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however. A general method to that effect was

outlined in Buchholz et al. (1991). As a matter of

fact, for that method only the knowledge of states in

the subspace of neutral states is required. Yet in this

approach one would need for the computation of,

say, elastic collision cross sections of charged

particles the vacuum correlation functions involving

at least eight local observables. This practical

disadvantage of increased computational complexity

of the method is offset by the conceptual advantage

of maki ng no appeal to quantities which are a priori

nonobservable.

Massless Particles

and Huygens’ Principle

The preceding general methods of scattering theory

apply only to massive particles. Yet taking advan-

tage of the salient fact that massless particles always

move with the speed of light, Buchholz succeeded in

establishing a scattering theory also for such

particles (Haag 1992). Moreover, his arguments

lead to a quantum version of Huygens’ princ iple.

As in the case of massive particles, one assumes

that there is a subspace H

Hcorresponding to a

representation of U(P

) of mass m = 0 and, for

simplicity, integer helicity; moreover, there must

exist local operators interpolating between the

vacuum and the single-particle states. These

assumptions cover, in particular, the important

examples of the photon and of Goldstone particles.

Picking any suitable local operator A interpolating

between  and some vector in H

, one sets, in

analogy to [4],

ðx

ð1=2Þ"ðx

Þðx

 x

Þ@

AðxÞ½22

Here g

) ¼

(1=jln tj) g((x

 t)=jln tj) with g as in

[4], and the solution of the Klein–Gordon equation

in [4] has been replaced by the fundamental solution

of the wave equation; furthermore, @

A(x) denotes

the derivative of A(x) with respect to x

. Then, once

again, the strong limit of A

 as t !1is P

A,

with P

the projection onto H

In order to establish the convergence of A

as in

the LSZ approach, one now uses the fact that these

operators are, at asymptotic times t, localized in the

complement of some forward, respectively, back-

ward, light cone. Because of locality, they therefore

commute with all operators which are localized in

the interior of the respective cones. More specifi-

cally, let OR

be the localization region of A and

let O



 R

be the two regions having a positive,

respectively, negative, timelike distance from all

points in O. Then, for any operator B which is

compactly localized in O



, respectively, one obtains

lim

t !1

B=lim

t !1

=BP

A. This

relation establishes the existence of the limits

in=out

¼ lim

t!1

½23

on the (by the Reeh–Schlieder property) dense sets of

vectors {B : B 2A(O



)} H. It requires some

more detailed analysis to prove that the limits have

all of the properties of a (smeare d) free massless

field, whose translates x 7!A

in=out

(x) satisfy the wave

equation and have c-number commutation relations.

From these free fields, one can then proceed to

asymptotic creation and annihilation operators and

construct asymptotic Fock spaces H

in=out

H of

massless particles and a corresponding scattering

matrix as in the massive case. The details of this

construction can be found in the original article, cf.

Haag (1992).

It also follows from these arguments that the

asymptotic fields A

in=out

of massless particles ema-

nating from a region O, that is, for which the

underlying interpolating operators A are locali zed in

O, commute with all operators localized in O



respectively. This result may be understood as an

expression of Huygens’ principle. More precisely,

denoting by A

in=out

(O) the algebras of bounded

operators generated by the asymptotic fields A

in=out

respectively, one arrives at the following quantum

version of Huygens’ principle.

Theorem 4 Consider a theory of massless particles

as described above and let A

in=out

(O) be the algebras

generated by massless asymptotic fields A

in=out

with

A 2A(O). Then

and

ðOÞ  AðO



out

ðOÞ  AðO

½24

Here the prime denotes the set of bounded operators

commuting with all elements of the respective

algebras (i.e., their commutant s).

Beyond Wigner’s Concept of Particle

There is by now ample evidence that Wigner’s

concept of particle is too narrow in order to cover

all particle-like structures appearing in quantum

field theory. Examples are the partons which show

up in nonabelian gauge theories at very small

spacetime scales as constituents of hadrons, but

which do not appear at large scales due to the

confining forces. Their mathematical description

462 Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools

requires a quite different treatment, which cannot be

discussed here. But even at large scales, Wigner’s

concept does not cover all stable particle-like

systems, the most prominent examples being parti-

cles carrying an abelian gauge charge, such as the

electron and the proton, which are inevitably

accompanied by infinite clouds of (‘‘on-shell’’)

massless particles.

The latter problem was discussed first by Schroer,

who coined the term ‘‘infraparticle’’ for such

systems. Later, Buchholz showed in full generality

that, as a consequence of Gauss’ law, pure states

with an abelian gauge charge can neither have a

sharp mass nor carry a unitary representatio n of the

Lorentz group, thereby uncoveri ng the simple origin

of results found by explicit computations, notably in

quantum electrodynamics ( Steinmann 2000). Thus,

one is faced with the question of an appropriate

mathematical characterization of infraparticles

which generalizes the concept of particle invented

by Wigner. Some significant steps in this direction

were taken by Fro¨ hlich, Morchio, and Strocchi, who

based a definition of infraparticles on a detailed

spectral analysis of the energy–momentum opera-

tors. For an account of these developments and

further references, cf. Haag (1992).

We outline here an approach, originated by Buch-

holz, which covers all stable particle-like structures

appearing in quantum field theory at asymptotic times.

It is based on Dirac’s idea of improper particle states

with sharp energy and momentum. In the standard

(rigged Hilbert space) approach to giving mathema-

tical meaning to these quantities, one regards them as

vector-valued distributions, whereby one tacitly

assumes that the improper states can coherently be

superimposed so as to yield normalizable states. This

assumption is valid in the case of Wigner particles but

fails in the case of infraparticles. A more adequate

method of converting the improper states into normal-

izable ones is based on the idea of acting on them with

suitable localizing operators. In the case of quantum

mechanics, one could take as a localizing operator any

sufficiently rapidly decreasing function of the position

operator. It would map the improper ‘‘plane-wave

states’’ of sharp momentum into finitely localized

states which thereby become normalizable. In quan-

tum mechanics, these two approaches can be shown to

be mathematically equivalent. The situation is differ-

ent, however, in quantum field theory.

