Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary

12 Pages·0.19 MB·English

by  T. P. Branson

VACUUM EXPECTATION VALUE ASYMPTOTICS
FOR SECOND ORDER DIFFERENTIAL
OPERATORS ON MANIFOLDS WITH BOUNDARY
Thomas P. Branson†, Peter B. Gilkey‡, Dmitri V. Vassilevich∗
0
0
0 Abstract. LetM beacompact Riemannianmanifoldwithsmoothboundary. We
2 studythevacuumexpectationvalueofanoperatorQbystudyingTrL2Qe−tD,where
D is an operator of Laplace type on M, and where Q is a second order operator
n with scalar leading symbol; we impose Dirichlet or modified Neumann boundary
a conditions.
J

4

2
v
8 §1 Introduction
7
LetM be a compactsmoothRiemannianmanifoldofdimensionm withsmooth
1
2 boundary ∂M. We say that a second order operator D on the space of smooth
0 sections C∞(V) of a smooth vector bundle over M has scalar leading symbol if
7 the leading symbol is hijI ξ ξ for some symmetric 2-tensor h. We say that D is
V i j
9 of Laplace type if hij is the metric tensor on the cotangent bundle. Let D be an
/
h operator of Laplace type. If the boundary of M is non-empty, we impose Dirichlet
t or Neumann boundary conditions B to define the operator D , see §4 for further
- B
p details. Let Q be an auxiliary second order partial differential operator on V with
e
scalarleadingsymbol;iftheorderofQisatmost1,thenthishypothesisissatisfied
h
trivially. As t↓0, there is an asymptotic expansion
:
v
i
X ∞
r (1.1) TrL2(Qe−tDB)∼ X an(Q,D,B)t(n−m)/2,
a n=−2
seeGilkey [8,Lemma 1.9.1]whereadifferentnumbering conventionwasused. The
invariants a (Q,D,B) are locally computable. We have a (Q,D,B) = 0 and
n −2
a (Q,D,B) = 0 if Q has order at most 1. If the boundary of M is empty, the
−1
boundary condition B plays no role and we drop it from the notation; in this case,
if n is odd, then a (Q,D)=0.
n
1991Mathematics Subject Classification. Primary58G25.
Keywords and phrases. Laplacian,spectralgeometry, heatequationasymptotics.
† Researchpartiallysupportedbyaninternational travelgrantoftheNSF(USA)
‡ ResearchpartiallysupportedbytheNSF(USA)andMPIM(Germany)
∗ ResearchpartiallysupportedbyGRACENAS(Russia)andDAAD(Germany)
BGVversion7cprinted1February2008
TypesetbyAMS-TEX
1
2 T. BRANSON, P. GILKEY, AND D. VASSILEVICH
Ourpaperismotivatedbyseveralphysicalexamples. First,consideraEuclidean
quantum field theory with a propagator D−1 depending on external fields. Typ-
ically, D is a second order differential operator of Laplace type. In the one–loop
approximation,the vacuum expectation value <Q> of a second order differential
operator is given by < Q >= TrL2(QD−1). By formal manipulations, this can be
represented in the form
<Q>∼R0∞dt TrL2(Qe−tD)∼R0∞dtP∞n=−2an(Q,D,B)t(n−m)/2.
Theseintegralsaredivergentatthelowerlimitandneedtoberegularized. Thiscan
bedonebyreplacing0by1/Λinthelimitsofintegration;Λiscalledtheultraviolet
cutoff parameter. The coefficients a (Q,D,B) define the asymptotics of <Q> as
n
Λ→∞. Thefirstmtermsaredivergentandareessentialforrenormalization. The
coefficients a (Q,D,B) also define large mass asymptotics of <Q> in the theory
n
of a massive quantum field; for details see for example [1].
A second example is provided by quantum anomalies. In the Fujikawa ap-
proach [6], the anomaly A is defined as A = lim Tr(Qe−D/Λ2), where Q is
Λ→∞
the generator of an anomalous symmetry transformation, and D is a regulator.
Usually divergent terms may be absorbed in renormalization and one has that
A ∼ a (Q,D,B). Other examples where these asymptotics arise naturally are
m−2
the study of the anomaly for an arbitrary local symmetry transformation, and in
the study of the vacuum expectation value of the stress-energy tensor.
