Polar duality is a well-known concept from convex geometry and analysis. In the present paper we study a symplectically covariant versions of polar duality, having in mind their applications to quantum harmonic analysis. It makes use of the standard symplectic form on phase space and allows a precise study of the covariance matrix of a density operator.

The concept of polar dual set in convex geometry corresponds to the concept of dual space in linear algebra. Given a convex body X in the Euclidean space Rn its polar dual is the set X of all p(Rn)* such that ⟨p, x⟩ ≤ ; here is a positive constant, usually taken to be one in the standard literature (we use for flexibility a parameter-dependent definition; in quantum mechanics would be Planck’s constant h divided by 2π; in harmonic analysis one would take = 1/2π while the standard choice in the theory of partial differential equations is = 1). We will most of the time identify the dual space (Rn)* with Rn itself, in which case the polar dual of X is identified, using the standard Euclidean structure (x, p) ⟼ p⋅x with the set
X={pRn:supxX(px)}.
(1)
We will study a variants of polar duality on the symplectic space (T*Rn,σ) where σ is the standard symplectic form j=1ndpjdxj; we will identify the cotangent bundle T*Rn with R2nRxn×Rpn. This variant is what we call “symplectic polar duality”: if ΩR2n is a convex body we define its symplectic polar dual by
Ω,σ={zR2n:supzΩσ(z,z)};
(2)
clearly Ω,σ = J) where J is the standard symplectic automorphism J(x, p) = (px). The interest of this notion comes (among other things) from the fact that it has the symplectic covariance property S,σ) = [S(Ω)],σ for every S ∈ Sp(n).

As we will see in the course of this paper, symplectic polar duality is closely related to difficult questions of positivity for trace class operators, and allows to express “quantization conditions” in an elegant and concise geometric way. On the other hand, Lagrangian polar duality allows a geometric redefinition of the notion of quantum state; these states are usually viewed as “wavefunctions” in physics; in our approach they appear as geometric objects defined in terms of convex products X×(X) whose factors are supported by transversal Lagrangian planes, and their functional aspects appear only as subsidiary through the use of the John ellipsoid (maximum volume ellipsoid).

The applications of concepts of convex geometry and analysis outside their original area is not new, see for instance Milman34 who applies such methods to probability theory; also see the treatise3 by Aubrun and Szarek.

Let us describe some highlights of this work, emphasizing what we hold for the most important results (our choice being of course somewhat subjective, and highly depending on the authors’ tastes).

The covariance matrix of a physical state (be it classical, or quantum) is a statistical object whose importance in information theory is crucial; it encodes the statistical properties of the state and its study is, as we will see, greatly facilitated by the polar duality approach. The central result is, no doubt, Theorem 17 who gives two criteria for what we call “quantum admissibility” of a phase space ellipsoid (the definition of this notion of admissibility is closely related to the uncertainty principle of quantum mechanics, an is rigorously defined in Definition 14). The first criterion say that an ellipsoid Ω is admissible if and only if it contains its symplectic polar dual Ω,σ; the second is of a more subtle nature; it shows that it is sufficient (and necessary) for admissibility that Ω,σF ⊂ Ω ∩ F for every symplectic subspace of (R2n,σ). This is a tomographic condition reminiscent of an old result by Narcowich35 concerning covariance and information ellipsoid in quantum mechanics. This analogy is made even more convincing in Theorem 27 where we give a dynamical description of quantum admissibility of covariance and information ellipsoids using the techniques we develop; in particular the role of the so fruitful notion of symplectic capacity is highlighted (symplectic capacities are strongly related to Gromov’s famous non-squeezing theorem). We take this opportunity to give a new functional-analytical characterization of those (classical, or quantum) state for which the covariance matrix is well-defined. This is done in terms of a class of modulation spaces.

Notation 1.

The standard symplectic form σ is written in matrix form as σ(z, z′) = Jz⋅zwhere J=0n×nIn×nIn×n0n×n is the standard symplectic matrix. The standard symplectic group Sp(n) is the group of automorphisms S of T*RnR2n such that S*σ = σ; in matrix notation: SSp(n) if and only if STJS = J (or, equivalently, SJST = J), ST the transpose of S). The unitary representation of the double cover of Sp(n) (the metaplectic group) is denoted by πMp: Mp(n) ⟶ Sp(n). The Lagrangian Grassmannian of (R2n,σ) is denoted Lag(n), thus Lag(n) if and only if is a linear subspace of R2n with dim  = n and σ| = 0. The elements of Lag(n) will be called Lagrangian planes.

For detailed treatments of the topics of convex geometry and analysis used in this article we refer to the treatise3 by Aubrun and. Szarek and to Vershynin’s online lecture notes.40 For a comprehensive study of convex geometry with applications to optimization theory see Boyd et al.8 

1. Definition

Let X and Y be convex sets in Rn; then,
(XY)=XY  ,  (XY)=XỸ
(3)
XYYX  ,  X closed (X)=X
(4)
AGL(n,R)(AX)=(AT)1X
(5)
(in the second formula (3) XỸ is the convex hull of XY). If A=ATGL(n,R) is positive definite then
{xRxn:Axx}={pRpn:A1pp}
(6)
hence, in particular,
BXn()=BPn(),BPn()=BXn()
(7)
where BXn() (resp. BPn()) is the ball in Rxn (resp. Rpn) with radius and centered at 0.

Remark 2.

The properties of polar duality is less transparent for convex bodies not centered at the origin and requires the use of the so-called Santaló point.37 

2. Projections and intersections

Polar duality exchanges the projection and the intersection operations.3 While this result seems to be well-known it seems difficult to find a detailed proof in the literature, so we prove this important result, following.40 For this we need the following elementary lemma:

Lemma 3.

Let X = {x: Ax⋅x ≤ 1} and P = {p: Bp⋅p ≤ 1} (A, B positive definite and symmetric) be two ellipsoids. We have XP if and only if AB−1, and X = P if and only if AB = In×n.

Proof.

We have X=A1/2(BXn()) and P=B1/2(BPn()) and the inclusion XP is thus equivalent to the inequality A1/2B−1/2 in the Löwner ordering, that is, to AB−1 with equality if and only if X = P.■

Polar duality exchanges the operations of intersection and orthogonal projection:

Proposition 4.
Let XRxn be a convex body containing 0 in its interior and F a linear subspace of Rxn; we have
(ΠFX)=XF ,  (XF)=ΠF(X)
(8)
where ΠF us the orthogonal projection in Rxn onto F. [In (ΠFX) and (XF) the polar duals are taken inside the subspace F equipped with the induced inner product.]

