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\section*{Dirac interaction picture}

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

\hypertarget{algebraic_qunantum_field_theory}{}\paragraph*{{Algebraic Qunantum Field Theory}}\label{algebraic_qunantum_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{InQuantumMechanics}{In quantum mechanics}\dotfill \pageref*{InQuantumMechanics} \linebreak
\noindent\hyperlink{in_quantum_field_theory}{In quantum field theory}\dotfill \pageref*{in_quantum_field_theory} \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}

The ``interaction picture'' in [[quantum physics]] is a way to decompose solutions to the [[Schrödinger equation]] and more generally the construction of [[quantum field theories]] into a [[free field theory]]-part and the [[interaction]] part that acts as a [[perturbation]] of the free theory. Therefore the interaction picture lends itself to the construction of [[perturbative quantum field theory]], and in fact the only mathematically rigorous such construction scheme that is known, namely \emph{[[causal perturbation theory]]}, proceeds this way.

[[dynamics|Dynamics]] in [[physics]] affects both [[observables]] and, dually, [[states]]; this is most well known in [[quantum physics]] but applies equally well to [[classical physics]]. The different ``pictures'' of physics differ in how the dynamics is explicitly formalized:

\begin{itemize}%
\item In the [[Schrödinger picture]], states are propagated through [[time]], while observables are held fixed; the axiomatic formalization of this is given by [[cobordism category]] [[representation]]s in [[FQFT]].


\item In the [[Heisenberg picture]], the dependence of observables on time (or more generally [[spacetime]]) is encoded, while the state is held fixed; the axiomatic formalization of this is given by the [[Haag–Kastler axioms]] of [[AQFT]].


\item The \textbf{Dirac (interaction) picture} is a mixture of these two approaches: dynamics is split into a free (or otherwise solvable) part and an [[interaction]] (often then treated as a [[perturbation theory|perturbation]]); one of these is taken to affect the states, the other the observables.



\end{itemize}
The pictures are named after those physicists who first used or popularised these approaches to quantum physics.

\hypertarget{InQuantumMechanics}{}\subsubsection*{{In quantum mechanics}}\label{InQuantumMechanics}

In [[quantum mechanics]], let $\mathcal{H}$ be some [[Hilbert space]] and let

\begin{displaymath}
H = H_{free} + V
\end{displaymath}
be Hermitian operator, thought of as a [[Hamiltonian]], decomposed as the [[sum]] of a free part ([[kinetic energy]]) and an interaction part ([[potential energy]]).

For example for a [[non-relativistic particle]] of [[mass]] $m$ propagating on the [[line]] subject to a [[potential energy]] $V_{pot} \colon \mathbb{R} \to \mathbb{R}$, then $\mathcal{H} = L^2(X)$ is the Hilbert space space of [[square integrable functions]] and

\begin{displaymath}
H 
   =  
  \underset{H_{free}}{\underbrace{\tfrac{- \hbar^2}{2m} \frac{\partial^2}{\partial^2 x}}}
    + 
  V
  \,,
\end{displaymath}
where $V = V_{pot}(x)$ is the operator of multiplying square integrable functions with the given potential energy function.

Now for

\begin{displaymath}
\itexarray{
    \mathbb{R} &\overset{\vert \psi (-)\rangle }{\longrightarrow}& \mathcal{H}
   \\
   t &\mapsto& \vert \psi(t) \rangle 
  }
\end{displaymath}
a one-parameter family of [[quantum states]], the [[Schrödinger equation]] for this state reads

\begin{displaymath}
\frac{d}{d t} \vert \psi(t) \rangle
  \;=\;
  \tfrac{1}{i \hbar} H \vert \psi\rangle
  \,.
\end{displaymath}
It is easy to solve this [[differential equation]] formally via its [[Green function]]: for $\vert \psi \rangle \in \mathcal{H}$ any state, then the unique solution $\vert \psi(-) \rangle$ to the Schr\"o{}dinger equation subject to $\vert \psi(0) \rangle = \vert \psi \rangle$ is

\begin{displaymath}
\vert \psi(t)\rangle_S \coloneqq \exp( \tfrac{t}{i \hbar} H ) \vert \psi \rangle
  \,.
\end{displaymath}
(One says that this is the solution ``in the [[Schrödinger picture]]'', whence the subscript.)

However, if $H$ is sufficiently complicated, it may still be very hard to extract from this expression a more explicit formula for $\vert \psi(t) \rangle$, such as, in the example of the free particle on the line, its expression as a function (``[[wave function]]'') of $x$ and $t$.

