On the Uniqueness of the Non-Abelian Gauge Theories in Epstein-Glaser Approach to ...

55 Pages·1998·0.72 MB·English

On the Uniqueness of the Non-Abelian Gauge Theories
in Epstein-Glaser Approach to Renormalisation Theory
D. R. Grigore 1
Dept. of Theor. Phys., Inst. Atomic Phys.
Bucharest-M(cid:21)agurele, P. O. Box MG 6, ROMA^NIA
Abstract
WegeneralisearesultofAsteandScharfsayingthat, undersomereasonableassumptions,
consistent with renormalisation theory, the non-Abelian gauge theories describes the only
possibility of coupling the gluons. The proof is done using Epstein-Glaser approach to
renormalisation theory.
1e-mail: [email protected],[email protected]
1 Introduction
Renormalisation theory has a long history and among the many achievements one can count
quantum electrodynamics (with the extremely good experimental veri(cid:12)cation) and, more re-
cently, the non-Abelian gauge theories which are used to describe electro-weak theory and
quantum chromodynamics. The major point is that these theories are renormalisable i.e. one
can get rid of the so-called ultra-violet divergences in a consistent way and obtain a well de-
(cid:12)ned series for the S-matrix. Using the traditional approach based on Feynman diagrams, this
assertion has been proved in second order of the perturbation theory by ’t Hooft and Veltman
and has been established in all orders of the perturbation theory in the extremely sophisticated
approach of Becchi, Rouet and Stora.
However, it was proved by Epstein and Glaser [28], [30] that the most natural and straight-
forward way of dealing with perturbation theory and constructing a S-matrix ful(cid:12)lling Bogo-
liubov axioms, is based on a direct exploitation of the causality axiom, and this can be done
using a technical device, called the distribution splitting. This analysis was performed for a
scalar (cid:12)eld and extended for quantum electrodynamics in an external (cid:12)eld in [26]. The full
analysis of the interacting QED was done by G. Scharf and collaborators and is presented in
a pedagogical manner in [43]. In recent years, the analysis was extended to the case of pure
non-Abelian gauge symmetries [15], [16], [18], [19], [31]-[34], [37], [39] and of the electro-weak
theory [27], [5]. Many other achievements of this approach are illustrated by the bibliography.
Recently Aste and Scharf [2] proved that the only possibility of non-trivial coupling between
r > 1 zero-mass vector (cid:12)elds is through the usual Yang-Mills recipe. In other words, the
existence of a compact semi-simple Lie group (to which the gauge principle is applied to obtain
a local gauge theory) should not be assumed from the beginning: it simply follows from the
consistency requirements of the theory. The mainhypothesis leading to this result was a certain
quantum form of the gauge principle invariance imposed to the S-matrix.
In this paper we will generalise this result and simplify somewhat they proof. Let us explain
inwhat sense ourapproach ismore general. The mainobstacle in constructing theperturbation
series for a zero-mass (cid:12)eld is the fact that, as it happens for the electromagnetic (cid:12)eld, one is
forced to use non-physical degrees of freedom for the description of the free (cid:12)elds [50], [46],
[36] in a Fock space formalism; one introduces in this Fock space a non-degenerate sesquilinear
form and the true physical Hilbert space is obtained by a certain process of factorisation.
More sophisticated, one can extend the Fock space to an auxiliary Hilbert space H including
gh
some (cid:12)ctious (cid:12)elds, called ghosts, and construct an supercharge (i.e. an operator Q verifying
Q2 = 0) such that the physical Hilbert space is H (cid:17) Ker(Q)=Im(Q) (see for instance [49]
phys
and references quoted there). We will present a careful analysis (which seems to be missing
from the literature) of the Fock space representation of the electromagnetic (cid:12)eld. The problem
that one faces in the attempt to construct the perturbative S matrix (cid:19)a la Epstein-Glaser is
that one can de(cid:12)ne Wick monomials (from which the S-matrix is built) only on the auxiliary
Fock space H ; so one must impose, beside the usual Bogoliubov axioms, the supplementary
gh
condition that the S matrix factorizes to H . We will prove that, the combination of these
phys
conditions leads uniquely to the pure Yang-Mills interaction. No other hypothesis are necessary
to prove this statement. It is necessary to study rigorously the S matrix only up to order 2
1
in the perturbative sense. The gauge invariance condition is a consequence of our analysis and
it is not necessary to impose it as an independent axiom. We also hope to present a simpler
approach to the question of unitarity of the S-matrix.