Proof.
It suffices to prove the first formula (8) since the second follows by duality:
XF=(X)F=(ΠFX)
and hence (XF) = ΠF(X). Let us next show that ΠF(X) ⊂ (XF). For pX we have, for every xXF,
xΠFp=ΠFxp=xp
hence ΠFp ∈ (XF). To prove the inclusion ΠF(X) ⊃ (XF) we note that it is sufficient, by the anti-monotonicity and reflexivity properties of polar duality, to prove that (ΠF(X))XF. Let x(ΠF(X)); we have x⋅ΠFp for every pX. Since xF (because the dual of a subset of F is taken inside F) we also have
xΠFp=ΠFxp=xp
from which follows that x(X)=X, which shows that xXF. This concludes the proof.■

3. John and Löwner ellipsoids

A fundamental tool in convex geometry and Banach space geometry is the John ellipsoid of a convex body ΩR2n. It is4,8 the (unique) ellipsoid ΩJohn of maximal volume contained in Ω; similarly the (unique) minimum enclosing ellipsoid is the Löwner ellipsoid ΩLöwner. We note that both the John and the Löwner ellipsoids transform covariantly under linear (and affine) transforms: if LGL(n,R) then
(L(X))John=L(XJohn),  (L(X))Löwner=L(XJohn).
(9)

Remark 5.

One can define the John ellipsoid in any finite-dimensional normed space, regardless of whether the space is Euclidean or not. The definition of the John ellipsoid in a normed space is the same as the one given earlier.

Polar duality interchanges the John and Löwner ellipsoids; we have the following duality relations hold for convex symmetric bodies:3 
(XJohn)=(X)Löwner,  (XLöwner)=(X)John.
(10)

4. Blaschke–Santaló inequality and Mahler volume

Let X be an origin symmetric convex body in Rxn. By definition, the Mahler volume (or volume product) of X is the product
υ(X)=Voln(X)Voln(X)
(11)
where Voln is the usual Euclidean volume on Rxn. The Mahler volume is a dimensionless quantity because of its rescaling invariance: we have υ(λX) = υ(X) for all λ > 0. More generally, the Mahler volume is invariant under linear automorphisms of Rxn: if LGL(n,R) we have
υ(LX)=Voln(LX)Voln(LT)1X
(12)
=Voln(X)Voln(X)=v(X).
(13)
A remarkable property of polar duality is the Blaschke–Santaló inequality:6 assume again that X is a centrally symmetric body; then
υ(X).Voln(Bn())2=(π)nΓn2+12
(14)
and equality is attained if and only if XRxn is an ellipsoid centered at the origin (see Ref. 5 for a proof using Fourier analysis). It is conjectured (the “Mahler conjecture”32) that one has the lower bound
υ(X)(4)nn!
(15)
with equality only when X is the hypercube C = [−1, 1]n. Bourgain and Milman7 have shown the existence, for every nN, of a constant Cn > 0 such that
Voln(X)Voln(X)Cnn/n!
(16)
and more recently Kuperberg31 has shown that one can choose Cn = (π/4)n.

Remark 6.
In view of the invariance property (12) this is equivalent to saying that the minimum is attained by any n-parallelepiped
X=[2σx1x1,2σx1x1]××[2σxnxn,2σxnxn].
(17)
This is related to the covariances of the tensor product ψ = ϕ1 ⊗⋅⋅⋅⊗ ϕn of standard one-dimensional Gaussians ϕj(x)=(π)1/4exj2/2; the function ψ is a minimal uncertainty quantum state in the sense that it reduces the Heisenberg inequalities to equalities. This observation might lead to aquantum proofof the Mahler conjecture.

Here is an elementary application of polar duality which highlights the role it plays in questions related to the uncertainty principle of quantum mechanics. We are following the presentation we gave in Ref. 20.

Hardy’s uncertainty principle27 in its original (one dimensional) formulation says that if the moduli of ψL1(R)L2(R) and of its Fourier transform, here defined by
ψ̂(p)=Fψ(p)=12πeipxψ(x)dx
are different from zero and satisfy estimates
|ψ(x)|CAea2x2 ,  |ψ̂(p)|CBeb2p2
(CA, CB > 0, a, b > 0), then we must have ab ≤ 1 and if ab = 1 we have ψ(x)=Cea2x2 for some complex constant C. We have proven a multidimensional version of this result (see Refs. 21 and, 23, Chapter 10). Let ψ,ψ̂L1(Rn)L2(Rn), ψ ≠ 0 where
ψ̂(p)=Fψ(p)=1(2π)n/2Rneipxψ(x)dx.
Then:

Proposition 7.
Let A,BSym(n,R) be positive definite and ψ as above. Assume that there exist a constants CA, CB > 0 such that
|ψ(x)|CAe12Axx  and  |Fψ(p)|CBe12Bpp.
(18)

(i) The eigenvalues λj, 1 ≤ jn, of AB are 1; (ii) If λj = 1 for all j, then ψ(x)=ke12Ax2 for some kC.

It turns out that this result can easily be restated in terms of polar duality:

Corollary 8.

The Hardy estimates are satisfied by a non-zero function ψL2(Rn) if and only if the ellipsoids X = {x: Ax⋅x} and P = {p: Bx⋅x} satisfy XP with equality X = P if and only ψ(x)=ke12Axx for some kC.

Proof.

This immediately follows from Lemma 3.■

We will apply the results above to sub-Gaussian estimates of the Wigner function in Sec. III C.

In what follows we identify the dual of the symplectic space (R2n,σ) with itself.

1. Symplectic covariance

Let J=0n×nIn×nIn×n0n×n be the standard symplectic matrix. Using the matrix formulation σ(z, z′) = Jz⋅z′ of the symplectic form it is straightforward to verify It is straightforward to verify that the symplectic polar dual
Ω,σ={zR2n:supzΩσ(z,z)}
(19)
of a convex body ΩR2n is related to the ordinary polar dual Ω by the formula
Ω,σ=(JΩ)=J(Ω).
(20)
It follows that:

Lemma 9.
Consider the phase space ellipsoid
ΩM={z:Mzzr2}
where M is symmetric and positive definite and r > 0. Then
ΩM,σ={z:JM1Jzz(/r)2}.
(21)

Proof.

The ordinary -polar dual ΩM of ΩM is defined by M−1zz ≤ (/r)2. In view of (20) ΩM,σ=J(ΩM) hence the result.

