www.elsevier.com/locate/anihpc
Planar binary trees and perturbative calculus of observables in classical field theory
Dikanaina Harrivel
LAREMA, UMR 6093, Université d’Angers, France Received 20 June 2005; accepted 16 September 2005
Available online 2 February 2006
Abstract
We study the Klein–Gordon equation coupled with an interaction term(2+m2)ϕ+λϕp=0. In the linear case (λ=0) a kind of generalized Noether’s theorem gives us a conserved quantity. The purpose of this paper is to find an analogue of this conserved quantity in the interacting case. We will see that we can do this perturbatively, and we define explicitly a conserved quantity, using a perturbative expansion based on Planar Trees and a kind of Feynman rule. Only the casep=2 is treated but our approach can be generalized to anyφp-theory.
©2006 Elsevier Masson SAS. All rights reserved.
Résumé
Nous étudions l’équation de Klein–Gordon couplée avec un terme d’interaction(2+m2)ϕ+λϕp=0. Dans le cas où l’équation est linéaire (λ=0), une généralisation du théorème de Noether nous donne une quantité conservée. Le but de cet article est de trouver un analogue de cette quantité dans le cas non-linéaire (λ=0). Nous verrons que pourλpetit, on peut définir explicitement une quantité conservée en utilisant un développement perturbatif basé sur les Arbres Plans et des règles de Feynman particulières.
Seul le casp=2 est traité mais notre approche peut être appliquée pour toutp2.
©2006 Elsevier Masson SAS. All rights reserved.
MSC:35C10; 35C15; 35L70; 05C90
Keywords:Klein–Gordon equation; Planar trees; Perturbatives theory; Conserved quantity
Introduction
In this paper, we study the Klein–Gordon equation coupled with a second order interaction term 2+m2
ϕ+λϕ2=0 (Eλ)
whereϕ:Rn+1→Ris a scalar field and2denotes the operator ∂(x∂20)2 −n i=1 ∂2
∂(xi)2. The constantmis a positive real number which is the mass andλis a real parameter, the “coupling constant”.
E-mail address:dika@tonton.univ-angers.fr.
0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2005.09.006
For any s∈Rwe define the hypersurface Σs ⊂Rn+1 by Σs := {x =(x0, . . . , xn)∈Rn+1; x0=s}. The first variablex0plays the role of time variable, and so we will denote it byt. Hence we interpretΣsas a space-like surface by fixing the time to be equal to some constants.
Whenλequals zero, (Eλ) becomes the linear Klein–Gordon equation(2+m2)ϕ=0. Then it is well known (see e.g. [14]) that for any functionψ which satisfies (2+m2)ψ =0 and for any solution ϕ of (Eλ) forλ=0 then assuming thatϕdecays sufficiently at infinity in space, we have for all(s1, s2)∈R2
Σs
1
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ=
Σs
2
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ. (∗)
This last identity can be seen as expressing the coincidence on the set of solutions of(E0)of two functionalsIψ,s1
andIψ,s2 where for all functionψ:Rn+1→Rand alls∈R, the functionalIψ,sis defined by ϕ−→
Σs
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ.
So (∗) says exactly that on the set of solutions of the linear Klein–Gordon equation, the functionalIψ,s does not depend on the times.
This could be interpreted as a consequence of a generalized version of Noether’s theorem, using the fact that up to a boundary term, the functional
K
1 2
∂ϕ
∂t 2
−|∇ϕ|2 2 −m2
2 ϕ2
dx
is infinitesimally invariant under the symmetryϕ→ϕ+εχ, whereχis a solution of (E0).
This property is no longer true whenλ=0 i.e. when Eq. (Eλ) is not linear. The purpose of this article is to obtain a result analogous to (∗) in the nonlinear (interacting) case. Another way to formulate the problem could be: if we only know the fieldϕand its time derivative on a surfaceΣs1 then how can we evaluateIψ,s2 fors2=s1?
We will see that the computation ofIψ,s2 can be done perturbatively whenλis small ands2 is close tos1. This perturbative computation takes the form of a power series over Planar Binary Trees, this notion will be explained in Section 2. Note that Planar Binary Trees appear in other works on analogous Partial Differential Equations studied by perturbation (see [4,6,16,7,2,5]) although the point of view differs with ours.
