Metaplectic formulation of the Wigner transform and applications
aa r X i v : . [ m a t h - ph ] J a n METAPLECTIC FORMULATION OF THE WIGNERTRANSFORM AND APPLICATIONS
NUNO COSTA DIAS, MAURICE A. DE GOSSON, AND JO ˜AO NUNO PRATA
Abstract.
We show that the cross Wigner function can be written inthe form W ( ψ, φ ) = ˆ S ( ψ ⊗ ˆ φ ) where ˆ φ is the Fourier transform of φ and ˆ S is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in thetime-frequency space). This formulation can be extended to genericWeyl symbols and yields an interesting fractional generalization of theWeyl-Wigner formalism. It also provides a suitable approach to studythe Bopp phase space representation of quantum mechanics, familiarfrom deformation quantization. Using the ”metaplectic formulation”of the Wigner transform we construct a complete set of intertwinersrelating the Weyl and the Bopp pseudo-differential operators. This isan important result that allows us to prove the spectral and dynamicalequivalence of the Schr¨odinger and the Bopp representations of quantummechanics. MSC [2000]: Primary 35S99, 35P05; Secondary 35S05, 45A751.
Introduction
It is well known that the cross Wigner function(1) W ( ψ, φ )( x, p ) = 1(2 π ) n Z e ip · ξ p ψ ( x − ξ p ) φ ( x + ξ p ) dξ p defined for arbitrary φ, ψ ∈ L ( R n ), satisfies the Moyal identity || W ( ψ, φ ) || L ( R n ) = 1(2 π ) n/ || φ || L ( R n ) || ψ || L ( R n ) . Hence, for || φ || L ( R n ) = 1, the mapping(2) W φ : L ( R n ) −→ L ( R n ); ψ W φ ψ = (2 π ) n/ W ( ψ, φ )called the windowed Wigner transform , is a non-surjective isometry L ( R n ) → L ( R n ).In this paper we will further show that W φ can be written in the form(3) W φ ψ = ˆ S ( ψ ⊗ ˆ φ )where ˆ S = e − iθ ˆ H is the unitary operator generated by the self-adjointHamiltonian(4) ˆ H = 2 ˆ ξ x · ˆ ξ p − (cid:16) ˆ x · ˆ ξ x (cid:17) + − (cid:16) ˆ p · ˆ ξ p (cid:17) + + 4ˆ x · ˆ p , and θ is a suitable value of the scalar parameter. In (4) the subscript +denotes symmetrization and(5) ˆ x = x · , ˆ ξ x = − i∂ x and ˆ p = p · , ˆ ξ p = − i∂ p are the fundamental operators in the phase space Schr¨odinger representa-tion of the Heisenberg algebra (thus acting on functions Ψ( x, p ) ∈ D ( ˆ H ) ⊂ L ( R n × R n )) [9].Since ˆ H is quadratic, the transformation ˆ S is metaplectic [12, 14, 16].Its projection onto the group of linear symplectic transformations Sp(4 n ) of R n × R n is a symplectomorphism generated by the classical Hamiltonian(6) H = 2 ξ x · ξ p − x · ξ x − p · ξ p + 4 x · p and consists of a rotation along an ellipse in the phase space.The ”metaplectic formulation” of the Wigner function can be extended togeneric Weyl operators. The resulting formalism yields new calculation toolsfor the Weyl-Wigner calculus and displays several interesting applications.We will use it to study a new topic and to revisit an old subject.The new topic is a fractional generalization of the Weyl-Wigner formalism.Using the metaplectic approach we construct a fractional version of both theWigner function and the Weyl symbols. Moreover, we generalize several ofthe key properties of the standard Weyl-Wigner framework to the fractionalcase. These include the regularity properties, inversion and kernel formulasand the composition formula [16, 25]. We also study the main features ofthe associated fractional quantization.The old topic is the Bopp representation of quantum mechanics [4, 5, 9,16]. This is a phase space operator representation that was recently used[9, 17] to prove precise spectral results for the deformation quantization ofBayen et al [1, 2, 10, 11]. A similar formulation was used in [15, 18] tostudy the spectral properties of generalized Landau operators [19], and in[7, 8] to address the spectral problem in noncommutative quantum mechan-ics. In another recent paper [3], a generalization of the Bopp representationwas used to address the problem of determining a consistent formulationof coupled classical quantum dynamics [6, 23]. The results of [3] suggestthat the phase space representations are more suitable to address this prob-lem than the standard configurational space ones. Here, we will focus onthe mathematical formalism and use the metaplectic representation of theWigner transform to determine a complete family of non-surjective isome-tries that intertwines the Weyl and the Bopp pseudo-differential operators.A straightforward consequence of this result is that the Schr¨odinger and theBopp representations of quantum mechanics (although not unitarily related)display equivalent spectral and dynamical properties.This paper is organized as follows: in the next section we present the meta-plectic formulation of the Wigner function (section 2.2) and Weyl symbols(section 2.3). In section 3 we define and study the fractional generalizationof the Weyl calculus. In section 4 we use the metaplectic formulation to ETAPLECTIC FORMULATION... 3 study the spectral and dynamical properties of the Bopp representation ofquantum mechanics.
Notation 1.
A generic point in the double phase space R n = R n ⊕ R n is denoted by z = ( x, p, ξ x , ξ p ) where ( x, ξ x ) ∈ R n and ( p, ξ p ) ∈ R n arecanonical conjugate pairs. The standard sympletic form on R n is σ ( z, z ′ ) = ξ x · x ′ + ξ p · p ′ − ξ ′ x · x − ξ ′ p · p and the corresponding sympletic group is Sp (4 n ) .The metaplectic group Mp (4 n ) is the unitary representation of the doublecover of Sp (4 n ) . Notation 2.
