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The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics

  • Paul SkerrittEmail author
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Abstract

Recently Ohsawa (Lett Math Phys 105:1301–1320, 2015) has studied the Marsden–Weinstein–Meyer quotient of the manifold \(T^*\mathbb {R}^n\times T^*\mathbb {R}^{2n^2}\) under a \(\mathrm {O}(2n)\)-symmetry and has used this quotient to describe the relationship between two different parametrisations of Gaussian wave packet dynamics commonly used in semiclassical mechanics. In this paper, we suggest a new interpretation of (a subset of) the unreduced space as being the frame bundle \({\mathcal {F}}(T^*\mathbb {R}^n)\) of \(T^*\mathbb {R}^n\). We outline some advantages of this interpretation and explain how it can be extended to more general symplectic manifolds using the notion of the diagonal lift of a symplectic form due to Cordero and de León (Rend Circ Mat Palermo 32:236–271, 1983).

Keywords

Frame bundle Dual pair Symplectic reduction Semiclassical mechanics Gaussian wave packet 

Mathematics Subject Classification

53D20 70G65 81Q20 81Q70 

1 Introduction

1.1 Motivation

The Gaussian wave packet ansatz is frequently used in the study of the time-dependent Schrödinger equation and its semiclassical limit. In particular, the wave function
$$\begin{aligned} \psi (x) = \exp \left\{ \frac{i}{\hbar } \left[ \frac{1}{2}(x-q)^\top ({\mathcal {A}}+i{\mathcal {B}})(x-q) + p^\top (x-q) + (\phi +i\delta ) \right] \right\} , \end{aligned}$$
parametrised by \((q,p)\in T^*\mathbb {R}^n\), \(\phi ,\delta \in \mathbb {R}\) and \({\mathcal {A}}+i{\mathcal {B}}\in \varSigma _n = \lbrace W\in \mathrm {M}_n(\mathbb {C})\,|\, W^\top =W, \mathrm {Im}\ W>0 \rbrace \), is well known to be an exact solution of the Schrödinger equation for quadratic Hamiltonians, provided the parameters satisfy certain ODEs [6].

Faou and Lubich [4, 9] have shown that these parameter ODEs constitute a Hamiltonian system, and recently, Ohsawa and Leok [13] have clarified the symplectic structure underlying these Hamiltonian dynamics. Their main observation is that \(\varSigma _n\) is a symplectic manifold, the well-studied Siegel upper half plane [17].

In [5], Hagedorn introduced another parametrisation of Gaussian wave functions, by replacing \({\mathcal {A}}+i{\mathcal {B}}\in \varSigma _n\) by \(PQ^{-1}\), where \(Q, P\in \mathrm {M}_n(\mathbb {C})\) satisfy certain algebraic relations. In this parametrisation, the ODEs governing the parameter evolution are somewhat simpler. In [12], Ohsawa explained how to interpret the relation between QP and \({\mathcal {A}}+i{\mathcal {B}}\) as an instance of symplectic reduction on the symplectic manifold \(T^*\mathbb {R}^n\times T^*\mathbb {R}^{n^2}\) under a certain right \(\mathrm {O}(2n)\)-action. The matrices QP are seen to be coordinates on a level set of the momentum map corresponding to the \(\mathrm {O}(2n)\)-action, and symplectic reduction at this momentum level gives the Siegel upper half plane \(\varSigma _n\). The dynamics in the unreduced and reduced spaces are then related by the quotient map in the usual way, explaining how the Hagedorn ODEs can also be interpreted as a Hamiltonian system.

1.2 Main results and outline

The purpose of the present paper is to reinterpret Ohsawa’s results and to suggest some possible extensions. Specifically, we interpret an open subset of Ohsawa’s unreduced space \(T^*\mathbb {R}^n\times T^*\mathbb {R}^{2n^2}\) as the frame bundle\({\mathcal {F}}(T^*\mathbb {R}^n)\simeq T^*\mathbb {R}^n \times \mathrm {GL}(2n,\mathbb {R})\) of \(T^*\mathbb {R}^n\) and give an intrinsic description of the symplectic structure on \({\mathcal {F}}(T^*\mathbb {R}^n)\). There are several advantages to this picture:
  • there exists a dual pair structure on \(\mathrm {GL}(2n,\mathbb {R})\), generated by left multiplication by \(\mathrm {Sp}(2n,\mathbb {R})\) and right multiplication by \(\mathrm {O}(2n)\) . The existence of this dual pair allows one to realise the symplectic reduced space of \({\mathcal {F}}(T^*\mathbb {R}^n)\) at all values of \(\mathrm {O}(2n)\)-momentum in terms of adjoint orbits of \({\mathfrak {sp}}(2n,\mathbb {R})\). An understanding of all reduced spaces will be important in any attempt to deduce the semiclassical theory described here using ideas from geometric quantisation.1

  • the physical picture of moving frames on \(T^*\mathbb {R}^n\) is intuitive and gives insight into the definition of the Hamiltonian on the unreduced space. Through it, we see that the dynamics of the semiclassical wave function are determined by the underlying classical dynamics in two ways: The variables \(z:=(q,p)\in T^*\mathbb {R}^n\) in the semiclassical wave function evolve according to an underlying classical Hamiltonian H on \(T^*\mathbb {R}^n\), modified by an order \(\hbar \) correction for non-quadratic Hamiltonians. Meanwhile, the variables \(E:= \begin{bmatrix} {\text {Re}}\,Q&\ {\text {Im}}\,Q \\ {\text {Re}}\,P&\ {\text {Im}}\,P\end{bmatrix}\in \mathrm {GL}(2n,\mathbb {R})\) evolve according to the linearisation of the Hamiltonian flow due to H about z. This view of the E-dynamics as linearised symplectic flow depends crucially on the interpretation of E as parametrising the possible frames at z.

  • the picture described in the previous point also has overlap with other work on semiclassical mechanics, in particular the nearby orbit approximation of Littlejohn [7].

  • the frame bundle picture allows a natural interpretation of the conserved quantity associated with the \(\mathrm {O}(2n)\)-invariance of the theory as simply the collection of symplectic products \(\lbrace \omega _{\mathbb {R}^{2n}}(E_a,E_b)\mid 1\le a,b\le 2n\rbrace \) corresponding to the frame \(E = \begin{bmatrix} E_1&E_2&\ldots&E_{2n}\end{bmatrix}\in \mathrm {GL}(2n,\mathbb {R})\). In particular, reduction at the particular value of momentum considered in this paper leads to conservation of symplectic frame.

  • the frame bundle picture can be extended to more general symplectic manifolds than \(T^*\mathbb {R}^n\), using the results of Cordero et al. [3] on the lifting of symplectic structures to frame bundles.

We now outline the structure of the paper. In Sect. 2, we describe a symplectic structure on the set \(\mathrm {M}_{2n}(\mathbb {R})\) of \(2n\times 2n\)-dimensional real matrices, demonstrate that the obvious left \(\mathrm {Sp}(2n,\mathbb {R})\)- and right \(\mathrm {O}(2n)\)-actions are Hamiltonian, and calculate the corresponding momentum maps and their Lie algebra-valued counterparts. In Sect. 3, we review some properties of dual pairs that will be needed in the sequel. In Sect. 4, we restrict our symplectic manifold to \(\mathrm {GL}(2n,\mathbb {R})\subset \mathrm {M}_{2n}(\mathbb {R})\) and demonstrate that the two group actions give a dual pair structure on this restricted space. In Sect. 5, we discuss reduction of \(\mathrm {GL}(2n,\mathbb {R})\) under the \(\mathrm {O}(2n)\)-action at a particular value of momentum, use the results of Sect. 3 to describe the reduced space as an adjoint orbit in \({\mathfrak {sp}}(2n,\mathbb {R})\), and give a geometric description of this adjoint orbit as the set of \(\omega _{\mathbb {R}^{2n}}\)-compatible complex structures, where \(\omega _{\mathbb {R}^{2n}}\) is the canonical symplectic structure on \(\mathbb {R}^{2n}\). In Sect. 6, we use the standard isomorphism of \(\omega _{\mathbb {R}^{2n}}\)-compatible complex structures with the positive Lagrangian Grassmannian to introduce coordinates on the adjoint orbit, and demonstrate that in these coordinates the symplectic projection and reduced symplectic form agree with those obtained by Ohsawa. In Sect. 7, starting with the cotangent symplectic structure \(\omega =\sum _{\alpha =1}^n dq^\alpha \wedge dp_\alpha \) and an arbitrary Hamiltonian H on \(T^*\mathbb {R}^n\), we describe the \(\hbar \)-diagonal symplectic structure \(\varOmega ^{\hbar }\) and \(\hbar \)-lifted Hamiltonian \(H^{\hbar }\) on \({\mathcal {F}}(T^*\mathbb {R}^n)\), which agree with those defined by Ohsawa on their common domain. We give a geometric interpretation of \(H^{\hbar }\), demonstrate that for quadratic H one obtains the dynamics of Hagedorn, and emphasise that the \(\mathrm {O}(2n)\)-invariance of \(H^{\hbar }\) leads to conservation of symplectic frame. We also describe the dynamics on the reduced space and show it reproduces those of Heller. Finally in Sect. 8, we explain how the results of Cordero et al. [3] can be used to extend the construction of a symplectic structure on the frame bundle to arbitrary symplectic manifolds.

2 The symplectic geometry of \(\mathrm {M}_{2n}(\mathbb {R})\)

In this section, we outline the symplectic structure on \(\mathrm {M}_{2n}(\mathbb {R})\), describe two natural symplectic group actions on \(\mathrm {M}_{2n}(\mathbb {R})\) and calculate the momentum maps associated with these actions.

2.1 The symplectic form on \(\mathrm {M}_{2n}(\mathbb {R})\)

Take \(\mathbb {R}^{2n}\) with its canonical symplectic structure
$$\begin{aligned} \omega _{\mathbb {R}^{2n}}(v,w) := v^\top \mathbb {J} w, \end{aligned}$$
where \(\mathbb {J} := \begin{bmatrix} 0_n&\quad I_n \\ -I_n&\quad 0_n \end{bmatrix}\). The 2n-fold direct sum of \((\mathbb {R}^{2n}, \omega _{\mathbb {R}^{2n}})\) is naturally isomorphic to the space \(\mathrm {M}_{2n}(\mathbb {R})\) of real \(2n\times 2n\) matrices with the symplectic form
$$\begin{aligned} \varOmega (E,F) := \sum _{a=1}^{2n}\omega _{\mathbb {R}^{2n}}(E_a,F_a) = {\text {Tr}}\,(E^\top \mathbb {J}F), \end{aligned}$$
where \(E_a\) denotes the ath column of \(E \in \mathrm {M}_{2n}(\mathbb {R})\), considered as a vector in \(\mathbb {R}^{2n}\). We can consider \((\varOmega , \mathrm {M}_{2n}(\mathbb {R}))\) as a symplectic manifold, using the canonical isomorphism \(T_E\mathrm {M}_{2n}(\mathbb {R})\simeq \mathrm {M}_{2n}(\mathbb {R})\), and we denote the induced symplectic form by \(\varOmega \) also. Explicitly, if \(V_E = \frac{d}{dt}\big \vert _{t=0}(E+tV)\) for \(V\in \mathrm {M}_{2n}(\mathbb {R})\) etc., then
$$\begin{aligned} \varOmega _E(V_E,W_E) := \varOmega (V,W) = {\text {Tr}}\,(V^\top \mathbb {J}W). \end{aligned}$$

2.2 Commuting symplectic actions

Let \(L_S:\mathrm {M}_{2n}(\mathbb {R})\rightarrow \mathrm {M}_{2n}(\mathbb {R})\) denote left multiplication by \(S\in \mathrm {Sp}(2n,\mathbb {R})\), and \(R_O:\mathrm {M}_{2n}(\mathbb {R})\rightarrow \mathrm {M}_{2n}(\mathbb {R})\) denote right multiplication by \(O\in \mathrm {O}(2n)\). Using the expressions for the differentials of these actions
$$\begin{aligned} T_EL_S(V_E) = (SV)_{SE}\qquad \text {and}\qquad T_ER_O(V_E) = (VO)_{EO}, \end{aligned}$$
it is straightforward to check that L and R define left and right symplectic actions, respectively, i.e. for any \(E\in \mathrm {M}_{2n}(\mathbb {R})\),
$$\begin{aligned} (L_S^*\varOmega )_E = \varOmega _E \qquad \text {and}\qquad (R_O^*\varOmega )_E = \varOmega _E. \end{aligned}$$
The momentum maps corresponding to these two symplectic actions have a standard form.

