Born-Jordan Quantization and the Uncertainty Principle

The Weyl correspondence and the related Wigner formalism lie at the core of traditional quantum mechanics. We discuss here an alternative quantization scheme, whose idea goes back to Born and Jordan, and which has recently been revived in another context, namely time-frequency analysis. We show that in particular the uncertainty principle does not enjoy full symplectic covariance properties in the Born and Jordan scheme, as opposed to what happens in the Weyl quantization.


Introduction
The problem of "quantization" of an "observable" harks back to the early days of quantum theory; mathematically speaking, and to use a modern language, it is the problem of assigning to a symbol a pseudo-differential operator in a way which is consistent with certain requirements (symmetries under a group of transformations, positivity, etc.). Two of the most popular quantization schemes are the Kohn-Nirenberg and Weyl correspondences. The first is widely used in the theory of partial differential equations and in time-frequency analysis (mainly for numerical reasons), the second is the traditional quantization used in quantum mechanics. Both are actually particular cases of Shubin's pseudo-differential calculus, where one can associate to a given symbol a an infinite family (A τ ) τ of pseudo-differential operators parametrized by a real number τ , the cases τ = 1 and τ = 1 2 corresponding to, respectively, Kohn-Nirenberg and Weyl operators. It turns out that each of Shubin's τ -operators can be alternatively defined in terms of a generalization Wig τ of the usual Wigner distribution by the formula A τ ψ, φ = a, Wig τ (ψ, φ) and this observation has recently been used by researchers in time-frequency analysis to obtain more realistic phase-space distributions (more about this in the discussion at the end of the paper). They actually went one step further by introducing a new distribution by averaging Wig τ for the values of τ in the interval [0,1]. This leads, via the analogue of the formula above to a third class of pseudo-differential operators, corresponding to the averaging of Shubin's operators A τ . It was noted by the present author that this averaged pseudo-differential calculus is actually an extension of one of the first quantization schemes discovered by Born, Jordan, and Heisenberg around 1927, prior to that of Weyl's.
The aim of the present paper is to give a detailed comparative study of the Weyl and Born-Jordan correspondences (or "quantization schemes") with an emphasis on the symplectic covariance properties of the associated uncertainty principles.

Notation 1
We write x = (x 1 , ..., x n ) and p = (p 1 , ..., p n ) and z = (x, p). In matrix calculations x, p, z are viewed as column vectors. The phase space R 2n ≡ R n × R n is equipped with the standard symplectic form σ(z, z ′ ) = px ′ − p ′ x; equivalently σ(z, z ′ ) = Jz · z ′ where J = 0 n×n I n×n −I n×n 0 n×n is the standard symplectic matrix. We denote by S(R 2n ) the Schwartz space of rapidly decreasing smooth functions and by S ′ (R 2n ) its dual (the tempered distributions).

Discussion of Quantization
After Werner Heisenberg's seminal 1925 paper [23] which gave rigorous bases to the newly born "quantum mechanics", Born and Jordan [4] wrote the first comprehensive exposition on matrix mechanics, followed by an article with Heisenberg himself [5]. These articles were an attempt to solve an ordering problem: assume that some quantization process associated to the canonical variables x (position) and p (momentum) two operators X and P satisfying the canonical commutation rule X P − P X = i . What should then the operator associated to the monomial x m p n be? Born and Jordan's answer was which immediately leads to the "symmetrized" operator 1 2 ( X P + P X) when the product is xp. In fact Weyl and Born-Jordan quantization lead to the same operators for all powers x m or p n , or for the product xp (for a detailed analysis of Born and Jordan's derivation see Fedak and Prentis [9], also Castellani [6] and Crehan [7]). Approximately at the same time Hermann Weyl had started to develop his ideas of how to quantize the observables of a physical system, and communicated them to Max Born and Pascual Jordan (see Scholz [28]). His basic ideas of a group theoretical approach were published two years later [33,34]. One very interesting novelty in Weyl's approach was that he proposed to associate to an observable of a physical system what we would call today a Fourier integral operator. In fact, writing the observable as an inverse Fourier transform he defined its operator analogue by which is essentially the modern definition that will be given below (formula (9). We will denote the Weyl correspondence by a Weyl ←→ A W or A W = Op(a). Weyl was led to this choice because of the immediate ordering problems that occurred when one considered other observables than monomials a(x, p) = x k or a(x, p) = p ℓ . For instance, using Schrödinger's rule what should the operator associated with a(x, p) = xp be? Weyl's rule immediately yields the symmetrized quantization rule a( X, P ) = 1 2 ( X P + P X) and one finds that more generally (McCoy [27], 1932) It turns out that the Weyl quantization rule (4) for monomials is a particular case of the so-called "τ -ordering": for any real number τ one defines this rule reduces to Weyl's prescription when τ = 1 2 . When τ = 1 one gets the "normal ordering" X n P n familiar from the elementary theory of partial differential equations, and τ = 0 yields the "anti-normal ordering" P n X n sometimes used in physics. We now make the following fundamental observation: the Born-Jordan prescription (1) is obtained by averaging the τ -ordering on the interval [0, 1] (de Gosson [18], de Gosson and Luef [21]). In fact and hence One interesting feature of the quantization rules above is the following: suppose that the operators X and P are such that then [ X m , P n ] is independent of the choice of quantization; in fact (see Crehan [7] and the references therein): In physics as well as in mathematics, the question of a "good" choice of quantization is more than just academic. For instance, different choices may lead to different spectral properties. The following example is due to Crehan [7]. Consider the Hamiltonian function The term that gives an ordering problem is evidently (p 2 +x 2 ) 3 ; Crehan then shows that the most general quantization invariant under the symplectic The eigenfunctions of H are those of the harmonic oscillator, and the corresponding eigenvalues are the numbers ..) which clearly shows the dependence of the spectrum on the parameters α and λ, and hence of the chosen quantization.