The norm and the inner product in L ( R n ) are denoted by || · || L ( R n ) and ( · , · ) L ( R n ) , or just by || · || and ( · , · ) whenever the dimensionof the functions’ domain is clear from the context. The distributional bracketis h· , ·i .For f : R n −→ C and g : R n −→ C , f ⊗ g denotes the tensor productfunction f ⊗ g : R n × R n −→ C ; ( x, y ) f ⊗ g ( x, y ) = f ( x ) g ( y ) Operators are denoted by roman letters with a hat (the exceptions are ˆ ξ x and ˆ ξ p ). The hat also denotes the Fourier transform (Notation 3) but itshould always be clear from the context what it refers to. Notation 3.
The partial Fourier and inverse Fourier transforms are writ-ten: ˆΨ( x, ξ p ) = F p → ξ p [Ψ( x, p )] = 1(2 π ) n/ Z R n e − iξ p · p Ψ( x, p ) dp and ˇΨ( x, p ) = F − ξ p → p [Ψ( x, ξ p )] = 1(2 π ) n/ Z R n e iξ p · p Ψ( x, ξ p ) dξ p and are defined as unitary operators in L ( R n ) (for the Fourier transformin L ( R n ) we use exactly the same notation). As usual they can be extendedby duality to S ′ ( R n ) . Notice that when we write F p → ξ p [Ψ( x, p )] , ( x, p ) isthe argument of the original function and the result is a function of ( x, ξ p ) .On the other hand, if we write F p → ξ p Ψ( x, ξ p ) then ( x, ξ p ) is already theargument of F p → ξ p Ψ . Metaplectic formulation of the Wigner transform
In this section we prove our main result. It states that W ( ψ, φ ) =(2 π ) − n/ ˆ S ( ψ ⊗ ˆ φ ) where ˆ S = e − iθ ˆ H is the unitary operator generated by theHamiltonian (4) for θ = √ arccos . We will also extend this formulationto generic Weyl symbols. NUNO COSTA DIAS, MAURICE A. DE GOSSON, AND JO ˜AO NUNO PRATA
The classical symplectomorphism.
The unitary transformation ˆ S is generated by the quadratic Hamiltonian (4) and thus belongs to the meta-plectic group Mp( R n ) [12, 13, 16, 21]. Its projection onto the group ofsymplectic transformations of ( R n , σ ) is a symplectomorphism S belongingto the one-parameter group of symplectomorphisms s ( θ ) generated by theclassical Hamiltonian (6). In order to prepare our main results, we nowdetermine the explicit form of s ( θ ) and S . Theorem 4.
Let s : R × R n −→ R n be the one-parameter group of sym-plectic transformations generated by the classical Hamiltonian H = 2 ξ x · ξ p − x · ξ x − p · ξ p + 4 x · p. For θ = √ arccos the symplectomorphism S = s ( θ ) is explicitly (7) S : R n −→ R n , x x/ ξ p / ξ x ξ x − pp p/ ξ x / ξ p ξ p − x . Moreover, S is the natural projection of ˆ S = e − iθ ˆ H ∈ Mp (4 n ) onto Sp (4 n ) .Proof. The map s ( θ ) = ( x ( θ ) , p ( θ ) , ξ x ( θ ) , ξ p ( θ )) is defined by the Hamiltonequations ˙ x = ∂H∂ξ x = 2 ξ p − x ˙ p = ∂H∂ξ p = 2 ξ x − p ˙ ξ x = − ∂H∂x = ξ x − p ˙ ξ p = − ∂H∂p = ξ p − x which decouple into two systems ( ˙ x = ∂H∂ξ x = 2 ξ p − x ˙ ξ p = − ∂H∂p = ξ p − x , ( ˙ p = ∂H∂ξ p = 2 ξ x − p ˙ ξ x = − ∂H∂x = ξ x − p with solutions ( k = √ x ( θ ) = x (0) (cid:2) cos kθ − k − sin kθ (cid:3) + 2 k − ξ p (0) sin kθξ p ( θ ) = ξ p (0) (cid:2) cos kθ + k − sin kθ (cid:3) − k − x (0) sin kθ and(9) p ( θ ) = p (0) (cid:2) cos kθ − k − sin kθ (cid:3) + 2 k − ξ x (0) sin kθξ x ( θ ) = ξ x (0) (cid:2) cos kθ + k − sin kθ (cid:3) − k − p (0) sin kθ . Hence, s ( θ ) is given explicitly by the equations (8,9). For θ = k − arccos = k − arcsin k these equations yield the transformation S exactly. ETAPLECTIC FORMULATION... 5
Since H is the Weyl symbol of ˆ H (given by (4)), the symplectomorphisms s ( θ ) are the projections onto Sp(4 n ) of the metaplectic operators ˆ U ( θ ) = e − iθ ˆ H . So, in particular, S = s ( θ ) is the projection of ˆ S = ˆ U ( θ ). (cid:3) Remark 5.
The family of transformations s ( θ ) is periodic. We have s ( θ + πk ) = s ( θ ) for all θ ∈ R . The orbits generated by s belong to the followinglevel surfaces (10) (cid:26) x + ξ p − xξ p = Const p + ξ x − pξ x = Const ′ and consist of a rotation along two ellipses in the x, ξ p and p, ξ x planes ofthe phase space. The main result.
Let ˆ T ( θ ) be the one-parameter (periodic) group ofunitary transformations defined by(11)ˆ T ( θ ) : L ( R n ) −→ L ( R n ); Φ( x, ξ p ) ˆ T ( θ )Φ( x, ξ p ) = Φ( x ( − θ ) , ξ p ( − θ ))where x ( − θ ) and ξ p ( − θ ) are given by (8). Let us also define ˆ T = ˆ T ( θ ).We start by proving the following Lemma 6.
