Linear Hamiltonians in generators of the real Jacobi group on the extended Siegel-Jacobi space and equations of motion attached

Abstract: Using the energy function on the extended Siegel-Jacobi upper half space of order $n$, $\tilde{\mathcal{X}}^J_n$, with $n\in \mathbb{N}$, the equations of motion in the variables $(x,y,q,p,κ)$ attached to linear Hamiltonians in the generators of the real Jacobi group $G^J_n(\mathbb{R})$ are presented, where $x,y$ are symmetric matrices in $\mathcal{M}(n,\mathbb{R})$ and $p,q$ are real $n$-vectors. The case $n=1$ is presented separately.

• The paper presents explicit classical equations of motion for linear Hamiltonians on extended Siegel-Jacobi spaces using advanced cosymplectic (ACOS) structures.
• It details the decomposition of energy functions into distinct Heisenberg and symplectic components, ensuring integrability via sum-of-squares formulations.
• The study bridges group-theoretical methods and cosymplectic geometry, paving the way for applications in geometric quantization and integrable dynamics.

Linear Hamiltonians and Equations of Motion on Extended Siegel-Jacobi Spaces

Introduction and Mathematical Setting

The paper addresses the explicit construction of classical equations of motion associated with linear Hamiltonians built from generators of the real Jacobi group $G^J_n$ on the extended Siegel-Jacobi spaces $\tilde{X}^J_n$ for arbitrary $n\in\mathbb{N}$. The Jacobi group, defined as $G^J_n = H_n \rtimes \mathrm{Sp}(n)$, encapsulates both the $(2n+1)$-dimensional Heisenberg group $H_n$ and the symplectic group $\mathrm{Sp}(n)$, providing a rich non-abelian symmetry structure relevant in several geometric, algebraic, and physical contexts, including coherent state theory and classical/quantum mechanics on phase spaces with symmetry.

The authors emphasize the Siegel-Jacobi upper half-space $\tilde{X}^J_n$, an extension of classical Siegel spaces tailored for the Jacobi group action, and systematically develop its geometric structure. The manifold is shown to admit an almost cosymplectic (ACOS) structure—a generalization relevant for odd-dimensional phase spaces where a symplectic two-form $\omega$ and a compatible one-form $\theta$ coexist, satisfying non-degeneracy but not necessarily closedness, allowing for a more flexible dynamical framework.

Energy Functions and Symplectic Structures

The central construction is the energy function (classical Hamiltonian) associated to a quantum Hamiltonian linear in the generators of $\tilde{X}^J_n$0. For $\tilde{X}^J_n$1, the Hamiltonian is parameterized as

$\tilde{X}^J_n$2

with complex/symmetric coefficients. The corresponding energy function on $\tilde{X}^J_n$3 splits into independent parts in the variables $\tilde{X}^J_n$4, reflecting Heisenberg and symplectic degrees of freedom and an auxiliary coordinate $\tilde{X}^J_n$5 from the extension.

For $\tilde{X}^J_n$6, analogous constructions generalize the Hamiltonian with operator-valued coefficients $\tilde{X}^J_n$7, $\tilde{X}^J_n$8, $\tilde{X}^J_n$9, and vectors $n\in\mathbb{N}$0. Hermiticity induces symmetry and anti-symmetry conditions on these matrices. The associated energy function is computed explicitly, and shown to decompose into:

• $n\in\mathbb{N}$1, a quadratic form in the Heisenberg variables $n\in\mathbb{N}$2, with explicit matrix structure.
• $n\in\mathbb{N}$3, a function on the symplectic variables $n\in\mathbb{N}$4, exploiting matrix identities and partial Cayley transforms.

The energy function admits a sum-of-squares form, pivotal for establishing integrability and evaluating dynamics.

ACOS and GTACOS Manifold Structures

The authors construct, for both $n\in\mathbb{N}$5 and $n\in\mathbb{N}$6, explicit ACOS structures $n\in\mathbb{N}$7 on extended Siegel-Jacobi spaces. The one-form $n\in\mathbb{N}$8 and symplectic two-form $n\in\mathbb{N}$9 are detailed in Darboux-like coordinates, distinguished by their dependence on both Heisenberg variables and auxiliary coordinates. For $G^J_n = H_n \rtimes \mathrm{Sp}(n)$0 arbitrary:

• $G^J_n = H_n \rtimes \mathrm{Sp}(n)$1
• $G^J_n = H_n \rtimes \mathrm{Sp}(n)$2

Consistency conditions for ACOS/GTACOS (generalized transitive almost cosymplectic) structures are proven, including non-vanishing of $G^J_n = H_n \rtimes \mathrm{Sp}(n)$3 and exactness of $G^J_n = H_n \rtimes \mathrm{Sp}(n)$4.