First definition
Let z 0 = (x 0 , p 0 ) and consider the "displacement" Hamiltonian function H z 0 = σ(z, z 0 ). The flow determined by the corresponding Hamilton equations is given by f t (z) = z + tz 0 ; for Ψ ∈ S ′ (R 2n ) we define T (z 0 )Ψ(z) = (f 1 ) * Ψ(z) = Ψ(z − z 0 ). The τ -quantization of H z 0 is the operator H z 0 = σ( Z, z 0 ), Z = ( X, P ); the solution of the corresponding Schrödinger equation at time t = 1 with initial condition ψ is given by the Heisenberg operator T (z 0 ) = e i σ( Z,z 0 ) ; its action on ψ ∈ S ′ (R n ) is explicitly given by Let a ∈ S ′ (R 2n ) be an observable (or "symbol"). By definition, the Weyl correspondence a Weyl ←→ A W is defined by where a σ = F σ a is the symplectic Fourier transform of a, that is for a ∈ S(R 2n ); informally The symplectic Fourier transform F σ is an involution (F 2 σ = I d ); it is related to the usual Fourier transform F on R 2n by the formula F σ a(z) = Fa(Jz), J the standard symplectic matrix. The action of the operator A W on a function ψ ∈ S ′ (R n ) is given by How can we modify this formula to define Born-Jordan quantization? An apparently easy answer would be to first define τ -quantization by replacing H z 0 by its τ -quantized version H z 0 ,τ , and then to average the associated operators T τ (z 0 ) thus obtained to get a " T BJ (z 0 ) operator" which would allow to define A BJ . However, such a procedure trivially fails, because all τ -quantizations of the displacement Hamiltonian H z 0 coincide with H z 0 as can be verified using the polynomial rule (5). There is however a simple way out of this difficulty; it consists in replacing, as we did in [21], and we define the Born-Jordan operator A BJ by This formula will be justified below.

Pseudo-differential formulation
There is another way to describe Born-Jordan quantization. Writing formula (12) in pseudo-differential form yields the usual formal expression for the Weyl correspondence (we assume for simplicity that a ∈ S(R 2n ) and ψ ∈ S(R n )). We now define the τ -dependent operatorà la Shubin [30]: the Born-Jordan operator A BJ with symbol a is then defined by the average which we can write, interchanging the order of the integrations, We have been a little bit sloppy in writing the (usually divergent) integrals above, but all three definitions become rigorous if we view the operators A W , A τ , and A BJ as being defined by the distributional kernels (F −1 2 is the inverse partial Fourier transform with respect to the second set of variables) and is the inverse Fourier transform in the second set of variables. We will give below an alternative rigorous definition, but let us first check that definition (18)-(22) coincides with the one given in previous subsection. Define the modified Heisenberg-Weyl operators These obey the same commutation rules as the usual Heisenberg operators T (z 0 ).
Proposition 2 Let a ∈ S ′ (R 2n ), ψ ∈ S(R n ). The Born-Jordan operator (17) is given by formula (14), that is In particular, A BJ is the Weyl operator with symbol Proof. (Cf. [21,18]). One verifies by a straightforward computation that the Shubin formula (16) can be rewritten as Let us now average in τ over the interval [0, 1]; interchanging the order of integrations and using the trivial identity hence formula (26). To prove the last statement we note that formula (26) can be rewritten where (a BJ ) σ = aΘ. Taking the inverse Fourier transform we get, noting because the function Θ is even. One easily verifies that the (formal) adjoint of A τ = Op τ (a) is given by and hence Born-Jordan operators thus share with Weyl operators the property of being (essentially) self-adjoint if and only if their symbol is real. This property makes Born-Jordan prescription a good candidate for physical quantization, while Shubin quantization should be rejected being unphysical for τ = 1 2 .

