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\begin{document}

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\section*{Euclidean field theory}

\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{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{wick_rotation_to_relativistic_field_theory}{Wick rotation to Relativistic Field theory}\dotfill \pageref*{wick_rotation_to_relativistic_field_theory} \linebreak
\noindent\hyperlink{temporal_compactification_to_thermal_relativistic_field_theory}{Temporal compactification to Thermal relativistic field theory}\dotfill \pageref*{temporal_compactification_to_thermal_relativistic_field_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{thermal_quantum_field_theory}{Thermal quantum field theory:}\dotfill \pageref*{thermal_quantum_field_theory} \linebreak


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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

In [[field theory]] one speaks of \emph{Euclidean field theory} if the underlying [[spaces]] on which the [[field (physics)|fields]] are defined are [[Riemannian manifolds]], as opposed to [[Lorentzian manifold|Lorentzian]] [[spacetimes]] used in [[relativistic field theory]], hence locally are \emph{[[Euclidean spaces]]} instead of [[Minkowski spacetimes]], whence the name ``Euclidean field theory.''

Concretely this means that in Euclidean field theory the [[local field theory|locality]] condition on the [[net of observables|net of]] [[quantum observables]] requires observables $A, B$ to [[commutator|commute]] as soon as their [[spacetime supports]] are [[disjoint subset|disjoint]] at all

\begin{displaymath}
supp(A) \cap supp(B) = \emptyset
  \;\;\;\;\Rightarrow\;\;\;\;
  [A,B] = 0
  \,.
\end{displaymath}
This is in contrast to the analogous condition in [[relativistic field theory]] whose \emph{[[causal locality]]} requires this implication only if the two [[supports]] are in addition [[spacelike]]-separated.

Equivalently this means that the [[n-point functions]] of Euclidean field theories are [[distributions of several variables]] with [[singularities]] on the [[fat diagonal]] (instead of on all of the relative [[light cones]], as for [[relativistic field theory]]). This means that Euclidean $n$-point functions [[restriction of distributions|restrict]] to [[non-singular distributions]] on the [[configuration spaces of points|configuration space of n points]], allowing to express [[correlators as differential forms on configuration spaces of points]]. Systematic discussion of [[perturbative quantum field theory|perturbative]] [[renormalization|renormalized]] Euclidean field theory from this perspective is due to \hyperlink{BergbauerBrunettiKreimer09}{Bergbauer-Brunetti-Kreimer 09}, \hyperlink{Berghoff14a}{Berghoff 14a}, \hyperlink{Berghoff14b}{Berghoff 14b}.

This Euclidean locality property applies in particular in [[statistical mechanics]], where the ``[[field (physics)|fields]]'' of the field theory are not thought of as encoding the [[spacetime]]-behaviour of [[fundamental particles]] as governed by [[quantum physics]], but instead the spatial [[expectation values]] (at any given time) of [[equilibrium]] [[thermodynamics|thermodynamic]]-processes governed by [[classical physics]].

An archetypical example of a Euclidean field theory in this thermodynamic sense is the [[Ising model]]. Generally, most [[2d conformal field theories]] considered are Euclidean field theories and (should) have the interpretation of describing [[thermodynamics|thermodynamical]] systems ``at criticality''.

However, other Euclidean [[2d CFTs]] are not necessarily regarded as thermodynamical systems, notably the [[worldsheet]]-field theories defining a [[string perturbation series]] in [[perturbative string theory]].

Another broad class of examples of Euclidean field theories are [[topological quantum field theories]] after their [[diffeomorphism]]-[[symmetry]] is partially [[gauge fixing|gauge fixed]] by a choice of [[Riemannian metric]]. The archetypical example here is [[perturbative quantum field theory|perturbative]] [[Chern-Simons theory]], see \href{Chern-Simons+theory#FeynmanPerturbationSeries}{there} for more.

\hypertarget{wick_rotation_to_relativistic_field_theory}{}\subsubsection*{{Wick rotation to Relativistic Field theory}}\label{wick_rotation_to_relativistic_field_theory}

Despite this superficially stark contrast between Euclidean and relativistic field theory, the two turn out to be tightly related to each other, at least under some conditions, in a subtle way that involves and generalizes the concept of [[analytic continuation]] from [[complex analysis]], here this is called \emph{[[Wick rotation]]}.

