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Symplectic Radon Transform and the Metaplectic Representation.

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We explore the symplectic Radon transform using metaplectic representation and its action on the Lagrangian Grassmannian. This research offers rigorous proofs for quantum systems, interpreting the transform as a generalized marginal distribution for Wigner transforms.

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Area of Science:

  • Quantum mechanics
  • Mathematical physics
  • Harmonic analysis

Background:

  • The symplectic Radon transform is a key tool in analyzing quantum states.
  • Understanding its properties is crucial for multi-dimensional quantum systems.

Purpose of the Study:

  • To rigorously study the symplectic Radon transform within the framework of metaplectic representation.
  • To analyze its action on the Lagrangian Grassmannian for multi-dimensional quantum systems.
  • To interpret the transform as a generalized marginal distribution for Wigner transforms.

Main Methods:

  • Utilizing the metaplectic representation of the symplectic group.
  • Applying the theory to the Lagrangian Grassmannian.
  • Developing rigorous mathematical proofs for general quantum systems.

Main Results:

  • Established rigorous proofs for the symplectic Radon transform in multi-dimensional quantum systems.
  • Interpreted the Radon transform of a quantum state as a generalized marginal distribution for its Wigner transform.
  • Identified the inverse Radon transform as a 'demarginalization process' for Wigner distributions.

Conclusions:

  • The study provides a deeper understanding of the symplectic Radon transform in quantum mechanics.
  • The interpretation offers new perspectives on Wigner distribution analysis.
  • This work advances the mathematical framework for quantum information and signal processing.