Inthispaper,wewillstudythe asymptoticsa (Q,D,B)inageneralmathemat-
n
ical framework. In §2, we review the geometry of operators of Laplace type and
derive some variational formulas. The operator D determines the metric g, a con-
nection ∇ on V, and an endomorphism E of V. Conversely, given these data, we
can define an operator of Laplace type D(g,∇,E); see Lemma 2.3 for details. Let
q be a symmetric 2-tensor and let q be a 1-form valued endomorphism of V. We
2 1
shalluseq todefineavariationofthe metricg(ε):=g+εq andweshalluseq to
2 2 1
defineavariationoftheconnection∇(ε):=∇+εq . LetQ :=∂ D(g(ε),∇,E)|
1 2 ε ε=0
and let Q := ∂ D(g,∇(ε),E)| . Let Q be a second order operator with scalar
1 ε ε=0
leading symbol. We may decompose Q = Q +Q +Q for Q ∈ End(V) and for
2 1 0 0
Q and Q defined by suitably chosen q and q . Since a (Q,D) = a (Q ,D),
2 1 2 1 n Pi n i
it suffices to compute the a (Q ,D). In Lemma 2.4, we will study the operators
n i
Q and Q and show that ∂ a (1,D(̺),B) = −a (∂ D(̺),D(̺),B) for any 1-
2 1 ̺ n+2 n ̺
parameter family of operators of Laplace type and fixed boundary condition B. In
§3 and §4 we use this variational formula and apply results of [2] and [5] to study
the invariants a (Q,D,B); in §3 we consider manifolds without boundary and in
n
§4 we consider manifolds with boundary.
An operator A is said to be of Dirac type if A2 is of Laplace type. Bransonand
Gilkey [2]studiedtheasymptoticsofTrL2(Ae−tA2)foranoperatorAofDiractype
on a closed manifold. In §5, we use the results of §3 to rederive these results and
to compute some additional terms in the asymptotic expansion. The numbering
convention we shall use in this paper differs from that used in [2]; the invariants
a (A,A2) of this paper were denoted by a (A,A2) in [2].
n n−1
§2 Geometry of operators of Laplace type
We adopt the following notational conventions. Greek indices µ, ν, etc. will
range from 1 through m = dim(M) and index local coordinate frames ∂ and dxν
ν
VACUUM EXPECTATION VALUE ASYMPTOTICS 3
for the tangent and cotangentbundles TM and T∗M. Roman indices i, j will also
rangefrom 1 throughm and index local orthonormalframes e andei for TM and
i
T∗M. We shall suppress the bundle indices for tensors arising from V. We adopt
the Einstein convention and sum over repeated indices. Let D be an operator of
Laplace type. This means that we can decompose D locally in the form
(2.1) D =−(gνµI ∂ ∂ +aσ∂ +b)
V ν µ σ
where a and b are local sections of TM ⊗End(V) and End(V) respectively. It is
importanttohaveamoreinvariantexpressionthanthatwhichisgiveninequation
(2.1). Let Γ be the Christoffel symbols of the Levi-Civita connection of the metric
g on M, let ∇ be anauxiliary connection on V, and let E ∈C∞(End(V)). Define:
D(g,∇,E):=−(Tr ∇2+E)
g
(2.2)
=−gµσ{I ∂ ∂ +2ω ∂ −Γ νI ∂ +∂ ω +ω ω −Γ νω }−E.
V µ σ µ σ µσ V ν µ σ µ σ µσ ν
We compare equations (2.1) and (2.2) to prove the following Lemma.
2.3 Lemma. If D is an operator of Laplace type, then there exists a unique con-
nection ∇ on V and a unique endomorphism E of V so that D =D(g,∇,E).
(1) If ω is the connection 1-form of ∇, then ω =g (aν +gµσΓ νI )/2.
δ νδ µσ V
(2) We have E =b−gνµ(∂ ω +ω ω −ω Γ σ).
µ ν ν µ σ νµ
Let D =D(g,∇,E). We use the Levi-Civita connection of the metric g and the
connection ∇ on V to covariantly differentiate tensors of all types. We shall let ‘;’
denote multiple covariantdifferentiation. Thus, for example, Df =−(f +Ef).
;kk
2.4 Lemma. Let D =D(g,∇,E) be an operator of Laplace type. Let q =q be
2 2,ij
a symmetric 2-tensor and let q =q be an endomorphism valued 1-tensor. Then
1 1,i
(1) Q f :=∂ D(g,∇+εq ,E)f| =−2q f −q f.
1 ε 1 ε=0 1,i ;i 1,i;i
(2) Q f :=∂ D(g+εq ,∇,E)f| =q f +(2q −q )f /2.
2 ε 2 ε=0 2,ij ;ij 2,ij;j 2,jj;i ;i
(3) Let D(̺) be a smooth 1-parameter family of operators of Laplace type and
fix B. Then a (∂ D(̺),D(̺),B)=−∂ a (1,D(̺),B).
n ̺ ̺ n+2
Proof. Fix a pointx ∈M; wemay assumethatx is inthe interiorofM. Choose
0 0
coordinatescenteredatx andalocalframeforV sothatg (x )=δ ,Γ(x )=0,
0 νµ 0 νµ 0
and so that ω(x )=0. We use equation (2.2) to compute:
0
∂ D(g,∇+εq ,E)(x )| =−gνσ(2q ∂ +∂ q )(x )
ε 1 0 ε=0 1,ν σ ν 1,σ 0
=(−2q ∇ −q )(x ),
1,i i 1,i;i 0
∂ {g+εq }νσ(x )| =−q (x ),
ε 2 0 ε=0 2,νσ 0
2∂ Γ(g+εq ) µ(x )| =(q +q −q )(x ),
ε 2 νσ 0 ε=0 2,µν;σ 2,µσ;ν 2,νσ;µ 0
∂ D(g+εq ,∇,E)(x )|
ε 2 0 ε=0
=(q (∂ ∂ +∂ ω )+∂ Γ(g+εq ) ν∂ )(x )| .