Formula (20) can easily be generalized to yield the following important symplectic covariance result:■

Proposition 10.
Let Ω be a symmetric convex body and SSp(n). (i) we have
(S(Ω)),σ=S(Ω,σ).
(22)
(ii) More generally for SSp(n) and F a linear subspace of R2n we have
S(ΩF),σ=(SΩSF),σ.
(23)

Proof.
(i) Using successively (20), the scaling property (5) in dimension 2n, and again (20), we have
S(Ω,σ)=SJ(Ω)=J(ST)1(Ω)=J(S(Ω))=(S(Ω)),σ.
(ii) Formula (23) follows from formula (22) since Ω ∩ F is convex and symmetric.■

2. Quantum blobs

Symplectic balls with radius are the only fixed ellipsoids for ordinary polar duality. They play an important role in various formulations of the uncertainty principle of quantum mechanics14,18,24 where they represent minimum uncertainty units; this motivates the following definition:

Definition 11.

A quantum blob QS(z0) is a symplectic ball with radius : QS(z0)=S(B2n(z0,)) for some SSp(n). When z0 = 0 we write QS = QS(0).

We will see later (Lemma 16) that a characteristic property of quantum blobs is that their orthogonal projections on symplectic planes can never become smaller than πℏ.

Proposition 12.

Let Ω be a centered ellipsoid in (R2n,σ). We have Ω = Ω,σ if and only if Ω is a quantum blob, i.e. if Ω = QS for some SSp(n).

Proof.
That QS,σ=QS is clear in view of (22):
QS,σ=S(B2n()),σ=S(B2n())=QS.
Suppose conversely that the ellipsoid Ω is defined by Mzz; then its symplectic polar dual Ω,σ is defined by −JM−1Jz⋅z (Lemma 9) and we have Ω = Ω,σ if and only if M = −JM−1J. This condition is trivially equivalent to MJM = J which implies M ∈ Sp(n) hence Ω=S(B2n()) with S = M−1/2 ∈ Sp(n).■

Proposition 4 relating orthogonal projections and intersections generalizes as follows to the case of symplectic polar duality:

Proposition 13.
Let ΩR2n be a centrally symmetric convex body and F a linear subspace of R2n. We have
(ΠFΩ),σ=Ω,σ(JF)  and (ΩF),σ=ΠJF(Ω,σ).
(24)

Proof.
Since Ω is symmetric we have J(Ω ∩ F) = −J(Ω ∩ F) hence the kernel of the projector −JΠFJ is F and its range is Jℓ so that −JΠFJ = ΠJF. This proves the first equality (24). We have, by definition, (Ω ∩ F),σ = J(Ω ∩ F). In view of formula (8) we have (Ω ∩ F) = ΠF) and hence (Ω ∩ F),σ = JΠF). Thus
(ΠFΩ),σ=JΠF(Ω)=(JΠFJ)(Ω,σ);
which is the second equality (24). It immediately follows from Proposition 4 noting that
(ΠFΩ),σ=J(ΩF)=Ω,σJF.
We finally note that the Blaschke–Santaló inequality (14) becomes in this context
Vol2n(Ω)Vol2n(Ω,σ).Vol2n(B2n())2
(25)
with equality if and only the convex set X is an ellipsoid. This follows from (14) noting that
Vol2n(Ω,σ)=Vol2n(JΩ)=Vol2n(Ω).

1. Definition and a necessary and sufficient condition

The following definition will be motivated below:

Definition 14.

Let ΩMR2n be the ellipsoid {z: Mz⋅ z} (M = Mt > 0). We will say that ΩM is quantum admissible if it contains a quantum blob QS=S(B2n()), SSp(n).

We are going to prove, using symplectic polarity, two simple but important necessary and sufficient conditions for an ellipsoid to be quantum admissible. We first recall the Williamson symplectic diagonalization result41 (see 16 and 28 for “modern” proofs). For every M = MT > 0, there exists S ∈ Sp(n) such that
M=STDS ,  D=Λσ0n×n0n×nΛσ
(26)
where Λσ=diag(λ1σ,,λnσ) the λjσ being the symplectic eigenvalues of M (i.e. the moduli of the eigenvalues of JMM1/2JM1/2). It is usual to rank the symplectic eigenvalues in non-increasing order
λmaxσ=λ1σλ2σλnσ=λminσ.
Note that the symplectic spectrum of M−1 is ((λ1σ)1,,(λnσ)1). It is usual to call the factorization (26) the “Williamson normal form of M.”

Remark 15.

The diagonalizing symplectic matrix S in (26) is not unique; see Son et al.39 for a detailed analysis of the set of diagonalizing symplectic matrices.

Recall from Proposition 12 that the equality ΩM,σ=ΩM occurs if and only if ΩM=S(B2n()) for some S ∈ Sp(n), i.e. if and only if Ω is a “quantum blob.” Below we state and prove a general criterion for admissibility which we glorify it by giving it the status of a theorem. Let us first introduce some preparatory material:

  • A two-dimensional subspace F of (R2n,σ) is called a symplectic plane if the restriction σ|F of symplectic form σ is non-degenerate; equivalently F has a basis {e1, e2} such that σ(e1, e2) = 1. In particular every plane Fj of conjugate coordinates xj, pj is symplectic; and for every symplectic plane F there exists Sj ∈ Sp(n) such that F = Sj(Fj).

  • We will use Gromov’s symplectic non-squeezing theorem;26 it says (in its simplest form) that no symplectomorphism f ∈ Symp(n) of (R2n,σ) can send a ball B2n(R) into a cylinder Zj2n(r):xj2+pj2r2 if r < R [we are denoting by Symp(n) the group of all symplectomorphisms28,36,42 of (R2n,σ)].

We will also need the following immediate consequence of Gromov’s theorem:

Lemma 16.
Let F be a symplectic plane in (R2n,σ) and fSymp(n). The area of the orthogonal projection ΠF of f[B2n(z0, r)] on F satisfies
Area(ΠFf(B2n(z0,r)))πr2.
(27)

Proof.
It is sufficient to suppose that z0 = 0 since areas are translation-invariant. Assume that ΠFj(S(B2n(r)))=πR2 with R < r. Then fB2n(r) must be contained in the cylinder Zj(R), but this contradicts Gromov’s non-squeezing theorem. (ii) Assume that R < r. Then, by (27),
πr2Area(ΠFS(B2n(R)))<Area(ΠFS(B2n(r))).

Let us now state and prove our theorem:

Theorem 17.
The ellipsoid ΩM is quantum admissible if and only if the two following equivalent conditions are satisfied: (i) We have the inclusion
ΩM,σΩM.
(28)
(ii) We have the inequality
Area(ΩM,σF)π
(29)
for every symplectic plane F in (R2n,σ).

