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\section*{adiabatic switching}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{adiabatic_switching}{Adiabatic switching}\dotfill \pageref*{adiabatic_switching} \linebreak
\noindent\hyperlink{adiabatic_limit}{Adiabatic limit}\dotfill \pageref*{adiabatic_limit} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{adiabatic_switching}{}\subsubsection*{{Adiabatic switching}}\label{adiabatic_switching}

In [[perturbative quantum field theory]], the term \emph{adiabatic switching} refers to considering a smooth transition between vanishing and non-vanishing [[interaction]] [[coupling constant|coupling]]: the interactions is slowly, hence (borrowing a term from [[thermodynamics]]) ``adiabatically'', switched on or off. This is mostly a mathematical device, not meant to directly reflect a physical situation of changing coupling, but it does serve to construct physical quantities. This is closely related to the role of [[operator-valued distributions]] which are quantities that give well defined [[linear operators]] (hence [[quantum observables]]) only when evaluated on any [[bump function]].

Originally adiabatic switching was considered (\hyperlink{LippmannSchwinger50}{Lippmann-Schwinger 50}) only in the [[time]]-direction (for a fixed choice of time on [[Minkowski spacetime]]) by multiplying the [[interaction]] term of the [[Lagrangian density]]/[[Hamiltonian]] by the [[exponential]] $\exp(- \epsilon {\Vert t \Vert})$ (for $\epsilon \in (0,\infty)$ a [[positive number|positive]] [[real number]] and for ${\Vert t\Vert}$ the [[absolute value]] of the time [[coordinate]]). Review is for instance in (\hyperlink{Strocchi13}{Strocchi 13, section 6.3}).

Using this, the \emph{Gell-Mann and Low formula} (\hyperlink{GellMannLow51}{Gell-Mann \& Low 51}, see \hyperlink{Molinari06}{Molinari 06}) expresses the [[eigenstates]] $\vert \psi \rangle$ of an interacting [[Hamiltonian]] $H = H_{free} + H_{int}$ in terms of the eigenstates $\vert \Psi_{free} \rangle$ of the free Hamiltonian by the ``adiabatic limit''

\begin{displaymath}
\vert \Psi^{\pm}_{int} \rangle 
  \;\propto\;
  \underset{\epsilon \to 0}{\lim}
  S_\epsilon(0, \pm \infty)
  \vert \Psi_{free} \rangle
\end{displaymath}
(if the [[limit of a sequence|limit]] exists) where $S_\epsilon$ denotes the [[S-matrix]] of the adiabatically switched Hamiltonian $H_\epsilon \coloneqq H_{free} + e^{- \epsilon {\Vert t\Vert}}H_{int}$.

More generally, one may consider adiabatic switching taking place not just in time, but in all of [[spacetime]]. This the basis of [[causal perturbation theory]] and [[locally covariant perturbative quantum field theory]]:

In the construction of [[perturbative quantum field theory]] via the method of [[causal perturbation theory]] the [[interaction]] terms $L_{int}$ used in the mathematical construction of the [[S-matrix]] are multiplied with a ``[[coupling constant]]'' $g$ which is in fact taken to be a [[smooth function]] of [[compact support]] on [[spacetime]], hence a [[bump function]]:

\begin{displaymath}
L_g = L_{free} + g L_{int}
  \,.
\end{displaymath}
This means that the the [[interaction]] as modeled by the [[S-matrix]]

\begin{displaymath}
S_g \coloneqq T \exp( \tfrac{i}{\hbar} \int_{X} g :L_{int}(x): )
\end{displaymath}
is non-trivial only on a [[compact topological space|compact]] [[subspace]] of [[spacetime]], towards its boundary it smoothly drops to zero. Hence outside this region the interaction is ``switched off''.

Since the actual interactions in [[physics]] are of course \emph{not} ``switched off'' anywhere, the use of an adiabatic switching is just an intermediate mathematical step. Originally in (\hyperlink{EpsteinGlaser73}{Epstein-Glaser 73}) the idea was that after having constructed the [[S-matrix]] for any adiabatic switching $g$, the [[limit of a sequence|limit]] (``adiabatic limit'') $g \to 1$ had to be taken to remove the switching in the end. Failure of this limit to exist is interpreted as ``[[infrared divergency]]'' of the [[perturbative quantum field theory]] (since the divergency comes from large scales, hence long [[wavelength]]).

But as observed in (\hyperlink{IlinSlavnov78}{Il'in-Slavnov 78}) and rediscovered in (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00}), an adiabatic switching map that is unity on a globally hyperbolic sub-spacetime $O \subset X$ is sufficient to compute the perturbative [[interacting field algebra]], hence the [[algebra of quantum observables]] $A(O)$ on that subspace, and the collection of all of these as $O$ ranges forms a [[causally local net of observables]] which fully captures the [[quantum field theory]] in the sense of the [[Haag-Kastler axioms]] (\href{S-matrix#PerturbativeQuantumObservablesIsLocalnet}{this prop.}). This perspective is now known as \emph{[[locally covariant algebraic quantum field theory]]}.

\hypertarget{adiabatic_limit}{}\subsubsection*{{Adiabatic limit}}\label{adiabatic_limit}

The [[limit of a sequence|limit]] of the [[perturbative S-matrix]] as the [[adiabatic switching]] is removed (if it exists) is called the \emph{adiabatic limit} or \emph{strong adiabatic limit}.

If one just asks that the corresponding limit exists for the [[n-point functions]] one speaks of a \emph{weak adiabatic limit}.

