Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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In this model, the mass is generated by a
topological mechanism since I
CS
possesses the usual
attributes for a topological entity: it is diffeomor-
phisms invariant without a metric tensor; when
the potentials are appropriately parametrized, it is
given by a surface term. (In the abelian case,
the appropriate parametrization is in terms of
Clebsch decomposition, A
= @
þ @
.) Most
importantly, in the nonabelian theory [66] changes
by 8
2
mn with three-dimensional gauge transforma-
tions carrying winding number n.Hence,for
consistency of the nonabelian quantum theory, m
must be quantized as n=4 (in units of h and the
coupling constant, which have been scaled to unity).
All this is a clear field-theoretic analog to the
quantum mechanics of the Dirac monopole, and
just as for the magnetic monopole, a Hamiltonian
argument for quantizing m can be constructed, as an
alternative to the above action-based derivation.
The time component of [64] relates the electric
and magnetic fields to the charge density:
D E mB ¼ ½67
In the abelian case, the first term involves a total
derivative and its spatial integral vanishes, leaving a
formula that identifies magnetic flux with a total
charge. At low energy, the mass term dominates the
conventional kinetic term in [64], and the flux–
charge relation becomes a local field-current
identity,
m
F
J
½68
These formulas have made Chern–Simons-modified
gauge theories relevant to issues in condensed matter
physics, for example, the quantum Hall effect. In the
abelian case, m need not be quantized.
Adding Fermions
Three-dimensional Dirac matrices are minimally rea-
lized by 2 2 Pauli matrices. As a consequence, a mass
term is not parity invariant; also, there is no
5
matrix,
since the product of the three Dirac (= Pauli) matrices
is proportional to I. While there are no chiral
anomalies, there is the so-called parity anomaly:
integrating a single doublet of massless SU(2) fermions
one obtains (A) det[
(i@
þ A
)], which should
preserve parity and gauge invariance.
Since there are no anomalies in current divergences,
(A) is certainly invariant against infinitesimal gauge
transformations. But for finite gauge transformations
(categorized by
3
(SU(2) = Z) one finds that (A)is
not invariant: when the gauge transformation belongs
to an odd-numbered homotopy class, (A)changes
sign. To regain gauge invariance, one must either work
with an even number of fermion doublets or, if only
one doublet (more generally, odd number) is to be
used, one must add to the gauge Lagrangian a parity-
violating Chern–Simons term with half the correctly
quantized coefficient, to neutralize the gauge non-
invariance of (A).
Alternatively, (A) can be regularized in a
gauge-invariant manner. But this requires massive,
Pauli–Villars regulator fields, which produce a parity-
violating expression for (A). One cannot avoid the
parity anomaly.
Adding Bosons
There are a variety of bosonic field models that one
may consider: Abelian or nonabelian; with conven-
tional kinetic term or supplemented by the Chern–
Simons topological mass; or, for low energy, no kinetic
term but only the Chern–Simons term, as in [68].
Abelian charged Bose fields in a Maxwell theory lead
to vortex solitons, based on
1
(U(1)) = Z. These are
just the instantons of the (1 þ 1)-dimensional bosonic
gauge theory discussed previously. With Maxwell
kinematics there are no charged vortices, but these
appear when the Chen–Simons mass is added; see [67].
Pure Chern–Simons kinematics, with no Maxwell
term, can produce completely integrable soliton
equations (Liouville, Toda) when the Bose field
dynamics is appropriately chosen.
Conclusion
Topological effects in field theory are associated with
the infinities and regularization that beset quantum
field theories. These give rise to the chiral anomaly,
parity anomaly (and scale symmetry anomalies, not
discussed here). Yet the anomalies themselves are finite
quantities that have topological significance (Atiyah–
Singer, Chern–Pontryagin, Chern–Simons). This para-
doxical pairing has not been understood. Nor can we
explain why the anomalies interfere in a topological
manner with symmetries associated with masslessness.
Although the range of topological effects in gauge
theory is large, and even larger in non-gauge theories
(sigma models, Skyrme models) the relevance to actual
fundamental physics is confined to the -angle phe-
nomenon, which is analyzed accurately and abstractly
by reference to
3
(G) and to the interplay with
fermions through the chiral anomaly. Instantons are
relevant only to an approximate, semiclassical discus-
sion. Although after much mathematical work, general
instanton configurations are well understood, only the
1-instanton solution enjoys physical significance.
Other topological entities that fascinate are either
nonexistent in fundamental physics or are relevant to
Nonperturbative and Topological Aspects of Gauge Theory 577
condensed matter physics (vortices, Chern–Simons
effects). But here too, we note that the funda-
mental equation of condensed matter physics – the
many-body Schro¨dinger equation – carries no evident
topological structure. Only the phenomenological
equations, which replace the fundamental one, give
rise to topological intricacies.
Acknowledgment
This work is supported in part by funds provided by
the US Department of Energy (DOE) under coop-
erative research agreement DE-FC02-94ER40818.
See also: Abelian and Nonabelian Gauge Theories Using
Differential Forms; Abelian Higgs Vortices; Anomalies;
BF Theories; Gauge Theories from Strings; Gauge
Theory: Mathematical Applications; Quantum Field
Theory: A Brief Introduction; Seiberg–Witten Theory.
Further Reading
Adler SL (1970) Perturbation theory anomalies. In: Deser S,
Grisaru M, and Pendleton H (eds.) Lectures on Elementary
Particles and Quantum Field Theory, vol. 1, pp. 3–164.
Cambridge: MIT Press.
Adler SL (2005) Anomalies to all orders (ArXiv: hep-th/0405040).
In: ‘t Hooft G (ed.) Fifty Years of Yang–Mills Theory.
