Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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If
1
=n
0
, then we have
Z
Hðx; u
k
Þdx ¼
Z
0
þ
Z
1
Z
0
Hðx; u
k
Þdx
Z
1
W
1
ðxÞdx !1 ½56
This contradicts [53], and we see that
k
= ku
k
k
D
is
bounded. Once we know that the
k
are bounded,
we can apply well-known theorems to obtain the
desired conclusion.
&
Remark 1 It should be noted that the crucial
element in the pr oof of Theorem 8 was [51].Ifwe
had been dealing with an ordinary Palais–Smale
sequence, we could only concl ude that
ku
k
k
2
D
ðf ð; u
k
Þ; u
k
Þ¼oð
k
Þ
which would imply only
Z
Hðx; u
k
Þdx ¼ oð
k
Þ
This would not contradict [56], and the argument
would not go through.
As another application, we wish to solve
x
00
ðtÞ¼r
x
Vðt; xðtÞÞ ½57
where
xðtÞ¼ðx
1
ðtÞ; ...; x
n
ðtÞÞ ½58
is a map from I = [0, 2]toR
n
such that each
component x
j
(t) is a periodic function in H
1
with
period 2, and the function
Vðt; xÞ¼Vðt; x
1
; ...; x
n
Þ
is continuous from R
nþ1
to R with a gradient
r
x
Vðt; xÞ¼ð@V=@x
1
; ...;@V=@x
n
Þ
2 CðR
nþ1
; R
n
Þ
½59
For each x 2 R
n
, the function V(t, x) is periodic in t
with period 2. We shall study this problem under
the following assumptions:
1.
0 Vðt; xÞCðjxj
2
þ 1Þ
t 2 I; x 2 R
n
2. There are constants m > 0, 3m
2
=2
2
such that
Vðt; xÞ; jxjm; t 2 I; x 2 R
n
3. There are constants >1=2andC such that
Vðt; xÞjxj
2
when
jxj > C; t 2 I; x 2 R
n
4. The function given by
Hðt; xÞ¼2Vðt; xÞr
x
Vðt; xÞx ½60
satisfies
Hðt; xÞWðt Þ2L
1
ðIÞ; jxjC ½61
t 2 I, x 2R
n
, and
Hðt; xÞ!1 as jxj!1 ½62
We have
Theorem 9 Under the above hypotheses, the
system [57] has a nonconstant solution.
Proof. Let X be the set of vector functions x(t)
described above. It is a Hilbert space with norm
satisfying
kxk
2
X
¼
X
n
j¼1
kx
j
k
2
H
1
We also write
kxk
2
¼
X
n
j¼1
kx
j
k
2
where kkis the L
2
(I) norm. Let
N ¼fxðtÞ2X : x
j
ðtÞconstant; 1 j ng
and M = N
?
. The dimension of N is n,and
X = M N. The following is easily proved.
Lemma 1 If x 2 M, then
kxk
2
1
6
kx
0
k
2
and
kxkkx
0
k
We define
GðxÞ¼kx
0
k
2
2
Z
I
Vðt; xðtÞÞdt; x 2 X ½63
For each x 2 X write x = v þ w,wherev 2 N, w 2 M.
For convenience, we shall use the following equivalent
norm for X:
kxk
2
X
¼kw
0
k
2
þkvk
2
If x 2 M and
kx
0
k
2
¼
2
¼
6
m
2
then Lemma 1 implies that kxk
1
m, and we have
by Hypothesis 2 that V(t, x) .
452 Saddle Point Problems
Hence,
GðxÞkx
0
k
2
2
Z
jxj<m
dt
2
2ð2Þ0 ½64
Note that Hypothesis 3 is equivalent to
Vðt; xÞjxj
2
C; t 2 I; x 2 R
n
½65
for some constant C. Next, let
yðtÞ¼v þ sw
0
where v 2 N, s 0, and
w
0
¼ðsin t; 0; ...; 0Þ
Then w
0
2 M, and
kw
0
k
2
¼kw
0
0
k
2
¼
Note that
kyk
2
¼kvk
2
þ s
2
¼ 2jvj
2
þ s
2
Consequently,
GðyÞ¼s
2
kw
0
0
k
2
2
Z
I
Vðt; yðtÞÞdt
s
2
2
Z
I
jyðt Þj
2
dt þ 2C
¼ s
2
2ðkvk
2
þ s
2
Þþ2C
ð1 2Þs
2
4jvj
2
þ 2C
!1as s
2
þjvj
2
!1
We also note that Hypothesis 1 implies
GðvÞ0; v 2 N ½66
Take
A ¼fv 2 N : kvkRg
[fsw
0
þ v : v 2 N; s 0; ksw
0
þ vk
X
¼ Rg
B ¼ @B
\ M; 0 <¼ 6m
2
= < R
where
B
¼fx 2 X : kxk
X
<g
By Example 2, A links B. Moreover, if R is
sufficiently large,
sup
A
G ¼ 0 inf
B
G ½67
Hence, we may conclude that there is a sequence
{x
(k)
} X such that
Gðx
ðkÞ
Þ!c 0; ð1 þkx
ðkÞ
k
X
ÞG
0
ðx
ðkÞ
Þ!0
Hence,
Gðx
ðkÞ
Þ¼k½x
ðkÞ
0
k
2
2
Z
I
Vðt; x
ðkÞ
ðtÞÞdt ! c 0 ½ 68
ðG
0
ðx
ðkÞ
Þ; zÞ=2 ¼ð½x
ðkÞ
0
; z
0
Þ
Z
I
r
x
Vðt; x
ðkÞ
Þzðt Þdt ! 0; z 2 X ½69
and