In quantum field theory, the appropriate localiz-

ing operators L are of the form [18]. They constitute

a (noncl osed) left ideal L in the C



-algebra A

generated by all local operators. Improper particle

states of sharp energy–momentum p can then be

defined as linear maps ji

: L!Hsatisfying

UðxÞjLi

¼ e

ipx

jLðxÞi

; L 2L ½25

It is instructive to (formally) replace L here by the

identity operator, making it clear that this relation

indeed defines improper states of sharp energy–

momentum.

In theories of massive particles, one can always find

localizing operators L 2L such that their images

jLi

2H are states with a sharp mass. This is the

situation covered in Wigner’s approach. In theories

with long-range forces there are, in general, no such

operators, however, since the process of localization

inevitably leads to the production of low-energy

massless particles. Yet improper states of sharp momen-

tum still exist in this situation, thereby leading to a

meaningful generalization of Wigner’s particle concept.

That this characterization of particles covers all

situations of physical interest can be justified in the

general setting of relativistic quantum field theory as

follows. Picking g

as in [4] and any vector  2H

with finite energy, one can show that the functionals



, t 2 R, given by



ðL



LÞ¼

ðx

Þh; ðL



LÞðxÞi; L 2L ½26

are well defined and form an equicontinuous family

with respect to a certain natural locally convex

topology on the algebra C= L



L. This family of

functionals therefore has, as t !1, weak- limit

points, denoted by . The functionals  are positive

on C but not normalizable. (Technically speaking,

they are weights on the underlying algebra A.) Any

such  induces a positive-semidefinite scalar product

on the left ideal L given by

i¼

ðL



Þ; L

; L

2L ½27

After quotienting out elements of zero norm and

taking the completion, one obtains a Hilbert space

and a linear map L 7!jLi from L into that space.

Moreover, the spacetime translations act on this

space by a unitary representation satisfying the

relativistic spectrum con dition.

It is instructive to compute these functionals and

maps in theories of massive particl es. Making use of

relation [21] one obtains, with a slight change of

notation,

i¼

dðpÞhp jL



jp i½28

where  is a measure giving the probability density

of finding at asymptotic times in state  a particle of

energy–momentum p. Once again, possible summa-

tions over different particle types and internal

degrees of freedom have been omitted here. Thus,

Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools 463

setting jLi

L jpi, one concludes that the map

L 7!jLi can be decomposed into a direct integral of

improper particle states of sharp energy–momen-

tum, ji=



d(p)

1=2

ji

. It is cruci al that this result

can also be established without any a priori input

about the nature of the particle content of the

theory, thereby providing evidence of the universal

nature of the concept of improper particle states of

sharp momentum, as outlined here.

Theorem 5 Consider a relativistic quantum field

theory satisfying the standing assumptions. Then the

maps L 7!jLi defined above can be decomposed into

improper particle states of sharp energy–momentum p,

ji ¼



dðpÞ

1=2

ji

½29

where  is some measure depending on the state 

and the respective time limit taken.

It is noteworthy that whenever the space of

improper particle states corresponding to fixed

energy–momentum p is finite dimensional (finite

particle multiplets), then in the corresponding Hilbert

space there exists a continuous unitary representation

of the little group of p. This implies that improper

momentum eigenstates of mass m = (p

)

1=2

> 0carry

definite (half)integer spin, in accordance with Wigner’s

classification. However, if m = 0, the helicity need not

be quantize d, in contrast to Wigner’s results.

Though a general scattering theory based on

improper particle states has not yet been developed,

some progress has been made in Buchholz et al.

(1991). There it is outlined how inclusive collision

cross sections of scattering states, where an unde-

termined number of low-energy massless particles

remains unobserved, can be defined in the presence

of long-range forces, in spite of the fact that a

meaningful scattering matrix may not exist.

Asymptotic Completeness

Whereas the description of the asymptotic particle

features of any relativistic quantum field theory can be

based on an arsenal of powerful methods, the question

of when such a theory has a complete particle

interpretation remains open to date. Even in concrete

models there exist only partial results, cf. Iagolnitzer

(1993) for a comprehensive review of the current state

of the art. This situation is in striking contrast to the

case of quantum mechanics, where the problem of

asymptotic completeness has been completely settled.

One may trace the difficulties in quantum field

theory back to the possible formation of superselection

sectors (Haag 1992) and the resulting complex particle

structures, which cannot appear in quantum-mechan-

ical systems with a finite number of degrees of freedom.

Thus, the first step in establishing a complete particle

interpretation in a quantum field theory has to be the

determination of its full particle content. Here the

methods outlined in the preceding section provide a

systematic tool. From the resulting data, one must then

reconstruct the full physical Hilbert space of the theory

comprising all superselection sectors. For theories in

which only massive particles appear, such a construc-

tion has been established in Buchholz and Fredenhagen

(1982), and it has been shown that the resulting Hilbert

space contains all scattering states. The question of

completeness can then be recast into the familiar

problem of the unitarity of the scattering matrix. It is

believed that phase space (nuclearity) properties of the

theory are of relevance here (Haag 1992).

However, in theories with long-range forces, where

a meaningful scattering matrix may not exist, this

strategy is bound to fail. Nonetheless, as in most high-

energy scattering experiments, only some very specific

aspects of the particle interpretation are really tested –

one may think of other meaningful formulations of

completeness. The interpretation of most scattering

experiments relies on the existence of conservation

laws, such as those for energy and momentum. If a

state has a complete particle interpretation, it ought to

be possible to fully recover its energy, say, from its

asymptotic particle content, that is, there should be no

contributions to its total energy which do not manifest

themselves asymptotically in the form of particles.