Singapore: Word Scientific.
Bertlmann RA (1996) Anomalies in Quantum Field Theory.
Oxford: Clarendon.
Coleman S (1985) Classical lumps and their quantum descendants
and the uses of instantons. In: Coleman S Aspects of
Symmetry, pp. 185–350. Cambridge: Cambridge University
Press.
Fujikawa K and Suzuki H (2004) Path Integrals and Quantum
Anomalies. Oxford: Oxford University Press.
Jackiw R (1977) Quantum meaning of classical field theory.
Reviews of Modern Physics 49: 681–706.
Jackiw R (1979) Introduction to the Yang–Mills quantum theory.
Reviews of Modern Physics 52: 661–673.
Jackiw R (1985) Field theoretic investigations in current algebra
and topological investigations in quantum gauge theories. In:
Treiman S, Jackiw R, Zumino B, and Witten E (eds.) Current
Algebra and Anomalies, pp. 81–359. Princeton: Princeton
University Press; Singapore: World Scientific.
Jackiw R (1995) Diverse Topics in Theoretical and Mathematical
Physics. Singapore: World Scientific.
Jackiw R (2005) Fifty years of Yang–Mills theory and our
moments of triumph (ArXiv: physics/0403109). In: ‘t Hooft G
(ed.) Fifty Years of Yang–Mills Theory. Singapore: World
Scientific.
Jackiw R and Pi S-Y (1992) Chern–Simons solitons. Progress of
Theoretical Physics 107: 1–40.
Rajaraman R (1982) Solitons and Instantons. Amsterdam:
North-Holland.
Shifman M (1994) Instantons in Gauge Theories. Singapore:
World Scientific.
‘t Hooft G (1976) Symmetry breaking through Bell–Jackiw
anomalies. Physical Review Letters 37: 8–11.
Weinberg S (1996) The Quantum Theory of Fields. vol. II, chs. 22
and 23. Cambridge: Cambridge University Press.
Normal Forms and Semiclassical Approximation
D Bambusi, Universita
´
di Milano, Milan, Italy
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Quantum mechanics was born at the beginning of the
twentieth century with the quantization rules for the
harmonic oscillator and for the hydrogen atom. Such
rules were almost immediately extended to more
general systems by the so-called Bohr–Sommerfeld
quantization rule: ‘‘the actions of the classical system
can assume only those values which are integer
multiples of h.’’ However, the actions are defined
only in some special situations and, moreover, at the
present time the Schro¨ dinger equation is the paradigm
of quantum mechanics. A question naturally arises: is
there any relation between the eigenvalues of the
Schro¨ dinger operator and the numbers obtained by
Bohr–Sommerfeld quantization rule (when available)?
According to common wisdom, the ‘ ‘Bohr–
Sommerfeld numbers’’ are a first approximation to the
eigenvalues of the Schro¨dinger operator in the so-called
semiclassical limit. However, precise mathematical
results on the subject were obtained only in the 1980s
and a good understanding of the problem has been
achieved only recently. In particular it is now clear how
to compute higher-order corrections to the eigenvalues:
this is done through suitable normal form procedures.
In the present article we will discuss the above
questions for the case of perturbed harmonic
oscillators, a case which, on the one hand, is
physically relevant and, on the other, is well under-
stood. We will only briefly discuss the quantization
of perturbations of integrable systems.
A Statement
On L
2
(R
n
), consider the Schro¨ dinger operator
^
H ¼
h
2
2
þ V ½1
where is the n-dimensional Laplacian and V is a
smooth real potential having an absolute nonde-
generate minimum at the origin. We are interested in
578 Normal Forms and Semiclassical Approximation
the eigenvalues of [1] close to zero. Introduce
coordinates adapted to the normal modes, namely
such that
VðxÞ¼
X
n
i¼1
!
2
i
x
2
i
2
þ Oðkxk
3
Þ
Assume
(H1) Nonresonance: There exist >0 and 2 R
such that, for any k 2 Z
n
{0} one has
j! kj
jkj
½2
(H2) V(x) > 0 for x 6¼ 0, and
lim inf
jxj!1
VðxÞ > 0
(H3) V 2 C
1
(R
n
) and for any r 0 there exists C
r
such that
@
jj
V
@x
ðxÞ
C
jj
hxi
m
; 8 2 N
n
where we used the notation hxi:= ð1 þkxk
2
Þ
1=2
.
Theorem 1 Assume that (H1)–(H3) hold. Then, for
any positive N, M there exist positive constants
h
N , M
,
N, M
, C
1
N , M
, C
2
N, M
, and a smooth function
Z
N ;M
ðI
1
; ...; I
n
;hÞ
such that, 80 <
N, M
and 0 < h h
N, M
, the
eigenvalues of [1] in [0, ) have the representation
k
¼ k þ
1
2
!h þZ
N ; M
k þ
1
2
h;h
þ R
NM
ðk; hÞ; k 2 N
n
; k
j
1 ½3
where
jR
NM
ðk; hÞj C
1
N; M
N
þ C
2
N; M
h
M
More precisely, for any k 2 N
n
such that
k þ
1
2
!h þZ
N; M
k þ
1
2
h;h
2½0;Þ½4
there exists an eigenvalue
k
2 [0, ) for which [3]
holds, and vice versa, for any eigenvalue in [0, )
there exists a k satisfying [3] and [4]. The function
Z
N , M
(I
1
, ..., I
n
;0) coincides with the classical
Birkhoff normal form of the system computed up
to order N.
The proof of the theorem is constructive, in the
sense that it provides an algorithm allowing to
construct explicitly, by elementary operations, the
function Z
N, M
. One could choose = (h) =h
with
some positive <1, obtaining a simpler statement
valid for the eigenvalues in [0, h
). It is also possible
to weaken the nonresonance condition (H1) to the
condition ! k 6¼ 0 for k 2 Z
n
{0}.
