First integrals for non linear hyperbolic equations
Dikanaina Harrivel
∗, Fr´ed´eric H´ elein
†April 20, 2007
Abstract
Given a solution of a nonlinear wave equation on the flat space-time (with a real analytic nonlinearity), we relate its Cauchy data at two different times by nonlinear representation formulas in terms of asymptotic series. We first show how to construct formally these series by mean of generating functions based on an algebraic framework inspired by the construction of Fock spaces in quantum field theory. Then we build an analytic setting in which all these constructions really make sense and give rise to convergent series.
It is well-known that, for hyperbolic wave equations like, for instance, the linear Klein–
Gordon equation on the space-time R×Rn:
∂2u
∂t2 −∆u+m2u= 0, (1)
one can construct first integrals, i.e. conserved quantities. One of the most important example is the energyEt[u] := 12R
Rn
(∂u∂t)2+|∇~u|2+m2u2
|x0=td~x, which is particularly useful in the analysis of the solutions of (1). (Here we denote by x= (x0, ~x)∈R×Rn a point in space-time, ∂u∂t := ∂x∂u0 and ∇~u := ∂x∂u1,· · · ,∂x∂un
.) By claiming that the family of functionals (Et)t∈R is a first integral we mean that, for any given solution u of (1), the value ofEt[u] does not depend ont. Equation (1) possesses however other conserved quantities such as
Itϕ[u] :=
Z
Rn
∂u
∂tϕ−u∂ϕ
∂t
|x0=t d~x, (2)
where ϕ is a fixed solution of (1). One interpretation of these functionals is based on Noether’s theorem: solutions to (1) are the critical points of the functional L0[u] :=
R
R×Rn 1 2
(∂u∂t)2− |∇~u|2−m2u2
dx and hence a first integral is associated to each sym- metry. The conservation of the energyEt is then a consequence of the invariance of this
∗Institut Camille Jordan, UMR CNRS 5208, Universit´e Claude Bernard Lyon 1, 13 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France,harrivel@math.univ-lyon1.fr
†Institut de Math´ematiques de Jussieu, UMR CNRS 7586 Universit´e Denis Diderot Paris 7, Case 7012, 2 place Jussieu 75251 Paris Cedex 5, France,helein@math.jussieu.fr
problem by translations in time, whereas the conservation of functionals Itϕ is due to the fact that the Lagrangian functional L is invariant up to a boundary term by the substi- tution u 7−→ u+sϕ. Another point of view, which is closer to differential Geometry, is to consider the set E0 of all solutions to (1) as a manifold (here we stay vague about the choice of the topology). Then to each time t we associate a system of coordinates on E0
which is nothing but the Cauchy data [u]t:= u(t,·),∂u∂t(t,·)
and the conservation of, say Itϕ, means that a functional Iϕ can be consistently constructed on E0 by using, for each time t, the expression Itϕ on the coordinate system u 7−→[u]t. Note that the functionals Iϕ play an important role in the quantization of equation (1), since if ϕ is a (complex) solution of (1) of the form e±ik·x, then the quantization ofIϕ leads to creation and anni- hilation operators (see [15], the resulting quantum fields are then of course free). In this paper we are interested in finding analogous conserved quantities for a more general, non linear Klein–Gordon equation:
∂2u
∂t2 −∆u+m2u+W(u,∇u) = 0, (3) whereW :R×Rn+1 −→Rand∇u:= ∂x∂u0,∂x∂u1,· · · ,∂x∂un
. We denote byEW the set of so- lutions of thenon linear Klein–Gordon equation (3). For instance ifW(y, z0, z1,· · ·, zn) = V′(y), where V : R −→ R is a real analytic function, solutions of (3) are the critical points of LV[u] := R
R×Rn 1 2
(∂u∂t)2− |∇~u|2−m2u2−V(u)
dx and in this case the en- ergy EtV[u] := 12R
Rn
(∂u∂t)2+|∇~u|2+m2u2+V(u)
|x0=t d~x is still a conserved quantity.
However, as soon as W is not linear, there is no way to find non trivial functions ϕ such the functionals Itϕ be first integrals of (3), as observed in [9, 11]. This is the reason why it was proposed in [16] to look at more general functionals, of the form
Ft[u]t = X∞
p=1
Ft,p[u]⊗tp, (4)
where [u]t denotes the Cauchy data at time t and, letting H to be the set of all possible values of Cauchy data at some time, each Ft,p is a linear functional on the p-th tensor product H⊗p :=H⊗ · · · ⊗H and [u]⊗tp := [u]t⊗ · · · ⊗[u]t∈H⊗p. Hence each functional u7−→ Ft,p[u]⊗tpis a homogeneous polynomial of degreepandFtis a series inu. In [13], the first Author proved that such series can be constructed in the case whereW(u,∇u) =u2. In the following we recall the content of this paper.
