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Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that “quantisation commutes with reduction” and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity.

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a $${{\mathrm{{{\mathrm{Spin}}}^c}}}$$ -Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zürich I:635-664, 2007) and proved by Ma and Zhang in (Acta Math 212:11-57, 2014). [ABSTRACT FROM AUTHOR]

In this note, we resume the geometric quantization approach to the motion of a charged particle on a plane, subject to a constant magnetic field perpendicular to the latter, by showing directly that it gives rise to a completely integrable system to which we may apply holomorphic geometric quantization. In addition, we present a variant employing a suitable vertical polarization and we also make contact with Bott's quantization, enforcing the property 'quantization commutes with reduction', which is known to hold under quite general conditions. We also provide an interpretation of translational symmetry breaking in terms of coherent states and index theory. Finally, we give a representation theoretic description of the lowest Landau level via the use of an -equivariant Dirac operator. [ABSTRACT FROM AUTHOR]

It is shown that the generalized Hopf map ℍ × ℍ → ℍ × ℝ × ℝ quaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S 7 → S 4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator. Furthermore it is shown how the Hopf fibration and associated twistor fibration play a role in the geometry of the Kepler problem and the rigid body problem. [ABSTRACT FROM AUTHOR]

We study the symplectic reduction of the phase space describing k particles in Rn with total angular momentum zero. This corresponds to the singular symplectic quotient associated to the diagonal action of On on k copies of T*Rn at the zero value of the homogeneous quadratic moment map. We give a description of the ideal of relations of the ring of regular functions of the symplectic quotient. Using this description, we demonstrate Z+-graded regular symplectomorphisms among the On- and Sn-symplectic quotients and determine which of these quotients are graded regularly symplectomorphic to linear symplectic orbifolds. We demonstrate that when n≤k, the zero fibre of the moment map has rational singularities and hence is normal and Cohen-Macaulay. We also demonstrate that for small values of k, the ring of regular functions on the symplectic quotient is graded Gorenstein. [ABSTRACT FROM AUTHOR]

We show that the Siegel upper half space $${\Sigma_{d}}$$ is identified with the Marsden-Weinstein quotient obtained by symplectic reduction of the cotangent bundle $${T^{*} \mathbb{R}^{2d^{2}}}$$ with O(2 d)-symmetry. The reduced symplectic form on $${\Sigma_{d}}$$ corresponding to the standard symplectic form on $${T^{*} \mathbb{R}^{2d^{2}}}$$ turns out to be a constant multiple of the symplectic form on $${\Sigma_{d}}$$ obtained by Siegel. Our motivation is to understand the geometry behind two different formulations of the Gaussian wave packet dynamics commonly used in semiclassical mechanics. Specifically, we show that the two formulations are related via the symplectic reduction. [ABSTRACT FROM AUTHOR]

Let K be a compact Lie group of positive dimension. We show that for most unitary K -modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When K is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian K -manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary SU 2 -modules yield symplectic quotients that are Z + -graded regularly symplectomorphic to a linear symplectic orbifold. We similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold. [ABSTRACT FROM AUTHOR]

Using explicit computations of Hilbert series, we prove that when d > 2, a d -dimensional symplectic quotient at the zero level of a unitary circle representation V such that V S 1 = { 0 } cannot be ℤ-graded regularly symplectomorphic to the quotient of a unitary representations of a finite group. [ABSTRACT FROM AUTHOR]

In this article, we first present the Kustaanheimo - Stiefel regularization of the spatial Kepler problem in a symplectic and quaternionic approach. We then establish a set of action-angle coordinates, the so-called LCF coordinates, of the Kustaanheimo - Stiefel regularized Kepler problem, which is consequently used to obtain a conjugacy relation between the integrable approximating 'quadrupolar' system of the lunar spatial three-body problem and its regularized counterpart. This result justifies the study of Lidov and Ziglin [14] of the quadrupolar dynamics of the lunar spatial three-body problem near degenerate inner ellipses. [ABSTRACT FROM AUTHOR]

