# nLab classical anomaly

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## Surveys, textbooks and lecture notes

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• Axiomatizations

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• Tools

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• Structural phenomena

• Types of quantum field thories

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• examples

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# Contents

## Idea

In the context of classical mechanics, a classical anomaly is a central extension of a Noether current algebra (e.g. Toppan 01).

## Definition

In terms of symplectic geometry this means the following (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 classical anomaly if it does not lift to a genuine $G$-Hamiltonian action, but only to a projective $G$-Hamiltonian action, hence to a Hamiltonian action of a central extension $\widehat G$ of $G$. Specifically, the 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

$\array{ \widehat G &\longrightarrow& QuantMorph(X,\omega) \\ \downarrow &(pb)& \downarrow \\ G &\stackrel{}{\longrightarrow}& HamSympl(X,\omega) } \,.$

This $\widehat G$ is the Heisenberg group of the given $G$-action (See also 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

$\array{ \widehat \mathfrak{g} &\longrightarrow& \mathfrak{pois}(X,\omega) \\ \downarrow && \downarrow \\ \mathfrak{g} &\stackrel{}{\longrightarrow}& \Gamma(T X)_{\omega} } \,.$

## Examples

### Mass-anomaly of Galilean action on non-relativistic mechanics

The canonical Galileo group-action on the phase space of non-relativistic classical mechanics has a classical anomaly, given by a group 2-cocycle proportional to the mass of the system, the Galileo 2-cocycle (e.g. Chen-Shaw-Yen 85, Azcárraga-Izquierdo 95 Marle 14).

higher and integrated :

( of of for $\mathbb{G}$-)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$
$n$geometrystructureunextended structureextension byquantum extension
$\infty$ of $(\Omega \mathbb{G})$- on $X$
1 under
1
2
2
$n$
$n$

(extension are listed for sufficiently connected $X$)

## References

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

A discussion under the term “classical central charge” is in

• J. D. Brown, Marc Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207. (web)

For a list of some examples and further pointers to the (historical) literature, see

• Francesco Toppan, On anomalies in classical mechanical systems, Journal of Nonlinear Mathematical Physics Volume 8, Number 3 (2001), 518–533 (pdf)

• Glenn Barnich, Cédric Troessaert, Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity (arXiv:1309.0794)

Discussion in terms of Heisenberg group extensions and generalization to higher symplectic geometry is in

Discussion in the context of formalization of classical field theory in cohesive homotopy theory is in

The example of the Galileo 2-cocycle is discussed for instance in

• Chen, Shaw, Yen, An example of a 2-cocycle, pdf

• Charles-Michel Marle, The manifold of Motions and the total mass of a mechanical system, 2014 (pdf)

and in the broader context of WZW model terms in

following

• José de Azcárraga, Wess-Zumino terms, extended algebras and anomalies in classical physics, Contemp.Math. 132 (1992) 75-98 (spire)

Last revised on January 10, 2017 at 16:07:56. See the history of this page for a list of all contributions to it.