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\begin{document}

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\section*{Aharonov-Bohm effect}

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

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

[[!include physicscontents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \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 \emph{Aharonov-Bohm effect} is a configuration of the [[electromagnetic field]] which has vanishing electric/magnetic [[field strength]] (vanishing [[Faraday tensor]] $F = 0$) and but is nevertheless non-trivial, in that the [[vector potential]] $A$ is non-trivial. Since the vector potential affects the [[phase|quantum mechanical phase]] on the [[wavefunction]] of [[electrons]] moving in an electromagnetic field, in such a configuration classical physics sees no effect, but the phase of quantum particles, which may be observed as a [[quantum interference|interference]] pattern on some screen, does.

More technically, a configuration of the [[electromagnetic field]] is generally given by a [[circle]]-[[principal connection]] and an Aharonov-Bohm configuration is one coming from a [[flat connection]], whose [[curvature]]/[[field strength]] hence vanishes, but which is itself globally non-trivial. This is only possible on [[spaces]] ([[spacetimes]]) which have a non-trivial [[fundamental group]], hence for instance it doesn't happen on [[Minkowski spacetime]].

In practice one imagines an idealized [[electric current]]-carrying [[solenoid]] in [[Euclidean space]]. Away from the solenoid itself the [[magnetic field]] produced by it gives such a configuration.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

Let $\mathbb{R}^2 - \{0\}$ be the [[plane]] with the origin removed, and consider the space $(\mathbb{R}^2 - \{0\}) \times \mathbb{R}$ (thought of as 3d [[Cartesian space]] with the z-axis removed) and [[spacetime]] $(\mathbb{R}^2 - \{0\}) \times \mathbb{R}^2$ (thought of as the previous configuration statically moving in time).

For the following argument only the topological structure of the space matters, and nothing needs to explicitly depend on the $z$-[[coordinate]] and the time-coordinate, so for notational simplicity we may suppress these and consider just $\mathbb{R}^2 - \{0\}$.

On this space minus the x-axis consider the [[polar coordinates]] $(\phi,r)$ with

\begin{displaymath}
x = r cos(\phi)\,,\;\;\; y = r sin(\phi)
  \,.
\end{displaymath}
Accordingly we have the [[differential 1-forms]]

\begin{displaymath}
\mathbf{d}x = cos(\phi)\mathbf{d}r - r sin(\phi) \mathbf{d}\phi
\end{displaymath}
\begin{displaymath}
\mathbf{d}y = sin(\phi)\mathbf{d}r + r cos(\phi) \mathbf{d}\phi
\end{displaymath}
hence

\begin{displaymath}
\begin{aligned}
    \mathbf{d}\phi 
     & = 
     \frac{1}{r}cos(\phi)\mathbf{d}y - \frac{1}{r}sin(\phi) \mathbf{d}x
     \\
     & = 
     \frac{1}{r^2} x \mathbf{d}y - \frac{1}{r^2} y \mathbf{d}x
  \end{aligned}
  \,.
\end{displaymath}
Here the expression on the right extends smoothly also to the $x$-axis and this extension we call

\begin{displaymath}
\theta \coloneqq 
  \frac{1}{r^2} x \mathbf{d}y - \frac{1}{r^2} y \mathbf{d}x
  \;\;
  \in \Omega^1(\mathbb{R}^2 - \{0\})
  \,.
\end{displaymath}
From the way this is constructed it is clear that $\theta$ is a closed differential form

\begin{displaymath}
\mathbf{d}\theta = 0
  \,.
\end{displaymath}
However, on $\mathbb{R}^2 - \{0\}$ this is not an exact form. In other words, if one regards $\theta$ as the [[vector potential]] being the configuration of an [[electromagnetic field]]

\begin{displaymath}
A \coloneqq \theta
\end{displaymath}
then:

\begin{enumerate}%
\item the [[field strength]] vanishes $F = \mathbf{d}A = 0$;


\item but there is no [[gauge transformation]] relating $A$ to the trivial field configuration.



\end{enumerate}
This is possible because $\mathbb{R}^2 - \{0\}$ is not [[simply connected topological space|simply connected]] and hence the [[Poincaré lemma]] does not apply.

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

\begin{itemize}%
\item [[fiber bundles in physics]]


\item [[Dirac charge quantization]], [[magnetic monopole]]



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

\begin{itemize}%
\item L. Mangiarotti, [[Gennadi Sardanashvily]], section 6.6 of \emph{Connections in Classical and Quantum Field Theory}, World Scientific, 2000

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect}{Aharonov-Bohm effect}}

\end{itemize}


\end{document}