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Hamiltonian form

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Definition

For ωΩ n+1(X)\omega \in \Omega^{n+1}(X) an n-plectic geometry, and for vΓ(X)v \in \Gamma(X) a vector field, a Hamiltonian form for vv is, if it exists, a differential form hΩ n(X)h \in \Omega^n(X) such that

ι vω=dh. \iota_{v} \omega = \mathbf{d} h \,.

For n=1n = 1 this reduces to the notion of a Hamiltonian function on a symplectic manifold.

If a Hamiltonian form for vv exists then vv is called a Hamiltonian vector field.

The Hamiltonian forms are the local classical observables/prequantum observables in higher prequantum field theory, often called local currents. They form the Poisson-bracket Lie n-algebra of local observables.

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)

Revised on September 17, 2013 21:46:39 by Urs Schreiber (145.116.129.172)