Roughly this says that [[propagators]] and hence [[n-point functions]] of [[relativistic field theory]] on [[Minkowski spacetime]] $\mathbb{R}^{d,1}$ (may) have [[analytic continuation]] to [[complex number|complex values]] of the [[time]]-[[coordinates]], such that replacing [[real number|real]] time with [[imaginary number|imaginary]] time turns these [[n-point functions]] into those of a [[Euclidean field theory]] on $\mathbb{R}^{d+1}$, and vice versa.

Precise formulation of the conditions that go into this [[Wick rotation]] between [[relativistic field theory]] and Euclidean field theory is the content of the [[Osterwalder-Schrader theorem]].

\hypertarget{temporal_compactification_to_thermal_relativistic_field_theory}{}\subsubsection*{{Temporal compactification to Thermal relativistic field theory}}\label{temporal_compactification_to_thermal_relativistic_field_theory}

In fact this relation goes deeper still: Under suitable conditions the Euclidean field theory not on $\mathbb{R}^{d+1}$ but on $\mathbb{R}^d \times S^1_\beta$, with the [[circle]]-[[product space|factor]] $S^1_{\beta}$ of [[length]] $\beta$, corresponds to relativistic field theory on [[Minkowski spacetime]] $\mathbb{R}^{d,1}$ in a [[vacuum state]] that represents [[thermal equilibrium]] at [[temperature]] $T \coloneqq 1/\beta$. (The previous case of Euclidean field theory on $\mathbb{R}^{d+1}$ may be thought of as the special case $\beta \to \infty$, hence $T \to 0$.)

This curious relation of [[Wick rotation]] with ``compact peridodic Euclidean time'' makes, when it applies, Euclidean field theory be a unification of [[relativistic field theory]] with [[statistical mechanics]]/[[thermodynamics]], then called \emph{thermal quantum field theory} or \emph{quantum statistical field theory} or similar.

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


\end{quote}
Notice that the evident [[symmetry breaking|breaking]] of [[Lorentz group|Lorentz symmetry]] on the right side of this correspondence is perfectly consistent with what happens on the left hand: A thermal vacuum state in Minkowski spacetime also singles out a preferred Lorentz frame.

The basic idea of this relation seems to go back to \hyperlink{Bloch58}{Bloch 58}. The physics literature often states this suggestively but informally in terms of [[path integral]]-imagery, see e.g. \hyperlink{Moore03}{Moore 03, section 1.1}.

The first precise formulation seems to be due to \hyperlink{HoeghKrohn74}{Høegh-Krohn 74} (in 1+1 dimensions) and a more comprehensive discussion in view of the [[Osterwalder-Schrader theorem]] for compact Euclidean time is due to \hyperlink{KleinLandau81}{Klein-Landau 81}.

The use of a ``periodic Euclidean time coordinate'' is also known as \emph{[[Matsubara formalism]]} (e.g. \hyperlink{LandsmanVanWert87}{Landsman-vanWert 87, section 2.3.1}) and specifically the condition that the periodicity has to be $\beta \coloneqq 1/T$ is known as the \emph{[[KMS conditions]]} for a \emph{[[KMS state]]} (For \emph{Kubo-Martin-Schwinger}, due to \hyperlink{Kubo57}{Kubo 57}, \hyperlink{MartinSchwinger59}{Martin-Schwinger 59} with its final form due to \hyperlink{HaagHugenholtzWinnink67}{Haag-Hugenholtz-Winnink 67}, see \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87, section 3.1}).

Beware that literature discussing the KMS-condition often does not make the periodicity of Euclidean time explicit, and vice versa. This is clarified in \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87, sections 2 and 3}.

General introduction to Euclidean and thermal field theory includes \hyperlink{Thoma00}{Thoma 00, section 2.2}, \hyperlink{PeetersZamaklar09}{Peeters-Zamaklar 09, section 1.3}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Ising model]]

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

\begin{itemize}%
\item [[instanton]], [[caloron]]


\item [[2d CFT]]


\item [[relativistic field theory]]


\item [[lattice gauge theory]]


\item [[infinite-temperature thermal field theory]]



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

\hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2}

General introduction to Euclidean field theory includes

\begin{itemize}%
\item [[Kasper Peeters]], [[Marija Zamaklar]], \emph{Euclidean Field Theory}, Lecture notes 2009-2011 (\href{http://maths.dur.ac.uk/users/kasper.peeters/eft.html}{web}, \href{http://maths.dur.ac.uk/users/kasper.peeters/pdf/eft.pdf}{pdf})

