Quantum Invariants

13 Pages·0.17 MB·English

by  Arthur Jaffe

Quantum Invariants
∗
Arthur Jaffe
8
9 Harvard University
9
1 Cambridge, MA 02138, USA

n
a January 1998
J

7
1

1 Abstract
v
19 Consider the partition function ZQ(a,g) = √1π −∞∞TrH(γU(g)ae−Q2+itda)e−t2dt. In this
1 paper we give an elementary proof that this is an Rinvariant. This is what we mean: assume
1 that Q is a self-adjoint operator acting on a Hilbert space , and that the operator Q is odd
0 Z H
with respect to a -grading γ of . Assume that a is an operator that is even with respect
98 to γ and whose squ2are equals I. SuHppose further that e−Q2 has a finite trace, and that U(g)
/ is a unitary group representation that commutes with γ, with Q, and with a. Define the
h
t differential da = [Q,a]. Then ZQ(a,g) is an invariant in the following sense: if the operator
-
p Q(λ)dependsdifferentiably onaparameter λ,andifd asatisfies asuitablebound,(wespecify
λ
e these regularity conditions completely in XI) then ZQ(λ)(a,g) is independent of λ. Once we
h
§
: havesetuptheproperframework, ashortcalculation in IXshowsthat∂ZQ(λ)/∂λ = 0. These
v §
i considerations apply to non-commutative geometry, to super-symmetric quantum theory, to
X
string theory, and to generalizations of these theories to underlying quantum spaces.
r
a
I Introduction
In an earlier paper [QHA] we have studied a class of geometric invariants that arise within the
framework of differential geometry and its non-commutative generalization [C1, C2, JLO]. By pair-
ing a cocycle τ with an operator-valued, even, square-root of unity a, we obtained a specific formula
for an invariant ZQ(a,g) = τJLO,a . In case that τ is the JLO-cochain [JLO], this invariant has the
numerical value 1 TrD(γU(g)Eae Q2+itda)e t2dt, see [QHA]. Here Q is a self adjoint operator
√π ∞ − −
−∞ H
and Hilbert space , and da = [Q,a]. We assume that the differential Q is odd with respect to the
R
H
Z -grading γ, while a is even. We also assume that g is an element of a group G of symmetries of
2
Q and of a. The invariant Z is not necessarily integer, but it is an integer-valued when g equals the
identity element.
∗
Work supported in part by the Department of Energy under Grant DE-FG02-94ER-25228and by the National
Science Foundation under Grant DMS-94-24344.
2 Arthur Jaffe
Inthisnotewe present analternative point ofview to [QHA].Rather thanrelatingZto a general
theory of invariants, to entire cyclic cohomology, and to K-theory, we start from the formula for
ZQ(λ)(a,g) above and ask the basic question: can one see directly, when Q = Q(λ) depends on a
parameter λ, that ZQ(λ)(a,g) is actually independent of λ?
Weanswer thisquestionaffirmatively, bystudying anewauxiliaryHilbertspace ˆ containing ,
and defining a new representation for Z as an expectation J(λ,a,g) on ˆ. We replaHce the operatHor
H
Q(λ) on with the operator q(λ,a) = Q(λ)+ηa on ˆ. We call q the extended supercharge. Here
ˆ differsHfrom by also containing the additional inHdependent fermionic coordinate η chosen so
H H
that η2 = I and ηQ(λ) + Q(λ)η = 0. We may interpret ηa as a connection associated with the
translation in the auxiliary direction t, paired with η. In VIII we define an expectation on ˆ,
§ H
namely
J(λ,a,g) = Ja , (I.1)
hh ii
where the notation is explained in (VIII.1–2). We also show that J(λ,a,g) and ZQ(λ)(a,g) agree.
Once we have defined the proper framework, we show with a short calculation in IX that Z
§
is constant. We present the algebraic aspects of the proof (because these are new) without any
analytic details. Of course the analytic details are absolutely crucial. Taking λ [0,1], we could set
∈
Q(λ) = λQ+(1 λ)Q˜ andinterpolatebetweenanyQandanyotherQ˜,aslongastheybothcommute
−
with the symmetry group U(g). Thus were algebraic arguments presented here valid without any
further assumptions, we would be in the unfortunate position of showing that all invariants for a
given a agree!