The one-parameter unitary evolution group ˆ U ( θ ) = e − iθ ˆ H isgiven explicitly by ˆ U ( θ ) : L ( R n ) −→ L ( R n )(12) ˆ U ( θ )Ψ( x, p ) = F − ξ p → p ˆ T ( θ ) F p → ξ p [Ψ( x, p )] Proof.
Since ˆ H is self-adjoint and S ( R n ) ⊂ D ( ˆ H ), the initial value problem(13) i ∂ Ψ ∂θ = ˆ H Ψ , Ψ( · ,
0) = Ψ ( · ) ∈ S ( R n )has the unique solution Ψ( x, p, θ ) = e − iθ ˆ H Ψ ( x, p )where e − iθ ˆ H : L ( R n ) −→ L ( R n ) is the strongly continuous unitary evo-lution group generated by ˆ H (see, for instance [Chapter 5,[22]]). Our taskis then to determine the solution of (13) explicitly.The Hamiltonian operator ˆ H (4) can be re-written as(14) ˆ H = F − ξ p → p (cid:2) − iξ p · ∂ x + ix · ∂ x − iξ p · ∂ ξ p + 4 ix · ∂ ξ p (cid:3) F p → ξ p . where the symmetric terms of (4) were calculated explicitly. Defining ˆΨ( x, ξ p , θ ) = F p → ξ p [Ψ( x, p, θ )], we obtain from eq.(13)(15) i ∂ ˆΨ ∂θ = (cid:2) − iξ p · ∂ x + ix · ∂ x − iξ p · ∂ ξ p + 4 ix · ∂ ξ p (cid:3) ˆΨˆΨ( · ,
0) = ˆΨ ( · ) ∈ S ( R n ) . NUNO COSTA DIAS, MAURICE A. DE GOSSON, AND JO ˜AO NUNO PRATA
The solution of this initial value problem is easily found to be:ˆΨ( x, ξ p , θ ) = ˆΨ ( x ( − θ ) , ξ p ( − θ )) = ˆ T ( θ ) ˆΨ ( x, ξ p ) . Moreover, if ˆΨ ∈ S ( R n ) then ˆΨ( x, ξ p , θ ) ∈ S ( R n ) for all θ .Consequently, the solution of eq.(13) isΨ( x, p, θ ) = F − ξ p → p h ˆΨ( x ( − θ ) , ξ p ( − θ )) i = F − ξ p → p ˆ T ( θ ) F p → ξ p [Ψ ( x, p )] . Let us then defineˆ U ( θ ) : L ( R n ) → L ( R n ); ˆ U ( θ )Ψ ( x, p ) := F − ξ p → p ˆ T ( θ ) F p → ξ p [Ψ ( x, p )] . We conclude that ˆ U ( θ ) is a linear and unitary (so continuous) operator andsatisfies ˆ U ( θ ) (cid:12)(cid:12)(cid:12) S ( R n ) = e − iθ ˆ H (cid:12)(cid:12)(cid:12) S ( R n ) . Since both operators are continuous (and S ( R n ) is dense in L ( R n )) wealso have in L ( R n ), ˆ U ( θ ) = e − iθ ˆ H which completes the proof. (cid:3) It follows that
Theorem 7.
Let ˆ S = ˆ U ( θ ) and θ = √ arccos (cid:0) (cid:1) . For ψ, φ ∈ L ( R n ) wehave (16) W ( ψ, φ ) = 1(2 π ) n/ ˆ S ( ψ ⊗ ˆ φ ) . Proof.
We notice thatˆ φ ( p ) = F y → p [ φ ( y )] = F − y → p [ φ ( y )]and so ψ ⊗ ˆ φ ( x, p ) = F − y → p [ ψ ⊗ φ ( x, y )] . We also notice from (8) and (11) thatˆ T Φ( x, ξ p ) = Φ( x ( − θ ) , ξ p ( − θ )) = Φ( x − ξ p / , x + ξ p / . Hence, we getˆ S ( ψ ⊗ ˆ φ )( x, p ) = F − ξ p → p ˆ T F p → ξ p F − y → p [ ψ ⊗ φ ( x, y )] = F − ξ p → p h ˆ T ψ ⊗ φ ( x, ξ p ) i = 1(2 π ) n/ Z R n e ip · ξ p ψ ( x − ξ p / φ ( x + ξ p / dξ p which is precisely (2 π ) n/ W ( ψ, φ ). (cid:3) ETAPLECTIC FORMULATION... 7
Metaplectic formulation of Weyl symbols.
Let L ( S ( R n ) , S ′ ( R n ))be the space of linear and continuous operators of the form S ( R n ) −→S ′ ( R n ). The Schwartz kernel theorem states that all operators ˆ a ∈ L ( S ( R n ) , S ′ ( R n ))admit a kernel representation(17) ˆ aψ ( x ) = h K a ( x, · ) , ψ ( · ) i where K a ∈ S ′ ( R n × R n ). The Weyl symbol of ˆ a is defined by(18) a ( x, p ) = Z R n e − ip · y K a ( x + y, x − y ) dy where the Fourier transform is taken in the distributional sense. We willdenote by ˆ a Weyl ←→ a the one-to-one association between pseudo-differentialoperators (i.e. linear and continuous operators from S ( R n ) to S ′ ( R n )) andWeyl symbols.From (17) we immediately obtain for all φ ∈ S ( R n )(19) h ˆ aψ, φ i = h K a ( x, y ) , ψ ⊗ φ ( y, x ) i and it is well known that this equation can be re-written in terms of thecross Wigner function and the Weyl symbol of ˆ a , yielding the relation [16](20) h ˆ aψ, φ i = h a, W ( ψ, φ ) i . Since the Wigner function W ( ψ, φ ) is (proportional to) the Weyl symbolof the operator with kernel ψ ⊗ φ , we may suspect that a metaplectic formu-lation is also possible for generic Weyl symbols. This result is the contentof the next theorem. Theorem 8.
