Symplectic Polar Duality

• Symplectic polar duality is an adaptation of classical polarity that uses a nondegenerate skew-symmetric form to create invariant dual constructions in symplectic spaces.
• It underpins key quantum mechanics applications by generating coordinate-free uncertainty principles and geometric state reconstructions via convex and Lagrangian dual pairs.
• It also informs operator theory, incidence geometry, and coding, enabling canonical decompositions, symplectic representations, and dual code analysis.

A symplectic polar dual is a geometric, algebraic, or analytic construct in which classical polar duality—typically defined on convex sets, subspaces, or functionals—is adapted to and made natural for symplectic geometry, often with a focus on invariance under the symplectic group and applications to quantum mechanics, information geometry, combinatorics, and representation theory. The symplectic form, a nondegenerate skew-symmetric bilinear form, fundamentally alters the polarity operation, leading to new dualities for subspaces, convex bodies, operator theory, algebraic codes, and the structure of geometric quantum states.

1. Symplectic Polar Duality: Definitions and Core Constructions

At its most fundamental, symplectic polar duality extends the classical notion of polarity—where for a convex body $X \subset \mathbb{R}^n$ containing the origin, the $h$-polar dual is defined as

$$X^h = \{ p \in \mathbb{R}^n : p \cdot x \leq h \;\; \forall x \in X \},$$

and for a subspace $W$, the usual dual is $$W^\perp = \{ x : \omega(x,w) = 0\;\forall w \in W\}$$ —to the symplectic category. This is achieved by utilizing the standard symplectic form

$$\omega(z, z') = p \cdot x' - x \cdot p' = z^\mathrm{T} J z', \quad J = \begin{bmatrix} 0 & I \ -I & 0 \end{bmatrix}$$

on phase space $\mathbb{R}^{2n}$.

The two principal variants are:

• Symplectic polar dual of a set $\Omega \subset \mathbb{R}^{2n}$ (Gosson et al., 2023):

$$\Omega^{h, \omega} = \{ z \in \mathbb{R}^{2n} : \sup_{z' \in \Omega} \omega(z, z') \leq h \}$$

This is fully covariant under the action of $Sp(n)$: $S(\Omega)^{h, \omega} = S(\Omega^{h, \omega})$.

• Lagrangian (or relative) symplectic polar duality (Gosson et al., 2023):

Given a Lagrangian frame $(\ell, \ell')$, for $X \subseteq \ell$,

$$X^{h}_{\ell'} = \{ z' \in \ell' : \sup_{z \in X} \omega(z, z') \leq h \}.$$

The resulting geometric quantum state in this formalism is $X \times X^{h}_{\ell'}$.

In phase space analysis and quantum mechanics (Gosson, 2013, Gosson, 2020, Gosson et al., 2022), these duals underlie refined uncertainty relations, geometric quantum state representations, and tomographic reconstructions.

2. Symplectic Polar Spaces and Incidence Geometry

A major manifestation of symplectic polar duality arises in the finite geometry of polar spaces, particularly those defined over $PG(V)$ via a nondegenerate alternating (symplectic) bilinear form $f$ (Cardinali et al., 2022, Saniga et al., 2021, Boutray et al., 2021):

• Symplectic polar space $S$: Points are projective points of $PG(V)$, lines and higher subspaces are totally isotropic with respect to $f$. The symplectic dual of a subspace $S \subset V$ is $$S^\perp = \{ x \in V : f(x, s) = 0,\;\forall s \in S\}$$.
• Incidence-theoretic characterizations (Cardinali et al., 2022):

Symplectic polar spaces may be characterized purely by their incidence properties, independent of explicit embeddings, by conditions on hyperplanes, traces of non-collinear points, and regularity of hyperbolic lines. In particular, for pairs of noncollinear points $a, b$, the trace $a^- \cap b^-$ (the hyperbolic line) must be contained in a singular hyperplane.

