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 |
| [a2] | R. Beals, C. Fefferman, "On local solvability of linear partial differential equations" Ann. of Math. , 97 (1973) pp. 482–498 |
| [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 |
| [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 |
| [a6] | C. Fefferman, D.H. Phong, "On positivity of pseudo-differential operators" Proc. Nat. Acad. Sci. USA , 75 (1978) pp. 4673–4674 |
| [a7] | L. Hörmander, "The Weyl calculus of pseudo-differential operators" Commun. Pure Appl. Math. , 32 (1979) pp. 359–443 |
| [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 |
| [a11] | A. Weil, "Sur certains groupes d'opérateurs unitaires" Acta Math. , 111 (1964) pp. 143–211 |
| [a12] | H. Weyl, "Gruppentheorie und Quantenmechanik" , S. Hirzel (1928) |
