nLab
quantomorphism group

Contents

Idea

Given a (pre)symplectic manifold (X,ω)(X,\omega), its quantomorphism group is the Lie group that integrates the Lie bracket inside the Poisson algebra of (X,ω)(X, \omega). This is a circle group-central extension of the group of Hamiltonian symplectomorphisms. It extends and generalizes the Heisenberg group of a symplectic vector space.

(Warning on terminology: A more evident name for the quantomorphism group might seem to be “Poisson group”. But this already means something different, see Poisson Lie group.)

Traditional construction

Over a symplectic manifold (X,ω)(X, \omega) an explicit construction of the corresponding quantomorphism group is obtained by choosing (PX,)(P \to X, \nabla) a prequantum circle bundle, regarded with an Ehresmann connection 1-form AA on PP, and then defining

QuantomorphismGroupDiff(P) QuantomorphismGroup \hookrightarrow Diff(P)

to be the subgroup of the diffeomorphism group PPP \stackrel{\simeq}{\to} P on those diffeomorphisms that preserve AA. In other words, the quantomorphism group is the group of equivalences of bundles with connection that need not cover the identity diffeomorphism on the base manifold XX.

Notice that the tuple (P,A)(P,A) is a regular contact manifold (see the discussion there), and so the quantomorphism group is equivalently that of contactomorphisms (P,A)(P,A)(P,A) \to (P,A) of weight 0.

In higher geometry

This perspective lends itself to a more abstract description in higher differential geometry: we may regard the prequantum circle bundle as being modulated by a morphism

:XBU(1) conn \nabla : X \to \mathbf{B} U(1)_{conn}

in the cohesive (∞,1)-topos H=\mathbf{H} = Smooth∞Grpd, with domain the given symplectic manifold and codomain the smooth moduli stack for circle bundles with connection. This in turn may be regarded as an object H /BU(1) conn\nabla \in \mathbf{H}_{/\mathbf{B}U(1)_{conn}} in the slice (∞,1)-topos. Then the quantomorphism group is the automorphism group

QuantMorph(X.)BU(1) connAut() \mathbf{QuantMorph}(X.\nabla) \coloneqq \underset{\mathbf{B}U(1)_{conn}}{\prod} \mathbf{Aut}(\nabla)

in H\mathbf{H}, or rather its differential concretification (FRS 13).

From this it is clear what the quantomorphism ∞-group of an n-plectic ∞-groupoid should be: for

:XB nU(1) conn \nabla : X \to \mathbf{B}^n U(1)_{conn}

the morphism modulating a prequantum circle n-bundle, the corresponding quantomorphism nn-group is again Aut()Aut(\nabla), now formed in H /B nU(1) conn\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}

Properties

Smooth structure

The quantomorphism group for a symplectic manifold may naturally be equipped with the structure of a group object in ILH manifolds (Omori, Ratiu-Schmid), as well as in convenient manifolds (Vizman, prop.).

Group extension

Proposition

For (X,ω)(X,\omega) a connected symplectic manifold there is a central extension of groups

1U(1)QuantomorphismGroup(X,ω)HamiltonianSymplectomorphisms(X,ω)1. 1 \to U(1) \to QuantomorphismGroup(X,\omega) \to HamiltonianSymplectomorphisms(X,\omega) \to 1 \,.

This is due to (Kostant). It appears also (Brylinski, prop. 2.4.5).

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)

Examples

Covering an affine symplectic group

Given a symplectic vector space (V,ω)(V,\omega) one may consider the restriction of its quantomorphism group to the affine symplectic group ASp(V,ω)ASp(V,\omega) (Robbin-Salamon 93, corollary 9.3)

ESp(V,ω) QuantMorph(V,ω) ASp(V,ω) HamSympl(V,ω) \array{ ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega) \\ \downarrow && \downarrow \\ ASp(V,\omega) &\hookrightarrow& HamSympl(V,\omega) }

Sometimes (e.g. Robbin-Salamon 93, p. 30) this ESp(V,ω)ESp(V,\omega) is called the extended symplectic group, but maybe to be more specific one should at the very least say “extended affine symplectic group” or “extended inhomogeneous symplectic group” (ARZ 06, prop. V.1).

