Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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We now quotient by the kernels of this pairing to
obtain B (V), B(V
) as two nondegenerately paired
braided groups. This quotient generates all the rela-
tions, which are very often but not necessarily
quadratic (in practice, one typically imposes only the
quadratic relations to have braided groups with a
possibly degenerate pairing). The construction is due to
the author. Moreover, we can define partial derivatives
on these braided groups by
a = 1 a þ e
i
@
i
a þ
for any a in the algebra, that is, as an infinitesimal
generator of translations under the braided group law;
similarly exp, indefinite and Gaussian integration,
Fourier transform, etc. The simplest example here is
B = C[x] viewed not as a usual Hopf algebra but as a
braided group in the category of Z-graded spaces with
(x x) = qx x. Also in this example the braided
addition law on C[x]is
x
n
¼
X
n
m¼0
n
m
q
x
m
x
nm
defined by [m]
q
, and the partial derivative defined
by it is the Jackson (1908) q-derivative
@f ðxÞ¼
f ðxÞf ðqxÞ
xð1 qÞ
while
e
q
(x) = e
q
(x) e
q
(x) if we allow power
series. Such objects occur in the theory of q-special
functions (see q-Special Functions).
Among deeper theorems (see Majid 1995, 2002),
there is a triangular decomposition
U
q
ðgÞ¼U
q
ðn
Þ T U
q
ðn
þ
Þ
where U
q
(n
þ
) is a braided group and U
q
(n
) is dually
paired to its opposite. T denotes the torus generators
{g
a
} in the section ‘‘q-D eformat ion envelo ping alge-
bras.’’ More genera lly, if g
0
g is a principal
embedding of Lie algebras (given by an inclusion of
Dynkin diagrams), then U
q
(g) = B
g
U
q
(g
0
) B
op
for some additive braided group of additional root
generators and its dual. The general construction
B
H B
op
here is ‘‘double bosonization’’ which
associates to dual braided groups B, B
in the
category of representations of some quasitriangular
Hopf algebra H, a new quasitriangular Hopf algebra.
The simplest example B = C[x] lives in the category of
representations of T = U
q
(1) in an algebraic form.
The dual is another braided line C[p] and
C[p]
U
q
(1) C[x] is a version of U
q
(sl
2
). In this
way, the braided line C[x] is at the root of all
q-deformation quantum groups.
An earlier theorem is that for any braided group B
covariant under a (co)quasitriangular H,wehaveits
‘bosonization’ B
H. There is a similar ‘‘biproduct’ ’
if B lives in the category of crossed modules for any
Hopf algebra H. These have been extensively applied
in physics notably in the construction of inhomoge-
neous quantum groups. Similar to C
2
q
(but as a
-algebra), there is a natural self-dual q-Minkowski
space B = R
1,3
q
which is covariant under
g
U
q
(so
1,3
),
and its bosonization is the q-Poincare´ plus dilations
group R
1,3
q
g
U
q
(so
1,3
). It is not possible to avoid the
dilation here. The double-bosonization extends this
to the q-conformal group U
q
(so
2,4
). The braided
adjoint action becomes the action of conformal
translations on R
1,3
q
. The construction of q-propaga-
tors and q-deformed physics on such q-Minkowski
space was achieved in the mid 1990s as one of the
main successes of the theory of braided groups.
This R
1,3
q
can be given also as a matrix of
generators, relations, -structure and, a second
braided coproduct:
¼ q
2
; ¼ q
2
; ¼
¼ þð1 q
2
Þð Þ
¼ þð1 q
2
Þ
¼ þð1 q
2
Þ
¼
¼
10
01
;
¼
This is in addition to the additive coproduct above.
It corresponds to the point of view of Minkowski
space as Hermitian 2 2 matrices. Note that
is
not a -algebra map in the usual sense and indeed
Hermitian matrices are not a group under multi-
plication, but this does form a natural braided -
bialgebra. If we quotient by the braided determinant
relation q
2
= 1, we have the unit hyperbo-
loid in R
1, 3
q
which turns out to be the braided group
B
q
[SU
2
] mentioned at the end of the previous
section (as obtained canonically from C
q
[SU
2
]). We
now have a braided antipode
S
¼
q
2
þð1 q
2
Þ q
2
q
2
This was the first nontrivial example of a braided
group (Majid 1991b) and we see that it has two
q !1 limits
Uðsu
2
Þ B
q
½SU
2
!C½Hyperboloid R
1;3
Because most constructions in physic s can be
uniformly deformed by such methods (including
the totally q-antisymmetric tensor), one finds that q
provides a new regulator in which infinities in
quantum field theory can be in principle be encoded
as poles at q = 1. That transmutation from the
696 Hopf Algebras and q-Deformation Quantum Groups
quantum group to its braided version unifies unitary
nonabelian symmetries with pseudo-Riemannian
geometry is another deeper aspect of relevance to
physics. In addition, q-constructions have their
original role in quantum integrable systems, at q a
root of unity and for infinite-dimensional (affine)
Lie algebra deformations.
Quasi-Hopf Algebras
Although the braided category of representations of
a quantum group has a trivial ‘‘associator’’
V, W, Z
:(V W) Z !V (W Z) between any
three objects, a general braided category and the
diagrammatic methods of ‘‘braided algebra’’ in the
last section do not require this (one simply translates
diagrams into algebra by inserting as needed). A
more general object that generates such categories as
its representations is a ‘‘quasi-Hopf algebra.’’ This is
a generalization of Hopf algebras in which the
coproduct : H !H H is not necessarily coasso-
ciative. Instead,
ðid Þ ¼ ðð idÞ Þ
1
ðid idÞ ¼ 1
234
ðid idÞðÞ
123
¼ðid
2
ÞðÞð id
2
ÞðÞ
for some invertible element 2H H H. The
numbers denote the position in the tensor product
and one says that is a 3-cocycle. The axioms for
the antipode and quasitriangular structure R are
also modified. The tensor product of representations
is given as usual by , and the braiding and
associator by the actions of R and .