Proof.
(i) Suppose that ΩM is quantum admissible; then there exists S ∈ Sp(n) such that QS=S(B2n())ΩM. By the anti-monotonicity of symplectic polar duality this implies that we have
ΩM,σQS,σ=QSΩM,
which proves the necessity of the condition. Suppose conversely that ΩM,σΩM. We have
ΩM,σ={zR2n:(JM1J)zz}
(30)
hence the inclusion ΩM,σΩM implies that M ≤ (−JM−1J) ( stands here for the Löwner ordering of matrices). Performing a symplectic diagonalization (26) of M and using the relations JS−1 = STJ, (ST)1J=JS this is equivalent to
M=STDSST(JD1J)S
that is to D ≤ −JD−1J. This implies that we have Λσ(Λσ)1 and hence λjσ1 for 1 ≤ jn; thus DI and M = STDSSTS. The inclusion S(B2n())ΩM follows and we are done. (ii) Suppose that ΩM is admissible and let ΠF be the orthogonal projection in R2n on F. By Proposition 13 we have
ΩM,σF=(ΠJFΩM),σ.
Since ΩM is an ellipsoid the symplectic version (25) of the Blaschke–Santaló inequality becomes the equality
Area(ΠJFΩM),σArea(ΠJFΩM)=(π)2
(31)
that is
Area(ΩM,σF)Area(ΠJFΩM)=(π)2.
The inequality (29) follows: since ΩM is admissible, it contains a quantum blob (B2n()) hence Area(ΠJFΩM) ≥ πℏ in view of Lemma 16. Assume conversely that Area(ΩM,σF)π for every symplectic plane F; by (31) we must then have Area(ΠJFΩM) ≥ πℏ for every F. Let us show that this implies that ΩM must be admissible. Since admissibility is preserved by symplectic conjugation we may assume, using a Williamson diagonalization (26), that M is of the diagonal type Λσ0n×n0n×nΛσ where Λσ=diag(λ1σ,,λnσ) the λjσ being the symplectic eigenvalues of M. The ellipsoid ΩM is thus given by
λ1σ(x12+p12)++λnσ(xn2+pn2).
Let us intersect ΩM with the symplectic plane F1 (the plane of coordinates x1, p1). It is the ellipse x12+p12/λ1σ which has area π/λ1σ. Now, ΩM is admissible if and only if λ1σ=λmaxσ1 which is equivalent to the condition Area(ΩMF) ≥ π, hat is to Area(ΩM,σF)π again in view of the Blaschke–Santaló equality (31).■

Condition (29) is truly remarkable; it shows that given an ellipsoid and its symplectic polar dual the datum of a sequence of two-dimensional conditions suffices to decide whether the ellipsoid is admissible or not. This “tomographic” property is related to a condition using the Poincaré invariant given by Narcowich35 we will briefly discuss in our study of covariance and information ellipsoids in Sec. V B.

Recall21 that the cross-Wigner function of a pair (ψ,ϕ)L2(Rn)×L2(Rn) it is defined by
W(ψ,ϕ)(z)=12πnRneipyψ(x+12y)ϕ(x12y)̄dy.
(32)
W(ψ, ϕ) is a continuous function satisfying the estimate
|W(ψ,ϕ)(z)|2πnψL2ϕL2.
(33)
When ψ = ϕ the function W(ψ, ψ) = is the usual Wigner function
Wψ(z)=12πnRneipyψ(x+12y)ψ(x12y)̄dy.
(34)
The Wigner functions of general Gaussian functions is well-known;16,21 if
ψA,B(x)=1πn/4(detA)1/4e12A+iBxx
(35)
where A,BSym(n,R), A > 0. Then16,21
WψA,B(z)=1πne1hGzz
(36)
where G ∈ Sp(n):
G=STSSp(n) ,  S=A1/20n×nA1/2BA1/2.
(37)
Explicitly
G=A+BA1BYBA1A1BA1.
(38)
Notice that, conversely, if ψL2(Rn) is such that W(z)=(π)ne1hMzz for some M = MT ∈ Sp(n), M > 0, then ψ = eψA,B (χR) where X and Y are determined by (38) and χ| = 1.

Sub-Gaussian estimates for the Wigner function refer to bounds on the magnitude of the Wigner function that ensure that it does not fluctuate too much. A function is sub-Gaussian if its tails decay faster than any Gaussian distribution. Using Proposition 7 one proves21,23 that:

Corollary 18.

Let ψL2(Rn), ψ ≠ 0, and assume that there exists C > 0 such that Wψ(z)Ce1Mzz where M = MT > 0. Then the symplectic eigenvalues λ1σλ2σλnσ of M are all 1. When λ1σ=λ2σ==λnσ=1 then the function ψ is a generalized Gaussian (35).

It follows from this result that the Wigner function can never have compact support: assume that there exists R > 0 such that (z) = 0 for |z| > R. Then, for every a > 0 there exists a constant C(a) > 0 such that Wψ(z)C(a)ea|z|2 for all zR2n. Choosing a large enough this contradicts the statement in Corollary 18 because as soon as a > 1 the symplectic eigenvalues of M = aIn×n are all equal to a.

Corollary 18 can be elegantly reformulated in terms of symplectic polar duality:

Proposition 19.

The Wigner function of ψL2(Rn) satisfies a sub-Gaussian estimate Wψ(z)Ce1Mzz if and only if the ellipsoid ΩM = {z: Mz z} is admissible: ΩM,σΩM, that is, ΩM contains a quantum blob S(B2n()), SSp(n).

Proof.

In view of Williamson’s diagonalization theorem 26 there exists S ∈ Sp(n) such that M = STDS where D=Λσ0n×n0n×nΛσ hence SM) = ΩD. Since S,σ) = [S(Ω)],σ (Proposition 10) it is sufficient to prove the result for M = D. Since ΩD,σ=ΩJD1J we thus have to show that the sub-Gaussian estimate is satisfied if and only if ΩJD1JΩD. This is equivalent to D ≤ −JD−1J (in the Löwner ordering) that is to Λσ(Λσ)1 which is possible if and only the symplectic eigenvalues λjσ are all 1. When the λjσ are all equal to one we have D = I2n × 2n., that is M = STS so that Wψ(z)=Ce1Mzz for some constant C.■

1. Definition using the Wigner function

Let Ls1(R2n) is the weighted L1-space defined by
Ls1(R2n)={ρ:R2nC:zsρL1(R2n)}
(39)
where z=(1+|z|2)1/2. In the definition below we are following our presentation of modulation spaces given in Ref. 19; see Refs. 13, 25, and 29 for a definitions used in time-frequency analysis [they are based on the short-time Fourier transform (Gabor transform)]. We denote by S(Rn) the Schwartz space of test functions decreasing rapidly to zero at infinity, together with their derivatives.

