Weyl quantization

Let be a classical Hamiltonian (cf. also Hamilton operator) defined on . The Weyl quantization rule associates to this function the operator defined on functions as

(a1)

For instance, , with

whereas the classical quantization rule would map the Hamiltonian to the operator . A nice feature of the Weyl quantization rule, introduced in 1928 by H. Weyl [a12], is the fact that real Hamiltonians get quantized by (formally) self-adjoint operators. Recall that the classical quantization of the Hamiltonian is given by the operator acting on functions by

(a2)

where the Fourier transform is defined by

(a3)

so that , with . In fact, introducing the one-parameter group , given by the integral formula

(a4)

one sees that

In particular, one gets . Moreover, since one obtains

yielding formal self-adjointness for real (cf. also Self-adjoint operator).

Wigner functions.

Formula (a1) can be written as

(a5)

where the Wigner function is defined as

(a6)

The mapping is sesquilinear continuous from to , so that makes sense for (here, and stands for the anti-dual):

The Wigner function also satisfies

and the phase symmetries are unitary and self-adjoint operators on . Also ([a10], [a12]),

where (here ). These formulas give, in particular,

where stands for the space of bounded linear mappings from into itself. The operator is in the Hilbert–Schmidt class (cf. also Hilbert–Schmidt operator) if and only if belongs to and . To get this, it suffices to notice the relationship between the symbol of and its distribution kernel :

The Fourier transform of the Wigner function is the so-called ambiguity function

(a7)

For , the Wigner function is the Weyl symbol of the operator (cf. also Symbol of an operator), where is the (Hermitian) dot-product, so that from (a5) one finds

As is shown below, the symplectic invariance of the Weyl quantization is actually its most important property.

Symplectic invariance.

Consider a finite-dimensional real vector space (the configuration space ) and its dual space (the momentum space ). The phase space is defined as ; its running point will be denoted, in general, by a capital letter (). The symplectic form (cf. also Symplectic connection) on is given by

(a8)

where stands for the bracket of duality. The symplectic group is the subgroup of the linear group of preserving (a8). With

for one has

so that the equation of the symplectic group is

One can describe a set of generators for the symplectic group , identifying with : the mappings

i) , where is an automorphism of ;

ii) and the other coordinates fixed;

iii) , where is symmetric from to . One then describes the metaplectic group, introduced by A. Weil [a11]. The metaplectic group is the subgroup of the group of unitary transformations of generated by

j) , where ;

jj) partial Fourier transformations;

jjj) multiplication by , where is a symmetric matrix. There exists a two-fold covering (the of both and is )

such that, if and , are in , while is their Wigner function, then

This is Segal's formula [a9], which can be rephrased as follows. Let and . There exists an in the fibre of such that

(a9)

In particular, the images by of the transformations j), jj), jjj) are, respectively, i), ii), iii). Moreover, if is the phase translation, , (a9) is fulfilled with and phase translation given by

If is the symmetry with respect to , in (a9) is, up to a unit factor, the phase symmetry defined above. This yields the following composition formula: with

(a10)

with an integral on . One can compare this with the classical composition formula,

(cf. (a2)) with

with an integral on . It is convenient to give an asymptotic version of these compositions formulas, e.g. in the semi-classical case. Let be a real number. A smooth function defined on is in the symbol class if

Then one has for and the expansion

(a11)

with . The beginning of this expansion is thus

where denotes the Poisson brackets and . The sums inside (a11) with even are symmetric in and skew-symmetric for odd. This can be compared to the classical expansion formula

with . Moreover, for in , the multiple composition formula gives

and if ,

Consider the standard sum of homogeneous symbols defined on , where is an open subset of ,

with smooth on and homogeneous in the following sense:

and , i.e. for all compact subsets of ,

This class of pseudo-differential operators (cf. also Pseudo-differential operator) is invariant under diffeomorphisms, and using the Weyl quantization one gets that the principal symbol is invariantly defined on the cotangent bundle whereas the subprincipal symbol is invariantly defined on the double characteristic set

of the principal symbol. If one writes

one gets and . Moreover,

Thus, if one defines the subprincipal symbol as the above analytic expression where is the classical symbol of , one finds that this invariant is simply the second term in the expansion of the Weyl symbol . In the same vein, it is also useful to note that when considering pseudo-differential operators acting on half-densities one gets a refined principal symbol invariant by diffeomorphism.

Weyl–Hörmander calculus and admissible metrics.

The developments of the analysis of partial differential operators in the 1970s required refined localizations in the phase space. E.g., the Beals–Fefferman local solvability theorem [a2] yields the geometric condition (P) as an if-and-only-if solvability condition for differential operators of principal type (with possibly complex symbols). These authors removed the analyticity assumption used by L. Nirenberg and F. Treves, and a key point in their method is a Calderón–Zygmund decomposition of the symbol, that is, a micro-localization procedure depending on a particular function, yielding a pseudo-differential calculus tailored to the symbol under investigation. Another example is provided by the Fefferman–Phong inequality [a6], establishing that second-order operators with non-negative symbols are bounded from below on ; a Calderón–Zygmund decomposition is needed in the proof, as well as an induction on the number of variables. These micro-localizations go much beyond the standard homogeneous calculus and also beyond the classes , previously called exotic. In 1979, L.V. Hörmander published [a7], providing simple and general rules for a pseudo-differential calculus to be admissible. Consider a positive-definite quadratic form defined on . The dual quadratic form with respect to the symplectic structure is

