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Intrinsic Quantization of Linear Hamiltonian Systems.
Luigi Accardi1, Carlo Pandiscia1
1Volterra Center, University of Roma Tor Vergata, Via Columbia 2, 00133 Roma, Italy.
Entropy (Basel, Switzerland)
|May 4, 2026
Summary
This study explores quantizing linear Hamiltonian systems by inducing a complex Hilbert space from classical dynamics. This approach recovers canonical quantization results, bridging analysis, geometry, and physics.
Area of Science:
- Mathematical Physics
- Quantum Mechanics
- Geometric Quantization
Background:
- Linear Hamiltonian systems offer a rich area for research in mathematical physics.
- Classical dynamics in these systems can be represented within a complex Hilbert space.
- This framework connects classical mechanics to quantum principles.
Purpose of the Study:
- To provide an overview of the quantization of linear Hamiltonian systems.
- To demonstrate how complex structures arise from classical systems.
- To link foundational concepts to modern symplectic geometry.
Main Methods:
- Inducing a complex structure and scalar product on phase space.
- Constructing a complex Hilbert space from classical dynamics.
- Applying Boson Fock quantization to unitary groups.
Main Results:
- Classical linear Hamiltonian systems naturally yield complex Hilbert spaces.
- Unitary dynamics in these systems are described by one-parameter groups.
- Boson Fock quantization unifies with canonical quantization.
Conclusions:
- The framework offers a consistent method for quantizing linear Hamiltonian systems.
- It highlights the interplay between analysis, geometry, and physics.
- This approach provides a valuable case study in theoretical physics development.
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