ðG
0
ðx
ðkÞ
Þ; x
ðkÞ
Þ=2 ¼k½x
ðkÞ
0
k
2
Z
I
r
x
Vðt; x
ðkÞ
Þx
ðkÞ
dt ! 0 ½70
If
k
¼kx
ðkÞ
k
X
C
then there is a renamed subsequence such that x
(k)
converges to a limit x 2 X weakly in X and
uniformly on I. From [69] we see that
ðG
0
ðxÞ; zÞ=2 ¼ðx
0
; z
0
Þ
Z
I
r
x
Vðt; xðtÞÞ zðtÞdt ¼ 0; z 2 X
from which we conclude easily that x is a solution of
[57]. From [68], we see that
GðxÞc 0
showing that x(t) is not a constant. For if c > 0 and
x 2 N, then
GðxÞ¼2
Z
I
Vðt; xðtÞÞdt 0
If c = 0, we see that x 2 B by Theorem 6. Hence,
x 2 M.If
k
¼kx
ðkÞ
k
X
!1
let
~
x
(k)
= x
(k)
=
k
. Then, k
~
x
(k)
k
X
= 1. Let
~
x
(k)
=
~
w
(k)
þ
~
v
(k)
,where
~
w
(k)
2 M and
~
v
(k)
2 N.Thereisarenamed
subsequence such that
~
x
(k)
converges uniformly in I to
a limit
~
x and k[
~
x
(k)
]
0
k!r and k
~
x
(k)
k!,wherer
2
þ
2
= 1. From [68] and [70], we obtain
k½
~
x
ðkÞ
0
k
2
2
Z
I
Vðt; x
ðkÞ
ðtÞÞdt=
2
k
! 0
and
k½
~
x
ðkÞ
0
k
2
Z
I
r
x
Vðt; x
ðkÞ
Þx
ðkÞ
dt=
2
k
! 0
Thus,
2
Z
I
Vðt; x
ðkÞ
ðtÞÞdt=
2
k
! r
2
½71
Saddle Point Problems 453
and
Z
I
r
x
Vðt; x
ðkÞ
Þx
ðkÞ
dt =
2
k
! r
2
½72
Hence,
Z
I
Hðt; x
ðkÞ
ðtÞÞdt=
2
k
! 0 ½73
By Hypothesis 3, the left-hand side of [71] is
2k
~
x
ðkÞ
k
2
4C=
2
k
Thus,
r
2
2
2
¼ 2ð1 r
2
Þ
showing that r > 0. Hence,
~
x(t) 6 0. Let
0
I
be the set on which [
~
x(t)] 6¼ 0. The measure of
0
is positive. Thus, jx
(k)
(t)j!1 as k !1 for
t 2
0
. Hence,
Z
I
Hðt; x
ðkÞ
ðtÞÞdt
Z
0
Hðt; x
ðkÞ
ðtÞÞdt þ
Z
In
0
WðtÞdt !1
contrary to Hypothesis 4. Thus, the
k
are bounded,
and the proof is complete.
&
Superlinear Problems
Consider the problem
u ¼ f ðx; uÞ; x 2 ; u ¼ 0on@ ½74
where R
n
is a bounded domain whose bound-
ary is a smooth manifold, and f (x, t) is a continuous
function on
R. This semilinear Dirichlet pro-
blem has been studied by many authors. It is called
‘‘sublinear’’ if there is a constant C such that
jf ðx; tÞj Cðjtjþ1Þ; x 2 ; t 2 R
Otherwise, it is called ‘‘superlinear’’. Assume
(a
1
) There are constants c
1
, c
2
0 such that
jf ðx; tÞj c
1
þ c
2
jtj
s
where 0 s < (n þ 2)=(n 2) if n > 2.
(a
2
) f (x, t) = o(jtj)ast ! 0.
(a
3
) Either
Fðx; tÞ=t
2
!1as t !1
or
Fðx; tÞ=t
2
!1as t !1:
We have
Theorem 10 Under hypotheses (a
1
)(a
3
) the
boundary-value problem
u ¼ f ðx; uÞ; x 2 ; u ¼ 0on@ ½75
has a nont rivial solution for almost every positive .
Unfortunately, this theorem does not give any
information for any specific . It still leaves open the
problem of solving [74]. For this purpose, we add
the assumption
(a
4
) There are constants >2, r 0 such that
Fðx; tÞtf ðx; tÞCðt
2
þ 1Þ; jtjr ½76
We have
Theorem 11 Under hypotheses (a
1
)(a
4
) problem
[74] has a nontrivial solution.
We also have
Theorem 12 If we replace hypothesis (a
4
) with
(a
0
4
) The function H(x, t) is convex in t,
then the problem [74] has at least one nontrivial
solution.
Weak Linking
It is not clear if it is possible for A to link B if neither is
contained in a finite-dimensional manifold. For
instance, if E = M N,whereM, N are closed
infinite-dimensional subspaces of E and B
R
is the ball
centered at the origin of radius R in E, it is unknown if
the set A = M \ @B
R
links B = N. (If either M or N is
finite dimensional, then A does link B.) Unfortunately,
this is the situation which arises in some important
applications including Hamiltonian systems, the wave
equation and elliptic systems, to name a few.