A theorem very close to [1] was proved by
Sjo¨ strand (1992) by a method different from the
one that will be presented here (see also Graffi and
Paul (1987)). In the analytic or Gevrey case (recall
that a C
1
function f(x) is Gevrey in some domain if
there exist constants C, such that, for all multi-
indexes 2 N
n
one has
@
jj
f
@x
C
jj
ð!Þ
in the whole domain), the error can be reduced to be
exponentially small with the parameters (Bambusi
et al. 1999). Previous results dealing with compact
perturbations of the harmonic oscillator were
obtained by Bellissard and Vittot (1990).Itis
possible to deal also with the resonant case in
which (H1) is violated. In this case the spectrum of
the complete system is qualitatively different from
the spectrum of the harmonic one. As discussed
later, the normal form allows one to compute the
main qualitative differences.
Birkhoff Normal Form
In this section we recall the procedure leading to
classical Birkhoff normal form, whose quantization
leads to the proof of Theorem 5.
Birkhoff’s Theorem
The operator [1] is the quantization of the classical
Hamiltonian
X
n
i¼1
2
i
2
þ VðxÞ½5
Denote
H
0
ð; xÞ :¼
X
n
j¼1
!
j
I
j
; I
j
:¼
2
j
þ !
2
j
x
2
j
2!
j
½6
then we have
Theorem 2 For any positive inte ger N 2 there
exist a neighborhood U
N
of the origin and a
canonical transformation T
N
: R
2n
U
N
! R
2n
which puts the system [5] in Birkhoff normal form
up to order N, namely such that
H T
N
¼ H
0
þ Z
N
þ R
N
½7
Normal Forms and Semiclassical Approximation 579
where Z
N
Poisson-commutes with H
0
, namely
{H
0
; Z
N
} 0 and R
N
is small, that is,
jR
N
ð; xÞj C
N
kð; xÞk
N þ1
½8
Moreover, if the frequencies are nonresonant, namely
! k 6¼ 0; 8k 2 Z
n
nf0g½9
the function Z
N
depends on the actions I
j
only.We
recall that the Poisson bracket of tw o functions f
and g is defined by
ff ; gg :¼
X
n
j¼1
@f
@
j
@g
@x
j
@f
@x
j
@g
@
j
¼fg; f g
and coincides with the Lie derivative of g with
respect to the Hamiltonian vect or field of f.
Remark 1 In the case where the frequencies fulfill
(H1) and the potential V is analytic (or of Gevrey
class) the remainder can be reduced to be exponen-
tially small with k(, x)k.
Scheme of the Proof
Make the rescaling =
0
, x = x
0
. In terms of the
primed variables, the Hamiltonian of the system [5]
takes the form
H
ð
0
; x
0
Þ¼H
0
ð
0
; x
0
ÞþWðx
0
Þ½10
with
Wðx
0
Þ :¼
Vðx
0
Þ
2
P
n
j¼1
!
2
j
ðx
0
j
Þ
2
=2
3
¼ W
3
ðx
0
ÞþW
4
ðx
0
Þþ ½11
and W
l
is the Taylor polynomial of order l of V.In
what follows we will omit primes from the scaled
variables.
Given an auxiliary Hamiltonian
3
, denote by
3
t
the flow of the corresponding Hamiltonian vector
field. We construct
3
so that H
3
is in normal
form up to order
2
.
Remark 2 Given a C
1
function g one has g
3
P
1
l = 0
l
g
l
, with
g
0
:¼ g ; g
l
¼
1
l
f
3
; g
l1
g; l 1 ½12
where denotes the fact that the left-hand side is
asymptotic to the right-hand side (a precise defini-
tion appears later in the article). If both g and
3
are
analytic then the series of g
3
can be shown to
converge in a neighborhood of the origin. Using [12]
to compute H
3
, we get
H
3
¼ H
0
þ ½ W
3
þf
3
; H
0
g þ Oð
2
Þ
So H
3
is in nor mal form up to O(
2
) provided
3
fulfills the so-called homological equation:
W
3
þf
3
; H
0
g¼Z
3
½13
where the unknown function Z
3
has to be in normal
form. Note that, since the operator
7!f; H
0
g
maps linearly polynomials of degree l into poly-
nomials of degree l, eqn [13] can be interpreted
as a linear equation in the finite-dimensional space
of polynomials of degree 3 in the phase-space
variables.
Lemma 1 The homological equation [13] admits a
solution (
3
, Z
3
).
Proof Introduce the canonical coordinates (, )by
j
:¼
1
ffiffiffi
2
p
j
ffiffiffiffiffi
!
j
p
þ ix
j
ffiffiffiffiffi
!
j
p
!
j
:¼
1
i
ffiffiffi
2
p
j
ffiffiffiffiffi
!
j
p
ix
j
ffiffiffiffiffi
!
j
p
!
½14
In these variables the unperturbed Hamiltonian H
0
reads H
0
=
P
j1
i!
j
j
j
and W
3
is transformed in a
different polynomial, again of third order.
The important fact is that in these coordinates the
eigenvectors of the linear operator {H
0
; .} are the
monomials
k
l
k
1
1
k
n
n
l
1
1
l
n
n
Indeed, one has {H
0
;
k
l
} = i! (k l)
k
l
.Asa
consequence, writing
W
3
ð;Þ¼
X
k; l
C
k; l
k
l
one can define the resonant set
R :¼fðk; lÞ: ! ðk lÞ¼0g
and
Z
3
ð;Þ :¼
X
k; l2R
C
k;l
k
l
3
ð;Þ :¼
X
k; l62R
C
k;l
i! ðk lÞ
k
l
½15
Going back to the original variables, one has the
solution of the homological equation.