This notion, due to Drinfeld (1990), arises when
one wishes to write down the quantum groups U
q
(g)
more explicitly as built on the algebras U(g) (recall
that they are isomorphic over formal power series).
Thus, for each semisimple g there is a natural
(quasitriangular) quasi-Hopf algebra (U(g), , R)
where U (g) has the usual Hopf algebra structure,
R is an exponential of the split Casimir (or inverse
Killing form) in g g and is constructed as a
solution of the Knizhnik–Zamolodchikov equations
coming out of conform al field theory. This is not
U
q
(g) but it has an equivalent braided category of
representations. Thus, there is an element F 2U(g)
2
(extended over formal power series) such that
F
¼ Fð ÞF
1
; R
F
¼ F
21
RF
1
F
F
12
ð idÞðFÞðid ÞðF
1
ÞF
1
23
¼ 1
recovers U
q
(g) as a quasitriangular Hopf algebra
built directly on the algebra U(g). The conjugation
operations here (and a similar process regarding the
antipode) are a ‘‘Drinfeld twist’’ of a quasi-Hopf
algebra, an d such twisting by any invertible F such
that
ð idÞF ¼ðid ÞF ¼ 1
(a cochain) does not change the representation
category up to equivalence. In the present case, the
twist transforms into
F
= 1, that is, into an
ordinary Hopf algebra isomorphic over formal
power series to U
q
(g). Note that in rational
conformal field theory the tensor product of
representations appears as a finite-dimensional
commutative associative algebra (the Verlinde alge-
bra) with integer structure constants N
ij
k
(this comes
from the operator-product expansion of primary
fields in the theory). This is because one has more
precisely a truncated representation category corre-
sponding to q a root of unity, and because we are
identifying equivalent representations (so N
ij
k
are
the multiplicity in the decomposition of a tensor
product of two representations). However, if one
wants to know the tensor product decomposition
more fully, not just its isomorphism class, this is
given in a choice of bases by recoupling matrices.
Computation in terms of these shows that the actual
tensor product is neither commu tative nor associa-
tive, but of the form above at least in the case of the
WZW model.
Hopf algebra theory typically extends to the
quasi-Hopf case. For example, given a quasi-Hopf
algebra H there is a quantum doubl e D(H) at least
in the finite-dimensional case, due to the author. An
example is to take H = C(G) and a 3-cocycle on G
in the usual sense
ðy; z; wÞðx; yz; wÞðx; y; zÞ
¼ ðx; y; zwÞðxy; z; wÞ
on elements of G and ( x,1,y) = 1. Then (C(G), )
can be viewed as a quasi-Hopf algebra. Its double
D
(G) is generated by C (G) as a sub-quasi-Hopf
algebra and by elements of G with
x y ¼ yx
X
s
s
ðy; xÞðsÞ;
s
x ¼ x
xsx
1
x ¼
X
ab¼s
ðx; x
1
ax; x
1
bxÞða; b; xÞ
ða; x; x
1
bxÞ
x
a
x
b
in terms of a basis {
s
}ofC(G), the product of G on
the right, and
ðx; yÞðsÞ¼
ðx; y; y
1
x
1
sxyÞðs; x; yÞ
ðx; x
1
sx; yÞ
Hopf Algebras and q-Deformation Quantum Groups 697
a 2-cocycle on G with values in C(G) (the algebra is
a cocycle semidirect product). There is a quasitrian-
gular structure R=
P
x
x. This quasi-Hopf
algebra first appeared in discrete topological quan-
tum field theory related to orbifolds in the work of
Dijkgraaf, Pasquier, and Roche.
There are further generalizations in the same spir it
and which are linked to con formal field theories of
more general type; for example, weak (quasi-) Hopf
algebras in which 1 6¼ 1 1 but is a projector.
These have been related to quantum groupoids.
Finally, we mention some applications of twisting
outside of the original context. First of all, we are not
limited to starting with U(g): starting with any Hopf
algebra or quasi-Hopf algebra H we can similarly twist
it to another one H
F
with the same algebra as H and
F
, R
F
,
F
given by conjugation as above. The
representation category remains unchanged up to
equivalence, so in some sense the twisted object is
equivalent. Moreover, if we start with a Hopf algebra
H and ask F to be a 2-cocycle in the sense
F
12
ð idÞðFÞðid ÞðF
1
ÞF
1
23
¼ 1
then H
F
will remain a Hopf algebra. It has
conjugated antipode (see Majid 1995 )
S
F
ðaÞ¼UðSaÞU
1
; U ¼ðid SÞðFÞ
Many Hopf algebras are twists of more standard
ones, for example, the multiparameter quantum
groups tend to be twists of the standard U
q
(g).
Likewise, ‘‘triangular’’ Hopf algebras (where
R
21
R= 1) tend to be twists of classical group or
enveloping algebras.
Asecondapplicationoftwistsisanapproachto
quantization. Although it can be applied to H itself, this
is more interesting if we think of H as a background
quantum group and ask to quantize objects covariant
under H.Forthesakeofdiscussion,westartwithH
an ordinary Hopf algebra. We twist this to H
F
and
denote by T the equivalence functor from representa-
tions of H to representations of H
F
. This functor acts
as the identity on all objects and all morphisms, but
comes with nontrivial isomorphisms c
V, W
: T (V)
T (W) !T(V W) for any two objects, compatible
with bracketting (see Majid 1995). Given any algebraic
construction covariant under H, we simply apply the
functor T to all aspects of the construction and obtain
an equivalent H
F
-covariant construction. As an exam-
ple, if A is an H-covariant algebra, then applying T to
its product we have T ():T (A A) !T(A). Using
c
A, A
we obtain a map
: TðAÞTðAÞ!TðAÞ
a b ¼ðF
1
. ða bÞÞ
in terms of the product in A. Thus, we have a new
algebra A
F
built on the same vector space as A but
with a modified product. This is called a
‘‘covariant twist’’ of an algebra and should not be
confused with the Drinfeld twist above. It is due to
the author in the early 1990s. If F is a 2-cocycle,
then A
F
remains associative. The transmutation
constr uction ment ioned in the sect ion ‘‘Self-dual
quan tum groups’’ or the passage from R
4
q
to R
1, 3
q
are
examples in quantum group theory. Other examples
include the standard Moyal product on R
n
, also
called noncommutative spacetime [x
, x
] = i
by
string theorists (see Bicrossproduct Hopf Algebras
and Noncommutative Spacet ime).