We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state (e.g. from $$q$$ and $$p$$ to $$q$$ or $$p$$ in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau's moment map, the Mardsen-Weinstein symplectic reduction, the symplectic 'category' introduced by Weinstein, and the conjecture (proposed by Guillemin and Sternberg) according to which 'quantization commutes with reduction'. By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit (the extreme cases of) this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points. [ABSTRACT FROM AUTHOR]

We study iteration maps of difference equations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and presymplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix representing the quiver is singular. The reduced iteration map is explicitly computed for several periodic quivers using either the presymplectic reduction or a Poisson reduction via log-canonical Poisson structures. [ABSTRACT FROM AUTHOR]

The authors examine the quantization commutes with reduction phenomenon for Hamiltonian actions of compact Lie groups on closed symplectic manifolds from the point of view of topological K-theory and K-homology. They develop the machinery of K-theory wrong-way maps in the context of orbifolds and use it to relate the quantization commutes with reduction phenomenon to Bott periodicity and the K-theory formulation of the Weyl character formula. [ABSTRACT FROM AUTHOR]

For Hamiltonian systems with spherical symmetry there is a marked difference between zero and non-zero momentum values, and amongst all relative equilibria with zero momentum there is a marked difference between those of zero and those of non-zero angular velocity. We use techniques from singularity theory to study the family of relative equilibria that arise as a symmetric Hamiltonian which has a group orbit of equilibria with zero momentum is perturbed so that the zero-momentum relative equilibrium are no longer equilibria. We also analyze the stability of these perturbed relative equilibria, and consider an application to satellites controlled by means of rotors. [ABSTRACT FROM AUTHOR]

This paper contains results on geometric Routh reduction and it is a continuation of a previous paper [7] where a new class of transformations is introduced between Lagrangian systems obtained after Routh reduction. In general, these reduced Lagrangian systems have magnetic force terms and are singular in the sense that the Lagrangian does not depend on some velocity components. The main purpose of this paper is to show that the Routh reduction process itself is entirely captured by the application of such a new transformation on the initial Lagrangian system with symmetry. [ABSTRACT FROM AUTHOR]

In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré–Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincaré–Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum. [ABSTRACT FROM AUTHOR]

We show that the n-dimensional MICZ-Kepler system arises from symplectic reduction of the 'Kepler problem' on the cone over the rotation group SO( n). As a corollary we derive an elementary formula for the general solution of the MICZ-Kepler problem. The heart of the computation is the observation that the additional MICZ-Kepler potential, | ϕ|/ r, agrees with the rotational part of the cone's kinetic energy. [ABSTRACT FROM AUTHOR]

Abstract: We develop differential and symplectic geometry of differentiable Deligne–Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks. [Copyright &y& Elsevier]

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. Guillemin and Sternberg (Invent Math 67:515-538, 1982, no. 3), showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M // G. This map, though, is not in general unitary, even to leading order in $${\hslash}$$. Hall and Kirwin (Commun Math Phys 275:401-422, 2007, no. 2), showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed $${\hslash}$$, becomes unitary in the semiclassical limit $${\hslash\rightarrow0}$$ ( cf. the work of Ma and Zhang (C R Math Acad Sci Paris 341:297-302, 2005, no. 5), and (Astérisque No. 318:viii+154, 2008). The unitarity of the classical Guillemin-Sternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M // G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as $${\hslash\rightarrow0}$$. [ABSTRACT FROM AUTHOR]

Abstract: We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field. [Copyright &y& Elsevier]

Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that "quantisation commutes with reduction" and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity. [ABSTRACT FROM AUTHOR]

Abstract: In order to analyse the dynamics of a given Hamiltonian system in the space defined as the Cartesian product of two spheres, we propose to combine Delaunay coordinates with Poincaré-like coordinates. The coordinates are of local character and have to be selected accordingly with the type of motions one has to take into consideration, so we distinguish the following types: (i) rectilinear motions; (ii) circular and equatorial motions; (iii) circular non-equatorial motions; (iv) non-circular equatorial motions; and (v) non-circular and non-equatorial motions. We apply the theory to study the dynamics of the reduced flow of a generalised Størmer problem that is modelled as a perturbation of the Kepler problem. After using averaging and reduction theories, the corresponding flow is analysed on the manifold , calculating the occurring equilibria and their stability. Finally, the flow of the original problem is reconstructed, concluding the existence of some families of periodic solutions and KAM tori. [Copyright &y& Elsevier]

