nLab quantomorphism group

Idea

Given a (pre)symplectic manifold $(X,\omega)$, its quantomorphism group is the Lie group that integrates the Lie bracket inside the Poisson algebra of $(X, \omega)$. 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 $(X, \omega)$ an explicit construction of the corresponding quantomorphism group is obtained by choosing $(P \to X, \nabla)$ a prequantum circle bundle, regarded with an Ehresmann connection 1-form $A$ on $P$, and then defining

$$QuantomorphismGroup \hookrightarrow Diff(P)$$

to be the subgroup of the diffeomorphism group $P \stackrel{\simeq}{\to} P$ on those diffeomorphisms that preserve $A$. 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 $X$.

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

This is an infinite-dimensional Lie group. References discussing its infinite-dimensional manifold-structure are collected below. But the group has immediately the structure of a group in diffeological spaces (making it a smooth group) (Souriau 79).

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

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

in the cohesive (∞,1)-topos $\mathbf{H} =$ 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 $\nabla \in \mathbf{H}_{/\mathbf{B}U(1)_{conn}}$ in the slice (∞,1)-topos. Then the quantomorphism group is the automorphism group

$$\mathbf{QuantMorph}(X,\nabla) \coloneqq \underset{\mathbf{B}U(1)_{conn}}{\prod} \mathbf{Aut}(\nabla)$$

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

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

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

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

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

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

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

higher and integrated Kostant-Souriau extensions:

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

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

$n$	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 $X$)

Examples

Covering an affine symplectic group

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

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

Sometimes (e.g. Robbin-Salamon 93, p. 30) this $ESp(V,\omega)$ 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 $V$ regarded as the translation group over itself is the Heisenberg group $Heis(V,\omega)$

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

The group $ESp(V,\omega)$ 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 $ESp$ is the semidirect product of the metaplectic group $Mp(V,\omega)$ with the Heisenberg group (ARZ 06, prop. V.1, see also Low 12)

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

Related concepts

• conserved current

• Hamiltonian action, classical anomaly

slice-automorphism ∞-groups in higher prequantum geometry

cohesive ∞-groups:	Heisenberg ∞-group	$\hookrightarrow$	quantomorphism ∞-group	$\hookrightarrow$	∞-bisections of higher Courant groupoid	$\hookrightarrow$	∞-bisections of higher Atiyah groupoid
L-∞ algebras:	Heisenberg L-∞ algebra	$\hookrightarrow$	Poisson L-∞ algebra	$\hookrightarrow$	Courant L-∞ algebra	$\hookrightarrow$	twisted vector fields

higher Atiyah groupoid

higher Atiyah groupoid:	standard higher Atiyah groupoid	higher Courant groupoid	groupoid version of quantomorphism n-group
coefficient for cohomology:	$\mathbf{B}\mathbb{G}$	$\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$	$\mathbf{B} \mathbb{G}_{conn}$
type of fiber ∞-bundle:	principal ∞-bundle	principal ∞-connection without top-degree connection form	principal ∞-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) doi:10.1007/978-1-4612-0281-3

• Bertram Kostant, Quantization and unitary representations, in Lectures in modern analysis and applications III. Lecture Notes in Math. 170 (1970), Springer Verlag, 87—208 doi:10.1007/BFb0079068

A textbook account is in

• Jean-Luc Brylinski, section II.4 Loop spaces, characteristic classes and geometric quantization, Birkhäuser (1993)

and in

• Rudolf Schmid, Infinite-dimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., Volume 1 (2004), 54-120. Project Euclid

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

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)

and in section 4.4.17 of

• differential cohomology in a cohesive topos

Another expository account, which spells out the details of the isomorphism from smooth functions with the Poisson bracket to infinitesimal quantomorphisms is given in

• Jennifer Vaughan?, Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization, PhD dissertation, University of Toronto, 2016.

Smooth manifold structure

The diffeological space-structure (diffeological group, smooth group structure) on the quantomorphism group is at least implicit in

• Jean-Marie Souriau, Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. (MathSciNet)

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 quantomorphism 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. doi:10.1007/PL00001656, arXiv:math/9910065

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

• 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)