If we do not demand that F is a cocycle, then the
algebra A
F
is still associative but in the target
category, which means
ða bÞc ¼ ððÞÞ
A;A;A
ðða bÞcÞ
Such objects are called ‘‘quasialgebras.’’ It may still
be that
A, A , A
happens to be trivial (
F
happens to
act trivially) so that A
F
remains associative. This
turns out frequently to be the case and many
quantizations in physics, including C
q
[G] but not
limited to q-examples, can be obtained in this way.
It means that although they are associative there is a
hidden nonassociativity which can surface in other
constructions involving . The physical application
here is with H = U(g) a classical enveloping algebra,
A functions on a classical manifold on which g acts,
and a cochain F. In general the resulting quasialge-
bra will not be associative but rather a quantization
of a ‘‘quasi-Poisson manifold’’ obeying
fa; fb; cgg þ cylic ¼ 2
~
nða b cÞ
Here
~
n is the trivector field for the action of the
lowest order part of
F
and the (quasi)Poisson
bivector is the leading-order part of F
21
F
1
.As
mentioned, there are many cases where
~
n (and the
action of the rest of
F
) happens to be trivial.
Finally, let us give a discrete example using such
quantum group meth ods. We consider H = C(G)
and F 2C(G G) a cochain. Twisting by this gives
H
F
= (C(G),
F
) a quasi-Hopf algebra where
F
ðx; y; zÞ¼
Fðy ; zÞFððx; yzÞ
Fðxy ; zÞFðx; yÞ
We take A = CG the group algebra. The action of
C(G) on it is the diagonal one. The modified algebra
A
F
therefore has product
x y ¼ F
1
ðx; yÞxy
in terms of the product in G, and will be a
quasialgebra if F is not a cocycle. For example, let
698 Hopf Algebras and q-Deformation Quantum Groups
G = (Z
2
)
3
which we write additively (so elements
are 3-vectors with values in Z
2
) and take
Fðx; yÞ¼ð1Þ
P
i<j
x
i
y
j
þy
1
x
2
x
3
þx
1
y
2
x
3
þx
1
x
2
y
3
then,
F
ðx; y; zÞ¼ð1Þ
xðy zÞ
Moreover, A
F
= O, the octonions (Albuquerque and
Majid 1999). So these are a nonassociative quanti-
zation of the classical discrete space (Z
2
)
3
. We see
that they are in fact associative up to sign and with
sign þ1 when the corresponding 3-vectors are
linearly independent.
Noncommutative Geometry
In this article, we have frequently encountered the
view of quantum groups and other noncommutative
algebras as by definition the coordinate algebras on
‘‘noncommutative spaces.’’ However , the ‘‘quantum
groups approach’’ to such noncommutative geome-
try that emerges has a somewhat different flavor
from other approaches, as we discu ss now.
In fact, the problem of geometry at such a level
was mentioned already by Dirac in the 1920s and
led to theorems of Gelfand and Naimark in the
1940s and 1950s whereby a noncommutative
C
-algebra should be viewed as a noncommutative
topological space, and of Serre and Swan in the
1960s whereby a finitely generated projective
module should be viewed as a vector bundle.
Algebraic K-theory led to further refinement of this
picture and particularly, in the 1980s, to A Connes’
formulation in terms of cyclic cohomology and
‘‘spectral triples’’ (see Noncommutative Geometry
and the Standard Model; Noncommutative Tori,
Yang–Mills and String Theory; Quantum Hall
Effect; Hopf Algebra Structure of Renormalizable
Quantum Field Theory; Path Integrals in Noncom-
mutative Geometry). The quantum groups approach
is less axiomatic, and consists of at least three
disparate elements.
The first layer of the quantum groups approach is
the theory of q-deformed groups and q-spaces on
which they act, using braided category methods
(such as braided linear spaces). The braided group
additive law leads to partial derivatives and these
define q-exterior algebras etc. This programme
covered during the 1990s most of what is needed
to q-deform physics in flat space at an algebraic
level. Formulas here tend to be complex but
controlled by R-matrices, and the correct R-matrix
formulas can be found systematically by working
with braided algebra as explained in the section
‘‘Braid ed group s and quan tum planes .’’ Fro m a
slightly different side, q-representation theory and
the further theory of q-homogeneous spaces is
intimately tied to a theory of q-special functions
(such as the q-exponential function in the section
‘‘ q-Defor mation enveloping algebras’’) of inte rest in
their own right (see q-Special Functions). The use of
-algebras in some cases completable to C
-algebras
is a point of contact with other approaches to
noncommutative geometry but problems emerge
when one considers the braiding. As a result, the
natural q-Poincare´ (plus dilation) quantum group is
not even a Hopf -algebra. Briefly, once one starts
to braid the constructions, one may need to
represent them with braided (not usual) Hilbert
spaces and q-analysis.
