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\begin{document}

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\section*{classical anomaly}

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

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

[[!include physicscontents]]

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{GalileoAnomaly}{Mass-anomaly of Galilean action on non-relativistic mechanics}\dotfill \pageref*{GalileoAnomaly} \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}

In the context of [[classical mechanics]], a \emph{classical anomaly} is a [[central extension]] of a [[Noether current]] algebra (e.g. \hyperlink{Toppan01}{Toppan 01}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In terms of [[symplectic geometry]] this means the following (\hyperlink{Arnold78}{Arnold 78, appendix 5.A}). Given a [[Lie group]] $G$ acting by [[symplectomorphisms]] on a [[phase space]] [[symplectic manifold]] $(X,\omega)$, this [[symmetry]] has a \emph{classical anomaly} if it does not lift to a genuine $G$-[[Hamiltonian action]], but only to a [[projective representation|projective]] $G$-[[Hamiltonian action]], hence to a [[Hamiltonian action]] of a [[central extension]] $\widehat G$ of $G$. Specifically, the \emph{classical anomaly} of the original symplectic $G$-action is the 2-[[cocycle]] which classifies this extension.

If the original $G$-action is by [[flows]] of [[Hamiltonian vector fields]] (just not with explicitly chosen [[Hamiltonians]]), then there is a universal classical anomaly given by the [[pullback]] of the [[quantomorphism group]] [[extension]]

\begin{displaymath}
\itexarray{
     \widehat G &\longrightarrow& QuantMorph(X,\omega)
     \\
     \downarrow &(pb)& \downarrow
     \\
    G &\stackrel{}{\longrightarrow}& HamSympl(X,\omega) 
  }
  \,.
\end{displaymath}
This $\widehat G$ is the [[Heisenberg group]] of the given $G$-action (See also \hyperlink{FiorenzaRogersSchreiber13}{Fiorenza-Rogers-Schreiber 13} for discussion in [[higher prequantum geometry]]).

Notice that on the infinitesimal level of [[Lie algebras]], using that the Lie algebra of the [[quantomorphism group]] is the [[Poisson Lie algebra]] $\mathfrak{pois}(X,\omega)$, this means that an infinitesimal action of a Lie algebra $\mathfrak{g}$ via [[Hamiltonian vector fields]] on $X$ has a classical anomaly if it lifts to an action with consistently chosen [[Hamiltonians]] -- also called a [[moment map]] -- only after passing to a central [[Lie algebra extension]]

\begin{displaymath}
\itexarray{
     \widehat \mathfrak{g} &\longrightarrow& \mathfrak{pois}(X,\omega)
     \\
     \downarrow && \downarrow
     \\
     \mathfrak{g} &\stackrel{}{\longrightarrow}& \Gamma(T X)_{\omega}
  }
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{GalileoAnomaly}{}\subsubsection*{{Mass-anomaly of Galilean action on non-relativistic mechanics}}\label{GalileoAnomaly}

The canonical [[Galileo group]]-[[action]] on the [[phase space]] of non-relativistic [[classical mechanics]] has a classical anomaly, given by a [[group cohomology|group 2-cocycle]] proportional to the [[mass]] of the system, the \emph{[[Galileo 2-cocycle]]} (e.g. \hyperlink{ChenShawYen85}{Chen-Shaw-Yen 85}, \hyperlink{AzcarragaIzquierdo95}{Azc\'a{}rraga-Izquierdo 95} \hyperlink{Marle14}{Marle 14}).

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

\begin{itemize}%
\item [[quantum anomaly]]


\item [[fiber bundles in physics]]



\end{itemize}
[[!include geometric quantization extensions - table]]

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

A textbook discussion of the concept is (without the terminology yet) is in appendix 5.A of

\begin{itemize}%
\item [[Vladimir Arnol'd]], \emph{[[Mathematical methods of classical mechanics]]}, Graduate texts in Mathematics 60 (1978)

\end{itemize}
A discussion under the term ``classical [[central charge]]'' is in

\begin{itemize}%
\item J. D. Brown, [[Marc Henneaux]], \emph{Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity}, Commun. Math. Phys. 104 (1986) 207. (\href{http://dx.doi.org/10.1007/BF01211590}{web})

\end{itemize}
For a list of some examples and further pointers to the (historical) literature, see

\begin{itemize}%
\item Francesco Toppan, \emph{On anomalies in classical mechanical systems}, Journal of Nonlinear Mathematical Physics Volume 8, Number 3 (2001), 518--533 ([[ToppanClassicalAnomaly.pdf:file]])

\end{itemize}
See also

\begin{itemize}%
\item [[Glenn Barnich]], C\'e{}dric Troessaert, \emph{Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity} (\href{http://arxiv.org/abs/1309.0794}{arXiv:1309.0794})

\end{itemize}
Discussion in terms of [[Heisenberg group]] [[group extensions|extensions]] and generalization to [[higher symplectic geometry]] is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory]]} (\href{http://arxiv.org/abs/1304.0236}{arXiv:1304.0236})

\end{itemize}
Discussion in the context of formalization of [[classical field theory]] in [[cohesive homotopy theory]] is in

\begin{itemize}%
\item [[Urs Schreiber]], section 2.14 of \emph{[[schreiber:Classical field theory via Cohesive homotopy types]]} (\href{https://dl.dropboxusercontent.com/u/12630719/classicalinhigher.pdf}{pdf})

\end{itemize}
The example of the [[Galileo 2-cocycle]] is discussed for instance in

\begin{itemize}%
\item Chen, Shaw, Yen, \emph{An example of a 2-cocycle}, \href{http://psroc.phys.ntu.edu.tw/cjp/v23/4.pdf}{pdf}


\item Charles-Michel Marle, \emph{The manifold of Motions and the total mass of a mechanical system}, 2014 (\href{http://charles-michel.marle.pagesperso-orange.fr/diaporamas/ManifoldMotionsMass.pdf}{pdf})



\end{itemize}
and in the broader context of [[WZW model]] terms in

\begin{itemize}%
\item [[José de Azcárraga]], Jos\'e{} M. Izquierdo, section 8.3 of \emph{[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]} , Cambridge monographs of mathematical physics, (1995)

\end{itemize}
following

\begin{itemize}%
\item [[José de Azcárraga]], \emph{Wess-Zumino terms, extended algebras and anomalies in classical physics}, Contemp.Math. 132 (1992) 75-98 (\href{https://inspirehep.net/record/320052?ln=de}{spire})

\end{itemize}
[[!redirects classical anomalies]]

[[!redirects Galileo 2-cocycle]]



\end{document}