The same analysis will be then applied to a theory of quarks and gluons. The result is that
the Dirac Fermions should form a multiplet of the semi-simple compact group which appears
naturally from the analysis of the pure Yang-Mills case.
We mention here that results of these type have been obtained quite a long time ago in [40]
and [10] from some arguments concerning the high energy behaviour of the S matrix elements
in the tree approximation. However, a rigorous basis of this analysis seems to be lacking.
In Section 2 we will present the general scheme of construction of a perturbation theory
in the causal approach of Epstein and Glaser giving all the relevant details about the Fock
space construction of the theory. In Section 3 we will study in detail the quantization of the
electromagnetic (cid:12)eld, because some subtle points are simply overlooked in the literature or
presented too summarily and not completely correct. We hope that we will be able to argue
convincinglythatadeepanclearunderstandingofthesepointsisessentialfortheunderstanding
ofquantumgaugetheories. InSection4wepresent ourmainresultconcerning theunicityofthe
Yang-Mills interaction and we generalise the analysis to the case when matter (cid:12)elds (quarks)
are present. In the last Section we indicate some further possible developments.
Regarding the level of rigor, we make the following comments: (a) we will work with the
formal notations for distributions for reasons of simplicity; (b) when working with Hilbert
spacesofL2(X;d(cid:11))onehastoconsidernotfunctionsbutclassesoffunctionwhichareidentically
almosteverywhere (a.e.); (c)domainproblemsforunboundedoperatorscanbe(cid:12)xedinstandard
ways [50], [42].
2
2 Perturbation Theory in Fock Spaces
2.1 Second Quantization
Here we give the main concepts and formul(cid:26) connected to the method of second quantization.
We follow essentially [48] ch. VII; more details can be found in [9] and [7].
The idea of the method of second quantization is to provide a canonical framework for
a multi-particle system in case one has a Hilbert space describing an \elementary" particle.
(One usually takes the one-particle Hilbert space H to be some projective unitary irreducible
representation of the Poincar(cid:19)e group, but this is not important for this subsection). Let H be
a (complex) Hilbert space; the scalar product on H is denoted by < (cid:1);(cid:1) >. One (cid:12)rst considers
the tensor algebra
T(H) (cid:17) (cid:8)1 H⊗n; (2.1.1)
n=0
where, by de(cid:12)nition, the term corresponding to n = 0 is the division (cid:12)eld C. The generic
element of T(H) is of the type (c;(cid:8)(1);(cid:1)(cid:1)(cid:1);(cid:8)(n);(cid:1)(cid:1)(cid:1));(cid:8)(n) 2 H⊗n; the element (cid:8) (cid:17) (1;0;:::)
0
is called the vacuum. Let us consider now the symmetrisation (resp. antisymmetrisation)
operators S(cid:6) de(cid:12)ned by
S(cid:6) (cid:17) (cid:8)1 S(cid:6) (2.1.2)
n=0 n
where S(cid:6) = 1 and S(cid:6); n (cid:21) 1 are de(cid:12)ned on decomposable elements in the usual way
0 n
X
1
S+(cid:30) ⊗(cid:1)(cid:1)(cid:1)⊗(cid:30) (cid:17) (cid:30) ⊗(cid:1)(cid:1)(cid:1)⊗(cid:30) (2.1.3)
n 1 n n! P(1) P(n)
P2Pn
and X
1
S−(cid:30) ⊗(cid:1)(cid:1)(cid:1)⊗(cid:30) (cid:17) (−1)jPj(cid:30) ⊗(cid:1)(cid:1)(cid:1)⊗(cid:30) ; (2.1.4)
n 1 n n! P(1) P(n)
P2Pn
here P is the group of permutation of the numbers 1;2;:::;n and jPj is the sign of the permu-
n
tationP. Oneextends theoperatorsS(cid:6) arbitraryelements ofT bylinearity andcontinuity; itis
n
convenient to denote the elements in de(cid:12)ned by these relations by (cid:30) _(cid:1)(cid:1)(cid:1)_(cid:30) and respectively
1 n
by (cid:30) ^(cid:1)(cid:1)(cid:1)^(cid:30) .