The second layer of the quantum groups approach
is based on ‘‘differential calculus’’ as a specification
of an exterior algebra of differential forms or
differential graded algebra (DGA). In general this is
a wild problem but, as in classical geometry, the
requirement of a quantum group covariance greatly
narrows the possible calculi, although no longer to
the point of uniqueness. The first examples of
covariant calculi on the quantum group C
q
[SU
2
]
were found by Woron owicz (1989). The bicovariant
one of these was cast in R-matrix form by Jurco
while the first actual classification results on the
moduli of irreducible calculi were obtained by the
author (the bicovariant ones are essentially in
correspondence with irreducible representations V,
with left-invariant differentials forming a braided
group of the form B(V V
)). Probably the most
interesting feature of this theory is that for all C
q
[G]
the bicovariant q-calculus cannot be of classical
dimensions. For example, for C
q
[SU
2
] the smallest
nontrivial calculus is four dimensional. The ‘‘extra
dimension’’ is a biinvariant 1-form which has the
property that [, a] = da for all a 2C
q
[SU
2
] and
which can be viewed as a spontaneously generated
time (see Bicrossproduct Hopf Algebras and Non-
commutative Spacetime). Quantum group methods
also provide DGAs on finite groups, this time
classified in the bicovariant case by nontrivial
conjugacy classes. These therefore provide Lie
structures on finite groups. One can go much further
and define quantum principal bundles (with quan-
tum groups as fiber) over general noncommutative
algebras (Brzezinski and Majid 1993), associated
bundles, frame bundles, and Riemannian geometry
of the algebra (see Quantum Group Differentials,
Bundles and Gauge Theory).
Again q-deformation provides key examples but
the theory may then be applied to other situations.
Hopf Algebras and q-Deformation Quantum Groups 699
For example, the permutation group S
3
has a natural
connected calculus with dimensions 1 : 3 : 4 : 3 : 1
(in other words the space has six points but each
point has the local structure of a 4-manifold in some
sense). It turns out to have a unique Levi-Civita type
connection r for its invariant metric, with constant
curvature. The use of DGAs here is in common with
other approaches (e.g., Connes 1994) and indeed
bundles a ssociated to quantum group principal
bundles and suitable connections can be shown to
be pro jective modules. The approaches diverge at
the level of spectral triples, however, and the
examples of ‘‘Dirac operators’’ that emerge from
quantum group methods do not usually obey the
required axioms.
A third established layer of the quantum groups
approach is to trade some of the noncommutativity
for nonassociativity, as in the dual version of
Drinfeld’s constr uction, that is, C
q
[G] in terms of
classical C[G] as a (co)quasi-Hopf algebra. The
general approach here is a quantization functor T
which provides all constr uctions but which will
typically bring out the underlying nonassociative
geometry even when the noncommutative covariant
algebras of interest is associative. For example,
applying the functor to the classical exterior algebra
(G) gives a bicovariant (C
q
[G]) of classical
dimensions but with nonassociative products (it is
a supercoquasi-Hopf a lgebra). As before, one may
then apply these quantum group methods to other
algebras not related to q-deformation.
Beyond these are many recent developments, some
of which are covered in other articles. Probably one
of the most interesting frontiers, at the time of
writing, is the exploration of links of both quantum
groups and noncommutative geome try to number
theory.
See also: Affine Quantum Groups; Axiomatic Approach
to Topological Quantum Field Theory; Bicrossproduct
Hopf Algebras and Noncommutative Spacetime; Braided
and Modular Tensor Categories; Classical r-Matrices, Lie
Bialgebras, and Poisson Lie Groups; Duality in
Topological Quantum Field Theory; Eight Vertex and
Hard Hexagon Models; Hopf Algebra Structure of
Renormalizable Quantum Field Theory; The Jones
Polynomial; Noncommutative Geometry and the
Standard Model; Noncommutative Tori, Yang–Mills and
String Theory; Path Integrals in Noncommutative
Geometry; q-Special Functions; Quantum Group
Differentials, Bundles and Gauge Theory; Quantum Hall
Effect; Symmetries in Quantum Field Theory of Lower
Spacetime Dimensions; Topological Quantum Field
Theory: Overview; von Neumann Algebras: Subfactor
Theory; Yang–Baxter Equations.
Further Reading
Albuquerque H and Majid S (1999) Quasialgebra structure of the
octonions. Journal of Algebra 220: 188–224.
Baxter RJ (1982) Exactly Solvable Models in Statistical
Mechanics. New York: Academic Press.
Brzezinski T and Majid S (1993) Quantum group gauge theory on
quantum spaces. Communications in Mathematical Physics
157: 591–638.
Connes A (1994) Noncommutative Geometry. San Diego:
Academic Press.
Drinfeld VG (1987) Quantum groups. In: Proceedings of the
ICM. American Mathematical Society.
Drinfeld VG (1990) Quasi-Hopf algebras. Leningrad Mathema-
tical Journal 1: 1419–1457.
Faddeev LD, Reshetikhin NYu, and Takhtajan LA (1990)
Quantization of Lie groups and Lie algebras. Leningrad
Mathematical Journal 1: 193–225.
Jackson FH (1908) On q-functions and a certain difference
operator. Transactions of the Royal Society of Edinburgh 46:
253–281.
Jimbo M (1985) A q-difference analog of U(g) and the Yang–
Baxter equation. Letters in Mathematical Physics 10: 63–69.
Larson RG and Radford DE (1988) Semisimple cosemisimple
Hopf algebras. American Journal of Mathematics 110:
187–195.
Lusztig G (1993) Introduction to Quantum Groups. Boston:
Birkha¨ser.
Majid S (1988) Hopf algebras for physics at the Planck scale.
Journal of Classical and Quantum Gravity 5: 1587–1606.
Majid S (1990) Quasitriangular Hopf algebras and Yang–Baxter
equations. International Journal of Modern Physics A 5: 1–91.
Majid S (1991a) Representations, duals and quantum doubles of
monoidal categories. Supplimento di Rendiconti del Circolo di
Palermo, Series II 26: 197–206.
Majid S (1991b) Examples of braided groups and braided
matrices. Journal of Mathematical Physics 32: 3246–3253.
