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\begin{document}

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\section*{quantum anomaly}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{AnomalousActionFunctional}{Anomalous action functional}\dotfill \pageref*{AnomalousActionFunctional} \linebreak
\noindent\hyperlink{fermionic_anomalies}{Fermionic anomalies}\dotfill \pageref*{fermionic_anomalies} \linebreak
\noindent\hyperlink{higher_gaugetheoretic_anomalies}{Higher gauge-theoretic anomalies}\dotfill \pageref*{higher_gaugetheoretic_anomalies} \linebreak
\noindent\hyperlink{AnomalousSymmetry}{Anomalous symmetry}\dotfill \pageref*{AnomalousSymmetry} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{anomalous_action_functional_2}{Anomalous action functional}\dotfill \pageref*{anomalous_action_functional_2} \linebreak
\noindent\hyperlink{SpinningParticlesAndSuperBranes}{Spinning particles and super-branes}\dotfill \pageref*{SpinningParticlesAndSuperBranes} \linebreak
\noindent\hyperlink{gravitational_anomaly}{Gravitational anomaly}\dotfill \pageref*{gravitational_anomaly} \linebreak
\noindent\hyperlink{axial_anomaly}{Axial anomaly}\dotfill \pageref*{axial_anomaly} \linebreak
\noindent\hyperlink{conformal_anomaly_of_the_string}{Conformal anomaly of the string}\dotfill \pageref*{conformal_anomaly_of_the_string} \linebreak
\noindent\hyperlink{freedwitten_anomaly}{Freed-Witten anomaly}\dotfill \pageref*{freedwitten_anomaly} \linebreak
\noindent\hyperlink{diaconescumoorewitten_anomaly}{Diaconescu-Moore-Witten anomaly}\dotfill \pageref*{diaconescumoorewitten_anomaly} \linebreak
\noindent\hyperlink{m5brane_anomaly}{M5-Brane anomaly}\dotfill \pageref*{m5brane_anomaly} \linebreak
\noindent\hyperlink{anomalous_symmetry_2}{Anomalous symmetry}\dotfill \pageref*{anomalous_symmetry_2} \linebreak
\noindent\hyperlink{conformal_anomaly}{Conformal anomaly}\dotfill \pageref*{conformal_anomaly} \linebreak
\noindent\hyperlink{other}{Other}\dotfill \pageref*{other} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{AnomalousActionFunction}{Anomalous action functional}\dotfill \pageref*{AnomalousActionFunction} \linebreak
\noindent\hyperlink{ReferencesGaugeAnomaly}{Gauge anomaly}\dotfill \pageref*{ReferencesGaugeAnomaly} \linebreak
\noindent\hyperlink{in_bvbrst_formulation}{In BV-BRST formulation}\dotfill \pageref*{in_bvbrst_formulation} \linebreak
\noindent\hyperlink{other_2}{Other}\dotfill \pageref*{other_2} \linebreak
\noindent\hyperlink{in_finitedimensional_quantum_mechanics}{In finite-dimensional quantum mechanics}\dotfill \pageref*{in_finitedimensional_quantum_mechanics} \linebreak


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

There are at least two things that are called \emph{quantum anomalies} in the context of [[quantum field theory]]

\begin{itemize}%
\item \textbf{anomalous action functional}: the [[action functional]] (in [[path integral|path integral quantization]]) is not a globally well defined [[function]], but instead a [[section]] of a [[line bundle]] on [[configuration space]];


\item \textbf{anomalous symmetry} (gauge anomaly): a symmetry of the [[action functional]] does not extend to a symmetry of the exponentiated action times the path integral measure; or equivalently the [[action]] of a group on classical [[phase space]] is not preserved by [[deformation quantization]].



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

\hypertarget{AnomalousActionFunctional}{}\subsubsection*{{Anomalous action functional}}\label{AnomalousActionFunctional}

There are two major kinds of [[action functionals]] that may be anomalous in that they are not actually [[function]]s/[[nonlinear functional|functional]]s on the configuration space of fields, but just [[section]]s of some [[line bundle]]:

\begin{itemize}%
\item theories with fermions (see e.g. [[spinors in Yang-Mills theory]]) whose [[action functional]] is given by a [[Dirac operator]], or else other fields whose action functional is given by a [[Fredholm operator]].