Even with the adiabatically switched S-matrix elements (not taking a limit) the [[local net of quantum observables]] is well defined (\href{S-matrix#PerturbativeQuantumObservablesIsLocalnet}{this prop.}), this is hence a [[functor]]

\begin{displaymath}
\mathcal{O} \mapsto \mathcal{A}(\mathcal{O})
\end{displaymath}
that assigns [[algebras of observables]] to [[causally closed subsets]] of [[spacetime]]. The [[colimit]] algebra

\begin{displaymath}
\mathcal{A} \coloneqq \underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim} \mathcal{A}(\mathcal{O})
\end{displaymath}
over this [[functor]] (in the sense of [[category theory]]) always exists. This is also called the \emph{algebraic adiabatic limit}.

(See around \hyperlink{Duch17}{Duch 17, section 4} for review of strong, weak and algebraic adiabaitc limit; and \hyperlink{Duch17}{Duch 17, chapter II} for results on the weak adiabatic limit)

Here

\begin{enumerate}%
\item the \emph{algebraic adiabatic} limit defines the \emph{[[quantum observables]]} in the limit;


\item the weak adiabatic limit may serve to define also the \emph{[[state on a star-algebra|states]]}, hence the [[interacting vacuum]] (\hyperlink{Duch17}{Duch 17, p. 113-114}).



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

The concept of adiabatic switching in the time direction was introduced in

\begin{itemize}%
\item B. A. Lippmann, [[Julian Schwinger]], Phys. Rev. 79, 469 (1950)

\end{itemize}
reviewed for instance in

\begin{itemize}%
\item [[Franco Strocchi]], sectioon 6.3 of \emph{An Introduction to Non-Perturbative Foundations of Quantum Field Theory}, Oxford University Press, 2013

\end{itemize}
and the corresponding formula for the interacting eigenstates in terms of the free ones is due to

\begin{itemize}%
\item [[Murray Gell-Mann]], F. Low, \emph{Bound states in quantum field theory} Phys. Rev. 84, 350 (1951)


\item [[Murray Gell-Mann]], M. L. Goldberger, Phys. Rev. 91 398 (1953)



\end{itemize}
see

\begin{itemize}%
\item Luca Guido Molinari, \emph{Another proof of Gell-Mann and Low's theorem}, Journal of Mathematical Physics 48, 052113, 2007 (\href{https://arxiv.org/abs/math-ph/0612030}{arXiv:math-ph/0612030})


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Gell-Mann_and_Low_theorem}{Gell-Mann and Low theorem}}



\end{itemize}
The generalization to switching in all space-time directions was considered for the construction of [[causal perturbation theory]] in

\begin{itemize}%
\item [[Henri Epstein]] and [[Vladimir Glaser]], \emph{[[The Role of locality in perturbation theory]]}, Annales Poincar\'e{} Phys. Theor. A 19 (1973) 211.

\end{itemize}
The observation that this in fact makes causal perturbation theory a tool for constructing [[local nets of observables]] for [[locally covariant perturbative quantum field theory]] is due to

\begin{itemize}%
\item V. A. Il'in and D. S. Slavnov, \emph{Observable algebras in the S-matrix approach}, Theor. Math. Phys. 36 (1978) 32.


\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], \emph{Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds}, Commun. Math. Phys. 208 : 623-661,2000 (\href{https://arxiv.org/abs/math-ph/9903028}{math-ph/9903028})



\end{itemize}
The term ``algebraic adiabatic limit'' for the resulting [[local net of observables]] (or its [[inductive limit]]) appears in

\begin{itemize}%
\item [[Klaus Fredenhagen]], [[Falk Lindner]], p. 7 of \emph{Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics}, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014 (\href{https://arxiv.org/abs/1306.6519}{arXiv:1306.6519})

\end{itemize}
The weak adiabatic limit in [[causal perturbation theory]] for massive fields was shown to exists in

\begin{itemize}%
\item \hyperlink{EpsteinGlaser73}{Epstein-Glaser 73}

\end{itemize}
Extension of this result to [[quantum electrodynamics]] and [[phi{\tt \symbol{94}}4 theory]] was given in

\begin{itemize}%
\item P. Blanchard and R. Seneor, \emph{Green’s functions for theories with massless particles (in perturbation theory)}, Ann. Inst. H. Poincaré Sec. A 23 (2), 147–209 (1975) (\href{http://www.numdam.org/item?id=AIHPA_1975__23_2_147_0}{Numdam})

\end{itemize}
See also

\begin{itemize}%
\item [[Günter Scharf]], section 3.11 of \emph{[[Finite Quantum Electrodynamics -- The Causal Approach]]}, Berlin: Springer-Verlag, 1995, 2nd edition

\end{itemize}
Further extension of the result is due to

\begin{itemize}%
\item [[Paweł Duch]], \emph{Massless fields and adiabatic limit in quantum field theory} (\href{https://arxiv.org/abs/1709.09907}{arXiv:1709.09907})

\end{itemize}
[[!redirects adiabatic switchings]]

[[!redirects Gell-Mann and Low formula]]

[[!redirects adiabatic limit]] [[!redirects adiabatic limits]]

[[!redirects algebraic adiabatic limit]] [[!redirects algebraic adiabatic limits]]

[[!redirects strong adiabatic limit]] [[!redirects strong adiabatic limits]]

[[!redirects weak adiabatic limit]] [[!redirects weak adiabatic limits]]



\end{document}