Majid S (1995) Foundations of Quantum Group Theory.
Cambridge: Cambridge University Press.
Majid S (2002) A Quantum Groups Primer, London Mathema-
tical Society Lecture Notes Series, vol. 291. Cambridge:
Cambridge University Press.
Manin YuI (1988) Quantum groups and non-commutative
geometry. Technical Report, Centre de Recherches Math,
Montreal.
Sweedler ME (1969) Hopf Algebras. New York: Benjamin.
Woronowicz SL (1989) Differential calculus on compact matrix
pseudogroups (quantum groups). Communications in Mathe-
matical Physics 122: 125–170.
700 Hopf Algebras and q-Deformation Quantum Groups
h-Pseudodifferential Operators and Applications
B Helffer, Universite
´
Paris-Sud, Orsay, France
ª 2006 Elsevier Ltd. All rights reserved.
From Classical Mechanics
to Quantum Mechanics
The initial goal of semiclassical mechanics was to
explore the correspondence principle, due to N Bohr
in 1923, which states that one should recover the
classical mechanics from the quantum mechanics as
the Planck constant h tends to zero. So we start with
a very brief presentation of these two theories.
Classical Mechanics
We start (with the Hamiltonian formalism) from a C
1
function p on R
2n
:(x, ) 7!p(x, ), which describes the
motion of the system under consideration and is called
the Hamiltonian. The variable x corresponds, in the
simplest case, to the position and to the momentum of
one particle. The evolution is then described, starting
at time 0 of a given point (y, ), by the so-called
Hamiltonian equations
dx
j
dt
¼ð@p=@
j
Þðxðt Þ;ðtÞÞ; for j ¼ 1; ...; n
d
j
dt
¼ð@p=@x
j
Þðxðt Þ;ðtÞÞ; for j ¼ 1; ...; n
½1
The classical trajectories are then defined as the
integral curves of a vector field defined on R
2n
called
the Hamiltonian vector field associated with p
and defined by H
p
= (@p=@, @p=@x). All these
definitions are more generally relevant in the
framework of symplectic geometry on a symplectic
manifold M (but we choose, for simplicity, to explain
the theory on R
2n
), which can be seen as the cotangent
vector bundle T
R
n
, and is the ‘‘local’’ model of the
general situation. This space is equipped naturally
with a symplectic structure defined by giving at each
point a nondegenerate 2-form, which is here
:=
P
j
d
j
^ dx
j
. This 2-form permits us to associate
canonically to a 1-form on T
R
n
x
avectorfieldon
T
R
n
x
. In this correspondence, if p is a function on
T
R
n
x
, H
p
is associated with the differential dp.
In this article, we consider the example of the
Hamiltonian p(x, ) =
2
þ V(x), also called the
Schro¨ dinger Ham iltonian, as the guiding example.
More specifically, the case of the harmonic oscilla-
tor, where V is given by V(x) =
P
n
j = 1
j
x
2
j
(with
j
> 0), is the most significant, which is the natural
approximation of a potential near its minim um,
when nondegenerate.
In the framework of the classical mechanics, the
main questions could be:
Are the trajectories bounded?
Are there periodic trajectories?
Is one trajectory dense in its energy surface?
Is the energy surface compact?
The solution of these questions could be very difficult.
Let us just mention the trivial fact that, if p
1
()is
compact for some , then, by the conservation of
energy law
pðxðtÞ ; yðtÞÞ ¼ pðy;Þ½2
the whole trajectory starting of one point (y, )
remains in the bounded set {p
1
(p(y, ))} in R
2n
. This
is in particular the case for the harmonic oscillator.
Quantum Mechanics
The quantum theory was born dynamics-wise around
1920.Itisstructurallyrelatedtotheclassical
mechanics in a way that we shall describe very briefly.
In quantum mechanics, our basic object will be a
(possibly nonbounded) self-adjoint operator defined
on a dense subspace of a Hilbert space H. In order to
simplify the presentation, we shall always take
H= L
2
(R
n
).
This operator can be associated with p by using
the techniques of quantization. We choose here to
present a procedure, called the Weyl quantization
procedure (which was already known in 1928),
which under suitable assumptions on p and its
derivatives, will be defined for u 2S(R
n
)by
p
w
ðx; hD
x
; hÞuðxÞ
¼ð2hÞ
n
ZZ
exp
i
h
ðx yÞ
p
x þ y
2
;;h
uðyÞdy d ½3
The operator p
w
(x, hD
x
, h) is called an h-pseudodiffer-
ential operator of Weyl symbol p.Onecanalsowrite
Op
w
h
(p) in order to emphasize that it is the operator
associated to p by the Weyl quantization. Here h is a
parameter which plays the role of the Planck constant.
Of course, one has to give a sense to these integrals
and this is the object of the theory of the oscillatory
integrals. If p = 1, we observe that, by Plancherel’s
formula,
uðxÞ¼ð2hÞ
n
ZZ
exp
i
h
ðx yÞ
uðyÞdy d ½ 4
h-Pseudodifferential Operators and Applications 701
the associated operator is nothing but the identity
operator. A way to rewrite any h-differential operator
P
jjm
a
(x)(hD
x
)
as an h-pseudodifferential opera-
tor is to apply it on both sides to [4]. In particular, we
observe that if the symbol is p(x, ) =
2
þ V(x), then
the operator associated with p by the h-Weyl
quantization is the Schro¨ dinger operator h
2
þ V.
Other interesting examples appear naturally in solid
state physics. Let us, for example, mention the Harper
operator H (or almost-Mathieu; see Helffer and
Sjo¨ strand (1989) and references therein), whose
symbol is the map (x, ) 7! cos þ cos x,andwhich
can also be defined, for u 2 L
2
(R), by
ðHuÞðxÞ¼
1
2
ðuðx þ hÞþuðx hÞÞ þ cos xuðxÞ
We shall later recall how to relate the properties
of p and those of the associated operator. More
precisely, we shall describe under which conditions
on p the operator p
w
(x, hD
x
; h) is semibounded,
symmetric, essentially self-adjoint, compact, with
compact resolvent, trace class, Hilbert–Schmidt
(see Robert (1987) for an extensive presentation).