Definition 20.

The modulation space Ms1(Rn) (sR) consists of all ψL2(Rn) such that W(ψ,ϕ)Ls1R2n for every ϕS(Rn). When s = 0 the space M01(Rn)=S0(Rn) is called the Feichtinger algebra.

It turns out that it suffices to check that condition W(ψ,ϕ)Ls1R2n holds for one function ϕ ≠ 0 (hereafter called “window” for it then it holds for all ϕS(Rn).The mappings ψψϕ,Ms1 defined by
ψϕ,Ms1=W(ψ,ϕ)Ls1=R2n|W(ψ,ϕ)(z)|zsdz
form a family of equivalent norms, and the topology on Ms1(Rn) thus defined makes it into a Banach space. We have the chain of inclusions
S(Rn)Ms1(Rn)Ls1(Rn)F(Ls1(Rn))
where F is the Fourier transform; it follows by Riemann–Lebesgue that in particular
Ms1(Rn)Ls1(Rn)C0(Rn).
Observe that
Ms1(Rn)Ms1(Rn)ss
and one proves25 that
s0Ms1(Rn)=S(Rn).

An essential property of the spaces Ms1(Rn) is their metaplectic invariance for s ≥ 0. Recall that the metaplectic group Mp(n) is the unitary representation in L2(Rn) of the double cover Sp2(n) of the symplectic group Sp(n) (see for instance Ref. 16 for a detailed study of the metaplectic representation). We will denote πMp the covering projection Mp(n) ⟶ Sp(n); it is a two-to-one epimorphism.

Proposition 21.

The modulation spaces Ms1(Rn), s ≥ 0, are invariant under the action of Mp(n): if ψMs1(Rn) and ŜMp(n) then ŜψMs1(Rn).

Proof.
By definition we have ψMs1(Rn) if and only if W(ψ,ϕ)Ls1R2n for every ϕS(Rn); similarly ŜψMs1(Rn) if and only W(Ŝψ,ϕ)Ls1R2n for every ϕ. Now, in view of the symplectic covariance of the cross-Wigner function16,19
W(Ŝψ,ϕ)=W(Ŝψ,Ŝ(Ŝ1ϕ))=W(ψ,Ŝ1ϕ)S1
(40)
where S=πMp(Ŝ). Since Ŝ1ϕS(Rn) there remains to prove that W(ψ,Ŝ1ϕ)S1Ls1R2n. Set ϕ=Ŝ1ϕ; we have det S = 1 hence
R2n|W(ψ,ϕ)(S1z)|zsdz=R2n|W(ψ,ϕ)(z)|Szsdz.
We have SzsCSzs for some constant CS > 0, thus
R2n|W(ψ,ϕ)(S1z)|zsdzCSR2n|W(ψ,ϕ)(z)|zsdz<
and we are done.■

Quantum mechanics is inherently probabilistic, and the behavior of quantum systems is described by mathematical entities known as quantum states. These states are represented by density operators (also called density matrices by physicists). An element ρ̂ of the algebra L1(Rn) of trace class operators on L2(Rn) is called a density operator if ρ̂0 and has trace Trρ̂=1; the passiveness implies in particular that ρ̂ is a self-adjoint compact operator so that there exists a sequence (αj) with αj ≥ 0, ∑jαj = 1 and an orthonormal stem (ψj), ψjL2(Rn) such that we have we have the spectral decomposition
ρ̂=jαjρ̂j,  αj0  ,jαj=1
(41)
where ρ̂j is the rank one orthogonal projection in L2(Rn) on the ray Cψj. It follows that that the Weyl symbol of the operator ρ̂ is19,
(2π)nρ=jαjWψj
(42)
(the function ρ is often called the Wigner distribution of ρ̂ in the physical literature). In Refs. 15 and 22 we introduced the notion of Feichtinger state:

Definition 22.

A density operator ρ̂L1(Rn) is called a aFeichtinger stateif each ψjMs1(Rn) for some s ≥ 0.

Feichtinger states satisfy the marginal properties: we have ρL1(R2n) and
Rnρ(x,p)dp=jαj|ψj(x)|2,Rnρ(x,p)dx=jαj|Fψj(x)|2.
(43)
The main interest of the notion of Feichtinger state in our context is that they also allow to define rigorously the covariance matrix Σcov of a density operator. The latter is defined –if it exists!– as being the symmetric 2n × 2 × n matrix
Σcov=R2n(zz̄)(zz̄)Tρ(z)dz
(44)
where z̄=R2nzρ(z)dz is the average (or mean value) vector. It is convenient to write the covariance matrix in n × n block-matrix form as
Σcov=ΣXXΣXPΣPXΣPP  ,  ΣPX=ΣXPT
(45)
where ΣXX=(σxjxk)1j,k,n, ΣPP=(σpjpk)1j,k,n, and ΣXP=(σxjpk)1j,k,n with
σxjxk=R2nxjxkρ(z)dz
(46)
and so on. We have the following conjugation result:

Proposition 23.

The covariance matrix Σcov of a Feichtinger state ρ̂ with s ≥ 2 is well-defined. If ŜMp(n) then conjugate state Ŝρ̂Ŝ1 is also a Feichtinger state with s ≥ 2 and the covariance matrix of Ŝρ̂Ŝ1 is SΣcovST.

Proof.
Since s ≥ 2 we have
R2n|ρ(z)|(1+|z|2)dz<.
(47)
It is no restriction to assume z̄=0; setting zα = xα if 1 ≤ αn and zα = pα if n + 1 ≤ α ≤ 2n we have Σ=(σαβ)1α,β2n where the integrals
σαβ=R2nzαzβρ(z)dz
are absolutely convergent in view of the trivial inequalities |zαzβ| ≤ 1 + |z|2. Let ŜMp(n); it follows from the standard properties of Weyl pseudodifferential calculus16,19,38 that the Weyl symbol of the conjugate Ŝρ̂Ŝ1 is(2πℏ)nρS−1. In view of Proposition 21 ρMs1(Rn) implies that ρSMs1(Rn) and the result follows from (44) by a simple calculation.■

