Born–Jordan Pseudodifferential Calculus, Bopp Operators and Deformation Quantization

There has recently been a resurgence of interest in Born–Jordan quantization, which historically preceded Weyl’s prescription. Both mathematicians and physicists have found that this forgotten quantization scheme is actually not only of great mathematical interest, but also has unexpected application in operator theory, signal processing, and time-frequency analysis. In the present paper we discuss the applications to deformation quantization, which in its traditional form relies on Weyl quantization. Introducing the notion of “Bopp operator” which we have used in previous work, this allows us to obtain interesting new results in the spectral theory of deformation quantization.


Introduction
Deformation quantization is a popular framework for quantum mechanics among mathematical physicists. It was suggested by Moyal [33] and Groenewold [26], and put on a firm mathematical ground by Bayen et al. [1,2]; later Kontsevich [29,30] extended the theory to Poisson manifolds. Roughly speaking, the idea is to "deform" classical (Hamiltonian) mechanics into quantum mechanics using a parameter (Planck's constant); this is achieved using the notion of "star product" or "Moyal product" of two functions on R 2n . The star product is defined in physics by the suggestive formula the exponential in the right hand side (the "Janus operator") is understood as a power series, the arrows indicating the direction in which the derivatives act. This formula was proposed for the first time by Groenewold in his seminal work [26] in 1946. A rigorous definition is the following: denoting by Weyl ←→ the Weyl correspondence between operators and symbols, assume that a, b ∈ S (R 2n ) and let A Weyl ←→ a and B Weyl ←→ b (A is sometimes called the "Weyl transform" of a). If the product C = AB is defined and C Weyl ←→ c then, by definition, c = a b.
We have shown in previous work [17,18] that we have where A is a pseudodifferential operator acting on distributions defined on R 2n ; formally We have called A the "Bopp pseudodifferential operator" with symbol a; it is the Weyl operator on T * R 2n ≡ R 2n × R 2n with symbol a(z, ζ) = a x − 1 2 ζ p , p + 1 2 ζ x (4) where (ζ x , ζ p ) are viewed as the dual variables of (x, p). This reformulation of the star product in terms of pseudodifferential operators is very fruitful; not only does it allow the study of the generalized eigenvalues and eigenfunctions of "stargenvalue" problems using standard pseudodifferential techniques, but it also leads to interesting regularity results in various functional spaces. The main observation, which leads to the theme of the present paper, is that the whole procedure heavily relies on the Weyl pseudodifferential calculus. From a physical point of view, this means that we are privileging Weyl quantization; technically this choice has many advantages because Weyl quantization is the simplest and most austere of all quantizations: using Schwartz's kernel theorem one shows that every continuous linear operator A: S(R n ) −→ S (R n ) can be viewed as a Weyl operator, and the Weyl correspondence is uniquely characterized by the property of symplectic covariance: if A Weyl ←→ a then SA S −1 Weyl ←→ a • S −1 for every metaplectic operator S ∈ Mp(n) with projection S ∈ Sp(n) (the symplectic group). However, in real life things are not always that simple. Just a couple of years before Weyl [38] defined the eponymous correspondence, Born and Jordan [7], elaborating on Heisenberg's 1925 "matrix mechanics" [27], proposed a quantization procedure having a firm physical motivation (conservation of energy); their approach culminated one year later in their famous "drei Männer Arbeit" [8] with Heisenberg. There are many good reasons to believe that the Born and Jordan quantization scheme is the right one in physics (Kauffmann [28]); in addition, some very recent work of Boggiatto and his collaborators [3][4][5] shows that the Wigner formalism corresponding to Born-Jordan quantization is much more adequate in signal analysis than the traditional Weyl-Wigner approach. It allows to damp the appearance of unwanted "ghost" frequencies in spectrograms; numerical experiments confirm these theoretical facts.
Vol. 84 (2016) Born-Jordan Pseudodifferential Calculus 465 In [16] the first of the authors has studied the properties of Born-Jordan pseudodifferential calculus; in the present paper we go one step further, and reformulate deformation quantization in terms of this calculus.
Notation. We will write z = (x, p) where x ∈ R n and p ∈ (R n ) * ≡ R n . Operators S(R n ) −→ S (R n ) are usually denoted by A, B, . . . while operators S(R 2n ) −→ S (R 2n ) are denoted by A, B, . . . The lower-case Greek letters ψ, φ, . . . stand for functions (or distributions) defined on R n while their uppercase counterparts ψ, φ, . . . denote functions (or distributions) defined on R 2n . The distributional bracket on R n is denoted by ·, · and that on R 2n by ·, · . We denote by σ = dp 1 ∧ dx 1 + · · · + dp n ∧ dx n the standard symplectic form on T * R n ≡ R n × R n ; in coordinates: σ(z, z ) = Jz · z where J = 0 n I n −I n 0 n is the standard symplectic matrix.