1 n
We now de(cid:12)ne the Bosonic (resp. Fermionic) Fock space according to:
F(cid:6)(H) (cid:17) S(cid:6)T(H); (2.1.5)
obviously we have:
F(cid:6)(H) = (cid:8)1 H(cid:6) (2.1.6)
n=0 n
where
H(cid:6) (cid:17) C; H(cid:6) (cid:17) S(cid:6)H⊗n (n (cid:21) 1) (2.1.7)
0 n n
are the so-called nth-particle subspaces.
The operations _ (resp. ^) make F(cid:6)(H) into associative algebras. One de(cid:12)nes in the
Bosonic (resp. Fermionic) Fock space the creation and annihilation operators as follow: let
(cid:30) 2 H be arbitrary. In the Bosonic case they are de(cid:12)ned on elements from 2 H+ by
n
p
A((cid:30))y (cid:17) n+1(cid:30)_ (2.1.8)
3
and respectively
1
A((cid:30)) (cid:17) p i (2.1.9)
(cid:30)
n
where i is the unique derivation of the algebra F+(H) verifying
(cid:30)
i 1 = 0; i =< (cid:30); > 1: (2.1.10)
(cid:30) (cid:30)
Remark 2.1 We note that the general idea is to associate to every element of the one-particle
space (cid:30) 2 H a couple of operators A]((cid:30)) acting in the Fock space F+(H).
As usual, we have the canonical commutation relations (CCR):
(cid:2) (cid:3) (cid:2) (cid:3)
y y y
[A((cid:30));A( )] = 0; A((cid:30)) ;A( ) = 0; A((cid:30));A( ) =< (cid:30); > 1: (2.1.11)
The operators A( ); A( )y are unbounded and adjoint one to the other.
In the Fermionic case we de(cid:12)ne these operators on elements from 2 H− by
n
p
A((cid:30))y (cid:17) n+1(cid:30)^ (2.1.12)
and respectively
1
A((cid:30)) (cid:17) p i (2.1.13)
(cid:30)
n
where i is the unique graded derivation of the algebra F−(H) verifying
(cid:30)
i 1 = 0; i =< (cid:30); > 1: (2.1.14)
(cid:30) (cid:30)
Now we have the canonical anticommutation relations (CAR):
(cid:8) (cid:9) (cid:8) (cid:9)
fA((cid:30));A( )g = 0; A((cid:30))y;A( )y = 0; A((cid:30));A( )y =< (cid:30); > 1: (2.1.15)
The operators A( ); A( )y are bounded and adjoint one to the other.
If U is a unitary (or antiunitary) operator on H, it lifts naturally to an operator Γ(U) on
the tensor algebra T(H), according to
Γ(U) (cid:17) (cid:8)1U⊗n (2.1.16)
0
or, more explicitly, on decomposable elements
Γ(U) ⊗(cid:1)(cid:1)(cid:1)⊗ = U ⊗(cid:1)(cid:1)(cid:1)⊗U : (2.1.17)
1 n 1 n
The operator Γ(U) leaves invariant the symmetric and resp. the antisymmetric algebras
F(cid:6)(H) and we have
Γ(U )A((cid:30))Γ(U ) = A(U (cid:30)): (2.1.18)
g g−1 g
4
2.2 Elementary Relativistic Free Particles
As we have anticipated in the previous subsection, on usually takes H to be the Hilbert space
of an unitary irreducible representation of the Poincar(cid:19)e group. We give below the relevant
formul(cid:26) for the scalar particle of mass m and for the photon. By comparison, one will be able
to see the origin of the di(cid:14)culties of the renormalisation theory for zero-mass particles.
According to [48], a scalar particle of mass m can be described in the Hilbert space H (cid:17)
L2(X+;C;d(cid:11)+) of Borel complex function (cid:30) de(cid:12)ned on the upper hyperboloid of mass m (cid:21) 0
m m
X+ (cid:17) fp 2 R4j kpk2 = m2g which are square integrable with respect to the Lorentz invariant
m
measure d(cid:11)+ (cid:17) dp . Here the conventions are the following: k (cid:1) k is the Minkowski norm
m 2!(p)
de(cid:12)ned by kpk2 (cid:17) p(cid:1)p and p(cid:1)q is the Minkowski bilinear form:
p(cid:1)q (cid:17) p q −p(cid:1)q: (2.2.1)
0 0
p
If p 2 R3 we de(cid:12)ne (cid:28)(p) 2 X+ according to (cid:28)(p) (cid:17) (!(p);p); !(p) (cid:17) p2 +m2:
m
The scalar product in H is:
Z
< (cid:30); >(cid:17) d(cid:11)+(cid:30)(p) (p): (2.2.2)
m
Xm+
The expression for the corresponding unitary irreducible representation of the Poincar(cid:19)e
group is:
(U (cid:30))(p) (cid:17) eia(cid:1)p(cid:30)((cid:3)−1 (cid:1)p) for (cid:3) 2 L"; (U (cid:30))(p) (cid:17) (cid:30)(I (cid:1)p); (2.2.3)
a;(cid:3) It s
here I ; I are the elements of the Lorentz group corresponding to the spatial and respectively
s t
temporal inversion. Also by ((cid:3);p) 7! (cid:3)(cid:1)p we denote the usual action of the Lorentz group on
R4 and C4. The couple (H;U) is called scalar particle.