Now the mean energy–momentum of a state  2His

given by h, Pi, P being the energy–momentum

operators, and the mean energy–momentum contained

in its asymptotic particle content is

d(p)p,where

is the measure appearing in the decomposition [29].

Hence, in case of a complete particle interpretation,

the following should hold:

h; Pi¼

dðpÞp ½30

Similar relations should also hold for other con-

served quantities which can be attributed to parti-

cles, such as charge, spin, etc. It seems that such a

weak condition of asymptotic completeness suffices

for a consistent interpretation of most scattering

experiments. One may conjecture that relation [30]

and its generalizations hold in all theories admitting

a local stress–energy tensor and local currents

corresponding to the charges.

See also: Algebraic Approach to Quantum Field Theory;

Axiomatic Quantum Field Theory; Dispersion Relations;

Perturbation Theory and its Techniques; Quantum

Chromodynamics; Quantum Field Theory in Curved

464 Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools

Spacetime; Quantum Mechanical Scattering Theory;

Scattering, Asymptotic Completeness and Bound States;

Scattering in Relativistic Quantum Field Theory: The

Analytic Program.

Further Reading

Araki H (1999) Mathematical Theory of Quantum Fields.

Oxford: Oxford University Press.

Buchholz D and Fredenhagen K (1982) Locality and the structure

of particle states. Communications in Mathematical Physics

84: 1–54.

Buchholz D, Porrmann M, and Stein U (1991) Dirac versus

Wigner: towards a universal particle concept in quantum field

theory. Physics Letters B 267: 377–381.

Dybalski W (2005) Haag–Ruelle scattering theory in presence of

massless particles. Letters in Mathematical Physics 72: 27–38.

Fredenhagen K, Gaberdiel MR, and Ruger SM (1996) Scattering

states of plektons (particles with braid group statistics) in (2 þ 1)

dinemsional quantum field theory. Communications in Mathe-

matical Physics 175: 319–336.

Haag R (1992) Local Quantum Physics. Berlin: Springer.

Iagolnitzer D (1993) Scattering in Quantum Field Theories.

Princeton, NJ: Princeton University Press.

Jost R (1965) General Theory of Quantized Fields. Providence,

RI: American Mathematical Society.

Steinmann O (2000) Perturbative Quantum Electrodynamics and

Axiomatic Field Theory. Berlin: Springer.

Streater RF and Wightman AS (1964) PCT, Spin and Statistics,

and All That. Reading, MA: Benjamin/Cummings.

Scattering in Relativistic Quantum Field Theory:

The Analytic Program

J Bros, CEA/DSM/SPhT, CEA Saclay,

Gif-sur-Yvette, France

Introduction to the Analytic Structures

of Quantum Field Theory

The importance of complex variables and of the

concept of analyticity in theoretical physics finds

one of its best illustrations in the analytic structure

of relativistic quantum field theory (QFT). The latter

have been investigated from several viewpoints in

the last 50 years, according to the successive

progress in QFT.

In the two main axiomatic frameworks of QFT,

namely the one based on Wightman axioms (for a

short presentation, see Dispersion Relat ions and also

Axiomatic Quantum Field Theory) and the Haag,

Kastler, and Araki theory of ‘‘local observables’’ (see

Algebraic Approach to Quantum Field Theory),

there are general justifications of analyticity proper-

ties for relevant ‘‘N-point structure functions’’ both

in complexified spacetime variables and in complex-

ified energy–momentum variables.

In the Wightman framework, relativistic quantum

fields are operator-valued distributions 

(x)onfour-

dimensional Minkowski spacetime that transform

covariantly under a unitary representation of the

Poincare´ group in the Hilbert space of states. The

basic quantities of QFT are (tempered) distributions

on R

of the form <, (x

) (x

)

>, which

depend on pairs of states , 

, belonging to the

Hilbert space of the QFT considered: they can be

called N-point structure functions of the field  ‘‘in

x-space,’’ namely in Minkowski spacetime (here, for

brevity, we assume that the system is defined in terms

of a single quantum field). In parallel, it is important to

consider the Fourier transform

(p) =

ipx

(x)dx of

the field in the Minkowskian energy–momentum

space (p  x ¼

 p  x denoting the Minkowskian

scalar product). The corresponding quantities

<,

(p

) 

(p

)

> , can then be called N-point

structure functions of the field  ‘‘in p-space,’’ namely

in energy–momentum space.

In the algebraic QFT framework, each basic

local observable B affil iated to a certain bounded

region of spacetime O generates a Haag–Kastler–

Araki quantum field B(x) by the action of

the translations of spacetime, namely B(x) ¼

U(x)BU(x)

1

. Here U(x) denotes the unitary repre-

sentation of the group of spacetime translations in

the Hilbert space of states: B(x) is affiliated to the

translated region O(x) = {y; y  x 2O}. Then again

one can consider N-point structure functions of the

theory of the form <, B (x

) B(x

)

> and

<,

B(p

) 

B(p

)

To summarize the situation as it occurs in both

cases, one can say the following:

1. A certain postulate of relativistic causality

implies the analyticity of structure functions of

a certain class, often called ‘‘Green functions,’’

in the complex energy–momentum variables

= p

þ iq

, in particular for purely imaginary

energies.

2. ‘‘Stability properties’’ of the states , 

such as a

‘‘bounded energy content’’ of these states imply

Scattering in Relativistic Quantum Field Theory: The Analytic Program 465

the analyticity of the previous structure functions

in the complex spacetime variables, in particular

for purely imaginary times.