We start with some ϕ ∈ E0 (i.e. a solution to the linear equation (1)) and we use its Cauchy data [ϕ]0 at time 0 to obtain a functional I0ϕ onE. Then the first result is that it is possible to build a formal series Ftϕ of the type (4) s.t. formally Ftϕ[u]t =I0ϕ[u] for all time t. The first term in the series is
Ft,1ϕ[u]t:=
Z
{t}×Rn
d~y Z
{0}×Rn
d~x u(y)
←→
∂
∂y0G(y−x)
←→
∂
∂x0ϕ(x),
where G is the unique tempered distribution on R×Rn which is the solution of ∂∂t2G2 −
∆G+m2G= 0 and such that G|x0=0 = 0 and ∂G∂t|x0=0 =δ0, the Dirac mass at the origin in Rn. Moreover we have introduced the notation ∂y←→∂0 meaning that
a(y)
←→
∂
∂y0b(y) := ∂a
∂y0(y)b(y)−a(y) ∂b
∂y0(y), for any functions a and b. However since ϕ ∈ E0, ϕ(y) = R
{0}×Rnd~xG(y−x)∂x←→∂0ϕ(x), so that
Ft,1ϕ[u]t= Z
{t}×Rn
d~y u(y)
←→
∂
∂y0ϕ(y) =Itϕ[u].
The second term Ft,2ϕ [u]t⊗[u]t is equal to Z Z
({t}×Rn)2
d~y1d~y2
Z
[0,t]×Rn
dz Z
{0}×Rn
d~x u(y1)u(y2)
←→
∂
∂y10
←→
∂
∂y20G(y1−z)G(y2−z)G(z−x)
←→
∂
∂x0ϕ(x).
These terms can alternatively be described through diagrams, more precisely trees, by using adapted Feynman rules:
Ft,1ϕ[u]t= y x
, Ft,2ϕ [u]⊗t2 = y1
AA AA AA
A y2
z
}} }} }} }
x
The variable xat the bottom horizontal line is associated with the insertion on the right hand side in the integral of ∂x←→∂0ϕ(x) and with an integration over ~x at time x0 = 0.
Similarly each variable yat the top horizontal line is associated with the insertion on the left hand side in the integral ofu(y)∂y←→∂0 and with an integration over~yat timey0 =t. The intermediate vertexz represents an integration over [0, t]×Rn and each intermediate edge is associated to the insertion of G(‘up’−‘down’), where ‘up’ and ‘down’ are the variables at the ends of the corresponding edge.
The second content of the result in [13] (still forW(u,∇u) =u2) is that one can choose suitable function spaces for respectively ϕ and u such that the preceding seriesconverges and defines a first integral. More precisely we letHs(Rn) be the Hilbert space of tempered distributions v ∈ S′(Rn) such that ǫsbv ∈ L2(Rn), where bv is the Fourier transform of v and ǫ(ξ) := p
m2 +|ξ|2. If s > n/2, ϕ is in C0(R, H−s(Rn))∩ C1(R, H−s−1(Rn)) and u is a solution to (3) which belongs to C0([0, T], Hs+2(Rn))∩ C1([0, T], Hs+1(Rn)) for some T, then, under a reasonable smallness assumption on T and ||u||, the functional Ftϕ is defined for all t∈[0, T] and the value of Ftϕ[u]t does not depend on t ∈[0, T].
The proof in [13], which uses a lot of combinatorics, can be extended without diffi- culties to nonlinearities of the type W(u,∇u) =up−1, for all integer p ≥3, by replacing
binary trees by (p−1)-nary trees. However it seems difficult to extend it to deal with more general nonlinearities or to systems of hyperbolic wave equations. The purpose of our paper is precisely to present a new principle which allows both to construct and to prove the convergence of functionals of the type (4) for more general nonlinearities. It is based on the following heuristic construction.
We let (φ⊲(x))x∈R×Rn and (φ⊳(x))x∈R×Rn be respectively families of ‘creation’ and ‘an- nihilation’ linear operators, acting on some infinite dimensional vector spaceF, a classical analogue of Fock spaces used in quantum fields theories. They are formally solutions of the linear Klein–Gordon equation (1), i.e. φ⊲ +m2φ⊲ = φ⊳ +m2φ⊳ = 0. The space F contains a particular vector |0i and its dual space F∗ another vector h0|, such that h0|0i = 1. Operators φ⊲(x) and φ⊳(x) obey to commutation relations [φ⊲(x), φ⊲(y)] = [φ⊳(x), φ⊳(y)] = 0 and [φ⊳(x), φ⊲(y)] = G(x−y) and their action on F and F∗ are such that φ⊳(x)|0i = 0 and h0|φ⊲(x) = 0. Using these rules, for any ϕ ∈ E0 and u ∈ EW, one can define formally the following expression
Ftϕ[u]t :=−h[u]t|Texp
− Z
0<z0<t
W(φ⊲(z),∇φ⊲(z))φ⊳(z)dz
|fϕi, (5) where we have set
h[u]t|:=h0|exp − Z
y0=t
u(y)
←→
∂
∂y0φ⊳(y)d~y
!