&
Definition 1 The function Z
3
solving [13] will be
called the resonant part of W
3
and will be denoted
by hW
3
i.
580 Normal Forms and Semiclassical Approximation
Using the fun ction
3
, one can transform the
Hamiltonian to the form
H
0
þ Z
3
þ
2
R
3
Remark 3 Equa tion [12] allows to construct
directly the Taylor expansion of R
3
in terms of the
Taylor expansion of W and of its Poiss on brackets
with
3
.
Iterating the construction (which however slightly
changes due to the presence of Z
3
), one gets the
proof of Theorem 2.
Remark 4 In the nonresonant case ! (k l) = 0
implies that k = l; therefore, the resonant part of a
polynomial is the sum of monomials of the form
k
k
¼ I
k
1
1
I
k
n
n
that is, it is a function of the actions only. Moreover,
in this case one has Z
3
= 0, while in general Z
4
6¼ 0.
Some Symbolic Calculus
To understand how to quantize the procedure of
Birkhoff normal form, we consider the classical–
quantum correspondence. It is well known that
there are different procedures in order to associate
an operator with a classical observable. Here we
concentrate on the Weyl quantization rule.
To a function f 2S(R
2n
) (Schwartz class), we
associate an operator
ˆ
f acting on functions 2
S(R
n
), which is defined by
½
^
f ðxÞ :¼
1
ð2hÞ
n
Z
R
n
R
n
f
x þ y
2
;
e
iðxyÞ
h
ðyÞdy d ½16
Definition 2 The operator [16] is called the Weyl
quantization of f and in turn f is called the symbol of
ˆ
f.
Using the method of oscillatory integrals, the
Weyl quantization rule can be extended to much
more general observables f. We recall that, roughly
speaking, the method of oscillatory integrals consists
in giving meaning to a formal expression of the form
[16] by using successive integration by parts (see,
e.g., Martinez (2001)).
Definition 3 A function f 2 C
1
(R
2n
) will be called
a smooth symbol of class S( hzi
m
) if, for any r 0,
there exists C
r
such that
@
jj
f
@z
ðzÞ
C
jj
hzi
m
; 8 2 N
2n
Where hzi is as defined earlier.
It is useful to extend such a definition to functions
explicitly depending also on h. This can be done in a
straightforward way by asking the constants C
r
to
be independent of h in a neighborhood of the origin.
Different classes of symbols can also be defined, but
for our purpose this class is enough.
Theorem 3 Let f 2 S(hzi
m
), m 2 R, and 2S(R
n
);
then the formal expression [16] is a well-defined
oscillatory integral.
Example 1 Under Weyl quantization rule, one has
^
j
¼ ih@
x
j
;
^
x
j
¼ x
j
ðmultiplication operatorÞ
d
j
x
j
¼
1
2
ð
^
j
^
x
j
þ
^
x
j
^
j
Þ
Definition 4 A sequence (f
j
)
j0
with f
j
2 S(hzi
m
)
will be called the asymptotic expansion of f 2
S(hzi
m
) if, for any integer N , there exist two positive
constants C
N
, h
N
such that
f ¼
X
N
j¼0
h
j
f
j
þ R
N
with jR
N
(z, h)jC
N
h
N þ1
hzi
m
, and h 2 (0, h
N
).
The key point for the quantization of the normal
form procedure is the following.
Theorem 4 Let f 2 S(hzi
m
1
) and g 2 S(hzi
m
2
); then
there exists a unique F 2 S(hzi
m
1
þm
2
) such that
^
F ¼
^
f
^
g ðoperator product!Þ
moreover, one has
F ¼exp
ih
2
ð@
x
@
@
y
@
Þ
ðf ðx;Þgðy;ÞÞj
y¼x;¼
½17
Finally, F admits an asymptotic expansion in h
which coincides with the formal expansion of [17].
The proof is obtained by using eqn [16] to
write down an expression for
ˆ
f
ˆ
g and obtain a
formula for F. Then, one shows that the formula
is well defined and therefore the result is not
formal.
Definition 5 In the above context, the symbol G of
i
h
^
f ;
^
g
hi
¼:
^
G
will be called the ‘‘Moyal bracket’’ of f and g and
will be denoted by {f ; g}
M
.
By formula [17], one has in particular
ff ; gg
M
¼ff ; ggþh
2
1
ðf ; gÞþOðh
4
Þ½18
Normal Forms and Semiclassical Approximation 581
where
1
ðf ; gÞ¼
1
24
@
3
f
@
3
@
3
g
@x
3
3
@
3
f
@
2
@x
@
3
g
@x
2
@
þ3
@
3
f
@@x
2
@
3
g
@x@
2
@
3
f
@x
3
@
3
g
@
3
where we used a vector notation for the derivatives.
If either f or g are polynomials of degree 2, then
ff ; gg
M
¼ff ; gg½19
Given a self-adjoint operator A and a smooth
function G : R ! R, it is well known how to
define by spectral theorem the operator G( A).
Suppose now that A =
ˆ
f for some symbol f.In
general, one has G(
ˆ
f) 6¼
d
G f .However,bysym-
bolic calculus (i.e., using e qn [17]), one has:
Lemma 2 Denote I
j
(x, ) = (!
2
j
x
2
j
þ
2
j
)=2!
j
. Then,
for any positive integer k there exis ts a function
F
k
(I
j
, h) such that
d
ðI
j
Þ
k
¼ F
k
ð
^
I
j
; hÞ
where the right-hand side is defined by spectral
calculus. Moreover, F
k
can be computed explicitly
by the recursion formula F
kþ1
= I
j
F
k
þ
F
k1
h
2
(k
2
k þ 1)=4.