In both cases, analyticity is obtained as a basic pro-

perty of the Fourier–Laplace transformation in several

variables. Let V

denote the forward cone of the

Minkowskian space (V

V



{x; x

x  x > 0,

> 0}) and let

f ðp þ iq Þ¼

iðpþiqÞx

f ðxÞdx ½1

gðx þ iyÞ¼ð2Þ

4

ipðxþiyÞ

gðpÞdp ½2

be the associated reciprocal Fourier formulas,

applied, respectively, to functions f (x) with support

contained in the translated forward cone V

= a þ

, a 2 V

(or in its closure), and to functions

g(p)

with support contained in the translated forward

cone V

= P þ V

, P 2 V

of energy–momentum

space (or in its closure). Then in view of the

convergence properties of the previous integrals, one

easily checks that

f (k) is holomorphic with possible

exponential increase in the imaginary directions

controlled by the bound e

qa

in the tube domain

= R

þ iV

; similarly, g(z) is holomorphic with

an increase controlled by the exponential bound e

yP

in the tube domain T



= R

þ iV



On the one hand, for each N the structure functions

<,

(p

)

(p

)

> (or <,

B(p

)

B(p

)

have conical support properties of the previous type in

the variables p

, as a consequence of the relativistic

shape of the energy–momentum spectrum. In both

axiomatic frameworks, in fact, one postulates that

there is a state of zero energy–momentum ,calledthe

vacuum, and that the energy–momentum spectrum ,

namely the joint spectrum of the generators P



of the

Lie algebra of the group U(x), is contained in the

closure of V

: this is the so-called spectral condition.

A more refined assumption introduced for the require-

ments in particle physics is that  contains discrete

parts localized on sheets of (mass-shell) hyperboloids

inside V

. These support properties in p-space imply

that the corresponding inverse Fourier transforms

<, (x

)(x

)

> are boundary values of holo-

morphic functions in appropriate tube domains of the

complex space variables (z

,..., z

On the other hand, in order to exhibit structure

functions with conical support properties in x-space,

one needs to build appropri ate algebraic combina-

tions of functions <, (x

) (x

)

> with

permuted argum ents in order to take the benefit of

the causality postulate, which is always formulated

in terms of the commutator of two field operators.

There are two versions of this postulate. In the

Wightman framework, causality is expressed by the

condition of local commutativity or microcausality,

½ðx

Þ; ðx

Þ ¼ 0 for ðx

 x

< 0 ½3

In the algebraic QFT framework, causality is

expressed by a similar property in terms of any

field B(x) generated by a local observable B ¼

B(0)

affiliated to a region of spacetime enclosed in a

given ‘‘double cone’’ O

= V

\ (V

). The corres-

ponding expression of causality is

½Bðx

Þ; Bðx

Þ ¼ 0

for ðx

 x

Þ=2ðV

[ðV

Þ½4

for all a such that a > 2b.

So, we see that basically, causality and spectral

condition generate analyticity respectively in com-

plexified p-space and x-space. However, the situa-

tion is more intricate, since for each N there are

always several holomorphic branches (two in the

case N = 2) in the variables (z

, ..., z

) and also in

the variables (k

, ..., k

): each of these two sets is

obtained essentially by permutations of the N vector

variables. The important point is that these various

branches can be seen to ‘‘communicate together,’’

thanks to the existence of ‘‘coincidence regions’’ of

their boundary values on the reals. Here again the

roles played by causal ity and stability are symmetric

(but inverted): while causality produces coincidence

regions for the holomorphic functions in complex

spacetime, spect ral conditions produce coincidence

regions for the holomorphic functions in complex

energy–momentum space.

In view of a basic theorem of several complex

variable analysis, called the edge-of-the-wedge the-

orem (see below in (4)), the two sets of commu-

nicating holomorphic branches actually define by

mutual analytic continuation two holomorphic

function H

, 

, ..., k

) and W

, 

, ..., z

)in

respective domains D

, 

and 

, 

. However, these

two primitive domains are not natural holomorphy

domains (a phenomenon which is particular to

complex geometry in several variables). The prob-

lem of finding their holomorphy envelopes, namely

the smallest domains

, 

and



, 

in which any

functions holomorphic in the primitive domains can

be analytically continued, is the idealistic purpose of

what has been called the analytic program of

axiomatic QFT. So, we see that there is an analytic

program in x-space and there is an analytic program

in p-space. In practice, except for the case N = 2,

where the complete answer is known, only a partial

knowledge of the holomorphy envelopes has been

obtained.

466 Scattering in Relativistic Quantum Field Theory: The Analytic Program

The analytic program in p-space, which is the

only one to be described in the rest of this article,

was often considered as physically more interesting,

in view of the fact that it aims to establish

analyticity properties of the scattering kernels on

the complex mass shell. As a matter of fact, an

important part of it concerns the derivation of the

analyticity domains of dispersion relations for two-

particle scattering amplitudes. This part is important

from the historical viewpoint as well as from

conceptual, physical, and pedagogical viewpoints

(the reader may find it useful to first check the

article Dispersion Relations, which illustrates how a

structure function of the form H

, 

, k

) can be

used for that purpose with a suitable choice of the

states  and 

). In the general development of the

analytic program (in x-space as well as in p-space),

it is recommended to consider the infinite set of

structure functions H

, 

, ..., k

) and

, 

, ..., z

) where  is the privileged

vacuum state of the theory, in view of the fact that

each of these sets characterizes entirely the field

theory considered.

Before shifting to the analytic program in p-space,

we would like to mention various points of interest

of the analytic program in x-space:

1. Various results of this program have been

extensively used for proving fundamental prop-

erties of QFT, such as the PCT-invariance

theorem, the spin–statistics connection, etc.