\item [[gauge theory|gauge theories]] with higher degree gauge fields ([[differential cohomology|differential cocycles]] of higher degree.)



\end{itemize}
\hypertarget{fermionic_anomalies}{}\paragraph*{{Fermionic anomalies}}\label{fermionic_anomalies}

The [[path integral]] for a [[quantum field theory]] with [[fermion]]s can be decomposed into a [[fermionic path integral]] (see there for more details) over the [[fermionic field]]s followed by that over the [[bosonic field]]s. The former, a [[Berezin integral]], is typically well defined for a fixed configuration of the bosonic fields, but does not produce a well defined function on the space of all bosonic fields: but a \emph{twisted function} , a [[section]] of some [[line bundle]] called a [[determinant line bundle]] or, in $8k+2$ dimensions, its [[square root]], the [[Pfaffian line bundle]].

So to even start making sense of the remaining path integral over the bosonic degree of freedom, this [[determinant line bundle]] or the corresponding [[Pfaffian line bundle]] has to be trivializable. Its non-trivializability is the \emph{fermionic anomaly} .

More in detail (\hyperlink{Freed86}{Freed 86}), the [[path integral]] over an [[Lagrangian]] of the form $(\overline \phi, D \phi)$ for

\begin{displaymath}
D \;\colon\; V \longrightarrow W
\end{displaymath}
a [[Fredholm operator]] computes the [[determinant]] of that operator. Formally this is a [[section]] of the [[determinant line bundle]] over the remaining [[field (physics)|fields]]

\begin{displaymath}
(det V)^\ast \otimes (det W)
  \simeq
  (det ker D)^\ast \otimes (det coker D)
  \,,
\end{displaymath}
where the left hand side makes sense and the equivalence holds for $V$ and $W$ finite dimensional, and where the right hand side is the definition of the expression for general [[Fredholm operators]]. ((\hyperlink{Freed86}{Freed 86, 1.}))

In more detail this [[determinant line bundle]] also carries a [[connection]] on a bundle. To make the formal [[path integral]], which is a [[section]] of this bundle, into an actual function, one this bundle with connection needs to be trivializable and trivialized. The [[obstruction]] to this is the anomaly.

\hypertarget{higher_gaugetheoretic_anomalies}{}\paragraph*{{Higher gauge-theoretic anomalies}}\label{higher_gaugetheoretic_anomalies}

For the moment see [[Green-Schwarz mechanism]] for more.

\hypertarget{AnomalousSymmetry}{}\subsubsection*{{Anomalous symmetry}}\label{AnomalousSymmetry}

\begin{quote}%
under construction


\end{quote}
Let

\begin{displaymath}
S : C \to \mathbb{R}
\end{displaymath}
be a (well defined) [[action functional]]. Write $P$ for its resolved [[covariant phase space]] in [[dg-geometry]] and

\begin{displaymath}
S^{BV} : P \to \mathbb{R}
\end{displaymath}
for the BV-action functional, both as given by [[BRST-BV formalism]].

If the action functional is local (comes from a [[Lagrangian]] on a [[jet bundle]]) the [[covariant phase space]] $P$ a priori only carries a [[presymplectic structure]]. But by BV-theory there exists an equivalent (homotopical) derived action functional $S_\Psi^{BV} : P \to \mathbb{R}$ such that $S_\Psi^{BV}$ does induce a genuine [[symplectic structure]] on the [[derived geometry|derived]] space $P$.

For ordinary [[Poisson manifold]]s and hence [[symplectic manifold]]s [[Maxim Kontsevich]]`s theorem says that their [[deformation quantization]] always exist. But if $S$ is the [[action functional]] of a [[gauge theory]] then $P$ is in general a nontrivial derived [[infinity-Lie algebroid]] (its function algebra has ``ghosts'' and ``ghosts of ghost'': the [[Chevalley-Eilenberg algebra]] generators) and the theorem does not apply. Instead, the quantization of the derived symplectic space $P$ exists only if the first and second [[infinity-Lie algebroid cohomology]] of $P$ vanishes:

These two cohomology groups

\begin{displaymath}
Anom_{gauge} = H^1(CE(P)) \oplus H^2(CE(P))
\end{displaymath}
are called the \textbf{gauge anomaly} of the system. Only if they vanish does the [[quantization]] of the [[gauge theory]] encoded by $S$ exist.