Let us express our main result. Without loss of generality we can suppose thats1=0. We denote byT (2)the set of Planar Binary Tree (see Section 2 for definition) and for eachb∈T (2)we write b ∈N∗the number of leaves ofb. Then for any functionsψ∈C2([0, T], H−q)(whereqis such thatq > n/2) which satisfies(2+m2)ψ=0, we explicitly construct a family of b -multilinear functionals(Ψ (b)←∂→s⊗ b )b∈T (2)acting onC2([0, T], Hq)and indexed by the setT (2)of Planar Binary Trees such that the following result holds;
Theorem 1. Let q ∈N be such that q > n/2, T >0 be a fixed time and ψ ∈C2([0, T], H−q+2) be such that (2+m2)ψ=0inH−q.
(i) For allϕinC2([0, T], Hq)ands∈ [0, T]the power series inλ
b∈T (2)
(−λ)b −1Ψ (b)←∂→s⊗ b , (ϕ, . . . , ϕ)
(S)
has a nonzero radius of convergenceR. More precisely we have
R
4CqM2Tϕ(s)
Hq + ∂ϕ
∂t(s)
Hq
−1
hereMandCqare some constants.
(ii) Letϕ∈C2([0, T], Hq)be such that(2+m2)ϕ+λϕ2=0. If the condition 8|λ|CqM2T ϕ C2([0,T],Hq)
1+ |λ|CqT ϕ C2([0,T],Hq)
<1
is satisfied then the power series(∗)converges and we have for alls∈ [0, T]
b∈T (2)
(−λ)|b|Ψ (b)←∂→s⊗ b , (ϕ, . . . , ϕ)
=
Σ0
∂ψ
∂t ϕ−ψ∂ϕ
∂t
.
The quantity ϕ C2([0,T],Hq)can be evaluated by using the initials conditions ofϕand using a perturbative expan- sion. More details will be available in a upcoming paper [10]. This result can be generalized forφp+1-theory,p3, but instead of Planar Binary Trees we have to consider Planarp-Trees.
Beside the fact that the functional
bλ|b|Ψ (b)←∂→s b provides us with a kind of generalized Noether’s theorem charge, it can also help us to estimate the local values of the fieldsϕand∂ϕ∂t. We just need to choose the test function ψsuch thatψ=0 on the surfaceΣ0and ∂ψ∂t |Σ0 is an approximation of the Dirac mass at the pointx0∈Σ0. One gets the value of ∂ϕ∂t at a pointx0∈Σ0by exchangingψand∂ψ∂t in the previous reasoning.
Another motivation comes from the multisymplectic geometry. One of the purpose of this theory is to give a Hamiltonian formulation of the (classical) field theory similar to the symplectic formulation of the one dimensional Hamiltonian formalism (the Hamilton’s formulation of Mechanics). If the time variable is replaced by several space–
time variables, the multisymplectic formalism is based on an analogue to the canonical symplectic structure on the cotangent bundle, a manifold equipped with amultisymplectic form. For an introduction to the multisymplectic geom- etry one can refer to [11] and for more complete informations one can read the papers of F. Hélein and J. Kouneiher [12,13]. Starting from a Lagrangian density which describes the dynamics of the field, one can construct a Hamiltonian function through a Legendre transform and obtain a geometric formulation of the problem. Note that this formalism differs from the standard Hamiltonian formulation of fields theory used by physicists (see e.g. [14]), in particular the multisymplectic approach is covariant i.e. compatible with the principles of special and general Relativity.
The main motivation of the multisymplectic geometry is quantization, but it requires as preliminary to define the observable quantities, and the Poisson Bracket between these observables. A notion of observable have been introduced in the seventies by the Polish school, see e.g. J. Kijowski [15], Tulcjiew [19]. For more informations one can read the papers of F. Hélein and J. Kouneiher [12] and [13]. In the problem which interests us in this paper these observable quantities are essentially the functionalsIψ,s. In order to be able to compute the Poisson bracket between two such observablesIψ1,s1 andIψ2,s2, we must be able to transportIψ1,s1 into the surfaceΣs2. Whenλ=0, the identity (∗) gives us a way to do this manipulation, but whenλ=0 this is no longer the case. So F. Hélein proposed an approach based on perturbation; the reader will find more details on this subject in his paper [11].