We now consider linking when both M and N are
infinite dimensional and G
0
has some additional
continuity property. A property that is very useful is
that of weak-to-weak continuity:
u
k
! u weakly in E
¼) G
0
ðu
k
Þ!G
0
ðuÞ weakly ½77
We make the following definition:
Definition 3 A subset A of a Banach space E links
a sub set B of E ‘‘weakly’’ if for every G 2 C
1
(E, R)
satisfying [77] and
a
0
:¼ sup
A
G b
0
:¼ inf
B
G ½78
there is a sequence {u
k
} E and a constant c such
that
b
0
c < 1½79
454 Saddle Point Problems
and
Gðu
k
Þ!c; G
0
ðu
k
Þ!0 ½80
We have the following counterpart of Theorem 7.
Theorem 13 Let E be a separable Hilbert space,
and let G be a continuous functional on E with a
continuous derivative satisfying [77]. Let N be a
closed subspace of E, and let Q be a bounded open
subset of N containing the point p. Let F be a
continuous map of E onto N such that
(i) Fj
Q
= I, and
(ii) For each finite-dimensional subspace S 6¼ {0} of
E containing p, there is a finite-dimensional
subspace S
0
6¼ {0} of N containing p such that
v 2
Q \ S
0
; w 2 S ¼) Fðv þ wÞ2S
0
½81
Set A = @Q, B = F
1
(p). If
a
1
¼ sup
Q
G < 1½82
and [22] holds, then there is a sequence { u
k
} E
such that [24] holds with a a
1
.
Theorem 13 states that if Q, F, p satisfy the
hypotheses of that theorem, then A = @Q links
B = F
1
(p) weakly. It follows from this theorem
that all sets A, B known to link when one of the
subspaces M, N is finite dimensional will link
weakly even when M, N are both infinite
dimensional.
Now we give some applications of Theorem 13 to
semilinear boundary-value problems. Let be a
domain in R
n
and let A be a self–adjoint operator in
L
2
() having 0 in its resolvent set (thus, there is an
interval (a, b) in its resolvent set satisfying
a < 0 < b). Let f (x, t) be a continuous function on
R such that
jf ðx; tÞj VðxÞ
2
jtjþWðxÞVðxÞ½83
x 2 , t 2 R, and
f ðx; tÞ=t !
ðxÞ as t !1 ½84
where V, W 2 L
2
(), and multiplication by V(x) >
0 is a compact operator from D = D (jAj
1=2
)to
L
2
(). Let
M ¼
Z
1
b
dEðÞD; N ¼
Z
a
1
dEðÞD
where {E()} is the spectral measure of A.ThenM, N
are invariant subspaces for A and D = M N.If
ðu; vÞ¼
Z
ð
þ
u
þ
u
Þv dx ½85
(u) = (u, u), then we assume that
ðvÞðAv; vÞ; v 2 N ½86
ðAw; wÞðwÞ; w 2 M ½87
We also assume that the only solution of
Au ¼
þ
u
þ
u
½88
is u 0, where u
= max {u, 0}. We have
Theorem 14 Under the above hypotheses there is
at least one solution of
Au ¼ f ðx; uÞ; u 2 DðAÞ½89
Next, we consider an application concerning
radially symmetric solutions for the problem
u
tt
u ¼ f ðt; x; uÞ; t 2 R; x 2 B
R
½90
uðt; xÞ¼0; t 2 R; x 2 @B
R
½91
uðt þ T; xÞ¼uð t; xÞ; t 2 R; x 2 B
R
½92
where B
R
= {x 2 R
n
: jxj < R}. We assume that the
ratio R=T is rational. Let
8R=T ¼ a=b ½93
where a , b are relatively prime positive integers. It
can be shown that
n 6 3 ðmodð4; aÞÞ ½94
implies that the linear problem corresponding to
[90]–[92] has no essential spectrum. If
n 3 ðmodð4; aÞÞ ½95
then the essential spectrum of the linear operator
consists of precisely one point
0
¼ðn 3Þðn 1Þ=4R
2
½96
Consider the case
f ðt; r; sÞ¼s þ pðt; r; sÞ½97
where is a point in the resolvent set, r = j xj, and
jpðt; r; sÞj Cðjsj
þ 1Þ; s 2 R ½98
for some number <1. We then have
Theorem 15 If [94] holds, then [90]–[92] have a
weak rotationally invar iant solution. If [95] holds
and
0
<, assume in addition that p(t, r, s) is
nondecreasing in s. If <
0
, assume that p(t, r , s) is
nonincreasing in s. Then [90]–[92] have a weak
rotationally invariant solution.
Saddle Point Problems 455
See also: Combinatorics: Overview; Homoclinic
Phenomena; Ljusternik–Schnirelman Theory; Minimax
Principle in the Calculus of Variations.
Further Reading
Ambrosetti A and Prodi G (1993) A primer of nonlinear analysis.
Cambridge Studies in Advanced Mathematics 34. Cambridge:
Cambridge University Press.
Chang KC (1993) Infinite Dimensional Morse Theory and
Multiple Solution Problems. Boston: Birkha¨ user.
Ekeland I and Temam R (1976) Convex Analysis and Variational
Problems. Amsterdam: North-Holland.
Ghoussoub N (1993) Duality and Perturbation Methods in Critical
Point Theory. Cambridge: Cambridge University Press.
Mawhin J and Willem M (1989) Critical Point Theory and
Hamiltonian Systems. Berlin: Springer.
Rabinowitz PR (1986) Minimax Methods in Critical Point Theory
with Applications to Differential Equations, Conf. Board of
Math. Sci. Reg. Conf. Ser. in Math. No. 65. Providence, RI:
American Mathematical Society.