The τ -Wigner distribution
In a recent series of papers Boggiatto and his collaborators [1,2,3] have introduced a τ -dependent Wigner distribution Wig τ (f, g) which they average over the values of τ in the interval [0, 1]. This procedure leads to an element of the Cohen class [22], i.e. to a transform of the type . From the point of view of time-frequency analysis this can be interpreted as the application of a filter to the Wigner transform.
Let us define the τ -Wigner cross-distribution Wig τ (ψ, φ) of a pair (ψ, φ) of functions in S(R n ): Choosing τ = 1 2 one recovers the usual cross-Wigner transform and when τ = 0 we get the Rihaczek-Kirkwood distribution well-known from time-frequency analysis [22]. The mapping Wig τ is a bilinear and continuous mapping S(R n )×S(R n ) −→ S(R 2n ). When ψ = φ one writes Wig τ (ψ, ψ) = Wig τ ψ; it is the τ -Wigner distribution considered by Boggiatto et al. [1,2,3]). It follows from the definition of Wig τ that we have in particular hence Wig τ ψ is not a real function in general if τ = 1 2 . and Proof. Formula (37) is straightforward. On the other hand hence formula (38). Notice that the right-hand sides of (37) and (38) are independent of the parameter τ .
In particular [1], the τ -Wigner distribution Wig τ ψ = Wig τ (ψ, ψ) satisfies the usual marginal properties: There is a fundamental relation between Weyl pseudo-differential operators and the cross-Wigner transform, that relation is often used to define the Weyl operator A W = Op W (a): for ψ, φ ∈ S(R n ). Not very surprisingly this formula extends to the case of τ -operators: Proposition 4 Let ψ, φ ∈ S(R n ), a ∈ S(R 2n ), and τ a real number. We have where ·, · is the distributional bracket on R 2n and A τ = Op τ (a).
Proof. By definition of Wig τ we have the equality (41) follows in view of definition (16) Formula (41) allows us to define A τ ψ = Op τ (a)ψ for arbitrary symbols a ∈ S ′ (R 2n ) and ψ ∈ S(R n ) in the same way as is done for Weyl pseudodifferential operators: choose φ ∈ S(R n ); then Wig τ (ψ, φ) ∈ S(R 2n ) and the distributional bracket a, Wig τ (ψ, φ) is thus well-defined. This defines A τ as a continuous operator S(R n ) −→ S ′ (R n ).

Averaging over τ
We define the (cross) Born-Jordan-Wigner (BJW) distribution of ψ, φ ∈ S(R n ) by the formula We set Wig BJ ψ = Wig BJ (ψ, ψ). The properties of the BJW distribution are readily deduced from those of the τ -Wigner distribution studied above. In particular, the marginal properties (37) and (38) are obviously preserved: and An object closely related to the (cross-)Wigner distribution is the (cross-)ambiguity function of a pair of functions ψ, φ ∈ S(R n ): It turns out that A(ψ, φ) and Wig τ (ψ, φ) are obtained from each other by a symplectic Fourier transform: (F σ is an involution), thus justifying the following definition: An important property is that the BJW distribution of a function ψ is real, as is the usual Wigner distribution. In fact using the conjugacy formula (36) we have Formula (41) relating Shubin's τ -operators to the τ -(cross) Wigner distribution carries over to the Born-Jordan case: for all ψ, φ ∈ S(R n ).
The following consequence of Proposition 4 will be essential in our study of the uncertainty principle: Proposition 6 Let Θ be defined by (27). We have and Amb BJ (ψ, φ) = (2π ) n Amb(ψ, φ)Θ Proof. In view of formula (28) in Proposition 2 and formula (42) above we have (the last equality because Θ is an even function). Since F σ Θ = FΘ because Θ is even and invariant under permutation of the x and p variables, this proves formula (46), and formula (47) follows, taking the symplectic Fourier transform of both sides.