In the first section, we begin the perturbative expansion by dealing with the linear case and the first order correction.
The second section introduces the Planar Binary Trees which allow us to define the corrections of higher order, and the statement of the main result is given. Finally the last section contains the proof of the theorem.
1. Perturbative calculus: beginning expansion
1.1. A simple case:λ=0
Let us consider the spaceSλ:= {solutions of (Eλ)}we will be more precise about topology in Section 1.3.
Consider the linear Klein–Gordon equation i.e.λ=0. LetT >0 ands∈ [0, T]be a fixed positive time (the negative case is similar) andψ:Rn+1→Ra regular function. Ifϕbelongs toS0then assuming thatϕand its derivatives decay sufficiently at infinity in space we have
Σs
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ−
Σ0
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ=
D
∂2ψ
∂t2 ϕ−ψ∂2ϕ
∂t2
dx (1.1)
whereDdenotes the setD:= [0, s] ×Rn⊂Rn+1. Sinceϕ∈S0we have
∂2ϕ
∂t2 =
i
∂2ϕ
∂z2i −m2ϕ
hence if one replaces∂2ϕ
∂t2 in the right-hand side of (1.1) and perform two integrations by parts, assuming that boundary terms vanish, one obtains
Σs
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ−
Σ0
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ=
D
ϕ
2+m2
ψ. (1.2)
Hence if we assume thatψsatisfies the linear Klein–Gordon equation(2+m2)ψ=0 then it follows that for allϕin S0and for alls∈ [0, T]the right-hand side of (1.2) vanishes.
We want to know how these computations are modified forϕ∈Sλwhenλ=0. Forϕ∈Sλwe have∂2ϕ
∂t2 =
i
∂2ϕ
∂z2i − m2ϕ−λϕ2. Hence instead of (1.2) one obtains
Σs
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ−
Σ0
∂ψ
∂t ϕ−ψ∂ϕ
∂t
dσ=λ
D
ψ ϕ2 (1.3)
whereψ is supposed to satisfy the equation(2+m2)ψ =0. The difference is no longer zero. However, one can remark that the difference seems1 to be of order λ. This is the basic observation which leads to the perturbative calculus.
1.2. First order correction: position of the problem
Lets be a nonnegative integer. In the previous section, it was shown that if one choose a functionψ such that (2+m2)ψ=0, then equality (1.3) occurs for allϕ∈Sλ. The purpose of this section is to search for a counter-term of orderλwhich annihilates the right-hand side of (1.3).
LetΨ(2)be a smooth functionΨ(2):Rn+1×Rn+1→Rthen for allϕ∈Sλconsider the quantity
Σs×Σs
Ψ(2) ←−−
∂
∂t1−
−−→∂
∂t1 ←−−
∂
∂t2−
−−→∂
∂t2
ϕ⊗ϕdσ1⊗dσ2. (1.4)
We need to clarify the notation←A−and−→B for some given operatorAandB. When the arrow is right to left (resp. left to right) the operator is acting on the left (resp. right). For instance we have∂ψ∂tϕ−ψ∂ϕ∂t =ψ (←∂t∂−−−→∂t∂)ϕ.