We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group SE(2), and we relate the cocycle in the description of this central extension to a certain curvature tensor. [ABSTRACT FROM AUTHOR]

Abstract: With an eye towards index theoretic applications we describe a Schubert like stratification on the Grassmannian of hermitian lagrangian spaces in . This is a natural compactification of the space of hermitian matrices. The closures of the strata define integral cycles, and we investigate their intersection theoretic properties. We achieve this by blending Morse theoretic ideas, with techniques from o-minimal (or tame) geometry and geometric integration theory. [Copyright &y& Elsevier]

Abstract: Using the analytic assembly map that appears in the Baum–Connes conjecture in noncommutative geometry, we generalise the Guillemin–Sternberg conjecture that ‘quantisation commutes with reduction’ to (discrete series representations of) semisimple groups G with maximal compact subgroups K acting cocompactly on symplectic manifolds. We prove this generalised statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements , the set of elements of with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that , for a compact Hamiltonian K-manifold N. The proof comes down to a reduction to the compact case. This reduction is based on a ‘quantisation commutes with induction’-principle, and involves a notion of induction of Hamiltonian group actions. This principle, in turn, is based on a version of the naturality of the assembly map for the inclusion . [Copyright &y& Elsevier]

Abstract: For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied. [Copyright &y& Elsevier]

The trigonometric Ruijsenaars–Schneider model is derived by symplectic reduction of Poisson–Lie symmetric free motion on the group U( n). The commuting flows of the model are effortlessly obtained by reducing canonical free flows on the Heisenberg double of U( n). The free flows are associated with a very simple Lax matrix, which is shown to yield the Ruijsenaars–Schneider Lax matrix upon reduction. [ABSTRACT FROM AUTHOR]

Abstract: Let be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map . If the map descends to the symplectic quotient to a map and the symplectic manifold M is exact and has no short loops, we prove that the Hofer norm of the induced map is bounded above by the Hofer norm of . [Copyright &y& Elsevier]

Abstract: We study reduction of generalized complex structures. More precisely, we investigate the following question. Let be a generalized complex structure on a manifold , which admits an action of a Lie group preserving . Assume that is a -invariant smooth submanifold and the -action on is proper and free so that is a smooth manifold. Under what condition does descend to a generalized complex structure on ? We describe a sufficient condition for the reduction to hold, which includes the Marsden–Weinstein reduction of symplectic manifolds and the reduction of the complex structures in Kähler manifolds as special cases. As an application, we study reduction of generalized Kähler manifolds. [Copyright &y& Elsevier]

Abstract: We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair , where is a Kähler manifold and is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold as the rank of a natural quotient of , and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKähler reduction as an equivariant version of hyperKähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKähler reduction induces hyperKähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKähler reduction. [Copyright &y& Elsevier]

In this paper we show that the classical symplectic stratification theorem of the reduced spaces of a canonical group action in a symplectic manifold can be obtained even in the absence of a momentum map by replacing this object by its natural generalization, the cylinder valued momentum map introduced by Condevaux, Dazord, and Molino. In the process of proving this result we will provide a normal form for the cylinder valued momentum map. [Copyright &y& Elsevier]

Abstract: This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kähler manifold M with a Hamiltonian torus action. In the seminal paper [V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (3) (1982) 515–538], Guillemin and Sternberg introduced an isomorphism between the invariant part of the quantum space associated to M and the quantum space associated to the symplectic quotient of M, provided this quotient is non-singular. We prove that this isomorphism is a Fourier integral operator and that the Toeplitz operators of M descend to Toeplitz operators of the reduced phase space. We also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced Toeplitz operator, a result related to the Riemann–Roch–Kawasaki theorem. [Copyright &y& Elsevier]

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far-reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example, an approach to non-commutative symplectic symmetric spaces. [ABSTRACT FROM AUTHOR]