More concretely, the function algebra on $P$ is a graded-commutative [[dg-algebra]] equipped with a graded Poisson bracket $\{-,-\}_{BV}$ and an element $Q \in C^\infty(P)$ (the BV-BRST charge) whose [[Hamiltonian vector field]] is the [[derivation]] that is the [[differential]] of the dg-algebra $C^\infty(X)$. If the gauge anomaly does not vanish, then, while the deformation quantization of the graded algebra $C^\infty(P)$ to a non commutative graded algebra with commutator $[-,-]$ will exist, it may happen that the image $S$ of $Q$ under the quantization no longer satisfies the [[quantum master equation]] $[S,S] = \hbar \Delta S$.

Therefore the [[derivation]] $[S,-]$ will not define a quantized [[differential]] and therefore the quantization of the graded-commutative [[dg-algebra]] $C^\infty(P)$ will only be a noncommutative algebra, not a non-commutative dg-algebra, hence will not be functions on a non-commutative space in [[derived geometry]].

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

\hypertarget{anomalous_action_functional_2}{}\subsubsection*{{Anomalous action functional}}\label{anomalous_action_functional_2}

\hypertarget{SpinningParticlesAndSuperBranes}{}\paragraph*{{Spinning particles and super-branes}}\label{SpinningParticlesAndSuperBranes}

The [[sigma-model]] for a [[supersymmetry|supersymmetric]] fundamental [[brane]] on a target space $X$ has an anomaly coming from the nontriviality of [[Pfaffian line bundle]]s associated with the [[fermion]]ic fields on the worldvolume. These anomalies disappear (i.e. these bundles are trivializable) when the structure group of the [[tangent bundle]] of $X$ has a sufficiently high lift through the [[Whitehead tower]] of $O(n)$.

\begin{itemize}%
\item \textbf{Spin structure} the worldline anomaly for the spinning particle/superparticle vanishes when $X$ has [[Spin structure]]

This is a classical result. A concrete derivation is in

\begin{itemize}%
\item [[Edward Witten]], \emph{Global anomalies in String theory} in \emph{Symposium on anomalies, geometry, topology} , World Scientific Publishing, Singapore (1985)

\end{itemize}

\item \textbf{String structure} the worldsheet anomaly for the spinning string/superstring in [[heterotic string theory]] vanishes (essentially) when $X$ has [[String structure]]

This is originally due to Killingback and Witten. A commented list of literature is \href{http://golem.ph.utexas.edu/string/archives/000572.html}{here}. Recently [[Ulrich Bunke]] gave the rigorous proof

\begin{itemize}%
\item [[Ulrich Bunke]], \emph{String structures and trivialisations of a Pfaffian line bundle} (\href{http://arxiv.org/abs/0909.0846}{arXiv})

\end{itemize}


\end{itemize}
in terms of [[differential cohomology]] in general and [[differential string structure]]s in particular.

\begin{itemize}%
\item \textbf{Fivebrane structure} the worldvolume anomaly for the super-5-brane in [[dual heterotic string theory]] vanishes (essentially) when $X$ has [[Fivebrane structure]]. See there.

\end{itemize}
\hypertarget{gravitational_anomaly}{}\paragraph*{{Gravitational anomaly}}\label{gravitational_anomaly}

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

\end{itemize}
\hypertarget{axial_anomaly}{}\paragraph*{{Axial anomaly}}\label{axial_anomaly}

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

\end{itemize}
\hypertarget{conformal_anomaly_of_the_string}{}\paragraph*{{Conformal anomaly of the string}}\label{conformal_anomaly_of_the_string}

The [[2d CFT]] on the [[worldsheet]] of the [[bosonic string]] (in flat space, without further background fields) has an anomaly unless the [[dimension|dimensional]] [[target space]] is $d = 26$.