Born-Jordan Versus Weyl
Let us quickly review the Born-Jordan and Weyl quantizations of monomials x m j p j . In what follows, the capital letters X j and P j denote operators acting on some space of functions or distributions on R d , and satisfying Born's commutation relations For instance, in traditional quantum mechanics d = n and X j is the operator of multiplication by x j while P j = −i ∂ xj , but there is no compelling reason for limiting ourselves to these operators. Keeping this in mind, the Weyl quantization of monomials is given by the rule while Born-Jordan quantization is given by The Weyl and Born-Jordan correspondences agree for all monomials which are at most quadratic, as well as for monomials of the type p j x m j or p j x j . They are however different as soon as we have l ≥ 2 and m ≥ 2 (Turunen [37]). It turns out that both rules can be obtained from the τ -correspondence, defined by where τ is a real number. The case τ = 1 2 yields the Weyl correspondence (6). Integrating the τ -correspondence over the interval [0, 1] and using the formula we get the Born-Jordan correspondence (7). Historically, things evolved the other way round: in [7] Born and Jordan were led to the eponymous correspondence (7) by a strict analysis of Heisenberg's [27] ideas. In their subsequent publication [8] with Heisenberg they showed that their constructions extend mutatis mutandis to systems with an arbitrary number of degrees of freedom.
In the general case one proceeds as follows (de Gosson [16]): let τ be a real parameter, and define the τ -pseudodifferential operator A τ = Op τ (a) with symbol a ∈ S (R 2n ) as being the operator S(R n ) −→ S (R n ) with distributional kernel is the inverse Fourier transform in the second set of variables. This defines the so-called Shubin τ -correspondence [34] A τ ←→ a by; it is easy to check that one recovers the correspondence (8) for monomials. For a ∈ S(R 2n ) and ψ ∈ S(R n ) the more suggestive formula holds (Shubin [34], §23), which can also be extended to more general settings. The choice τ = 1 2 leads to the usual Weyl operators: A = Op BJ (a) with The Born-Jordan pseudodifferential operator A = Op BJ (a) is obtained by averaging the Shubin operators A τ over τ ∈ [0, 1]: it is thus the operator S(R n ) −→ S (R n ) with kernel (Heuristically, the Weyl operator (10) is obtained by approximating the integral in (12) using the midpoint rule). One verifies by a direct calculation, that the correspondence A BJ ←→ a reduces to the Born-Jordan rules (7) for polynomials x m j p j . As already mentioned, the Born-Jordan and Weyl correspondences agree for all quadratic polynomials in the variables x j , p j . More generally Vol. 84 (2016) Born-Jordan Pseudodifferential Calculus 467 (de Gosson [16]) both quantizations are also identical for symbols arising from physical Hamiltonians of the type where A j and V are real C ∞ functions.

Harmonic Analysis of A BJ
It is usual to write Weyl operators A = Op W (a) in the form of operator valued integrals where a σ is the "twisted symbol" of A: for a function (or distribution) ψ on R n . Similarly, the Shubin operator A τ = Op τ (a) can be written (de Gosson [16]) where T τ (z 0 ) is the modified Heisenberg-Weyl operator defined by

