White noise analysis on manifolds and the energy representation of a gauge group

0.12 MB·English

by  Takahiro Hasebe

White Noise Analysis on manifolds and the
energy representation of a gauge group
Takahiro Hasebe
Research Institute for Mathematical Sciences, Kyoto University
9 Kyoto 606-8502, Japan
0
0
2
The energy representation of a gauge group on a Riemannian manifold has
t
c been discussed by several authors. In Ref. 12, Y. Shimada has shown the
O
irreducibility with the use of White Noise Analysis, where the compactness of
9 the Riemannian manifold is assumed. In this paper we extend its technique to
the noncompact case.

]
h
p 1 Introduction
-
h
t TheenergyrepresentationofagaugegrouponaRiemannianmanifoldhasbeen
a
m discussed by several authors, for instance, in Refs. 1, 2, 4, 5, 7 and 13. Their
methods are essentially to reduce the problem to the estimate of the support
[
of a Gaussian measure in an infinite dimensional space. The best result in

2 this direction seems to be the one in Ref. 13. After all these contributions,
v
the irreducibility in two dimensions has remained unsettled yet. One of the
9
difficultiesistheconformalinvarianceoftheenergyrepresentation,i.e.,whenwe
2
3 transformthe Riemannianmetricg(x)intoeρ(x)g(x), the energyrepresentation
1 remains unchanged. For a historicalsurvey of this line of research,we refer the
5. reader to Ref. 1.
0 Incontrasttotheabove,Y.Shimadahasrecentlyshowntheirreducibilityof
8 a gauge group on a compact Riemannian manifold in Ref. 12, applying White
0
Noise Analysis. Unfortunately, there is a mistake in the proof of Lemma 4.6
:
v in Ref. 12. Shimada has used the relation dΦs,t = β(exp(Φs,t)) in the proof
i of the equation (4.28), but this relation does not hold when the Lie group G is
X
non-abelian.
r
a The author could not find a way to overcome this mistake. However, Shi-
mada’s approachis still of importance when we wantan analysis of white noise
indexed by manifolds, or an analysis of the energy representation. Hence we
extend this technique to include some class of noncompact manifolds in the
presence of a weight function.
2 Preliminaries
2.1 Notation
In this section, we explain the notation frequently used throughout this paper.
1
• (M,g) denotes a Riemannian manifold M equipped with a Riemannian
metric g
• ∇ denotes the Levi-Civita connection on (M,g)
• dv = |g|dx is the Riemannian measure on (M,g)
p
• G,g denote a compact, semisimple Lie group and its Lie algebra, respec-
tively
• B(·,·) means the Killing form of g
• N:={0,1,2,3,···}
• Γ (T M):= the set of all smooth sections of the cotangent bundle on M
c ∗
with supports compact
• C (M):= thesetofallsmoothreal-valuedfunctions withsupportscom-
c∞
pact
• h·,·i is the natural bilinear form induced by g or g ⊗(−B) on tensor
x x x
productsoftangentandcotangentspacesatx,andLiealgebrag,depend-
ing on the context. When the complexification gC is considered, h·,·ix is
the natural inner product which is antilinear in the left and linear in the
right
• h·,·i0 is an inner product on Γc(T∗M) or Γc(T∗M)⊗gC determined by
hf,gi := hf,gi dv(x)
0 M x
R
• for each n≥1, ∇ :Γ (T M n)−→Γ (T M n 1) is the adjoint opera-
∗ c ∗ ⊗ c ∗ ⊗ −
tor of ∇ with respect to the inner product h·,·i
0
• foreachn≥0,∆:=−∇ ∇iscalledtheBochnerLaplacianonΓ (T M n)
∗ c ∗ ⊗
1
• |ω| :=hω,ωi2
x x
• C (M):={h∈C (M);sup |(∇mh)(x)| <∞ for all m∈N}
b∞ ∞ x M x
∈
• Γb(X) denotes the boson Fock space on X, where X is a Hilbert space
• L(F ,F ) is the set of all continuous linear operators from a topological
1 2
vector space F to a topological vector space F
1 2
2.2 White Noise Analysis
We explain White Noise Analysis needed in this paper. Let X be a complex
Hilbert space equipped with an inner product h·,·i and H be a self-adjoint
0
operator defined on a dense domain D(H) in X. Assume that H has {λ }
j ∞j=1
as eigenvalues, and {e } as corresponding eigenvectors.