But assume that the analogous expression for $H_{free}$ alone is well understood, hence that the operator

\begin{displaymath}
U_{S,free}(t_1, t_2) \coloneqq \exp\left({\tfrac{t_2 - t_1}{i \hbar} H_{free}}\right)
\end{displaymath}
is sufficiently well understood. The ``[[interaction picture]]'' is a way to decompose the Schr\"o{}dinger equation such that its dependence on $V$ gets separated from its dependence on $H_{free}$ in a way that admits to treat $H_{int}$ in [[perturbation theory]].

Namely define analogously

\begin{equation}
\begin{aligned}
    \vert \psi(t)\rangle_I
     &\coloneqq  
     \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \vert \psi(t)\rangle_S
    \\
    & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) 
        \exp\left({ \tfrac{+ t}{i \hbar} H} \right)\vert \psi \rangle 
    \\
    & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) 
        \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right)\vert \psi \rangle   \end{aligned} 
  \,.
\label{StateInTheInteractionPicture}\end{equation}
This is called the solution of the Schr\"o{}dinger equation ``in the [[interaction picture]]'', whence the subscript. Its definition may be read as the result of propagating the actual solution $\vert \psi(-)\rangle_S$ at time $t$ back to time $t = 0$, but using just the free Hamiltonian, hence with ``the interaction switched off''.

Notice that if the operator $V$ were to commute with $H_{free}$ (which it does not in all relevant examples) then we would simply have $\vert \psi(t)\rangle_I = \exp( \tfrac{t}{i \hbar } V ) \vert \psi\rangle$, hence then the solution \eqref{StateInTheInteractionPicture} in the interaction picture would be the result of ``propagating'' the initial conditions using \emph{only} the interaction. Now since $V$ may not be assumed to commute with $H_{free}$, the actual form of $\vert \psi(-) \rangle_{I}$ is more complicated. But \emph{infinitesimally} it remains true that $\vert \psi(-)\rangle_I$ is propagated this way, not by the plain operator $V$, though, but by $V$ viewed in the [[Heisenberg picture]] of the free theory. This is the content of the differential equation \eqref{DifferentialEquationInInteractionPicture} below.

But first notice that this will indeed be useful: If an explicit expression for the ``state in the [[interaction picture]]'' \eqref{StateInTheInteractionPicture} is known, then the assumption that also the operator $\exp\left({\tfrac{t}{i \hbar} H_{free}}\right)$ is sufficiently well understood implies that the actual solution

\begin{displaymath}
\vert \psi(t) \rangle_S = \exp\left({\tfrac{t}{i \hbar} H_{free}}\right) \vert \psi(t) \rangle_I
\end{displaymath}
is under control. Hence the question now is how to find $\vert \psi(-)\rangle_I$ given its value at some time $t$. (It is conventional to consider this for $t \to \pm \infty$, see \eqref{SMatrixInQuantumMechanics} below.)

Now it is clear from the construction and using the [[product law]] for [[differentiation]], that $\vert \psi(-)\rangle_S$ satisfies the following [[differential equation]]:

\begin{equation}
\frac{d}{d t} \vert \psi(t) \rangle
  \;=\;
  V_I(t) \vert \psi(t)\rangle_I
  \,,
\label{DifferentialEquationInInteractionPicture}\end{equation}
where

\begin{displaymath}
V_I(t)
    \coloneqq 
  \exp\left( -\tfrac{t}{i \hbar} H_{free} \right)
    V
  \exp\left( +\tfrac{t}{i \hbar} H_{free} \right)
\end{displaymath}
is known as the [[interaction]] term $V$ ``viewed in the interaction picture''. But in fact this is just $V$ ``viewed in the [[Heisenberg picture]]'', but for the \emph{free} theory. By our running assumption that the free theory is well understood, also $V_I(t)$ is well understood, and hence all that remains now is to find a sufficiently concrete solution to equation \eqref{DifferentialEquationInInteractionPicture}. This is the heart of working in the interaction picture.