Though the analytic part of the argument is crucial, it also remains identical to our paper
[QHA]. In that other work, we formulate precisely two regularity conditions: first the regularity of
Q(λ) with respect to λ, and secondly the regularity of a with respect to Q(λ). We call the latter
the fractional differentiability properties of a. The conditions we give are useful because they are
easy to verify in a large set of examples.
Under these regularity conditions, ZQ(λ)(a,g) is once-differentiable in λ. Furthermore, the re-
sulting λ-derivative of Z equals the expression that we would obtain by interchanging the order of
differentiating and the order of taking traces or integrals in the definition of Z. The analysis to
establish these facts is lengthy, but can be taken over directly from [QHA]. For the convenience of
the reader, in XI, we summarize the analytic hypotheses used in argument.
§
In X we consider a different but related case with two differentials Q and Q , but where only
1 2
§
Q is invariant under the symmetry group G. Instead we assume that Q2 + Q2 is also invariant,
1 1 2
and that Q2 Q2 commutes with all relevant operators. We show in this case that an expectation
1 − 2
(X.4) has a representation similar to (I.1) and also is an invariant with respect to λ.
II The Supercharge
Our basic framework involves an odd, self adjoint operator Q on a Z -graded Hilbert space . This
2
H
means we have a self-adjoint operator γ on for which γ2 = I. Thus splits into the direct sum
H H
= of eigenspaces of γ. The statement that Q is odd means Qγ +γQ = 0. In terms of
+
H H ⊕H−
Quantum Invariants 3
the direct sum decomposition,
0 Q I 0
Q = ∗+ and γ = . (II.1)
Q 0 0 I
(cid:18) + (cid:19) (cid:18) − (cid:19)
The operator Q is the supercharge1 and its square
Q Q 0
H = Q2 = ∗+ + (II.2)
0 Q Q
(cid:18) + ∗+(cid:19)
will be referred to as the Hamiltonian. We let xγ = γxγ denote the action of γ on operators. We
say that the operator x is even (bosonic) if x = xγ and odd (fermionic) if xγ = x. We define the
−
graded differential
dx = Qx xγQ . (II.3)
−
We suppose that there is a compact Lie group G with a continuous unitary representation U(g)
on such that
H
U(g)γ = γU(g) , and U(g)Q = QU(g) . (II.4)
Denote the action of U(g) on the operator x by
x xg = U(g)xU(g) 1 (II.5)
−
→
III The Observables
We also consider an algebra of bounded operators A on with the properties that each a A is
H ∈
even and invariant. In other words, each a A commutes with γ and with U(g) for all g G.
∈ ∈
We also consider Mat (A), the set of n n matrices with matrix elements (or entries) in A. If
n
×
a,b Mat (A) are matrices with entries a ,b A, we use the shorthand ab to denote the matrix
n ij ij
∈ ∈
with entries n a b A.
k=1 ik kj ∈
The differential of elements of A is
P
da = Qa aQ = [Q,a] (III.1)
−
is always defined as a quadratic form on . We make precise the nature of the differentiabil-
H
ity of A, in XI. The operators A and the derivative d are the fundamental building blocks of
§
non-commutative geometry; here A generalizes the notion of functions on a manifold M, and Q
generalizes a Dirac operator on a bundle over M.
1
We are not concernedwith the basic structure of or Q, aside from the possibility to perform the construction
H
in V.
§
4 Arthur Jaffe
IV The Invariant ZQ(λ)(a,g)
In [QHA] we gave a simple formula for an invariant. Let Q(λ) depend on a real parameter λ. We
denote the graded commutator (II.3) of Q(λ) with x by
d x = Q(λ)x xγQ(λ) . (IV.1)
λ
−
For a A define
∈ 1
ZQ(λ)(a,g) = ∞ Tr γU(g)ae−Q(λ)2+itdλa e−t2dt . (IV.2)
√π H
Z−∞ (cid:16) (cid:17)
More generally, we let a Mat (A). In this case
n
∈
1
ZQ(λ)(a,g) = √π ∞ Tr Cn2 γU(g)ae−Q(λ)2+itdλa e−t2dt , (IV.3)
Z−∞ H⊗ (cid:16) (cid:17)
where we extend γ,U(g),Q(λ)2 to diagonal n n matrices.
×
Theorem I. For a A, assume a2 = I. Furthermore assume that Q = Q(λ) and d a = [Q(λ),a]
λ
∈
satisfy the regularity hypotheses given in XI. Then ZQ(λ)(a,g) is independent of λ.