\end{itemize}
A systematic development of Euclidean [[renormalization|renormalized]] [[perturbative quantum field theory]] with [[correlators as differential forms on configuration spaces of points]] is described in

\begin{itemize}%
\item [[Christoph Bergbauer]], [[Romeo Brunetti]], [[Dirk Kreimer]], \emph{Renormalization and resolution of singularities} (\href{https://arxiv.org/abs/0908.0633}{arXiv:0908.0633})


\item [[Christoph Bergbauer]], \emph{Renormalization and resolution of singularities}, talks as IHES and Boston, 2009 (\href{http://www.emg.uni-mainz.de/Dateien/bergbauer.pdf}{pdf})


\item [[Marko Berghoff]], \emph{Wonderful renormalization}, 2014 (\href{http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf}{pdf}, \href{https://doi.org/10.18452/17160}{doi:10.18452/17160})


\item [[Marko Berghoff]], \emph{Wonderful compactifications in quantum field theory}, Communications in Number Theory and Physics Volume 9 (2015) Number 3 (\href{https://arxiv.org/abs/1411.5583}{arXiv:1411.5583})



\end{itemize}
\hypertarget{thermal_quantum_field_theory}{}\subsubsection*{{Thermal quantum field theory:}}\label{thermal_quantum_field_theory}

A good introduction 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}, )

\end{itemize}
The idea of Wick rotating thermal relativistic field theory to compact periodic Euclidean time 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}
A systematic discussion of the [[Osterwalder-Schrader theorem]] on [[Wick rotation]] for the case of thermal field theory/periodic Euclidean time 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}
See also

\begin{itemize}%
\item \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87, section 2}

\end{itemize}
The formulation of the KMS condition is due to

\begin{itemize}%
\item R. Kubo \emph{Statistical-Mechanical Theory of Irreversible Processes I. General Theory and Simple Applications to Magnetic and Conduction Problems}, Journal of the Physical Society of Japan 12, 570-586 1957


\item Paul C. Martin, [[Julian Schwinger]], \emph{Theory of Many-Particle Systems. I}, Physical Review 115, 1342-1373 (1959)



\end{itemize}
and found its final, now generally accepted, form in

\begin{itemize}%
\item [[Rudolf Haag]], N. M. Hugenholtz, M. Winnink, \emph{On the equilibrium states in quantum statistical mechanics}, Comm. Math. Phys. Volume 5, Number 3 (1967), 215-236 (\href{https://projecteuclid.org/euclid.cmp/1103840050}{euclid:1103840050})

\end{itemize}
Review of thermal field theory via Euclidean field theory includes

\begin{itemize}%
\item \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87, sections 2 and 3}


\item [[Klaas Landsman]], Ch.G.van Weert, \emph{Real- and imaginary-time field theory at finite temperature and density}, Physics Reports Volume 145, Issues 3–4, January 1987, Pages 141-249 ()


\item [[Jean Zinn-Justin]], \emph{Quantum Field Theory at Finite Temperature: An Introduction} (\href{https://arxiv.org/abs/hep-ph/0005272}{arXiv:hep-ph/0005272})


\item Markus H. Thoma, \emph{New Developments and Applications of Thermal Field Theory} (\href{https://arxiv.org/abs/hep-ph/0010164}{arXiv:hep-ph/0010164})


\item Yuhao Yang, \emph{An Introduction to Thermal Field Theory}, 2011 (\href{https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2011/Yuhao-Yang-Dissertation.pdf}{pdf})


\item Yi-Cheng Huang, \emph{Field Theory in the Imaginary-time Formulation} (\href{https://arxiv.org/abs/1311.1990v4}{arXiv:1311.1990v4})


\item [[Roberto Longo]], \emph{Kubo-Martin-Schwinger, Non-Equilibrium Thermal states, and Conformal Field Theory}, 2016 (\href{https://www.mat.uniroma2.it/~longo/Slides_files/Harvard16.pdf}{pdf})



\end{itemize}
Further discussion:

\begin{itemize}%
\item Christian D. Jäkel, \emph{The Reeh–Schlieder property for thermal field theories}, Journal of Mathematical Physics 41, 1745 2000 (\href{https://doi.org/10.1063/1.533208}{doi:10.1063/1.533208})


\item Christian D. Jäkel, Florian Robl, \emph{The relativistic KMS condition for the thermal $n$-point functions of the $P(\phi)_2$ model} (\href{https://arxiv.org/abs/1103.3609}{arXiv:1103.3609})