j ∞j=1
Hypothesis
· {e } is a CONS of X
j ∞j=1
· 1<λ ≤λ ≤···ր∞
1 2
2
Then we can construct a nuclear countably Hilbert space as follows (for
details, the reader is referred to Ref. 9). For p ∈ R, we can define an inner
product hx,yi := hHpx,Hpyi on D(Hp). Then D(Hp) becomes a Hilbert
p 0
space, which we write as E . Let E := ∩ E be a nuclear countably Hilbert
p p 0 p
≥
space equipped with the projective limit topology and let E be its dual with
∗
the strongdualtopology. Thus we obtaina Gelfand tripleE ⊂X ⊂E . In the
∗
same way, we construct a Gelfand triple (E) ⊂ Γb(X) ⊂ (E)∗ in terms of the
self-adjoint operator Γb(H).
2.3 The energy representation of a gauge group
First we define an inner product hf,gi := hf,gi eρ(x)dv with ρ∈C (M)
ρ,0 M x ∞
for f,g ∈ Γc(T∗M) or Γc(T∗M)⊗gC. WeRsimply write hf,gi0 for ρ = 0 in
accordance with the notation in section 2.
Let H(M;gC)ρ be the completion of the space Γc(T∗M)⊗gC by the inner
product h·,·i . This space is physically the one-particle state space.
ρ,0
For ψ ∈ Cc∞(M;G), the right logarithmic derivative β(ψ) ∈ Γc(T∗M)⊗gC
is defined as
(β(ψ))(x) :=dψxψ(x)−1 =Rψ(x)−1dψx. (1)
β satisfies
β(ψφ)=V(ψ)β(φ)+β(ψ). (2)
The latter equality is said to be the Maurer-Cartancocycle condition.
For ψ ∈Cc∞(M;G) and f ∈H(M;gC)ρ, let
(V(ψ)f)(x):=[idT∗M ⊗Ad(ψ(x))]f(x), x∈M, (3)
x
thenV isaunitaryrepresentationofthegaugegroupC (M;G)ontheHilbert
c∞
space H(M;gC)ρ.
Let U be a unitary representation of the gauge group on the boson Fock
space Γb(H(M;gC)ρ) determined by
1
U(ψ)exp(f):=exp − |β(ψ)|2 −hβ(ψ),V(ψ)fi exp(V(ψ)f+β(ψ)) (4)
2 ρ,0 ρ,0
(cid:16) (cid:17)
for f ∈ H(M;gC)ρ and ψ ∈ Cc∞(M;G). We call this representation the
(weighted) energy representation or, if we emphasize the weight function, the
energy representation with the weight function ρ.
Itisimportantthatthisrepresentationis,aseasilychecked,notaprojective
representation since the Maurer-Cartancocycle β is real.
Note. As we stated in Introduction, the energy representation is conformally
invariant in two dimensions. This is understood as follows. Let M be a d-
dimensional Riemannian manifold. If the Riemannian metric g is transformed
into eρg, dv and h·,·i on T M are transformed correspondingly:
x x∗
dv −→ed2ρdv (5)
h·,·i −→e ρ(x)h·,·i . (6)
x − x
Hence,the innerproducth·,·i remainsinvariantifandonlyifd=2. Because
ρ,0
of the existence of this conformal invariance in two dimensions, the proof of
irreducibility is difficult. The details are in Refs. 2 and 13.
3
3 Several conditions for a self-adjoint operator
Inthefollowing,weshowseveralconditionsinordertouseWhiteNoiseAnalysis
on a Riemannian manifold. For this purpose we introduce a function W which
tendstoinfinityininfinitedistances. Thisfunctionandapproximatelyconstant
functionsmakethemanifoldbehaveasifitiscompact. Herethephrase”asifit
is compact” means that we can use constant functions, which will be shown in
propositions 2 and 3. We prove in Theorem 1 that the energy representationis
irreduciblefor aRiemannianmanifolddiffeomorphicto someRiemannianman-
ifold equipped with such a function W and approximately constant functions.