1. A necessary condition for positivity

Assume from now on that ρ̂ is a Feichtinger state with s ≥ 2, guaranteeing the existence of the covariance matrix Σcov. It i a well-known property in harmonic analysis that the positivity condition ρ̂0 implies that Σ must satisfy the algebraic condition
Σ+i2J is positive semidefinite
(48)
which we write for short Σ+i2J0. (Note that the matrix Σ+i2J is selfadjoint since Σ is symmetric and J* = −J.) This condition implies in particular that Σ is positive definite35,19 and hence invertible. This (highly nontrivial) result can be proven by various methods, one can for instance use the notion of η-positivity due to Kastler30 together with a variant of Bochner’s theorem on the Fourier transform of a probability measure; for a simpler approach using methods from harmonic analysis see our recent paper10 with Cordero and Nicola. The condition (48) actually first appeared as a compact formulation of the uncertainty principle in Arvind et al..2 We have rigorously shown in Refs. 16 and 17 that (48) is equivalent to the Robertson–Schrödinger inequalities
σxjxjσpjpjσxjpj2+142  for 1jn
(49)
which form the textbook statement of the complete uncertainty principle of quantum mechanics.33 While condition (48) is generally only a necessary condition for the positivity of a trace class operator, it is also sufficient for operators with Gaussian Weyl symbol (2πℏ)nρ where
ρ(z)=1(2π)ndetΣe12Σ1(zz0)(zz0)
(50)
(Σ = ΣT > 0 playing the role of a covariance matrix Σcov; see for instance 15, 21, 25, and 28).

Symplectic capacities were defined by Ekeland and Hofer11,12 (see 9 and 28 for review of that notion). They are closely related to Gromov’s symplectic non-squeezing theorem shortly discussed above; in fact the existence of a single symplectic capacity is equivalent to Gromov’s theorem.

Gromov’s theorem ensures us of the existence of symplectic capacities. A (normalized) symplectic capacity on (R2n,σ) associates to every subset ΩR2n a number c(Ω) ∈ [0, + ∞] such that the following properties hold (Ref. 28, see Ref. 24 for a review):
  • SC1 Monotonicity: If Ω ⊂ Ω′ then c(Ω) ≤ c(Ω′);

  • SC2 Conformality: For every λR we have c(λΩ) = λ2c(Ω);

  • SC3 Symplectic invariance: c[f(Ω)] = c(Ω) for every f ∈ Symp(n);

  • SC4 Normalization: For 1 ≤ jn,

c(B2n(r))=πr2=c(Zj2n(r))
(51)
where Zj2n(r) is the cylinder with radius r based on the xj, pj plane.

It follows that the symplectic capacity of a quantum blob QS(z0)=S(B2n()) is c(QS(z0)) = πℏ.

There exist symplectic capacities, cmax and cmin, such that cminccmax for every symplectic capacityc, they are defined by
cmax(Ω)=inffSymp(n){πr2:f(Ω)Zj2n(r)}
(52)
cmin(Ω)=supfSymp(n){πr2:f(B2n(r))Ω}.
(53)
That cmin and cmin indeed are symplectic capacities follows from the axioms (SC1)–(SC4). Note that the conformality and normalization properties (SC2) and (SC4) show that for n > 1 symplectic capacities are not related to volume; they have the dimension of an area,1 or equivalently, that of an action. For instance, the Hofer–Zehnder capacity 28 is characterized by the property that when Ω is a compact convex set in (R2n,σ) with smooth boundary Ω then
cHZ(Ω)=γminpdx
(54)
where pdx = p1dx1 + ⋅⋅⋅ + pndxn and γmin is the shortest positively oriented Hamiltonian periodic orbit carried by Ω.

One also has the weaker notion of linear (or affine) symplectic capacity, obtained by replacing condition (SC3) with

SC3lin Linear symplectic invariance: c[S(Ω)] = c(Ω) for every S ∈ Sp(n) and c(Ω+z)=c(Ω) for every zR2n.

The corresponding minimal and maximal linear symplectic capacities cminlin and cmaxlin are then given by
cminlin(Ω)=supSSp(n){πR2:S(B2n(z,R)),zR2n}
(55)
cmaxlin(Ω)=inffSp(n){πr2:S(Ω)Zj2n(z,r),zR2n}.
(56)

1. The case of ellipsoids

It turns out that all symplectic capacities (linear as well as non-linear) agree on ellipsoids. Assume that
ΩM={zR2n:Mzzr2}
where MSym+(2n,R), and let λmaxσ=λ1σλ2σλnσ be the symplectic eigenvalues of M. If in particular Ω = ΩM: Mz⋅z then
c(ΩM)=π/λmaxσ.
(57)
We have in particular
cHZ(ΩM)=γminpdx=π/λmaxσ.
This is easy to verify using the calculations in the Proof of Theorem 17(ii): reducing ΩM to Williamson normal form
λ1σ(x12+p12)++λnσ(xn2+pn2)
the shortest Hamiltonian orbit carried by ΩM is given by Hamilton’s equations for the Hamiltonian function H1(x1,p1)=λ1σ(x12+p12) with the condition H1(x1, p1) = . One verifies that this periodic solution verifies
γminpdx=π/λ1σ=π/λmaxσ=c(ΩM).

For the symplectic polar dual ellipsoid we have the following result, which yields a Blaschke–Santaló type inequality for symplectic capacities of ellipsoids:

Proposition 24.
Let ΩM be as above and let ΩM,σ be its symplectic polar dual. (i) We have
c(ΩM)=π/λmaxσ   and  c(ΩM,σ)=πλminσ
(58)
where λmaxσ (resp. λminσ) is the largest (rep. smallest)symplectic eigenvalue of M. (ii) In particular
c(ΩM)c(ΩM,σ)(π)2
(59)
with equality if and only if ΩM=λB2n() for some λ > 0.

Proof.
(i) The formula c(ΩM)=πr2/λmaxσ is easily proven using a symplectic diagonalization (26) of M which reduces the problem to the case of an ellipsoid with axes contained in the conjugate xj, pj planes (see Refs. 28 and 16 for details). The symplectic polar dual of ΩM is
ΩM,σ={zR2n:JM1Jzz}
[formula (21) in Lemma 9]. Set N = −JM−1J: we have JN = M−1J hence the eigenvalues of JN are those of M−1/2JM−1/2 so the symplectic eigenvalues of N are the inverses of those of M; the second formula (58) follows. (ii) Formula (59) is obvious; that we have equality if and only if all ΩM=λB2n() follows from the fact that if the symplectic eigenvalues of M are all equal then by Williamson’s theorem M is a scalar multiple of a matrix STS.■

1. Covariance and information ellipsoids

We are going to express condition (48) in a simple geometric way using symplectic polarity. By definition, the covariance ellipsoid of a Feichtinger state ρ̂ is
Ωcov={zR2n:12Σcov1zz1}
(60)
where Σcov is the covariance matrix of ρ̂. The symplectic polar dual of Ωcov is the ellipsoid
Ωcov,σ={zR2n:12JΣJzz1}.
By definition the associated information (or precision) ellipsoid is
Ωinfo={zR2n:12Σcovzz1}
by the equality Ωcov,σ=J(Ωinfo), that is, Ωinfo=J(Ωcov) where Ωcov is the ordinary polar dual of Ωcov. It turns out that Ωcov and Ωinf o are Legendre duals of each other.35 Consider in fact the quadratic forms w(z)=12Σcov1zz and wσ(z)=12Σcovzz. The Legendre transform of w(z) is defined by
w(z)=zzw(z)
where z is expressed in terms of z′ by solving the equation z=zw(z)=Σcov1z hence w′(z′) = wσ(z′).