As a consequenc e of this fact and of the fact that
[
ˆ
I
j
,
ˆ
I
l
] = 0, one has that the Weyl quantization of a
polynomial function of the actions is a function of
the action operators.
Semiclassical Normal Form
Let be a smooth symbol such that
ˆ
is self-adjoint,
and consider the group of unitary operators
X
: = exp ((i=h)
ˆ
). Let g be a smooth symbol;
apply the unitary transformation X
to
ˆ
g,namely
compute X
ˆ
gX
1
. Noting that (on a suitable domain)
d
d
ðX
^
gX
1
Þ¼X
i
h
½
^
;
^
gX
1
one has (formally!) the expansion of X
ˆ
gX
1
in :
X
^
gX
1
¼
X
l0
l
d
g
q; l
where
d
g
q; 0
:¼
^
g;
d
g
q; l
¼
1
l
i
h
½
^
;
d
g
q; l1
; l 1 ½20
(Such a series can be interpreted as an asymptotic
expansion provided one restricts the domain at each
step of the approximation.) Equivalently, the symbol
of X
ˆ
gX
1
is formally given by
P
l
l
g
q, l
with
g
q; 0
:¼ g; g
q; l
:¼
1
l
f; g
q; l1
g
M
; l 1 ½21
from which one sees a remarkable similitude with
the classical equation. Moreover, [21] converges to
[12] when h ! 0.
Applying the unita ry transformation generated by
ˆ
to the Hamiltonian operator
ˆ
H
(cf. eqn [10]), one
has X
ˆ
H
X
1
=
c
H
1
q
with
H
1
q
¼ H
0
þ ½W
3
þf; H
0
g
M
þOð
2
Þ½22
H
0
þ ½W
3
þf; H
0
g þ Oð
2
Þ½23
where we used the fact that H
0
is a quadratic
polynomial, so that [19] holds.Itisthusclearthat
Lemma 1 allows to solve also the quantum homo-
logical equation appearing in this context and to
determine the symbol of the operator generating the
unitary transformation putting the Hamiltonian opera-
tor in normal form up to corrections of order
2
.
Moreover, one can compute in terms of Moyal
brackets (of polynomials!) the expansion of the symbol
of the new remainder and of the normal form. Iterating
the construction, one generates a well-defined semi-
classical normal form of the quantum system.
Example 2 Denote by Z
q, l
, l = 1, 2 ..., the term
added to the semiclassical normal form at the lth
step of the iterative constr uction. Explicitly, the first
terms are given by
Z
q;1
¼ W
3
hi
¼ Z
3
½24
Z
q; 2
¼ W
4
hi
þ
1
2
f
3
; W
3
g
M
þ
1
2
f
3
; Z
3
g
M
½25
Z
q; 3
¼ W
5
hi
þf
4
; Z
3
g
M
þ
1
3
f
3
; H
2
g
M
þ
1
2
f
3
; W
3;1
g
M
þf
3
; W
4
g
M
½26
where, according to Definition 1, h. i is the resonant
part of its argument,
j
is (formally) the symbol of
the operator generating the jth unitary transforma-
tion, and
H
2
:¼
1
2
f
3
; Z
3
W
3
g
M
; W
3;1
:¼f
3
; W
3
g
M
Note that all the Moyal brackets involved contain
polynomials of degree at most 4, so that they can be
computed exactly using formula [18] which in this
case does not contain corrections of order h
4
.
The problem in making previous construction
rigorous is that all the series involved are in general
582 Normal Forms and Semiclassical Approximation
divergent. Mo reover, it is not possible to show that
the remainders appearing when truncating such
series are small in a reasonable sense. Nevertheless,
it is possible, using the tools of microlocal analysis,
to show that the semiclassical normal form contains
essentially all the information on the part of the
spectrum close to zero.
The precise relation between the spectrum of
the original Hamiltonian and the spectrum of the
semiclassical normal form is captured by the
following definition.
Let H
1
(, h), H
2
(, h) be two families of self-adjoint
operators; set Spec
(H
1, 2
):= Spec(H
1, 2
) \ [0, ).
Definition 6 We say that
Spec
ðH
1
Þ¼Spec
ðH
2
Þmodð
1
þðh=Þ
1
Þ
if for any N, M > 0 there exist C
1
N, M
and C
2
N, M
such
that for any
1
2 Spec
(H
1
) there exists
2
2
Spec
(H
2
) such that
1
=
2
þ R
N, M
with
jR
N
jC
1
N;M
N
þ C
2
N ;M
ðh=Þ
M
½27
and conversely. Equation [27] has to hold for any
couple (h, ) with and (h=) small enough.
Theorem 5 Assume (H2) and (H3); assume also:
(H1
0
) There exist >0 and 2 R such that, for any
k 2 Z
n
, one has
either ! k ¼ 0 or j! kj
jkj
½28
Then there exists a polynomial function Z
q
such
that one has
Spec
ð
^
HÞ
¼ Spec
ð
^
H
0
þ
c
Z
q
Þmod
1
þ
h
1
½29
The polynomial Z
q
coincides with the semiclassical
normal form defined at the beginning of the
section.
Scheme of the proof It consists of six steps.
