nLab
classical anomaly

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Symplectic geometry

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 GG acting by symplectomorphisms on a phase space symplectic manifold (X,ω)(X,\omega), this symmetry has a classical anomaly if it does not lift to a genuine GG-Hamiltonian action, but only to a projective GG-Hamiltonian action, hence to a Hamiltonian action of a central extension G^\widehat G of GG. Specifically, the classical anomaly of the original symplectic GG-action is the 2-cocycle which classifies this extension.

If the original GG-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

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

This G^\widehat G is the Heisenberg group of the given GG-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 𝔭𝔬𝔦𝔰(X,ω)\mathfrak{pois}(X,\omega), this means that an infinitesimal action of a Lie algebra 𝔤\mathfrak{g} via Hamiltonian vector fields on XX 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

𝔤^ 𝔭𝔬𝔦𝔰(X,ω) 𝔤 Γ(TX) ω. \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 Kostant-Souriau extensions:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

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)

See also

  • 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)

Revised on January 10, 2017 16:07:56 by Urs Schreiber (87.191.41.20)