2,νσ ν σ ν σ ε 2 µµ ν 0 ε=0
The first two assertions now follow. We use [8, Lemma 1.9.3] to see that the
asymptotic series of the variation is the variation of the asymptotic series. We
4 T. BRANSON, P. GILKEY, AND D. VASSILEVICH
equatecoefficientsinthefollowingtwoasymptoticexpansionstocompletetheproof:
Pn∂̺an(1,D(̺),B)t(n−m)/2 ∼∂̺TrL2(e−tD(̺)B)
=TrL2(−t∂̺D(̺)e−tD(̺)B)
∼− a (∂ D(̺),D(̺),B)t(k+2−m)/2. (cid:3)
Pk k ̺
To use Lemma 2.4, we shall need some variational formulas. Let R δ be the
µνσ
curvature of the Levi-Civita connection with the sign convention that the Ricci
tensor is given by ρ := R µ and the scalar curvature is given by τ := gνσρ .
νσ µνσ νσ
Let ∆ =δd be the scalar Laplacian, let dvol be the Riemannian measure, and let
0
F be the curvature tensor of ∇.
µν
2.5 Lemma. Let ∇(ε) := ∇+εq and g(̺) := g+̺q . Let F := (∂ | F) ,
1 2 ij ε ε=0 ij
R δ :=(∂ | R) δ, and let D :=∂ | ∆ . Then
µνσ ̺ ̺=0 µνσ ̺ ̺=0 0
(1) F =q −q and ∂ | (∇F) =F +[q ,F ].
ij 1,j;i 1,i;j ε ε=0 ij;k ij;k 1,k ij
(2) Df =q f +(2q −q )f /2.
2,ij ;ij 2,ij;j 2,jj;i ;i
(3) ∂ dvol| =q dvol/2.
̺ ̺=0 2,ii
(4) (∂ Γ) σ| =gσγ(q +q −q )/2.
̺ µν ε=0 2,µγ;ν 2,νγ;µ 2,µν;γ
(5) R δ =gδγ(q +q −q −q −q R β
µνσ 2,µσ;γν 2,νγ;σµ 2,µγ;σν 2,νσ;γµ 2,σβ µνγ
−q R β)/2.
2,γβ µνσ
(6) ∂ | τ =−q ρ +R .
̺ ̺=0 2,ij ij kiik
(7) ∂ | |ρ|2 =2R ρ −2q ρ ρ .
̺ ̺=0 kijk ij 2,ij ik jk
(8) ∂ | |R|2 =2R R −2q R R .
̺ ̺=0 ijkl ijkl 2,jn ijkl inkl
Proof. The assertion (1) is immediate from the definition. Assertion (2) follows
fromLemma2.4. Assertions(3)and(4)arestraightforwardcalculations. Assertion
(5) follows from assertion (4) and from the identity1:
q −q =−q R ρ−q R ρ.
2,σγ;νµ 2,σγ;µν 2,γρ µνσ 2,σρ µνγ
Raising and lowering indices does not commute with varying the metric so we
emphasizethatthetensorRisthevariationofatensoroftype(3,1). Theremaining
assertions now follow. (cid:3)
§3 Manifolds without boundary
Lemma 2.4 reduces the computation of a (Q,D) to the special cases a (Q ,D)
n n i
for i = 0,1,2. Recall that a (Q,D) = 0 for n odd; a (Q,D) = 0 if ord(Q) ≤ 1.
n −2
If P is a scalar invariant, let P[M]:= P(x)dvol(x). Let tr be the fiber trace.
RM V
We refer to Gilkey [7] for the proof of the following result:
3.1 Theorem. Let M be a compact Riemannian manifold without boundary, let
D be an operator of Laplace type, and let Q ∈End(V). Then
0
(1) a (Q ,D)=(4π)−m/2tr {Q }[M].
0 0 V 0
1WearegratefultoArkadyTseytlinwhopointedoutasignerrorinthisidentityintheprevious
versionofthepaper
VACUUM EXPECTATION VALUE ASYMPTOTICS 5
(2) a (Q ,D)=(4π)−m/26−1tr {Q (6E+τ)}[M].
2 0 V 0
(3) a (Q ,D)=(4π)−m/2360−1tr {Q (60E +60τE+180E2
4 0 V 0 ;kk
+12τ +5τ2−2|ρ|2+2|R|2+30F F )}[M].