§
The main point of this paper is to present a new, elementary proof of Theorem I.
V The Extended Supercharge q
In order to exhibit our proof, we introduce a new Hilbert space ˆ on which the operators Q,γ,A,
H
and U(g) also act. In addition, on ˆ there are two additional self adjoint operators η and J, both
H
of which have square one,
η2 = J2 = I , (V.1)
and such that
[η,x] = [J,x] = 0 for x = γ,a A, or U(g) , (V.2)
∈
and also
ηJ +Jη = ηQ+Qη = [J,Q] = 0 . (V.3)
Let Γ = γJ denote a Z -grading on ˆ, and for x acting on ˆ let
2
H H
xΓ = ΓxΓ . (V.4)
The operator η is our auxiliary fermionic coordinate, and J = ( I)Nη is the corresponding Z
2
−
grading. 2
2
Supposethat = isatensorproductofbosonicandfermionicFockspaces,thatQislinearinfermionic
b f
H H ⊗H
creation or annihilation operators, and that γ =( I)Nf. This would be standard in the physics of supersymmetry.
∗ −
Suppose in addition that η =b+b denotes one fermionic degree of freedom independent of those in and acting
f
on the two-dimensional space . Then take ˆ = ( ) and J = ( I)b∗b, with Q,γ,a, anHd U(g) acting
η b f η
H H H ⊗ H ∧H −
on ˆ in the natural way. This gives a realization of (V.1–3) on ˆ.
H H
Quantum Invariants 5
Given a A, define the extended supercharge q = q(λ,a) by
∈
q = q(λ,a) = Q(λ)+ηa , (V.5)
and also let
h = h(λ,a) = q(λ,a)2 = Q(λ)2 +a2 ηd a . (V.6)
λ
−
Note that
qΓ = q , and hΓ = h . (V.7)
−
We use the notation d to denote the Γ-graded commutator on ˆ,
q
H
d x = qx xΓq . (V.8)
q
−
If we need to emphasize the dependence of q on λ or a, then we write d x. We continue to
q(λ,a)
reserve d or d to denote the γ-graded commutator (IV.1).
λ
ˆ
VI Heat Kernel Regularization on
H
ˆ ˆ
Let us introduce the heat kernel regularizations X of X on . Let X = x ,...,x denote
n n n 0 n
an ordered set of (n+1) linear operators x acting on ˆ. We cHall the x verti{ces and X }a set of
j j n
H
vertices. Choose a A and let q(λ,a) = Q(λ) + ηa, and h = h(λ,a) = q(λ,a)2. Define the heat
∈
kernel regularization Xˆ (λ,a) of X = x ,...,x (λ,a) by
n n 0 n ∧
{ }
Xˆ (λ,a) = x e s0hx e s1h x e snhδ(1 s s )ds ds . (VI.1)
n 0 − 1 − n − 0 n 0 n
··· − −···− ···
Zsj>0
Note that if T is any operator on ˆ that commutes with h = q2, then
H
x ,...,x T,x ,...,x (λ,a) = x ,...,x ,Tx ,...,x (λ,a) . (VI.2)
0 j j+1 n ∧ 0 j j+1 n ∧
{ } { }
Furthermore T = Jη anti-commutes with q(λ,a) and commutes with h(λ,a) for all a.
Proposition II.(VertexInsertion)LetX = x ,...,x denote a setof vertices possiblydepending
n 0 n
{ }
on λ. Then with the notation Q˙ = ∂Q(λ)/∂λ, we have
∂ n
x ,...,x (λ,a) = x ,...,x ,d Q˙,x ,...,x (λ,a)
0 n ∧ 0 j q j+1 n ∧
∂λ{ } − { }
j=0
X
n ∂x
j
+ x ,..., ,...,x (λ,a) . (VI.3)
0 n ∧
{ ∂λ }
j=0
X
Here
d Q˙ = d Q˙ = d Q˙ +η[a,Q˙] . (VI.4)
q q(λ,a) λ
6 Arthur Jaffe
Proof. By differentiating Xˆ defined in (VI.1), we obtain two types of terms. Differentiating the
n
x ’s gives the second sum in (VI.3). (This sum is absent if the x ’s are λ-independent.) The other
j j
terms arise from differentiating the heat kernels. We use the identity
∂ s ∂h
e sh = e uh e (s u)hdu . (VI.5)
− − − −
∂λ − ∂λ
Z0
Note that
∂h ∂ ∂q ∂q ∂q
= (q2) = q + q = d = d Q˙ .
q q
∂λ ∂λ ∂λ ∂λ ∂λ!
