look for some replacement axiom that is easier.

The naive hope is that $E$ alone (That is, $\check{E}$ without the explicit analyticity condition) would be sufficient to imply the Wightman axioms. As Prov. Legolasov mentioned in his comment, Osterwalder and Schrader initially claimed such an equivalence in the first of their two papers. However there was a subtle error involving the theory of distributional Laplace transforms that invalidated their proof.

In their second paper, they proposed a replacement set of axioms $E'$ which supplement $E$ by what they call a linear growth condition on the Schwinger functions as a function of $n$. This axiom is more practical to check, and it was proven to imply analyticity in the sense required for the rest of the OS-proof to go through. However, $W$ does not imply $E'$.

In summary, we have $E'\implies W\iff \check{E}$.

$E, E'$, and $\check{E}$ can all be referred to as 'Osterwalder-Schrader axioms', but $E'$ or variants of it are the most useful ones.