Solutions to equations of the ``[[parallel transport]]''-type such as \eqref{DifferentialEquationInInteractionPicture} are given by [[time-ordered product|time-ordering]] of Heisenberg picture operators, denoted $T$, applied to the naive exponential solution as above. This is known as the \emph{[[Dyson formula]]}:

\begin{displaymath}
\vert \psi(t)\rangle_I
  \;=\;
  T\left(
    \exp\left(
      \int_{t_0}^t
        V_I(t)
        \tfrac{d t}{i \hbar}
    \right)
  \right)
  \vert \psi(t_0)\rangle
  \,.
\end{displaymath}
Here [[time-ordered product|time-ordering]] means

\begin{displaymath}
T( V_I(t_1) V_I(t_2) )
  \;\coloneqq\;
  \left\{
    \itexarray{
      V_I(t_1) V_I(t_2) &\vert& t_1 \geq t_2
      \\
      V_I(t_2) V_I(t_1) &\vert& t_2 \geq t_2
    }
  \right.
  \,.
\end{displaymath}
(This is abuse of notation: Strictly speaking time ordering acts on the [[tensor algebra]] spanned by the $\{V_I(t)\}_{t \in \mathbb{R}}$ and has to be \emph{folllowed} by taking tensor products to actual products. )

In applications to [[scattering]] processes one is interest in prescribing the [[quantum state]]/[[wave function]] far in the past, hence for $t \to - \infty$, and computing its form far in the future, hence for $t \to \infty$.

The operator that sends such ``asymptotic ingoing-states'' $\vert \psi(-\infty) \rangle_I$ to ``asymptic outgoing states'' $\vert \psi(+ \infty) \rangle_I$ is hence the [[limit of a sequence|limit]]

\begin{equation}
S 
  \;\coloneqq\;
  \underset{t \to \infty}{\lim}
  T\left(
    \exp\left(
      \int_{-t}^t
        V_I(t)
        \tfrac{d t}{i \hbar}
    \right)
  \right)  
  \,.
\label{SMatrixInQuantumMechanics}\end{equation}
This limit (if it exists) is called the \emph{scattering matrix} or \emph{S-matrix}, for short.

\hypertarget{in_quantum_field_theory}{}\subsubsection*{{In quantum field theory}}\label{in_quantum_field_theory}

In [[perturbative quantum field theory]] the broad structure of the interaction picture in quantum mechanics (\hyperlink{InQuantumMechanics}{above}) remains a very good guide, but various technical details have to be generalized with due care:

\begin{enumerate}%
\item The algebra of operators in the [[Heisenberg picture]] of the free theory becomes the \emph{[[Wick algebra]]} of the [[free field theory]] (taking into account ``normal ordering'' of field operators) defined on \emph{[[microcausal functionals]]} built from [[operator-valued distributions]] with constraints on their [[wave front set]].


\item The [[time-ordered products]] in the [[Dyson formula]] have to be refined to [[causal locality|causally ordered]] products and the resulting product at coincident points has to be defined by [[point-extension of distributions]] -- the freedom in making this choice is the [[renormalization]] freedom (``conter-terms'').


\item The sharp interaction cutoff in the [[Dyson formula]] that is hidden in the integration over $[t_0,t]$ has to be smoothed out by [[adiabatic switching]] of the interaction (making the whole S-matrix an [[operator-valued distribution]]).



\end{enumerate}
Together these three point are taken care of by the axiomatization of the ``[[adiabatic switching|adiabatically switched]] [[S-matrix]]'' according to \textbf{[[causal perturbation theory]]}.

The analogue of the limit $t \to \infty$ in the construction of the [[S-matrix]] (now: [[adiabatic limit]]) in general does not exist in field theory (``infrared divergencies''). But in fact it need not be taken: The field algebra in a bounded region of [[spacetime]] may be computed with any adiabatic switching that is constant on this region. Moreover, the algebras assigned to regions of spacetime this way satisfy [[causal locality]] by the causal ordering in the construction of the S-matrix. Therefore, even without taking the adiabtic limit in [[causal perturbation theory]] one obtains a field theory in the form of a \emph{[[local net of observables]]}. This is the topic of \textbf{[[locally covariant perturbative quantum field theory]]}.

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

\begin{itemize}%
\item [[interaction]]


\item [[interacting field algebra]]


\item [[interacting field theory]]


\item [[S-matrix]]


\item [[Møller operator]]



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

For instance

\begin{itemize}%
\item [[Eberhard Zeidler]], section 7.19.3 of \emph{Quantum field theory. A bridge between mathematicians and physicists -- volume I} Springer (2009) (\href{http://www.mis.mpg.de/zeidler/qft.html}{web})

\end{itemize}
[[!redirects Dirac picture]] [[!redirects interaction picture]] [[!redirects Dirac interaction picture]] [[!redirects Dirac (interaction) picture]]



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