and |fϕi:=
Z
x0=0
φ⊲(x)
←→
∂
∂x0ϕ(x)d~x|0i (6) and the symbol T in (5) forces the exponential on his right to be time ordered. Hence if we write
Ds:=
Z
{s}×Rn
W(φ⊲(z),∇φ⊲(z))φ⊳(z)d~z:=
Z
Rn
W(φ⊲(s, ~z),∇φ⊲(s, ~z))φ⊳(s, ~z)d~z, (7) then
Texp
− Z
0<z0<t
W(φ⊲(z),∇φ⊲(z))φ⊳(z)dz
=Texp
− Z t
0
Dsds
:=
X∞ k=0
(−1)k Z
0<s1<···<sk<t
Dsk· · ·Ds1ds1· · ·dsk.
It is clear that, if the expression (5) has some meaning, then it depends only on [u]t and should be of the form (4). We will see that actually, under some hypothesis, it defines a time independant functional.
For s > n/2 we define EWs+1 to be the set of pairs ((t, t), u), where t < 0 < t and u ∈ C0((t, t), Hs+1(Rn))∩ C1((t, t), Hs(Rn)) is a weak solution of the non linear Klein–
Gordon equation (3). (Note that the conditions > n/2 implies thatuisC1 on (t, t)×Rn.) For all time t∈(t, t) we note||[u]t||s+1 :=||u(t,·)||Hs+1+||∂u∂t(t,·)||Hs. Our main result is
Theorem 0.1 Assume that W :R×Rn+1 −→R is an entire real analytic function and let s > n/2. Then there exists an entire complex analytic vector field X =X(z)dzd on C, such that ∀z ∈ C, X(z) = P∞
k=0akzk, with ak ≥ 0, ∀k ∈ N, which depends only on W and s such that the following holds. Let ((t, t), u)∈ EWs+1 such that
sup
0≤t<t
etX(||[u]t||s+1)<+∞,
then, for any weak solution ϕ ∈ C0(R, H−s(Rn))∩ C1(R, H−s−1(Rn)) of the linear Klein–
Gordon equation (1), there exists a continuous functional Ftϕ on Hs+1(Rn)×Hs(Rn) of the form (4), which satisfies
∀t∈[0, t), Ftϕ[u]t=I0ϕ[u]0. (8) Moreover Ftϕ[u]t is given by the expression (5).
Actually a large part of this paper is devoted to the construction of a framework in which (5) really makes sense. Once this is done Theorem 0.1 will follow relatively easily. Fur- thermore we prove a more general result which apply to nonlinear equations of the type u+m2u+V(u,∂u∂t) = 0, where s is arbitrary and V : Hs+1(Rn)×Hs(Rn)−→ Hs(Rn) is a real analytic functional on a neighbourhood of 0 in Hs+1(Rn)×Hs(Rn) (if V is a local functional on u and its space-time derivative, i.e. of the form u 7−→ W ◦(u,∇u), where W :Rn+2 −→ R is real analytic, and if s > n/2, this implies Theorem 0.1). Note that, as particular examples of applications of Theorem 0.1, we can choose ϕ such that [ϕ]0 = (0,−δ~x), for some x = (0, ~x), then it turns out that Ftϕ[u]t = u(x). Similarly by choosing [ϕ]0 = (δ~x,0) we obtain Ftϕ[u]t = ∂u∂t(x). Furthermore one can develop the expression given in (5) by using a ‘Wick theorem’ (as explained in Section 1 on a simple example) and then recover an expansion where each term is constructed by using adapted Feynman rules from a Feynman diagram modelled on a tree.
The paper is organized as follows. In the first section we expound heuristically the ba- sic ideas behind the construction of the formula (5) and we prove formally that it gives us a conserved quantities. We also explain how to recover the series of [13] described previ- ously by a kind of Wick theorem. Roughly speaking our Fock spaceF is composed of real analytic functionals on the infinite dimensional space E0s+1,|0i is the constant functional which takes the value 1 on eachϕ ∈ E0s+1,h0|is the ‘Dirac mass’ at the origin inE0s+1,φ⊲ is the multiplication by a linear functional andφ⊳ is a derivation on it. In the second section we introduce the analytic setting and the consistent definitions. The major difficulty is that the operators built fromφ⊳ cannot be bounded and hence their exponentials or time ordered exponentials cannot make sense as bounded operators from a function space to itself. This is the reason why by Fwe actually mean a family
Fr,F(1)r
0<r<∞ of normed spaces linked together by the dense inclusions FR ⊂ F(1)R ⊂ Fr ⊂ F(1)r if r < R and each symbol Ds (as defined in (7)) represents actually a family of bounded operators fromF(1)r
toFr. In particular, for any r∈(0,∞),X(r) (whereX is the vector field in Theorem 0.1)
controls the norm of the operator Ds from F(1)r to Fr. Once these required constructions are done we conclude the fourth section by the proof of Theorem 0.1. In the last section we discuss briefly how to extend Theorem 0.1 to nonlinear systems of hyperbolic equations.
