nLab classical anomaly

Contents

Idea

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

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} } \,.

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 Kostant-Souriau extensions:

(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\infty$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega \mathbb{G})$ -flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$ )

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)