;kk ij ij
(4) a (Q ,D)=(4π)−m/2tr Q /7!(18τ +17τ τ −2ρ ρ
6 0 V(cid:8) 0 ;iijj ;k ;k ij;k ij;k
−4ρ ρ +9R R +28ττ −8ρ ρ
jk;n jn;k ijkl;n ijkl;n ;nn jk jk;nn
+24ρ ρ +12R R +35/9τ3−14/3τρ2
jk jn;kn ijkℓ ijkℓ;nn
+14/3τR2−208/9ρ ρ ρ −64/3ρ ρ R
jk jn kn ij kl ikjl
−16/3ρ R R −44/9R R R
jk jnℓi knℓi ijkn ijℓp knℓp
−80/9R R R )+360−1Q (8F F +2F F
ijkn iℓkp jℓnp 0 ij;k ij;k ij;j ik;k
+6F F +6F F −12F F F −6R F F
ij;kk ij ij ij;kk ij jk ki ijkn ij kn
−4ρ F F +5τF F +6E +30EE +30E E
jk jn kn kn kn ;iijj ;ii ;ii
+30E E +60E3+12EF F +6F EF +12F F E
;i ;i ij ij ij ij ij ij
+10τE +4ρ E +12τ E −6E F +6F E
;kk jk ;jk ;k ;k ;j ij;i ij;i ;j
+30EEτ +12Eτ +5Eτ2−2Eρ ρ +2ER2)}[M].
;kk jk jk
Next, we study the invariants a (Q ,D).
n 1
3.2 Theorem. Let M be a compact Riemannian manifold without boundary, let
D be an operator of Laplace type, and let Q =∂ D(g,∇+εq ,E)| . Then
1 ε 1 ε=0
(1) a (Q ,D)=0.
0 1
(2) a (Q ,D)=−(4π)−m/2360−1tr {60F F }[M].
2 1 V ij ij
(3) a (Q ,D)=−(4π)−m/2360−1tr {−8F F −8F q F
4 1 V ij;k ij;k ij;k 1,k ij
+8F F q +4F F +4F q F −4F F q
ij;k ij 1,k ij;j ik;k ij;j 1,k ik ij;j ik 1,k
−36F F F −12R F F −8ρ F F +10τF F
ij jk ki ijkn ij kn jk jn kn kn kn
−60E q E+60E Eq +30EF F +30EF F }[M].
;k 1,k ;k 1,k ij ij ij ij
Proof. We use Lemma 2.4 and Theorem 3.1. Note that ∆ (f)=−f is indepen-
0 ;kk
dent of the connection ∇ for f ∈C∞(M). We compute:
(1) ∂ tr (60E )| =60∂ tr (E) | =−60∂ ∆ tr (E)=0.
ε V ;kk ε=0 ε V ;kk ε=0 ε 0 V
(2) ∂ tr (30F2)| =tr (60F F ).
ε V ε=0 V ij ij
(3) ∂ tr (8F F +6F F +6F F )|
ε V ij;k ij;k ij;kk ij ij ij;kk ε=0
=∂ {tr (−4F F )+tr (6F F ) }|
ε V ij;k ij;k V ij ij ;kk ε=0
=tr (−8F F −8F q F +8F F q ).
V ij;k ij;k ij;k 1,k ij ij;k ij 1,k
(4) ∂ tr (30EE +30E E+30E E )|
ε V ;ii ;ii ;i ;i ε=0
=∂ {tr (−30E E )+tr (30EE) }|
ε V ;i ;i V ;ii ε=0
=tr (−60E q E+60E Eq ). (cid:3)
V ;k 1,k ;k 1,k
We conclude this section by studying a (Q ,D). The following result is a con-
n 2
sequence of Lemmas 2.4, Lemma 2.5, and Theorem 3.1. We omit the formula for
a (Q ,D) in the interests of brevity.
4 2
6 T. BRANSON, P. GILKEY, AND D. VASSILEVICH
3.3 Theorem. Let M be a compact Riemannian manifold without boundary, let
D be an operator of Laplace type on C∞(V), and let Q =∂ D(g+εq ,∇,E)| .
2 ε 2 ε=0
(1) a (Q ,D)=−1a (q ,D).
−2 2 2 0 2,ii
(2) a (Q ,D)=−1a (q ,D)−(4π)−m/26−1tr {(−q ρ +R )I }[M].
0 2 2 2 2,ii V 2,ij ij kiik V
(3) a (Q ,D)=−1a (q ,D)−(4π)−m/2360−1 −Dtr (60E+12τI )
2 2 2 4 2,ii (cid:8) V V
+tr {10(−q ρ +R )τI −2(2R ρ −2q ρ ρ )I
V 2,ij ij kiik V kijk ij 2,ij ik jk V
+2(2R R −2q R R )I +60(−q ρ +R )E
ijkl ijkl 2,jn ijkl inkl V 2,ij ij kiik
−60q F F [M].
2,ik ij ik(cid:9)
§4 Manifolds with boundary
We now suppose M has smooth non-empty boundary ∂M. Near ∂M, let e be
i
a localorthonormalframe for TM where we normalizethe choice so that e is the
m
inward unit normal. We let indices a,b,... range from 1 through m−1 and index
the resulting orthonormal frame for the tangent bundle T(∂M) of the boundary.