\end{itemize}
With an eye towards [[lattice gauge theory]]:

\begin{itemize}%
\item Guy Moore, \emph{Informal lectures on lattice gauge theory}, 2003 (\href{https://theorie.ikp.physik.tu-darmstadt.de/qcd/moore/latt_lectures.pdf}{pdf})

\end{itemize}
In application to [[black hole thermodynamics]]:

\begin{itemize}%
\item \hyperlink{FullingRuijsenaars87}{Fulling-Ruijsenaars 87, section 4}


\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})


\item [[Valeri Frolov]], Andrei Zelnikov, section F4.4 of \emph{Introduction to black hole physics}, Oxford 2011



\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}
The expansion of thermal field theory around the [[infinite-temperature thermal field theory|infinite-temperature-limit]] (i.e. around $\beta = 1/T = 0$, i.e. [[KK-reduction]] in compact/periodic Euclidean time) is discussed in

\begin{itemize}%
\item [[Paul Ginsparg]], \emph{First and second order phase transitions in gauge theories at finite temperature}, Nuclear Physics B Volume 170, Issue 3, 15 December 1980, Pages 388-408 ()


\item Thomas Appelquist, Robert D. Pisarski, \emph{High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics}, Phys. Rev. D 23, 2305 (1981) (\href{https://doi.org/10.1103/PhysRevD.23.2305}{doi:10.1103/PhysRevD.23.2305})


\item Sudhir Nadkarni, \emph{Dimensional reduction in finite-temperature quantum chromodynamics}, Phys. Rev. D 27, 917 (1983) (\href{https://doi.org/10.1103/PhysRevD.27.917}{doi:10.1103/PhysRevD.27.917})


\item Sudhir Nadkarni, \emph{Dimensional reduction in finite-temperature quantum chromodynamics. II}, Phys. Rev. D 38, 3287 (1988) (\href{https://doi.org/10.1103/PhysRevD.38.3287}{doi:10.1103/PhysRevD.38.3287})


\item Alexander N Jourjine, \emph{Quantum field theory in the infinite temperature limit}, Annals of Physics Volume 155, Issue 2, July 1984, Pages 305-332 ()


\item [[Klaas Landsman]], \emph{Limitations to dimensional reduction at high temperature}, Nuclear Physics B Volume 322, Issue 2, 14 August 1989, Pages 498-530 ()


\item T. Reisz, \emph{Realization of dimensional reduction at high temperature}, Z. Phys. C - Particles and Fields (1992) 53: 169 (\href{https://doi.org/10.1007/BF01483886}{doi:10.1007/BF01483886})


\item Eric Braaten, \emph{Solution to the Perturbative Infrared Catastrophe of Hot Gauge Theories}, Phys. Rev. Lett. 74, 2164 (1995) (\href{https://doi.org/10.1103/PhysRevLett.74.2164}{doi:10.1103/PhysRevLett.74.2164})


\item K. Kajantiea, M. Laine, K. Rummukainen, M. Shaposhnikov, \emph{Generic rules for high temperature dimensional reduction and their application to the standard model}, Nuclear Physics B Volume 458, Issues 1–2, 1 January 1996, Pages 90-136 ()



\end{itemize}
and specifically with an eye to discussion of the [[quark-gluon plasma]] in

\begin{itemize}%
\item Jean-Paul Blaizot, Edmond Iancu, Anton Rebhan, \emph{Thermodynamics of the high temperature quark gluon plasma}, Quark–Gluon Plasma 3, pp. 60-122 (2004) (\href{https://arxiv.org/abs/hep-ph/0303185}{arXiv:hep-ph/0303185}, \href{http://inspirehep.net/record/615570}{spire:615570})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Thermal_quantum_field_theory}{Thermal quantum field theory}}


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Matsubara_frequency}{Matsubara frequency}}


\item Wolfram Math World, \emph{\href{http://mathworld.wolfram.com/KMSCondition.html}{KMS condition}}



\end{itemize}
[[!redirects Euclidean field theories]]

[[!redirects thermal quantum field theory]] [[!redirects thermal quantum field theories]]

[[!redirects thermal field theory]] [[!redirects thermal field theories]]

[[!redirects Matsubara formalism]] [[!redirects Matsubara formalisms]]

[[!redirects Matsubara frequency]] [[!redirects Matsubara frequencies]]

[[!redirects KMS condition]] [[!redirects KMS conditions]]



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