Explicitly
d Q˙ = (Q+ηa)Q˙ +Q˙(Q+ηa) = d Q˙ +η[aQ˙] .
q λ
ˆ
Inserted back intothedefinitionofX , we observe thatthedifferentiationoftheheat kernel between
n
vertex j and vertex j+1 produces one new d Q˙ vertex at position j+1. This completes the proof
q
−
of (VI.3).
Define the action of the grading Γ on sets of vertices X by
n
X XΓ = xΓ,xΓ,...,xΓ . (VI.6)
n → n { 0 1 n}
Since q2 = (qΓ)2, the regularization X Xˆ commutes with the action of Γ, namely
n n
→
Xˆ (λ,a) Γ = XΓ ∧(λ,a) . (VI.7)
n n
(cid:16) (cid:17) (cid:16) (cid:17)
It is also convenient to write explicitly differential of Xˆ ,
n
d Xˆ (λ,a) = qXˆ (λ,a) Xˆ (λ,a)Γq = qx ,x ,...,x (λ,a) xΓ,...,xΓq (λ,a) . (VI.8)
q n n − n { 0 1 n}∧ −{ 0 n }∧
Note that
n
d Xˆ (λ,a) = xΓ,xΓ,...,xΓ ,d x ,...,x (λ,a) . (VI.9)
q n { 0 1 j 1 q j n}∧
−
j=0
X
One other identity we mention is
Proposition III. (Combination Identity) The heat kernel regularizations satisfy
n
x ,x ,...,x (λ,a) = x ,x ,...,x ,I,x ,...,x (λ,a) . (VI.10)
0 1 n ∧ 0 1 j j+1 n ∧
{ } { }
j=0
X
Quantum Invariants 7
Proof. The jth term on the right side of (VI.9) is
x ,...,x ,I,x ,...,x (λ,a)
0 j j+1 n ∧
{ }
= x e s0h x e (sj+sj+1)h x e sn+1hδ(1 s s )ds ds .(VI.11)
0 − j − n − 0 n+1 0 n+1
··· ··· − −···− ···
Zsj>0
Change the s-integration variables to s = s ,s = s ,...,s = s +s , s = s ,...,s = s ,
′0 0 ′1 1 ′j j j+1 ′j+1 j+2 ′n n+1
ands = s . ThischangehasJacobian1,andtheresultingintegrandhastheformoftheintegrand
′n+1 j
for x ,...,x with variables s ,...,s , namely
{ 0 n}∧ ′0 ′n
Zs′0,s′1,...,s′n>0ds′0···ds′n(cid:18)Z ds′n+1x0e−s′0h···xne−s′nhδ(1−s′0 −···−s′n)(cid:19) , (VI.12)
with the integrand depending on the variable s only through the restriction of the range of the
′n+1
s integral. The original domain of integration restricts s to the range 0 s s , so
′n+1 ′n+1 ≤ ′n+1 ≤ ′j
the dependence of the integrand on s is the characteristic function of the interval [0,s ]. Thus
′n+1 ′j
performing the s integration produces a factor s in the s ,...,s -integrand. Add the similar
′n+1 ′j ′0 ′n
results for 0 j n to give the factor s +s + s . But the delta function in (VI.12) restricts
≤ ≤ ′0 ′1 ··· ′n
this sum to be 1, so the integral of the sum is exactly x ,...,x (λ,a).
0 n ∧
{ }
ˆ
VII Expectations on
H
Let a A satisfy a2 = I, and let Xˆ = Xˆ (λ,a) denote the heat kernel regularization of X . We
n n n
∈
define the expectation
1
Xˆ = ∞ Tr ΓU(g)Xˆ (λ,ta) dt . (VII.1)
n ˆ n
DD EEλ,a,g √4π Z−∞ H(cid:16) (cid:17)
Here we choose a2 = I to ensure that the t2 term in h provides a gaussian convergence factor to the
t-integral. This integral represents averaging over a’s whose squares are multiples of the identity.