Let f ∈ C∞(V). Let S be an endomorphism of V defined on ∂M. The Neumann
boundaryoperatorisdefinedbyB+f :=(∇ +S)f| andtheDirichletboundary
S m ∂M
operator is defined by B−f = f| ; we set S = 0 with Dirichlet boundary condi-
S ∂M
tionstohaveauniformnotation. LetL =(∇ e ,e )bethesecondfundamental
ab ea b m
form. Let ‘:’ denote multiple covariant differentiation tangentially with respect to
the Levi-Civita connection of the metric on the boundary and the connection ∇
on V. The difference between ‘;’ and ‘:’ is given by the second fundamental form.
For example, E and E agree since there are no tangential indices in E to be
;a :a
differentiatedwhile E =E −L E . There aresome newfeatures herewhich
;ab :ab ab ;m
arenotpresentinthecaseofmanifoldswithoutboundaryinthefollowingformulas.
Theinvariantsa (Q,D,B)arenon-zeroforoddnandthenormalderivativesofQ
n 0
enter. We still have a (Q,D,B)=0 if ord(Q)≤1.
−2
4.1 Theorem. Let M be a compact Riemannian manifold with smooth boundary,
let D be an operator of Laplace type and let Q ∈End(V). Then
0
(1) a (Q ,D,B±)=(4π)−m/2tr {Q }[M].
0 0 S V 0
(2) a (Q ,D,B±)=±(4π)−(m−1)/24−1tr {Q }[∂M].
1 0 S V 0
(3) a (Q ,D,B±)=(4π)−m/26−1 tr {Q (6E+τ)}[M]
2 0 S (cid:8) V 0
+tr {Q (2L +12S)+(3+,−3−)Q }[∂M] .
V 0 aa 0;m (cid:9)
(4) a (Q ,D,B±)=(4π)−(m−1)/2384−1tr {Q ((96+,−96−)E+(16+,−16−)τ
3 0 S V 0
+(8+,−8−)R +(13+,−7−)L L +(2+,10−)L L +96SL
amam aa bb ab ab aa
+192S2)+Q ((6+,30−)L +96S)+(24+,−24−)Q }[∂M].
0;m aa 0;mm
(5) a (Q ,D,B±)=(4π)−m/2360−1 tr {Q (60E +60τE+180E2+30F F
4 0 S (cid:8) V 0 ;kk ij ij
+12τ +5τ2−2ρ2+2R2)}[M]
;kk
+tr {Q (240+,−120−)E +(42+,−18−)τ +24L +120EL
V 0 ;m ;m aa:bb aa
+20τL +4R L −12R L +4R L +360(SE+ES)
aa amam bb ambm ab abcb ac
+21−1{(280+,40−)L L L +(168+,−264−)L L L
aa bb cc ab ab cc
+(224+,320−)L L L }+120Sτ +144SL L +48SL L
ab bc ac aa bb ab ab
+480S2L +480S3+120S +Q ((180+,−180−)E+(30+,−30−)τ
aa :aa 0;m
VACUUM EXPECTATION VALUE ASYMPTOTICS 7
+(84+,−180−)/7·L L +(84+,60−)/7·L L +72SL +240S2
aa bb ab ab aa
+Q (24L +120S)+(30+,−30−)Q }[∂M] .
0;mm aa 0;iim (cid:9)
4.2 Theorem. Let M be a compact Riemannian manifold with smooth boundary,
let D be an operator of Laplace type, and let Q =∂ D(g,∇+εq ,E)| . Then
1 ε 1 ε=0
(1) a (Q ,D,B±)=0.
−1 1 S
(2) a (Q ,D,B±)=−(4π)−m/26−1tr ((−12+,0−)q )[∂M].
0 1 S V 1,m
(3) a (Q ,D,B±)=−(384)−1(4π)−(m−1)/2tr ((−96+,0−)q L
1 1 S V 1,m aa
−384Sq )[∂M].
1,m
(4) a (Q ,D,B±)=−(4π)−m/2360−1{tr (60F F )[M]
2 1 S V ij ij
+tr {(−720+,0−)q E+(−120+,0−)q τ+(−144+,0−)q L L
V 1,m 1,m 1,m aa bb
+(−48+,0−)q L L −960Sq L −1440S2q )[∂M]}.
1,m ab ab 1,m aa 1,m
(5) a (Q ,D,B−)=5760−1(4π)(m−1)/2tr {240F F −720F F }[∂M].
3 1 V ab ab am am
(6) If the boundary of M is totally geodesic, then
a (Q ,D,B±)=−5760−1(4π)(m−1)/2tr (−1440E q
3 1 S V ;m 1,m
+1440(q E−Eq )S+240F F −960τSq −240ρ Sq
1,m 1,m ab ab 1,m mm 1,m
+180F F −270τ q +720S q
am am ;m 1,m :a 1,m:a
+720S (q S−Sq )−2880E(Sq +q S)−5760q S3)[∂M].