LetM beaRiemannianmanifoldequippedwithaRiemannianmetricg and
W be a positive smooth function. Let L2(T M) denote the completion of the
∗
spaceΓ (T M)withrespecttothenorminducedbyg. Notethatthequadratic
c ∗
form Q(f,f)= (|∇f|2 +W|f|2)dv(x) with domain(Q) ={f :Q(f,f)<∞}
M x x
is a nonnegativeR, symmetric closed form. Hence there is a self-adjoint operator
denoted by H = −∆+W such that Q(f,g) = hHf,gi . First we consider the
0
following condition on (M,g) and W.
(a) W ∈C (M), W ≥1;
∞
the spectrum of H = −∆ + W is discrete (denoted by {λ } ) and
n ∞n=1
satisfies 1<λ ≤λ ≤···≤λ ≤···ր∞);
1 2 n
there exists p≥ 0 such that H p belongs to the Hilbert-Schmidt class.
−
The condition (a) suffices for compact manifolds. Using this, we can introduce
the family of seminorms {| · | } defined on Γ (T M) (see section 2). In
p p 0 c ∗
≥
order to deal with noncompact manifolds, however, (a) is not sufficient. Below
we introduce a few more conditions. Here we define an important family of
seminorms {|·|′m}m N defined by |f|′m = mn=0|Wm∇nf|0, f ∈Γc(T∗M).
∈
P
(b) the two families of seminorms {|·|p}p 0 and {|·|′m}m N define the same
topology on E; ≥ ∈
(c) there exists a sequence {ψ } of smooth functions with supports com-
n ∞n=1
pact, which enjoys the following properties:
· ψ (x)−→ 1 as n−→∞, for all x∈M,
n
· for every m∈N there exists C =C(m) independent of n such that
sup |∇mψ (x)| ≤C(m) for all n≥1.
x M n x
∈
(b) implies that the space {f ∈ L2(T M);Wn∇mf ∈ L2(⊗mT M) for all
∗ ∗
n,m ∈ N} coincides with the space E. (b) is true if the Riemannian manifold
M is Rd or compact, with the function W taken as |x|2+1 and 2 respectively.
Here ∆ means the Bochner Laplacian −∇ ∇. For a proof of the compact case,
∗
we refer the reader to Ref. 11. The Euclidean case is well known. However,
for convenience and in order to understand the reason why the condition (b) is
nontrivialinageneralRiemannianmanifold,weprovethisfactfortheEuclidean
case. This fact for the one-dimensional case can be seen in Ref. 10 without a
proof.
4
Let A := 1 x + ∂ , A := 1 x − ∂ , N := A A and N :=
j √2(cid:16) j ∂xj(cid:17) ∗j √2(cid:16) j ∂xj(cid:17) j ∗j j
d N . It holds that [A ,A ] = δ and −∆+|x|2 +1 = 2N +d+1. Let
j=1 j j ∗k jk
PA♯ denote either A or A , and let W = |x|2+1. We show that there is some
j j ∗j
C =C(m)>0 such that for f ∈C (Rd) and j ,··· ,j ∈{1,2,··· ,d},
c∞ 1 m
|A♯j1···A♯jmf|0 ≤C(m)|(2N +d+1)m2 f|0. (7)
The proof of (7) results from the canonical commutation relations. Once (7)
is proved, the relations xj = Aj√+2A∗j and ∂∂xj = Aj√−2A∗j lead to the validity of
condition (b).
The above argument depends on the properties special to the number op-
erator and creation, annihilation operators on Rd. On a general Riemannian
manifold, we do not know how to verify (b) (under some mild condition on the
manifold), even if the function W is found to satisfy the condition (a).