Some comments on our method: series expansions of solution to nonlinear ordinary differential equations (ODE) have a long history and they have different formulations which are of course related, depending on their use. We can mention Lie series defined by K.T. Chen [5], the Chen–Fliess series [7] introduced in the framework of control theory by M. Fliess (or some variants like Volterra series or Magnus expansion [17]) which are extensively used in control theory [1, 18, 8] but also in the study of dynamical systems and in numerical analysis. Other major tools are Butcher series which explain the struc- ture of Runge–Kutta methods of approximation of the solution of an ODE. They have been introduced by J.C. Butcher [4] and developped by E. Hairer and G. Wanner [12]
which explain that Runge–Kutta methods are gouverned by trees. Later on C. Brouder [2, 3] realized that the structure which underlies the original Butcher’s computation is exactly the Hopf algebra defined by D. Kreimer in his paper about the renormalization theory [10]. Concerning analogous results on nonlinear partial differential equations, it seems that the fact that one can represent solutions or functionals on the set of solu- tions by series indexed by trees is known to physicists since the work of J. Schwinger and R. Feynman (and Butcher was also aware of that in his original work), although it is difficult to find precise references in the litterature (however for instance a formal se- ries expansion is presented in [6]). However, to our knowledge, the only previous rigorous result (i.e. with a proof of convergence of the series) is the result in [13] already discussed.
Notations — In the following we will denote by M :=Rn+1 the Minkowski space. We fix a space-time splitting M = R× Rn and we note t = x0 ∈ R the time coordinate and ~x = (x1,· · · , xn) ∈ Rn the space coordinates. We denote by := ∂t∂22 −∆ the d’Alembertian, where ∆ := Pn
i=1 ∂2
(∂xi)2 is the Laplace operator. We define the Fourier transform on smooth fastly decreasing functionsf ∈ S(Rn) by
fˆ(ξ) = 1
√2πn Z
Rn
f(~x)e−i~x·ξd~x.
And we extend it to the Schwartz classS′(Rn) of tempered distributions by the standard duality argument. Then, fors ∈R, we let
Hs(Rn) :={ϕ ∈ S′(Rn)| ǫsϕb∈L2(Rn)}, whereǫ(ξ) :=p
|ξ|2+m2 and we set||ϕ||Hs :=||ǫsϕb||L2. We denote byGthe distribution in C∞(R,S′(Rn)) whose spatial Fourier transform is given by
G(t, ξ) =b 1
√2πn sinp
|ξ|2+m2t
p|ξ|2+m2 = 1
√2πn
sin (ǫ(ξ)t) ǫ(ξ) .
Note that G∈ Cℓ(R, H−s+1−ℓ(Rn)) (for s > n/2) is nothing but the fundamental solution of
G+m2G= 0,
with the initial conditions G(0,·) = 0 and ∂G∂t(0,·) =δ0 that we already encountered.
1 A formal description of the classical Fock space
This part is essentially heuristic and provides the basic ideas which will become rigorous in the following sections.
1.1 A formal algebra
We first define a formal algebra A which helps us to define the generating function (5) (leaving aside the delicate question whether such an algebra exists). We let A be the algebra spanned over R by the symbols:
(φ⊲(x), φ⊳(y))x,y∈M, where we assume the following properties.
(i) ∀x, y ∈M, [φ⊳(y), φ⊲(x)] =G(y−x)
(ii) ∀x, y ∈M, [φ⊲(y), φ⊲(x)] = [φ⊳(y), φ⊳(x)] = 0
(iii) φ⊳(x) andφ⊲(x) depend smoothly onx and φ⊲+m2φ⊲ = 0 and φ⊳+m2φ⊳ = 0.
We further require the existence of representations of A on two vector spaces F and F∗ with the following properties. We denote by|fielements inFandhv|elements inF∗ and, for any ψ ∈ A we write |fi 7−→ ψ|fi its action on F and hv| 7−→ hv|ψ its action on F∗. We assume that there exists a pairingF∗×F∋(hv|,|fi)7−→ hv|fi ∈Rsuch that for any element ψ ∈ A on F∗ we have ∀hv| ∈ F∗, ∀|fi ∈ F, (hv|ψ)|fi = hv|(ψ|fi) =: hw|ψ|fi. Lastly we assume that there exist particular vectors|0i ∈F\ {0}andh0| ∈F∗\ {0}, such that
(iv) ∀y∈M, φ⊳(y)|0i= 0 (v) ∀x∈M, h0|φ⊲(x) = 0 (vi) h0|0i= 1.
A simple consequence of properties (i) and (ii) is the following result, analogous to a special case of the ‘Wick theorem’ used in quantum field theory.
