Relation between functional measures and states in AQFT

Let $$(M,g)$$ be a globally hyperbolic spacetime and $$\phi$$ a KG field. In AQFT we consider the algebra of observables $$\mathfrak{A}$$ generated by $$\phi(f)$$ where $$f\in C^\infty_0(M)$$ is a test function.

A state is then a linear functional $$\omega : \mathfrak{A}\to\mathbb{C}$$ which satisfies $$\omega(1)=1$$ and $$\omega(a^\ast a)\in [0,+\infty)$$ for all $$a\in \mathfrak{A}$$.

Now, a state allows for the computation of the $$n$$-point functions $$W_n(f_1,\dots,f_n)=\langle \phi(f_1)\cdots \phi(f_n)\rangle_\omega=\omega(\phi(f_1)\cdots \phi(f_n)).\tag{1}$$

Now in usual treatments, one writes via functional integration

$$\langle \operatorname{T}[\phi(x_1)\cdots \phi(x_n)]\rangle=\int \varphi(x_1)\cdots \varphi(x_n)\dfrac{e^{iS[\varphi]}\mathfrak{D}\varphi}{Z}\tag{2}$$

Now if we make sense of $$Z^{-1}e^{-S[\phi]}\mathfrak{D}\phi$$ as a functional measure after passing to Euclidean time (which is possible at least for the free theory), there seems to be a one-to-one relation between time-ordered correlators and functional measures. In other words, there seems to be a one-to-one relation between the restriction of $$\omega$$ to time-ordered products and functional measures.

This makes me wonder, can this be extended to the state $$\omega$$ itself? Is there some sort of correspondence between states and functional measures so that, for instance, a gaussian state would be in correspondence to a gaussian measure?

Or is this relation between time-ordered products and measures not even one-to-one or even well-defined?

What is the relation between algebraic states and functional measures, if there is any relation?

• Very interesting question. I am however wondering if this can be made sufficiently rigorous even for the vacuum state: passing from Euclidean time to Lorentzian is not completely trivial, since the Osterwalder-Schroder theorem contains some rather non-trivial condition on Euclidean correlators that is needed to ensure Wightman functions are distributions (I have in mind the O-S II paper). I don't know anything about whether this condition can be interpreted naturally in terms of functional measures. – Peter Kravchuk Jul 6 at 19:24
• Identity (2) is wrong because the left-hand side is not symmetric if interchanging $f_k$ and $f_k$ (a causal propagator takes place), whereas the right-hand side is symmetric by construction. Passing to the Euclidean time is a problem in a generic curved globally hyperbolic spacetime... Maybe in static spacetimes the question could make some sense, but it is not completely clear how the correlation functions are related with the algebraic state. For Gaussian states it seems to make sense if the states are static. – Valter Moretti Jul 7 at 12:33
• @ValterMoretti, I think the question is interesting even in flat space. I also do not understand your comment about identity (2) because there are no $f_k$ in it. – Peter Kravchuk Jul 7 at 22:51
• Yes, you are right sorry, just replace $f_k$ for $x_k$ in my commnent. Indeed, I include Minkowki spacetime in the class of static spacetimes. – Valter Moretti Jul 8 at 5:15
• What I wanted to remark is that, in the Lorenzian case, the left-hand side of (2) needs a T product. – Valter Moretti Jul 8 at 5:26