(1) Make the unitary transformation (U )(x):=
n=4
(
1=2
x) which transforms the Hamiltonian
operator [1] into the Weyl quantized of
H
:= (H
0
þ
1=2
W), but a Weyl quantization
where h is substituted by h
0
:=h=. (2) Mak e a cutoff
of H
, namely, fix R and consider a smooth function
t such that t(s ) 1 for jsjR, t(s) 0 for jsj2R,
define a(x, ):= W(x)t(k(, x)k). (3) Compare the
spectrum of the Hamiltonian
ˆ
H
with the spectrum
of H
t
:= H
0
þ a. By microlocal analysis, one has
that, in any fixed bounded interval such spectra
coincide modulo h
1
(see, e.g., Martinez (2001)).
(4) Rescale back the variables, namely apply the
transformation U
1
to H
t
. (5) Apply the normal
form algorithm to the so-obtained Hamiltonian
showing that all the series involved are convergent
in suitable norms. (6) Use again microlocal analysis to
show that the spectrum of the semiclassical normal
form coincides with the spectrum of the normalized
operator with compactly supported symbol.
&
Remark 5 Fix an arbitrary 1 >>0 and link to
h by :=h
. Then one obtai ns a simplified statement
according to which the spectrum of [1] in [0, h
]
coincides modulo h
1
with the spectrum of
ˆ
H
0
þ
b
Z
q
in the same interval.
Remark 6 In the case where the frequencies are
nonresonant one has that the symbol of the normal
form depends on the actions only. By Lemma 2 one
has that also the qua ntization of the normal form is
a function of the action operators only (explicitly
computable), and therefore the spectrum of the
normal form is given by a quantization formula as
claimed in Theorem 1.
The Resonant Case
In the case where the frequencies are nonresonant,
due to the particular structure of the normal form,
one obtains a very precise infor mation on the
spectrum. In the case where there are some
resonances, the situation is more difficult. In order
to illustrate what happens we concentrate on the
completely resonant case, that is, the case where all
the frequencies are integer multiples of a single
fundamental frequency .
In this case, the eigenvalues of
ˆ
H
0
form a subset of
Nh þ 1=2ðÞj!jh and are degenerate. One expects the
nonlinear part to break such a degeneracy and to
transform each eigenvalue in a small band. One can use
the normal form to study the structure of the so-
obtained band. To this end, the most relevant contribu-
tion is due to the first nonvanishing term of the normal
form. For the sake of definiteness, we assume that this is
the term of order 4, namely Z
4
.Denote
N :¼Z
4
H
1
0
ð1Þ
; BðEÞ E
1
3
h; E þ
1
3
h
Theorem 6 Fix 1 >
1
> 1=2, then, provided his
small enough, one has
Specð
^
HÞ\ðh; h
1
Þ
[
E2Specð
^
H
0
Þ
BðEÞ½30
Moreover, denote by
E þ
1
ðE; hÞE þ
m
ðE; hÞ½31
Normal Forms and Semiclassical Approximation 583
the eigenvalues of
ˆ
HinB(E) counted with multi-
plicity, then
1
ðE; hÞ¼E
2
Min N þ E
2
ðOðh=EÞþOðE
1=2
ÞÞ ½32
and similarly
E
2
m
ðE; hÞ¼Max N þE
2
ðOðh=EÞþOðE
1=2
ÞÞ ½33
This statement is due to Bambusi, Charles, and
Tagliaferro (see Bambusi 2004); for previous results,
see Vu
˜
Ng
o
_
c (1998).
Equation [30] shows that the spectrum has a band
structure, while eqns [32] and [33] allow one to
compute the minimum and the maximum of each band.
The idea of the proof is as follows. First forget high-
order terms of the normal form, whose effect is included
in the error terms. Then, due to the commutation
property of the normal form with
ˆ
H
0
, one has that Z
4
restricts to an operator acting on the eigenspaces of
ˆ
H
0
.
On the classical side, one has that by Marsden–
Weinstein procedure Z
4
defines a classical Hamiltonian
system on the manifold obtained by symplectic reduc-
tion of the original phase space. By the methods of
geometric quantization, it turns out that the quantum
operator acting on an eigenspace of
ˆ
H
0
is a Toeplitz
operator whose principal symbol is exactly the above
reduced classical Hamiltonian. Then, the proof follows
by classical properties of Toeplitz operators.
We point out that results of this kind are useful in
the computation of the molecular spectra (Michel
and Zhilinskii 2001, Zhilinskii 2001 ).
Quantization of KAM Tori
In this section we present a result on the quantiza-
tion of KAM tori. It allows one to construct part of
the spectrum of a close-to-integrable system.
We recall that a classical Hamiltonian system with n
degrees of freedom is said to be integrable if it has n
integrals of motion independent and in involution. If the
energy surface is compact, then, by Arnol’d–Liouville
theorem there exists a canonical transformation T
0
:
R
n
T
n
D T
n
! R
2n
introducing action-angle
variables, namely such that, denoting by K
0
the original
integrable Hamiltonian, K
0
T
0
is independent of the
angles 2 T
n
.Here,D is an open bounded domain.
Consider now a close-to-integrable analytic
Hamiltonian system, namely a Hamiltonia n system
with Hamiltonian
K ¼ K
0
þ K
1
where is a small parameter. We assume that,
denoting again by T
0
the canonical transformation
introducing action-angle variables for the system K
0
,
one has that both K
0
T
0
and K
1
T
0
are real
analytic on D T
n
. Then, the KAM theory applies.
To state the corresponding result, denote by D
0
D
a domain whose closure is contained in D.
Theorem 7 Assume that 8I 2 D one has
det
@
2
ðK
0
T
0
Þ
@I
2
6¼ 0 ½34
then there exists a positive constant
and, for any
with j j <
, there exis ts a Gevrey canonical
transformation T
: D
0
T
n
! R
2n
and a Cantor
set D
D
0
with the following properties:
K T
¼ ZðIÞþRðI;;Þ½35
where R(I, , ) vanishes at infinite order on D
, that is,
for any multi-index there exists C
jj
such that one has
@
jj
R
@ðI;Þ
ðI;;Þ
C
jj
exp
c
jI D
j
½36
with a suitable >0 and jI D
j denoting the
distance from D
. Moreover, as tends to zero, the
measure of D
tends to the measure of D
0
.