If we assume thatΨ(2) satisfies the boundary condition∀α∈ {0,1}2,(∂|α|Ψ(2)/∂tα)|Σ0×Σs =0 then for allϕ in Sλwe have
Σs×Σs
Ψ(2) ←−−
∂
∂t1−
−−→∂
∂t1 ←−−
∂
∂t2−
−−→∂
∂t2
ϕ⊗ϕ=
D×Σs
∂
∂t1
Ψ(2) ←−−
∂2
∂t12−
−−→∂2
∂t12 ←−−
∂
∂t2−
−−→∂
∂t2
ϕ⊗ϕ
hereDdenotes the setD:= [0, s] ×Rn. Assume further that we have∀α=(α1, α2)∈ {0,2} × {0,1},
∂|α|Ψ(2)
∂tα
D×Σ0
=0
then we can do the same operation for the second variablet2and finally we get
Σs×Σs
Ψ(2) ←−−
∂
∂t1−
−−→∂
∂t1
←−−
∂
∂t2−
−−→∂
∂t2
ϕ⊗ϕ=
D×D
Ψ(2) ←−−
∂2
∂t12−
−−→∂2
∂t12 ←−−
∂2
∂t22−
−−→∂2
∂t22
ϕ⊗ϕ.
Now sinceϕbelongs toSλ we have ∂∂t2ϕ2 =
i
∂2ϕ
∂z2i −m2ϕ−λϕ2, hence one can replace the second derivatives with respect to time ofϕand then perform integrations by parts in order to obtain
D×D
dx1dx2 2 i=1
ϕ(xi)Pi+λϕ2(xi)
Ψ(2)(x1, x2) (1.5)
1 Do not forget that the situation is actually more complicated because sinceϕsatisfies Eq. (Eλ), the fieldϕdepends onλ.
wherePi denotes the operator P :=2+m2 acting on the i-th variable. Here we have assumed that there are no boundary terms in the integrations by parts.
Using (1.5) and (1.3) one obtains that for allϕinSλwe have
Σs
∂ψ
∂t ϕ−ψ∂ϕ
∂t
+λ
Σs2
Ψ(2) ←−−
∂
∂t1−
−−→∂
∂t1
←−−
∂
∂t2−
−−→∂
∂t2
ϕ⊗2−
Σ0
∂ψ
∂t ϕ−ψ∂ϕ
∂t
=λ
D×D
ϕ⊗2P1P2Ψ(2)+
D
ϕ2ψ
+λ2· · ·. (1.6)
Hence if we choose a functionΨ(2)such that
P1P2Ψ(2)(x1, x2)= −δ(x1−x2)ψ (x1) (1.7)
whereδ is the Dirac operator then the first order term in the right-hand side of (1.6) vanishes. But because of the hyperbolicity of the operatorP, it seems difficult to control the regularity of such a functionΨ(2). Hence we need to allowΨ(2)be in a distribution space.
1.3. Function space background
Here we define the function spaces which will be used in the following. Let fix some timeT >0.
Letq∈Zthen we denote byHq(Rn)(or simplyHq) the Sobolev space Hq
Rn :=
f∈L2
Rn 1+ |ξ|2q/2f (ξ )ˆ ∈L2 Rn
.
Then it is well known (see e.g. [3,17,1]) thatHq endowed with the norm f Hq :=
Rn(1+ |ξ|2)q| ˆf|2(ξ )dξ is a Hilbert Space. Moreover one can see in every classical text book (see e.g. [1]) the following result
Theorem 1.1. If q > n/2 then Hq is a Banach Algebra i.e. there exists some constant Cq >0 such that for all (f, g)∈(Hq)2,f g∈Hqand
f g Hq Cq f Hq g Hq.
In the rest of the paper we fix some integerq∈Nsuch thatq > n/2.
Definition 1.1.Letk∈N∗be a positive integer, then we denote byEk∗the space defined by Ek∗:=C1
[0, T]k,k H−q
where for all Banach spaceBand for allk∈N∗, we denote byk
B∗the space ofk-linear continuous forms overB. ThenEk∗together with the norm · k∗defined by
U k∗:= max
α∈{0,1}k
sup
t∈[0,T]k (f1,...,fk)∈(Hq)k
fj H q1
∂|α|U
∂tα (t ), (f1, . . . , fk)
is a Banach Space, here·,·denotes the duality brackets. For allk∈N∗, we denote by(E1∗)⊗k the space of finite sum of decomposable elements where a decomposable elementU ofEk∗is such that there exists(U1, . . . , Uk)∈E1∗ such thatU=U1⊗ · · · ⊗Uki.e. for all(f1, . . . , fk)∈(Hq)k and for allt=(t1, . . . , tk)∈ [0, T]k
U (t ), (f1, . . . , fk)
= U (t1), f1
· · ·U (tk), fk
.