Lemma 1.1 For all k ∈N∗, ∀y1,· · · , yk ∈M, ∀x∈M, [φ⊳(y1)· · ·φ⊳(yk), φ⊲(x)] =
Xk j=1
G(yj−x)φ⊳(y1)· · ·φ\⊳(yj)· · ·φ⊳(yk), (9)
where φ⊳(y1)· · ·φ\⊳(yj)· · ·φ⊳(yk) :=φ⊳(y1)· · ·φ⊳(jj−1)φ⊳(jj+1)· · ·φ⊳(yk).
Proof — We prove (9) by recursion on k. For k = 1 this identity is nothing but (i).
Assume that (9) has been proved for k pointsyj’s, then,∀y1,· · · , yk+1 ∈M,∀x∈M, (by denoting φ⊳j :=φ⊳(yj) andφ⊲ :=φ⊲(x) for short)
φ⊳1· · ·φ⊳k+1φ⊲ = φ⊳1 φ⊳2· · ·φ⊳k+1φ⊲
=φ⊳1
"
φ⊲φ⊳2· · ·φ⊳k+1+ Xk+1
j=2
G(yj−x)φ⊳2· · ·φb⊳j· · ·φ⊳k+1
#
= φ⊲φ⊳1φ⊳2· · ·φ⊳k+1+G(y1−x)φ⊳2· · ·φ⊳k+1+φ⊳1 Xk+1
j=2
G(yj−x)φ⊳2· · ·φb⊳j· · ·φ⊳k+1.
and the result follows for k+ 1 points.
1.2 Time invariance of the functional defined by (5)
We will see that, formally, the functional defined by (5) is time independent. We first introduce more concise notations, setting
Z
t
u←→∂ φ⊳ :=
Z
y0=t
u(y)
←→
∂
∂y0φ⊳(y)d~y, Z t
0
W(φ⊲,∇φ⊲)φ⊳ :=
Z
0<z0<t
W(φ⊲(z),∇φ⊲(z))φ⊳(z)dz, and
|fϕi:=
Z
0
φ⊲←→
∂ ϕ|0i:=
Z
x0=0
φ⊲(x)
←→∂
∂x0ϕ(x)d~x|0i, so that we may write
Ftϕ[u]t=−h0| exp
− Z
t
u←→∂ φ⊳
Texp
− Z t
0
W(φ⊲,∇φ⊲)φ⊳
|fϕi. (10) As a preliminary we derive some relations which are consequences of (9). It first implies that for all k ∈N,
"Z
t
u←→
∂ φ⊳ k
, φ⊲(x)
#
=k Z
y0=t
u(y)
←→
∂
∂y0G(y−x) Z
t
u←→
∂ φ⊳ k−1
. (11)
This follows from (9) by writing Z
t
u←→
∂ φ⊳ k
= Z
{y10=···=y0k=t}
u1· · ·uk←→
∂1 · · ·←→
∂kφ⊳1· · ·φ⊳k d~y1· · ·d~yk,
with the convention that uj := u(yj) and φ⊳j := φ⊳(yj). Then (11) implies that for any analytic function of one variable f,
f
Z
t
u←→
∂ φ⊳
, φ⊲(x)
= Z
y0=t
u(y)
←→
∂
∂y0G(y−x) f′ Z
t
u←→
∂ φ⊳
. (12)
But assuming that x0 =t and that f(ζ) =e−ζ, this gives
exp
− Z
t
u←→∂ φ⊳
, φ⊲(x)
=u(x) exp
− Z
t
u←→∂ φ⊳
if x0 =t, (13) because of the Cauchy conditions on Gat time 0. Lastly we deduce from (13) that
exp
− Z
t
u←→
∂ φ⊳
, Z
t
W(φ⊲,∇φ⊲)φ⊳
= Z
t
Wf(u,∇u)φ⊳ exp
− Z
t
u←→
∂ φ⊳
, (14) where Wf(u,∇u) = W(u,∇u)−W(0,0). Now assuming that the expression in (10) is differentiable with respect to time t,
dFtϕ[u]t
dt = h0|d dt
Z
t
u←→
∂ φ⊳
exp
− Z
t
u←→
∂ φ⊳
Texp
− Z t
0
W(φ⊲,∇φ⊲)φ⊳
|fϕi +h0| exp
− Z
t
u←→
∂ φ⊳ Z
t
W(φ⊲,∇φ⊲)φ⊳
Texp
− Z t
0
W(φ⊲,∇φ⊲)φ⊳
|fϕi. We can use (14) and the fact thath0|R
tW(φ⊲,∇φ⊲)φ⊳ =h0|R
tW(0,0)φ⊳ to transform the second term in the right hand side and get that dFtϕdt[u]t is equal to
h0| d
dt Z
t
u←→
∂ φ⊳
+ Z
t
W(u,∇u)φ⊳
exp
− Z
t
u←→
∂ φ⊳
Texp
− Z t
0
W(φ⊲,∇φ⊲)φ⊳
|fϕi. But by using the fact that φ⊳ is a solution of the linear Klein–Gordon equation and an integration by parts in space variables we obtain
d dt
Z
t
u←→
∂ φ⊳
= Z
t
u←→
∂ 2φ⊳ = Z
Rn
∂2u
∂t2φ⊳−u∂2φ⊳
∂t2 = Z
Rn
∂2u
∂t2φ⊳−u ∆φ⊳−m2φ⊳
= Z
Rn
∂2u
∂t2φ⊳− ∆u−m2u φ⊳ =
Z
Rn
(u+m2u)φ⊳. Hence dFtϕdt[u]t is equal to
h0| Z
t
(u+m2u+W(u,∇u))φ⊳exp
− Z
t
u←→
∂ φ⊳
Texp
− Z t
0
W(φ⊲,∇φ⊲)φ⊳
|fϕi, which vanishes if u+m2u+W(u,∇u) = 0.