A particular consequence is that the set D
is
foliated in invariant tori. From the proof, it also
turns out that the motion on each torus is
quasiperiodic with frequencies fulfilling the assump-
tion (H1) stated earlier. Moreover, the tori are
linearly stable and even more : they are stable in an
exponential sense (namely, a solution starting O()
close to a torus takes at least a time O( exp (c=
)) to
double its distance from the torus).
Quantizing the normalizing transformation T
by
using the theory of Fourier integral operators, one
can also put the quantum Hamiltonian in a suitab le
normal form which allows to deduce some spectral
information on the system.
To fix ideas we restrict to the case where K is a
natural system, namely it has the form (3.1), and is
close to integrable in the above sense. Fix two
parameters E
1
< E
2
; assume (1) that K
1
([1, E
2
þ
]) is compact for some positive and (2) that the
domain D
0
can be constructed in such a way that
T
0
: D
0
T
n
! K
1
0
([E
0
, E
1
]) is a bijection and,
moreover, the KAM condition [34] holds. Denote
by 2 Z
n
the Maslov class of the tori of K
0
(see,
e.g., Lazutkin (1993)) and, having fixed some 0 <
<1, define the set of indexes
I :¼ k 2 Z
n
: jD
hðk þ =4Þj h
fg
½37
Theorem 8 There exist positive constants h
, c, C,
and <1, and a function K
q
: D
0
(0, h
) ! R
with the following property: for any k 2I there
exists at least one eigenvalue of
ˆ
K in the interval
584 Normal Forms and Semiclassical Approximation
Z
q
ðhðkþ=4Þ; hÞCe
c=h
;
h
Z
q
ðhðkþ=4Þ; hÞþCe
c=h
i
½38
One can also show that a large part of the
spectrum is constructed in this way. This is obtained
by compar ing the semiclassical estimate of the
number of eigenval ues in [E
1
, E
2
] to the number of
eigenvalues thus constructed.
Theorem 8 is due to Popov (2000); the quantiza-
tion of KAM tori was initiated by Lazutkin and
widely developed by Colin de Verdie` re, who obtained
a result similar to Theorem 8 for the case where K is
C
1
and describes the geodesic flow on a compact
Riemannian manifold (Colin de Verdie` re 1977).
See also: Central Manifolds, Normal Forms;
h-Pseudodifferential Operators and Applications; Optical
Caustics; Quantum Mechanics: Foundations;
Schro
¨
dinger Operators; Stationary Phase Approximation.
Further Reading
Bambusi D (2004) Semiclassical normal forms. In: Multiscale
Methods in Quantum Mechanics, pp. 23–39. Trends in
Mathematics. Boston, MA: Birkha¨user.
Bambusi D, Graffi S, and Paul T (1999) Normal forms and
quantization formulae. Communications in Mathematical
Physics 207: 173–195.
Bellissard J and Vittot M (1990) Heisenberg’s picture and
noncommutative geometry of the semiclassical limit in
quantum mechanics. Annales de l’Institut Henri Poincare
´
.
Physique The
´
orique 52: 175–235.
Colin de Verdie` re Y (1977) Quasi Modes sur les varie´te´s
Riemanniennes. Inventiones Mathematichaes 3: 15–52.
Graffi S and Paul T (1987) The Schro¨ dinger equation and
canonical perturbation theory. Communications in Mathema-
tical Physics 108: 25–40.
Lazutkin V (1993) KAM Theory and Semiclassical Approxima-
tions to Eigenfunctions. Berlin: Springer.
Martinez A (2001) An Introduction to Semiclassical and Micro-
local Analysis. New York: Springer.
Michel L and Zhilinskii BI (2001) Symmetry, invariants, and
topology. I–III. Physics Reports 341: 11–84, 175–264.
Popov G (2000) Invariant tori effective stability and quasimodes
with exponentially small error terms I–II. Annales Henri
Poincare´ 1: 223–248, 249–279.
Robert D (1987) Autour de l’approximation semiclassique. Basel:
Birkha¨user.
Sjo¨ strand J (1992) Semi-excited levels in non-degenerate potential
wells. Asymptotic Analysis 6: 29–43.
Vu
˜
Ng
o
_
c S (1998) Sur le spectre des syste`mes comple`tement
inte´grables semi-classiques avec singularite´s. Ph.D. thesis,
Institut Fourier.
Zhilinskii BI (200 1) Symmetry, invariants, and topology. II:
Physics Reports 341: 85–171.
N-Particle Quantum Scattering
D R Yafaev, Universite
´
de Rennes, Rennes, France
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The present article relies heavily on Quantum
Mechanical Scattering Theory in this Encyclopedia
and can be considered as its continuation. We use
here freely the notation and results discussed in this
article.
An important problem of scattering theory con-
cerns the Schro¨ dinger H operator of N, N 3,
interacting particles. Since the potential energy of
pair interactions between particles depends on their
relative positions only, it does not tend to zero at
infinity in the configuration space of a system, even
if the center-of-mass motion is removed. This is
qualitatively different from the two-particle case.