Proposition 1. (i) The Born-Jordan operator A BJ = Op BJ (a) is given by
where Θ is defined by (ii) The twisted Weyl symbol a W σ of A BJ is given by the explicit formula (iii) The operator A BJ is hence a continuous operator S(R n ) −→ S (R n ) for every a ∈ S (R 2n ).
Remark 2. Notice that Θ(z) = sinc(px/2 ) where sinc(t) = (sin t)/t is the cardinal sine function familiar from signal analysis. IEOT Proof. The statement (ii) immediately follows from formula (19) taking the representation (14) of Weyl operators into account. The proof of formula (19) goes as follows (cf. [16], Proposition 11): integrating both sides of the equality (17) with respect to the parameter τ ∈ [0, 1] one gets It immediately follows from formula (20) that since Θ(z 0 ) = 0 for all z 0 = (x 0 , p 0 ) such that p 0 x 0 = 2Nπ for some integer N ∈ Z we see that an arbitrary continuous operator A : S(R n ) −→ S (R n ) is not in general a Born-Jordan operator: every such operator A has indeed a twisted Weyl symbol a W σ in view of Schwartz's kernel theorem, but because of zeroes of we cannot in general expect the Eq. (21) to be solved for a σ . This property of Born-Jordan operators really distinguishes them among all traditional pseudodifferential operators: the Born-Jordan "correspondence" is neither surjective, nor injective. Keeping this caveat in mind, we will still write symbolically a BJ −→ A or A = Op BJ (a).

Composition and Adjoints of Born-Jordan Operators.
Let A : S(R n ) −→ S (R n ) and B : S(R n ) −→ S(R n ) be two continuous operators; their product AB is well-defined, and its Weyl symbol can be explicitly determined in terms of those of A and B. In fact if A = Op W (a) and There are several ways to rewrite this formula; performing elementary changes of variables we have which is well-known in the literature. For our purposes, it will be more tractable to use the following formula, which gives the twisted symbol of the compose in terms of the twisted symbols of the factors: Vol. 84 (2016) Born-Jordan Pseudodifferential Calculus 469 Proposition 3. Let A = Op BJ (a) and B = Op BJ (b) be two Born-Jordan pseudodifferential operators; we suppose that C = AB is defined as an operator where Θ is defined by (20).
In particular, the Born-Jordan operator A is formally self-adjoint if and only its symbol is real. (25) is an immediate consequence of formulas (24) and of (21) since c σ = a b. The statement (ii) follows, using again (21). (iii) The adjoint of the τ -pseudodifferential operator [15,34]); it follows that Remark 4. Note that χ, and hence c, are not uniquely defined by the relation c W σ (z) = χ(z)Θ(z) since Θ(z) = 0 for infinitely many values of z. On the other hand, it is not obvious that an arbitrary Weyl operator can be written as a Born-Jordan operator. That this is however the case has been proven recently in Cordero et al. [11] using techniques from distribution theory (the Paley-Wiener theorem).

Bopp Calculus
Setting v = z 0 , z + 1 2 u = z in the formula (23) and introducing the notation we can rewrite formula (22) as The restrictions of the operators T (z 0 ): S (R 2n ) −→ S (R 2n ) to L 2 (R 2n ) are unitary, and satisfy the same commutation relations as the Heisenberg-Weyl operators. In [17] we have proven the following result: We now introduce the following elementary operators (called "Bopp shifts" following Bopp [6]; also see Kubo [31]) acting on phase space functions and distributions: These operators satisfy Born's commutation relations (5), and we can thus define the extended quantization rule corresponding to (6)- (7), respectively. The Weyl and Born-Jordan symbols of X j and P j being, respectively, x j − 1 2 ζ p,j and p j + 1 2 ζ x,j formula (29) suggests the notation

The Born-Jordan Starproduct
In the Born-Jordan case we would like to define Bopp quantization using a procedure extending the natural correspondence induced by the monomial rule (7). We will proceed as follows: returning to formula (17) we define the phase-space τ -operator by where T τ (z 0 ) is defined in terms of the operator (26) by In analogy with formula (2) we now define the "Born-Jordan starproduct" ,BJ : Definition 6. Let a ∈ S (R 2n ). The Bopp-Born-Jordan (BBJ) operator with symbol a is the operator defined by the integral where A τ is the pseudodifferential operator (32). Let b ∈ S(R 2n ). We set Vol. 84 (2016) Born-Jordan Pseudodifferential Calculus 471 In view of formula (19) the BBJ operator has the explicit expression where Θ ∈ L ∞ (R 2n ) ∩ C ∞ (R 2n ) is given by (20).