Schechter M (1999) Linking Methods in Critical Point Theory.
Boston: Birkha¨user.
Schechter M (1986) Spectra of Partial Differential Operators,
2nd edn. Amsterdam: North-Holland.
Struwe M (1996) Variational Methods. Berlin: Springer.
Willem M (1996) Minimax Theorems. Boston: Birkha¨ user.
Scattering in Relativistic Quantum Field Theory: Fundamental
Concepts and Tools
D Buchholz, Universita
¨
tGo
¨
ttingen,
Go
¨
ttingen, Germany
S J Summers, University of Florida,
Gainesville, FL, USA
ª 2006 Elsevier Ltd. All rights reserved.
Physical Motivation and Mathematical
Setting
The primary connection of relativistic quantum field
theory to experimental physics is through scattering
theory, that is, the theory of the collision of elementary
(or compound) particles. It is therefore a central topic
in quantum field theory and has attracted the attention
of leading mathematical physicists. Although a great
deal of progress has been made in the mathematically
rigorous understanding of the subject, there are
important matters which are still unclear, some of
which will be indicated below.
In the paradigmatic scattering experiment, several
particles, which are initially sufficiently distant from
each other that the idealization that they are not
mutually interacting is physically reasonable,
approach each other and interact (collide) in a region
of microscopic extent. The products of this collision
then fly apart until they are sufficiently well separated
that the approximation of noninteraction is again
reasonable. The initial and final states of the objects in
the scattering experiment are therefore to be modeled
by states of noninteracting, that is, free, fields, which
are mathematically represented on Fock space. Typi-
cally, what is measured in such experiments is the
probability distribution (cross section) for the transi-
tions from a specified state of the incoming particles to
a specified state of the outgoing particles.
It should be mentioned that until the late 1950s,
the scattering theory of relativistic quantum particles
relied upon ideas from nonrelativistic quantum-
mechanical scattering theory (interaction representa-
tion, adiabatic limit, etc.), which were invalid in the
relativistic context. Only with the advent of axio-
matic qua ntum field theory did it become possible to
properly formulate the concepts and mathematical
techniques which will be outlined here.
Scattering theory can be rigorously formulated
either in the context of quantum fields satisfying
the Wightman axioms (Streater and Wightman 1964)
or in terms of local algebras satisfying the Haag–
Kastler–Araki axioms (Haag 1992). In brief, the
relation between these two settings may be described
as follows: in the Wightman setting, the theory is
formulated in terms of operator-valued distributions
on Minkowski space, the quantum fields, which act on
the physical state space. These fields, integrated with
test functions f having support in a given region O of
spacetime (only four-dimensional Minkowski space
R
4
will be treated here), (f ) =
R
d
4
xf(x)(x), form
under the operations of addition, multiplication, and
Hermitian conjugation a polynomial
-algebra P(O)of
unbounded operators. In the Haag–Kastler–Araki
setting, one proceeds from these algebras to algebras
A(O) of bounded operators which, roughly speaking,
are formed by the bounded functions A of the
operators (f ). This step requires some mathematical
care, but these subtleties will not be discussed here. As
the statements and proofs of the results in these two
frameworks differ only in technical details, the theory
is presented here in the more convenient setting of
algebras of bounded operators (C
-algebras).
Central to the theory is the notion of a particle,
which, in fact, is a quite complex concept, the full
nature of which is not completely understood, cf.
456 Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools
below. In order to maintain the focus on the
essential points, we consider in the subsequent
sections primarily a single massive particle of integer
spin s, that is, a boson. In standard scattering theory
based upon Wigner’s characterization, this particle
is simply ident ified with an irreducible unitary
representation U
1
of the identity component P
"
þ
of
the Poincare´ group wi th spin s and mass m > 0. The
Hilbert sp ace H
1
upon which U
1
(P
"
þ
) acts is called
the one-particle space and determines the possible
states of a single particle, alone in the universe.
Assuming that configurations of several such parti-
cles do not interact, one can proceed by a standard
construction to a Fock space describing freely
propagating multiple particle states,
H
F
¼
M
n2N
0
H
n
where H
0
= C and H
n
is the n-fold symmetrized direct
product of H
1
with itself. This space is spanned by
vectors
1
n
,where denotes the symme-
trized tensor product, representing an n-particle state
wherein the kth particle is in the state
k
2
H
1
, k = 1, ..., n. The representation U
1
(P
"
þ
) induces
a unitary representation U
F
(P
"
þ
)onH
F
by
U
F
ðÞ
1
n
ðÞ
¼
:
U
1
ðÞ
1
U
1
ðÞ
n
½1
In interacting theories, the states in the correspond-
ing physical Hilbert space H do not have such an a
priori interpretation in physical terms, however. It is
the primary goal of scattering theory to identify in H
those vectors which describe, at asymptotic times,
incoming, respectively, outgoing, configurations of
freely moving particles. Mathematically, this amounts
to the construction of certain specific isometries
(generalized Møller operators),
in
and
out
,mapping
H
F
onto subspaces H
in
Hand H
out
H,respec-
tively, and intertwining the unitary actions of the
Poincare´ group on H
F
and H. The resulting vectors
1
n
ðÞ
in=out
¼
:
in=out
1
n
ðÞ2H½2
are interpreted as incoming and outgoing particle
configurations in scattering processes wherein the
kth particle is in the state
k
2H
1
.