Let ˆ a : S ( R n ) −→ S ′ ( R n ) be a generic pseudo-differentialoperator, and let K a ∈ S ′ ( R n × R n ) be the kernel of ˆ a . Then the Weylsymbol a of ˆ a is given by (21) a ( x, p ) = (2 π ) n/ ˆ S F − y → p [ K a ( x, y )] where ˆ S = ˆ U ( θ ) is given by (12).Proof. We note that for arbitrary K a ∈ S ′ ( R n × R n ) we have F p → ξ p F − y → p [ K a ( x, y )] = K a ( x, ξ p ) . We also note from (12) that ˆ S can be trivially extended to S ′ ( R n ). Hence,from (12)ˆ S F − y → p [ K a ( x, y )] = F − ξ p → p h ˆ T K a ( x, ξ p ) i = F − ξ p → p h K a ( x − ξ p , x + ξ p ) i = (cid:18) π (cid:19) n/ Z R n e − ip · ξ p K a ( x + ξ p , x − ξ p ) dξ p = (cid:18) π (cid:19) n/ a ( x, p )which concludes the proof. (cid:3) NUNO COSTA DIAS, MAURICE A. DE GOSSON, AND JO ˜AO NUNO PRATA A Fractional generalization of the Wigner function andWeyl symbols
In this section we use the metaplectic formulation of the Wigner functionand Weyl symbols to obtain a natural fractional generalization for boththese objects. We also extend the main properties of the Wigner transformand Weyl symbols to the fractional case. These results end up yieldingsimple, alternative proofs for several known results about the Weyl-Wignerformalism.3.1.
Main definitions and regularity results.
A natural definition ofthe fractional cross Wigner function is
Definition 9.
Let ψ, φ ∈ L ( R n ) and let ˆ U ( θ ) be the unitary operator givenby (12). The fractional cross Wigner function W θ ( ψ, φ ) is defined by (22) W θ ( ψ, φ )( x, p ) := (2 π ) − n/ ˆ U ( θ ) ψ ⊗ ˆ φ ( x, p ) for θ ∈ R . Since ˆ U ( θ ) is periodic, the family of quasi-distributions ( W θ ( ψ, φ )) θ ∈ R is also periodic. We have W θ ( ψ, φ ) = W θ + π √ ( ψ, φ ) for all θ ∈ R . Someimportant elements of this family are W ( ψ, ψ ) = (2 π ) − n/ ψ ⊗ ˆ ψ , which is e ip · x times the anti-standard or Kirkwood quasi-distribution [20]. We alsohave W θ ( ψ, ψ ) = (2 π ) − n/ ψ ⊗ ˆ ψ (check eq.(37)) which is e − ip · x times thestandard-ordered quasi-distribution [20]. Finally, W θ ( ψ, φ ) = W ( ψ, φ ) isthe cross Wigner function.The metaplectic transformation ˆ U ( θ ) ∈ Mp(4 n ) is a unitary operator L ( R n ) −→ L ( R n ) and a continuous mapping S ( R n ) −→ S ( R n ) thatextends by duality to a continuous mapping S ′ ( R n ) −→ S ′ ( R n ). Hence, Theorem 10.
The sesquilinear map W θ is a continuous mapping of theforms: W θ : S ( R n ) × S ( R n ) −→ S ( R n ) W θ : L ( R n ) × L ( R n ) −→ L ( R n ) and extends to a continuous mapping of the form W θ : S ′ ( R n ) × S ′ ( R n ) −→ S ′ ( R n ) . Proof.
Let M ( ψ, φ ) = ψ ⊗ ˆ φ . Since the Fourier transform F y → p is a con-tinuous mapping S ( R n ) −→ S ( R n ), L ( R n ) −→ L ( R n ) and extends byduality onto a continuous mapping S ′ ( R n ) −→ S ′ ( R n ), the sesquilinearmap M can also be realized as a continuous mapping of any of the forms S ( R n ) ×S ( R n ) −→ S ( R n ), L ( R n ) × L ( R n ) −→ L ( R n ) and also S ′ ( R n ) ×S ′ ( R n ) −→ S ′ ( R n ). The result follows from W θ = (2 π ) − n/ ˆ U ( θ ) M . (cid:3) ETAPLECTIC FORMULATION... 9
We also have for all θ ∈ R ( W θ ( ψ , φ ) , W θ ( ψ , φ )) L ( R n ) = 1(2 π ) n ( ψ , ψ ) L ( R n ) ( φ , φ ) L ( R n ) which is the fractional generalization of the Moyal identity. Hence, for φ = φ = φ such that || φ || L ( R n ) = 1, the fractional windowed Wigner transform W θφ · = (2 π ) n/ W θ ( · , φ ) is a non-surjective isometry L ( R n ) −→ L ( R n ).Just as in the case of the cross Wigner function, we may define a fractionalgeneralization of Weyl symbols. Definition 11.
Let ˆ a : S ( R n ) −→ S ′ ( R n ) be an arbitrary pseudo-differentialoperator and let K a ( x, y ) ∈ S ′ ( R n × R n ) be its kernel. The θ -Weyl symbolof ˆ a is defined by (23) a θ ( x, p ) := (2 π ) n/ ˆ U ( θ ) F − y → p [ K a ( x, y )] . As an example, let a θ ˆ ρ be the θ -Weyl symbol of the rank one operator ˆ ρ defined for fixed ψ, φ ∈ L ( R n ) byˆ ρ : S ( R n ) −→ S ′ ( R n ) , ξ ˆ ρξ = ( ξ, φ ) L ( R n ) ψ. We have, as in the standard Weyl-Wigner formalism, a θ ˆ ρ = (2 π ) n W θ ( ψ, φ ).3.2. Inversion and Kernel formulas.