The Functions Amb BJ and Wig BJ
In what follows ·, · denotes the distributional bracket on R 2n . The Weyl correspondence between symbols and operators can be defined using the Wigner formalism. In fact, given a symbol a ∈ S(R 2n ) one can show (see e.g. [15], §10.1) that the operator A Weyl ←→ a is the only operator such that where Wig(ψ, φ) is the cross-Wigner distribution (or function) of ψ, φ ∈ S(R n ): Noting that (Aψ|φ) L 2 = Aψ, φ and that Wig(ψ, φ) ∈ S(R 2n ) formula (37) allows to extend the definition of the operator A to the case where a ∈ S (R 2n ). In view of Plancherel's theorem we can rewrite (37) as where f ∨ (z) = f (−z). Since the symplectic Fourier transform of the cross-Wigner transform is the cross-ambiguity function [14,15,23] Amb  (15) is involutive: F 2 σ = I d and satisfies the following variant of the Plancherel identity where ., . denotes the scalar product on R 2n : Proposition 7. Let a ∈ S (R 2n ) and ψ, φ ∈ S(R n ); we have where Amb BJ (ψ, φ) and Wig BJ (ψ, φ) are defined by where Θ σ = F σ Θ is the symplectic Fourier transform of Θ. IEOT Proof. In view of formula (19) we have hence formula (42) since Θ(−z) = Θ(z). By the second equality in Plancherel's formula (41), we have, since F σ is involutive, which is formula (43).
As the usual cross-Wigner transform, Wig τ satisfies a Moyal identity (or "orthogonality relation" as it is sometimes called): Boggiatto et al. [5] have shown that for every τ ∈ R and for all functions ψ, ψ , φ, φ in L 2 (R n ). However, the Moyal identity does not hold for Wig BJ . Here is why: let Q(ψ, φ) = Wig(ψ, φ) * θ (θ ∈ S (R n )) be an element of the Cohen class. The Moyal identity is satisfied if and only if the Fourier transform θ of the Cohen kernel θ satisfies | θ(z)| = (2π ) n (Cohen [12,13]). In the Born-Jordan case the Fourier transform of the Cohen kernel is the function Θ(z) = sinc(px/2π ) which does not satisfy this condition.

Intertwiners
We are going to show that the usual Born-Jordan operator A BJ = Op BJ (a) and the corresponding BBJ operator A BJ = Op BJ (a) are intertwined by a family of linear mappings L 2 (R n ) −→ L 2 (R 2n ). This important result will allow us to study the regularity and spectral properties of the BBJ operators.
We will call U φ and U φ,(τ ) the Born-Jordan and τ -intertwiner, respectively The reason for this terminology will become clear in a moment.

The Intertwining Property
Recall that we defined (formula (26)) the unitary operator T (z 0 ) : We will need the following property of the cross-Wigner transform: The interest of the definition of the mapping U φ comes from their intertwining properties: Proposition 11. Let A BJ = Op BJ (a) and A BJ = Op BJ (a). The following intertwining properties hold for all φ ∈ S(R n ).

Proof. Let Ψ ∈ S(R 2n ). In view of formula (36) we have
and hence, for Ψ = U φ ψ, We have In view of formula (53) we have and hence which proves the first formula (54). The second formula follows from the equalities Vol. 84 (2016) Born-Jordan Pseudodifferential Calculus 475