If, in a theory, the equality H
in
= H
out
holds, then
every incoming scattering state evolves, after the
collision processes at finite times, into an outgoing
scattering state. It is then physically meaningful to
define on this space of states the scattering matrix,
setting S =
in
out
. Physical data such as collision
cross sections can be derived from S and the corre-
sponding transition amplitudes h(
1
m
)
in
,
(
0
1
0
n
)
out
i, respectively, by a standard proce-
dure. It should be noted, however, that neither the
above physically mandatory equality of state spaces nor
the more stringent requirement that every state has an
interpretation in terms of incoming and outgoing
scattering states, that is, H= H
in
= H
out
(asymptotic
completeness), has been fully established in any inter-
acting relativistic field theoretic model so far. This
intriguing problem will be touched upon in the last
section of this article.
Before going into details, let us state the few
physically motivated postulates entering into the
analysis. As discussed, the point of departure is a
family of algebras A(O), more precisely a net,
associated with the ope n subregions O of Min-
kowski space and acting on H. Restricting attention
to the case of bosons, we may assume that this net is
local in the sense that if O
1
is spacelike separated
from O
2
, then all elements of A(O
1
) commute with
all elements of A(O
2
). (In the presence of fermions,
these algebras contain also fermionic operators
which anticommute.) This is the mathematical
expression of the principle of Einstein causality.
The unitary representation U of P
"
þ
acting on H is
assumed to satisfy the relativistic spectrum condition
(positivity of energy in all Lorentz frames) and, in
the sense of equality of sets, U()A(O)U()
1
=
A(O) for all 2P
"
þ
and regions O, where O
denotes the Poincare´ transformed region. It is also
assumed that the subspace of U(P
"
þ
)-invariant
vectors is spanned by a single unit vector ,
representing the vacuum, which has the Reeh–
Schlieder property, that is, each set of vectors
A(O) is dense in H. These standin g assumptions
will subsequently be amended by further conditions
concerning the particle content of the theory.
Haag–Ruelle Theory
Haag and Ruelle were the first to establish the
existence of scattering states within this general
framework (Jost 1965); further substantial improve-
ments are due to Araki and Hepp (Araki 1999). In all
of these investigations, the arguments were given for
quantum field theories with associated particles (in
the Wigner sense) which have strictly positive mass
m > 0andforwhichm is an isolated eigenvalue of
the mass operator (upper and lower mass gap).
Moreover, it was assumed that states of a single
particle can be created from the vacuum by local
operations. In physical terms, these assumptions
allow only for theories with short-range interactions
and particles carrying strictly localizable charges.
In view of these limitations, Haag–Ruelle theory
has been developed in a number of different
directions. By now, the scattering theory of massive
particles is under complete con trol, including also
Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools 457
particles carrying nonlocalizable (gauge or topo-
logical) charges and particles having exotic statistics
(anyons, plektons) which can appear in theories in
low spacetime dimensions. Due to constraints of
space, these results must go without further men-
tion; we refer the interested reader to the articles
Buchholz and Fredenhagen (1982) and Fredenhagen
et al. (1996). Theories of massless particles and of
particles carrying charges of electric or magnetic
type (infraparticles) will be discussed in subsequent
sections.
We outline here a recent generalization of Haag–
Ruelle scattering theory prese nted in Dybalski
(2005), which covers massive particles with localiz-
able charges without relying on any further con-
straints on the mass spectrum. In particular, the
scattering of electrically neutral, stable particles
fulfilling a sharp dispersion law in the presence of
massless particles is included (e.g., neutral atoms in
their ground states). Mathematically, this assump-
tion can be expressed by the requirement that there
exists a subspace H
1
Hsuch that the restriction of
U(P
"
þ
)toH
1
is a representation of mass m > 0. We
denote by P
1
the projection in H onto H
1
.
To establish notation, let O be a bounded space-
time region and let A 2A(O) be any operat or such
that P
1
A 6¼ 0. The existence of such localized (in
brief, local) operators amounts to the assumption
that the particle carries a localizable charge. That
the particle is stable, that is, completely decouples
from the underlying continuum states, can be cast
into a condition first stated by Herbst: for all
sufficiently small >0
kE
ð1 P
1
ÞAkc
½3
for some constants c, >0, where E
is the projec-
tion onto the spectral subspace of the mass operator
corresponding to spectrum in the interval (m ,
m þ ). In the case originally considered by Haag
and Ruelle, where m is isolated from the rest of the
mass spectrum, this condition is certainly satisfied.
Setting A(x) ¼
:
U(x)AU(x)
1
, where U(x) is the
unitary implementing the spacetime translation
x = (x
0
, x) (the velocity of light and Planck’s
constant are set equal to 1 in what follows), one
puts, for t 6¼ 0,
A
t
ðf Þ¼
Z
d
4
xg
t
ðx
0
Þf
x
o
ðxÞAðxÞ½4
Here x
0
7!g
t
(x
0
) ¼
:
g((x
0
t)=jtj
)=jtj
induces a
time averaging about t, g being any test function
which integrates to 1 and whose Fourier transform
has compact support, and 1=(1 þ ) <<1 with
as above. The Fourier transform of f
x
0
is given by
f
f
x
0
(p) ¼
:
e
f (p)e
ix
0
!( p)
, where f is some test function
on R
3
with
e
f (p) having compact support, and
!(p) = (p
2
þ m
2
)
1=2
. Note that (x
0
, x) 7!f
x
0
(x)isa
solution of the Klein–Gordon equation of mass m.