Let, as before,(24) W θφ : L ( R n ) −→ L ( R n ); φ W θφ ψ = (2 π ) n/ W θ ( ψ, φ )be the θ -windowed Wigner transform defined for θ ∈ R and some window φ ∈ S ( R n ) such that || φ || = 1. For θ = θ we write W φ ≡ W θ φ as in eq.(2).Let also ( W θφ ) ∗ : L ( R n ) −→ L ( R n ) be the mapping(25) ( W θφ ) ∗ Ψ( · ) := Z R n ˆ φ ( p ) ˆ U − ( θ )Ψ( · , p ) dp . Then
Theorem 12.
The mapping ( W θφ ) ∗ is the adjoint of W θφ and satisfies:(i) ( W θφ ) ∗ W θφ = 1 ;(ii) W θφ ( W θφ ) ∗ = P θφ where P θφ is the orthogonal projection onto the rangeof W θφ ;(iii) ( W θφ ) ∗ extends to a continuous operator S ′ ( R n ) −→ S ′ ( R n ) definedby (26) h ( W θφ ) ∗ Ψ , ξ i = h Ψ , ˆ U − ( θ ) ξ ⊗ ˆ φ i , ∀ ξ ∈ S ( R n ) . Proof.
The adjoint of W θφ is defined by(( W θφ ) ∗ Ψ , ξ ) = (Ψ , W θφ ξ ) , ∀ ξ ∈ L ( R n ) and since (cid:16) Ψ , W θφ ξ (cid:17) = (cid:16) Ψ , ˆ U ( θ ) ξ ⊗ ˆ φ (cid:17) = (cid:16) ˆ U − ( θ )Ψ , ξ ⊗ ˆ φ (cid:17) = Z R n ξ ( x ) (cid:20)Z R n ˆ φ ( p ) ˆ U − ( θ )Ψ( x, p ) dp (cid:21) dx , we have ( W θφ ) ∗ Ψ = Z R n ˆ φ ( p ) ˆ U − ( θ )Ψ( x, p ) dp as claimed.(i) Let now Ψ = W θφ ψ = ˆ U ( θ )( ψ ⊗ ˆ φ ). Then( W θφ ) ∗ Ψ = Z R n ˆ φ ( p ) ˆ U − ( θ ) ˆ U ( θ )( ψ ⊗ ˆ φ )( x, p ) dp = Z R n ˆ φ ( p ) ˆ φ ( p ) ψ ( x ) dp = ψ ( x ) . (ii) Conversely, let Ψ ∈ L ( R n ) and let P θφ = W θφ ( W θφ ) ∗ . Then, of course P θφ P θφ = W θφ ( W θφ ) ∗ W θφ ( W θφ ) ∗ = P θφ and if Ψ ∈ Ran W θφ then Ψ = W θφ ψ for some ψ ∈ L ( R n ) and P θφ Ψ = W θφ ( W θφ ) ∗ W θφ ψ = W θφ ψ = Ψ . Hence, the claim.(iii) As a distribution ( W θφ ) ∗ Ψ is completely defined by its action on ar-bitrary test functions ξ ∈ S ( R n ). For Ψ ∈ S ( R n ) and φ ∈ S ( R n ), we havefrom (25) h ( W θφ ) ∗ Ψ , ξ i = h ˆ U − ( θ )Ψ , ξ ⊗ ˆ φ i = h Ψ , ˆ U − ( θ ) ξ ⊗ ˆ φ i . Since ˆ U − ( θ ) : S ′ ( R n ) −→ S ′ ( R n ), this formula extends trivially to Ψ ∈S ′ ( R n ). (cid:3) All these considerations are of course valid for the standard windowedWigner transform, which is just a particular element of the family W θφ , θ ∈ R .To proceed let us generalize the Kernel formula (20) for the fractional case.Let the kernel of ˆ a : S ( R n ) −→ S ′ ( R n ) be given by K a ( x, y ) ∈ S ′ ( R n × R n )so that ˆ aψ ( x ) = h K a ( x, y ) , ψ ( y ) i . We then have(27) h ˆ aψ, φ i = h K a ( x, y ) , ψ ( y ) φ ( x ) i and it is well known that this formula can be re-expressed in terms of theWigner function as in (20). This is because(28) h K a ( x, y ) , ψ ( y ) φ ( x ) i = h a ( x, p ) , W ( ψ, φ )( x, p ) i . ETAPLECTIC FORMULATION... 11
Let also ˆ b = ˆ a † be the formal adjoint of ˆ a , defined by(29) h ˆ bψ, φ i = h ˆ aφ, ψ i , ∀ ψ, φ ∈ S ( R n ) . It is well known that K b ( x, y ) = K a ( y, x ). This relation follows immediatelyfrom eq.(29) by taking into account that h ˆ bψ, φ i = h K b ( x, y ) , ψ ( y ) φ ( x ) i and that h ˆ aφ, ψ i = h K a ( y, x ) , ψ ( y ) φ ( x ) i . We can now state the following
Theorem 13.