This is discussed as a condition of trivialization of a bundle in (\hyperlink{Freed86}{Freed 86, section 2}). A brief summary is stated \href{http://mathoverflow.net/a/99667/381}{this comment on MO}.

For more see at [[conformal anomaly]] for more.

\hypertarget{freedwitten_anomaly}{}\paragraph*{{Freed-Witten anomaly}}\label{freedwitten_anomaly}

see at \emph{[[Freed-Witten anomaly]]}.

\hypertarget{diaconescumoorewitten_anomaly}{}\paragraph*{{Diaconescu-Moore-Witten anomaly}}\label{diaconescumoorewitten_anomaly}

see at \emph{[[Diaconescu-Moore-Witten anomaly]]}

\hypertarget{m5brane_anomaly}{}\paragraph*{{M5-Brane anomaly}}\label{m5brane_anomaly}

see at \emph{\href{M5-brane#AnomalyCancellation}{M5-brane anomaly}}

\hypertarget{anomalous_symmetry_2}{}\subsubsection*{{Anomalous symmetry}}\label{anomalous_symmetry_2}

\hypertarget{conformal_anomaly}{}\paragraph*{{Conformal anomaly}}\label{conformal_anomaly}

For the moment see [[Liouville cocycle]].

\hypertarget{other}{}\subsubsection*{{Other}}\label{other}

\begin{itemize}%
\item [[RR-field tadpole]]

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

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


\item [[fiber bundles in physics]]



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

\begin{itemize}%
\item \hyperlink{AnomalousActionFunction}{References on anomalous action functionals}


\item \hyperlink{ReferencesGaugeAnomaly}{References on gauge anomalies}



\end{itemize}
\hypertarget{AnomalousActionFunction}{}\subsubsection*{{Anomalous action functional}}\label{AnomalousActionFunction}

The original articles on anomalous action functionals are

\begin{itemize}%
\item [[Luis Alvarez-Gaumé]] and [[Edward Witten]], \emph{Gravitational Anomalies} Nucl. Phys. B234 (1984) 269.


\item [[Luis Alvarez-Gaumé]] and [[Paul Ginsparg]], \emph{The structure of gauge and gravitational anomalies} , Ann. Phys. 161 (1985) 423. (\href{http://inspirehep.net/record/202565/?ln=en}{spire})


\item [[Edward Witten]], \emph{Global gravitational anomalies} , Commun. Math. Phys. 100 (1985) 197. (\href{http://projecteuclid.org/euclid.cmp/1103943444}{EUCLID})



\end{itemize}
A survey of these results is in the slides

\begin{itemize}%
\item Paolo Di Vecchia, \emph{Green-Schwarz anomaly cancellation} (2010) (\href{http://www.college-de-france.fr/media/par_ele/UPL47331_DiVecchia.4.pdf}{pdf})

\end{itemize}
The mathematical formulation of this in terms of [[index theory]] is due to

\begin{itemize}%
\item [[Michael Atiyah]], [[Isadore Singer]], \emph{Dirac operators coupled to vector potentials}, Proc. Nat. Acad. Sci. USA \textbf{81}, 2597-2600 (1984)


\item [[Jean-Michel Bismut]] and [[Daniel Freed]], \emph{The analysis of elliptic families. I. Metrics and connections on determinant bundles} , Comm. Math. Phys. 106 (1986), no. 1, 159--176.


\item [[Jean-Michel Bismut]] and [[Daniel Freed]], \emph{The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem} , Comm. Math. Phys. 107 (1986), no. 1, 103--163.