With these assumptions, it follows by a straight-
forward application of the harmonic analysis of
unitary groups that in the sense of strong conver-
gence A
t
(f ) !P
1
A(f ) and A
t
(f )
!0ast !1,
where A(f ) =
R
d
3
xf(x)A(0, x). Hence, the opera-
tors A
t
(f ) may be thought of as creation operators
and their adjoints as annihilation operators. These
operators are the basic ingredients in the construc-
tion of scattering states. Choosing local operators
A
k
as above and test functions f
(k)
with disjoint
compact supports in momentum space,
k = 1, ..., n, the scattering states are obtained as
limits of the Haag–Ruelle approximants
A
1t
ðf
ð1Þ
ÞA
nt
ðf
ðnÞ
Þ ½5
Roughly speaking, the operators A
kt
(f
(k)
) are loca-
lized in spacelike separated regions at asymptotic
times t, due to the support properties of the Fourier
transforms of the functions f
(k)
. Hence they com-
mute asymptotically because of locality and, by the
clustering properties of the vacuum state, the above
vector becomes a product state of single-particle
states. In order to prove convergence, one proceeds,
in analogy to Cook’s method in quantum-mechanical
scattering theory, to the time derivatives,
@
t
A
1t
ðf
ð1Þ
ÞA
nt
ðf
ðnÞ
Þ
¼
X
k6¼l
A
1t
ðf
ð1Þ
Þ½@
t
A
kt
ðf
ðkÞ
Þ; A
lt
ðf
ðlÞ
ÞA
nt
ðf
ðnÞ
Þ
þ
X
k
A
1t
ðf
ð1Þ
Þ_
k
A
nt
ðf
ðnÞ
Þ@
t
A
kt
ðf
ðkÞ
Þ ½6
where
k
_
denotes omission of A
kt
(f
(k)
). Employing
techniques of Araki and Hepp, one can prove that
the terms in the first summation on the right-hand
side (RHS) of [6], involving commutators, decay
rapidly in norm as t approaches infinity because of
locality, as indicated above. By applying condition
[3] and the fact that the vectors @
t
A
kt
(f
(k)
) do not
have a compone nt in the single-particle space H
1
,
the terms in the second summation on the RHS of
[6] can be shown to decay in norm like jtj
(1þ)
.
Thus, the norm of the vector [6] is integrable in t,
implying the existence of the strong limits
P
1
A
1
ðf
ð1Þ
Þ P
1
A
n
ðf
ðnÞ
Þ
in=out
¼
:
lim
t!1
A
1t
ðf
ð1Þ
ÞA
nt
ðf
ðnÞ
Þ ½7
458 Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools
As indicated by the notation, these limits depend
only on the single-particle vectors P
1
A
k
(f
(k)
) 2H
1
,
k = 1, ..., n, but not on the specific choice of
operators and test functions. In order to establish
their Fock structure, one employs results on cluster-
ing properties of vacuum correlation functions in
theories without strictly positive minimal mass.
Using this, one can compute inner products of
arbitrary asymptotic states and verify that the maps
P
1
A
1
ðf
ð1Þ
Þ P
1
A
n
ðf
ðnÞ
Þ
7! P
1
A
1
ðf
ð1Þ
Þ P
1
A
n
ðf
ðnÞ
Þ
in=out
½8
extend by linearity to isomorphisms
in=out
from the
Fock space H
F
onto the subspaces H
in=out
H
generated by the collision states. Moreover, the
asymptotic states transform under the Poincare´
transformations U(P
"
þ
)as
UðÞ P
1
A
1
ðf
ð1Þ
Þ P
1
A
n
ðf
ðnÞ
Þ
in=out
¼ U
1
ðÞP
1
A
1
ðf
ð1Þ
Þ
U
1
ðÞP
1
A
n
ðf
ðnÞ
Þ
in=out
½9
Thus, the isomorphisms
in=out
intertwine the action
of the Poincare´ group on H
F
and H
in=out
.We
summarize these results, which are vital for the
physical interpretation of the underlying theory, in
the following theorem.
Theorem 1 Consider a theory of a particle of mass
m> 0 which satisfies the standing assumptions and
the stability condition [3]. Then there exist canoni-
cal isometries
in=out
, mapping the Fock space H
F
based on the single-particle space H
1
onto subspaces
H
in=out
H of incoming and outgoing scattering
states. Moreover, these isometries intertwine the
action of the Poincare´ transformations on the
respective spaces.
Since the scattering states have been identified
with Fock space, asymptotic creation and annihila-
tion operators act on H
in=out
in a natural manner.
This point will be explained in the following section.
LSZ Formalism
Prior to the results of Haag and Ruelle, an axiomatic
approach to scat tering theory was developed by
Lehmann, Symanzik, and Zimmermann (LSZ),
based on time-ordered vacuum expectation values
of quantum fields. The relative advantage of their
approach with respect to Haag–Ruelle theory is that
useful reduction formulas for the S-matrix greatly
facilitate computations, in particular in perturba-
tion theory. Moreover, these formulas are the
starting point of general studies of the momentum
space an alyticity properties of the S-matrix (disper-
sion relations), as outlined in Dispersion Relations
(cf. also Iagolnitzer (1993)). Wit hin the present
general setting, the LSZ method was esta blished by
Hepp.