For the photon, such a simple description of the Hilbert space as a space of functions
is no longer available. However, we can obtain a description of this type if one considers a
factorisation procedure [48]. Let us consider the Hilbert space H (cid:17) L2(X+;C4;d(cid:11)+) with the
0 m
scalar product Z
< (cid:30); >(cid:17) d(cid:11)+ < (cid:30)(p); (p) > (2.2.4)
0 C4
X+
P 0
where < u;v > (cid:17) 4 u v is the usual scalar product from C4. In this Hilbert space we have
C4 i=1 i i
the following (non-unitary) representation of the Poincar(cid:19)e group:
(U (cid:30))(p) (cid:17) eia(cid:1)p(cid:3)(cid:1)(cid:30)((cid:3)−1 (cid:1)p) for (cid:3) 2 L"; (U (cid:30))(p) (cid:17) (cid:30)(I (cid:1)p): (2.2.5)
a;(cid:3) It s
Let us de(cid:12)ne on H the operator g by
(g (cid:1)(cid:30))(p) (cid:17) g (cid:1)(cid:30)(p) (2.2.6)
and following non-degenerate sesquilinear form:
((cid:30); ) (cid:17) − < (cid:30);g (cid:1) >; (2.2.7)
5
here g 2 L" is the Minkowski matrix with diagonal elements 1;−1;−1;−1 and the operator g
is appearing in (2.2.6) also called a Krein operator. Explicitly:
Z Z
((cid:30); ) (cid:17) d(cid:11)+((cid:30)(p); (p)) = d(cid:11)+g(cid:22)(cid:23)(cid:30) (p) (p); (2.2.8)
0 0 (cid:22) (cid:23)
X+ X+
0 0
the indices (cid:22);(cid:23) take the values 0;1;2;3 and the summation convention over the dummy indices
is used.
Then one easily establishes that we have
(U (cid:30);U ) = ((cid:30); ); for (cid:3) 2 L"; (U (cid:30);U ) = ((cid:30); ): (2.2.9)
a;(cid:3) a;(cid:3) It It
We have now two elementary results:
Lemma 2.2 Let us consider the following subspace of H:
H0 (cid:17) f(cid:30) 2 Hj p(cid:22)(cid:30) (p) = 0g: (2.2.10)
(cid:22)
Then the sesquilinear form ((cid:1);(cid:1))j is positively de(cid:12)ned.
H
Lemma 2.3 Let us consider the following subspace of H0:
H00 (cid:17) f(cid:30) 2 H0j k(cid:30)k = 0g: (2.2.11)
Then
H00 (cid:17) f(cid:30) 2 Hj there exists (cid:21) : X+ ! C s:t: (cid:30)(p) = p(cid:21)(p)g: (2.2.12)
0
Then we have the following result:
Proposition 2.4 The representation (2.2.9) of the Poincar(cid:19)e group leaves invariant the sub-
spaces H0 and H00 and so, it induces an representation in the Hilbert space
H (cid:17) (H0=H00) (2.2.13)
photon
(here by the overline we understand completion). The factor representation, denoted also by
U is unitary and irreducible. By restriction to the proper orthochronous Poincar(cid:19)e group it is
equivalent to the representation H[0;1] (cid:8)H[0;−1].