Then using the fact that the space of compactly supported smooth functions is dense inHq, one can easily prove the following property
Property 1.1.For allkinN∗,(E1∗)⊗kis a dense subspace ofEk∗.
We will denote byEthe space defined byE:=C2([0, T], Hq). ThenEis a Banach space and we can see naturally Ek∗as a subspace ofk
E∗the space ofk-linear continuous form overE;∀U∈Ek∗and∀ϕ=(ϕ1, . . . , ϕk)∈Ek U, ϕ :=
T
0
dt1· · · T
0
dtk U (t1, . . . , tk),
ϕ1(t1), . . . , ϕk(tk) .
Now let us generalize the expression (1.4) for the elements ofEk∗.
Definition 1.2.LetUbelong toE1∗ands∈ [0, T], then we denote byU←∂→s the continuous linear form overEdefined by∀ϕ∈E
U←∂→s, ϕ :=
∂U
∂t (s), ϕ(s)
−
U (s),∂ϕ
∂t(s)
. (1.8)
Then using the Property 1.1 one can easily prove the following property
Property 1.2.Letk∈N∗ands∈ [0, T]then there exists an unique operatorEk∗→k
E∗,U→U←∂→s⊗ksuch that for any decomposable elementU=U1⊗ · · · ⊗Ukof(E1∗)⊗k and for allϕ=(ϕ1, . . . , ϕk)∈Ek
U←∂→s⊗k, ϕ :=
k
j=1
Uj←∂→s, ϕj .
1.4. Resolution of the first order correction
Let us introduce the perturbative calculus by dealing with the first order correction. In this section we will define a functionalΨ(2)such that
ψ←∂→s, ϕ
+λΨ(2)←∂→s⊗2, (ϕ, ϕ)
− ψ←∂→0, ϕ
=O λ2
(1.9) for allϕ∈Esolution of (Eλ).
Proposition–Definition 1.1.Let Υ:E1∗→E2∗ be the operator2 defined by∀ψ∈E1∗,∀t=(t1, t2)∈ [0, T]2 and
∀(f1, f2)∈(Hq)2
Υ ψ (t1, t2), (f1, f2) :=
T
0
dτ ψ (τ ), (G∗f1)(t1−τ )(G∗f2)(t2−τ )
where for allf ∈Hq,t∈ [0, T],(G∗f )(t )denotes the element ofHqsuch that∀k∈Rn (G∗f )(t )(k):=θ (t )sin(t ωk)
ωk f (k)¯ˆ (1.10)
whereθdenote the Heaviside function3and whereωk:=(m2+ |k|2)1/2for allk∈Rn.
Remark 1.1.One can seeΥ ψ as a distributionΥ ψ∈2
D((O, T )×Rn)and we have the following expression forΥ ψ
Υ ψ (x1, x2)=
P+
dy Gret(x1−y)Gret(x2−y)ψ (y) (1.11)
2 Υ ψ=(G⊗G)∗Δ0ψwhereΔ0is a generalized coproduct.
3 θ (t)=0 ift <0 and 1 otherwise.
whereP+= {x∈Rn+1|x0>0}and whereGret(z)denotes the retarded Green function of the Klein–Gordon operator Gret(z):= 1
(2π )nθ z0
Rn
dnksin(z0ωk) ωk eik.z herez¯denotes the spatial part ofz∈Rn+1i.e.z=(z0,z).¯
One can verify thatΥ is well defined and we have the following result
Proposition 1.1.Letλbe a real number ands∈ [0, T]a fixed time. Letψ∈C2([0, T], H−q+2)⊂E1∗be such that
∂2ψ
∂t2 −ψ+m2ψ=0. Ifϕ∈Eis a solution of Eq.(Eλ)then the following inequality holds ψ←∂→s, ϕ
−λ(Υ ψ )←∂→s⊗2, (ϕ, ϕ)
− ψ←∂→0, ϕλ2 s2Cq2
m ϕ 3E+ |λ|s3Cq3 3m2 ϕ 4E
ψ 1∗. (1.12)
Proof of Proposition 1.1. Letψ∈C2([0, T], H−q+2)be such that ∂∂t2ψ2 −ψ+m2ψ=0 inH−q andϕ∈Ebe a solution of Eq. (Eλ). Then sinceψandϕareC2the functionf:t→ ψ←∂→t, ϕis derivable with respect tot and
f(t )= ∂2ψ
∂t2(t ), ϕ(t )
−
ψ (t ),∂2ϕ
∂t2(t )
.