• Applications to quantum contextuality:

In the geometry $W(2N-1,2)$, points encode $N$-qubit Pauli observables; two observables commute if and only if their images are collinear (Saniga et al., 2021, Boutray et al., 2021). The symplectic polar dual relates to the commutant structure, and the analysis of subspace duals underpins contextuality proofs and geometric classification.

• Duality in Veldkamp space (Saniga et al., 2021):

Veldkamp spaces, consisting of geometric hyperplanes as points, realize a “dual” geometry to the original polar space, central in the classification of observable-based proofs of the Kochen–Specker theorem.

3. Symplectic Polar Duality in Convex and Quantum Geometry

Convex duality in the symplectic category has found critical applications in quantum information (Gosson et al., 2022, Gosson et al., 2023, Gosson, 2020):

• Quantum polarity and uncertainty (Gosson, 2020, Gosson, 2013):

Interpreting position uncertainty via a convex $X$ and its momentum dual $X^h$, so that the pair $(X, P)$ is a quantum dual pair if $X^h \subseteq P$. The saturated case ($X^h = P$) recovers minimum-uncertainty states. Symplectic capacity measurements and polar duality produce coordinate-free versions of the uncertainty principle, with direct links to symplectic non-squeezing and the Blaschke–Santaló and Mahler inequalities.

• Covariance ellipsoids and quantum blobs (Gosson et al., 2022):

The covariance ellipsoid $\Omega = \{z : z^\mathrm{T} M^{-1} z \leq h\}$, under projections to Lagrangian subspaces, gives orthogonal “quantum dual pairs.” The product $X \times X^h$ supports a quantum blob—a minimal, symplectically-invariant phase space region compatible with the strong Robertson–Schrödinger uncertainty, characterized reflexively by $Q^{h,\omega} \subset Q$.

• Geometric quantum states (Gosson et al., 2023):

Generalized Gaussians correspond bijectively to pairs of dual convex bodies $X \subset \ell$, $X^h_{\ell'} \subset \ell'$, with the product $X \times X^h_{\ell'}$ capturing the covariance structure of a quantum state in a manifestly symplectic-covariant fashion.

• Symplectic tomography (Gosson et al., 2022):

Polar duality enables the reconstruction of quantum states from marginal (projected) data, by relating the duals (or intersections) of covariance ellipsoids to the full state.

4. Symplectic Polar Duals in Linear Algebra and Coding Theory

Symplectic duality appears in operator theory, coding theory, and combinatorics:

• Canonical decomposition and operator polarizations (Malagón, 2017):

Any self-adjoint operator $f$ on a symplectic vector space $(V, \omega)$ decomposes, relative to a Darboux basis, as

$$A = \begin{bmatrix} B & 0 \ 0 & B^T \end{bmatrix},$$

with $B$ acting on a Lagrangian subspace $U$, $B^T$ on its symplectic dual $U^*$. Thus the operator splits into a part and its explicit symplectic polar dual action.

• Symplectic analogs of polar decomposition (Teretenkov, 2019):

Any invertible matrix $X \in \mathbb{R}^{2n \times 2n}$ can be factored as $X = H T$ (Hamiltonian times anti-symplectic), $X = M S$ (skew-Hamiltonian times symplectic), or $X = R D S$ (symmetric times a signature times symplectic), providing symplectic “polar” forms. These have direct utility in analyzing bosonic Gaussian channels, preserving the phase space structure and offering numerically stable canonical forms aligned with symplectic symmetry.

• Coding theory and incidence (González et al., 2022, Segovia et al., 2013):

The dual code of an affine symplectic Grassmann code mirrors the duality in the underlying symplectic polar Grassmannian: $$W^\perp = \{v \in V : B(v, w) = 0 \;\forall w \in W\}$$. This duality informs the minimum-weight codewords and automorphism structure, with combinatorial formulas for the dimension of universal embeddings tightly bound to the enumeration of nonmaximal isotropic subspaces.