Let ˆ a : S ( R n ) −→ S ′ ( R n ) be an arbitrary pseudo-differentialoperator and let ˆ b be its formal adjoint. Let b θ be the θ -Weyl symbol of ˆ b .Then (30) h ˆ aψ, φ i = h b θ , W θ ( ψ, φ ) i for all ψ, φ ∈ S ( R n ) .Proof. We have from (27) h ˆ aψ, φ i = h K a ( y, x ) , ψ ( x ) φ ( y ) i = h ˆ U ( θ ) F − y → p (cid:2) K a ( y, x ) (cid:3) , ˆ U ( θ ) F − y → p (cid:2) ψ ( x ) φ ( y ) (cid:3) i = (2 π ) n/ h ˆ U ( θ ) F − y → p (cid:2) K a ( y, x ) (cid:3) , W θ ( ψ, φ )( x, p ) i . To complete the proof we just notice that K b ( x, y ) = K a ( y, x ) and use thedefinition of θ -Weyl symbol (23). (cid:3) Finally, notice that in the Weyl case θ = θ we have b θ = a θ = a and soformula (30) reduces to the standard formula (20).3.3. θ -Weyl calculus and quantization. From the definitions 9 and 11we immediately realize that W θ ( ψ, φ ) = ˆ U ( θ − θ ) W ( ψ, φ ) a θ = ˆ U ( θ − θ ) a . (31)where a ≡ a θ and a θ are the Weyl and the θ -Weyl symbols of an arbitrarypseudo-differential operator ˆ a , respectively. These formulas allow us to relatethe standard and the fractional Weyl calculus. Let us denote by ˆ a θ ←→ a θ theassociation between the θ -symbol a θ = ˆ U ( θ − θ ) a and the pseudo-differentialoperator ˆ a Weyl ←→ a . Then Theorem 14.
Let ˆ a : S ( R n ) −→ S ′ ( R n ) and ˆ b : S ( R n ) −→ S ( R n ) be twoarbitrary pseudo-differential operators. Let ˆ a θ ←→ a θ and ˆ b θ ←→ b θ . Then ˆ a ˆ b θ ←→ a θ ∗ θ b θ where the product ∗ θ is given by (32) a θ ∗ θ b θ = ˆ U ( θ − θ ) h(cid:16) ˆ U − ( θ − θ ) a θ (cid:17) ∗ M (cid:16) ˆ U − ( θ − θ ) b θ (cid:17)i . Here ∗ M is the standard Moyal product [16] a ∗ M b ( z ) = (cid:0) π (cid:1) n Z R n e i σ ( z ′ ,z ′′ ) a ( z + z ′ ) b ( z − z ′′ ) dz ′ dz ′′ where z = ( x, p ) are canonical coordinates, and σ is the standard symplecticform on R n .Proof. Let ˆ a Weyl ←→ a and ˆ b Weyl ←→ b . Then [16]ˆ a ˆ b Weyl ←→ a ∗ M b . The association between an arbitrary ˆ a and its θ -symbol a θ is defined byˆ a θ ←→ a θ = ˆ U ( θ − θ ) a where ˆ a Weyl ←→ a . Hence, the θ -symbol of ˆ a ˆ b is a θ ∗ θ b θ = ˆ U ( θ − θ )( a ∗ M b ) . Using a = ˆ U − ( θ − θ ) a θ and b = ˆ U − ( θ − θ ) b θ we get (32). (cid:3) It easily follows from (32) that ∗ θ is non-commutative, distributive andassociative. Let us prove this last property. Since ∗ M is associative we have(let ˆ U = ˆ U ( θ − θ ))( a θ ∗ θ b θ ) ∗ θ c θ = ˆ U hh(cid:16) ˆ U − a θ (cid:17) ∗ M (cid:16) ˆ U − b θ (cid:17)i ∗ M (cid:16) ˆ U − c θ (cid:17)i = ˆ U h(cid:16) ˆ U − a θ (cid:17) ∗ M h(cid:16) ˆ U − b θ (cid:17) ∗ M (cid:16) ˆ U − c θ (cid:17)ii = a θ ∗ θ ( b θ ∗ θ c θ ) . (33)To proceed let us determine the θ -symbol of the adjoint operator. Let, asbefore, ˆ b = ˆ a † be the formal adjoint of ˆ a . Then the Weyl symbols of ˆ a andˆ b satisfy a = b and we have for the θ -symbols(34) b θ + α = ˆ U ( α ) b = ˆ U ( α ) a = ˆ U − ( α ) a = a θ − α . Hence, and in general, the θ -symbol of a self-adjoint operator is not real.The exception is the standard Weyl case.Using the result (34) we can re-write the kernel formula (30) in the form(35) h ˆ aψ, φ i = h a θ − α , W θ + α ( ψ, φ ) i for all α ∈ R .Finally, let us remark on a few features of the fractional Weyl quantiza-tion. The next result is a trivial consequence of (35) Corollary 15.
The expectation value of a quantum observable ˆ a in a state ψ ∈ S ( R n ) is given in terms of its θ -symbol by (36) h ˆ aψ, ψ i = Z R n a θ ( z ) W θ ( ψ, ψ )( z ) dz . ETAPLECTIC FORMULATION... 13 where the integral is interpreted in the distributional sense.Proof.
