## 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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