:a 1,a 1,a 1,m 1,m 1,m
Proof. IfQ =q I forq ∈C∞(M)isascalaroperator,thenTheorem4.1follows
0 0 V 0
from Branson and Gilkey [2]. If Q is not a scalar operator, we must worry about
0
the lack of commutativity; the only point at which this enters is in the coefficient
of tr (Q SE) and tr (Q ES). We express
V 0 V 0
a (Q ,D,B )=(4π)−m/2360−1tr (C Q SE+C Q ES)[∂M]+other terms;
4 0 S V 1 0 2 0
the sum C +C =720 is determined by the scalar case. If D, Q , and S are real,
1 2 0
then TrL2(Q0e−tD) is real; this shows that C1 and C2 are real. If Q0 D, and S are
self-adjoint,TrL2(Q0e−tD) is realsotrV(Q0(C1SE+C2ES))[∂M]is real;this now
shows C =C and completes the proof of Theorem 4.1.
1 2
Tokeepboundaryconditionsconstant,letS(ε):=S−εq so∂ S| =−q .
1,m ε ε=0 1,m
Assertions(1)-(5)ofTheorem4.2nowfollowdirectlyfromLemma2.4,fromLemma
2.5, and from Theorem 4.1. In [5], we showed that
a (1,D,B±)=±5760−1(4π)(m−1)/2tr {(360E +1440E S
5 S V ;mm ;m
+720E2+240E +240τE+120F F +48τ +20τ2−8ρ2
:aa ab ab ;ii
+8R2−120ρ E−20ρ τ +480τS2+(90+,−360−)F F
mm mm am am
+12τ +24ρ +15ρ +270τ S+120ρ S2+960S S
;mm mm:aa mm;mm ;m mm :aa
+600S S +16R ρ −17ρ ρ −10R R +2880ES2
:a :a ammb ab mm mm ammb ammb
+1440S4)+E}[∂M]
The variation of the terms other than E gives rise to the expressions listed in
Theorem4.2(5,6). TheremaindertermE isgivenbelow. Itvanishesiftheboundary
is totally geodesic and involves 40 undetermined coefficients.
E =d±L E +d±L τ +d±L R +d+L S +d+L S
1 aa ;m 2 aa :m 3 ab ammb;m 4 aa :bb 5 ab :ab
+d+L S +d+L S +d+L S+d+L S+d±L L
6 aa:b :b 7 ab:a :b 8 aa:bb 9 ab:ab 10 aa:b cc:b
8 T. BRANSON, P. GILKEY, AND D. VASSILEVICH
+d±L L +d±L L +d±L L +d±L L
11 ab:a cc:b 12 ab:a bc:c 13 ab:c ab:c 14 ab:c ac:b
+d±L L +d±L L +d±L L +d±L L +d±L L
15 aa:bb cc 16 ab:ab cc 17 ab:ac bc 18 aa:bc bc 19 bc:aa bc
+1440+L SE+d+L Sρ +240+L Sτ +d+L Sρ
aa 20 aa mm aa 21 ab ab
+d+L SR +(195+,105−)L L E+(30+,150−)L L E
22 ab mabm aa bb ab ab
+(195+/6,105−/6)L L τ +(5+,25−)L L τ +d±L L ρ
aa bb ab ab 23 aa bb mm
+d±L L ρ +d±L L ρ +d±L L R +d±L L ρ
24 ab ab mm 25 aa bc bc 26 aa bc mbcm 27 ab ac bc
+d±L L R +d±L L R +d+L S3+d+L L S2
28 ab ac mbcm 29 ab cd acbd 30 aa 31 aa bb
+d+L L S2+d+L L L S+d+L L L S+d+L L L S
32 ab ab 33 aa bb cc 34 aa bc bc 35 ab bc ac
+d±L L L L +d±L L L L +d±L L L L
36 aa bb cc dd 37 aa bb cd cd 38 ab ab cd cd
+d±L L L L +d±L L L L
39 aa bc cd db 40 ab bc cd da
The variation of E is zero for Dirichlet boundary conditions or if the boundary is
totally geodesic. (cid:3)
To study a (Q ,D,B±) we need some additional formulas.
n 2 S
4.3 Lemma.
(1) Let g(̺):=g+̺q , N :=∂ | e (̺), and L :=∂ | L . Then
2 ̺ ̺=0 m αβ ̺ ̺=0 αβ
a) N =−q e −q e /2.
2,am a 2,mm m
b) L =(q +q −q −q L )/2.
ab 2,am;b 2,bm;a 2,ab;m 2,mm ab
(2) q =q −L q +L q .
2,am;a 2,am:a aa 2,mm ab 2,ab
(3) R =q −q .
kiik 2,ki;ki 2,ii;kk
Proof. Let 1 ≤ α,β ≤ m−1. Let y = (yα) be local coordinates on the boundary
∂M centered at y . We suppose g (y ) = δ +O(|y|2). Introduce coordinates
0 αβ 0 αβ
x = (y,xm) so the curves t 7→ (y,t) are unit speed geodesics perpendicular to the
boundary. Then g = 1 and g = 0; ∂ is the inward geodesic normal vector
mm αm m
fieldforg. LetN(̺)be theinwardgeodesicnormalvectorfieldforthemetricg(̺).