5. Symplectic Polar Duals in Representation Theory and Quantization

• Symplectic polar representations (Geatti et al., 2016, Kristel et al., 2023):

Polar symplectic representations $(G, V)$ admit Cartan subspaces $c$ such that $V = c \oplus (\mathfrak{g}\cdot v)$, with $c$ isotropic and $$(\mathfrak{g} \cdot v)^\omega \subset \mathfrak{g} \cdot v$$. The geometry of orbits, moment maps, and the corresponding dual spaces interlocks through symplectic duality, especially as it pertains to coisotropicity and visible actions.

Polarizations—splittings of a symplectic vector space $(H, \omega)$ into Lagrangian subspaces $L^+, L^-$ such that $H_\mathbb{C} = L^+ \oplus L^-$—are such that $L^- = \left(L^+\right)^\omega$. These grassmannians of Lagrangian polarizations encode Fock space structures, representation equivalences, and moduli of complex structures.

• Symplectic dual pairs in infinite dimensions (Haller et al., 17 May 2024):

For the group of volume-preserving diffeomorphisms $Diff(M, \mu)$, one constructs symplectic dual pairs using cotangent bundles over spaces of embeddings $T^*Emb(S, M)$. The duality is realized via commuting Hamiltonian group actions with moment maps whose infinitesimal orbits are mutual symplectic orthogonals. Reduction along one action yields nonlinear Grassmannians of augmented submanifolds as explicit models for coadjoint orbits, with the symplectic polar relation reflected in the orthogonality of tangent spaces to group orbits.

6. Symplectic Polar Duality and Quantum Information Geometry

The recent formalism of Geometric Quantum States encapsulates quantum states (notably generalized Gaussians) as explicit pairs in symplectic polar duality:

• A convex body $X \subset \ell$ and its Lagrangian polar dual $X^h_{\ell'}$ give rise to the product set $X \times X^h_{\ell'} \subset \mathbb{R}^{2n}$ (Gosson et al., 2023).
• This representation separates state data along transversal Lagrangian subspaces, is covariant under $Sp(n)$, and connects with symplectic capacities and topological quantum invariants.
• The formalism yields symplectically invariant uncertainty principles and tomographic conditions; states are identified as quantum admissible if $(X^h)^{h} = X$ (reflexivity), with unique minimum-uncertainty states corresponding to quantum blobs (Gosson et al., 2022).

7. Summary Table: Symplectic Polar Duality Realizations

Context	Symplectic Dual Construct	Invariance/Significance
Convex body in phase space	$\Omega^{h,\omega}$, $X^h$	$Sp(n)$ covariance, quantum uncertainty
Projective geometry/polar spaces	$S^\perp$, $W^*$	Incidence, commutation relations, contextuality
Operator on symplectic space	Block-diagonal via polarization $(B, B^T)$	Self-adjoint structure, dual action
Symplectic reduction	Dual pairs of commuting moment maps	Coadjoint orbits, foliation structure
Quantum state geometry	$(X, X^h_{\ell'})$ pairs	Geometric quantum state realization
Coding theory	Code dual analogous to symplectic dual space	Weight structure, combinatorial symmetry
Representation theory	Lagrangian grassmannians, moment maps	Fock spaces, equivalence of representations

References to Key Papers

• Geometric quantization and quantum state polar duality: (Gosson, 2020, Gosson et al., 2022, Gosson et al., 2023)
• Polar spaces, combinatorics, coding: (Cardinali et al., 2022, Saniga et al., 2021, González et al., 2022, Segovia et al., 2013)
• Operator theory, matrix polar decomposition: (Malagón, 2017, Teretenkov, 2019)
• Representation theory and symplectic actions: (Geatti et al., 2016, Kristel et al., 2023)
• Quantum information and symplectic geometry: (Gosson, 2013)
• Infinite-dimensional dual pairs: (Haller et al., 17 May 2024)

Concluding Perspective

Symplectic polar duals articulate the interplay between geometry, algebra, and analysis underpinned by symplectic forms. Across phase space convexity, projective geometries, operator theory, code theory, quantum information, and infinite-dimensional reduction, the duality is symplectically covariant, often reflexive, intimately connected with invariants (such as symplectic capacity), and forms the geometric skeleton for uncertainty, commutation, and duality phenomena in both classical and quantum regimes.