Proposition 2.1

  1. (i)
    The momentum map \({\text {J}}_{\mathrm {Sp}}:\mathrm {M}_{2n}(\mathbb {R})\rightarrow {\mathfrak {sp}}(2n,\mathbb {R})^*\) corresponding to the left \(\mathrm {Sp}(2n,\mathbb {R})\)-action is given by
    $$\begin{aligned} \langle {\text {J}}_{\mathrm {Sp}}(E), \zeta \rangle = \frac{1}{2}\varOmega (\zeta E,E). \end{aligned}$$
     
  2. (ii)
    The momentum map \({\text {J}}_{\mathrm {O}}:\mathrm {M}_{2n}(\mathbb {R})\rightarrow {\mathfrak {o}}(2n)^*\) corresponding to the right \(\mathrm {O}(2n)\)-action is given by
    $$\begin{aligned} \langle {\text {J}}_{\mathrm {O}}(E),\xi \rangle = \frac{1}{2}\varOmega (E\xi ,E). \end{aligned}$$
     

Proof

Both results follow from the general expression for the momentum map of a linear symplectic action on a symplectic vector space—see, for example, [11, Section 12.4, Example (a)]. \(\square \)

Remark 2.2

Both momentum maps are easily seen to be equivariant,
$$\begin{aligned} {\text {J}}_{\mathrm {Sp}}(SE) = \mathrm {Ad}_{S^{-1}}^*({\text {J}}_{\mathrm {Sp}}(E)) \qquad \text {and}\qquad {\text {J}}_{\mathrm {O}}(EO) = \mathrm {Ad}_O^*({\text {J}}_{\mathrm {O}}(E)) \end{aligned}$$
(again, a general result for linear symplectic actions).

2.3 Lie algebra-valued momentum maps

For any (real) Lie subalgebra \({\mathfrak {g}}\subset \mathfrak {gl}(N,\mathbb {C})\), consider the trace form \(\langle \langle \cdot ,\cdot \rangle \rangle :{\mathfrak {g}}\times {\mathfrak {g}}\rightarrow \mathbb {C}\) defined by
$$\begin{aligned} \langle \langle \xi ,\zeta \rangle \rangle := \frac{1}{2}{\text {Tr}}\,(\xi \zeta ). \end{aligned}$$
If \({\mathfrak {g}}\) is invariant under conjugate transpose, then \(\langle \langle \cdot ,\cdot \rangle \rangle \) is non-degenerate, as is its real part, since
$$\begin{aligned} \langle \langle \xi ,\xi ^\dagger \rangle \rangle = {\text {Re}}\,\langle \langle \xi ,\xi ^\dagger \rangle \rangle >0\text { for }\xi \ne 0. \end{aligned}$$
In particular, it is non-degenerate and real-valued on \({\mathfrak {sp}}(2n,\mathbb {R})\) and \({\mathfrak {o}}(2n)\). Using the trace form to identify Lie algebras with their duals, we can write down Lie algebra-valued versions of the momentum maps discussed in the previous section.

Proposition 2.3

  1. (i)
    The \({\mathfrak {sp}}(2n,\mathbb {R})\)-valued momentum map \({\text {j}}_{\mathrm {Sp}}:\mathrm {M}_{2n}(\mathbb {R})\rightarrow {\mathfrak {sp}}(2n,\mathbb {R})\) corresponding to \({\text {J}}_{\mathrm {Sp}}\) is given by
    $$\begin{aligned} {\text {j}}_{\mathrm {Sp}}(E) = -EE^\top \mathbb {J}. \end{aligned}$$
     
  2. (ii)
    The \({\mathfrak {o}}(2n)\)-valued momentum map \({\text {j}}_{\mathrm {O}}:\mathrm {M}_{2n}(\mathbb {R})\rightarrow {\mathfrak {o}}(2n)\) corresponding to \({\text {J}}_{\mathrm {O}}\) is given by
    $$\begin{aligned} {\text {j}}_{\mathrm {O}}(E) = -E^\top \mathbb {J}E. \end{aligned}$$
     

Proof

  1. (i)
    Denoting the natural pairing of \({\mathfrak {sp}}(2n,\mathbb {R})^*\) with \({\mathfrak {sp}}(2n,\mathbb {R})\) by \(\langle \cdot ,\cdot \rangle \), we have that for \(\zeta \in {\mathfrak {sp}}(2n,\mathbb {R})\),
    $$\begin{aligned} \langle {\text {J}}_{\mathrm {Sp}}(E),\zeta \rangle&= \frac{1}{2}\varOmega (\zeta E,E) = \frac{1}{2}{\text {Tr}}\,(E^\top \zeta ^\top \mathbb {J} E) \\&= \frac{1}{2}{\text {Tr}}\,(EE^\top \zeta ^\top \mathbb {J}) = -\frac{1}{2}{\text {Tr}}\,(EE^\top \mathbb {J}\zeta ) \qquad \text {since }\zeta \in {\mathfrak {sp}}(2n,\mathbb {R})\\&= \langle \langle -EE^\top \mathbb {J},\zeta \rangle \rangle . \end{aligned}$$
    Also
    $$\begin{aligned} (-EE^\top \mathbb {J})^\top \mathbb {J} + \mathbb {J}(-EE^\top \mathbb {J}) = \mathbb {J}EE^\top \mathbb {J} - \mathbb {J}EE^\top \mathbb {J} = 0. \end{aligned}$$
    So \(-EE^\top \mathbb {J}\in {\mathfrak {sp}}(2n,\mathbb {R})\), and the result follows.
     
  2. (ii)

    Similar.\(\square \)

     

Remark 2.4

Again, the Lie algebra-valued momentum maps are equivariant
$$\begin{aligned} {\text {j}}_{\mathrm {Sp}}(SE) = \mathrm {Ad}_S({\text {j}}_{\mathrm {Sp}}(E)) \qquad \text {and}\qquad {\text {j}}_{\mathrm {O}}(EO) = \mathrm {Ad}_{O^{-1}}({\text {j}}_{\mathrm {O}}(E)). \end{aligned}$$

3 Mutually transitive actions and dual pairs

In Sect. 4, we will demonstrate that the left \(\mathrm {Sp}(2n,\mathbb {R})\)-action and right \(\mathrm {O}(2n)\)-action define a dual pair structure on a suitable subset of \(\mathrm {M}_{2n}(\mathbb {R})\). In anticipation, we here introduce the notion of mutually transitive actions. We then explain how mutual transitivity allows us to view reduced spaces of one action as coadjoint orbits of the other. Finally, we indicate the relation of mutual transitivity to the more standard notion of a dual pair.

A fuller treatment of dual pairs and related concepts can be found in [8, Section IV.7], [14, Chapter 11], and [2]. We also refer to [18] for a more thorough discussion of mutual transitivity.

3.1 Mutually transitive actions

Let \((N,\varOmega )\) be a symplectic manifold, and let \(\varPhi _1:G_1\times N \rightarrow N\) and \(\varPhi _2:G_2\times N\rightarrow N\) be symplectic actions. We assume N, \(G_1\) and \(G_2\) are all finite-dimensional.

Definition 3.1

We say the actions \(\varPhi _1, \varPhi _2\) are mutually transitive if the following three properties hold:
  • \(\varPhi _1\) and \(\varPhi _2\) commute,

  • \(\varPhi _1\) and \(\varPhi _2\) are Hamiltonian actions, with corresponding equivariant momentum maps \({\text {J}}_1\) and \({\text {J}}_2\),

  • each level set of \({\text {J}}_1\) is a \(G_2\)-orbit and vice versa, i.e. for any \(x\in N\),
    $$\begin{aligned} {\text {J}}_1^{-1}({\text {J}}_1(x)) = G_2\cdot x \qquad \text {and}\qquad {\text {J}}_2^{-1}({\text {J}}_2(x)) = G_1\cdot x. \end{aligned}$$

Denoting the coadjoint orbit in \({\mathfrak {g}}_i^*\) through \(\mu _i\) by \({\mathcal {O}}_{\mu _i}\), we then have

Proposition 3.2

Let \(\varPhi _1, \varPhi _2\) be mutually transitive actions. Then for all \(x\in N\),
$$\begin{aligned} {\text {J}}_1^{-1}({\mathcal {O}}_{{\text {J}}_1(x)}) = {\text {J}}_2^{-1}({\mathcal {O}}_{{\text {J}}_2(x)}). \end{aligned}$$

Proof

$$\begin{aligned} \mathrm {J}_1^{-1}({\mathcal {O}}_{{\text {J}}_1(x)})&= G_1\cdot {\text {J}}_1^{-1}({\text {J}}_1(x)) \quad \,\text {since}\, {\text {J}}_1 \,\text {is}\, G_1\,\text {-equivariant}\, \\&= G_1\cdot (G_2\cdot x) \quad \hbox { since }\, \varPhi _2 \hbox { is transitive on the fibres of }\, {\text {J}}_1\\&= G_2\cdot (G_1\cdot x) \quad \text {since}\, \text {the actions}\, \varPhi _1\, \text {and}\, \varPhi _2 \, \text { commute}\\&= G_2\cdot {\text {J}}_2^{-1}({\text {J}}_2(x)) \quad \hbox { since }\, \varPhi _1 \hbox { is transitive on the fibres of }\, {\text {J}}_2\\&= {\text {J}}_2^{-1}({\mathcal {O}}_{{\text {J}}_2(x)}) \quad \text {since}\, {\text {J}}_2\, \text {is}\, G_2\text {-equivariant}. \end{aligned}$$
\(\square \)

Corollary 3.3

( [2, Theorem 2.8(i)]) There exists a one-to-one correspondence between coadjoint orbits in \({\text {J}}_1(N)\) and \({\text {J}}_2(N)\) given by
$$\begin{aligned} {\mathcal {O}}_{\mu _1}\mapsto {\text {J}}_2({\text {J}}_1^{-1}({\mathcal {O}}_{\mu _1})) { ={\text {J}}_2({\text {J}}_1^{-1}(\mu _1))} \end{aligned}$$
or equivalently
$$\begin{aligned} \qquad \qquad {\mathcal {O}}_{{\text {J}}_1(x)}\mapsto {\mathcal {O}}_{{\text {J}}_2(x)} \qquad \text {for }x\in N. \end{aligned}$$

3.2 The relation between coadjoint orbits and reduced spaces

From now on, we assume that all group actions \(\varPhi _i\) are both free and proper (see [18] for a discussion when these conditions do not hold).

For \(\mu _1\in {\mathfrak {g}}_1^*\), let \((G_1)_{\mu _1}\) denote the coadjoint stabiliser of \(\mu _1\) and \(N_{\mu _1}\) the quotient space \({\text {J}}_1^{-1}(\mu _1)/(G_1)_{\mu _1}\) at \(\mu _1\). It is well known that \(N_{\mu _1}\) may be given a smooth structure making \(\sigma _1:{\text {J}}_1^{-1}(\mu _1)\rightarrow N_{\mu _1}\) a submersion and a symplectic structure \(\varOmega _{N_{\mu _1}}\) satisfying
$$\begin{aligned} i_{\mu _1}^*\varOmega = \sigma _1^*\varOmega _{N_{\mu _1}}, \end{aligned}$$
(3.1)
where \(i_{\mu _1}:{\text {J}}_1^{-1}(\mu _1)\rightarrow N\) is the inclusion (see, for example, [14, Theorem 6.1.1 (i)]). The pair \((N_{\mu _1}, \varOmega _{N_{\mu _1}})\) is called the reduced space or Marsden–Weinstein–Meyer quotient.

For mutually transitive actions, it turns out that the reduced space under one action is symplectomorphic to a coadjoint orbit of the other action:

Proposition 3.4

([2, Theorem 2.8(iii)]) Let \(\varPhi _1, \varPhi _2\) be mutually transitive actions on N. Then any reduced space under the \(G_1\)-action is symplectomorphic to a coadjoint orbit in \({\text {J}}_2(N)\subset {\mathfrak {g}}_2^*\) and similarly with 1 and 2 interchanged. Explicitly, for \(x\in N\),
$$\begin{aligned} N_{{\text {J}}_1(x)} \simeq {\mathcal {O}}_{{\text {J}}_2(x)}, \qquad N_{{\text {J}}_2(x)} \simeq {\mathcal {O}}_{{\text {J}}_1(x)}, \end{aligned}$$
via a resp. \(G_2\)- and \(G_1\)-equivariant symplectomorphism.
In fact, the symplectomorphism from \(N_{{\text {J}}_1(x)}\) to \({\mathcal {O}}_{{\text {J}}_2(x)}\) has a simple interpretation: Since the \(G_1\)- and \(G_2\)-actions on N commute, the \(G_2\)-action drops to the reduced space \(N_{{\text {J}}_1(x)}\). The resulting momentum map \(\check{{\text {J}}}_2:N_{{\text {J}}_1(x)}\rightarrow {\mathcal {O}}_{{\text {J}}_2(x)}\) makes the following diagram commuteand can be shown to be a symplectomorphism (in particular, \(\check{{\text {J}}}_2\) is \(G_2\)-equivariant).