We are going to prove that the covariance ellipsoid of a Feichtinger state always is quantum admissible:

Proposition 25.

The covariance ellipsoid Ωcov of a Feichtinger state ρ̂ satisfies the condition Ωcov,σΩcov and is hence quantum admissible.

Proof.
Setting set M=2Σ1 the covariance ellipsoid is then defined by ΩM: Mz z and we have to prove that the positivity condition Σ+i2J0 is equivalent to the condition ΩM,σΩM. In view of Propositions 17 it is sufficient to prove that this conditions hold if and only if ΩM contains a quantum blob QS=S(B2n()), S ∈ Sp(n). Performing a symplectic diagonalization M = STDS (26) of M the condition Σ+i2J=M1+iJ0 implies that D−1 + iJ ≥ 0, that is
D1+iJ=(Λσ)1iIn×niIn×n(Λσ)10
where Λσ=diag(λ1σ,,λnσ) the λjσ being the symplectic eigenvalues of M. The eigenvalues of D−1 + iJ are the real numbers λj=(λjσ)1±1 and the condition D−1 + iJ ≥ 0 thus implies that we must have λjσ1 for 1 ≤ jn. It follows that the ellipsoid ΩD: Dz⋅z contains the ball B2n() and hence ΩM contains the quantum blob S(B2n()) where S ∈ Sp(n) is the diagonalizing matrix. The result now follows applying Propositions 17.■

Remark 26.
Assume that the Wigner distribution of ρ̂ is a Gaussian (50) with z0 = 0. If Ω,σ = Ω it follows from Propositions 17) that Ω is a quantum blob S(B2n()) and hence
ρ(z)=1(2π)ne12Gzz ,  G=STS
(61)
for some SSp(n). Then16,19 ρ=W(Ŝ1ϕ0) where
ϕ0(x)=(π)ne|x|2/2
and ŜMp(n) has projection πMp(Ŝ)=S.

2. A dynamical characterization of admissibility

Let Ω be an ellipsoid in R2n with smooth boundary Ω. We assume that Ω is the energy hypersurface of some (quadratic) Hamiltonian function HC(R2n,R): i.e. Ω = {z: H(z) = E} for some ER. We ask now when Ω can be viewed as the covariance ellipsoid of a quantum state; the following is in a sense a restatement of Theorem 17, but we give an independent proof here:

Theorem 27.
The ellipsoid Ω is a quantum covariance ellipsoid Ωcov (resp. an information ellipsoid Ωinf o) if and only if the following equivalent conditions are satisfied: (i) We have
γpdxc(Ω)π
(62)
for every periodic Hamiltonian orbit γ carried by Ω; if we have equality for the shortest orbit then Ω is a quantum blob. (ii) Let Ω* be the Legendre transform of Ω; we have
c(Ω*)4π/.
(63)
(iii) Let F be an arbitrary two-dimensional subspace of R2n an let the ellipse γF*=Ω*F be positively oriented. We have
γF*pdx4π.
(64)

Proof.
(i) The inequality (62) follows from (54) and (58). (ii) If Ω is defined by 12Σ1zz1 then Ω* is defined by 12Σzz1. Setting M=2Σ1 the ellipsoid Ω* is given by Nz z where N=24M1. The symplectic spectrum of N is thus 24((λnσ)1,,λ1σ)1) where (λ1σ,,λnσ) is the symplectic spectrum of M (recall our convention to rank symplectic eigenvalues in non-increasing order). It follows that the symplectic capacity of Ω* is
c(Ω*)=π24(λnσ)114π
the last inequality because Ω is quantum admissible if and only if λnσ=λmaxσ1. To prove the action inequality (64) we can proceed as follows: suppose first that F is a null space for the symplectic form [i.e. F has a basis {e1, e2} such that σ(e1, e2) = 0]. The, by Stokes’s theorem we have
γF*pdx=Ω*Fσ=0
(65)
so that (64) is trivially verified. Assume next that F is a symplectic plane. Then, by formula (29) in Theorem 17 we have
Area(ΩM,σF)π
(66)
but this is precisely (64) since M=2Σ1.■

The first author (MdG) has been financed by the Grant Nos. P 33447 N and PAT 2056623 of the Austrian Research Foundation FWF.

The authors have no conflicts to disclose.

Maurice de Gosson: Funding acquisition (equal); Investigation (equal); Project administration (equal). Charlyne de Gosson: Data curation (equal); Project administration (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