The proof follows from eqs.(34) and (35) by noticing that for ˆ a self-adjoint we have ˆ b = ˆ a and so a θ − α = a θ + α . Notice also that in the standardWeyl case, we have in addition a θ ≡ a = a . (cid:3) From eq.(35) we can also determine the marginal probability distributionsfor the fractional Wigner function. Let, for instance, ˆ a be the projectionoperator given formally by | x >< x | where x ∈ R n . We have explicitlyˆ a : S ( R n ) −→ S ′ ( R n ); ψ ψ ( x ) δ ( x − x )where δ is the Dirac delta distribution. The Weyl symbol of ˆ a can be easilycalculated from eq.(18): a ( x, p ) = δ ( x − x ). Hence, the marginal probabilitydistribution for the position observable is simply P ( x ) = h ˆ aψ, ψ i = Z R n Z R n δ ( x − x ) W ( ψ, ψ )( x, p ) dxdp = Z R n W ( ψ, ψ )( x , p ) dp . In the fractional case, we also have from eq.(31) and eq.(36) P ( x ) = Z R n Z R n (cid:16) ˆ U − ( θ − θ ) δ ( x − x ) (cid:17) W θ ( ψ, ψ )( x, p ) dxdp . However, in general, this formula does not further simplify as in the case ofstandard Wigner functions.Another important feature of the Wigner function W ( ψ, ψ ) is that it isa real quasidistribution. This property is also not shared by the fractionalWigner functions. For the θ -cross Wigner function we get from (31)(37) W θ + α ( ψ, φ ) = W θ − α ( φ, ψ )which yields a reality condition only for the case α = 0 and φ = ψ .4. The Bopp representation of quantum mechanics
In this section we show that the windowed Wigner transform W φ inter-twines the Schr¨odinger and the Bopp representations of quantum mechanicsand discuss the implications of this result for the spectral and dynamicalproperties of operators in the two representations. This subject was previ-ously studied in several papers [9, 17, 18]. Here, we present an alternativeapproach using the metaplectic formalism developed in the previous sections.Let ˆ a : S ( R n ) −→ S ′ ( R n ) be a generic pseudo-differential operator. Interms of its Weyl symbol a ∈ S ′ ( R n × R n ), the operator can be written [24](38) ˆ aψ ( x ) = 1(2 π ) n Z R n e i ( x − x ′ ) · ξ x a ( 12 ( x + x ′ ) , ξ x ) ψ ( x ′ ) dx ′ dξ x where the integral is well defined for a ∈ S ( R n × R n ) and should other-wise be interpreted in the distributional sense. The operator ˆ a is formallyˆ a = a ( x, − i∂ x ) and the mapping a ˆ a yields a precise definition of the standard Schr¨odinger representation of quantum mechanics.A trivial extension of ˆ a to phase space functions is given by the operatorˆ A : S ( R n × R n ) −→ S ′ ( R n × R n ) of the form:(39) ˆ A Ψ( x, p ) = 1(2 π ) n Z R n e i ( x − x ′ ) · ξ x a ( 12 ( x + x ′ ) , ξ x )Ψ( x ′ , p ) dx ′ dξ x . Then
Theorem 16.
The operator ˆ A is a pseudo-differential operator with Weylsymbol (40) A ( x, p ; ξ x , ξ p ) = a ( x, ξ x ) . Moreover, for an arbitrary wave function Ψ( x, p ) = ψ ⊗ ˆ φ ( x, p ) ∈ S ( R n ) ,we have (41) ˆ A ( ψ ⊗ ˆ φ ) = (ˆ aψ ) ⊗ ˆ φ. Proof.
Let ˆ A be the pseudo-differential operator with Weyl symbol A of theform (40). Then ˆ A Ψ( x, p ) is explicitlyˆ A Ψ( x, p )= 1(2 π ) n Z R n e i [( x − x ′ ) · ξ x +( p − p ′ ) · ξ p ] A ( x + x ′ , p + p ′ , ξ x , ξ p )Ψ( x ′ , p ′ ) dx ′ dp ′ dξ x dξ p = 1(2 π ) n Z R n e i [( x − x ′ ) · ξ x +( p − p ′ ) · ξ p ] a ( 12 ( x + x ′ ) , ξ x )Ψ( x ′ , p ′ ) dx ′ dp ′ dξ x dξ p = 1(2 π ) n Z R n e i ( x − x ′ ) · ξ x a ( 12 ( x + x ′ ) , ξ x )Ψ( x ′ , p ) dx ′ dξ x which coincides with the expression (39) exactly.The intertwining relation (41) follows directly from (39) for Ψ( x, p ) = ψ ( x ) ˆ φ ( p ). (cid:3) Just like ˆ a , the operator ˆ A is also formally ˆ A = a ( x, − i∂ x ) (but now thefundamental operators x · and − i∂ x act on phase space functions). The map-ping a ˆ A yields the phase space Schr¨odinger representation of quantummechanics. The spectral and dynamical properties of the operators ˆ a and ˆ A are equivalent. This is an important property that we now discuss in somedetail. For complete proofs the reader should refer to [9].For an arbitrary window φ ∈ S ( R n ) such that || φ || L ( R n ) = 1, let T φ : S ′ ( R n ) −→ S ′ ( R n ); ψ Ψ = T φ ψ = ψ ⊗ ˆ φ , and let T ∗ φ : S ′ ( R n ) −→ S ′ ( R n ); Ψ ψ = T ∗ φ Ψbe defined by h T ∗ φ Ψ , ξ i = h Ψ , ξ ⊗ ˆ φ i , ∀ ξ ∈ S ( R n ) . ETAPLECTIC FORMULATION... 15
These two maps coincide exactly with W θφ and ( W θφ ) ∗ (given by eq.(24) andeq.(26), respectively) for θ = 0. Hence, they satisfy the properties stated inTheorem 12 (i) and (ii). They also satisfy the intertwining relations(42) T φ ˆ a = ˆ AT φ , T ∗ φ ˆ A = ˆ aT ∗ φ . The first relation is valid in S ( R n ) and is just a restatement of eq.(41). Thesecond relation is valid in S ( R n ) and was proved in [9].The next theorem is a consequence of these intertwining relations. Itshows that the operators ˆ a and ˆ A have equivalent spectral properties. Theproof can also be found in [9]. Theorem 17.
Let ˆ a and ˆ A be the pseudo-differential operators given byeqs.(38,39), respectively. Let φ ∈ S ( R n ) be such that || φ || L ( R n ) = 1 . Then(i) The eigenvalues of ˆ a and ˆ A are the same.(ii) If ψ λ is an eigenfunction of ˆ a then Ψ λ = T φ ψ λ is an eigenfunction of ˆ A (associated with the same eigenvalue).(iii) Conversely, let Ψ λ be an eigenfunction of ˆ A . If ψ λ = T ∗ φ Ψ λ = 0 then ψ λ is an eigenfunction of ˆ a (associated with the same eigenvalue).(iv) If ( ψ λ ) λ is an orthonormal basis of eigenfunctions of ˆ a and ( φ γ ∈S ( R n )) γ is an orthonormal basis of L ( R n ) then ( T φ γ ψ λ ) γ,λ is a completeset of eigenfunctions of ˆ A and forms an orthonormal basis of L ( R n ) . The next theorem concerns the dynamical properties.