3.3 The relation to dual pairs

In this subsection, we make contact with the notion of dual pair (in the sense of Weinstein [19]).

Definition 3.5

([1, Appendix E]) Let N be a symplectic manifold and \(P_1, P_2\) the Poisson manifolds. A pair of Poisson maps
$$\begin{aligned} P_1 {\mathop {\longleftarrow }\limits ^{{\text {J}}_1}} N {\mathop {\longrightarrow }\limits ^{{\text {J}}_2}} P_2 \end{aligned}$$
is called a full dual pair if \({\text {J}}_1, {\text {J}}_2\) are submersions, and
$$\begin{aligned} (\ker T{\text {J}}_1)^\varOmega = \ker T{\text {J}}_2. \end{aligned}$$

Proposition 3.6

If \(\varPhi _1, \varPhi _2\) are mutually transitive and free, then their momentum maps \({\text {J}}_1, {\text {J}}_2\) form a full dual pair.

Proof

Let \(x\in N\). The identity \(G_1\cdot x = {\text {J}}_2^{-1}({\text {J}}_2(x))\) implies that
$$\begin{aligned} {\mathfrak {g}}_1\cdot x = T_x({\text {J}}_2^{-1}({\text {J}}_2(x))). \end{aligned}$$
A standard result says that the former equals \((\ker T_x{\text {J}}_1)^\varOmega \). Also freeness of \(\varPhi _1, \varPhi _2\) implies that \({\text {J}}_1, {\text {J}}_2\) are submersions onto their images [14, Corollary 4.5.13]. Then \({\text {J}}_2^{-1}({\text {J}}_2(x))\) is an embedded submanifold [14, Section 1.1.13], and so,
$$\begin{aligned} T_x({\text {J}}_2^{-1}({\text {J}}_2(x))) = \ker T_x{\text {J}}_2. \end{aligned}$$
The result follows. \(\square \)

4 The dual pair structure on \(\mathrm {GL}(2n,\mathbb {R})\subset \mathrm {M}_{2n}(\mathbb {R})\)

The subset \(\mathrm {GL}(2n,\mathbb {R})\) of \(\mathrm {M}_{2n}(\mathbb {R})\) is open, and hence, \(\varOmega \) restricts to an non-degenerate form on \(\mathrm {GL}(2n,\mathbb {R})\). Denote the restriction of \(\varOmega \) and the momentum maps to \(\mathrm {GL}(2n,\mathbb {R})\) by the same symbols for simplicity.

With this restriction, we can demonstrate that the left \(\mathrm {Sp}(2n,\mathbb {R})\)- and right \(\mathrm {O}(2n)\)-actions are mutually transitive (Definition 3.1). For this, it is more convenient to work with the equivalent double fibration

Proposition 4.1

  1. (i)

    The left \(\mathrm {Sp}(2n,\mathbb {R})\)-action acts transitively on the fibres of \({\text {j}}_{\mathrm {O}}\).

     
  2. (ii)

    The right \(\mathrm {O}(2n)\)-action acts transitively on the fibres of \({\text {j}}_{\mathrm {Sp}}\).

     

Proof

  1. (i)
    Using Proposition 2.3, we have that for any \(E, E'\in \mathrm {GL}(2n,\mathbb {R})\),
    $$\begin{aligned} {\text {j}}_{\mathrm {O}}(E')&= {\text {j}}_{\mathrm {O}}(E) \\&\iff -(E')^\top \mathbb {J} E' = -E^\top \mathbb {J} E \\&\iff (E'E^{-1})^\top \mathbb {J} (E'E^{-1}) = \mathbb {J}\\&\iff E' = SE \qquad \text {for some }S\in \mathrm {Sp}(2n,\mathbb {R}). \end{aligned}$$
     
  2. (ii)

    Similar.\(\square \)

     

Proposition 4.2

The left \(\mathrm {Sp}(2n,\mathbb {R})\)-action and right \(\mathrm {O}(2n)\)-action are free and proper.

Proof

We prove the result for the \(\mathrm {Sp}(2n,\mathbb {R})\)-action only; the \(\mathrm {O}(2n)\) case is similar. Freeness is trivial. To prove properness, suppose we have two convergent sequences \((E_k)\) and \((S_k\cdot E_k)\) in \(\mathrm {GL}(2n,\mathbb {R})\) with \(E_k\rightarrow E\) and \(S_k E_k\rightarrow E\). Then \(S_k = S_k E_k E_k^{-1} \rightarrow E E^{-1} = I\). \(\square \)

Remark 4.3

For any \(E\in \mathrm {M}_{2n}(\mathbb {R})\), it remains true that \(E\cdot \mathrm {O}(2n)\subset {\text {j}}_{\mathrm {Sp}}^{-1}({\text {j}}_{\mathrm {Sp}}(E))\) and \(\mathrm {Sp}(2n,\mathbb {R})\cdot E \subset {\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(E))\). However, restriction to \(\mathrm {GL}(2n,\mathbb {R})\subset \mathrm {M}_{2n}(\mathbb {R})\) is necessary to get equality in the second inclusion. For example,
$$\begin{aligned} {\text {j}}_{\mathrm {O}}^{-1}(0) = \lbrace E\in \mathrm {M}_{2n}(\mathbb {R})\,|\, \mathrm {span}\,E \text { is an isotropic subspace of }\mathbb {R}^{2n}\rbrace , \end{aligned}$$
and this is not simply an orbit of \(\mathrm {Sp}(2n,\mathbb {R})\) (since, for example, the \(\mathrm {Sp}(2n,\mathbb {R})\)-action preserves the dimension of \(\mathrm {span}\,E\)). It is, however, possible to prove that \(E\cdot \mathrm {O}(2n)= {\text {j}}_{\mathrm {Sp}}^{-1}({\text {j}}_{\mathrm {Sp}}(E))\) for all\(E\in \mathrm {M}_{2n}(\mathbb {R})\).
With the restriction from \(\mathrm {M}_{2n}(\mathbb {R})\) to \(\mathrm {GL}(2n,\mathbb {R})\), the momentum map \({\text {j}}_{\mathrm {Sp}}\) acquires a nice interpretation. As before, let \(E_a\) denote the ath column of E, so \(\lbrace E_1,\ldots E_{2n}\rbrace \) defines a basis of \(\mathbb {R}^{2n}\). Let g(E) denote a metric on \(\mathbb {R}^{2n}\) with respect to which the basis \(\lbrace E_a\rbrace \) is orthonormal. Then any basis related to E by the right \(\mathrm {O}(2n)\)-action defines the same metric, i.e.
$$\begin{aligned} g(E\cdot O) = g(E) \qquad \text {for all }O\in \mathrm {O}(2n), \end{aligned}$$
and in fact, the set of metrics on \(\mathbb {R}^{2n}\) is in one-to-one correspondence with the \(\mathrm {O}(2n)\)-orbits of \(\mathrm {GL}(2n,\mathbb {R})\). We then have the following result:

Proposition 4.4

The metric g(E) and canonical symplectic form \(\omega _{\mathbb {R}^{2n}}\) are related by
$$\begin{aligned} \omega _{\mathbb {R}^{2n}}(u,v) = g(E)({\text {j}}_{\mathrm {Sp}}(E)u,v). \end{aligned}$$

Proof

If \(\lbrace s_a \rbrace \) is the standard basis of \(\mathbb {R}^{2n}\) (so \(E_a = \sum _{b=1}^{2n}E_{ba}s_b\)), the definition of g(E) implies that
$$\begin{aligned} g(E)(s_a,s_b) = [(E E^\top )^{-1}]_{ab}. \end{aligned}$$
From this, it follows that
$$\begin{aligned} g(E)({\text {j}}_{\mathrm {Sp}}(E)u,v)&= ({\text {j}}_{\mathrm {Sp}}(E)u)^\top (E E^\top )^{-1} v = u^\top (-E E^\top \mathbb {J})^\top (E E^\top )^{-1}v \\&= u^\top \mathbb {J} v = \omega _{\mathbb {R}^{2n}}(u,v). \end{aligned}$$
\(\square \)
Inverting this identity to give
$$\begin{aligned} g(E)(u,v) = \omega _{\mathbb {R}^{2n}}(({\text {j}}_{\mathrm {Sp}}(E))^{-1}u,v), \end{aligned}$$
we see that \({\text {j}}_{\mathrm {Sp}}(E)\) defines the metric g(E), and so essentially corresponds to it. In the next section, we will consider this correspondence in the particular case that the ordered basis \((E_a)\) forms a symplectic frame of \((\mathbb {R}^{2n}, \omega _{\mathbb {R}^{2n}})\).

5 Reduction

Following Ohsawa [12], we consider in this section the Marsden–Weinstein quotient for the right \(\mathrm {O}(2n)\)-action at a particular value of momentum. Using the correspondence between reduced spaces and coadjoint orbits provided by the dual pair structure, we will give a geometric characterisation of this space. Using this characterisation, we will explain how to naturally introduce a global coordinate chart on the reduced space, reproducing Ohsawa’s description of the space as the Siegel upper half plane.

5.1 Reduction through a general point

It will be more convenient to work with the Lie algebra-valued momentum maps \({\text {j}}_{\mathrm {O}}\) and \({\text {j}}_{\mathrm {Sp}}\). For \(\xi \in {\mathfrak {g}}\), now let \(G_\xi \) denote the adjoint stabiliser of \(\xi \) and \({\mathcal {O}}_\xi \subset {\mathfrak {g}}\) the adjoint orbit through \(\xi \). Writing \(N=\mathrm {GL}(2n,\mathbb {R})\), the reduced space through \(E\in \mathrm {GL}(2n,\mathbb {R})\) is
$$\begin{aligned} N_{{\text {j}}_{\mathrm {O}}(E)} = \frac{{\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(E))}{\mathrm {O}(2n)_{{\text {j}}_{\mathrm {O}}(E)}}. \end{aligned}$$
Using the trace form to transfer the Poisson structure from \({\mathfrak {sp}}(2n,\mathbb {R})^*\) to \({\mathfrak {sp}}(2n,\mathbb {R})\), Proposition 3.4 tells use there exists a \(\mathrm {Sp}(2n,\mathbb {R})\)-equivariant symplectomorphism
$$\begin{aligned} \check{{\text {j}}}_{\mathrm {Sp}}: N_{{\text {j}}_{\mathrm {O}}(E)} \rightarrow {\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(E)}. \end{aligned}$$

5.2 Reduction through the identity

We consider the reduced space for the left \(\mathrm {O}(2n)\)-action through \(I=I_{2n}\in \mathrm {GL}(2n,\mathbb {R})\). From Proposition 2.3(ii), we see that the level set of \({\text {j}}_{\mathrm {O}}\) through I is the set
$$\begin{aligned} {\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(I)) = \lbrace E\in \mathrm {GL}(2n,\mathbb {R})\,|\, -E^\top \mathbb {J} E = -\mathbb {J} \rbrace = \mathrm {Sp}(2n,\mathbb {R}), \end{aligned}$$
which corresponds to the set of symplectic frames of \((\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\). Meanwhile,
$$\begin{aligned} \mathrm {O}(2n)_{{\text {j}}_{\mathrm {O}}(I)} = \lbrace O\in \mathrm {O}(2n)\,|\, \mathrm {Ad}_O(-\mathbb {J}) = -\mathbb {J} \rbrace = \mathrm {Sp}(2n,\mathbb {R})\cap \mathrm {O}(2n)\simeq \mathrm {U}(n), \end{aligned}$$
and we see that the reduced space \(N_{{\text {j}}_{\mathrm {O}}(I)}\) is isomorphic to \(\mathrm {Sp}(2n,\mathbb {R})/\mathrm {U}(n)\), as originally obtained by Ohsawa [12].

5.3 Geometric interpretation of the adjoint orbit \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\)

As explained above, there is a \(\mathrm {Sp}(2n,\mathbb {R})\)-equivariant map
$$\begin{aligned} \check{{\text {j}}}_{\mathrm {Sp}}:N_{{\text {j}}_{\mathrm {O}}(I)} \rightarrow {\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}. \end{aligned}$$
We now give an alternative geometric characterisation of \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\) in terms of complex structures on \(\mathbb {R}^{2n}\).

Definition 5.1

Let \(j:\mathbb {R}^{2n}\rightarrow \mathbb {R}^{2n}\) be a complex structure on \(\mathbb {R}^{2n}\), i.e. an \(\mathbb {R}\)-linear map satisfying \(j^2=-I\). The map j is called \(\omega _{\mathbb {R}^{2n}}\)-compatible if it satisfies the two properties
  1. (i)

    j is symplectic (i.e. \(j\in \mathrm {Sp}(2n,\mathbb {R})\)), and

     
  2. (ii)

    the bilinear form \(g_j(u,v):= \omega _{\mathbb {R}^{2n}}(u,jv)\) is positive definite.

