Symplectic Covariance Properties for Shubin and Born-Jordan Pseudo-Differential Operators

Among all classes of pseudo-differential operators only the Weyl operators enjoy the property of symplectic covariance with respect to conjugation by elements of the metaplectic group. In this paper we show that there is, however, a weaker form of symplectic covariance for Shubin's {\tau}-dependent operators, in which the intertwiners no longer are metaplectic, but still are invertible non-unitary operators. We also study the case of Born--Jordan operators, which are obtained by averaging the {\tau}-operators over the interval [0,1] (such operators have recently been studied by Boggiatto and his collaborators). We show that metaplectic covariance still hold for these operators, with respect top a subgroup of the metaplectic group.


Introduction
In the early years of quantum mechanics physicists were confronted with an ordering problem: assume that some quantization process associated to the real variables x (position) and p (momentum) two operators X and P satisfying the canonical commutation rule X j P j − P j X j = i . What should then be the operator associated to the monomial x m p n ? The first to give a mathematically motivated answer was Weyl [20]; he was was developing his ideas on a group theoretical approach to quantization which lead to the prescription It turns out that the Weyl ordering is a particular case of the more general "τ -ordering": for any real number τ one defines which reduces to Weyl's prescription when τ = 1 2 . We will from now assume that X j f = x j f and P j f = −2πi∂ x j f . The τ -ordering (2) is itself a particular case of the Shubin pseudo-differential calculus (Shubin [17]): given a symbol a the τ -pseudo-differential operator A τ = Op τ (a) is formally defined by A τ f (x) = e 2πip(x−y) a(τ x + (1 − τ )y, p)f (y)dpdy; for τ = 1 2 we recover the Weyl correspondence. Using Schwartz's kernel theorem it is not difficult to show that for every continuous linear operator A : S(R n ) −→ S ′ (R n ) and for every τ ∈ R there exists a ∈ S ′ (R 2n ) such that A = Op τ (a); the τ -operators are thus of a very general nature. Now, it is (reasonably) well-known (Stein [18], Wong [21]) that among all τ -operators only the Weyl operators enjoy a symmetry property known as "symplectic covariance": If Op τ (a • S) = S Op τ (a) S −1 for every S ∈ Mp(2n, R) then τ = 1 2 .
Here Mp(2n, R) is the metaplectic group and S the projection of S ∈ Mp(2n, R) on the symplectic group Sp(2n, R). Symplectic covariance in the sense above is thus a characteristic property of Weyl pseudo-differential calculus. In fact, one shows more generally (Stein [18], §12.7, Wong, Chapter 30) that: Let a −→ Op(a) be a linear mapping from S ′ (R 2n ) to the space of linear operators that is continuous in the topology of Then a −→ Op(a) is the Weyl correspondence so the property of symplectic covariance really singles out Weyl operators among all possible "quantization schemes".
The principal aim of this paper is to report on the fact that there exists a weaker form of symplectic covariance for τ -operators extending which reduces to the case above when τ = 1 2 . In fact, we will show in Proposition 4 that to each S ∈ Sp(2n, R) one can attach an invertible operator These operators are in general not unitary, and do thus not generate a copy of Mp(2n, R). As a consequence of our constructions we will be able to establish a similar property for Born-Jordan pseudo-differential operators. These operators were very recently introduced in de Gosson and Luef [12] where it was remarked that the constructions of Boggiatto and his collaborators [1, 2, 3] of a certain pseudo-differential class was related to a quantization procedure going back to Born and Jordan [4] and historically anterior to the work of Weyl [20]. Born and Jordan's quantization is based on the prescription an elementary calculation shows that this correspondence is obtained by averaging the τ -ordering (2) over the interval [0,1]. This suggests to define more generally the Born-Jordan pseudo-differntial operator with symbol a by the formula We will see that the symplectic covariance formula (3) can be used to derive a similar formula for A BJ .
In a recent paper de Gosson and Luef [12] have shown that this calculus corresponds to a generalization of an early quantization scheme due to Born and Jordan, and which has been largely superseded by the more elegant Weyl quantization procedure. Both Weyl and Born-Jordan quantization hark back to the early years of quantum mechanics.

Notation 1
The Euclidean scalar product of two vectors u and v on R m is denoted indifferently u · v or by uv. When X is a symmetric matrix we will often write Xu 2 for Xu · u. The standard symplectic form σ on the corresponding symplectic group is Sp(2n, R). We denote by S(R n ) the Schwartz space of rapidly decreasing functions on R n and by S ′ (R n ) its dual (the tempered distributions). The normalizations we use correspond to that familiar from the theory of pseudo-differential operators; for instance the Fourier transform of f ∈ S(R n ) is (it corresponds to the choice = 1/2π in the quantum-mechanical literature).
Acknowledgements. This work has been financed by the Austrian Research Agency FWF (Projektnummer P20442-N13).