1.3 Expanding the generating functions F
t[u]
tThe ‘Wick theorem’ (9) together with rules (iv) and (v) allow us also to recover an expansion of the functional Ftϕ[u]t defined by (5) in terms of finite integrals defined by trees by using Feynman rules. To illustrate this we consider the simplest case, i.e. when W(u,∇u) =λu2, for some λ∈R, and consider
Ftϕ[u]t =−h0| exp
− Z
t
u←→
∂ φ⊳
Texp
− Z t
0
λ(φ⊲)2φ⊳ Z
0
φ⊲←→
∂ ϕ|0i. (15) First note that all expressions of the form h0|φ⊳1· · ·φ⊳pφ⊲1· · ·φ⊲q|0i (where φ⊳i, φ⊲j denote respectivelyφ⊳(yi),φ⊲(xj) for some pointsyi, xj ∈M) vanish unlessp=q, as can be shown by repeated applications of (9) and of (iv) and (v). Hence for instance the coefficient of the 0th power of λ in the series (15) is
Ft,1ϕ[u]t=−h0| exp
− Z
t
u←→∂ φ⊳
1 Z
0
φ⊲←→∂ ϕ|0i=h0| Z
t
uy←∂→yφ⊳y Z
0
φ⊲x←∂→xϕx|0i, where we set uy := u(y), ←→
∂y := ∂y←→∂0, φ⊳y := φ⊳(y), φ⊲x := φ⊲(x), etc. However it follows from (i), (iv) and (v) thath0|φ⊳yφ⊲x|0i=h0|Gyx+φ⊲xφ⊳y|0i=Gyx (whereGyx:=G(y−x)).
Hence
Ft,1ϕ[u]t= Z
y0=t
Z
x0=0
uy←→
∂yGyx←→
∂xϕx=Itϕ[u]t. The coefficient of the first power of λ is
Ft,2ϕ[u]⊗t2 = −h0| exp
− Z
t
u←→
∂ φ⊳ − Z t
0
(φ⊲)2φ⊳ Z
0
φ⊲←→
∂ ϕ|0i
= h0|1 2
Z
t
Z
t
u1u2←→
∂1←→
∂2φ⊳1φ⊳2 Z t
0
(φ⊲z)2φ⊳z Z
0
φ⊲x←→
∂xϕ|0i, where, for a = 1,2, ua := u(ya), ←→
∂a := ∂y←→∂0
a, φ⊳a :=φ⊳(ya) and φ⊲z := φ⊲(z), etc. Denoting also by Gaz :=G(ya−z), for a = 1,2, we have by (9) [φ⊳1φ⊳2, φ⊲z] =G1zφ⊳2+G2zφ⊳1, which implies by (v) that h0|φ⊳1φ⊳2φ⊲z =G1zh0|φ⊳2+G2zh0|φ⊳1. A second application of (9) and (v) gives then
h0|φ⊳1φ⊳2(φ⊲z)2 =G1zh0|φ⊳2φ⊲z+G2zh0|φ⊳1φ⊲z = (G1zG2z+G2zG1z)h0|. We hence deduce that
Ft,2ϕ[u]⊗t2 = Z
t
Z
t
u1u2←→
∂1←→
∂2
Z t 0
G1zG2zh0|φ⊳z Z
0
φ⊲x←→
∂xϕ|0i,
= Z
t
Z
t
u1u2←→
∂1←→
∂2
Z t 0
G1zG2z
Z
0
Gzx←→
∂xϕ,
where we set Gzx:=G(z−x) and we further use (9) and (iv). We hence recover the same expression forFt,2ϕ [u]⊗t2 as the one obtained in [13] and expounded in the introduction.
1.4 How to construct the representation of A on F and F
∗The heuristic idea to constructAand its representationFis the following. We still denote by E0 the space of solutions of the linear Klein–Gordon equation (1) ϕ+m2ϕ = 0 on M and we letF to be the set of analytic functionals |fi:E0 −→R, i.e. such that
∀ϕ∈ E0, |fi(ϕ) = X∞
p=0
|fpi(ϕ⊗ · · · ⊗ϕ) = X∞
p=0
|fpi ϕ⊗p
, (16)
where we can view each |fpi as a symmetric p-multilinear functional on (E0)p or, alterna- tively, as a linear form on (E0)⊗p. Lastly we let A to be the algebra of linear operators acting on F and F∗ the dual space of F.