     
Denote the set of \(\omega _{\mathbb {R}^{2n}}\)-compatible complex structures on \(\mathbb {R}^{2n}\) by \({\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\).

Proposition 5.2

The adjoint orbit \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\!\subset \!{\mathfrak {sp}}(2n,\mathbb {R})\) is precisely the set \({\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\).

Proof

An arbitrary element of \(\zeta \in {\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\) is of the form \(\zeta = \mathrm {Ad}_S[{\text {j}}_{\mathrm {Sp}}(I)] = \mathrm {Ad}_S[-\mathbb {J}]\). Since \((-\mathbb {J})^2 = -I\), we have \(\zeta ^2 = -I\), and since \(-\mathbb {J}\in \mathrm {Sp}(2n,\mathbb {R})\), we have \(\zeta = \mathrm {Ad}_S(-\mathbb {J}) \in \mathrm {Sp}(2n,\mathbb {R})\). Finally, writing \(\zeta = -SS^\top \mathbb {J}\), we see that for \(u\in \mathbb {R}^{2n}-\lbrace 0\rbrace \),
$$\begin{aligned} g_\zeta (u,u) = \omega _{\mathbb {R}^{2n}}(u,\zeta u) = u^\top \mathbb {J} (-SS^\top \mathbb {J})u = u^\top (S^\top \mathbb {J})^\top (S^\top \mathbb {J})u>0. \end{aligned}$$
So we have that \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I_{2n})}\subset {\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\).
Conversely, suppose \(j\in {\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\). Then
$$\begin{aligned} g_j(ju,jv) = \omega _{\mathbb {R}^{2n}}(ju,j^2v) = \omega _{\mathbb {R}^{2n}}(u,jv) = g_j(u,v), \end{aligned}$$
the second identity following since j is symplectic. Hence, j is orthogonal with respect to the metric \(g_j\). A standard theorem guarantees the existence of a real orthonormal basis \(\lbrace u_\alpha , v_\alpha \,|\,\alpha =1,\ldots , n\rbrace \) of \(\mathbb {R}^{2n}\) such that \(j(u_\alpha - iv_\alpha ) = e^{i\theta _\alpha }(u_\alpha - iv_\alpha )\). The identity \(j^2 = -I\) implies that \(e^{i\theta _\alpha } = \pm i\) for all \(\alpha \), and by switching \(u_\alpha \) and \(v_\alpha \) if necessary, we can choose \(e^{i\theta _\alpha } = i\) for all \(\alpha \). So
$$\begin{aligned} j u_\alpha = v_\alpha , \qquad jv_\alpha = -u_\alpha . \end{aligned}$$
It can be verified that \((u_1,\ldots , u_n,v_1,\ldots , v_n)\) forms a symplectic basis for \((\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\). For example,
$$\begin{aligned} \omega _{\mathbb {R}^{2n}}(u_\alpha , v_\beta ) = \omega _{\mathbb {R}^{2n}}(u_\alpha , ju_\beta ) = g_j(u_\alpha , u_\beta ) = \delta _{\alpha \beta } \end{aligned}$$
by orthonormality. Taking \(E = \begin{bmatrix}u_1&\ldots&u_n&v_1&\ldots&v_n \end{bmatrix}\in \mathrm {Sp}(2n,\mathbb {R})\), the above relations can be written \(jE = -E\mathbb {J}\), which implies
$$\begin{aligned} j = -E\mathbb {J} E^{-1} = \mathrm {Ad}_E[-\mathbb {J}] =\mathrm {Ad}_E[{\text {j}}_{\mathrm {Sp}}(I)]. \end{aligned}$$
Thus \({\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\subset {\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\). \(\square \)
Adapting diagram (3.2) to our case, we obtainwith \(\check{{\text {j}}}_{\mathrm {Sp}}\) an \(\mathrm {Sp}(2n,\mathbb {R})\)-equivariant symplectomorphism. An equivalent diagram was recently obtained independently by Ohsawa and Tronci [15].

6 The upper half plane coordinates

As demonstrated in the previous section, the adjoint orbit \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\) can be naturally viewed as the space of \(\omega _{\mathbb {R}^{2n}}\)-compatible complex structures \({\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\). It is a standard result that \({\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\) has another natural interpretation, that of the negative Lagrangian Grassmannian \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\). In this section, we outline this correspondence and explain how it can be used to introduce a coordinate system on \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\).

6.1 The negative Lagrangian Grassmannian

Consider the complexification \((\mathbb {R}^{2n})\otimes _{\mathbb {R}}\mathbb {C} \simeq \mathbb {C}^{2n}\) of \(\mathbb {R}^{2n}\). Real linear functions on \(\mathbb {R}^{2n}\) extend to complex linear functions on \(\mathbb {C}^{2n}\) in the obvious way, and we will generally use the same symbol to denote both a function and its complex extension.

Define a sesquilinear form \(s:\mathbb {C}^{2n}\times \mathbb {C}^{2n}\rightarrow \mathbb {C}\) by
$$\begin{aligned} s(w,z) := -i\omega _{\mathbb {R}^{2n}}(w,\overline{z}). \end{aligned}$$

Definition 6.1

The negative Lagrangian Grassmannian\({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\) is the set of (complex) Lagrangian subspaces of \(\mathbb {C}^{2n}\) on which s restricts to a negative definite form,
$$\begin{aligned} {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}}):= \lbrace \varGamma \text { a subspace of }\mathbb {C}^{2n}\,|\,\varGamma \text { is Lagrangian}, s\vert _{\varGamma }<0 \rbrace . \end{aligned}$$

Proposition 6.2

([16, Chapter II, Lemma 7.1]) There is a bijection between \({\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\) and \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\).

Proof

(Sketch) For any \(j\in {\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\), j symplectic implies that it is diagonalisable, and \(j^2=-I\) implies that its eigenvalues are \(\pm i\). The \(-i\)-eigenspace is simply
$$\begin{aligned} \varGamma ^j = \frac{1}{2}(I +ij)\mathbb {R}^{2n} \subset \mathbb {C}^{2n}. \end{aligned}$$
It is straightforward to verify that \(\varGamma ^j\in {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\), and so, we obtain a map
$$\begin{aligned} \varGamma ^\cdot :{\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\rightarrow {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}}). \end{aligned}$$
Conversely, suppose \(\varGamma \in {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\). For any \(w\in \varGamma -\lbrace 0 \rbrace \), the negative definite condition implies that \(-i\omega (w,\overline{w})<0\). Since \(\varGamma \) is Lagrangian, it follows that \(\overline{w}\not \in \varGamma \). Hence, \(\varGamma \cap {\overline{\varGamma }} = \lbrace 0 \rbrace \), and \(\mathbb {C}^{2n} = \varGamma \oplus {\overline{\varGamma }}\). Define the complex linear map \(j_{\varGamma }:\mathbb {C}^{2n}\rightarrow \mathbb {C}^{2n}\) by
$$\begin{aligned} j_{\varGamma }(w) = {\left\{ \begin{array}{ll} -iw &{} w\in \varGamma \\ iw &{} w\in {\overline{\varGamma }} \end{array}\right. }. \end{aligned}$$
Since it commutes with complex commutation, \(j_\varGamma \) restricts to a real linear map \(j_\varGamma :\mathbb {R}^{2n}\rightarrow \mathbb {R}^{2n}\). It is straightforward to verify that \(j_\varGamma \in {\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\), and so, we obtain a map
$$\begin{aligned} j_\cdot : {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\rightarrow {\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}}). \end{aligned}$$
It may be checked that \(\varGamma ^\cdot :{\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\rightarrow {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\) and \(j_\cdot :{\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\rightarrow {\mathcal {J}}(\mathbb {R}^{2n},\omega _{\mathbb {R}^{2n}})\) are inverses of each other. \(\square \)
Following Proposition 6.2, we may extend the commutative diagram (5.1) toWe note that for \(S\in \mathrm {Sp}(2n,\mathbb {R})\),
$$\begin{aligned} \varGamma ^{\mathrm {Ad}_S j} = \frac{1}{2}(I+i\,\mathrm {Ad}_S j)\mathbb {R}^{2n} = \frac{1}{2}(I+i\,\mathrm {Ad}_S j)S\mathbb {R}^{2n} =S\left( \frac{1}{2}(I+ij)\mathbb {R}^{2n} \right) = S\varGamma ^j, \end{aligned}$$
where we have used the invertibility of S to write \(\mathbb {R}^{2n} = S\mathbb {R}^{2n}\) in the second equality. Hence, \(\varGamma ^\cdot \) is \(\mathrm {Sp}(2n,\mathbb {R})\)-equivariant with respect to the natural left action of \(\mathrm {Sp}(2n,\mathbb {R})\) on \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\).

6.2 Coordinates on \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\)

In order to introduce coordinates on \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\), we first introduce the two complex Lagrangian subspaces of \(\mathbb {C}^{2n}\)
$$\begin{aligned} \varGamma _1 = \mathrm {span}_{\mathbb {C}}\lbrace e_1, \ldots , e_n\rbrace \qquad \text {and}\qquad \varGamma _2 = \mathrm {span}_{\mathbb {C}}\lbrace e_{n+1},\ldots ,e_{2n}\rbrace , \end{aligned}$$
where \(\lbrace e_1,\ldots , e_{2n}\rbrace \) is the canonical basis of \(\mathbb {R}^{2n}\). Note that \(\mathbb {C}^{2n} = \varGamma _1\oplus \varGamma _2\), and denote the direct sum projections by \(p_i:\mathbb {C}^{2n}\rightarrow \varGamma _i\). We use the following lemma.

Lemma 6.3

Let \(\varGamma \in {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\). Then the projection \(p_i:\mathbb {C}^{2n}\rightarrow \varGamma _i\) restricts to a bijection on \(\varGamma \), for \(i=1,2\).

Proof

Since all three are Lagrangian, \(\varGamma , \varGamma _1, \varGamma _2\) all have complex dimension n. So we just need to prove that \(p_i\vert _{\varGamma }:\varGamma \rightarrow \varGamma _i\) is injective.

Suppose \(p_1(w)=0\) for \(w\in \varGamma \). Then \(w\in \varGamma _2\), and so, \(\overline{w}\in {\overline{\varGamma }}_2 = \varGamma _2\). Since \(\varGamma _2\) is Lagrangian, we have that \(-i\omega _{\mathbb {R}^{2n}}(w,\overline{w}) = 0\), i.e. \(s(w,w) = 0\). But since \(s\vert _\varGamma <0\), this implies that \(w=0\).

The same argument shows that \(p_2\vert _\varGamma :\varGamma \rightarrow \varGamma _2\) is a bijection. \(\square \)

In light of Lemma 6.3, for any \(\varGamma \in {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\) we may define a map \({\widehat{W}}_\varGamma :\varGamma _1\rightarrow \varGamma _2\) by
$$\begin{aligned} {\widehat{W}}_\varGamma = (p_2\vert _\varGamma )\circ (p_1\vert _\varGamma )^{-1}. \end{aligned}$$
Then for any \(w\in \varGamma \),
$$\begin{aligned} w = (p_1+p_2)\vert _\varGamma (w) = (I+{\widehat{W}}_\varGamma )((p_1\vert _\varGamma )(w)), \end{aligned}$$
demonstrating that
$$\begin{aligned} \varGamma = (I+{\widehat{W}}_\varGamma )\varGamma _1. \end{aligned}$$
(6.2)
It follows from identity (6.2) that the map \(\varGamma \mapsto {\widehat{W}}_\varGamma \) is injective. We now use this representation of \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\) as linear maps to introduce coordinates on \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\). The following lemma is straightforward to verify.

Lemma 6.4

([16, Chapter II, Lemma 7.2’]) Let \({\widehat{W}}:\varGamma _1\rightarrow \varGamma _2\) be a linear map, and define the matrix \(W\in \mathrm {M}_n(\mathbb {C})\) by
$$\begin{aligned} {\widehat{W}} (e_\alpha ) = \sum _{\beta =1}^n W_{\beta \alpha }\, e_{n+\beta }\qquad \text {for }\alpha =1,\ldots ,n. \end{aligned}$$
Then
$$\begin{aligned} (I+{\widehat{W}})\varGamma _1 \in {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\iff W^\top = W \text { and }\mathrm {Im}\,W>0. \end{aligned}$$
The set
$$\begin{aligned} \varSigma _n = \lbrace W\in \mathrm {M}_{n}(\mathbb {C}) \,|\, W^\top = W,\, {\text {Im}}\,W>0 \rbrace \end{aligned}$$
arising in Lemma 6.4 is traditionally referred to as the Siegel upper half plane. It has a well-known symplectic structure. In Sect. 6.5, we show this structure is the pushforward of the usual Kostant–Kirillov–Souriau symplectic structure on \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\) under the bijection \(\varGamma ^\cdot \) of diagram (6.1).