But sinceψandϕsatisfy ∂∂t2ψ2 −ψ+m2ψ=0 and ∂∂t2ϕ2 −ϕ+m2ϕ= −λϕ2we have f(t )= ψ (t ),
−m2 ϕ(t )
−
ψ (t ),∂2ϕ
∂t2(t )
=λψ (t ), ϕ2(t ) . Hence we finally get for alls∈ [0, T]in
ψ←∂→s, ϕ
− ψ←∂→0, ϕ
=f (s)−f (0)=λ s
0
ψ (τ ), ϕ2(τ )
dτ (1.13)
and we recover the identity (1.3).
Now let us study the term of order one of the left-hand side of (1.12). Using Definition 1.1 ofΥ one can show easily that it is given by the expression
(Υ ψ )←∂→s⊗2, (ϕ, ϕ)
= s
0
dτ
Rn
dk1
Rn
dk2M(s, τ, k1)M(s, τ, k2)ψ (τ, kˆ 1+k2) (1.14)
where∀(t, τ )∈ [0, T]2and∀k∈Rn, the quantityM(t, τ, k)is given by M(t, τ, k):=cos
(t−τ )ωk ˆ
ϕ(t )(k)−sin((t−τ )ωk) ωk
∂ϕ(t )
∂t (k). (1.15)
The identity (1.14) can be seen as(Υ ψ )←∂→s⊗2, (ϕ, ϕ) =u(s)whereu:[0, T] →Ris the continuous function given by
u(t ):=
t
0
dτ
Rn
dk1
Rn
dk2M(t, τ, k1)M(s, τ, k2)ψ(τ, k1+k2).
Then in view of definition (1.15) ofM(t, τ, k)one can see thatuis derivable with respect totand sinceu(0)=0 we getu(s)=s
0u(t )dt which leads to
(Υ ψ )←∂→s⊗2, (ϕ, ϕ)
= s
0
dt
(Rn)2
dk1dk2ϕ(t )(k 1)M(s, τ, k2)ψ (t )(k 1+k2)
− s
0
dt t
0
dτ
(Rn)2
dk1dk2
sin((t−τ )ωk1)
ωk1 P ϕ(t )(k 1)M(s, τ, k2)ψ (τ )(k1+k2) (1.16) whereP denotes the Klein–Gordon operatorP :=2+m2.
Then one can see the identity (1.16) as(Υ ψ )←∂→s⊗2, (ϕ, ϕ) =v(s)+w(s)where the functionsv, w:[0, T] →R are defined by
v(t )= t
0
dt1
(Rn)2
dk1dk2ϕ(t1)(k1)M(t, τ, k2)ψ (t1)(k1+k2),
w(t )= − t
0
dt1 min(t,t 1)
0
dτ
(Rn)2
dk1dk2
sin((t1−τ )ωk1)
ωk1 P ϕ(t1)(k1)M(t, τ, k2)ψ (τ )(k1+k2).
Then one can see thatvandware derivable with respect tot and that v(t )=
(Rn)2
dk1dk2ϕ(t )(k 1)ϕ(t )(k 2)ψ (t )(k 1+k2)
− t
0
dt1
(Rn)2
dk1dk2
sin((t−t1)ωk2) ωk2
ϕ(t1)(k1)P ϕ(t )(k2)ψ (t1)(k1+k2)
and
w(t )= s
0
dt1 min(t 1,t )
0
dτ
(Rn)2
dk1dk2
sin((t1−τ )ωk1) ωk1
sin((t−τ )ωk2)
ωk2 P ϕ(t1)(k1)P ϕ(t )(k2)ψ (τ )(k1+k2)
+ s
t
dt1
(Rn)2
dk1dk2
sin((t1−t )ωk1)
ωk1 P ϕ(t1)(k1)ϕ(t )(k 2)ψ (t1)(k1+k2).