Notice that the further restriction to VV regarded as the translation group over itself is the Heisenberg group Heis(V,ω)Heis(V,\omega)

Heis(V,ω) ESp(V,ω) QuantMorph(V,ω) V ASp(V,ω) HamSympl(V,ω) \array{ Heis(V,\omega) &\hookrightarrow& ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega) \\ \downarrow && \downarrow && \downarrow \\ V &\hookrightarrow& ASp(V,\omega) &\hookrightarrow& HamSympl(V,\omega) }

The group ESp(V,ω)ESp(V,\omega) is that of those quantomorphisms which come from quadratic Hamiltonians. Those elements covering elements in the symplectic group instead of the affine symplectic group come from homogeneously quadratic Hamiltonians (e.g. Robbin-Salamon 93, prop. 10.1). In fact ESpESp is the semidirect product of the metaplectic group Mp(V,ω)Mp(V,\omega) with the Heisenberg group (ARZ 06, prop. V.1, see also Low 12)

ESp(V,ω)Heis(V,ω)Mp(V,ω). ESp(V,\omega) \simeq Heis(V,\omega) \rtimes Mp(V,\omega) \,.

slice-automorphism ∞-groups in higher prequantum geometry

cohesive ∞-groups:Heisenberg ∞-group\hookrightarrowquantomorphism ∞-group\hookrightarrow∞-bisections of higher Courant groupoid\hookrightarrow∞-bisections of higher Atiyah groupoid
L-∞ algebras:Heisenberg L-∞ algebra\hookrightarrowPoisson L-∞ algebra\hookrightarrowCourant L-∞ algebra\hookrightarrowtwisted vector fields

higher Atiyah groupoid

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid version of quantomorphism n-group
coefficient for cohomology:B𝔾\mathbf{B}\mathbb{G}B(B𝔾 conn)\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})B𝔾 conn\mathbf{B} \mathbb{G}_{conn}
type of fiber ∞-bundle:principal ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

References

General

Original accounts are

  • Jean-Marie Souriau, Structure des systemes dynamiques Dunod, Paris (1970)

    Translated and reprinted as (see section V.18 for the quantomorphism group):

    Jean-Marie Souriau, Structure of dynamical systems - A symplectic view of physics, Brikhäuser (1997)

  • Bertram Kostant, Quantization and unitary representations, in Lectures in modern analysis and applications III. Lecture Notes in Math. 170 (1970), Springer Verlag, 87—208

A textbook account is in section II.4 of

  • Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993)

and in

  • Rudolf Schmid, Infinite-dimensional Lie groups with applications to mathematical physics

The description in terms of automorphism in the slice \infty-topos over the moduli stack of (higher) connections is in

and in section 4.4.17 of

Smooth manifold structure

The ILH group structure on the quantomorphism group is discussed in

  • H. Omori, Infinite dimensional Lie transformation groups, Springer lecture notes in mathematics 427 (1974)

  • T. Ratiu, R. Schmid, The differentiable structure of three remarkable diffeomorphism groups, Math. Z. 177 (1981)

The regular convenient Lie group structure is discussed in

  • Cornelia Vizman, Some remarks on the quantomorphism group (pdf)

A metric-structure on quantomorphisms groups is discussed in

  • Y. Eliashberg,; L. Polterovich, Partially ordered groups and geometry of contact transformations. Geom.Funct.Anal.10(2000),no.6, 1448-1476.

Examples

The quantomorphisms over elements of the symplectic group of a symplectic vector space are discussed in

  • Irving Segal, Transforms for operators and symplectic automorphisms over a locally compact abelian group, Math. Scand. 13 (1963) 31-43 www.mscand.dk/article/download/10686/8707

  • Joel Robbin, Dietmar Salamon, Feynman path integrals on phase space and the metaplectic representation in Dietmar Salamon (ed.), Symplectic Geometry, LMS Lecture Note series 192 (1993) (pdf)

  • Sergio Albeverio, J. Rezende and J.-C. Zambrini, Probability and Quantum Symmetries. II. The Theorem of Noether in quantum mechanics, Journal of Mathematical Physics 47, 062107 (2006) (pdf)

  • Stephen G. Low, Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group, J. Math. Phys. 55, 022105 (2014) (arXiv:1207.6787)

Revised on January 2, 2015 14:18:10 by Urs Schreiber (127.0.0.1)