1.
S.
Artstein-Avidan
,
V.
Milman
, and
Y.
Ostrover
, “
The M-ellipsoid, symplectic capacities and volume
,”
Comment. Math. Helv.
83
,
359
369
(
2008
).
2.
B. D.
Arvind
,
B.
Dutta
,
N.
Mukunda
, and
R.
Simon
, “
The real symplectic groups in quantum mechanics and optics
,”
Pramana
45
(
6
),
471
497
(
1995
).
3.
G.
Aubrun
and
S. J.
Szarek
,
Alice and Bob meet Banach
,
Mathematical Surveys and Monographs
(
American Mathematical Society
,
2017
), Vol.
223
.
4.
K. M.
Ball
, “
Ellipsoids of maximal volume in convex bodies
,”
Geom. Dedicata.
41
(
2
),
241
250
(
1992
).
5.
G.
Bianchi
and
M.
Kelly
, “
A Fourier analytic proof of the Blaschke–Santaló inequality
,”
Proc. Am. Math. Soc.
143
(
11
),
4901
4912
(
2015
).
6.
W.
Blaschke
, “
Über affine Geometrie VII: Neue extremeingenschaften von ellipse und ellipsoid
,
Ber. Verh. S Achs. Akad. Wiss., Math. Phys. Kl.
69
,
412
420
(
1917
).
7.
J.
Bourgain
and
V.
Milman
, “
New volume ratio properties for convex symmetric bodies
,”
Invent. Math.
88
,
319
340
(
1987
).
8.
S.
Boyd
,
S. P.
Boyd
, and
L.
Vandenberghe
,
Convex Optimization
(
Cambridge University Press
,
2004
).
9.
K.
Cieliebak
,
H.
Hofer
,
J.
Latschev
, nd
F.
Schlenk
, “
Quantitative symplectic geometry
,” arXiv:math/0506191 (
2005
).
10.
E.
Cordero
,
M.
de Gosson
, and
F.
Nicola
, “
On the positivity of trace class operators
,”
Adv. Theor. Math. Phys.
23
(
8
),
2061
2091
(
2019
).
11.
I.
Ekeland
and
H.
Hofer
, “
Symplectic topology and Hamiltonian dynamics
,”
Math. Z.
200
(
3
),
355
378
(
1989
).
12.
I.
Ekeland
and
H.
Hofer
, “
Syplectic topology and Hamiltonian dynamics II
,”
Math. Z.
203
,
553
567
(
1990
).
13.
H. G.
Feichtinger
, “
Modulation spaces: Looking back and ahead
,”
Sampling Theory Signal Image Process.
5
(
2
),
109
140
(
2006
).
14.
M.
de Gosson
, “
Phase space quantization and the uncertainty principle
,”
Phys. Lett. A
317
(
5–6
),
365
(
2003
).
15.
C.
de Gosson
and
M.
de Gosson
, “
On the non-uniqueness of statistical ensembles defining a density operator and a class of mixed quantum states with integrable Wigner distribution
,”
Quantum Rep.
3
(
3
),
473
481
(
2021
).
16.
M.
de Gosson
,
Symplectic Geometry and Quantum Mechanics
(
Springer Science & Business Media
,
2006
), Vol.
166
.
17.
M.
de Gosson
, “
The symplectic camel and the uncertainty principle: The tip of an iceberg?
,”
Found. Phys.
39
,
194
(
2009
).
18.
M.
de Gosson
, “
Quantum blobs
,”
Found. Phys.
43
(
4
),
440
457
(
2013
).
19.
M.
de Gosson
,
Symplectic Methods in Harmonic Analysis and in Mathematical Physics
(
Birkhäuser
,
2011
).
20.
M.
de Gosson
, “
Two geometric interpretations of the multidimensional Hardy uncertainty principle
,”
Appl. Comput. Harmonic Anal.
42
(
1
),
143
153
(
2017
).
21.
M.
de Gosson
,
The Wigner Transform
,
Advanced Texts in Mathematics
(
World Scientific
,
2017
).
22.
M.
de Gosson
,
Quantum Harmonic Analysis, an Introduction
(
De Gruyter
,
2021
).
23.
M.
de Gosson
and
F.
Luef
, “
Quantum states and Hardy’s formulation of the uncertainty principle: A symplectic approach
,”
Lett. Math. Phys.
80
,
69
82
(
2007
).
24.
M.
de Gosson
and
F.
Luef
, “
Symplectic capacities and the geometry of uncertainty: The irruption of symplectic topology in classical and quantum mechanics
,”
Phys. Rep.
484
,
131
179
(
2009
).
25.
K.
Gröchenig
,
Foundations of Time-Frequency Analysis
(
Birkhäuser
,
Boston
,
2000
).
26.
M.
Gromov
, “
Pseudo holomorphic curves in symplectic manifolds
,”
Inventiones Math.
82
(
2
),
307
347
(
1985
).
27.
G. H.
Hardy
, “
A theorem concerning Fourier transforms
,”
J. London Math. Soc.
s1-8
,
227
231
(
1933
).
28.
H.
Hofer
and
E.
Zehnder
,
Symplectic Invariants and Hamiltonian Dynamics
,
Birkhäuser Advanced Texts (Basler LehrbüCher)
(
Birkhäuser Verlag
,
1994
).
29.
M. S.
Jakobsen
, “
On a (no longer) new Segal algebra: A review of the Feichtinger algebra
,”
J. Fourier Anal. Appl.
24
(
6
),
1579
1660
(
2018
).
30.
D.
Kastler
, “
The C*-algebras of a free Boson field
,”
Commun. Math. Phys.
1
,
14
48
(
1965
).
31.
G.
Kuperberg
, “
From the Mahler conjecture to Gauss linking integrals
,”
Geom. Funct. Anal.
18
(
3
),
870
892
(
2008
).
32.
K.
Mahler
, “
Ein Übertragungsprinzip für konvexe Körper
,”
Časopis pro Pěstování Mat. Fyz.
68
(
3
),
93
102
(
1939
).
33.
A.
Messiah
,
Quantum Mechanics
(
North Holland, Amsterdam
,
1970
), Vol.
1
, translated by G. M. Temmer.
34.
V. D.
Milman
, “
Geometrization of probability
,”
Geom. Dyn. Groups Spaces
265
,
647
(
2008
).
35.
F. J.
Narcowich
, “
Conditions for the convolution of two Wigner functions to be itself a Wigner function
,”
J. Math. Phys.
30
(
11
),
2036
2041
(
1988
).
36.
L.
Polterovich
,
The Geometry of the Group of Symplectic Diffeomorphisms
(
Birkhäuser
,
2012
).
37.
L. A.
Santaló
, “
Un invariante a n para los cuerpos convexos del espacio de n dimensiones
,”
Portugaliae. Math.
8
,
155
161
(
1949
).
38.
M. A.
Shubin
,
Pseudodifferential Operators and Spectral Theory
(
Springer-Verlag
,
1987
) [original Russian edition in Nauka, Moskva. 1978].
39.
N. T.
Son
,
P.-A.
Absil
,
B.
Gao
, and
T.
Stykel
, “
Computing symplectic eigenpairs of symmetric positive-definite matrices via trace minimization and Riemannian optimization
,”
SIAM J. Matrix Anal. Appl.
42
,
1732
1757
(
2021
).
40.
R.
Vershynin
, “
Lectures in geometric functional analysis
” (unpublished) (2011), available https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf (
2011
), pp.
3-3
.
41.
J.
Williamson
, “
On the algebraic problem concerning the normal forms of linear dynamical systems
,”
Am. J. Math.
58
(
1
),
141
163
(
1936
).
42.
E.
Zehnder
, “
Lectures on dynamical systems: Hamiltonian vector fields and symplectic capacities
,”
EMS Textbooks in Mathematics
(
European Mathematical Society
,
2010
).