A good part of these can be found in the

books by Streater and Wightman (1980) and by

Jost (1965).

2. The funct ions H

and W

are holomorphic in

their respective p-space and x-spac e ‘‘Euclidean

subspaces.’’ To make this clear, let us assume

that a Lorentz frame has been chosen once for

all; the linear subspace of complex spacet ime

(resp. energy–momentum) vectors of the form

z = (iy

, x) (resp. k = (iq

, p)) is called the ‘‘Eucli-

dean subspace’’ of the corresponding complex

Minkowskian space, in view of the fact that the

quadratic form z

z  z = (y

þ x

) (resp.

k  k = (q

þ p

)) has a definite (negative)

sign on that subspace. Then it has been estab-

lished that (for each N) the restrictions of H

and W

to the corresponding N-vector Euclidean

subspaces are the Fourier transforms of each

other. This fact participates in the foundation of

the Euclidean formulation of QFT or ‘‘QFT at

imaginary times’’; the latter has provided many

important results in QFT, in particular for the

rigorous study of field models (initiated by

Glimm and Jaffe in the 1970s).

3. A more recent extension of QFT called thermal

QFT (TQFT), which aims to study the behavior of

quantum fields in a thermal bath, can be described

in terms of a modified analytic program. In the

latter, the spectral condition is replaced by the

so-called KMS condition, which prescribes x-space

analyticity properties of a particular type for the

structure functions W

: it requires analyticity

together with periodicity conditions with respect

to imaginary times, the period being the inverse of

the temperature (see Thermal Quantum Field

Theory). The usual analytic structure for the

theories with vacuum and spectral conditions is

recovered in the zero-temperature limit.

4. In more recent investigations concerning quan-

tum fields on (holomorphic) curved spacetim es,

analyticity properties of the structure functions

similar to those of thermal QFT can be estab-

lished. This is the case in particul ar with de Sitter

spacetime, for which a notion of ‘‘temperature of

geometrical origin’’ is most simply exh ibited.

In this article, an account of the general analytic

program of axiomatic QFT in complex energy–

momentum space will be presented; it will describe

some of the methods which have been used for

establishing analyticity properties of the N-point

structure functions of QFT and corresponding proper-

ties of the (n !n

)-particle collision processes, for all

n, n

such that n  2, n

 2, n þ n

= N. (For a more

detailed study, in particular concerning the microlocal

methods, see the book by Iagolnitzer (1992)).

Concerning the important case N = 4, this article

gives complements to the results described in the

article Dispersion Relations. In fact, the program

allows one to justify other important analytic

structures of the four-point functions and of two-

particle scattering functions. They concern



the field-theoretical basis of analyticity in the

complexified variable of angular momentum, first

introduced and developed in potential theory

(Regge 1959);



the Bethe–Salpeter (BS-) type structure (based on

the additional postulate of asymptotic complete-

ness), which is a relativistic field-theoretical gen-

eralization of the Lippmann–Schwinger structure

of nonrelativistic scattering theory (for Schro¨dinger

equations with Yukawa-type potentials).

The latter allows one to introduce the concept of

composite particle in the field-theoretical framework

(including bound states and unstable particles or

‘‘resonances’’) and also the concept of ‘‘Regge

particle,’’ thanks to complex angular momentum

analysis.

Scattering in Relativistic Quantum Field Theory: The Analytic Program 467

Various Aspects of the General

Analytic Program of QFT in Complex

Energy–Momentum Space

The N-Point Structure Functions of QFT

It is proved in the Wightman QFT axiomatic frame-

work that any QFT is completely characterized by the

(infinite) sequence of its ‘‘N-point functions’’ or

‘‘vacuum expectation values’’ (also called ‘‘Wightman

functions’’)

ðx

; ...; x

Þ¼

< ; ðx

Þðx

Þ >

which are tempered distributions on R

satisfying a

set of general properties that can be split up into

linear and nonlinear conditions. (This is known as

the Wightman reconstruction theorem).

Linear conditions Each individual N-point func-

tion satisfies three sets of linear conditions which

result, respectively, from :

1. Poincare´ invariance: typically, for every Poincare´

transformation g of Minkowski spacetime

ðx

; ...; x

Þ¼W

ðgx

; ...; gx

in particular, the W

are invariant under space-

time translations and therefore defined on the

quotient subspace R

4(N1)

of the differ-

ences x

 x

2. Microcausality: support conditions on commu-

tator functions of the following form:

ðj;jþ1Þ

ðx

;...;x

Þ¼

ðx

;...;x

jþ1

;...;x

W

ðx

;...;x

jþ1

;

;...;x

Þ¼0

in the region of R

defined by (x

x

jþ1

)

< 0.

3. Spectral condition: support conditions on the

Fourier transform

, ..., p

) =  (p

þþ

) 

, ..., p

N1

)ofW

, which assert that

, ..., p

N1

) = 0 if either one of the follow-

ing conditions is fulfilled: p

þþp

62 , for

j = 1, ..., N  1.

For each N, one can then construct a set of

distributions R

()

, ..., x

), called ‘‘generalized

retarded functions’’ (Araki, Ruelle, Steinmann,

1960 (see Iagolnitzer (1992, ref. [EGS])) which are

appropriate linear combinations of multiple com-

mutator functions built from W

and multiplied by

products of Heaviside step-functions (x

j,0

 x

k,0

)of

the differences of time coordinates. Each of these

distributions R

()

, ..., x

) has its support con-

tained in a convex salient cone C



. This construction

can be seen as a generalization of the decomposition

[23] of the commutator C

, 

in the article

Dispersion Relations. Then in view of the Laplace-

transform theorem in several variables, the Fourier

transform

()

, ..., p

) =  (p

þþp

) 

()

([p]

)issuchthat

()

([p]

) is the boundary value

of a holomorphic function

(), (c)

([k]

) defined in a

tube T



= R

4(N1)

þ i



. Here [k]

= [p]

þ i[q]

belongs to a 4(N  1)-dimensional complex linear

space M

(c)

: this is the set of complex vectors

[k]

, ..., k

)suchthatk

þþk

= 0.



the dual cone of C



in the real (4(N  1)-dimen-

sional) [q]

-space. Geometrically, each cone



defined in terms of a certain ‘‘cell’’ of [q]

-space

which is defined by prescribing consistent conditions

of the form "

2 V

with q

j2J

and "

= 1

for all proper subsets J of the set {1, 2, ..., N}.

