[Paper Review] Classifying Linear Canonical Relations

Jonathan Lorand|arXiv (Cornell University)|Aug 19, 2015

Mathematical and Theoretical Analysis8 references3 citations

TL;DR

This paper classifies linear canonical relations—lagrangian subspaces of $V \oplus V^{-}$—up to symplectic conjugation, using a decomposition into coisotropic subspaces and induced symplectic maps. It provides a partial normal form by reducing the problem to classifying coisotropic pairs and identifying invariant structures in the induced symplectic map on reduced spaces.

ABSTRACT

In this Master's thesis, we consider the problem of classifying, up to conjugation by linear symplectomorphisms, linear canonical relations (lagrangian correspondences) from a finite-dimensional symplectic vector space to itself. We give an elementary introduction to the theory of linear canonical relations and present partial results toward the classification problem. This exposition should be accessible to undergraduate students with a basic familiarity with linear algebra.

Motivation & Objective

• To classify linear canonical relations from a finite-dimensional symplectic vector space to itself, up to conjugation by linear symplectomorphisms.
• To develop a systematic framework for understanding the structure of linear canonical relations using coisotropic subspaces and symplectic reduction.
• To provide a partial solution to the classification problem by decomposing relations into elementary types and identifying invariants.
• To lay the foundation for a full normal form by analyzing the induced symplectic map on reduced spaces after coisotropic decomposition.
• To connect the classification to broader contexts in geometric quantization and categorical structures in classical and quantum mechanics.

Proposed method

• Uses the Witt-Artin decomposition to decompose the symplectic vector space into isotropic, coisotropic, and Lagrangian subspaces.
• Applies the Benenti-Tulczyjew theorem to represent a linear canonical relation as a pair of coisotropic subspaces and an induced symplectic isomorphism between their quotients.
• Reduces the classification problem to classifying coisotropic pairs up to symplectic isomorphism, using dimension and rank invariants of intersections.
• Constructs a matrix representation of the linear canonical relation in split coordinates, with blocks corresponding to Lagrangian, symplectic, and mixed components.
• Decomposes the relation into a direct sum of three parts: a Lagrangian part, a symplectic pair part, and a residual part governed by the induced map $\phi_{L_0}$ on $F \oplus H$.
• Proposes a block matrix normal form using canonical bases for $Q^{n_i}$, $P^{n_j}$, and an undetermined form for $\phi_{L_0}$, suggesting a path toward a full classification.

Experimental results

Research questions

• RQ1 What invariants uniquely classify linear canonical relations up to symplectic conjugation?
• RQ2 How can a linear canonical relation be decomposed into simpler, canonical components using coisotropic subspaces?
• RQ3 What is the structure of the induced symplectic map $\phi_L$ on the reduced space after coisotropic reduction?
• RQ4 How does the decomposition of the symplectic vector space into $D \oplus E \oplus F \oplus G \oplus H$ affect the classification of the relation?
• RQ5 Can a complete normal form be constructed by combining known normal forms for symplectic matrices with the coisotropic decomposition?

Key findings

• A linear canonical relation $L \subset V \oplus V^{-}$ is fully determined by a pair of coisotropic subspaces $A = \text{dom}(L)$ and $B = \text{ran}(L)$, together with a symplectic isomorphism $\phi_L: A/A^\perp \to B/B^\perp$.
• The classification reduces to classifying coisotropic pairs $(A,B)$ up to symplectic isomorphism, with invariants derived from the dimensions and ranks of intersections of $A$, $B$, and their annihilators.
• The relation $L$ admits a decomposition $L = L_D \oplus L_E \oplus L_0$, where $L_D$ corresponds to a Lagrangian subspace, $L_E$ to a symplectic pair, and $L_0$ to the residual part governed by $\phi_{L_0}$.
• A candidate normal form is given by a block matrix: $\left(\begin{array}{cc}Q^{n_1}&0\\ 0&Q^{n_1}\end{array}\right) \oplus \left(\begin{array}{cc}Q^{n_2}&0\\ 0&P^{n_2}\end{array}\right) \oplus \left(\begin{array}{cccc}Q^{n_4}&\mathbb{R}^{2(n_3+n_4)}&0\\ 0&[\phi_{L_0}]&Q^{n_5}\end{array}\right)$, with $[\phi_{L_0}]$ still to be determined.
• The residual part $L_0$ is characterized by $\ker(L) = G_1$, $\text{hal}(L) = H_1$, and $\phi_{L_0}: F \oplus H \to F \oplus G$, which inherits the structure of the original relation.
• The approach reveals that the classification is obstructed by the interplay between the normal form of $\phi_{L_0}$ and the splitting $F \oplus H$, suggesting that alternative normal forms for symplectic matrices may be more suitable.