\end{itemize}
and a clear comprehensive account of the situation (topological anomaly, geometric anomaly) is in

\begin{itemize}%
\item [[Raphael Flauger]], \emph{Anomalies and the Atiyah-Singer Index Theorem} (\href{http://www.ma.utexas.edu/~dafr/Index/Flauger.pdf}{pdf})


\item [[Daniel Freed]], \emph{Determinants, torsion, and strings}, Comm. Math. Phys. Volume 107, Number 3 (1986), 483-513. (\href{http://projecteuclid.org/euclid.cmp/1104116145}{Euclid})



\end{itemize}
Slick formulation of these anomalies as [[invertible topological field theories]] is discussed in

\begin{itemize}%
\item [[Daniel Freed]], \emph{Anomalies and Invertible Field Theories}, talk at \href{http://scgp.stonybrook.edu/events/event-pages/string-math-2013}{StringMath2013} (\href{https://arxiv.org/abs/1404.7224}{arXiv.1404.7224})

\end{itemize}
A physicists' monograph is

\begin{itemize}%
\item Reinhold A. Bertlmann, \emph{Anomalies in quantum field theory}, Oxford Science Publ., 1996, 2000

\end{itemize}
A clear description of the quantum anomalies for higher gauge theories is in

\begin{itemize}%
\item [[Dan Freed]], \emph{[[Dirac charge quantization and generalized differential cohomology]]} (\href{http://arxiv.org/abs/hep-th/0011220}{arXiv})

\end{itemize}
As an application of this, a detailed discussion of the cancellation of the anomaly of the [[supergravity C-field]] in 11-dimensional [[supergravity]] is in

\begin{itemize}%
\item [[Dan Freed]], [[Greg Moore]], \emph{Setting the quantum integrand of M-theory} (\href{http://arxiv.org/abs/hep-th/0409135}{arXiv:hep-th/0409135})

\end{itemize}
The role of [[spin structure]]s as the anomaly cancellation condition for the spinning particle is discussed in

\begin{itemize}%
\item [[Edward Witten]], \emph{Global anomalies in String theory} in \emph{Symposium on anomalies, geometry, topology} , World Scientific Publishing, Singapore (1985)

\end{itemize}
The anomaly line bundles for [[self-dual higher gauge theory]] is discussed in

\begin{itemize}%
\item [[Samuel Monnier]], \emph{The anomaly line bundle of the self-dual field theory} (\href{http://arxiv.org/abs/1109.2904}{arXiv:1109.2904})

\end{itemize}
Discussion in the context of [[extended topological field theory]] includes

\begin{itemize}%
\item [[Samuel Monnier]], \emph{Hamiltonian anomalies from extended field theories}, (\href{http://arxiv.org/abs/1410.7442}{arxiv/1410.7442})

\end{itemize}
Anomaly field theories are discussed in

\begin{itemize}%
\item [[Samuel Monnier]], \emph{A Modern Point of View on Anomalies}, (\href{http://arxiv.org/abs/1903.02828}{arxiv/1903.02828})

\end{itemize}
\hypertarget{ReferencesGaugeAnomaly}{}\subsubsection*{{Gauge anomaly}}\label{ReferencesGaugeAnomaly}

The original work on the [[chiral anomaly]] is due to

\begin{itemize}%
\item [[Stephen Adler]]. \emph{Axial-Vector Vertex in Spinor Electrodynamics} Physical Review 177 (5): 2426. (1969)


\item [[John Bell]], [[Roman Jackiw]], \emph{A PCAC puzzle: $\pi$0$\rightarrow$$\gamma$$\gamma$ in the $\sigma$-model``. Il Nuovo Cimento A 60: 47. (1969)}



\end{itemize}
See also

\begin{itemize}%
\item L. Faddeev and S. Shatashvili, ``Algebraic and Hamiltonian Methods in the theory of Nonabelian Anomalies,'' Theor. Math. Fiz., 60 (1984) 206; english transl. Theor. Math. Phys. 60 (1984) 770.


\item B. Zumino, ``Chiral anomalies and differential geometry,'' in Relativity, Groups and Topology II, proceedings of the Les Houches summer school, B.S. DeWitt and R. Stora, eds. North-Holland, 1984.



\end{itemize}
\hypertarget{in_bvbrst_formulation}{}\paragraph*{{In BV-BRST formulation}}\label{in_bvbrst_formulation}

General discussion in the context of [[BRST-BV formalism]] (breaking of the [[quantum master equation]] by quantum corrections) is discussed in

\begin{itemize}%
\item W. Troost, P. van Nieuwenhuizen, A. van Proeyen, \emph{Anomalies and the Batalin-Vilkovisky lagrangian formalism} (\href{http://adsabs.harvard.edu/abs/1990NuPhB.333..727T}{web})


\item [[Paul Howe]], [[Ulf Lindström]], P. White, \emph{Anomalies And Renormalization In The BRST-BV Framework} , Phys. Lett. B246 (1990) 430.