Symplectic (non-)covariance
he symplectic group Sp(2n, R) is by definition the group of all linear automorphisms s of R 2n which preserve the symplectic form σ(z, z ′ ) = Jz · z ′ ; equivalently s T Js = J. The group Sp(2n, R) is a connected Lie group, and its double covering Sp 2 (2n, R) has a faithful (but reducible) representation by a group of unitary operator, the metaplectic group Mp(2n, R) (see Folland [10], de Gosson [15,16]). That group is generated by the operators J, M L,m , and V −P defined by J = e −inπ/4 F and where L ∈ Gℓ(n, R) and P ∈ Sym(n, R); the integer m corresponds to the choice of an argument of det L. Denoting by π Mp the covering projection Mp(2n, R) −→ Sp(2n, R) we have π Mp ( J) = J and Let now A W = Op W (a) be an arbitrary Weyl operator, and s ∈ Sp(2n, R). We have where S ∈ Mp(2n, R) is anyone of the two metaplectic operators such that π Mp ( S) = s. This property is really characteristic of the Weyl correspondence; it is proven [15,16] using the identity where T (z) is the Heisenberg operator. One can show that if a pseudodifferential correspondence a ←→ Op(a) (admissible in the sense above, or not) is such that Op(a • s −1 ) = S Op(a) S −1 then it must be the Weyl correspondence. For Born-Jordan correspondence we do still have a residual symplectic covariance, namely:

The Uncertainty Principle
We begin by reviewing the notion of density matrix (or operator) familiar from statistical quantum mechanics. The notion goes back to John von Neumann [32] in 1927, and is intimately related to the notion of mixed state (whose study mathematically belongs to the theory of C * -algebras via the GNS construction). This will provide us with all the necessary tools for comparing the uncertainty relations in the Weyl and Born-Jordan case.

Density matrices
A density matrix on a Hilbert space H is a self-adjoint positive operator on H with trace one. In particular, it is a compact operator. Physically density matrices represent statistical mixtures of pure states, as explicitly detailed below. We will need the two following results: where Π j is the orthogonal projection of H on H j and m j = dim H j It is a consequence of the spectral decomposition theorem for compact operators (for a detailed proof see e.g. [16], §13.1).
Proof. We have P ψ φ = (φ|ψ) L 2 ψ hence the kernel of P ψ is K ψ = ψ ⊗ ψ. Using a Fourier transform formula (21) implies that the τ -symbol ρ τ of P ψ is given by Setting τ = 1 2 we get formula (51). Formula (52) is obtained by integrating ρ τ (z) with respect to τ ∈ [0, 1]. That ρ W and ρ BJ are real follows from the fact that both the Wigner and the WBJ distribution are real.
The following result describes density matrices in both the Weyl and Born-Jordan case in terms of the Wigner formalism: Proposition 10 Let ρ be a density matrix on L 2 (R n ). There exists an orthonormal system (ψ j ) j≥1 of L 2 (R n ) and a sequence of non-negative numbers (λ j ) j≥1 such that j≥1 λ j = 1 and the symbols ρ W and ρ BJ being given by and Proof. Taking H = L 2 (R n ) in Lemma 8 we can write ρ = j α j Π j where each Π j is the projection operator on a finite dimensional space H j ⊂ L 2 (R n ), and two spaces H j and H ℓ are orthonormal if j = ℓ. For each index j let us choose an orthonormal basis B j = (ψ j+1 , ..., ψ j+m j ) of H j ; the union B = ∪ j B j is then an orthonormal basis of ⊕ j H j , and we have, using Lemma 8 where Π ψ k is the orthogonal projection on the ray {λψ j : λ ∈ C}. Since each index α j is repeated m j times due to the expression between brackets, this can be rewritten with the λ j = m j α j summing up to one. In view of Lemma 9 the Weyl (resp. Born-Jordan) symbol of Π ψ j is (2π ) n Wig W ψ (resp. (2π ) n Wig BJ ψ) hence the result.
Notice that the orthonormal bases B j in the proof can be chosen arbitrarily; the decompositions (54) and (55) are therefore not unique. (In Physics, one would say that a mixed quantum state can be written in infinitely many way as a superposition of pure states, a pure state being a density operator with symbol a Wigner function).