This is the expression of the microcausality postu-

late (summarized in [3] or [4]) in complex energy–

momentum space. Concerning the difference

between the two formulations [3] and [4], one can

see that there is no geometrical difference concern-

ing the analyticity domains, but differences for the

type of increase of the structure functions in their

tube domains: in the case of [3], they are bounded

by powers of the energy–momenta, while in the case

of [4] they may have an exponential increase

governed by factors of the type e

qa

For each N, the linear space generated by all the

distributions

()

([ p]

) is constrained by a set of

linear relations (called Steinmann relations) which

result from algebraic expressions of discontinuities

of the following type, called (generalized) ‘‘absorp-

tive parts,’’

ðÞ

ð½p

Þ

ð

ð½p

¼< ; ½

ð

ð½p

ðJ

Þ;

ð

ð½p

ðJ

Þ > ½5

for all pairs of adjacent cells (, 

)

( J

, J

)

in the

following sense:  and 

only differ by changing the

value of "

= "

,(J

, J

) denoting any given

partition of the set {1, 2, ..., N}. In [5], the symbols

(

)

denote generalized retarded operators of lower

order and the argument [ p]

(J)

stands for the set of

independent 4-momenta { p

; j 2 J}. Formula [5] may

be seen as an N-point generalization of formula [26]

of Dispersion Relations for the case when the state

=

is replaced by .

Then by applying to [5] the same argument based

on spectral condition as in the exploitation of

eqn [26] in Dispersion Relations, one concludes

that the two distributions

()

and

(

)

coincide on

an open set R

, 

of the form p

= p

< M

, where

j2J

=  p

. It then follows from the gen-

eral ‘‘oblique edge-of-the-wedge theorem’’ (Epstein,

1960; see below) that the two corresponding

holomorphic functions

(), (c)

([k]

) and

(

), (c)

([k]

)

468 Scattering in Relativistic Quantum Field Theory: The Analytic Program

have a common analytic continuation in the union of

their tubes together with a certain complex ‘ ‘connecting

set,’ ’ bordered by R

, 

. Since this argument applies to

all pairs (, 

)

( J

, J

)

, the following important property

holds (see Iagolnitzer (1992,refs.[B2],[EGS])):

Theorem 1

(i) All the holomorphic functions

(), (c)

([k]

)

admit a common analytic continuation

([k]

), call ed the N -point structure function

(or Green function) of the given quantum field

in complex e nergy–momentum space. It is

holomorphic in a ‘‘primitive domain’’ D

(c)

, which is the union of all tubes T



together with complex ‘‘connecting sets ’’ bor-

dered by all the coincidence regions R

, 

defined previously.

(ii) For each N the complex domain D

contains the

whole Euclidean subspace E

of M

(c)

, which is

the set of all complex vect ors [k]

= (k

, ..., k

)

such that k

= (k

j,0

, k

); k

j,0

= iq

j,0

, k

= p

for

j = 1, 2, ..., N.(This Euclidean subspace depends

on the choice of a given Lorentz frame in

Minkowski spacetime.)

Positivity Conditions The Hilbert space framework

which underlies the axioms of QFT implies (an

infinite set of) positivi ty inequalities on the N-point

structure functions of the fields. As a typical

example related to the previous formula [5] when

j= jJ

j= N=2 (for N even), one can mention the

positive-definiteness property of the absorptive parts

for appropriate pairs of adjacent cells (

, 



)

, J

)

, which simply expres ses the positivity of

the following Hilbertian squared norm:



f ð½p

ðJ

Þf ð½p

ðJ

Þ½

ðÞ

ð½p



ð

ð½p

Þd½p

ðJ

d½p

ðJ



f ð½p

ðJ

ð

ð½p

ðJ

Þ > d½p

ðJ



 0 ½6

Scattering Kernels of General (n !n

)-Particle

Collisions and General Reduction Formulas

The presentation of (2 !2)-particle scattering ker-

nels in the article Dispersion Relations can be

generalized to arbitrary (n !n

)-particle collision

processes, involving n incoming massive particles

(n  2) and n

outgoing massive particles (n

 2).

The big ‘‘scattering matrix’’ or ‘‘S-matrix’’ in the

Hilbert space of states is the collection of all partial

scattering matrices S

n, n

or of the equivalent kernels

n, n

n,in

; p

,out

), defined by a straightforward gen-

eralization of formula [20] of the quoted article:

n;n

n;in

;

;out

n;n

n;in

ðp

n;in

;out

ðp

;out

 S

n;n

ðp

n;in

; p

;out

Þ

ðp

n;in

Þ

ðp

;out

Þ½7

Here we have considered for simplicity the case of

collisions involving a single type of particle with

mass m. In the arguments of the wave packets, the

kernel, and the measures (

, 

), p

n,in

and p

,out

respectively, denote the sets of incoming and

outgoing 4-momenta (p

, ..., p

) and (p

, ..., p

)

which all belong to the physical mass shell

= {p; p 2 V

, p

= m

}. By supplementing these

mass-shell constraints with the relativistic law of

conservation of total energy–momentum p

þþ

= p

þþp

, one obtains the definition of the

mass-shell manifold M

n, n

of (n !n

)-particle colli-

sion processes.