ExpandN(̺)(y )=∂ +̺(cm∂ +cβ∂ )+O(̺2). We provethe firstassertionby
0 m m β
solving the equations
0=g(̺)(N(̺),∂ )(y )=̺(cα+q )(y )+O(̺2)
α 0 2,αm 0
1=g(̺)(N(̺),N(̺))(y )=1+̺(2cm+q )(y )+O(̺2)
0 2,mm 0
toseecα(y )=−q (y )andcm(y )=−q /2. WeuseLemma2.5tocompute
0 2,mα 0 0 2,mm
the variation of the Christoffel symbols and complete the proof by computing:
L =Γ(ρ) ig(N(̺),∂ )
αβ αβ i
L (y )=(Γ˙ m−q L /2)(y ).
αβ 0 αβ 2,mm αβ 0
The second assertion is immediate, the third follows from Lemma 2.5. (cid:3)
Dirichletboundaryconditionsareunchangedbyavariationofthemetricg. The
following result follows from Lemma 2.4, Theorem 4.1, and Lemma 4.3. We omit
the formula for a in the interests of brevity.
2
4.4 Theorem. Let M be a compact Riemannian manifold with smooth boundary,
let D be an operator of Laplace type, and let Q :=∂ | D(g+εq ,∇,E). Then
2 ε ε=0 2
(1) a (Q ,D,B−)=−(4π)−m/2tr {q /2}[M].
−2 2 V 2,ii
VACUUM EXPECTATION VALUE ASYMPTOTICS 9
(2) a (Q ,D,B−)=(4π)−(m−1)/24−1tr {q /2}[∂M].
−1 2 V 2,aa
(3) a (Q ,D,B−)=−(4π)−m/26−1 tr {q (6E+τ)/2−q ρ
0 2 (cid:8) V 2,ii 2,ij ij
+R }[M] +tr {q L +2L −2q L }[∂M] .
kiik V 2,aa aa aa 2,ab ab (cid:9)
(4) a (Q ,D,B−)=(4π)−(m−1)/2384−1tr {q (96E+16τ +8R
1 2 V 2,aa amam
+7L L −10L L )/2+16(−q ρ +R )
aa bb ab ab 2,ij ij kiik
+8(R −q R +2R ) +7(2L L −2q L L )
amam 2,ab ambm amaN aa bb 2,ab ab cc
−10(2L L −2q L L )}[∂M].
ab ab 2,ac ab cb
(5) Suppose for simplicity that the metric g is flat, i.e. that R =0. Then
ijkl
a (Q ,D,B−)=−(4π)−m/2360−1{−Dtr (60E)+tr (60R E
2 2 V V ijji
+30q E +90q E2+15q Ω2)}[M]
2,ii ;kk 2,ii 2,ii
−(4π)−m/2360−1tr {60E q +120E q −18R
V ;m 2,mm ;a 2,am ikki;m
+L (20R +4R )+12R L +4R L
aa ikki bmbm ambm ab abcb ac
+q (−60E +60EL +12L
2,dd ;m aa aa:bb
+20/21L L L +44/7L L L +160/21L L L )
aa bb cc ab ab cc ab bc ac
(120E+40/7L L +88/7L L )(L −q L )
aa bb ab ab cc 2,cd cd
+88/7(2L L L −2q L L L )
ab ab cc 2,bd ab ad cc
+320/7(L L L −q L L L )+12L q }[∂M].
ab bc ca 2,ad ab bc cd bb:c 2,aa:c
We now study Neumann boundary conditions. The situation is quite different
as Neumann boundaryconditions arenotinvariantunder generalperturbations of
the metric; if q 6= 0 on ∂M, B+(̺) will involve tangential derivatives regardless
am S
of how S is varied. Thus Lemma 2.4 is not directly applicable. Nevertheless, we
can still compute the first three terms in the asymptotic expansion.
4.5 Theorem. Let M be a compact Riemannian manifold with smooth boundary,
let D be an operator of Laplace type, and let Q :=∂ D(g+εq ,∇,E)| . Then
2 ε 2 ε=0
(1) a (Q ,D,B+)=−(4π)−m/2tr {q /2}[M].
−2 2 S V 2,ii
(2) a (Q ,D,B+)=−(4π)−(m−1)/24−1tr {q /2}[∂M].
−1 2 S V 2,aa
(3) a (Q ,D,B+)=−(4π)−m/26−1tr {q (6E+τ)/2−q ρ }[M]
0 2 S V 2,ii 2,ij ij
−(4π)−m/26−1tr {q L −q L −6q S+6q }[∂M].