6.3 The action of \(\mathrm {Sp}(2n,\mathbb {R})\) in upper half plane coordinates

There is a natural action of \(\mathrm {Sp}(2n,\mathbb {R})\) on \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\), given by \((S,\varGamma ) \mapsto S\varGamma \). We wish to express this action in terms of upper half plane coordinates, i.e. to express \(W_{S\varGamma }\) in terms of S and \(W_\varGamma \).

Since \(\varGamma _1 = \mathrm {span}_{\mathbb {C}}\lbrace e_1,\ldots , e_n\rbrace \), we see from Eq. (6.2) that a basis for \(\varGamma \in {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\) is given by
$$\begin{aligned} e_\alpha ' := (I+{\widehat{W}}_\varGamma )e_\alpha = e_\alpha + \sum _{\beta =1}^n(W_\varGamma )_{\beta \alpha }e_{n+\beta },\qquad \alpha =1,\ldots , n. \end{aligned}$$
It follows that \(S\varGamma \) has basis
$$\begin{aligned} e_\alpha '' = Se_\alpha ' = Se_\alpha + \sum _{\beta =1}^n(W_\varGamma )_{\beta \alpha }\, Se_{n+\beta },\qquad \alpha = 1,\ldots , n. \end{aligned}$$
Writing \(S=\begin{bmatrix} A&\quad B \\ C&\quad D \end{bmatrix}\), this becomes
$$\begin{aligned} e_\alpha '' = \sum _{\gamma =1}^n\left[ (A+BW_\varGamma )_{\gamma \alpha }e_\gamma + (C+DW_\varGamma )_{\gamma \alpha }e_{n+\gamma }\right] ,\qquad \alpha =1,\ldots ,n. \end{aligned}$$
Now recall that the coordinates \(W_{S\varGamma }\) of \(S\varGamma \) are simply the matrix elements of \({\widehat{W}}_{S\varGamma } = (p_2\vert _{S\varGamma })\circ (p_1\vert _{S\varGamma })^{-1}:\varGamma _1\rightarrow \varGamma _2\). We have
$$\begin{aligned} p_1(e_\alpha '')&= \sum _{\gamma =1}^n (A+BW_\varGamma )_{\gamma \alpha }e_\gamma \\&\implies (p_1\vert _{S\varGamma })^{-1}(e_\delta ) = \sum _{\alpha =1}^n[(A+BW_\varGamma )^{-1}]_{\alpha \delta }\,e_\alpha '' \\&\implies {\widehat{W}}_{S\varGamma } (e_\delta ) \!=\! (p_2\vert _{S\varGamma })\left( (p_1\vert _{S\varGamma })^{-1}(e_\delta )\right) \!=\!\sum _{\alpha =1}^n[(A+BW_\varGamma )^{-1}]_{\alpha \delta }(p_2(e_\alpha ''))\\&= \sum _{\gamma =1}^n [(C+DW_\varGamma )(A+BW_\varGamma )^{-1}]_{\gamma \delta }\, e_{n+\gamma }. \end{aligned}$$
It follows that for \(S=\begin{bmatrix} A&B \\ C&D\end{bmatrix}\),
$$\begin{aligned} W_{S\varGamma } = (C+DW_\varGamma )(A+BW_\varGamma )^{-1}. \end{aligned}$$
(6.3)

6.4 The projection in upper half plane coordinates

Referring again to diagram (6.1), we have demonstrated the existence of an \(\mathrm {Sp}(2n,\mathbb {R})\)-equivariant map
$$\begin{aligned} \varGamma ^{{\text {j}}_{\mathrm {Sp}}(\cdot )}:{\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(I))\rightarrow {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}}). \end{aligned}$$
The set \({\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(I))\subset \mathrm {GL}(2n,\mathbb {R})\) is simply \(\mathrm {Sp}(2n,\mathbb {R})\), and we have shown that the Siegel upper half plane provides a global coordinate chart on the manifold \({\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\). Using the results of the previous section, we can derive an expression for the projection \(\varGamma ^{{\text {j}}_{\mathrm {Sp}}(\cdot )}\) in terms of the coordinates on its domain and range manifolds.
First note that \(\varGamma ^{{\text {j}}_{\mathrm {Sp}}(I)} = \frac{1}{2}(I+i{\text {j}}_{\mathrm {Sp}}(I))\mathbb {R}^{2n} = \frac{1}{2}(I-i\mathbb {J})\mathbb {R}^{2n}\) has a basis
$$\begin{aligned} \lbrace e_\alpha + i e_{n+\alpha }\,|\,\alpha =1,\ldots n\rbrace . \end{aligned}$$
It follows easily from Eq. (6.2) that
$$\begin{aligned} W_{\varGamma ^{{\text {j}}_{\mathrm {Sp}}(I)}} = iI\in \mathrm {M}_n(\mathbb {C}). \end{aligned}$$
By equivariance of \(\varGamma ^{{\text {j}}_{\mathrm {Sp}}(\cdot )}\), we have that \(\varGamma ^{{\text {j}}_{\mathrm {Sp}}(S)} = S\varGamma ^{{\text {j}}_{\mathrm {Sp}}(I)}\) for any \(S\in {\text {j}}_{\mathrm {Sp}}^{-1}({\text {j}}_{\mathrm {Sp}}(I))\), and so, \(W_{\varGamma ^{{\text {j}}_{\mathrm {Sp}}(S)}} = W_{S\varGamma ^{{\text {j}}_{\mathrm {Sp}}(I)}}\). Writing \(S = \begin{bmatrix}Q_1&Q_2 \\ P_1&P_2\end{bmatrix}\) for the coordinates on \({\text {j}}_{\mathrm {Sp}}^{-1}({\text {j}}_{\mathrm {Sp}}(I))\), and using Eq. (6.3), this becomes
$$\begin{aligned} W_{\varGamma ^{{\text {j}}_{\mathrm {Sp}}(S)}} = (P_1+P_2W_{\varGamma ^{{\text {j}}_{\mathrm {Sp}}(I)}})(Q_1+Q_2W_{\varGamma ^{{\text {j}}_{\mathrm {Sp}}(I)}})^{-1} = (P_1+iP_2)(Q_1+iQ_2)^{-1}. \end{aligned}$$
In summary, the projection written in (global) coordinates is
$$\begin{aligned} \varGamma ^{{\text {j}}_{\mathrm {Sp}}(\cdot )}:\begin{bmatrix} Q_1&Q_2 \\ P_1&P_2 \end{bmatrix} \mapsto (P_1+iP_2)(Q_1+iQ_2)^{-1}. \end{aligned}$$
(6.4)

6.5 Upper half plane coordinates on \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\)

Using \(\varGamma ^\cdot :{\mathcal {O}}_{{\text {j}}_{\mathrm {O}}(I)}\rightarrow {\mathcal {G}}_-(\mathbb {C}^{2n},\omega _{\mathbb {R}^{2n}})\) to pull coordinates back to \({\mathcal {O}}_{{\text {j}}_{\mathrm {O}}(I)}\), we may instead interpret (6.4) as the coordinate expression for the momentum map \({\text {j}}_{\mathrm {Sp}}:{\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(I))\rightarrow {\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\)
$$\begin{aligned} {\text {j}}_{\mathrm {Sp}}:\begin{bmatrix} Q_1&Q_2 \\ P_1&P_2 \end{bmatrix} \mapsto (P_1+iP_2)(Q_1+iQ_2)^{-1}. \end{aligned}$$
(6.5)
This is in agreement with [12, Equation (21)], which was obtained by a method involving the Iwasawa decomposition of \(\mathrm {Sp}(2n,\mathbb {R})\).
For \(\eta \in {\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\), let \(W_\eta \in \varSigma _n\) denote the corresponding coordinate. Then Eq. (6.3) becomes
$$\begin{aligned} W_{\mathrm {Ad}_S\eta } = (C+DW_\eta )(A+BW_\eta )^{-1}. \end{aligned}$$
(6.6)
It was shown by Siegel (see, for example, [17, Chapter 6, Section 3]) that \(\varSigma _n\) is a Kähler manifold, with Hermitian form \(H_{\varSigma _n}={\text {Tr}}\,({\mathcal {B}}^{-1}d\overline{W} \otimes {\mathcal {B}}^{-1}dW)\) and symplectic form \(\varOmega _{\varSigma _n} = \mathrm {Im}\, H^{\varSigma _n}\), i.e.
$$\begin{aligned} \varOmega _{\varSigma _n} = {\text {Tr}}\,({\mathcal {B}}^{-1} d{\mathcal {A}}\otimes {\mathcal {B}}^{-1} d{\mathcal {B}} - {\mathcal {B}}^{-1}d{\mathcal {B}} \otimes {\mathcal {B}}^{-1} d{\mathcal {A}}). \end{aligned}$$
(We remind that \({\mathcal {A}}:= {\text {Re}}\, W,\, {\mathcal {B}}:= {\text {Im}}\, W\).) Siegel also proves that the forms \(H_{\varSigma _n}\), and so \(\varOmega _{\varSigma _n}\), are invariant under pullback by the \(\mathrm {Sp}(2n,\mathbb {R})\)-action \(W\mapsto (C+DW)(A+BW)^{-1}\).
Correspondingly, the standard Kostant–Kirillov–Souriau symplectic form \(\varOmega ^+_{\mathcal {O}}\) on the adjoint orbit \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)} = {\mathcal {O}}_{-\mathbb {J}}\), given by
$$\begin{aligned} (\varOmega ^+_{\mathcal {O}})_{\eta }(\mathrm {ad}_\xi \eta , \mathrm {ad}_\zeta \eta ) = \langle \langle \eta , [\xi ,\zeta ]\rangle \rangle , \end{aligned}$$
for \(\eta \in {\mathcal {O}}_{-\mathbb {J}},\ \xi ,\zeta \in {\mathfrak {sp}}(2n,\mathbb {R})\), is invariant under the adjoint \(\mathrm {Sp}(2n,\mathbb {R})\)-action.

Hence, if we can show that \(\varOmega _{\varSigma _n}\) and \(\varOmega ^+_{\mathcal {O}}\) are proportional at \(-\mathbb {J}\) (corresponding to coordinate \(W=iI\)), then they are proportional on the entire adjoint orbit \({\mathcal {O}}_{-\mathbb {J}}\).

To see this is the case, let \(\xi = \begin{bmatrix} \xi _{11}&\xi _{12} \\ \xi _{21}&\xi _{22} \end{bmatrix}\in {\mathfrak {sp}}(2n,\mathbb {R})\). From (6.6), we deduce
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\bigg \vert _{t=0} W_{\mathrm {Ad}_{\exp t\xi }(-\mathbb {J})} = (\xi _{21}+i\xi _{22}) - i(\xi _{11}+i\xi _{12}), \end{aligned}$$
implying that
$$\begin{aligned} \mathrm {ad}_\xi (-\mathbb {J}) = (\xi _{12}+\xi _{21})\frac{\partial }{\partial {\mathcal {A}}} \bigg \vert _{iI} + (\xi _{22}-\xi _{11})\frac{\partial }{\partial {\mathcal {B}}}\bigg \vert _{iI}. \end{aligned}$$
(6.7)
Note that \(\xi ^\top \mathbb {J} + \mathbb {J}\xi = 0 \implies \xi _{12}^\top = \xi _{12},\, \xi _{21}^\top = \xi _{21},\,\xi _{22}^\top = -\xi _{11}\), and so, both coefficient matrices in (6.7) are symmetric.
We have that
$$\begin{aligned}&(\varOmega ^+_{\mathcal {O}})_{-\mathbb {J}}(\mathrm {ad}_\xi (-\mathbb {J}),\, \mathrm {ad}_\zeta (-\mathbb {J})) = \langle \langle -\mathbb {J}, [\xi ,\zeta ]\rangle \rangle \nonumber \\&\quad = \frac{1}{2}{\text {Tr}}\,\left( -(\xi _{21}\zeta _{11} + \xi _{22}\zeta _{21}) + (\xi _{11}\zeta _{12} + \xi _{12}\zeta _{22}) - (\xi \leftrightarrow \zeta ) \right) . \end{aligned}$$
(6.8)
Meanwhile, at \(W=iI\) the symplectic form on \(\varSigma _n\) simplifies to
$$\begin{aligned} (\varOmega _{\varSigma _n})_{iI} = {\text {Tr}}\,(d{\mathcal {A}}\otimes d{\mathcal {B}} - d{\mathcal {B}}\otimes d{\mathcal {A}})_{iI}, \end{aligned}$$
and so, (6.7) gives
$$\begin{aligned} (\varOmega _{\varSigma _n})_{iI}(\mathrm {ad}_\xi (-\mathbb {J}), \mathrm {ad}_\zeta (-\mathbb {J})) = {\text {Tr}}\,((\xi _{12}+\xi _{21})(\zeta _{22}-\zeta _{11}) - (\xi _{22}-\xi _{11})(\zeta _{12}+\zeta _{21})). \end{aligned}$$
Comparison with (6.8) shows that \((\varOmega ^+_{\mathcal {O}})_{-\mathbb {J}} = \frac{1}{2}(\varOmega _{\varSigma _n})_{iI}\), and therefore,
$$\begin{aligned} \varOmega ^+_{\mathcal {O}} = \frac{1}{2}\varOmega _{\varSigma _n}. \end{aligned}$$

7 Dynamics

In this section, we finally come to the main point of the previous constructions, namely to describe the dynamics of semiclassical Gaussian wave packets in terms of Hamiltonian dynamics on the frame bundle of \(T^*\mathbb {R}^n\), and its symplectic reduction.