The representation space X(G) = Hom(π, G)/G of the fundamental group π of a Riemann surface Σ of genus g ≥ 2 is the symplectic reduction of the extended moduli space defined in [6]. Using this description, we study the local structure of X(G) and show that the assumptions of the splitting theorem [11, Theorem 7.7] are satisfied. Hence the middle perversity intersection cohomology is canonically isomorphic to a subspace of the equivariant cohomology $H^*_{G}(\mathrm{Hom}(\pi,G))$ which can be computed quite explicitly. The case when G = SU(2) is discussed in detail. [ABSTRACT FROM AUTHOR]

In recent years, methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this Letter it is shown that the latter method is actually related to (and may be derived from) a particular case of the former if one regards dual of Lie algebroids as special Poisson manifolds. The core of the proof is the fact, discussed in the second part of this Letter, that coisotropic submanifolds of a (twisted) Poisson manifold are in one-to-one correspondence with possibly singular Lagrangian subgroupoids of source-simply-connected (twisted) symplectic groupoids. [ABSTRACT FROM AUTHOR]

The classical field theories that underlie the quantum treatments of the electromagnetic, weak, and strong forces share a peculiar feature: specifying the initial state of the field determines the evolution of some degrees of freedom of the theory while leaving the evolution of some others wholly arbitrary. This strongly suggests that some of the variables of the standard state space lack physical content—intuitively, the space of states of such a theory is of higher dimension than the corresponding space of genuine physical possibilities. The structure of such theories can helpfully be characterized in terms of the action of symmetry groups on their space of states; and the conceptual problems surrounding their strange behavior can be sharpened in light of the observation that it is usually possible to eliminate the redundant variables associated with these symmetries—which turn out to be precisely those variables whose evolution is unconstrained by the dynamical laws of the theory. This paper discusses this approach, uses it to frame questions about the interpretation of classical gauge theories, and to reflect (pessimistically) on our prospects of reaching satisfactory answers to these questions. [Copyright &y& Elsevier]

Suppose we are given a compact Riemannian manifold ( Q,g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by isometries. The natural question arises: will the geodesic flow on Q/G equipped with the submersion metric be integrable? Under one natural assumption, we prove that the answer is affirmative. New examples of manifolds with completely integrable geodesic flows are obtained. [ABSTRACT FROM AUTHOR]

The Serret-Andoyer transformation is a classical method for reducing the free rigid body dynamics, expressed in Eulerian coordinates, to a 2-dimensional Hamiltonian flow. First, we show that this transformation is the computation, in 3-1-3 Eulerian coordinates, of the symplectic (Marsden-Weinstein) reduction associated with the lifted left-action of SO(3) on T* SO(3)—a generalization and extension of Noether's theorem for Hamiltonian systems with symmetry. In fact, we go on to generalize the Serret-Andoyer transformation to the case of Hamiltonian systems on T* SO(3) with left-invariant, hyperregular Hamiltonian functions. Interpretations of the Serret-Andoyer variables, both as Eulerian coordinates and as canonical coordinates of the co-adjoint orbit, are given. Next, we apply the result obtained to the controlled rigid body with momentum wheels. For the class of Hamiltonian controls that preserve the symmetry on T* SO(3), the closed-loop motion of the main body can again be reduced to canonical form. This simplifies the stability proof for relative equilibria , which then amounts to verifying the classical Lagrange-Dirichlet criterion. Additionally, issues regarding numerical integration of closed-loop dynamics are also discussed. Part of this work has been presented in LumBloch:97a. [ABSTRACT FROM AUTHOR]

This paper presents a new derivation of the twist mapping in the planar restricted problem. It differs from other treatments in the use of a novel canonical transformation which allows for the utilization of symplectic reduction techniques. [ABSTRACT FROM AUTHOR]

This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and this class has the property that it is closed under Routh reduction; (ii) we construct a transformation relating the magnetic Lagrangian system obtained after two subsequent Routh reductions and the magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full symmetry group.

We present two versions of the Egorov theorem for orbifolds. The first one is a straightforward extension of the classical theorem for smooth manifolds. The second one considers an orbifold as a singular manifold, the orbit space of a Lie group action, and deals with the corresponding objects in noncommutative geometry.