We shall reserve the name of scattering kernel (or

scattering amplitude), denoted by T

n, n

n,in

; p

, out

to the so-called ‘‘connected component’’ of the

S-matrix kernel S

n, n

n,in

; p

, out

). By analogy with

the definition of T in terms of S for the two-particle

collision processes (see Dispersion Relations) T

n, n

defined by a recursive algorithm, which amounts to

subtract from S

n, n

all the compon ents of the

(n !n

)-collision processes that are decomposable

into independent collision processes involving smal-

ler number of particles, according to all admissible

partitions of the numbers n and n

For any given N, let us consider all the ‘ ‘affiliated’’

scattering kernels T

n, n

such that n þ n

= N and whose

corresponding collision processes, also called

‘ ‘channels,’’ are deduced from one another by the

relevant exchange of incoming particles a nd

outgoing antiparticles (e.g., 

þ 

!



þ 

, 

þ 

!

þ 

,and



! 

þ 

). There exist general reduc-

tion formulas according to which all these scattering

kernels are restrictions to the mass-shell manifold M

(N)

of appropriate boundary values of the (so-called)

‘ ‘amputated N-point function’ ’

, ..., k

) ¼

 m

) (k

 m

)  H

, ..., k

). More pre-

cisely, these reduction formulas can be written as

follows:

n;n

ðp

n;in

; p

;out

ðÞ

ðNÞ

ðÞ

ðp

; ...; p

ðÞ

ðNÞ

½8

In the latter,

()

denotes a certain boundary value of

on the reals: it is equal to a generalized retarded

function

()

([p]

) which depends in a specific way

on a region of the mass shell, called M

()

(N)

, in which

Scattering in Relativistic Quantum Field Theory: The Analytic Program 469

the (n !n

)-channel is considered. The important

thing to be noted in [8] is the sign convention which

attributes the notation p

to the momentum of any

incoming particle and therefore implies that p

belongs to the negative sheet of hyperboloid H



H

. This is the price to pay for expressing

symmetrically the energy–momentum conservation

law as p

þ p

þþp

= 0 (according to the QFT

formalism), but it also displays, as a nice feature,

the fact that all the affiliated scattering kernels

n, n

such that n þ n

= N are located on the

various connected components of the mass shell

(N)

2 H

; j = 1, 2, ..., N): the choice of the

sheet H



or H

of H

is exactly linked to the

incoming or outgoing character of the particle

considered.

Remark 1 The reduction formulas are more usually

expressed in terms of the Fourier transforms of the

(connected parts of the) N-point amputated chronolo-

gical functions 

([p]

)(see Scattering in Relativistic

Quantum Field Theory: Fundamental Concepts and

Tools). As a matter of fact, the latter coincide with the

boundary values

()

([p]

)ofH

in the corresponding

relevant regions M

()

(N)

Remark 2 Coming back to the case of two-particle

scattering amplitudes (i.e., n = n

= 2, N = 4), one

can see that the general study presented here implies

the consideration of the four-point function

, k

), which is a holomorphic function

of three independent complex 4-momenta (since

þ k

= 0). In that case, the domain D

contains 32 tubes T



which are specified by triplets

of conditions such as q

2 V

, q

2 V

, q

2 V

,or

q

2 V

, q

þ q

2 V

, q

þ q

2 V

, and those

obtained by permutations of the subscripts

(1, 2, 3, 4) and also by a global substitut ion of the

cone V



to V

Remark 3 The logical path from the postulates of

QFT to the analyticity properties of two-particle

scattering amplitudes that has been followed in the

article Dispersion Relations can be seen as a partial

exploitation of the general analyticity properties of

the four-point function: one was specially interested

there in the analyticity properties of H

in a single

4-momentum k

= k

(at fixed real values of

= p

). The ‘‘partial reduction formula’’ [27] of

Dispersion Relations corresponds to the restriction

of eqn [8] (for N = 4) to the linear submanifold

= p

, p

= p

). It may also be worthwhile to

stress the fact that, in spite of the exponential

bounds on H

implied by the postulates of algebraic

QFT, it has been possible to prove that the

scattering function is still bounded by a power of s

in its cut-plane (or crossing) domain; the dispersion

relations with two subtractions are still justified in

that case (Epstein, Glaser, Martin, 1969 (see Martin

(1969, preprint))).

Off-Shell Character of D

: Nontriviality of the

Analytic Structure of the Scattering Kernels

One can now see that for each value of N(N  4)

the situation created by complex geometry in the

space C

4(N1)

of [k]

is a mere generalization of the

one described in a simple situation in the article

Dispersion Relations.

1. There exists a funda mental (3N  4)-dimensional

complex submanifold, namely the complex mass

shell M

(c)

(N)

defined by the equations k

= m

;

j = 1, ..., N, which connects together the various

real mass-shell components M

()

(N)

interpreted as

the various physical regions of a set of affiliated

(n !n

)-collision processes. The problem of

proving the ‘‘analyticity of (n !n

)-scattering

functions’’ thus amounts to constructing such

holomorphic functions on the complex manifold

(c)

(N)

, whose boundary values on the various real

regions M

()

(N)

would reproduce the relevant

scattering kernels T

n, n

(p

n,in

; p

, out

2. All the tubes T



which generate the primitive

domain D

are off-shell domains, namely their

intersections with M

(c)

(N)

are empty. This simply

comes from the fact that the conditions q

2 V



(included in thei r definition) and k

= m

> 0 are

incompatible. One can also check that adding the

coincidence regions R

, 

between adjacent tubes

does not improve the situation. However, one

can state as a relevant scope the following

program.