A general uncertainty principle
In what follows the notation is indifferently the Weyl or the Born-Jordan correspondence. Both have the property: If the symbol a is real, then the operator A is essentially selfadjoint (in which case we call it an observable).
Notice that the Shubin correspondence does not have this property for τ = 1 2 since A * τ = A 1−τ . Let be a density matrix on L 2 (R n ). We assume that ρ = (2π ) n Op(ρ). The function ρ is a real function on phase space R 2n and we have Observe that we do not in general have ρ ≥ 0. Let A be an observable. Its expectation value with respect to ρ is by definition the real number where it is assumed that the integral on the right side is absolutely convergent. We will write, with some abuse of notation, (see the discussion in de Gosson [16], §12.3, of the validity of various "trace formulas"). In view of formulas (40) and (45) we have either λ j a, Wig BJ ψ j (when a ←→ A is the Born-Jordan correspondence) and hence where A j = ( Aψ j |ψ j ) L 2 (recall that (ψ j ) j≥1 is an orthonormal system). If A 2 also is an observable and if A 2 = Tr(ρ A 2 ) exists, then the number is the variance of A; its positive square root Var ρ A is called "standard deviation". More generally consider a second observable B; then the covariance of the pair ( A, B) with respect to ρ is defined by It is in general a complex number, and we have The covariance has the properties of a complex scalar product; it therefore satisfies the Cauchy-Schwarz inequality The following lemma will be useful in the proof of the uncertainty inequalities below: is a real number. In particular the Heisenberg inequality holds.
Proof. Replacing A and B with A − A and B − B it is sufficient to prove (67) when A = B = 0. We thus have to prove the inequality Noting that definition (68) can be rewritten we remark that in view of formulas (65) and (66) in the lemma above we have hence the proof of (70) is reduced to the proof of the inequality which is just the Cauchy-Schwarz inequality (64) for covariances.

Weyl vs Born-Jordan
Let us discuss the similarities and differences between the uncertainty principles associated with the Weyl and Born-Jordan correspondences. First, as already observed in the Introduction, the Weyl and Born-Jordan quantizations of monomials x m j p n j (and hence of their linear combinations) are identical when m + n ≤ 2. This implies, in particular, that if the symbols a and b are, respectively, multiplication by the coordinates x j and p j then the corresponding operators A and B are, in both cases given by X j = x j and P j = −i ∂ x j . It follows that Var ρ X j and Var ρ P j satisfy the usual Robertson-Schrödinger inequalities We mention that the inequalities (74) can be rewritten in compact form as where ≥ 0 means "semi-definite positive", J is the standard symplectic matrix, and is the statistical covariance matrix. The formulation (75) of the Robertson-Schrödinger inequalities clearly shows one of the main features, namely the symplectic covariance of these inequalities, which we have used in previous work [17,20] to express the uncertainty principle in terms of the notion of symplectic capacity, which is closely related to Gromov's non-squeezing theorem from symplectic topology. This has also given us the opportunity to discuss the relations between classical and quantum mechanics in [19].
Let S ∈ Mp(2n, R), S = π Mp ( S) and set A ′ = S A S −1 , B ′ = S B S −1 ; we are assuming that A, B correspond, as in the proof of Proposition 12, to an arbitrary quantization scheme a ←→ A. We have quite generally, using the cyclicity of the trace, with ρ ′ = S ρ S −1 . Suppose now that the operator correspondence a ←→ A is the Weyl correspondence; then, by Proposition 7 we have Sρ S −1 = Op(ρ• s −1 ) and the inequalities (67) become

Discussion
There is an old ongoing debate in quantum mechanics on which quantization scheme is the most adequate for physical applications; an interesting recent contribution is that of Kauffmann [25], who seems to favor the Born-Jordan correspondence. The introduction of the τ -Wigner and Born-Jordan distributions has been motivated in time-frequency analysis by the fact that the usual cross-Wigner distribution gives raise to disturbing ghost frequencies; it was discovered by Boggiatto and his collaborators [1,2,3] that these ghost frequencies were attenuated by averaging over τ .
The study of uncertainties for non-standard situations has been tackled (from a very different point of view) by Korn [26]; also see the review paper [11] by Folland and Sitaram, which however unfortunately deliberately ignores the fundamental issue of covariance. Gibilisco and his collaborators [12,13,14] give highly nontrivial refinements of uncertainty relations using convexity properties, and studied the notion of statistical covariance in depth.
We mention that in a very well written thesis, published as a book, Steiger [31] has given an interesting historical review and analysis of the evolution of the uncertainty principle; in addition he compares the interest of several different formulations, and gives a clever elementary derivation of the Robertson-Schrödinger inequalities for operators. The work also contains Matematica codes for the computation of (co-)variances.