\item J. Paris, W. Troost, \emph{Higher loop anomalies and their consistency conditions in nonlocal regularization} , Nucl. Phys. B482 (1996) 373 (\href{http://arxiv.org/abs/hep-th/9607215}{arXiv:hep-th/9607215})


\item [[Glenn Barnich]], \emph{Classical and quantum aspects of the extended antifield formalism} (\href{http://arxiv.org/abs/hep-th/0011120}{arXiv:hep-th/0011120})



\end{itemize}
The fact that the anomaly sits in degree-1 BRST cohomology corresponds to the consistency condition discussed in

\begin{itemize}%
\item [[Julius Wess]] and B. Zumino, \emph{Consequences of anomalous Ward identities} , Phys. Lett. B37 (1971) 95 
\item R. Stora, \emph{The Wess Zumino consistency condition: a paradigm in renormalized perturbation theory}, Fortsch. Phys. 54:175-182 (2006) \href{http://dx.doi.org/10.1002/prop.200510266}{doi}

\end{itemize}
Discussion of special applications in

\begin{itemize}%
\item F. De Jonghe, J. Paris and W. Troost, \emph{The BPHZ renormalised BV master equation and Two-loop Anomalies in Chiral Gravities} , Nucl. Phys. B476 (1996) 559 \href{http://arxiv.org/abs/hep-th/9502140}{arXiv:hep-th/9603012}


\item J. Paris, \emph{Nonlocally regularized antibracket - antifield formalism and anomalies in chiral $W(3)$ gravity} , Nucl. Phys. B450 (1995) 357 (\href{http://arxiv.org/abs/hep-th/9502140}{arXiv:hep-th/9502140})


\item R. Amorim, N.R.F.Braga, R. Thibes, \emph{Axial and gauge anomalies in the field antifield quantization of the generalized Schwinger model} (\href{http://arxiv.org/abs/hep-th/9712014}{arXiv:hep-th/9712014})



\end{itemize}
Discussion in the context of [[AQFT]] with [[functional analysis]] taken into account is in

\begin{itemize}%
\item section 5.3.3. in [[Katarzyna Rejzner]], \emph{Batalin-Vilkovisky formalism in locally covariant field theory} PhD thesis, Hamburg (2011) (\href{http://arxiv.org/abs/1111.5130}{arXiv:1111.5130})


\item [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], \emph{Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory} (\href{http://arxiv.org/abs/1110.5232}{arXiv:1110.5232})



\end{itemize}
\hypertarget{other_2}{}\paragraph*{{Other}}\label{other_2}

An interpretation of gauge anomalies as failures of [[Hamiltonian]]s to have [[self-adjoint extension]]s is in

\begin{itemize}%
\item J. G. Esteve, \emph{Origin of the anomalies: the modified Heisenberg equation} (\href{http://arxiv.org/abs/hep-th/0207164}{arXiv:hep-th/0207164})

\end{itemize}
\hypertarget{in_finitedimensional_quantum_mechanics}{}\subsubsection*{{In finite-dimensional quantum mechanics}}\label{in_finitedimensional_quantum_mechanics}

\begin{itemize}%
\item A. P. Balachandran, Amilcar R. de Queiroz, \emph{Mixed states from anomalies}, \href{http://arxiv.org/abs/1108.3898}{arxiv/1108.3898}
\item Carlos Alcalde, Daniel Sternheimer, \emph{Analytic vectors, anomalies and star representations}, Lett. Math. Phys. \textbf{17} (1989), no. 2, 117--127. \href{http://www.ams.org/mathscinet-getitem?mr=993017}{MR90h:22012}, \href{http://dx.doi.org/10.1007/BF00402326}{doi} (the last section has also the field theory case)

\end{itemize}
[[!redirects anomaly]] [[!redirects anomalies]] [[!redirects quantum anomalies]]

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