The pseudo-differential operators A τ
The τ -pseudo-differential operator A τ = Op τ (a) with symbol a ∈ S ′ (R n ) is, by definition, the operator with distributional kernel where F −1 2 is the inverse Fourier transform in the second set of variables. We can thus write formally (Shubin [17]) One easily verifies using this expression that the (formal) adjoint of A τ = Op τ (a) is given by Op τ (a) * = Op 1−τ (a).

The operators T τ (z)
Let T (z 0 ) be the Heisenberg operator: it is defined for f ∈ S ′ (R n ) by where z 0 = (x 0 , p 0 ). Let τ be a real parameter and set, more generally, that is, equivalently, We have T 1/2 (z 0 ) = T (z 0 ), and It is immediate to check the following relations: For many purposes it is useful to write formula (6) in terms of the operators T τ (z): where a σ is the symplectic Fourier transform of a, that is Following the usage in the theory of Weyl operators, we will call a σ the "twisted symbol of A τ ". The distributional kernel of A τ can then be written it is often more suitable for calculations than formula (5).

A composition formula
The τ -operators can be composed exactly in the same way as usual Weyl operators: Proposition 2 Let A τ and B τ be given by Proof. We have and hence, using formula (13) The composition formula (17) follows making the change of variables z = z 0 + z 1 , z ′ = z.

Relation with the τ -Wigner transform
Boggiatto and his collaborators [1,2,3] have recently introduced a τ -dependent Wigner transform W τ (f, g) related with the Shubin τ -pseudo-differential calculus. Averaging over τ in the interval [0, 1] leads to an element of the Cohen class, i.e. to a transform of the type Following result relates the operator A τ to the τ -Wigner transform: Proposition 3 Let f, g ∈ S(R n ). We have the formula Proof. We have hence the equality (18) in view of (6). Formula (18) yields an alternative definition of the operator A τ f for an arbitrary symbol a ∈ S ′ (R n ) and f ∈ S(R n ): choose g ∈ S(R n ); thenW τ (f, g) ∈ S(R 2n ) and the distributional bracket a, W τ (f, g) is thus defined; by definition A τ is the the continuous operator S(R n ) −→ S ′ (R n ) defined by the right hand-side of (18).
We notice that The τ -dependent Wigner transform W τ ψ = W τ (φ, ψ) satisfies the same marginal properties as the ordinary Wigner transform: (see Boggiatto et al. [1]). In the case τ = 1 the transform W τ reduces to the Rihaczek distribution, and when τ = 1 we get the dual Rihaczek distribution.

The symplectic Cayley transform
We will use the following notation: Let S ∈ Sp (0) (2n, R); by definition the symplectic Cayley transform (introduced in de Gosson [8,9,10,11]) of S is the symmetric matrix given by (the symmetry of M (S) is readily verified using the relation S T JS = SJS T = J, which is equivalent to S ∈ Sp(2n, R)). The mapping M (·) is a bijection Sp (0) (2n, R) −→ Sym (0) (2n, R) and the inverse of that bijection is given by

The intertwining operators R τ (S)
We will need the following well-known generalization of the Fresnel formula (see e.g. Folland [6], Appendix A): let X be a real invertible matrix of dimension m; then: where sign X is the difference between the number of > 0 and < 0 eigenvalues of X. Using this formula and the two lemmas above we set out to study the operators defined for S ∈ Sp (0) (2n, R).
and we have (iii) The operator (26) satisfies Proof. (i) It is equivalent to show that the operators By repeated use of formula (13) one gets where the constant K and the phase Φ are given by A straightforward calculation shows that It follows that Applying the Fresnel formula (25) with X = M + M ′ to the formula above and replacing K with its value we get hence c σ (z) = δ(z) by the Fourier inversion formula, which proves our claim. Let us finally show that R τ (S −1 ) = R 1−τ (S) * . We have hence, using formula (7) for the adjoint of a τ -operator, which is the same thing as R τ (S −1 ) = R 1−τ (S) * . Notice that formula (30) shows that the operators R τ (S) are unitary if and only if τ = 1 2 (the Weyl case) see de Gosson [8,9,10,11].

Application to the τ -Wigner function
The usual cross-Wigner function W (f, g) has the following well-known (and very useful) property of symplectic covariance: for all f, g ∈ S(R n ) and S ∈ Sp(2n, R) we have where S ∈ Mp(2n, R) is any of the two metaplectic operators which cover S. In the τ -dependent case this result must be modified as follows: Proposition 5 Let S ∈ Sp (0) (2n, R) and f, g ∈ S(R n ). We have Proof. Let A τ = Op τ (a). Recall that (Op τ (a)f |g) L 2 = a|W τ (f, g) (formula (18). In view of the second equality (30) we have, using On the other hand, using the intertwining property (28), we have (the last identity using the change of variables z −→ S −1 z and the fact that det S = 1). Formula (34) follows.