Theorem 18.
Let ˆ a and ˆ A be given by eqs.(38,39), respectively. Let φ ∈S ( R n ) satisfy || φ || L ( R n ) = 1 . Then:(i) If ψ ( x, t ) is the solution of the initial value problem (43) i ∂ψ∂t = ˆ aψ , ψ ( · ,
0) = ψ ( · ) ∈ S ( R n ) then Ψ = T φ ψ is the solution of (44) i ∂ Ψ ∂t = ˆ A Ψ , Ψ( · ,
0) = Ψ ( · ) ∈ S ( R n ) where Ψ = T φ ψ .(ii) Conversely, if Ψ is the solution of the initial value problem (44) then ψ = T ∗ φ Ψ is the solution of (43). The proof follows directly from the intertwining relations and the factthat the time derivative operator commutes with both T φ and T ∗ φ [9].The results of the previous theorems can be extended to any other opera-tor that is unitarily related with ˆ A . This is the case of the Bopp operators,formally defined by the mapping(45) a ( x, ξ x ) ˆ A B = a ( x + i ∂ p , p − i ∂ x ) . This map attributes to each phase space symbol a ∈ S ′ ( R n ), an operatorˆ A B acting on wave functions Ψ( x, p ) with support on the phase space. TheBopp representation is closely related with the deformation quantization ofBayen et. al [1, 2, 10, 11] and was used, in this context, to prove somegeneral spectral results for the stargenvalue equation [9, 17].We now prove that the Bopp representation can be precisely defined bythe mapping(46) a ˆ A B = ˆ S ˆ A ˆ S − where a ∈ S ′ ( R n ), ˆ A is given by eq.(39) and ˆ S = ˆ U ( θ ) by eq.(12). Let usconsider first the general result Proposition 19 (Metaplectic covariance property) . Let ˆ A Weyl ←→ A be apseudo-differential operator with Weyl symbol A ∈ S ′ ( R n ) , let ˆ S ∈ Mp (4 n ) and let S ∈ Sp (4 n ) be its natural projection onto Sp (4 n ) . Then (47) ˆ A S = ˆ S ˆ A ˆ S − Weyl ←→ A S = A ◦ S − . For a proof see [16, 25]. We then have:
Theorem 20.
The operators ˆ A B (given by eq.(46)) are pseudo-differentialoperators with Weyl symbols (48) A B ( x, p ; ξ x , ξ p ) = a ( x − ξ p , p + ξ x . Proof.
It follows from Proposition 19 and the definition of ˆ A B (46) thatˆ A B Weyl ←→ A B = A ◦ S − where A Weyl ←→ ˆ A is given by (40) and S − is the inverseof the projection of ˆ S onto Sp(4 n ). Hence S − = s ( − θ ) (cf. Theorem 4)which can be easily determined from (8) and (9). We have(49) S − : R n −→ R n , x x − ξ p / ξ x ξ x / pp p − ξ x / ξ p ξ p / x . The result (48) then follows from A B = A ◦ S − by taking into account(40). (cid:3) For the fundamental operators the mapping (46) yields: a ( x, ξ x ) = x = ⇒ ˆ A = x · = ⇒ ˆ A B = x + i ∂ p a ( x, ξ x ) = ξ x = ⇒ ˆ A = − i∂ x = ⇒ ˆ A B = p − i ∂ x and more generally, we have formally a ( x, ξ x ) ˆ A B = a ( x + i ∂ p , p − i ∂ x ) . ETAPLECTIC FORMULATION... 17
Finally, it follows from ˆ A B = ˆ S ˆ A ˆ S − that the spectral and dynamicalproperties of the Bopp and the phase space Schr¨odinger representations areequivalent. In view of Theorems 17, 18 this equivalence can be extended tothe standard
Schr¨odinger representation. Combining (42) with ˆ A B = ˆ S ˆ A ˆ S − we obtain the intertwining relationsˆ A B ˆ ST φ = ˆ ST φ ˆ a , T ∗ φ ˆ S − ˆ A B = ˆ aT ∗ φ ˆ S − which can be re-written in terms of the windowed Wigner transform W φ ≡ W θ φ and its adjoint W ∗ φ ≡ (cid:16) W θ φ (cid:17) ∗ (given respectively by (24) and (25,26)for θ = θ ) ˆ A B W φ = W φ ˆ a , W ∗ φ ˆ A B = ˆ aW ∗ φ . The Theorems 17 and 18 are then valid ipsis verbis for the pair of operatorsˆ a and ˆ A B if we substitute the maps T φ and T ∗ φ by the maps W φ and W ∗ φ ,respectively. Acknowledgements . We thank the anonymous referee for a detailed read-ing of the paper and for many useful insights and remarks. Maurice deGosson has been financed by the Austrian Research Agency FWF (Pro-jektnummer P20442-N13). Nuno Costa Dias and Jo˜ao Nuno Prata havebeen supported by the grant PTDC/MAT/099880/2008 of the PortugueseScience Foundation (FCT).
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Author’s addresses: • Jo˜ao Nuno Prata and
Nuno Costa Dias:
Departamento deMatem´atica. Universidade Lus´ofona de Humanidades e Tecnologias.Av. Campo Grande, 376, 1749-024 Lisboa, Portugal and Grupo deF´ısica Matem´atica, Universidade de Lisboa, Av. Prof. Gama Pinto2, 1649-003 Lisboa, Portugal • Maurice de Gosson:
Universit¨at Wien, Fakult¨at f¨ur Mathematik–NuHAG, Nordbergstrasse 15, 1090 Vienna, Austria
E-mail address : [email protected] E-mail address : [email protected] E-mail address ::