For simplicity of discussion, we consider again a
single particle type of mass m > 0 and integer spin s,
subject to condition [3]. According to the results of
the preceding section, one then can consistently
define asymptotic creation ope rators on the scatter-
ing states, setting
Aðf Þ
in=out
P
1
A
1
ðf
ð1Þ
Þ P
1
A
n
ðf
ðnÞ
Þ
in=out
¼
:
lim
t!1
A
t
ðf Þ P
1
A
1
ðf
ð1Þ
Þ P
1
A
n
ðf
ðnÞ
Þ
in=out
¼
P
1
Aðf Þ P
1
A
1
ðf
ð1Þ
Þ
P
1
A
n
ðf
ðnÞ
Þ
in=out
½10
Similarly, one obtains the corresponding asymptotic
annihilation operators,
Aðf Þ
in=out
P
1
A
1
ðf
ð1Þ
Þ P
1
A
n
ðf
ðnÞ
Þ
in=out
¼ lim
t!1
A
t
ðf Þ
P
1
A
1
ðf
ð1Þ
Þ
P
1
A
n
ðf
ðnÞ
Þ
in=out
¼0 ½11
where the latter equality hold s if the Fourier trans-
forms of the functions f ,f
(1)
,...,f
(n)
, have disjoint
supports. We mention as an aside that, by replacing
the time-averaging function g in the definition of
A
t
(f ) by a delta function, the above formulas still
hold. But the convergence is then to be understood
in the weak Hilbert space topology. In this form, the
above relations were anticipated by LSZ (asymptotic
condition).
It is straightforward to proceed from these
relations to reduction formulas. Let B be any local
operator. Then one has, in the sense of matrix
elements between outgoing and incoming scattering
states,
BAðf Þ
in
Aðf Þ
out
B ¼ lim
t!1
BAðf
t
ÞAðf
t
ÞBðÞ
¼ lim
t!1
Z
d
4
xf
t
ðxÞBAðxÞ
Z
d
4
xf
t
ðxÞAðxÞB
½12
f
t
(x)¼
:
g
t
(x
0
)f (x
0
)(vec(x)). Because of the (essential)
support properties of the functions f
t
, the contribu-
tions to the latter integrals arise, for asymptotic t,
from spacetime points x where the localization
Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools 459
regions of A(x) and B have a negative timelike (first
term), respectively, positive timelike (second term)
distance. One may therefore proceed from the
products of these operators to the time-ordered
products T(BA(x)), where T(BA(x))= A(x)B if the
localization region of A( x) lies in the future of that
of B, and T(BA(x))= BA(x) if it lies in the past. It is
noteworthy that a precise definition of the time
ordering for finite x is irr elevant in the present
context – any reasonable interpolation between the
above relations will do. Similarly, one can define
time-ordered products for an arbitrary number of
local operators. The preceding limit can then be
recast into
lim
t!1
Z
d
4
xðf
t
ðxÞf
t
ðxÞÞTðBAðxÞÞ ½13
The latter expression has a particularly simple form in
momentum space. Proceeding to the Fourier trans-
forms of f
t
and noticing that, in the limit of large t,
f
f
t
ðpÞ
e
f
t
ðpÞ
= p
0
!ðpÞðÞ
!2 i
e
f ðpÞðp
0
!ðpÞÞ ½14
one gets
BAðf Þ
in
Aðf Þ
out
B
¼2i
Z
d
3
p
e
f ðpÞ p
0
!ðpÞðÞ
TðB
e
AðpÞÞ
p
0
¼!ðpÞ
½15
Here T(B
e
A(p)) denotes the Fourier transform of
T(BA(x)), and it can be shown that the restriction of
(p
0
!(p))T(B
e
A(p)) to the manifold {p 2 R
4
: p
0
=
!(p)} (the ‘‘mass shell’’) is meaningful in the sense of
distributions on R
3
. By the same token, one obtains
Aðf Þ
out
B BAðf Þ
in
¼2i
Z
d
3
p
e
f ðpÞ p
0
!ðpÞðÞTð
f
A
ðpÞBÞ
p
0
¼!ðpÞ
½16
Similar relations, involving an arbitrary number of
asymptotic creation and annihilation operators, can
be established by analogous considerations. Taking
matrix elements of these relations in the vacuum state
and recalling the action of the asymptotic creation
and annihilation operators on scattering states, one
arrives at the following result, which is central in all
applications of scattering theory.
Theorem 2 Consider the theory of a particle of
mass m > 0 subject to the conditions stated in the
preceding sections and let f
(1)
, ..., f
(n)
be any family
of test functions whose Fourier transforms have
compact and nonoverlapping supports. Then
P
1
A
1
ðf
ð1Þ
Þ P
1
A
k
ðf
ðkÞ
Þ
out
;
D
P
1
A
kþ1
ðf
ðkþ1Þ
Þ P
1
A
n
ðf
ðnÞ
Þ
in
¼ð2iÞ
n
Z
Z
d
3
p
1
d
3
p
n
g
f
ð1Þ
ðp
1
Þ
g
f
ðkÞ
ðp
k
Þ
g
f
ðkþ1Þ
ðp
kþ1
Þ
g
f
ðnÞ
ðp
n
Þ
Y
n
i¼1
p
i
0
!ðp
i
ÞðÞ; T
f
A
1
ðp
1
Þ
D
f
A
k
ðp
k
Þ
g
A
kþ1
ðp
kþ1
Þ
f
A
n
ðp
n
Þ
E
j¼1;...;n
p
j
0
¼!ðp
j
Þ
½17
in an obvious notation.
Thus, the kernels of the scattering amplitudes in
momentum space are obtained by restricting the (by
the factor
Q
n
i = 1
(p
i
0
!(p
i
))) amputated Fourier
transforms of the vacuum expectation values of the
time-ordered products to the positive and negative
mass shells, respectively. These are the famous LSZ
reduction formulas, which provide a convenient link
between the time-ordered (Green’s) functions of a
theory and its asymptotic particle interpretation.