7.1 The Gaussian wave packet ansatz

Consider the time-dependent Schrödinger equation on \(\mathbb {R}^n\),
$$\begin{aligned} i\hbar \frac{\partial \psi }{\partial t}(x,t) = -\frac{\hbar ^2}{2m}\varDelta \psi (x,t) + V(x)\,\psi (x,t), \end{aligned}$$
(7.1)
where the potential V(x) is at most quadratic in x. Following [9], we recall here the Gaussian wave packet ansatz, in both the Heller [6] and Hagedorn [5] parametrisations.
In Heller’s parametrisation, the wave function is written
$$\begin{aligned} \begin{aligned} \psi (x,t)&= \left( \frac{\det {\mathcal {B}}(t)}{(\pi \hbar )^n} \right) ^{\frac{1}{4}} \exp \bigg \lbrace \frac{i}{\hbar } \bigg [\frac{1}{2}(x-q(t))^\top ({\mathcal {A}}(t)+i{\mathcal {B}}(t))(x-q(t)) \\&\quad +\, p(t)^\top (x-q(t)) + \phi (t) \bigg ] \bigg \rbrace , \end{aligned} \end{aligned}$$
(7.2)
with \((q,p)\in T^*\mathbb {R}^n\simeq \mathbb {R}^{2n}\), \(\phi \) a real phase, and \({\mathcal {A}}+i{\mathcal {B}}\in \varSigma _n\) with \({\mathcal {A}},{\mathcal {B}}\) real. Substitution of the ansatz into Eq. (7.1) shows that it produces a solution provided the following equations are satisfied
$$\begin{aligned} {\dot{q}}&= \frac{p}{m}, \qquad {\dot{p}} = -D_q V, \nonumber \\ {\dot{{\mathcal {A}}}}&= -\frac{1}{m}({\mathcal {A}}^2 - {\mathcal {B}}^2) - D^2_q V, \quad {\dot{{\mathcal {B}}}}= -\frac{1}{m}({\mathcal {A}}{\mathcal {B}}+ {\mathcal {B}}{\mathcal {A}}),\nonumber \\ {\dot{\phi }}&= \frac{p^2}{2m} - V(q) - \frac{\hbar }{2m}{\text {Tr}}\,{\mathcal {B}}. \end{aligned}$$
(7.3)
Here, \(D^2V\) denotes the Hessian of V.
In Hagedorn’s parametrisation, the wave function is
$$\begin{aligned} \begin{aligned} \psi (x,t)&= \frac{1}{(\pi \hbar )^{\frac{n}{4}}} \frac{1}{(\det Q(t))^{\frac{1}{2}}} \exp \bigg \lbrace \frac{i}{\hbar } \bigg [\frac{1}{2}(x-q(t))^\top P(t)Q(t)^{-1}(x-q(t)) \\&\quad +\, p(t)^\top (x-q(t)) + S(t) \bigg ] \bigg \rbrace , \end{aligned} \end{aligned}$$
(7.4)
for some appropriate choice of the square root in the denominator, with \((q,p)\in \mathbb {R}^{2n}\), S a real phase, and \(Q, P\in \mathrm {M}_n(\mathbb {C})\) satisfying
$$\begin{aligned} Q^\top P - P^\top Q = 0 \qquad \text {and}\qquad Q^\dagger P - P^\dagger Q = 2iI_n. \end{aligned}$$
Substitution into Eq. (7.1) yields
$$\begin{aligned} {\dot{q}} = \frac{p}{m}, \, {\dot{p}} = -D_q V, \, {\dot{Q}} = \frac{P}{m}, \, {\dot{P}} = -D^2_q V\,Q, \, {\dot{S}} = \frac{p^2}{2m} - V(q). \end{aligned}$$
(7.5)
The two parametrisations (7.2) and (7.4) are related by
$$\begin{aligned} W = PQ^{-1}, \qquad S = \phi - \frac{\hbar }{2}\mathrm {arg}(\det Q). \end{aligned}$$
(7.6)

7.2 Description as Hamiltonian dynamics on the frame bundle

Given a symplectic manifold \((M,\omega )\), recall that the a frame at \(z\in M\) is an ordered basis of \(T_z M\), and the frame bundle\({\mathcal {F}}(M)\) consists of all such ordered bases as z ranges over M. \({\mathcal {F}}(M)\) is naturally a right principal \(\mathrm {GL}(2n,\mathbb {R})\)-bundle, with group action
$$\begin{aligned} (v_1,\ldots , v_{2n})\cdot E = \left( \sum _{a=1}^{2n} v_a E_{a1}, \sum _{a=1}^{2n} v_a E_{a2}, \ldots , \sum _{a=1}^{2n} v_a E_{a,2n} \right) . \end{aligned}$$
(7.7)
We say that a frame \((v_a)\) at \(z\in M\) is symplectic if
$$\begin{aligned} \omega _z(v_a,v_b) = \mathbb {J}_{ab}\qquad a,b=1,\ldots ,2n, \end{aligned}$$
and a local section \(\varvec{s}:U\subset M\rightarrow {\mathcal {F}}(M)\) of the frame bundle is symplectic if \(\varvec{s}(z)\) is a symplectic frame for each \(z\in U\).
Consider now the case when \(M=T^*\mathbb {R}^n\) and \(\omega = \sum _{\alpha =1}^n dq^\alpha \wedge dp_\alpha \) is the usual cotangent bundle symplectic form. We will write the natural coordinates on \(T^*\mathbb {R}^n\) as either \((z_1,\ldots , z_{2n})\) or \((q^1,\ldots , q^n, p_1,\ldots , p_n)\), depending on situation. The bundle \({\mathcal {F}}(T^*\mathbb {R}^n)\) has a global symplectic frame \(\varvec{s}:T^*\mathbb {R}^n\rightarrow {\mathcal {F}}(T^*\mathbb {R}^n)\), given by
$$\begin{aligned} \varvec{s}(z) := \left( \frac{\partial }{\partial q^1}\bigg \vert _z, \ldots , \frac{\partial }{\partial q^n}\bigg \vert _z,\frac{\partial }{\partial p_1}\bigg \vert _z, \ldots ,\frac{\partial }{\partial p_n}\bigg \vert _z\right) = \left( \frac{\partial }{\partial z_1}\bigg \vert _z, \ldots , \frac{\partial }{\partial z_{2n}}\bigg \vert _z\right) , \end{aligned}$$
which induces a global trivialisation \(\varLambda :T^*\mathbb {R}^n \times \mathrm {GL}(2n,\mathbb {R})\rightarrow {\mathcal {F}}(T^*\mathbb {R}^n)\) of the frame bundle, given by
$$\begin{aligned} \varLambda (z,E) = \varvec{s}(z)\cdot E = \left( \sum _{a=1}^{2n}E_{a1}\frac{\partial }{\partial z_a}\bigg \vert _z, \ldots , \sum _{a=1}^{2n}E_{a,2n}\frac{\partial }{\partial z_a}\bigg \vert _z \right) . \end{aligned}$$
We define the \(\hbar \)-diagonal lifted symplectic form \(\varOmega ^{\hbar }\) on \(T^*\mathbb {R}^n\times \mathrm {GL}(2n,\mathbb {R})\) by
$$\begin{aligned} \varOmega ^{\hbar }_{(z,E)}((v_z, V_E), (w_z,W_E)) := \omega _z(v_z,w_z) + \frac{\hbar }{2}\varOmega _E(V_E,W_E). \end{aligned}$$
(7.8)
By analogy with [12], for any Hamiltonian \(H:T^*\mathbb {R}^n\rightarrow \mathbb {R}\) we define the lifted Hamiltonian on the frame bundle \(H^{\hbar }:T^*\mathbb {R}\times \mathrm {GL}(2n,\mathbb {R})\rightarrow \mathbb {R}\) to be
$$\begin{aligned} H^{\hbar }(z,E) := H(z) + \frac{\hbar }{4}{\text {Tr}}\,(E^\top D^2_z H E), \end{aligned}$$
(7.9)
where \(D^2_zH\) denotes the Hessian of H evaluated at z.
We can give a more geometric interpretation to (7.9) as follows: For any \(E\in \mathrm {GL}(2n,\mathbb {R})\), let g(E) denote the metric on \(T^*\mathbb {R}^n\) with respect to which the global frame \(\varLambda (z,E) = \varvec{s}(z)\cdot E\) is orthonormal, i.e. the vector fields
$$\begin{aligned} e_a(z) = \sum _{b=1}^{2n}E_{ba}\frac{\partial }{\partial z_b},\quad a=1,\ldots ,2n, \end{aligned}$$
form an orthonormal basis at each point. Explicitly
$$\begin{aligned} g(E)\left( \frac{\partial }{\partial z_a}, \frac{\partial }{\partial z_b}\right) = [(EE^\top )^{-1}]_{ab}. \end{aligned}$$
(7.10)
Then the Laplacian \(\varDelta _{g(E)}:C^\infty (T^*\mathbb {R}^n)\rightarrow C^\infty (T^*\mathbb {R}^n)\) associated with the metric g(E) is given for \(H\in C^\infty (T^*\mathbb {R}^n)\) by
$$\begin{aligned} \varDelta _{g(E)}H = \sum _{a=1}^{2n} e_a (e_a H) = \sum _{a,b,c=1}^{2n}E_{ba}E_{ca}\frac{\partial ^2 H}{\partial z_b\partial z_c} = {\text {Tr}}\,(E^\top D^2H E), \end{aligned}$$
and so, (7.9) can be alternatively written as
$$\begin{aligned} H^{\hbar }(z,E) = H(z) + \frac{\hbar }{4}(\varDelta _{g(E)}H)(z). \end{aligned}$$
(7.11)
To find the Hamiltonian vector field corresponding to \(H^{\hbar }\), note that
$$\begin{aligned} d_{(z,E)}H^{\hbar } = (d_1)_z \left( H + \frac{\hbar }{4} \varDelta _{g(E)}H\right) + (d_2)_E \left( \frac{\hbar }{4}\varDelta _{g(\cdot )}H(z)\right) , \end{aligned}$$
where \(d_1, d_2\) denote the exterior derivatives in \(T^*\mathbb {R}^n\) and \(\mathrm {GL}(2n,\mathbb {R})\), respectively. The first term can be written
$$\begin{aligned} i_{X_{H+ \frac{\hbar }{4}\varDelta _{g(E)}H}}\omega . \end{aligned}$$
The second term satisfies
$$\begin{aligned} (d_2)_E\left( \frac{\hbar }{4}\varDelta _{g(\cdot )}H(z)\right) (V_E)&= \frac{\hbar }{4} {\text {Tr}}\,\left( V^\top D^2_zH\,E + E^\top D^2_zH\, V \right) \\&= \frac{\hbar }{2}{\text {Tr}}\,\left( E^\top D^2_zH\,V \right) \\&= \frac{\hbar }{2}\varOmega _E((\mathbb {J}D^2_z H)\cdot E, V_E) \\&= \frac{\hbar }{2}\left( i_{(\mathbb {J}D^2_zH)\cdot E}\varOmega _E \right) (V_E). \end{aligned}$$
Here we have used the identity \(\zeta \cdot E = (\zeta E)_E\). Overall
$$\begin{aligned} d_{(z,E)} H^{\hbar } = i_{(X_{H+\frac{\hbar }{4}\varDelta _{g(E)}H}(z), (\mathbb {J}D^2_ZH)\cdot E)}\varOmega ^{\hbar }_{(z,E)}, \end{aligned}$$
i.e.
$$\begin{aligned} X_{H^{\hbar }}(z,E) = \left( X_{H+\frac{\hbar }{4}\varDelta _{g(E)}H}(z), (\mathbb {J}D^2_zH)\cdot E \right) . \end{aligned}$$
(7.12)
In the situation where \(H:T^*\mathbb {R}^n\rightarrow \mathbb {R}\) is at most quadratic in the coordinates \((q^1,\ldots , q^n,p_1,\ldots , p_n)\), the Hessian \(D^2_zH\), and hence \(\varDelta _{g(E)}H\), will be constant, and the Hamiltonian vector field \(X_{H^{\hbar }}\) simplifies to
$$\begin{aligned} X_{H^{\hbar }}(z,E) = \left( X_H(z), (\mathbb {J}D^2_zH)\cdot E \right) = \left( \mathbb {J}D_zH, (\mathbb {J}D^2_zH)\cdot E \right) . \end{aligned}$$
In this case, motion on the frame bundle consists of classical motion on the base space \(T^*\mathbb {R}^n\) and the corresponding induced linearised motion on the frame. This picture of semiclassical wave mechanics has much in common with the nearby orbit approximation discussed by Littlejohn [7, Section 7]— semiclassical motion consists of the classical motion, plus motion of a frame moving along the classical trajectory, and describing to first order the classical flow relative to it.
In particular, taking the standard Hamiltonian
$$\begin{aligned} H(q,p) = \frac{p^2}{2m} + V(q) \end{aligned}$$
and writing \(E = \begin{bmatrix} Q_1&Q_2 \\ P_1&P_2 \end{bmatrix}\) yields the first four equations of the Hagedorn equations (7.5).