These expectations can be considered as (n+1)-multilinear expectations on sets X of vertices.
n
Wesometimes suppress theλ-or a-or g-dependence oftheexpectations, orthe n-dependence ofsets
of vertices. Furthermore, where confusion does not occur we omit the that we use to distinguish
∧
a set of vertices X from the heat kernel regularization of the set. Thus at various times we denotes
Xˆ by X , or when we wish to clarify the dependence on n,a, or g with some subset of
n
a,g hh ii
DthDeseEinEdices, or even as one of the following:
Xˆ = X = X = X = X , (VII.2)
n λ,a,g hh ii hh iin hh iin,a hh iin,a,g
DD EE
etc.
Proposition IV. With the above notation, we have the identities
(Γ-invariance) X = XΓ , (VII.3)
hh iin n
DD EE
8 Arthur Jaffe
n
(differential) d X = xΓ,xΓ,...,xΓ ,d x ,...,x , (VII.4)
hh q iin 0 1 j−1 q j n
jX=0DDn oEE
(cyclic symmetry) x ,x ,...,x = xg−1Γ,x ,x ,...,x , (VII.5)
hh{ 0 1 n}ii n 1 2 n−1
DDn oEE
and
(combination identity)
n
x ,x ,...,x = x ,x ,...,x ,I,x ,...,x . (VII.6)
hh{ 0 1 n}iin hh{ 0 1 j j+1 n}iin+1
j=0
X
Also, in case Q = Qg and a = ag, then q = qg and we have
(infinitesimal invariance) d X = 0 . (VII.7)
q(λ,ta)
DD EE
Proof. The symmetry (VII.3) is a consequence of the fact that Γ2 = I, and Γ commutes with
U(g) and with q2. The expectation of (VI.9) completes the proof of (VII.4). The proof of (VII.5)
involves cyclicity of the trace. The identity (VII.6) is the expectation of (VI.10). To establish
(VII.7), note that every Xˆ can be decomposed uniquely as Xˆ = Xˆ++Xˆ , where Xˆ Γ = Xˆ .
n n n n− n± ± n±
The symmetry (VII.3) ensures that d X+ = 0. On the other hand, qΓ = (cid:16)q, tog(cid:17)ether with
q(λ,ta) n −
cyclicity of the trace and qg = q ensDuDres that EE
d X = q(λ,ta)X + X q(λ,ta)
q(λ,ta) n− n− n−
DD EE = DDq(λ,ta)X EE+DDq(λ,ta)g−1ΓEXE = 0 .
n− n−
DD EE DD EE
Except in (VII.7), we have implicitly assumed that the vertices x in X are t-independent. In
j n
case that X has one factor linear in t, the heat kernel regularizations of the following agree,
n
tx0,x1,...,xn ∧(λ,ta) = x0,x1,...,txj,...,xn ∧(λ,ta) , (VII.8)
{ } { }
for any j = 0,1,...,n. We then obtain an interesting relation for expectations,
Proposition V. (Integration by parts) Let a2 = I. Then
n 1
tx ,x ,...,x = x ,...,x ,ηd a,x ,...,x . (VII.9)
hh{ 0 1 n}iin 2 hh{ 0 j λ j+1 n}iin+1
j=0
X
Proof. In order to establish (VII.9), we collect together the terms exp( s t2) that occur in
j
−
x0,...,xn ∧(ta). Since the integrand for the heat kernel regularization has a δ-function restricting
{ }
the variables s to satisfy s + +s = 1, we obtain the factor exp( t2). Write
j 0 n
··· −
1 d
te t2 = e t2
− −
−2dt
(cid:16) (cid:17)
Quantum Invariants 9
and integrate by parts in t. The resulting derivative involves the t-derivative of each heat kernel
exp (s q(λ,ta)2) with the quadratic term in t removed from q2. Note that
j
−
d s d
e st2 e s(q2 t2) = e uq2 (q2 t2) e (s u)q2du
− − − − − −
dt −Z0 dt − !
s
= e uq2ηd ae (s u)q2du .
− λ − −
Z0
Here we use (V.6) with ta replacing a and with a2 = I in order to evaluate the t derivative of q2 t2.
−
Thus each derivative introduces a new vertex equal to 1ηd a, and the proof of (VII.9) is complete.