Hence sincev(0)=w(0)=0 and using the fact thatP ϕ(t )= −λϕ2(t )and in view of (1.10) we finally get (Υ ψ )←∂→s⊗2, (ϕ, ϕ)
= s
0
dτ ψ (τ ), ϕ2(τ ) +2λ
s
0
dt s
0
dτ ψ (τ ), G∗
ϕ2(t ) (t−τ )
ϕ(τ )
+λ2 s
0
dt1
s
0
dt2
s
0
dτ ψ (τ ), G∗
ϕ2(t1)
(t1−τ ) G∗
ϕ2(t2)
(t2−τ ) .
Hence (1.13) and the last identity leads to ψ←∂→s, ϕ
−λ(Υ ψ )←∂→s⊗2, (ϕ, ϕ)
− ψ←∂→0, ϕ
= −2λ2 s
0
dt s
0
dτ ψ (τ ), G∗
ϕ2(t ) (t−τ )
ϕ(τ )
−λ3 s
0
dt1
s
0
dt2
s
0
dτ ψ (τ ), G∗
ϕ2(t1)
(t1−τ ) G∗
ϕ2(t2)
(t2−τ )
. (1.17)
Now to complete the proof it suffices to estimate the right-hand side of (1.17). Using the definition (1.10) of G∗f (t )one can easily prove the following lemma
Lemma 1.4.1.Iff be inHq then for all(t, τ )∈ [0, T]2we have(G∗f )(t−τ )∈Hq and (G∗f )(t−τ ) Hq
1
mθ (t−τ ) f Hq.
Hence using Lemma 1.4.1 and Property 1.1 one get
s
0
dt s
0
dτ ψ (τ ), G∗
ϕ2(t ) (t−τ )
ϕ(τ ) s2Cq2
2m ψ 1∗ ϕ 3E and
s
0
dt1
s
0
dt2
s
0
dτ ψ (τ ), G∗
ϕ2(t1)
(t1−τ ) G∗
ϕ2(t2)
(t2−τ ) s3Cq3
3m2 ψ 1∗ ϕ 4E. Then inserting these two inequalities in (1.17) we finally get (1.12). 2
Hence we found a counter-term which annihilates the term (1.3) of order one with respect to λ. But some extra new terms of high order have been introduced. Thus we need to find a functionalλ2Ψ(3)in order to delete the terms of orderλ2, and then an other functionalλ3Ψ(4)for those of order three etc. In order to picture all these extra terms, it will be suitable to introduce the following object: thePlanar Binary Tree.
2. Planar Binary Tree
APlanar Binary Tree(PBT) is a connected oriented tree such that each vertex has either 0 or two sons. The vertices without sons are called theleavesand those with two sons are theinternal vertices. For each Planar Binary Tree, There are an unique vertex which is the son of no other vertex, this vertex will be called theroot. Since a Planar Binary Tree is oriented, one can define an order on the leaves. In the rest of the paper we choose to arrange the leaves from left to right.
We will denote byT (2)the set of Planar Binary Tree. Let denote by|b|the number of internal vertices of a Planar Binary Treeband b the leave’s number ofb. Then one can easily show that we have b = |b| +1. Let denote by
◦the unique Planar Binary Tree with no internal vertex.
If b1 and b2 are two Planar Binary Trees, then we denote byB+(b1, b2) the Planar Binary Tree obtained by connecting a new root tob1on the left and tob2on the right
B+(b1, b2)= .
Then one can easily show that|B+(b1, b2)| = |b1| + |b2| +1 and B+(b1, b2) = b1 + b2 , and for allb∈T (2), b= ◦, there is an unique couple(b1, b2)∈T (2)2such thatb=B+(b1, b2). For further details on the Planar Binary Trees, one can consult [9,18,8] or [20].