• The definition of φ⊲(x) — To any pointx∈M we associate a particular (linear) functional on E0, the evaluation at x:
|xi: E0 −→ R
ϕ 7−→ |xi(ϕ) :=ϕ(x).
And we let φ⊲(x) to be the operator of multiplication of functionals in F by the functional|xi:
φ⊲(x) : F −→ F
|fi 7−→ |xi|fi, i.e. ∀ϕ ∈ E0, (φ⊲(x)|fi) (ϕ) =ϕ(x) (|fi(ϕ)).
• The definition of φ⊳(y) — We remark that, for any y ∈ M, the distribution Γy :x 7−→G(y−x) is a weak solution of the linear Klein–Gordon equation, hence an element of E0. Thus, for any |fi ∈ F, we can formally define the variational derivative
δ|fi δΓy
(ϕ) := lim
ε→0
|fi(ϕ+εΓy)− |fi(ϕ)
ε .
and we let
φ⊳(y) : F −→ F
|fi 7−→ δδΓ|fyi.
An alternative, more algebraic definition ofφ⊳(y) for |fi given by (16) is (φ⊳(y)|fi) (ϕ) =
X∞ p=1
p|fpi(Γy⊗ϕ⊗ · · · ⊗ϕ).
Then we observe that by using the Leibniz rule φ⊳(y)φ⊲(x)|fi= δ(φ⊲(x)|fi)
δΓy
= δ(|xi|fi) δΓy
= δ|xi
δΓy|fi+|xiδ|fi δΓy
.
But since |xi is a linear functional δδΓ|xyi =|xi(Γy) =G(y−x), hence φ⊳(y)φ⊲(x)|fi=G(y−x)|fi+φ⊲(x)δ|fi
δΓy
=G(y−x)|fi+φ⊲(x)φ⊳(y)|fi. We thus deduce that
[φ⊳(y), φ⊲(x)] = G(y−x).
Moreover it is obvious that (still at a formal level) [φ⊳(y), φ⊳(y′)] = [φ⊲(x), φ⊲(x′)] = 0.
Lastly we let |0i ∈F to be the constant functional E0 ∋ϕ 7−→1 and h0| ∈F∗ to be h0|: F −→ R
|fi 7−→ |fi(0),
i.e. h0| plays the role of the Dirac mass at 0 ∈ E0. Then obviously (iv), (v) and (vi) are satisfied. In this representation we remark that an interpretation of h[u]t| in (5) is possible. Indeed if we think the operatorR
tu←→
∂ φ⊳ as a constant vector field on the infinite dimensional space E0, then the exponential exp
−R
tu←→
∂ φ⊳
acts on F by translation, i.e. through
exp
−R
tu←→
∂ φ⊳
|fi
(ϕ) = |fi(ϕ +ϕ[u]t), where ϕ[u]t ∈ E0 is actually the solution of (1) which has the same Cauchy data at time t as u (in the notations of Paragraph 2.2.2, ϕ[u]t := −u←→
♯tG). Hence we deduce that h[u]t| := h0|exp
−R
tu←→
∂ φ⊳ is the Dirac mass at ϕ[u]t ∈ E0. These facts will be proved in Corollary 3.3.
1.5 Towards a well-defined theory
All the preceding constructions are completely formal, as long as we do not precise any topology on F, F∗ and A. Moreover we would like to find topologies in such a way that the operators φ⊲ and φ⊳ are simultaneously well-defined. Here the main difficulty occurs, indeed:
(a) on the one hand, in order to defineφ⊲(x), we need to make sense of|xi:ϕ7−→ϕ(x) as a continuous operator and this requires the functions ϕ in E0 to be continuous.
Actually a careful inspection of the formal computations done in section 1.2 reveals that one also needs to define ∂φ∂x⊲0, which means that we actually need that ϕ be of class C1 with respect to time;
(b) on the other hand the definition of φ⊳(y) is consistent if we can differentiate with respect to Γy, i.e. if Γy ∈ E0. However Γy is only a distribution and hence this is in conflict with the first requirement.
The key observation to avoid these difficulties is that, in the definition of (5) and in the formal computations done in section 1.2 we only need to define
U(t) :=
Z
t
u←→
∂ φ⊳, V(t) :=
Z
t
(u+m2u)φ⊳ and D(t) :=
Z
t
W(φ⊲,∇φ⊲)φ⊳
as continuous maps of the time t, with values in a set of continuous operators and to assume that Uis derivable with respect to time, with ddtU =V. Hence we will assume that functions in E0 are sufficiently smooth, so that operatorsφ⊲ and ∂φ∂t⊲ will be well-defined.
However we will be able to make sense ofUand Vifuis sufficiently smooth and to define D again if functions in E0 of sufficiently smooth functions ϕ. Note that, on the infinite dimensional manifold E0, U and V can be viewed as tangent vector fields with constant coefficients, whereas D is a tangent vector field with non constant analytic coefficients.