The operators R τ (S) as pseudo-differential operators
Following result identifies R τ (S) as a τ -pseudo-differential operator: where s σ (z) is given by the formula For practical calculations formula (15) is useful; it immediately yields: The distributional kernel of the operator R τ (S) satisfies e 2πi(τ x+y)p e iπM (S)z 2 dp; this formula can be used in principle for the calculation of explicit expressions for the operators R τ (S). Let us give an example. Choosing S = J we have M (S) = 1 2 I; a straightforward computation using (37) yields Notice that when τ = 1 2 we get hence R 1/2 (J) is, up to a factor, the usual Fourier transform. In fact we have R 1/2 (J) ∈ Mp(2n, R); it is the metaplectic operator J with projection J on the symplectic group (see e.g. de Gosson [8], Folland [6]). This is not pure coincidence, in fact: Corollary 7 For S ∈ Sp (0) (2n, R) the operators R(S) = R 1/2 (S) are, up to a unimodular factor i ν(S) elements of the metaplectic group Mp(2n, R). In fact, when ν(S) is, modulo 2, the Conley-Zehnder of a any path joining the identity to S then i ν(S) R(S) ∈ Mp(2n, R).
Proof. In [8,9,10,11] we have shown that when ν(S) is the Conley-Zehnder [5] index (which we discuss below). The result follows since T 1/2 (z) = T (z).  [13] on "matrix mechanics" and proposed the quantization rule which coincides with (1) when m+ℓ ≤ 2. We now make the following fundamental remark: the Born-Jordan prescription (41) is obtained by averaging the τ -ordering (2) on the interval [0, 1]; this is immediately seen using the property of the beta function. This suggests to study, more generally, the pseudodifferential operators

Definition of Born-Jordan operators
In [1] Boggiatto et al. define a transform Q : S(R n ) × S(R n ) −→ S(R n ) by integrating over [0, 1] the τ -cross Wigner transforms (19); we will use the notation Q = W BJ ; thus, for f, g ∈ S(R n ): For a ∈ S ′ (R n ) these authors define an operator, which we denote A BJ , by the formula f, g ∈ S(R n ) (cf. (18)). We will call A BJ the Born-Jordan operator with symbol a and write A BJ = Op BJ (a). Using the representation (14) of the τ -operators we have where T BJ (z) is the unitary operator defined by We notice that it immediately from the relations (12)-(13) that: Proposition 8 The Born-Jordan operator A BJ = Op BJ (a) is given by where Θ is the real function defined by The operator A BJ is a continuous operator S(R n ) −→ S ′ (R n ) for every a ∈ S ′ (R 2n ). That this formula really defines a continuous operator S(R n ) −→ S ′ (R n ) follows from the fact that Θ ∈ L ∞ (R 2n ).
Proof. Integrating both sides of formula (9) in the interval [0, 1] we have T BJ (z) = Θ(z) T (z) hence the expression (48). In view of the relation (7) between a τ -operator and its adjoint we have hence the Born-Jordan operators share with Weyl operators the property that they are (formally) self-adjoint if and only if their symbol is real. This makes Born-Jordan operators good candidates for quantization. The reader is urged to notice that while every Born-Jordan operator is a Weyl operator, the converse property is not true because an arbitrary distribution b σ ∈ S ′ (R 2n ) cannot in general be written in the form a σ Θ (see de Gosson and Luef [12]; also the discussion in Kauffmann [14]).

Reduced metaplectic covariance
The intertwining properties for τ operators do not carry over to the Born-Jordan case; it is meaningless to expect a relation like R BJ (S) T BJ (z) = T BJ (Sz)R BJ (S) which would lead to a symplectic covariance property of the type (28). The good news is, however, that Born-Jordan operators enjoy a symplectic covariance property for operators belonging to a subgroup of the standard metaplectic group Mp(2n, R). Recall that Mp(2n, R) is generated by the modified Fourier transform J = e i nπ 4 F , the multiplication operators V −P f = e iπP The proof above shows that the essential step consists in noting that Θ(S −1 z) = Θ(z) when S = J or S = M L . It is clear that this property fails if one takes S = V P with P = 0, so we cannot expect to have full symplectic covariance for Born-Jordan operators. Such a property is anyway excluded in view of our discussion in the Introduction to this paper. symplectic covariance is characteristic of Weyl calculus.