Proof of (7). We show (7) by an example.
|A1A∗1A2A2f|20 = hA1A∗1A1A∗1A∗2A∗2A2A2f,fi0
= hA A A A A A A A f,fi +hA A A A A A f,fi
∗1 1 1 ∗1 ∗2 ∗2 2 2 0 1 ∗1 ∗2 ∗2 2 2 0
= hA∗1A1A∗1A1A∗2A∗2A2A2f,fi0+hA∗1A1A∗2A∗2A2A2f,fi0
+hA A A A A A f,fi +hA A A A f,fi
∗1 1 ∗2 ∗2 2 2 0 ∗2 ∗2 2 2 0
= hA A A A A A A A f,fi −hA A A A A A f,fi
∗1 1 ∗1 1 ∗2 2 ∗2 2 0 ∗1 1 ∗1 1 ∗2 2 0
+2hA A A A A A f,fi −2hA A A A f,fi
∗1 1 ∗2 2 ∗2 2 0 ∗1 1 ∗2 2 0
+hA A A A f,fi −hA A f,fi
∗2 2 ∗2 2 0 ∗2 2 0 (8)
= hN2N2f,fi −hN2N f,fi +2hN N2f,fi
1 2 0 1 2 0 1 2 0
−2hN N f,fi +hN2f,fi −hN f,fi
1 2 0 2 0 2 0
≤ hN2N2f,fi +hN2N f,fi +2hN N2f,fi
1 2 0 1 2 0 1 2 0
+2hN N f,fi +hN2f,fi +hN f,fi
1 2 0 2 0 2 0
≤ h(2N +d+1)4f,fi
0
= |(2N +d+1)2f|2.
0
Remark. If we replace the Schr¨odinger operator with an elliptic operator,
propositions and theorems in this section still hold. However, for simplicity,
we restrict ourselves to Schr¨odinger operators.
An example of manifolds where the condition (c) fails to hold is R2\(0,0). In
fact, if we try to make a sequence {ψ } ⊂C (R2\(0,0)) which tends to 1
n ∞n=1 c∞
pointwise,thenderivativesofψ tendtoinfinitynear(0,0)asntendstoinfinity.
n
Nextwe considerthe weightedrepresentationcase. Put∇ρ :=e−ρ2 ◦∇◦eρ2,
5
then we have ∇∗ρ =e−ρ2 ◦∇∗◦eρ2, in fact for all f,g ∈Γc(T∗M),
hf,∇∗ρgiρ,0 =h∇ρf,giρ,0
= h∇ f,gi eρ(x)dv
ρ x
Z
M
= he−ρ2∇(eρ2f),gixeρ(x)dv
Z
M
(9)
ρ ρ
= h∇(e2f),e2gixdv
Z
M
ρ ρ
= h(e2f),∇∗(e2g)ixdv
Z
M
ρ ρ
= hf,e−2∇∗(e2g)iρ,0,
where ∗ on the left and right hand sides mean adjoints in Γ (T M) and
c ∗ ρ
Γc(T∗M) respectively. Let Hρ := ∇∗ρ∇ρ +W and eρ,n := e−ρ2en , then we
have H e = λ e . Correspondingly, | · | and | · | are replaced with
ρ ρ,n n ρ,n p ′m
|f| := |Hpf| and |f| := m |Wm∇nf| respectively. With the
ρ,p ρ ρ,0 ′ρ,m n=0 ρ ρ,0
above replacements, we can prove tPhe following properties easily:
(a) the spectrum of H =∇ ∇ +W is {λ } and there exists p≥ 0 such
′ ρ ∗ρ ρ n ∞n=1
that H p belongs to the Hilbert-Schmidt class,
ρ−
(b′) thetwofamiliesofseminorms{|·|ρ,p}p 0 and{|·|′ρ,m}m N definethesame
≥ ∈
topology on E ,
ρ
where E is defined in the same way as E. (a) is obvious. The remaining (b)
ρ ′ ′
is easily checked; for instance,
|Wm∇nρf|ρ,0 =|e−ρ2Wm∇n(eρ2f)|ρ,0
=|Wm∇n(eρ2f)|0
≤C|eρ2f|p (∃C >0,∃p≥0) (10)
=C|Hpf|
ρ ρ,0
=C|f| .
ρ,p
The next theorem is essentially due to Ref. 12, but the proof is changed
slightly in the present case. This theorem allows us to differentiate the repre-
sentation of the gauge group.