By de(cid:12)nition, the couple (H ;U) is called photon.
photon
6
2.3 Free Fields and Wick products
Let us apply the second quantization procedure to the scalar particle, i.e. we consider that,
in the general scheme from the (cid:12)rst subsection, the one-particle subspace H is the Hilbert
space corresponding to the scalar particle and we consider Bose statistics i.e. the Bosonic Fock
space. Then one can canonically identify the nth-particle subspace F+(H) with the set of Borel
functions (cid:8)(n) : (X+)(cid:2)n ! C which are square integrable with respect to the product measure
m
d((cid:11)+)(cid:2)n and verify the symmetry property
m
(cid:8)(n)(p ;:::;p ) = (cid:8)(n)(p ;:::;p ); 8P 2 P : (2.3.1)
P(1) P(n) 1 n n
On the dense domain of test functions one can de(cid:12)ne the annihilation operators:
p
(A(k)(cid:8))(n)(p ;:::;p ) (cid:17) n+1(cid:8)(n+1)(k;p ;:::;p ): (2.3.2)
1 n 1 n
Working with the formalism of rigged Hilbert spaces one can also make sense of the creation
operators:
(cid:0) (cid:1) Xn
1
A(k)y(cid:8) (n)(p ;:::;p ) (cid:17) 2!(k)p (cid:14)(k −p )(cid:8)(n−1)(p ;:::;p^;:::;p ) (2.3.3)
1 n i 1 i n
n
i=1
P Q
where the Bourbaki conventions (cid:17) 0; (cid:17) 1 are used.
; ;
These operators verify the canonical commutation relations (see (2.1.11)):
(cid:2) (cid:3) (cid:2) (cid:3)
[A(k);A(k0)] = 0; A(k)y;A(k0)y = 0; A(k);A(k0)y = 2!(k)(cid:14)(k −k0)1 (2.3.4)
and can be used to express the creation and annihilation operators A((cid:30)); A((cid:30))y de(cid:12)ned in
the preceding section; namely we have:
Z Z
A((cid:30)) = d(cid:11)+(k)(cid:30)(k)A(k); A((cid:30))y = d(cid:11)+(k)(cid:30)(k)A(k)y: (2.3.5)
m m
Xm+ Xm+
Now we de(cid:12)ne the scalar free (cid:12)eld. Let f 2 S(R4) be a test function. We de(cid:12)ne the Fourier
transform with the convention:
Z
1
f~(p) (cid:17) e−ip(cid:1)xf(x): (2.3.6)
(2(cid:25))2
R4
We also de(cid:12)ne the functions (cid:12)
(cid:12)
f (k) (cid:17) f~((cid:6)k)(cid:12) : (2.3.7)
(cid:6)
Xm+
One can give now the expression of the free scalar (cid:12)eld of mass m on the dense domain of
the test functions according to:
Z
p p
(’(f)(cid:8))(n)(p ;(cid:1)(cid:1)(cid:1);p ) = 2(cid:25)[ n+1 d(cid:11)+(k)f (k)(cid:8)(n+1)(k;;p ;(cid:1)(cid:1)(cid:1);p )+
1 n m + 1 n
Xm+
Xn
1
p f (p )(cid:8)(n−1)(p ;(cid:1)(cid:1)(cid:1);p^;(cid:1)(cid:1)(cid:1);p )]: (2.3.8)
− i 1 i n
n
i=1
7
Then one can extend it to a selfadjoint operator on the Fock space. One can de(cid:12)ne the
scalar (cid:12)eld in a (cid:12)xed point as follows. First one de(cid:12)nes the negative frequency part by:
Z
p
(’ (x)(cid:8))(n)(p ;(cid:1)(cid:1)(cid:1);p ) (cid:17) (2(cid:25))−3=2 n+1 d(cid:11)+(k)e−ix(cid:1)k(cid:8)(n+1)(k;;p ;(cid:1)(cid:1)(cid:1);p ) (2.3.9)
− 1 n m 1 n
Xm+
as a legitimate operator in the Fock space. Working with rigged Hilbert spaces one can de(cid:12)ne
the positive frequency part:
Xn
1
(’ (x)(cid:8))(n)(p ;(cid:1)(cid:1)(cid:1);p ) (cid:17) (2(cid:25))−3=2p eix(cid:1)pi(cid:8)(n−1)(p ;(cid:1)(cid:1)(cid:1);p^;(cid:1)(cid:1)(cid:1);p ): (2.3.10)
+ 1 n 1 i n
n
i=1
Then the expression
’(x) (cid:17) ’ (x)+’ (x) (2.3.11)
+ −