But before looking at a more general situation, let
us consider the case of the Schro¨ dinger operator
S
h
= h
2
þ V(x). If V is (say, continuous)
bounded from below, S
h
,whichisaprioridefined
on S(R
n
) as a differential operator, admits a unique
self-adjoint extension on L
2
(R
n
). We are first
interested in the nature of the spectrum. If
VðxÞ!þ1 as jxj!1, one can show that S
h
,
more precisely its s elf-adjoint realization, has
compact resolvent and its spectrum consists of a
sequence o f eigenvalues tending to 1. We are next
interested in the asymptotic behavior of these
eigenvalues.
In the case of the harmonic operator, corresponding
to the potential
VðxÞ¼
X
n
j¼1
j
x
2
j
ðwith
j
> 0Þ
the criterion of compact resolvent is satisfied and the
spectrum is described as the set of
ðhÞ¼
X
n
j¼1
ffiffiffiffiffi
j
p
ð2
j
þ 1Þh
for 2 N
n
.
In this case we also have a complete description
of the normalized associa ted eigenfunctions which
are constructed recursively starting from the first
eigenfunction corresponding to
0
(h) =
P
j
ffiffiffiffiffi
j
p
h:
0
ðx; h Þ¼
Y
n
j¼1
ffiffiffiffiffi
j
p
h
1=4
!
exp
1
2h
X
j
ffiffiffiffiffi
j
p
x
2
j
!
½5
The eigenfunction
0
is strictly positive and decays
exponentially. Moreover (and here we enter in the
semiclassical world), the local decay in a fixed closed
set avoiding {0} (which is measured by its L
2
-norm) is
exponentially small as h !0. In particular, this says
that the eigenfunction lives asymptotically in the set
{V(x) (h)}. This last set can also be understood as
the projection by the map (x, ) 7!x of the energy
surface, which is classically attached to the eigenvalue
(h), that is, {(x, ) 2 R
2n
jp(x, ) = (h)}. This is a
typical semiclassical statement, which will be true in
full generality.
From Quantum Mechanics to Classical
Mechanics: Semiclassical Mechanics
Before describing the mathematical tools involved in
the exploration of the corre spondence principle, let
us describe a few results which are typical in the
semiclassical context. They concern Weyl’s asymp-
totics and the localization of the eigenfunctions.
Weyl’s asymptotics We start with the case of the
Schro¨ dinger operator S
h
, but we emphasize that the
h-pseudodifferential techni ques are not limited to
this situation.
We assume that V is a C
1
-function on R
n
which
is semibounded and satisfies
inf V <
lim
jxj!1
VðxÞ
The Weyl theorem (which is a basic theorem in
spectral theory) implies that the essential spectrum is
contained in
lim
jxj!1
VðxÞ; þ1
"#
It is also clear that the spectrum is contained in
[inf V, þ1]. In the interval
I = inf V;
lim
jxj!1
VðxÞ
"#
the spectrum is discrete, that is, it has only isolated
eigenvalues with finite multiplicity. For any E in I,itis
consequently interesting to look at the counting func-
tion N
h
(E) of the eigenvalues contained in [inf V, E],
N
h
ðEÞ¼]f
j
ðhÞ;
j
ðhÞEg½6
The main semiclassical result is then
Theorem 1 With the previous assumptions, we have:
lim
h!0
h
n
N
h
ðEÞ¼ð2Þ
n
Z
VðxÞE
ðE VðxÞÞ
n=2
dx
702 h-Pseudodifferential Operators and Applications
The main term in the expansion of N
h
(E), which
will be denoted by
W
h
ðEÞ :¼ð2hÞ
n
Z
VðxÞE
ðE VðxÞÞ
n=2
dx
is called the Weyl term. It has an analog for the
analysis of the counting function for Laplacians on
compact manifolds (see Quantum Ergodicity and
Mixing of Eigenfunctions and references therein), but
let us emphasize that here E is fixed and that one
looks at the asymptotics as h !0. In the other case, h
is fixed and one looks at the asymptotics as E !þ1
(note that on a compact manifold and for the
Laplacian, the formula N
h
(E) = N
1
(E=h
2
) permits
switching between these cases).
Although this formula is rather old (first as a
folk theorem), many efforts have been made by
mathematicians for analyzing the remainder (see
Robert (1987), Ivrii (1998) and references therein)
N
h
(E) W
h
(E), whose behavior is again related t o
classical analysis. When E is not a critical value of
V, h
nþ1
(N
h
(E) W
h
(E))canbeshowntobe
bounded but it appears to be o(1)ifthemeasure
of the periodic points for the flow is 0 (see Ivrii
(1998)).
Beyond the analysis of the counting function,
one is also interested (e.g., in questions concerning
the ground-state energy of an atom with a large
number of particles, N, satisfying the Pauli exclu-
sion principle (see Stability of Matter)) in other
quantities like the Riesz means, which are defined,
for a given s 0, by
N
s
h
ðEÞ¼
X
j
ðE
j
Þ
s
þ
The case s = 0 corresponds to the counting function.
It is then natural to ask for the asymptotic behavior
as h !0 of these functions.
We have, for example, the following result
(Helffer–Robert, Ivrii–Sigal, and Ivrii; see Robert
(1987) and Ivrii (1998)), which is written here in a
more Hamiltonian version, when E is not a critical
value of V,
N
s
h
ðEÞ¼ð2hÞ
n
Z
p
E
ðx;Þ0
ðp
E
ðx;ÞÞ
s
dx d
!
þOðh
inf ð1þ s;2Þ
Þ
with p
E
(x, ) =
2
þ V(x) E.