V 2,aa bb 2,ab ab 2,mm 2,aa
Proof. Define ord(q ) = 0, ord(E) = 2, ord(R ) = 2, ord(F) = 2, ord(L) = 1,
2,ij ijkl
andord(S)=1. Increasetheorderby1foreachexplicitcovariantderivativewhich
is present. Dimensional analysis then shows the interior integrands in the formula
fora (Q ,D,B)arehomogeneousofordern+2whilethe boundaryintegrandsare
n 2
homogeneous of degree n+1. We use H. Weyl’s theorem to write a spanning set
for the set of invariants and express:
(4.6) a (Q ,D,B+)=−(4π)−m/2tr (b q /2)[M]
−2 2 S V 1 2,ii
(4.7) a (Q ,D,B+)=−(4π)−(m−1)/24−1tr (c q /2+c q )[∂M].
−1 2 S V 1 2,aa 2 2,mm
(4.8) a (Q ,D,B+)=−(4π)−m/26−1{tr (q (6b E+b τ)/2
0 2 S V 2,ii 2 3
−b q ρ )[M]+tr (c q L +c q L +c q L
4 2,ij ij V 3 2,aa bb 4 2,ab ab 5 2,mm aa
+c q S+c q S+c q +c q )[∂M]}.
6 2,mm 7 2,aa 8 2,mm;m 9 2,aa;m
10 T. BRANSON, P. GILKEY, AND D. VASSILEVICH
Product formulas then show the constants are independent of the dimension m;
theseinvariantsformabasisfortheintegralinvariantsandareuniquelydetermined
for m large. A word of explanation for the formula in equation (4.8) is in order.
We can integrate by parts to replace the interior integrals q and q by
2,ij;ij 2,ii;jj
boundary integrals of q , q , and q . Since q [∂M] = 0, we
2,mm;m 2,am;a 2,aa;m 2,am:a
use Lemma 4.3 to omit the variable q . If we take ∂M empty, the boundary
2,am;a
condition plays no role and R [M] = 0 (see equation (4.10) below). We use
kiik
Theorem 3.3 to see b =b =b =b =1; this completes the proof of assertion(1)
1 2 3 4
and the first part of assertion (3).
Theq variablesdonotappearinequations(4.6),(4.7),and(4.8). Thuswe
2,am;∗
maytakeavariationwithq =0on∂M. Thisisanessentialsimplificationsince
2,am
it means N(̺)= gmm(̺)1/2∂ . Thus the boundary conditions do not involve any
m
tangentialderivatives. WesetS(̺)=gmm(̺)1/2S. Thentheboundaryconditionis
preserved;∇ +S(̺)=gmm(̺)1/2(∇ +S). Wehave∂ S(̺)| =−q S/2.
N(̺) m ̺ ̺=0 2,mm
ByLemma2.4,a (Q ,D,B+)=−(4π)−(m−1)/24−1tr (q /2)[∂M]. Thisshows
−1 2 S V 2,aa
c =1 and c =0 and completes the proof of assertion (2).
1 2
We use Lemma 2.4 and Theorem 4.1 to see:
a (Q ,D,B+)=
0 2 S
(4.9) −(4π)−(m−1)/26−1 tr (q (6E+τ)/2−q ρ +R )[M]
(cid:8) V 2,ii 2,ij ij kiik
+tr (q (L +6S)−2q L +2L −6q S)[∂M]
V 2,aa bb 2,ab ab aa 2,mm (cid:9)
We have that
R [M]=(q −q )[M]=(−q +q )[∂M]
kiik 2,ki;ki 2,ii;kk 2,am;a 2,aa;m
=(L q −L q +q )[∂M]
aa 2,mm ab 2,ab 2,aa;m
(4.10)
2L [∂M]=(2q −q −q L )[∂M]
aa 2,am;a 2,aa;m 2,mm aa
=(−3L q +2L q −q )[∂M].
aa 2,mm ab 2,ab 2,aa;m
We use equation (4.10) to compare equations (4.8) and (4.9). This shows
c =1, c =−1, c =−2, c =−6, c =6, c =0, c =0. (cid:3)
3 4 5 6 7 8 9
§5 Operators of Dirac Type
In this section, we study the invariants a (HA,A2), where M is a closed man-
n
ifold, A is an operator of Dirac type, and H is a smooth endomorphism; we refer
to Branson and Gilkey [4] for a discussion of the case of manifolds with boundary.
We begin with a technical result:
5.1 Lemma. Let A=γν∂ −ψ bean operator of Dirac type and let D =A2 be the
ν
associated operator of Laplace type. Let D =D(g,∇,E) and let H ∈C∞End(V).
(1) ω =g (−γσ∂ γν +ψγν +γνψ+gσρΓ ν)/2.
µ νµ σ σρ
(2) Let φ:=ψ+γνω . Then A=γν∇ −φ, and φ is invariantly defined.
ν ν
(3) γ +γ =0.
i;j j;i
(4) E =−ψ2+γµ∂ ψ−gνµ(∂ ω +ω ω −ω Γ σ)=−γ γ F /2+γ φ −φ2.
µ µ ν ν µ σ νµ i j ij i ;i
(5) Let q :=−Hγ /2, and Q :=−Hφ−H γ /2, then HA=Q +Q .
1,i i 0 ;i i 1 0