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\begin{document}

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\section*{Osterwalder-Schrader theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{aqft}{}\paragraph*{{AQFT}}\label{aqft}

[[!include AQFT and operator algebra contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{axioms_of_euclidean_field_theory}{Axioms of euclidean field theory}\dotfill \pageref*{axioms_of_euclidean_field_theory} \linebreak
\noindent\hyperlink{the_theorem}{The theorem}\dotfill \pageref*{the_theorem} \linebreak
\noindent\hyperlink{related_theorems}{Related theorems}\dotfill \pageref*{related_theorems} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Osterwalder-Schrader theorem} (\hyperlink{OsterwalderSchrader73}{Osterwalder-Schrader 73}) states precise conditions under which [[Wick rotation]] between [[relativistic field theory]] and [[Euclidean field theory]] works.

Rough idea: The [[Wightman axioms]] describe how the algebra of observables of a [[quantum field theory]] on [[Minkowski spacetime]] is generated by quantum fields. The Wightman reconstruction theorem asserts that knowing all correlation functions of all fields in the vacuum state is equivalent to knowing the quantum fields. The \emph{Osterwalder--Schrader theorem} states conditions that correlation functions on Euclidean spacetime have to satisfy to be equivalent to the correlation functions of a Wightman QFT on Minkowski spacetime.

In this sense the Osterwalder--Schrader theorem states and proves conditions that assure that the [[Wick rotation]] is a well defined isomorphism of quantum field theories on Minkowski and on Euclidean spacetime.

\hypertarget{axioms_of_euclidean_field_theory}{}\subsection*{{Axioms of euclidean field theory}}\label{axioms_of_euclidean_field_theory}

The axioms of euclidean field theory are the euclidean analogue of the [[Wightman axioms]] on Minkowski spacetime. The axioms may be formulated for tempered distributions, but we follow the lines of Glimm and Jaffe and define them for $\mathcal{D}'(\mathbb{R}^d)$, the space of distributions that is dual to the space of all smooth functions with compact support, $\mathcal{D}(\mathbb{R}^d)$. In the original paper of Osterwalder and Schrader the axioms are given in terms of the Schwinger functions. Here the axioms given in a form more directly related to the measure on field space and its characteristic function, rather than the Schwinger functions themselves. This form was first presented by Fr\"o{}hlich. We define the generating functional on $\mathcal{D}(\mathbb{R}^d)$

\begin{displaymath}
S(f) := \integral e^{i \phi(f)} d\mu
\end{displaymath}
as the inverse Fourier transform of a Borel probability measure $d\mu$ on $\mathcal{D}'(\mathbb{R}^d)$.

\begin{itemize}%
\item \textbf{OS0 (analyticity)}: For every finite set of test functions $f_1, f_2,...f_n$ and complex numbers $z:= (z_1, z_2, ...z_n)$ the function

\begin{displaymath}
z \mapsto S(\sum_{k=1}^n z_k f_k)
\end{displaymath}
is entire analytic on $\mathbb{C}^n$.


\item \textbf{OS1 (regularity)}: For some p with $1 \le p \le 2$ and some constant c the following inequality holds for all test functions f:

\begin{displaymath}
| S(f) | \le exp(c \| f \|_{L_1} + \| f\|^p_{L_p})
\end{displaymath}

\item \textbf{OS2 (invariance)}: S is invariant under euclidean symmetries E of $\mathcal{R}^d$ (translations, rotations, reflections), that is S(f) = S(Ef) for all symmetries E and test functions f.


\item \textbf{OS3 ([[reflection positivity]])} We define exponential functionals on $\mathcal{D}'(\mathbb{R}^d)$ via

\begin{displaymath}
A(\phi) := \sum_{k=1}^n c_k exp(\phi(f_k))
\end{displaymath}
Let $\mathcal{A}$ be the set of all these functionals, by axiom OS0 this is a subset of $L_2(d\mu)$. Euclidean symmetries act on $\mathcal{D}'(\mathbb{R}^d)$ via duality, that is $E\phi(f) = \phi(Ef)$, and thus define an unitary continuous action on $L_2(d\mu)$. Let $\mathcal{A}^+ \subset \mathcal{A}$ be the set of functionals with $supp(f_i) \subset \mathbb{R}^d_+$ where $\mathbb{R}^d_+ := \{(x, t): t \gt 0 \}$. Let $\theta: (x, t) \mapsto (x, -t)$ be the time reflection. Then the content of the axiom is:

\begin{displaymath}
0 \le \langle \theta A, A\rangle_{L_2}
\end{displaymath}

\item \textbf{OS4 (ergodicity)}: the time translation subgroup acts ergodically on the measure space $(\mathcal{D}'(\mathbb{R}^d), d\mu)$.


\item \textbf{theorem (Schwinger functions)}: A measure that satisfies OS0 has moments of all order, the nth moment has a density $S \in \mathcal{D}'(\mathbb{R}^{nd})$. These distributions are called Schwinger functions.



\end{itemize}
\hypertarget{the_theorem}{}\subsection*{{The theorem}}\label{the_theorem}

One possible formulation: To every measure satisfying the axioms stated above there is a Wightman field such that the Schwinger and Wightman functions are related by:

\begin{displaymath}
\integral \phi_E(x_1, t_1) \cdots \phi_E(x_n, t_n) = \langle \Omega, \phi_M(x_1, i t_1) \cdots \phi_M(x_n, i t_n) \Omega \rangle
\end{displaymath}
$\phi_E$ is a Schwinger function, $\phi_M$ is a Wightman field and $\Omega$ is the vacuum vector of the Wightman fields. See theorem 6.15 in the book by Glimm and Jaffe (see references).

\hypertarget{related_theorems}{}\subsection*{{Related theorems}}\label{related_theorems}

\begin{itemize}%
\item [[Reeh-Schlieder theorem]]


\item [[spin-statistics theorem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original article is

\begin{itemize}%
\item [[Konrad Osterwalder]], [[Robert Schrader]], \emph{Axioms for Euclidean Green's functions}, Comm. Math. Phys. Volume 31, Number 2 (1973), 83-112 (\href{https://projecteuclid.org/euclid.cmp/1103858969}{Euclid:1103858969})

\end{itemize}
Discussion for compact/periodic Euclidean time, as needed for [[thermal quantum field theory]] is in

\begin{itemize}%
\item Abel Klein, Lawrence Landau, \emph{Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle}, Pacific Journal of Mathematics, Vol. 94, No. 2, 1981 (\href{https://msp.org/pjm/1981/94-2/p12.xhtml}{DOI: 10.2140/pjm.1981.94.341}, \href{https://msp.org/pjm/1981/94-2/pjm-v94-n2-p12-s.pdf}{pdf})

\end{itemize}
Exposition is in

\begin{itemize}%
\item Dirk Schlingemann, \emph{From euclidean field theory to quantum field theory} (\href{http://arxiv.org/abs/hep-th/9802035}{arXiv:hep-th/9802035})

\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[James Glimm]], [[Arthur Jaffe]], \emph{[[Glimm-Jaffe|Quantum physics: a functional integral point of view]]}

\end{itemize}
[[!redirects Osterwalder-Schrader theorem]] [[!redirects Osterwalder–Schrader theorem]] [[!redirects Osterwalder--Schrader theorem]] [[!redirects Osterwalder-Schrader axioms]] [[!redirects Osterwalder--Schrader axioms]]



\end{document}