Theorem 1.LetρbeasmoothfunctiononM. Assumethattheconditions(a)
and (b) hold. Let ψt(x) := exp(tΨ(x)) for Ψ ∈ Cc∞(M;g). Then {V(ψt)}t R
∈
is a regular one-parameter subgroup of GL(E ), namely, for any p ≥ 0 there
ρ
exists q ≥0 such that
V(ψ )f −f
lim sup t −V′(Ψ)f −→0, (11)
t→0f∈Eρ,|f|ρ,q≤1(cid:12)(cid:12)(cid:12) t (cid:12)(cid:12)(cid:12)ρ,p
(cid:12) (cid:12)
where
(V′(Ψ)f)(x):=[idT∗M ⊗ad(Ψ(x))]f(x), x∈M, f ∈Eρ. (12)
x
6
Proof. We only prove the case ρ=0. The proof for nonzero ρ is the same. By
the condition (b), it is sufficient to prove for seminorms |·| ,m∈N . First we
′m
show that for Ψ ∈ C (M;g) and m ∈ N, there exists C = C(Ψ,m) > 0 such
c∞
that
|V (Ψ)f| ≤C(Ψ,m)|f| , f ∈E . (13)
′ ′m ′m ρ
The above inequality is obtained by the direct calculation. We frequently use
the same C or C(Ψ,m) in different lines.
m
|V′(Ψ)f|′m = |Wm∇n(Ψf −fΨ)|0
nX=0
m n
≤C(m) (|Wm∇kΨ∇n kf| +|Wm∇n kf∇kΨ| )
− 0 − 0
nX=0Xk=0 (14)
m n
≤C(Ψ,m) (|Wm∇n−kf|0+|Wm∇n−kf|0)
nX=0Xk=0
≤C(Ψ,m)|f|′m.
In particular, V (Ψ) belongs to L(E ,E ). Hence,
′ ρ ρ
V(ψt)f(x)=[idT∗M ⊗Ad(exp(tΨ(x)))]f(x)
x
=[idT∗M ⊗exp(tad(Ψ(x)))]f(x)
x
∞ 1
= idT∗M ⊗ (tad(Ψ(x)))k f(x)
Xk=0h x k! i
∞ 1
= (tV (Ψ))kf (x),
′
k!
Xk=0h i
and it holds that
(cid:12)V(ψt)tf −f −V′(Ψ)f(cid:12)′ ≤t ∞ k1!tk−2|V′(Ψ)kf|′m
(cid:12) (cid:12)m Xk=2
(cid:12) (cid:12)
(cid:12) (cid:12) ∞ 1
≤t tk 2C(Ψ,n)k|f|
k! − ′m
Xk=2
≤texp(C(Ψ,m))|f| .
′m
(15)
Then the conclusion of the proposition follows immediately, q.e.d.
The next result is useful in the analysis of energy representation. This is a
consequence of the conditions (b) and (c).
Proposition 2. Let ρ be a smooth function on M. Let Ψ be an element in
C (M;g). The operator
b∞
f 7−→V (Ψ)f (16)
′
belongs to L(E ,E ) and there exists a sequence {Ψ } of g-valued smooth
ρ ρ n ∞n=1
functions with supports compact such that
7
V (Ψ−Ψ )f −→0 in E as n−→∞, for all f ∈E .
′ n ρ ρ
Remark.Proposition2enablesustomakeuseofconstantfunctionsinC (M;g),
b∞
which are useful in calculations of commutants of the representation.
Proof. Again we prove only for ρ =0. Let Ψ be a fixed element in C (M;g)
b∞
and {ψ } be the sequence in the condition (c). We define Ψ := ψ Ψ ∈
n ∞n=1 n n
C (M;g).
c∞
|V (Ψ−Ψ )f| =|Hp((Ψ−Ψ )f −f(Ψ−Ψ ))|
′ n p n n 0
≤|Hp((Ψ−Ψ )f)| +|Hp(f(Ψ−Ψ ))|
n 0 n 0
1 1
=hH2p((Ψ−Ψ )f),(Ψ−Ψ )fi2 +hH2p(f(Ψ−Ψ )),f(Ψ−Ψ )i2
n n 0 n n 0
1 1 1 1
≤|(Ψ−Ψ )f|2 |(Ψ−Ψ )f|2 +|f(Ψ−Ψ )|2 |f(Ψ−Ψ )|2.