It turns out that asymptotically (for large times
t !þ1or t !1)anN-particle system splits up
into clusters,
C
1
[[C
n
¼f1; ...; Ng; C
k
\C
l
¼;if k 6¼ l ½1
Particles from the same cluster C
k
, k = 1, ..., n, form
a bound state, and different clusters do not interact
with each other. In particular, if n = 1and
C
1
= {1, 2, ..., N}, then we have a bound state of
the system. In another extreme case n = N, all
particles are free. The asymptotic evolution deter-
mined by clusters C
1
, ..., C
n
where n 2, and bound
states of all these clusters is called a scat tering
channel. Physically it is natural to expect that the list
of all such channels is exhaustive, that is, no other
scattering process is pos sible. This statement
is called asymptotic completeness.
We emphasize that an N-particle system may be in
different scattering states as t !þ1and t !1and
different rearrangement processes are possible. For
example, a three-particle system may asymptotically
consist of free particles or a pair of particles may be in
a bound state, whereas the third particle may be
asymptotically free. If particles are free at both 1
and þ1, then one speaks about elastic scattering; we
have a capture if particles free at 1 form a bound
state of a couple after the interaction; an opposite
process, when a bound state at 1 gives three free
particles, is known as a breakup. It is also possible that
a bound state of one couple yields a bound state of
N-Particle Quantum Scattering 585
another pair (a rearrangement) or a bound state of a
couple transforms into another bound state of the
same couple (an excitation). All these processes are
described by the scattering operator. On the contrary,
if the whole system forms a bound state at 1 (i.e.,
n = 1), then it remains in the same state for all t.
As far as monographic literature on N-particle
scattering is concerned, we mention Derezin
´
ski and
Ge´rard (1997), Faddeev (1965), Reed and Simon
(1979), and Yafaev (2000).
Setting the Scattering Problem
Let us recall the definition of the N-particle
Schro¨ dinger operator (Hamiltonian)
H ¼ H
0
þ V ½2
If the configuration space of each particle is R
d
,then
the operator H acts in the space L
2
(R
dN
). The operator
of kinetic energy (the ‘‘unperturbed’’ Hamiltonian) is
H
0
¼
X
N
j¼1
ð2m
j
Þ
1
x
j
½3
where x
j
and m
j
are the position and mass of the
particle labeled by j. The operator of potential energy
of pair interactions of particles (the perturbation) V is
the operator of multiplication by the function
VðxÞ¼
X
i<j
V
ij
ðx
j
x
i
Þ; i; j ¼ 1; ...; N ½4
Set = (ij), x
= x
j
x
i
. It is assumed that the
functions V
(x
) tend to zero sufficiently rapidly as
jx
j!1 in R
d
. However, the function V(x) 6!0
as jxj!1 in R
dN
if at least one of the distances
jx
i
x
j
j between particles remains bounded. This
difficulty is manifest even for two particles (N = 2),
but in this case it disappears if the motion of the
center of mass of the system is removed.
This means the following. Let the subspace X
cm
of
R
dN
be distinguished by the condition
X
N
j¼1
m
j
x
j
¼ 0 ½ 5
and let X
cm
be the orthogonal complement to X
cm
in
the space R
dN
endowed with the scalar product
hx; yi¼2
X
N
j¼1
m
j
hx
j
; y
j
i
R
d
½6
Then
L
2
ðR
dN
Þ¼L
2
ðX
cm
ÞL
2
ðX
cm
Þ
Denote by x
cm
, x
cm
the orthogonal projections of
x 2 R
dN
on the subspaces X
cm
, X
cm
, respectively, so
that x = (x
cm
, x
cm
). Clearly, the vector x
cm
has
components
x
cm
¼ M
1
X
N
j¼1
m
j
x
j
; M ¼
X
N
j¼1
m
j
Let T(p), (T(p)f )(x
1
, ..., x
N
) = f (x
1
þ p, ..., x
N
þ p),
be the operator of common translations of particles.
The operator H commutes with T(p), that is,
T(p)H = HT(p), for all p 2 R
d
. It follows that
H ¼ K I þ I H; K ¼ð2MÞ
1
x
cm
½7
where K is the kinetic energy operator of the center-
of-mass motion.
The operator
H ¼ H
0
þ V ½8
acts in the space H= L
2
(X
cm
). Here V is again the
operator of multiplication by function [4]. The
precise form of the differential operator H
0
depends
on the choice of coordinates in X
cm
. For example, if
N = 2 and x = x
2
x
1
, then H
0
= (2m)
1
x
where
m = m
1
m
2
(m
1
þ m
2
)
1
. In the case N = 3, a natural
choice of coordinates in X
cm
is given by one of the
three sets of Jacobi variables:
x
12
¼ x
2
x
1
x
12
¼ x
3
ðm
1
þ m
2
Þ
1
ðm
1
x
1
þ m
2
x
2
Þ
and similarly for x
13
, x
13
and x
23
, x
23
. In coordinates
x
, x
the operator of kinetic energy is determined
by the formula
H
0
¼ð2m
Þ
1
x
ð2m
Þ
1
x
where, for example,
ðm
12
Þ
1
¼ m
1
1
þ m
1
2
; m
1
12
¼ðm
1
þ m
2
Þ
1
þ m
1
3
If N = 2, then V(x) !0asjxj!1, x 2 X
cm
, but this
is no longer true for N 3. According to eqn [7] the
spectral and scattering theories for the operator H
reduce to those for the operator H. However, for
N 3, this reduct ion is not really helpful.
Let us now consider a breakup a = {C
1
, ..., C
n
}
of an N-particle system into clusters C
1
, ..., C
n
,
1 n =: #(a) N satisfying conditions [1].If
interactions between different clusters are neglected,
we obtain the operator
H
a
¼ H
0
þ V
a
; V
a
¼
X
n
l¼1
X
2C
l
V
½9
In particular, H
a
= H
0
if #(a) = N and H
a
= H if
#(a) = 1. Let the operator of common translations
586 N-Particle Quantum Scattering