Proposition–Definition 2.1.There is a unique family(Υ (b))b∈T (2)of operatorsΥ (b):E1∗→E b ∗such that Υ (◦):=id
∀(b1, b2)∈T (2)2; Υ
B+(b1, b2) :=
Υ (b1)⊗Υ (b2)
◦Υ (2.1)
where forU:E1∗→Ek∗andV:E1∗→El∗,U⊗Vdenotes the unique functional fromE2∗toE(k+l)∗such that for all U=U1⊗U2∈(E1∗)⊗2,U⊗V(U )=U(U1)⊗V(U2).
We postpone the proof of Proposition 2.1 until the next section. Letψbelong toE1∗, then we consider the family (Ψ (b))b∈T (2)defined by
Ψ (b):=Υ (b)(ψ )∈E b ∗.
Then using Remark 1.1 we can see that formally the functionalsΨ (b), b∈T (2), can be constructed using the following rules:
1. attach to each leaf ofbthe space–time variablex1, x2, . . . , xb with respect to the order of the leaves.
2. for each internal vertex attach a space–time integration variableyi∈Rn+1and integrate this variable overP+. 3. for each line between the verticesvandwwhere the depth ofvis lower than thew’s, put a factorGret(av−aw)
whereav(resp.aw) is the space–time variable associated withv(resp.w).
4. finally multiply byψ (ar)wherear is the space–time variable attached to the root of the Planar Binary Treeb.
To fix the ideas, let us treat an example. Letb∈T (2)be the Planar Binary Tree described by the following graph
b= .
Then using Definition 2.1 we haveΨ (b)=(id⊗Υ )◦Υ ψand forx=(x1, x2, x3)∈([0, T] ×Rn)3,Ψ (b)(x)is given by the following
Ψ (b)(x)=
P+
dy1dy2Gret(x1−y2)Gret(y1−y2)Gret(x2−y1)Gret(x3−y1)ψ (y2).
Theorem 2.1.
(i) Letψ∈E1∗andϕbe inEands∈ [0, T], then the power series inλ
b∈T (2)
(−λ)|b|Ψ (b)←∂→s⊗ b , (ϕ, . . . , ϕ)
(∗) has a nonzero radius of convergenceR. More precisely we have
R
4CqM2Tϕ(s)
Hq + ∂ϕ
∂t(s)
Hq
−1
hereMis defined byM:=max(m1,1)andCqis the constant of Theorem1.1.
(ii) Letϕ∈E be such that(2+m2)ϕ+λϕ2=0and ψ∈C2([0, T], H−q+2)⊂E1∗be such that(2+m2)ψ=0 inH−q. If the condition
8|λ|CqM2T ϕ E
1+ |λ|CqT ϕ E
<1 (2.2)
is satisfied then the power series(∗)converges and we have for alls∈ [0, T]
b∈T (2)
(−λ)|b|Ψ (b)←∂→s⊗ b , (ϕ, . . . , ϕ)
= ψ←∂→0, ϕ .
Remark.Note that it is possible to control the norm ϕ E with the norm of initial datas using some perturbative expansion. More precisely for any(ϕ0, ϕ1)∈(Hq)2,λ∈RandT ∈Rsuch thatT|λ| (ϕ0, ϕ1) is small enough, it is possible to construct a solutionϕ∈C2([0, T], Hq)of (Eλ) such thatϕ(0,·)=ϕ0and∂ϕ∂t(0,·)=ϕ1. Then one can control ϕ E using (ϕ0, ϕ1) . A proof of this result, based on a remark of Christian Brouder [5] will be expounded in a forthcoming paper.
Let us comment this last proposition. First of all, using Definition 1.2 of←∂→s, one can remark that the power series (∗) depends only onϕ(s,·)and∂ϕ∂t(s,·). Hence the theorem answers the original question.
We have written the solution forsnonnegative, but the study can be done in the same way for negatives. Finally the result exposed in Proposition 2.1 can be generalized toφp-theory i.e. for the equation(2+m2)ϕ+λϕp=0, p2. But the set of Planar Binary Trees must be replaced by T(p), the set ofPlanarp-Treesi.e. oriented rooted trees