n 2p n 0 n 2p n 0
Schwarz’s inequality was used in the last line. Remembering that there exist
C >0 and m∈N such that |h| ≤C|h| for all h∈E and, by the condition
2p ′m ρ
(c), that for every l ∈N there exists C =C(l) independent of n such that
sup |∇l(Ψ−Ψ )(x)| ≤C(l)
n x
x M
∈
for all n≥1, we have
1 1 1 1
|V′(Ψ−Ψn)f|p ≤C|(Ψ−Ψn)f|′m2|(Ψ−Ψn)f|02 +C|(f(Ψ−Ψn))|m′2|f(Ψ−Ψn)|02
m 1
2 1
=C |Wm∇k((Ψ−Ψ )f)| |(Ψ−Ψ )f|2
(cid:18) n 0(cid:19) n 0
Xk=0
m 1
2 1
+C |Wm∇k(f(Ψ−Ψ ))| |f(Ψ−Ψ )|2
(cid:18) n 0(cid:19) n 0
kX=0
1 1
≤C|(Ψ−Ψ )f|2 +C|f(Ψ−Ψ )|2.
n 0 n 0
(17)
Applying Lebesgue’s bounded convergence theorem, we get the desired result,
q.e.d.
4 Summary
We haveconsideredhow to apply White Noise Analysis to the energyrepresen-
tation of a gauge group. An interesting, expected development in the future is
a construction of a function W on a general Riemannian manifold.
Acknowledgments
The author would like to thank Prof. I. Ojima, Mr. H. Ando, Mr. R. Harada
andMr. H.Saigoforinspiringdiscussionsaboutthistopic. HealsothankProf.
8
T.HidaforgivinghiminstructionsaboutWhiteNoiseAnalysisduringhisvisits
to RIMS.
References
1. S. Albeverio, R. Høegh-Krohn, J. Marion, D. Testard and B. Torr´esani,
Noncommutative Distributions. Unitary Representation of Gauge Groups
and Algebras. Marcel Dekker, 1993.
2. S.Albeverio,R.Høegh-KrohnandD.Testard,Irreducibilityandreducibil-
ityfor the energyrepresentationofthe gaugegroupofmappingsofa Rie-
mannian manifold into a compact semisimple Lie group, J. Func. Anal.
41 (1981) 378-396.
3. I. Chavel, Riemannian Geometry. A Modern Introduction, Second Edi-
tion. Cambridge University Press, 2006.
4. I.M.Gelfand,M.I.GraevandA.M.Verˇsik,Representationsofthegroup
ofsmoothmappingsofamanifoldXintoacompactLiegroup,Compositio
Math. 35 (1977) 299-334.
5. I.M.Gelfand, M.I.GraevandA.M.Verˇsik,Representationsofthegroup
of functions taking values in a compact Lie group, Compositio Math. 42
(1980/1981)217-243.
6. T. Hida, H. -H. Kuo, J. Potthoff, and L. Streit, White Noise. An infinite
dimensional calculus. Kluwer, Dordrecht, 1993.
7. R. S. Ismagilov, Unitary representations of the group C (M,G), G =
0∞
SU , Mat. Sb. 100 (1976) 117-131; English transl. in Math. USSR-Sb.
2
142 (1976) 105-117.
8. A. Knapp, Lie groups beyond an introduction, 2nd ed. Birkh¨auser, 2002.
9. N. Obata, White Noise Calculus and Fock Space, Lecture Notes in Math,
Vol 1577. Springer-Verlag,1994.
10. M. Reed and B. Simon, Methods of Modern Mathmatical Physics. vol.1,
Functional Analysis. Academic Press, 1972.
11. G. Salomonsen, Equivalence of Sobolev spaces, Res. Math. 39 (2001)
115-130.
12. Y. Shimada, On irreducibility of the energy representation of the gauge
groupand the white noise distribution theory, Inf. Dim. Anal. Quantum
Probab. Rel. Topics. 8 (2005) 153-177.
13. N.Wallach,Onthe irreducibilityandinequivalenceofunitaryrepresenta-
tions of gauge groups, Compositio Math. 64 (1987) 3-29.
9