Properties of Intertwiners
We begin by considering the τ -intertwiners.
Proof. (i) Taking φ = φ with ||φ|| = 1 in Moyal's formula (51) we have hence U φ,(τ ) is an isometry. By definition of the adjoint we have (cf. formula (46)). Recalling the classical formula (f * g|h) = (f |g ∨ * h) and noting that F σ Θ ∨ (τ ) = F σ Θ (τ ) , the formula above becomes Taking the definition (38) of Wig(ψ, φ) into account, we get which is formula (55). IEOT We would now like to extend this result to the intertwiners U φ . However, the proof of part (i) of Proposition 12 relies on the Moyal identity (51), since the latter allows to derive (56). However, as we have remarked above, the Moyal identity does not hold for the transform Wig BJ (ψ, φ). We must thus expect a somewhat weaker result. We will need the following lemma, which is a kind of interpolation result: Lemma 13. Let τ and τ be two real numbers and two windows φ and φ . There exists a constant C φ,φ > 0 such that Proof. This amounts to establishing the existence of a constant C φ,φ > 0 such that Using Cauchy-Schwarz's inequality we have Applying Moyal's identity to the terms in the right-hand side we have hence the inequality (58) with C φ,φ = ||φ|| ||φ ||.

The Modulation Spaces M q s (R n )
The theory of modulation spaces goes back to Feichtinger [20,21]; for a detailed exposition see Gröchenig [24]. The traditional definition of these functional spaces makes use of the short-time Fourier transform (or Gabor transform) familiar from time-frequency analysis; we will replace the latter by the cross-Wigner transform whose symplectic symmetries are more visible; that both definitions are equivalent was proven in de Gosson and Luef [18] and de Gosson [15]. We will use the notation z s = (1 + |z| 2 ) s/2 for z ∈ R 2n ; here s is any nonnegative real number. It follows from Peetre's inequality that the function z −→ z s is submultiplicative: Let q be a real number ≥ 1, or ∞. We denote by L q s (R 2n ) the space of all Lebesgue-measurable functions Ψ on R 2n such that · s Ψ ∈ L q s (R 2n ). When q < ∞ the formula Let now φ be a fixed element of S(R n ), hereafter to be called a "window". For q < ∞ the modulation space M q s (R n ) is the vector space consisting of all ψ ∈ S (R n ) such that Wig(ψ, φ) ∈ L q s (R 2n ) where by (38) is the cross-Wigner transform [14,15]; equivalently One shows that in both cases the definitions are independent of the choice of the window φ, and that the || · || φ M q s (1 ≤ q ≤ ∞) form a family of equivalent norms on M q s (R n ), which becomes a Banach space for the topology thus defined; in addition M q s (R n ) contains S(R n ) as dense subspace. The class of modulation spaces M q s (R n ) contain as particular cases many of the classical function spaces. For instance ) which is the Sobolev-like space Q s (R 2n ) studied by Shubin [34], p. 45. We also have A particularly interesting example of modulation space is obtained by choosing q = 1 and s = 0; the corresponding space M 1 0 (R n ) is often denoted by S 0 (R n ), and is called the Feichtinger algebra [21] (it is an algebra both for pointwise product and for convolution). We have the inclusions (63)

Metaplectic and Heisenberg-Weyl Invariance Properties
Recall that the Wigner transform and the Heisenberg-Weyl operators satisfy for all ψ, φ ∈ S (R n ). Let (R 2n , σ) be the standard symplectic space. We denote by Sp(n) be the symplectic group of (R 2n , σ): we have S ∈ Sp(n) if and only if S is a linear automorphism of R 2n such that S * σ = σ. Equivalently, The symplectic group has a unique (connected) covering group of order two; the latter has a true representation as a group Mp(n) of unitary operators on L 2 (R n ); this group is called the metaplectic group. The covering projection Π : Mp(n) −→ Sp(n) is uniquely defined up to inner automorphisms; we calibrate this projection so that we have Π( J) = J where J ∈ Mp(n) is the modified Fourier transform defined by (we refer to [14,15,23] for detailed studies of the metaplectic representation  [18], Gröchenig [24]).
A remarkable property of the Feichtinger algebra is that it is the smallest Banach space invariant under the action of the Heisenberg-Weyl operators (16) and of the metaplectic group.
By definition, M ∞,1 s (R 2n ) consists of all a ∈ S (R 2n ) such that there exists a function Φ ∈ S(R 2n ) for which (R 2n ) contain many of the usual pseudodifferential symbol classes and we have the inclusion where C 2k+1 b (R 2n ) is the vector space of all functions which are differentiable up to order 2n + 1 with bounded derivatives. In fact, for every window Φ there exists a constant C Φ > 0 such that