In the general case (7.12) where H is not quadratic, there is an additional correction term in the motion on the base space, leading to a deviation from strictly classical motion. This deviation has recently been proposed as a means of introducing quantum tunnelling into semiclassical quantum mechanics [13].

7.3 \(\mathrm {O}(2n)\)-invariance and conservation of symplectic frame

The right \(\mathrm {GL}(2n,\mathbb {R})\)-action (7.7) on the bundle \({\mathcal {F}}(T^*\mathbb {R}^n) \simeq T^*\mathbb {R}^n\times \mathrm {GL}(2n,\mathbb {R})\) restricts to a right \(\mathrm {O}(2n)\)-action, and the lifted Hamiltonian \(H^{\hbar }\) is easily seen to be invariant under this restricted action. By Noether’s theorem, the corresponding momentum map \({\text {j}}_{\mathrm {O}}\circ \pi _2:T^*\mathbb {R}^n\times \mathrm {GL}(2n,\mathbb {R})\rightarrow {\mathfrak {o}}(2n)\) is conserved under the dynamical evolution corresponding to \(H^{\hbar }\), i.e.
$$\begin{aligned} {\text {j}}_{\mathrm {O}}(E) = -E^\top \mathbb {J} E \end{aligned}$$
is conserved. In particular, the previously considered level set
$$\begin{aligned} {\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(I)) = \lbrace E\in \mathrm {GL}(2n,\mathbb {R})\,|\,-E^\top \mathbb {J}E = -\mathbb {J} \rbrace \end{aligned}$$
simply corresponds to the collection of symplectic frames at a point, and conservation of \({\text {j}}_{\mathrm {O}}\) says that such a frame remains symplectic under dynamical evolution.

7.4 Description in the reduced space

Since we have a Hamiltonian \(H^{\hbar }\) that is invariant under the right \(\mathrm {O}(2n)\)-action, the dynamics on \({\mathcal {F}}(T^*\mathbb {R}^n)\) will drop to the symplectic quotient. Taking the symplectic quotient at momentum \({\text {j}}_{\mathrm {O}}(I) = -\mathbb {J}\), and using the notation of diagram (6.1), gives
$$\begin{aligned} \frac{({\text {j}}_{\mathrm {O}}\circ \pi _2)^{-1}({\text {j}}_{\mathrm {O}}(I))}{\mathrm {O}(2n)_{{\text {j}}_{\mathrm {O}}(I)}} = T^*\mathbb {R}^n \times \frac{{\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(I))}{\mathrm {O}(2n)_{{\text {j}}_{\mathrm {O}}(I)}} = T^*\mathbb {R}^n\times N_{{\text {j}}_{\mathrm {O}}(I)}, \end{aligned}$$
and we have the corresponding extension of diagram (5.1)
The map \({\text {id}}\times \check{{\text {j}}}_{\mathrm {Sp}}\) is a symplectomorphism, and so, we can equivalently describe the reduced dynamics on \(T^*\mathbb {R}^n\times {\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\). Rewriting Eq. (7.9) as
$$\begin{aligned} H^{\hbar }(z,E) = H(z) + \frac{\hbar }{4}{\text {Tr}}\,(-EE^\top \mathbb {J}^2 D^2_z H) = H(z) + \frac{\hbar }{4}{\text {Tr}}\,({\text {j}}_{\mathrm {Sp}}(E)\,\mathbb {J}D^2_zH), \end{aligned}$$
we see that
$$\begin{aligned} H^{\hbar } = \check{H}^{\hbar }\circ ({\text {id}}\times {\text {j}}_{\mathrm {Sp}}), \end{aligned}$$
where \(\check{H}^{\hbar }: T^*\mathbb {R}^n\times {\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\rightarrow \mathbb {R}\) is the reduced Hamiltonian
$$\begin{aligned} \check{H}^{\hbar }(z,\zeta ) := H(z) + \frac{\hbar }{4}{\text {Tr}}\,(\zeta \, \mathbb {J}D^2_zH). \end{aligned}$$
(7.13)
In fact, from the expression (7.10) for the metric g(E), we see that
$$\begin{aligned} g(E)\left( \frac{\partial }{\partial z_a},\frac{\partial }{\partial z_b}\right) = [({\text {j}}_{\mathrm {Sp}}(E)\mathbb {J})^{-1}]_{ab}, \end{aligned}$$
and so, the metric g(E) drops to a metric \(\check{g}(\zeta )\) on \(T^*\mathbb {R}^n\), depending on \(\zeta \in {\mathfrak {sp}}(2n,\mathbb {R})\),
$$\begin{aligned} \check{g}(\zeta )\left( \frac{\partial }{\partial z_a},\frac{\partial }{\partial z_b} \right) = [(\zeta \mathbb {J})^{-1}]_{ab}. \end{aligned}$$
Then the reduced Hamiltonian (7.13) can be written, analogously to (7.11), as
$$\begin{aligned} \check{H}^{\hbar }(z,\zeta ) := H(z) + \frac{\hbar }{4} (\varDelta _{\check{g}(\zeta )}H)(z). \end{aligned}$$
(7.14)
Since \(H^{\hbar } = \check{H}^{\hbar }\circ ({\text {id}}\times {\text {j}}_{\mathrm {Sp}})\), and \({\text {id}}\times {\text {j}}_{\mathrm {Sp}}\) is a Poisson map, the Hamiltonian vector fields
$$\begin{aligned} X_{H^{\hbar }}(z,E) = \left( X_{H+ \frac{\hbar }{4}\varDelta _{g(E)} H}(z), (\mathbb {J} D^2_zH)\cdot E \right) \end{aligned}$$
and
$$\begin{aligned} X_{\check{H}^{\hbar }}(z,\zeta ) = \left( X_{H+\frac{\hbar }{4}\varDelta _{\hat{g}(\zeta )}H}(z), \mathrm {ad}_{\mathbb {J} D^2_z H} \zeta \right) \end{aligned}$$
are \(({\text {id}} \times {\text {j}}_{\mathrm {Sp}})\)-related,
$$\begin{aligned} T({\text {id}}\times {\text {j}}_{\mathrm {Sp}})\circ X_{H^{\hbar }} = X_{\check{H}^{\hbar }}\circ ({\text {id}}\times {\text {j}}_{\mathrm {Sp}}). \end{aligned}$$
Again specialising to quadratic Hamiltonians, these \(({\text {id}}\times {\text {j}}_{\mathrm {Sp}})\)-related vector fields are
$$\begin{aligned} X_{H^{\hbar }}(z,E) = (X_H(z), (\mathbb {J}D^2_zH)\cdot E) \end{aligned}$$
and
$$\begin{aligned} X_{\check{H}^{\hbar }}(z,\zeta ) = (X_H(z), \mathrm {ad}_{\mathbb {J}D^2_zH}\zeta ). \end{aligned}$$
The unreduced dynamics, generated by \(X_{H^{\hbar }}\) and reproducing the Hagedorn equations (7.5), will drop to the reduced dynamics generated by \(X_{\check{H}^{\hbar }}\), and these will reproduce the Heller equations (7.3), by virtue of the relationship (6.5) expressing \({\text {j}}_{\mathrm {Sp}}\) in coordinates on \({\text {j}}_{\mathrm {O}}^{-1}({\text {j}}_{\mathrm {O}}(I))\) and \({\mathcal {O}}_{{\text {j}}_{\mathrm {Sp}}(I)}\).

8 Generalisation to other symplectic manifolds

Now that we have outlined the construction of a Hamiltonian system on the frame bundle of \(T^*\mathbb {R}^n\), an obvious question is whether this construction can be extended to more general symplectic manifolds. In this section, we suggest a generalisation by employing the results of Cordero and de León [3]. Essentially they show that the frame bundle can be provided with a symplectic structure if it has a trivialisation compatible with the symplectic structure, encoded in the form of a symplectic connection. Their construction may be seen as the symplectic analogue of the Sasaki–Mok metric on the frame bundle of a Riemannian manifold [10].

8.1 The almost-symplectic form on the frame bundle

Let \((M,\omega )\) be an arbitrary 2n-dimensional symplectic manifold, and let \(\pi :{\mathcal {F}}(M)\rightarrow M\) denote its frame bundle, which is a principal right \(\mathrm {GL}(2n,\mathbb {R})\)-bundle in the usual way. For a frame \(\varvec{e} = (e_1,\ldots , e_{2n}) \in {\mathcal {F}}(M)\), and \(\xi \in {\mathfrak {gl}}(2n,\mathbb {R})\), define
$$\begin{aligned} \varvec{e}\cdot \xi := \frac{\mathrm {d}}{\mathrm {d}s}\bigg \vert _{s=0} \varvec{e}\cdot \exp (s\xi ) \in V_{\varvec{e}}{\mathcal {F}}(M) \subset T_{\varvec{e}}{\mathcal {F}}(M), \end{aligned}$$
where \(V_{\varvec{e}}\) denotes the vertical space at \(\varvec{e}\in {\mathcal {F}}(M)\). Given a connection 1-form \(A\in \varOmega ^2({\mathcal {F}}(M),{\mathfrak {gl}}(2n,\mathbb {R}))\), Cordero and de León [3] define two natural lifts of the symplectic 2-form \(\omega \) to the frame bundle \({\mathcal {F}}(M)\):
  • the vertical lift \(\omega ^{\mathrm {ver}} = \pi ^*\omega \);

  • the horizontal lift \(\omega ^{\mathrm {hor}}\), given by
    $$\begin{aligned} \omega ^{\mathrm {hor}}_{\varvec{e}} (X_{\varvec{e}}, Y_{\varvec{e}}) := \left( \bigoplus _{a=1}^{2n} \omega \right) _{\varvec{e}} (\varvec{e}\cdot A_{\varvec{e}}(X_{\varvec{e}}), \varvec{e}\cdot A_{\varvec{e}}(X_{\varvec{e}})). \end{aligned}$$
In the definition of the horizontal lift, we have used the canonical isomorphism \(\oplus _{a=1}^{2n}T_z M \simeq V_{\varvec{e}}{\mathcal {F}}(M)\) (for \(z=\pi (\varvec{e}))\) given by
$$\begin{aligned} (v_1, \ldots , v_{2n}) \in \bigoplus _{a=1}^{2n} T_z M \longleftrightarrow \frac{d}{ds}\Bigg \vert _{s=0}(e_1 + sv_1,\ldots e_{2n}+sv_{2n})\in V_{\varvec{e}}{\mathcal {F}}(M), \end{aligned}$$
and here, \(\varvec{e}\cdot A_{\varvec{e}}(X_{\varvec{e}})\) is the infinitesimal generator at \(\varvec{e} \in {\mathcal {F}}(M)\) corresponding to \(A_{\varvec{e}}(X_{\varvec{e}})\in {\mathfrak {gl}}(2n,\mathbb {R})\). We then define the \(\hbar \)-diagonal lift\(\omega ^{\hbar }\) of the symplectic form \(\omega \) by
$$\begin{aligned} \omega ^{\hbar } := \omega ^{\mathrm {ver}} + \frac{\hbar }{2}\omega ^{\mathrm {hor}}. \end{aligned}$$
This is essentially the diagonal lift defined in [3], up to a factor of \(\frac{\hbar }{2}\).
Denote the horizontal lift of a vector field \(X\in {\mathfrak {X}}(M)\) to \({\mathcal {F}}(M)\) by \(X^{\mathrm {hor}}\) and the infinitesimal generator on \({\mathcal {F}}(M)\) corresponding to \(\xi \in {\mathfrak {gl}}(2n,\mathbb {R})\) by \(\xi _{{\mathcal {F}}(M)}\). Then it is straightforward to verify
  • \(\omega ^{\hbar }(X^{\mathrm {hor}}, Y^{\mathrm {hor}}) = \omega (X,Y)\circ \pi \),

  • \(\omega ^{\hbar }(Z^{\mathrm {hor}},\xi _{{\mathcal {F}}(M)}) = 0\) and

  • $$\begin{aligned} \omega ^{\hbar }(\xi _{{\mathcal {F}}(M)}, \zeta _{{\mathcal {F}}(M)})(\varvec{e})= & {} \frac{\hbar }{2}\sum _{a=1}^{2n}\omega _z((\varvec{e}\cdot \xi )_a, (\varvec{e}\cdot \zeta )_a) \\= & {} \frac{\hbar }{2}\sum _{a,b,c=1}^{2n}\omega _z(e_b,e_c)\,\xi _{ba}\zeta _{ca}. \end{aligned}$$
Using the non-degeneracy of \(\omega \) to infer that \(\omega _z(e_b,e_c)\) is an invertible matrix in the last identity, it follows that the 2-form \(\omega ^{\hbar }\) is non-degenerate, and so defines an almost-symplectic form on \({\mathcal {F}}(M)\).