Asymptotic Particle Counters
The preceding construction of scattering states
applies to a signi ficant class of theories; but even if
one restricts attention to the case of massive
particles, it does not cover all situations of physical
interest. For an essential input in the construction is
the existence of local operators interpolating
between the vacuum and the single-particle states.
There may be no such operators at one’s disposal,
however, either because the particle in question
carries a nonlocalizable charge, or because the given
family of operators is too small. The latter case
appears, for example, in gauge theories, where in
general only the observables are fixed by the
principle of local gauge invariance, and the physical
particle content as well as the corresponding inter-
polating operators are not known from the outset.
As observables create from the vacuum only neutral
states, the above construction of scattering states
then fails if charged particles are present. Never-
theless, thinking in physical terms, one would exp ect
that the observables contain all relevant information
in order to determine the features of scattering
states, in particular their collision cross section. That
this is indeed the case was first shown by Araki and
Haag (Araki 1999).
In scattering experiments, the measured data are
provided by detectors (e.g., particle counters) and
460 Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools
coincidence arrangements of detectors. Essential
features of detectors are their lack of response in
the vacuum state and their macroscopic localization.
Hence, within the present mathematical setting, a
general detector is represented by a positive operator
C on the physical Hilbert space H such that C=0.
Because of the Reeh–Schlieder theorem, these con-
ditions cannot be satisfied by local operators.
However, they can be fulfilled by ‘‘almost-local’’
operators. Examples of such operators are easy to
produce, putting C = L
L with
L ¼
Z
d
4
xfðxÞAðxÞ½18
where A is any local operator and f any test function
whose Fourier transform has compact support in the
complement of the closed forw ard light cone (and
hence in the complement of the energy momentum
spectrum of the theory). In view of the properties of
f and the invariance of under translations , it
follows that C = L
L annihilates the vacuum and
can be approximated with arbitrary precisi on by
local operators. The algebra generated by these
operators C will be denoted by C.
When preparing a scattering experiment, the first
thing one must do with a detector is to calibrate it,
that is, test its respons e to sources of single-particle
states. Within the mathematical setting , this
amounts to computing the matrix elements of C in
states 2H
1
:
h; Ci¼
ZZ
d
3
p d
3
q ðpÞðqÞhpjCjqi½19
Here p 7!(p) is the momentum space wave func-
tion of , hjCji is the kernel of C in the single-
particle space H
1
, and we have omitted (summations
over) indices labeling internal degrees of freedom of
the particle, if any. The relevant information about
C is encoded in its kernel. As a matter of fact, one
only needs to know its restriction to the diagonal,
p 7!hpjCjpi. It is called the sensitivity function of C
and can be shown to be regular under quite general
circumstances (Araki 1999, Buchholz and Fredenhagen
1982).
Given a state 2H for which the expectation
value h, C(x)) i differs significantly from 0, one
concludes that this state deviates from the vacuum
in a region abou t x. For finite x, this does not mean,
however, that has a particle interpretation at x.
For that spacetime point may, for example, be just
the location of a collision center. Yet, if one
proceeds to asymptotic times, one expects, in view
of the spreading of wave packets, that the prob-
ability of finding two or more particles in the same
spacetime region is dominated by the single-particle
contributions. It is this physical insight which
justifies the expectation that the detectors C(x)
become particle counters at asymptotic times.
Accordingly, one considers for asymptotic t the
operators
C
t
ðhÞ¼
:
Z
d
3
xhðx=tÞCðt; xÞ½20
where h is any test function on R
3
. The ro le of the
integral is to sum up all single-particle contributions
with velocities in the support of h in order to
compensate for the decreasing probability of finding
such particles at asymptotic times t about the
localization center of the detector. That these ideas
are consistent was demonstrated by Araki and Haag,
who established the following result (Araki 1999).
Theorem 3 Consider, as before, the theory of a
massive particle. Let C
(1)
, ..., C
(n)
2Cbe any family
of detector operators and let h
(1)
, ..., h
(n)
be any
family of test functions on R
3
. Then, for any state
out
2H
out
of finite energy,
lim
t!1
out
; C
ð1Þ
t
ðh
ð1Þ
ÞC
ðnÞ
t
ðh
ðnÞ
Þ
out
DE
¼
Z
Z
d
3
p
1
d
3
p
n
out
;
out
ðp
1
Þ
out
ðp
n
Þ
out
hi
Y
n
k¼1
hðp
k
=!ðp
k
ÞÞhp
k
jC
ðkÞ
jp
k
i½21
where
out
(p) is the momentum space density (the
product of creation and annihilation operators) of
outgoing particles of momentum p, and (summa-
tions over) possible indices labeli ng internal degrees
of freedom of the particle are omitted . An analogous
relation holds for incoming scattering states at
negative asymptotic times.
This result shows, first of all, that the scattering
states have indeed the desired interpretation with
regard to the observables, as anticipated in the
preceding sections. Since the assertion holds for all
scattering states of finite energy, one may replace in the
above theorem the outgoing scattering states by any
state of finite energy, if the theory is asymptotically
complete, that is, H= H
in
= H
out
. Then choosing, in
particular, any incoming scattering state and making
use of the arbitrariness of the test functions h
(k)
as well
as the knowledge of the sensitivity functions of the
detector operators, one can compute the probability
distributions of outgoing particle momenta in this state,
and thereby the corresponding collision cross sections.
The question of how to construct certain specific
incoming scattering states by using only local
observables was not settled by Araki and Haag,
Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools 461