There is a perfect correspondence between $G^J_n = H_n \rtimes \mathrm{Sp}(n)$5 and $G^J_n = H_n \rtimes \mathrm{Sp}(n)$6 cases. The manuscript meticulously tracks conventions (signs, normalization factors) required to establish this correspondence, enabling generalization from the extended Siegel-Jacobi half-plane to higher-dimensional spaces.

Explicit Equations of Motion

Hamilton's equations are rigorously derived for both $G^J_n = H_n \rtimes \mathrm{Sp}(n)$7 and arbitrary $G^J_n = H_n \rtimes \mathrm{Sp}(n)$8, in the context of ACOS manifolds. The vector field structure incorporates the Reeb vector and nontrivial coupling of the variables induced by the cosymplectic geometry. For $G^J_n = H_n \rtimes \mathrm{Sp}(n)$9, the equations take the form: $(2n+1)$0 which is interpreted as a system coupling the Heisenberg and symplectic variables, with $(2n+1)$1 mediating the cosymplectic structure.

For $(2n+1)$2, all partial derivatives of the energy function with respect to symmetric matrix variables and vectors are computed explicitly using advanced matrix calculus (vectorization and half-vectorization rules). The resulting system generalizes Hamiltonian dynamics to extended Siegel-Jacobi spaces, accounting for manifold dimension and cosymplectic effects: $(2n+1)$3

Compatibility with $(2n+1)$4 solutions is verified, showing necessary adjustments due to conventions.

Theoretical and Practical Implications

The results furnish an integrable framework for classical dynamics on extended Siegel-Jacobi spaces endowed with Jacobi group symmetry. The explicit equations accommodate both standard symplectic and cosymplectic variable couplings, generalizing phase space Hamiltonian dynamics to settings with additional structure and symmetry. Direct expressions for energy functions and motion equations facilitate further exploration of geometric quantization, representation theory of the Jacobi group, and applications in quantum mechanics, most notably systems modeled by coherent states and group-theoretical splittings.

Advanced cosymplectic geometry, as formalized in this work, offers flexible methods to approach classical/quantum systems with odd-dimensional phase spaces—including contact Hamiltonian formulations and dissipative dynamics. The explicit connection established between ACOS/GTACOS structures and classical dynamics on Siegel-Jacobi spaces sets a theoretical foundation for further exploration of symplectic reduction, geometric quantization, and integrable systems with super-symmetry or nontrivial topology.

The systematic matrix calculus employed for the $(2n+1)$5 case is essential for practical computations in high-dimensional settings, providing formulae amenable to symbolic or numerical implementation.

Future Directions

Potential generalizations include non-linear Hamiltonians, incorporation of additional symmetry groups (e.g., extended Jacobi or other semi-direct products), and quantization schemes on the extended Siegel-Jacobi space. The flexibility of cosymplectic structures paves the way for investigations into contact geometry, singular Lagrangian systems, and dissipative Hamiltonian flows.

Further connections to physical models—quantum optics, quantum field theory, and integrable systems—are suggested by the coherent-state framework and the explicit group-theoretical parametrization. Extensions to consider higher-order group actions, multi-variable coherent states, or non-trivial fibre bundles could yield new insights into geometric phases, Berry connections, and holomorphic quantization.

Conclusion

The paper presents a comprehensive framework for classical dynamics generated by linear Hamiltonians in the Jacobi group generators, acting on extended Siegel-Jacobi spaces for arbitrary $(2n+1)$6. Explicit formulae for energy functions, ACOS/GTACOS structures, and the equations of motion are provided, establishing rigorous connections between cosymplectic geometry and integrable systems with Jacobi group symmetry. These results enrich the theoretical toolbox for geometric mechanics, quantization, and representation theory, and facilitate practical computation for high-dimensional phase spaces.