3. Linear program (so-called because it only rel ies

on the linear conditions presented in the section

‘‘N-point structure functions of QFT’’): find parts

of the holomorphy envelope of D

(possibly

improved by the exploitation of the Steinmann

relations) whose intersections with the complex

mass shell M

(c)

(N)

are nonempty. In the best case,

show that such intersections can exist which

connect two different regions M

()

(N)

together,

which means ‘‘proving the crossing property

between these two regions.’’

4. We shall see in the following that, except for the

case N = 4, the results of this linear program

have been rather disappointing as far as reaching

the complex mass shell is concerned; however,

other interesting analytic structures also coming

from positivity conditions and from the addi-

tional postulate of asymptotic completeness have

been investigated under the general name of

470 Scattering in Relativistic Quantum Field Theory: The Analytic Program

nonlinear program. The ‘‘synergy’’ created by the

combination of these two programs remains, to a

large extent, to be explored.

Results of Analytic Completion

in the ‘‘Linear Program’’

We can only outline here some of the geometrical

methods which allow one to compute parts of the

holomorphy envelopes of the domains D

. One

important method, which may be used after apply-

ing suitable conformal mappin gs, reduces to the

following basic theorem.

The tube theorem The holomorphy envelope of a

‘‘tube domain’’ of the form T

= R

þ iB, where B is

an arbitrary domain in R

called the basis of the

tube, is the convex tube T

= R

þ i

B, where

B is the

convex hull of B.

The opposite or oblique edge-of-the-wedge theo-

rem (Epstein 1960 (see Streater and Wightman

(1980, ch. 2, ref. 18))) is a refined local version of

the tube theorem, in which the basis B is of the form

B = C

[ C

, where C

, C

are two disjoint (opposite

or nonopposite) cones with apex at the origin

and where T

is replaced by a pair of ‘‘local tubes’’

(loc)

, T

(loc)

). Here the adjec tive ‘‘local’’ means that

the real parts of the variables are confined in a given

open set U (which can be arbitrarily small). The

connectedness of T

is now replaced by the

consideration of any pair of functions (f

, f

)

holomorphic in these local tubes whose boundary

values on their common real set U coincide. The

result is that f

and f

admit a common analytic

continuation f in a local tube T

(loc)

, where C is the

convex hull of C

[ C

. In the case of opposite cones

= C

), f is then analytic in the real set U, while

in the general oblique case f is only analytic in a

complex connecting set bordered by U (namely a set

which connects T

(loc)

and T

(loc)

). There exists an

extended version of the edge-of-the-wedge theorem

in which the boundary values of f

and f

are only

defined as distributions.

For simplicity, we shall just give a very rough

classification of the type of results obtained. We

shall distinguish:



analyticity domains in the space of several

(possibly all) variables: they can be of global

type or of microlocal type, namely restricted to

complex neighborhoods of real points;



analyticity domains in special families of one-

dimensional complex manifolds; and



combinations of one-dimensional results which

generate domains in several variables by a refined

use of the tube theorem, called the Malgrange–

Zerner ‘‘flat tube theorem,’’ or ‘‘flat edge-of-the-

wedge theorem.’’ In the latter, the local tubes

(loc)

and T

(loc)

of f

and f

reduce to one-variable

domains of the upper half-plane in separate

variables z

= x

þ iy

, z

= x

þ iy

but with a

common range of real parts (x

, x

) 2U. The data

, x

) and f

, z

) have coinciding boundary

values (f

, x

) = f

, x

)) in the limit (y

!0,

!0). The result is again the existence of a

common analytic continuation to f

and f

, which

is a function of two complex variables f (z

, z

)in

the intersection of the quadrant (y

> 0, y

> 0)

with a complex neighborhood of U. (Note that

this result of complex analysis still holds when the

real boundary values of the holomorphic func-

tions have singularities, namely are only defined

in the sense of distributions).

Global analyticity properties The following prop-

erty (discovered by Streater for three-point func-

tions) looks like an extension of the tube theorem.

The holomorphy envelope of the union of two tubes



, T



corresponding to adjacent pairs of cells

(, 

)

, J

)

together with a complex connecting set

bordered by R

, 

= {[ p]

; p

< m

} is the convex

hull T

, 

of the union of these tubes minus the

following analytic hypersurface 

which can be

called ‘‘a cut’’: 

= {[k]

: k

= m

þ ,   0}. The

interest of this result (although it remains by itself an

off-shell result) is that it can generate larger cut-

domains by additional analytic completions, which

may have intersections with the complex mass shell

(see below for the case N = 4).

Microlocal analyticity prop erties In the case of the

four-point function

, it is possible to consider

opposite cut-domains of the previous type, for which



= 

{1, 2}

is the energy-cut of the channel (1, 2 !

3, 4), and for which the spectral conditions presc ribe

an ‘‘edge-of-the-wedge situation’’ in the neighbor-

hood of the corresponding mass-shell component

(1, 2 !3, 4)

. The result is that H

is proved to be

holomorphic in a full complex cut-neighborhood of

(1, 2 !3, 4)

in the ambient complex energy–momen-

tum space. The intersection of this local domain

with the complex mass shell M

(c)

(4)

is of course a full

complex cut-neighborhood of M

(1, 2 !3, 4)

in M

(c)

(4)

,and

this proves that the corresponding scattering amplitude

is the boundary value of an analytic scattering function

defined as the restriction

F(s, t) ¼

(c)

(4)

:itis

holomorphic in a domain of complex (s, t)space

deprived from the s–cut.

In the general case N > 4, the results are less

spectacular, although a more sophisticated microlocal

method involving a ‘‘generalized edge-of-the-wedge

Scattering in Relativistic Quantum Field Theory: The Analytic Program 471