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Gaussian distributions, Jacobi group, and Siegel-Jacobi space. (English) Zbl 1323.53018

The family \(\mathcal{N}\) of Gaussian distribution functions \[ p(x;\mu,\sigma):=\frac{1}{\sqrt{2\pi}\sigma}\exp\{-\frac{(x-\mu)^2}{2\sigma^2}\}, ~x,\mu, \sigma\in \mathbb{R}, \mu>0, \] is considered as a 2-dimensional statistical manifold. It is proved that as manifold the tangent bundle \(T\mathcal{N}\) is the Siegel-Jacobi space [J.-H. Yang, J. Number Theory 127, No. 1, 83–102 (2007; Zbl 1133.32011)], denoted by \(\mathbb{S}^J\). The Siegel-Jacobi space is a homogeneous Kähler manifold associated to the Jacobi group \(G^J\) [M. Eichler and D. Zagier, The theory of Jacobi forms. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.10018)]. The points of \(\mathbb{S}^J\) are in \(\mathbb{H}_1\times \mathbb{R}^2\), where \(\mathbb{H}_1\) denotes the (Siegel) upper half plane \(\{\tau\in\mathbb{C}|\operatorname{Im} \tau >0\}\). The starting point is P. Dombrowski’s construction [J. Reine Angew. Math. 210, 73–88 (1962; Zbl 0105.16002)] of dually flat structures and Kähler geometry. A dualistic structure \((h,\nabla,\nabla^*)\) is introduced on the manifold \(M\) with affine connection \(\nabla\). Following R. Cirelli et al. [J. Math. Phys. 31, No. 12, 2891–2897 (1990; Zbl 0850.70206)], the space \(\mathcal{K}(N)\) of Kähler functions \(f:N\rightarrow \mathbb{R}\) is introduced as consisting of those functions whose Hamiltonian vector fields \(X_f=i_{\omega}d f\) are Killing vectors on a Kähler manifold \(N\). The definition of statistical manifold, Fisher metric and \(\nabla^{\alpha}\)-connections are recalled [S.-i. Amari and H. Nagaoka, Methods of information geometry. Transl. from the Japanese by Daishi Harada. Providence, RI: AMS, American Mathematical Society; Oxford: Oxford University Press (2000; Zbl 0960.62005)].
In Molitor’s approach, the Siegel-Jacobi space is endowed with a particular homogeneous metric, denoted \(g_{KB}\), used by E. Kähler [Mathematische Werke. Mathematical works. Edited by Rolf Berndt and Oswald Riemenschneider. Berlin: Walter de Gruyter (2003; Zbl 1039.01013)] and R. Berndt [Prog. Math. 51, 21–32 (1984; Zbl 0551.10021)], expressed in the variables \((u,v,x,y)\in\mathbb{R}^4\), where \((\tau,z)\in\mathbb{H}\times \mathbb{C}\), \(\tau=u+i v; z=x + i y\). The author expresses in the coordinates \(\theta, \dot{\theta}\) for \(T\mathcal{N}\) the Fisher metric, the Christoffel symbols and \(g\), \(J\), \(\omega\), and it is proved that as Kähler manifolds \(T\mathcal{N}\simeq \mathbb{S}^J\), which is the main result obtained in the paper. It is proved that the group of holomorphic isometries of \(T\mathcal{N}\) is the affine symplectic group \(G^{AS}\). This is a long proof based on the 2-dimensional eikonal equation.
A description of the group of isometries of \(T\mathcal{N}\) as \(\mathrm{SL}^{\pm}(2,\mathbb{R})\ltimes\mathbb{R}^2\) is proved. Let \(\xi_M\) be the fundamental vector field associated with the element \(\xi\) of the Lie algebra \(\mathfrak{g}\) of the Lie group \(G\) acting on the manifold \(M\). Let \(\mathfrak{as}\) denote the Lie algebra of \(G^{AS}\) and \(\iota(M)\) the space of Killing vector fields of a Riemannian manifold \(M\). Then it is proved that the map \(\mathfrak{as}\rightarrow \iota(T\mathcal{N})\), \(\xi\rightarrow \xi_{T\mathcal{N}}\) is an anti-isomorphism of Lie algebras. In the section “Kähler functions and momentum map”, a linear map \(\psi: \mathfrak{g}^J\rightarrow \mathbb{C}^{\infty}(\mathbb{S}^J)\) is explicitly defined, taking for the Jacobi Lie algebra \(\mathfrak{g}^J\) the base chosen by R. Berndt and R. Schmidt in their monograph [Elements of the representation theory of the Jacobi group. Basel: Birkhäuser (1998; Zbl 0931.11013)], while for \(\mathbb{S}^J\simeq T\mathcal{N}\) the symplectic coordinates \((\eta,\dot{\theta})\) are used. It is proved that: The map \(\mathbf{J}: \mathbb{S}^J\rightarrow (\mathfrak{g}^J)^*\), \(\mathbf{J}(p)(L):=\psi(L)(p)\), \(p\in\mathbb{S}^J,~L\in\mathfrak{g}^J\) is an equivariant momentum map; the map \(\psi: \mathfrak{g}^J\rightarrow \mathcal{K}(\mathbb{S}^J)\) is a Lie algebra isomorphism; as symplectic manifold, \(\mathbb{S}^J\) is a coadjoint orbit of the Jacobi group \(G^J\).
In the last section “Gaussian distributions: extrinsic geometry”, \(T\mathcal{N}\) is regarded as a submanifold of the infinite-dimensional projective space \(\mathbb{P}(\mathcal{H})\), endowed with the Fubini-Study metric \(g_{FS}\) and Kähler two-form \(\omega_{FS}\). If \(T: \mathbb{S}^J\hookrightarrow \mathbb{P}(\mathcal{H})\), \(T(\tau,z)=\left[ \exp(\frac{i}{2}(\tau x^2-zx))\right]\), then it is proved that the map \(T\) is a smooth immersion, \(T^*\omega_{FS}=\frac{1}{4} \omega_{KB}\) and \(T^*g_{FS}=\frac{1}{4}g_{KB}+\frac{1}{4}S\), where the \(S\) is a tensor field of symmetric forms on \(\mathbb{S}^J\) explicitly constructed. Also, the representation \(Q\) of the Lie algebra \(\mathfrak{g}^J\) is defined on the set of differential operators of degree at most two with polynomial coefficients of one variable, a realisation of the quantum forced oscillator. It is proved that \(-\frac{i}{2}Q\) is a unitary representation of the Lie algebra \(\mathfrak{g}^J\) – the infinitesimal Schrödinger-Weil representation. The paper ends with some observations about the Schrödinger equations in the case of linear Hamiltonian in the generators of the Jacobi algebra \(\mathfrak{g}^J\) via the integral curves of Hamiltonian vector fields of Kähler functions \(\mathrm{J}^L:\mathbb{S}^J\rightarrow\mathbb{R}\), \(L\in\mathfrak{g}^J\). I mention that Proposition 4.4 in the paper of Molitor is Proposition 4.1 in the paper [A. Gheorghe, “Quantum and classical Lie systems for extended symplectic groups”, Romanian J. Phys. 58 1436–1445 (2013)].

MSC:

53B35 Local differential geometry of Hermitian and Kählerian structures
62B10 Statistical aspects of information-theoretic topics
94A17 Measures of information, entropy
11F50 Jacobi forms
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References:

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