geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
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.)
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
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).
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
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
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
the morphism modulating a prequantum circle n-bundle, the corresponding quantomorphism $n$-group is again $Aut(\nabla)$, now formed in $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$
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.).
For $(X,\omega)$ a connected symplectic manifold there is a central extension of groups
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 $\mathbb{G}$-principal ∞-connection)
(extension are listed for sufficiently connected $X$)
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)
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 $Heis(V,\omega)$
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)
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: | 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 |
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 section II.4 of
and in
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
The diffeological space-structure (diffeological group, smooth group structure) on the quantomorphism group is at least implicit in
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 quantomorphism groups is discussed in
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 https://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)