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\section*{Wick rotation}

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

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!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{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{method}{Method}\dotfill \pageref*{method} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

What is called \emph{Wick rotation} (after [[Gian-Carlo Wick]]) is a method in [[physics]] for finding a construction in [[relativistic field theory]] on [[Minkowski spacetime]] or, more generally, on [[Lorentzian manifolds]], from a related construction in [[Euclidean field theory]] on [[Riemannian manifolds]] in a way that involves or generalizes the idea of [[analytic continuation]] for the \emph{[[time]]} [[coordinates]].

\begin{quote}%
The [[complex plane]] for a complexified [[time]] [[coordinate]] $t + i s$. After Wick rotation it is the [[imaginary part]] $s$ that replaces the ``[[real part|real]]'' time $t$.

graphics grabbed from \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87}


\end{quote}
This is motivated by the observation that the [[Minkowski metric]] (with the $-1,1,1,1$ convention) and the four-Euclidean metric are equivalent if the [[time]] components of either are allowed to have [[imaginary part|imaginary]] values. Hence Wick rotation, when it applies, involves [[analytic continuation]] of [[n-point functions]] to [[complex number|complex valued]] [[time]] [[coordinates]].

For [[relativistic field theory|relativistic]]/[[Euclidean field theory|Euclidean]] [[quantum field theory]] on [[Minkowski spacetime]] $\mathbb{R}^{d,1}$/[[Euclidean space]] $\mathbb{R}^{d+1}$ Wick rotation is rigorously understood and takes the form of a [[theorem]] relating the [[Wightman axioms]] for [[n-point functions]] in [[relativistic field theory]] to the \emph{[[Osterwalder-Schrader axioms]]} for [[correlators]] in [[Euclidean field theory]].

See for instance \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87, section 2} for a clear account.

More generally, in this context Wick rotation applies to relativistic field theory in [[thermal equilibrium]] [[states]] at [[positive number|positive]] [[temperature]] $T$ (``[[KMS states]]''), and then relates this to [[Euclidean field theory]] with compact/periodic Euclidean time of length $\beta = 1/T$ (see \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87, section 3}):

\begin{displaymath}
\itexarray{
  \left.
  \itexarray{
     \text{relativistic field theory}
     \\
     \text{on Minkowski spacetime}
     \\
     \mathbb{R}^{d,1}
     \\
     \text{in a thermal equilibrium state}
     \\
     \text{at temperature}\; T
  }
  \right\}
  &
  \;\;\;\;
  \overset{ \text{Wick rotation} }{\leftrightarrow}
  \;\;\;\;
  &
  \left\{
  \itexarray{
    \text{Euclidean field theory}
    \\
    \text{on Euclidean space}
    \\
    \mathbb{R}^d \times S^1_{\beta}
    \\
    \text{with compact/periodic Euclidean time}
    \\
    \text{of length} \; \beta = 1/T
  }
  \right.
  \\
  \phantom{A}
  \\
  \underset{
    {
    \text{equal-time n-point function} 
    \atop 
    \text{of relativistic fields}
    }
    \atop
    \text{ in thermal equilibrium state } \; \vert T\rangle
  }{
  \underbrace{
    \left\langle  T\vert  :\mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) : \vert T \right\rangle_{\mathbb{R}^{d,1}}
  }}
  &\;=\;&
  \underset{
    \text{correlator of Euclidean fields}
    \atop
    \text{ for "Euclidean time" of periodicity}\; \beta = 1/T
  }{
  \underbrace{
  \left\langle 0 \vert \mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t)  \vert 0 \right\rangle_{\mathbb{R}^{d} \times S^1_{\beta}}
  }}
  }
\end{displaymath}
In this form, Wick rotation is also known as \emph{[[thermal quantum field theory]]}. See there for more.

\begin{quote}%
graphics grabbed form \hyperlink{FrolovZelnikov11}{Frolov-Zelnikov 11}


\end{quote}
Wick rotation also applies on suitable [[black-hole]]-[[spacetimes]] spring thermodynamics, such as the [[Bekenstein-Hawking entropy]], find elegant explanations, at least at the level of the manipulation of formulas (see e.g. \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87, section 4}).

\hypertarget{example}{}\subsubsection*{{Example}}\label{example}

Consider the Minkowski metric with the $-1,1,1,1$ convention for the tensor:

$d s^{2}= -(d t)^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}$

and the four-dimensional Euclidean metric:

$d s^{2}= d \tau^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}$.

Notice that if $d t = i\cdot d \tau$, the two are equivalent.

\hypertarget{method}{}\subsubsection*{{Method}}\label{method}

A typical method for employing Wick rotation would be to make the substitution $t=i\tau$ in a problem in Minkowski space. The resulting problem is in Euclidean space and is sometimes easier to solve, after which a reverse substitution can (sometimes) be performed, yielding a solution to the original problem.

Technically, this works for any four-vector comparison between Minkowski space and Euclidean space, not just for space-time intervals.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[reflection positivity]]

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

Wick rotation between [[relativistic field theory]] in terms of [[Wightman axioms]] for [[n-point functions]] on [[Minkowski spacetime]] and [[Euclidean field theory]] in terms of [[Osterwalder-Schrader axioms]] for [[correlators]] on [[Euclidean space]] is due to

\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}
The generalization of this for [[positive number|positive]] [[thermal equilibrium]] [[vacuum staes]], relating to [[thermal quantum field theory]] with compact/periodic Euclidean time is discussed 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}
The idea here apparently goes back to

\begin{itemize}%
\item Claude Bloch, \emph{Sur la détermination de l'état fondamental d'un système de particules}, Nucl. Phys. 7 (1958) 451

\end{itemize}
This has maybe first been made precise, for the case of 1+1 dimensions, in

\begin{itemize}%
\item [[Raphael Høegh-Krohn]], \emph{Relativistic Quantum Statistical Mechanics in two-dimensional Space-Time}, Communications in Mathematical Physics 38.3 (1974): 195-224 (\href{https://www.duo.uio.no/bitstream/handle/10852/44072/1973-22.pdf}{pdf})

\end{itemize}
Good review on the relation to [[thermal quantum field theory]] and [[black hole thermodynamics]] is in

\begin{itemize}%
\item S.A. Fulling, S.N.M. Ruijsenaars, \emph{Temperature, periodicity and horizons}, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 (\href{https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf}{pdf}, )


\item [[Gary Gibbons]], Malcolm J. Perry, \emph{Black Holes and Thermal Green Functions}, Vol. 358, No. 1695 (1978) (\href{https://www.jstor.org/stable/79482}{jstor:79482})



\end{itemize}
Discussion of thermal Wick rotation on global [[anti-de Sitter spacetime]] (which is already periodic in \emph{real} time) is in

\begin{itemize}%
\item B. Allen, A. Folacci, [[Gary Gibbons]], \emph{Anti-de Sitter space at finite temperature}, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 ()

\end{itemize}
See also

\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})


\item [[Graeme Segal]], \emph{Wick rotation and the positivity of energy in quantum field theory} (\href{https://www.youtube.com/watch?feature=player_embedded&v=vTvXHL6ZJik}{video})


\item [[Edward Witten]], \emph{The Feynman $i \epsilon$ in String Theory} (\href{http://arxiv.org/abs/1307.5124}{arXiv:1307.5124})



\end{itemize}


\end{document}