8.2 Conditions for the lifted form to be closed

In order for the almost-symplectic form \(\omega ^{\hbar }\) to be a symplectic form, it must satisfy the additional condition \(d\omega ^{\hbar } = 0\). The conditions for this to be the case are computed in [3].

Proposition 8.1

([3, Proposition 6.1])
  1. (i)

    \(d\omega ^{\hbar }(X^{\mathrm {hor}}, Y^{\mathrm {hor}}, Z^{\mathrm {hor}}) = (d\omega (X,Y,Z))\circ \pi \);

     
  2. (ii)

    \(d\omega ^{\hbar }(X^{\mathrm {hor}}, Y^{\mathrm {hor}}, \xi _{{\mathcal {F}}(M)}) = \frac{\hbar }{2} \omega ^{\mathrm {ver}}(F^A(X^{\mathrm {hor}}, Y^{\mathrm {hor}})_{{\mathcal {F}}(M)}, \xi _{{\mathcal {F}}(M)})\);

     
  3. (iii)

    \(d\omega ^{\hbar }(X^{\mathrm {hor}},\xi _{{\mathcal {F}}(M)}, \zeta _{{\mathcal {F}}(M)}) = \frac{\hbar }{2}(\nabla ^A_X\omega )^{\mathrm {ver}}(\xi _{{\mathcal {F}}(M)},\zeta _{{\mathcal {F}}(M)})\);

     
  4. (iv)

    \(d\omega ^{\hbar }(\xi _{{\mathcal {F}}(M)}, \zeta _{{\mathcal {F}}(M)},\chi _{{\mathcal {F}}(M)}) = 0\).

     
Here, \(F^A\in \varOmega ^2({\mathcal {F}}(M),{\mathfrak {gl}}(2n,\mathbb {R}))\) denotes the curvature 2-form of A, and \(\nabla ^A\) denotes the covariant derivative associated with A.

Corollary 8.2

$$\begin{aligned} d\omega ^{\hbar } = 0 \iff d\omega = 0, \quad F^A = 0, \quad \nabla ^A\omega = 0, \end{aligned}$$
i.e. \(\omega ^{\hbar }\) is closed if and only if \(\omega \) is closed and A is a flat, symplectic (i.e. \(\omega \)-preserving) connection.

8.3 Consequences of the condition

Assuming the conditions in Corollary 8.2 hold, the frame bundle \({\mathcal {F}}(M)\) has a particularly simple description. Firstly, \(F^A=0\) implies that the horizontal distribution on \({\mathcal {F}}(M)\) is involutive and hence integrable. If we further assume the absence of monodromy,2 this implies the existence of a global horizontal section \(\varvec{s}:M\rightarrow {\mathcal {F}}(M)\) through any point of \({\mathcal {F}}(M)\). Since the connection A is symplectic, the frames \(\varvec{s}(z) = (s_1(z),\ldots , s_{2n}(z))\) are such that \(\omega _z(s_a(z), s_b(z))\) is independent of z. Let us arrange for these frames to be symplectic, i.e.
$$\begin{aligned} \omega _z(s_a(z),s_b(z)) = \mathbb {J}_{ab}. \end{aligned}$$
Then \({\mathcal {F}}(M)\) is trivial, and we have a bundle morphism \(\varLambda :M\times \mathrm {GL}(2n,\mathbb {R})\rightarrow {\mathcal {F}}(M)\) given by
$$\begin{aligned} \varLambda (z,E) = \varvec{s}(z)\cdot E. \end{aligned}$$
We use \(\varLambda \) to pull back the symplectic form \(\omega ^{\hbar }\). Firstly,
$$\begin{aligned} \varOmega ^{\mathrm {ver}} := \varLambda ^*\omega ^{\mathrm {ver}} = (\pi \circ \varLambda )^*\omega = \pi _1^*\omega , \end{aligned}$$
where \(\pi _1:M\times \mathrm {GL}(2n,\mathbb {R})\rightarrow M\) is projection onto the first factor. Also, defining \(\varOmega ^{\mathrm {hor}} := \varLambda ^*\omega ^{\mathrm {hor}}\), and using
$$\begin{aligned} T_{(z,E)}\varLambda (v_z, V_E) = T_z \varvec{s}(v_z)\cdot E + \varvec{s}(z) \cdot V, \end{aligned}$$
we get
$$\begin{aligned} \varOmega ^{\mathrm {hor}}_{(z,E)} ((v_z, V_E), (w_z,W_E))&= \omega ^{\mathrm {hor}}_{\varvec{s}(z)\cdot E} ( \varvec{s}(z)\cdot V, \varvec{s}(z)\cdot W )\\&\qquad \text {since}\, T_z\varvec{s}(v_z)\cdot E\,\text {is horizontal}\\&= \sum _{a,b,c=1}^{2n}\omega _z((s_b(z),s_c(z))V_{ba}W_{ca} \\&= {\text {Tr}}\,(V^\top \mathbb {J} W). \end{aligned}$$
Thus we have shown
$$\begin{aligned} \varLambda ^*\omega ^{\mathrm {hor}} =: \varOmega ^{\mathrm {hor}} = \pi _2^*\varOmega , \end{aligned}$$
where \(\pi _2: M\times \mathrm {GL}(2n,\mathbb {R})\rightarrow \mathrm {GL}(2n,\mathbb {R})\) is projection onto the second factor and \(\varOmega \) is the symplectic 2-form on \(\mathrm {GL}(2n,\mathbb {R})\) defined in Sect. 2.
Overall, we obtain the symplectic form \(\varOmega ^{\hbar } := \varLambda ^*\omega ^{\hbar }\) on \(M\times \mathrm {GL}(2n,\mathbb {R})\), given explicitly by
$$\begin{aligned} \varOmega ^{\hbar } = \varOmega ^{\mathrm {ver}} + \frac{\hbar }{2}\varOmega ^{\mathrm {ver}} = \pi _1^*\omega + \frac{\hbar }{2}\pi _2^*\varOmega = \omega \oplus \frac{\hbar }{2}\varOmega , \end{aligned}$$
i.e.
$$\begin{aligned} \begin{aligned} \varOmega ^{\hbar }_{(z,E)}((v_z,V_E), (w_z,W_E))&= \omega _z(v_z,w_z) + \frac{\hbar }{2}\varOmega _E(V_E, W_E) \\&= \omega _z(v_z,w_z) + \frac{\hbar }{2} {\text {Tr}}\,(V^\top \mathbb {J} W). \end{aligned} \end{aligned}$$
(8.1)
We see that Eq. (8.1) agrees with Eq. (7.8), which corresponds to applying the above construction with A the obvious global flat connection on \(T^*\mathbb {R}^n\), and global horizontal section
$$\begin{aligned} \varvec{s}(z) = \left( \frac{\partial }{\partial q^1}\bigg \vert _z, \ldots , \frac{\partial }{\partial q^n}\bigg \vert _z,\frac{\partial }{\partial p_1}\bigg \vert _z, \ldots ,\frac{\partial }{\partial p_n}\bigg \vert _z\right) . \end{aligned}$$

8.4 The frame bundle Hamiltonian

Again assuming the conditions in Corollary 8.2 hold, and given a Hamiltonian \(H:M\rightarrow \mathbb {R}\), definition (7.11) of the lifted Hamiltonian carries over directly. To recap, a choice of connection \(A\in \varOmega ^1({\mathcal {F}}(M),{\mathfrak {gl}}(2n,\mathbb {R}))\) and global horizontal section \(\varvec{s}:M \rightarrow {\mathcal {F}}(M)\) induces a global bundle morphism \(\varLambda (z,E) = \varvec{s}(z)\cdot E\). For every \(E\in \mathrm {GL}(2n,\mathbb {R})\), we define g(E) to be the metric on M with respect to which the frame \(\varLambda (z,E)\) is orthonormal. The lifted Hamiltonian \(H^{\hbar }:M\times \mathrm {GL}(2n,\mathbb {R})\rightarrow \mathbb {R}\) is then defined by
$$\begin{aligned} H^{\hbar }(z,E) := H(z) + \frac{\hbar }{4}\left( \varDelta _{g(E)}H\right) (z). \end{aligned}$$
As before, this may also be written
$$\begin{aligned} H^{\hbar }(z,E) := H(z) + \frac{\hbar }{4} {\text {Tr}}\, (E^\top D^2_z H E), \end{aligned}$$
where now \(D^2H:M\rightarrow {\mathfrak {gl}}(2n,\mathbb {R})\) is given by
$$\begin{aligned} (D^2_zH)_{ab} := s_a(z) \left( s_b H \right) . \end{aligned}$$
Note, however, that by contrast to Eq. (7.9), here \(D^2H\) is not necessarily symmetric, since \(s_b\) is also a function of z. Using a similar analysis to that in Sect. 7 gives the Hamiltonian vector field generated by \(H^{\hbar }\) as
$$\begin{aligned} X_{H^{\hbar }} (z,E) = \left( X_{H+\frac{\hbar }{4}\varDelta _{g(E)}}(z), \frac{1}{2}\mathbb {J}(D^2_zH + (D^2_zH)^\top )\cdot E \right) . \end{aligned}$$
(8.2)
Symmetry of \(D^2H\) may be recovered by imposing a further condition on the connection A, viz. that it be torsionless. For then, applying the torsionless condition to the individual (covariantly constant) vectors of the global frame \(\varvec{s}\) yields
$$\begin{aligned} 0 = \nabla ^A_{s_a} s_b - \nabla ^A_{s_b} s_a - [s_a, s_b] \implies [s_a, s_b] = 0, \end{aligned}$$
and it follows that
$$\begin{aligned} (D^2_zH)_{ab} = s_a(z) \left( s_b H\right) = s_b(z) \left( s_a H\right) = (D^2_zH)_{ba}. \end{aligned}$$
Hence, Eq. (8.2) reduces to Eq. (7.12).
From here on, the analysis is identical to Sect. 7. In particular, \(\varLambda \) intertwines the right \(\mathrm {O}(2n)\)-actions on the frame bundle \({\mathcal {F}}(M)\) and \(M\times \mathrm {GL}(2n,\mathbb {R})\)
$$\begin{aligned} \varLambda (z,E)\cdot O = (\varvec{s}(z)\cdot E)\cdot O = \varvec{s}(z)\cdot (EO) = \varLambda (z,EO), \end{aligned}$$
and the latter action has a momentum map \({\text {j}}_{\mathrm {O}}:M\times \mathrm {GL}(2n,\mathbb {R})\rightarrow {\mathfrak {o}}(2n)\) as before. Also the (\(\varvec{s}\)-dependent) left \(\mathrm {Sp}(2n,\mathbb {R})\)-action
$$\begin{aligned} S\cdot (\varvec{s}(z)\cdot E) = \varvec{s}\cdot (SE) \end{aligned}$$
is intertwined with the left \(\mathrm {Sp}(2n,\mathbb {R})\)-action on \(M\times \mathrm {GL}(2n,\mathbb {R})\) and gives a momentum map \({\text {j}}_{\mathrm {Sp}}:M\times \mathrm {GL}(2n,\mathbb {R})\rightarrow {\mathfrak {sp}}(2n,\mathbb {R})\) as before.

Footnotes

  1. 1.

    Although we consider one particular adjoint orbit in this paper, it should be possible to associate semiclassical wave functions with arbitrary integral adjoint orbits. We hope to explore this possibility in future work.

  2. 2.

    This is the case, for example, if M is simply connected—if not, we can always go to universal cover.

Notes

Acknowledgements

The author is grateful to Tomoki Ohsawa, Cesare Tronci, Manuel de León, Cornelia Vizman, Darryl Holm and Martin Wolf for useful discussions, and to the reviewer for helpful comments. This work was supported by the Leverhulme Trust Research Project Grant 2014-112 and the Institute of Mathematics and its Applications Small Grant Scheme.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SurreyGuildfordUK

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