(a12)

Define an admissible metric on the phase space as a mapping from to the set of positive-definite quadratic forms on , , such that the following three properties are fulfilled:

(uncertainty principle) For all ,

(a13)

there exist some positive constants , , such that, for all ,

(a14)

there exist some positive constants , , such that, for all ,

(a15)

Property (a13) is clearly related to the uncertainty principle, since for each one can diagonalize the quadratic form in a symplectic basis so that

where is a set of symplectic coordinates. One then gets

Condition (a13) thus means that , which can be rephrased in the familiarly vague version as

in the -balls. This condition is relevant to any micro-localization procedure. When , one says that the quadratic form is symplectic. Property (a14) is called slowness of the metric and is usually easy to verify. Property (a15) is the temperance of the metric and is more of a technical character, although very important in handling non-local terms in the composition formula. In particular, this property is useful to verify the assumptions of Cotlar's lemma. Moreover, one defines a weight as a positive function on such that there exist positive constants , so that for all ,

(a16)

and

(a17)

Eventually, one defines the class of symbols as the -functions on the phase space such that

(a18)

It is, for instance, easily checked that

with and that this metric is an admissible metric when , . The metric defining satisfies (a13)–(a14) but fails to satisfy (a15). Indeed, there are counterexamples showing that for the classical and the Weyl quantization [a4] there are symbols in whose quantization is not -bounded. In fact, one of the building block for the calculus of pseudo-differential operators is the -boundedness of the Weyl quantization of symbols in , where is an admissible metric. One defines the Planck function of the calculus as

(a19)

and notes that from (a13), . One obtains the composition formula (a11) with , and . In particular, one obtains, with obvious notations,

The Fefferman–Phong inequality has also a simple expression in this framework: Let be a non-negative symbol in , then the operator is semi-bounded from below in . The proof uses a Calderón–Zygmund decomposition and in fact one shows that , where is the Planck function related to the admissible metric defined by

On the other hand, if is an admissible metric and uniformly with respect to a parameter , the following metric also satisfies (a13)–(a15): , with

One gets in this case that uniformly. A key point in the Beals–Fefferman proof of local solvability under condition (P) can be reformulated through the construction of the previous metric. Sobolev spaces related to this type of calculus were studied in [a1] (cf. also Sobolev space). For an admissible metric and a weight , the space is defined as

It can be proven that a Hilbertian structure can be set on , that and that for and another weight, the mapping

is continuous.

Further developments of the Weyl calculus were explored in [a3], with higher-order micro-localizations. Several metrics

are given on the phase space. All these metrics satisfy (a13)–(a14), but, except for , fail to satisfy globally the temperance condition (a15). Instead, the metric is assumed to be (uniformly) temperate on the -balls. It is then possible to produce a satisfactory quantization formula for symbols belonging to a class . A typical example is given in [a5], with applications to propagation of singularities for non-linear hyperbolic equations:

where is defined on , and on

It is then possible to quantize functions homogeneous of degree in the variable , and in the variable , so as to get composition formulas, Sobolev spaces, and the standard pseudo-differential apparatus allowing a commutator argument to work for propagation results.

References

[a1]	J.-M. Bony, J.-Y. Chemin, "Espaces fonctionnels associés au calcul de Weyl–Hörmander" Bull. Soc. Math. France , 122 (1994) pp. 77–118 Zbl 0798.35172
[a2]	R. Beals, C. Fefferman, "On local solvability of linear partial differential equations" Ann. of Math. , 97 (1973) pp. 482–498 MR0352746 Zbl 0256.35002
[a3]	J.-M. Bony, N. Lerner, "Quantification asymtotique et microlocalisations d'ordre supérieur" Ann. Sci. Ecole Norm. Sup. , 22 (1989) pp. 377–483
[a4]	A. Boulkhemair, "Remarque sur la quantification de Weyl pour la classe de symboles " C.R. Acad. Sci. Paris , 321 : 8 (1995) pp. 1017–1022 MR1360564 Zbl 0842.35144
[a5]	J.-M. Bony, "Second microlocalization and propagation of singularities for semi-linear hyperbolic equations" K. Mizohata (ed.) , Hyperbolic Equations and Related Topics , Kinokuniya (1986) pp. 11–49 MR925240
[a6]	C. Fefferman, D.H. Phong, "On positivity of pseudo-differential operators" Proc. Nat. Acad. Sci. USA , 75 (1978) pp. 4673–4674 MR0507931 Zbl 0391.35062
[a7]	L. Hörmander, "The Weyl calculus of pseudo-differential operators" Commun. Pure Appl. Math. , 32 (1979) pp. 359–443 MR517939 Zbl 0388.47032
[a8]	L. Hörmander, "The analysis of linear partial differential operators III-IV" , Springer (1985)
[a9]	I. Segal, "Transforms for operators and asymptotic automorphisms over a locally compact abelian group" Math. Scand. , 13 (1963) pp. 31–43
[a10]	A. Unterberger, "Oscillateur harmonique et opérateurs pseudo-différentiels" Ann. Inst. Fourier , 29 : 3 (1979) pp. 201–221 MR0552965 Zbl 0396.47027
[a11]	A. Weil, "Sur certains groupes d'opérateurs unitaires" Acta Math. , 111 (1964) pp. 143–211 MR0165033
[a12]	H. Weyl, "Gruppentheorie und Quantenmechanik" , S. Hirzel (1928) MR0450450 Zbl 54.0954.03