Given a (pre)symplectic manifold , its quantomorphism group is the Lie group that integrates the Lie bracket inside the Poisson algebra of . 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.)
Over a symplectic manifold an explicit construction of the corresponding quantomorphism group is obtained by choosing a prequantum circle bundle, regarded with an Ehresmann connection 1-form on , and then defining
to be the subgroup of the diffeomorphism group on those diffeomorphisms that preserve . 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 .
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 the cohesive (∞,1)-topos 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 in the slice (∞,1)-topos. Then the quantomorphism group is the automorphism group
the morphism modulating a prequantum circle n-bundle, the corresponding quantomorphism -group is again , now formed in
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.).
higher and integrated Kostant-Souriau extensions:
|geometry||structure||unextended structure||extension by||quantum extension|
|higher prequantum geometry||cohesive ∞-group||Hamiltonian symplectomorphism ∞-group||moduli ∞-stack of -flat ∞-connections on||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-plectic geometry||Lie n-algebra||Hamiltonian vector fields||line Lie n-algebra||Poisson Lie n-algebra|
|smooth n-group||Hamiltonian n-plectomorphisms||circle n-group||quantomorphism n-group|
(extension are listed for sufficiently connected )
Sometimes (e.g. Robbin-Salamon 93, p. 30) this 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).
The group 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 is the semidirect product of the metaplectic group with the Heisenberg group (ARZ 06, prop. V.1, see also Low 12)
|cohesive ∞-groups:||Heisenberg ∞-group||quantomorphism ∞-group||∞-bisections of higher Courant groupoid||∞-bisections of higher Atiyah groupoid|
|L-∞ algebras:||Heisenberg L-∞ algebra||Poisson L-∞ algebra||Courant L-∞ algebra||twisted vector fields|
|higher Atiyah groupoid:||standard higher Atiyah groupoid||higher Courant groupoid||groupoid version of quantomorphism n-group|
|coefficient for cohomology:|
|type of fiber ∞-bundle:||principal ∞-bundle||principal ∞-connection without top-degree connection form||principal ∞-connection|
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
The description in terms of automorphism in the slice -topos over the moduli stack of (higher) connections is in
and in section 4.4.17 of
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
A metric-structure on quantomorphisms groups is 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
Stephen G. Low, Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group, J. Math. Phys. 55, 022105 (2014) (arXiv:1207.6787)