Uncertainty principle and Weyl term The Weyl
term can be heuristically understood in the follow-
ing way. According to the uncertainty principle, a
‘‘quantum’’ particle should occupy at least a volume
of order h
n
in the phase space with the measure
dx d (proportional to (
P
n
j = 1
d
j
^ dx
j
)
n
). This
guess is a consequence of the inequal ity
h
2
kuk
2
Z
R
ðx x
0
Þ
2
juj
2
dx
!
1=2
Z
R
h
i
d
dx
0
u
2
dx
!
1=2
; 8u 2SðRÞ
expressing the noncommutation of the operat ors
((h=i)d/dx
0
) and (multiplication by) (x x
0
).
When kuk= 1 and x
0
(mean position) and
0
(mean
momentum) are defined by x
0
:=
R
R
xjuj
2
dx and
0
:= (h=i)
R
R
u
0
(x)
u(x)dx, this inequality expresses
the impossibility for a quantum particle to have a
simultaneous small localization in position and
momentum.
Consequently, the maximal number of ‘‘quantum’’
particles which can live in the region {p
E
(x, ) 0} is
approximately (up to some universal multiplicative
constant) the volume of this region divided by (2h)
n
.
Lieb–Thirring inequalities and Scott’s conjecture In
the case of regular potentials, we have seen that the
quantity h
n
N
s
h
(E) was asymptotically equal as h !0
to L
cl
s, n
(
R
V(x)E
(E V(x))
sþn=2
dx). For other ques-
tions occurring in atomic phy sics (see Stability of
Matter), one is more interested in the exis tence of
universal constants M
s, n
such that
h
n
N
s
h
ðEÞM
s;n
Z
VðxÞE
ðE VðxÞÞ
sþn=2
dx
!
for any V and any h.
The best M
s, n
(which exists if s þ n=2 > 0) is
denoted by L
s, n
(for s = 0; this is called the
Cwickel–Lieb–Rozenblium inequality). The semi-
classical result gives the inequality L
s, n
L
cl
s, n
.
A still open question is the so-called Lieb–Thirring
conjecture: do we have L
1, 3
= L
cl
1, 3
? This is related to
the question of the stability of the matter (see Stability
of Matter). The last results in this direction have been
obtained quite recently by A Laptev and T Weidl,
who show, for example, the equality for s 3=2.
The control, when s = 1, of a second term (for more
singular potentials) for N
s
h
(E) was the object of the
Scott conjecture, which was solved recently in many
important cases by Hughes, Siedentop–Weikard,
Ivrii–Sigal, and Feffermann–Secco (see Ivrii (1998),
Stability of Matter, and references therein).
Localization of the eigenfunctions The localization
property was already observed on the specific case of
the harmonic oscillator. But this was a consequence
h-Pseudodifferential Operators and Applications 703
of an explicit description of the eigenfunctions. This
is quite important to have a good description of the
decay of the eigenfunctions (as h ! 0) outside the
classically permitted region without having to know
an explicit formula. Var ious approaches can be used.
The first one fits very well in the case of the
Schro¨ dinger operator (more generally to h-pseudodiffer-
ential operators with symbols admitting holomorphic
extensions in the variable) and gives exponential
decay. This is based on the so-called Agmon estimates
(developed in the semiclassical context by Helffer–
Sjo¨ strand and Simon). We shall not say more about
this approach, which is the starting point of the analysis
of the tunneling (see Helffer (1988), Dimassi and
Sjo¨ strand (1999),andMartinez (2002)).
The second one is an elementary application
of the h-pseudodifferential formalism which will
be described later and leads , for example, to the
following statement. Let E in I and let ((h
j
),
(h
j
)
(x)) be a sequence of spectral pairs in I
L
2
(R
n
), where h
j
!0asj !þ1, (h
j
) !E, and
x 7!
(h
j
)
(x)isanL
2
-normalized eigenfunction
associated with (h
j
). Let be a relatively compact
set in R
n
such that
V
1
ð1; EÞ \
¼;
Then, there exists, for all integer N, a constant C
N,
such that
k
ðh
j
Þ
k
L
2
ðÞ
C
N;
h
N
j
A third one uses the notion of frequency set and
will be discussed later (see also the book of Martinez
(2002) for what can be done with the Fourier–Bros–
Iagolnitzer transform as developed by J Sjo¨ strand).
Brief Introduction to the
h-Pseudodifferential Calculus
For fixed h, the pseudod ifferential calculus has a long
story starting in its modern form in the 1960s. A
rather achieved version of the calculus is presented in
Ho¨ rmander (1984). We will emphasize here on the
semiclassical aspect of the calculus, that is, on the
dependence of the calculus on the pa rameter h > 0.
h-Pseudodifferential Calculus
Basic calculus: the class S
0
We shall mainly discuss
the most simple one called the S
0
calculus. Let us
first say that the S
0
calculus is sufficient once we
have suitably (micro)-localized the problem (e.g., by
the functional calculus). Note that it is also
sufficient for the local analysis of many problems
occurring on compact manifolds.
This class of symbols p is simply defined by the
conditions:
j@
x
@
pðx;Þj C
;
½7
for all (, ) 2 N
n
N
n
. The symbols can possibly
be h-dependent. With this symbol, one can associate
an h-pseudodifferential operator by [3]. This opera-
tor is a continuous operator on S(R
n
) but can also
be defined by duality on S
0
(R
n
).
The first basic analytical result is the Calderon–
Vaillancourt theorem (see Ho¨ rmander (1984)) estab-
lishing the L
2
-continuity. We also mention that if p
is in L
2
(R
2n
), the associated operator is Hilbert–
Schmidt. One can also give conditions on p implying
the trace-class property (replace the uniform control
in [7] by a control in L
1
).