2 λ
J
VIII The Functional (λ,a,g)
Let us consider a single vertex and X = x = Ja, where a A, and its expectation
0 0
∈
J(λ,a,g) = Ja . (VIII.1)
hh ii
Explicitly
1
J(λ,a,g) = ∞ Tr γU(g)ae q(λ,ta)2 dt . (VIII.2)
ˆ −
√4π Z−∞ H(cid:16) (cid:17)
This functional allows us to recover the functional Z.
Theorem VI. Let a satisfy a2 = I. Then
J(λ,a,g) = ZQ(λ)(a,g) . (VIII.3)
Proof. Let h = h tηda, where h = Q(λ)2 + t2. The Hille-Phillips perturbation theory for
0 0
−
semi-groups can be written
e q(λ,ta)2 = e h0+tηda
− −
= e h0 + ∞ tn e s0h0ηd ae s1h0ηd a ηd ae snh0δ(1 s s s )ds ds ds .
− − λ − λ λ − 0 1 n 0 1 n
··· − − −···− ···
n=1 Zsj>0
X
(VIII.4)
In the nth term we collect all factors of η on the left. Note that η commutes with a and h ,
0
and it anti-commutes with d a. Therefore the result of collecting the factors of η on the left is
λ
ηn( 1)n(n 1)/2. If n is odd, then ηn = η and Tr (η) = 0. Thus only even n terms contribute to
(VI−II.2). F−or even n, ηn( 1)n(n 1)/2 = ( 1)n/2I aHnηd Tr (I) = 2. Thus (VIII.2) becomes
− − − Hη
1
J(λ,a,g) = ∞ dt ∞ ( t2)n a,d a,...,d a e t2 , (VIII.5)
√π − h{ λ λ }i2n −
Z−∞ nX=0
10 Arthur Jaffe
where we use expectations on similar to on ˆ (but without the t-integration) and
h in H hh iin H
defined by
x ,...,x = Tr γU(g)x e s0Q(λ)2 x e snQ(λ)2 δ(1 s s s )ds ds ds .
h{ 0 n}in Zsj>0 H(cid:16) 0 − ··· n − (cid:17) − 0− 1−···− n 0 1··· n
(VIII.6)
But using the Hille-Phillips formula once again, (VIII.6) is just
1
∞dtTr γU(g)ae−Q(λ)2+itdλa e−t2 = ZQ(λ)(a,g) . (VIII.7)
√π H
Z∞ (cid:16) (cid:17)
(Here we use the symmetry of (VIII.7) under γ to justify vanishing of terms involving odd powers of
d a.) Thus we can prove that ZQ(λ)(a,g) is independent of λ by showing that J(λ,a,g) is constant
λ
in λ.
J
IX (λ,a,g) Does Not Depend on λ
WenowproveTheoremI.Calculate∂J/∂λusing(VI.3),inthesimplecaseofonevertexindependent
of λ. Thus
∂ ∂
J(λ,a,g) = Ja = Ja,d Q˙ . (IX.1)
q(λ,ta)
∂λ ∂λ hh ii −
DDn oEE
Using the identity (VII.7) in the form
˙ ˙ ˙
0 = d Ja,Q = d (Ja),Q + Ja,d Q , (IX.2)
q(λ,ta) q(λ,ta) q(λ,ta)
DD n oEE DDn oEE DDn oEE
we have
∂
J(λ,a,g) = d (Ja),Q˙ . (IX.3)
q(λ,ta)
∂λ
DDn oEE
It is at this point that we have used qg = q, namely the invariance of both Q and a under U(g). To
evaluate (IX.3), note that
d (Ja) = [q(λ,ta),Ja] = Jd a 2tJη . (IX.4)
q(λ,ta) λ
−
Here we use the assumption a2 = I. From Proposition V we therefore infer
∂
J(λ,a,g) = Jd a,Q˙ 2 tJη,Q˙
λ
∂λ −
DDn oEE DDn oEE
˙ ˙ ˙
= Jd a,Q Jη,ηd a,Q Jη,Q,ηd a . (IX.5)
λ λ λ
− −
DDn oEE DDn oEE DDn oEE
Since Jη commutes with h = q2, and since JηQ˙ = Q˙Jη, use (VI.2) to establish
−
˙ ˙ ˙ ˙
Jη,ηd a,Q + Jη,Q,ηd a = I,Jd a,Q I,Q,Jd a
λ λ λ λ
−
DDn oEE DDn oEE = DDnJd a,Q˙,IoEE+DDnJd a,I,Q˙oEE .(IX.6)
λ λ
DDn oEE DDn oEE