is the called real scalar (cid:12)eld in the point x. One can justify this de(cid:12)nition by making sense of
the formula:
Z Z
p (cid:2) (cid:3)
’(f) = dxf(x)’(x) = 2(cid:25) d(cid:11)+(k) f (k)A(k)+f (k)Ay(k =
m + −
R4 X0+ p (cid:2) (cid:3)
y
2(cid:25) A(f )+A (f ) : (2.3.12)
+ −
The attribute real is due to the equation:
(cid:3)
’(x) = ’(x); (2.3.13)
the attribute free to the equation:
(cid:3)’(x) = 0 (2.3.14)
and the attribute scalar to the transformation properties with respect to the Poincar(cid:19)e group:
if we de(cid:12)ne the natural extension of the representation (2.2.9) to the Fock space:
U (cid:17) Γ(U ); 8g 2 P (2.3.15)
g g
then we have:
U ’(x)U−1 = ’((cid:3)(cid:1)x+a); 8(cid:21) 2 L"; U ’(x)U = ’(I (cid:1)x) (2.3.16)
a;(cid:3) a;(cid:3) It It s
Moreover, we have the important causality property:
[’(x);’(y)] = 0; for (x−y)2 (cid:20) 0: (2.3.17)
An important generalisation of the notion of free (cid:12)eld is given by the concept of Wick
monomials. According to the rigourous treatment of [50] one can make sense of the following
expressions:
W (x) (cid:17) ’ (x)r’ (x)s; r;s 2 N: (2.3.18)
rs + −
8
Indeed, if one formally integrates with a test function f 2 S(R4) the expression de(cid:12)ned
above, one gets the following expression:
Z
W (f) (cid:17) f(x)W (x) (2.3.19)
rs rs
R4
for which the following explicit formul(cid:26) are available. For n < r:
(W (f)(cid:8))(n) = 0 (2.3.20)
rs
and for n (cid:21) r:
p
(n−r+s)!n!
(Wrs(f)(cid:8))(n) = (2(cid:25))2−32(r+s) (n−r)! (cid:2)
!
Z
Ys Xs Xr
S d(cid:11)+(k )f~ (cid:28)(k )− (cid:28)(p ) (cid:8)(n−r+s)(k ;(cid:1)(cid:1)(cid:1);k ;p ;(cid:1)(cid:1)(cid:1);p ); (2.3.21)
n m j j i 1 s r+1 n
(Xm+)(cid:2)s j=1 j=1 i=1
here the operator S symmetrizes in the variables p ;(cid:1)(cid:1)(cid:1);p :
n 1 n
The central result making the expressions above legitimate operators in the Fock space is
the following lemma [46].
Lemma 2.5 In the conditions above the following function
!
Z
Ys Xs Xr
F(p ;(cid:1)(cid:1)(cid:1);p ) (cid:17) d(cid:11)+(k )f~ (cid:28)(k )− (cid:28)(p ) (cid:8)(n−r+s)(k ;(cid:1)(cid:1)(cid:1);k ;p ;(cid:1)(cid:1)(cid:1);p )
1 n m j j i 1 s r+1 n
(Xm+)(cid:2)s j=1 j=1 i=1
(2.3.22)
is a test function.
Now one de(cid:12)nes Wick products to be the expressions
Z
X
: ’l : (f) (cid:17) W (f) = f(x) : ’(x)l : (2.3.23)
rs
R4
r+s=l
with the usual de(cid:12)nition for : ’(x)l : namely one puts all the creation operators at the left of
the annihilation operators. In a similar way one can de(cid:12)ne Wick product with some derivative
on the various factors from : ’(x)l :.
Now we remind Wick theorem. Suppose that A (x);:::;A (x);B (x);:::;B (x) are the
1 n 1 m
free scalar (cid:12)eld or derivatives of it. Then we have the following formula:
: A (x):::A (x) :: B (y):::B (y) :=: A (x):::A (x)B (y):::B (y) : +
1 n 1 m 1 n 1 m
miXn(n;m) X Yp
< (cid:8)0;Ait(x)Bjt(y)(cid:8)0 >: Ai01(x):::Ai0n−p(x)Bj10(y):::Bjm0 −p(y) : (2.3.24)
p=1 partitionst=1
where fi ;:::;i g; fi0;:::;i0 g is a partition of 1;:::;n, and fj ;:::;j g; fj0;:::;j0 g
1 p 1 n−p 1 p 1 m−p
is a partition of 1;:::;m; the sum runs over all such partitions.
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