The second important property is the existence of
a calculus. If a is in S
0
and b is in S
0
then the
composition a
w
(x, hD
x
) b
w
(x, hD
x
) of the two
operators is a pseudodifferential operator associated
with an h-dependent symbol c in S
0
:
a
w
ðx; hD
x
Þb
w
ðx; hD
x
Þ¼c
w
ðx; hD
x
; hÞ
We see here that we immediately meet symbols
admitting expansions in powers of h, which we shall
call regular symbols, in the sense that they admit
expansions of the type
aðx;; hÞ
X
j
a
j
ðx;Þh
j
bðx;; hÞ
X
j
b
j
ðx;Þh
j
In this case the Weyl symbol c of the composition
has a similar expansion:
cðx;; hÞ exp
ih
2
ðD
x
D
D
y
D
Þ
ðaðx;; hÞbðy;; hÞÞ
x¼y; ¼
The symbol a
0
is called the principal symbol. At the
level of principal symbols, the rule is simply that
the principal symbol of a
w
b
w
is the product of
the principal symbols of a
w
and b
w
: c
0
= a
0
b
0
.
Another important property is the following corre-
spondence between commutator of two operators
and Poisson brackets. The principal symbol of the
commutator (1=h)(a
w
b
w
b
w
a
w
)is(1=i){a
0
, b
0
},
where {f , g} is the Poisson bracket of f and g:
ff ; ggðx;Þ¼H
f
g
¼
X
j
@
j
f @
x
j
g @
x
j
f @
j
g
704 h-Pseudodifferential Operators and Applications
About global classes The class S
0
is far from being
sufficient for analyzing the global spectral problem
and we refer the reader to Ho¨ rmander (1984) or
Robert (1987) for an extensive presentation of the
theory and for the discussion of other quantizations.
Our initial operators (think of the harmonic oscilla-
tor) do not belong to these classes of pseudodiffer-
ential operators. We are consequently obliged to
construct more general classes including these
examples in order to realize this localization. Once
such a class is introduced, one of the main points to
consider is the existence of a quasi-inverse (or
parametrix) for a suitably defined elliptic operator
of positive order. Following Beals–Feffermann
(see also the most general Ho¨ rmander calculus
in Ho¨ rmander (1984) and references therein), we
introduce a scale function (possibly h-depen dent ;
typically, m(x, ; h) = h
m
0
(x, )) (x, ) 7!m(x, ; h)
and C
1
strictly positive weight functions and
such that 1. All these functions are
strictly positive and should satisfy additional
conditions on their variation and growth. The
class of symbols S
reg
(m, , )isdefinedby
jD
x
D
pðx;; hÞj C
;
mðx;; hÞðx;Þ
jj
ðx;Þ
jj
These apparently complicated estimates permit
actually the control of the variation of the symbol
in reference balls defined by
2
ðx
0
;
0
Þjx x
0
j
2
þ
2
ðx
0
;
0
Þj
0
j
2
c
Elliptic theory As noted above, the main point is to
have a large class of invertible operators, such that
the inverses are also in the class. This is what we call
an elliptic theory and the typical statement is:
Theorem 2 Let P be an h-pseudodifferential operator
associated with a symbol p in S
reg
(m, , ). We assume
that it is elliptic in the sense that 1/p belongs to
S
reg
(1=m, , ). Then there exists an h-pseudodifferen-
tial operator Q with symbol in S
reg
(1=m, , ), such that
QP ¼ I þ R; PQ ¼ I þ S
The remainders R and S are pseudodifferential
operators with symbols in
\
N
S
h
N
;;
!
These remainders are called ‘‘regularizing.’’ Note
that this notion depends strongly on the choice of the
class of pseudodifferential operators! When ==1,
we are just inverting modulo a remainder whose norm
in L(L
2
)isO(h
1
) (or simply O(h) at the first step).
With other weights like ==
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þjxj
2
þjj
2
q
,we
invert P modulo a remainder, which has, in addition, a
distribution kernel in the Schwartz space S( R
n
R
n
).
The invertibility modulo a compact operator (which
implies the Fredholm property) is a consequence of the
assumption
lim
jxjþjj!þ1
ðx;Þðx;Þ¼þ1
The proof is rather easy, once the formalism of
composition and the notion of principal symbol have
been understood. One can indeed start from the
operator Q
0
of symbol 1=p and observe that Q
0
P = I þ
R
1
holds, with R
1
in Op
w
(S((h=), , )). The
operator (I þ R
1
)
1
Q
0
(
P
j0
(1)
j
R
j
1
)Q
0
gives
essentially the solution.
Essential Self-Adjointness and Semiboundedness
We now sketch two applications of this calculus in
spectral theory. We shall usually consider in our
applications an h-pseudodifferential operator P,
whose Weyl symbol p is regular, that is, admitting
an asymptotic expansion:
ðH0Þ pðx;; hÞ
X
j0
h
j
p
j
ðx;Þ
(We refer to Robert (1987), Ho¨ rmander (1984), and
Dimassi and Sjo¨ strand (1999) for a more precise
formulation). Moreover, we a ssume that
ðH1Þðx;Þ7!pðx;; hÞ2R
This implies, as can be immediately seen from [3],
that p
w
is symmetric (= formally self-adjoint):
hp
w
u; vi
L
2
¼hu; p
w
vi
L
2
; 8u; v 2SðR
n
Þ
The third assumption is that the principal symbol is
bounded from below (and there is no restriction to
assume that it is positive)
ðH2Þ p
0
ðx;Þ0
This assumption implies that the operator itself is
bounded from below. This result belongs to the
family of the so-called Garding inequalities. More
precisely, the assumption (there are other quantiza-
tions, e.g., the anti-Wick quantization, for which
this result becomes trivial, the difference between
the two quantizations being O(h)) will basically
give, if m 1, the existence of a constant C such
that, for any u 2S(R
n
),
hPu; ui
L
2
L
2
Chjjujj
2
h-Pseudodifferential Operators and Applications 705