2 The analytic setting
In this section we define precisely the ‘Fock space’ F and introduce the operators U, V and D as well as the exponential of these operators in order to make sense of (5).
2.1 Functions spaces
First we define more precisely the spaceE0 of solution of the linear Klein–Gordon equation (1), in particular we describe the topology of this space. Fors∈R we define E0s+1 by
E0s+1 :={ϕ ∈ C0(R, Hs+1(Rn))∩ C1(R, Hs(Rn))| ϕ+m2ϕ= 0}.
Each map ϕ ∈ E0s+1 is characterized by its Cauchy data [ϕ]0 = (ϕ0, ϕ1) ∈ Hs+1(Rn)× Hs(Rn) at time t= 0. Indeed one recovers ϕ from (ϕ0, ϕ1) through the relation
ˆ
ϕ(t, ξ) = ˆϕ0(ξ) cosǫ(ξ)t+ ˆϕ1(ξ)sinǫ(ξ)t
ǫ(ξ) (ǫ(ξ) :=p
|ξ|2+m2),
where ˆϕ0 and ˆϕ1 are the spatial Fourier transform of respectively ϕ0 and ϕ1. Note that the quantity
||ϕ||2Es+1 :=||ϕ(t,·)||2Hs+1+||∂ϕ
∂t(t,·)||2Hs
is independant of t∈R.
Definition 2.1 For eachp∈N we let(Np
E0s+1)∗ be the space of linear applications|fpi: Np
E0s+1 :=E0s+1⊗· · ·⊗E0s+1 −→Rwhich are continuous, i.e. such that there exists a con- stantC >0such that ∀ϕ1,· · · , ϕp ∈ E0s+1, ||fpi(ϕ1⊗ · · · ⊗ϕp)| ≤C||ϕ1||E0s+1· · · ||ϕp||E0s+1. We denote by ⌈fp⌋ the optimal value of C in this inequality, so that we have:
||fpi(ϕ1⊗ · · · ⊗ϕp)| ≤ ⌈fp⌋ ||ϕ1||E0s+1· · · ||ϕp||E0s+1, ∀ϕ1,· · ·, ϕp ∈ E0s+1. (17) We let S(Np
E0s+1)∗ be the subspace of |fpi ∈ (Np
E0s+1)∗ which are symmetric, i.e. for all permutation σ ∈Sp,
|fpi(ϕσ(1)⊗ · · · ⊗ϕσ(p)) =|fpi(ϕ1⊗ · · · ⊗ϕp), ∀ϕ1,· · · , ϕp ∈ E0s+1. (18) We can now define our ‘Fock space’:
Definition 2.2 For any s∈R and r∈(0,+∞), we let Fr be the space of formal series
|fi= X∞
p=0
|fpi, where each |fpi ∈S(Np
E0s+1)∗ and such that the quantity Nr(|fi) :=
X∞ p=0
⌈fp⌋rp (19)
is finite. Then (Fr, Nr) is a Banach space.
Actually we can identify any |fpi ∈S(Np
E0s+1)∗ with a continuous homogeneous polyno- mial map of degreepfromE0s+1toRby the relation|fpi(ϕ) :=|fpi(ϕ⊗· · ·⊗ϕ) =|fpi(ϕ⊗p) and hence series |fi ∈Fr with convergent analytic series on BEs+1
0 (0, r), the ball of radius r in E0s+1, by the relation
|fi(ϕ) = X∞
p=0
|fpi(ϕ⊗p).
Let us define |0ito be the constant functional over E0s+1 which is equal to 1:
|0i:ϕ∈ E0s+1 7−→1.
Then |0ibelongs to Fr for all r >0 andNr(|0i) = 1.
We will also extend the definition (19) to a complex variable z:
Nz(|fi) :=
X∞ p=0
⌈fp⌋zp, ∀z such that|z| ≤r,
which gives us, for any fixed |fi ∈ Fr, a holomorphic function on the ball BC(0, r)⊂C.
We define, for k∈N, Nr(k)(|fi) := dk
dzkNz(|fi)|z=r and F(k)r :={|fi ∈Fr| Nr(k)(|fi)<+∞}, so that for instance Nr(1)(|fi) := P∞
p=1p⌈fp⌋rp−1. We set F∞ := ∩r>0Fr and Fpol :=
{|fi=PN
p=0|fpi|N ∈N,|fpi ∈S(Np
E0s+1)∗}. Note that, using in particular the obvious inequality r < R =⇒ Nr(ϕ)< NR(ϕ), we have the dense inclusions
∀r, R∈(0,∞), s.t. r < R,∀k ∈N, Fpol ( F∞ ( F(k+1)R ( F(k)R ⊂FR( Fr.
Since the space Fr is an (infinite dimensional) topological vector space for all r > 0, we can consider its topological dual space, denoted by (Fr)∗. For example we can consider the linear formh0| over Fr defined by
h0|:Fr ∋ |fi 7−→ h0|fi:=|fi(0),
Then one can easily see that h0| belongs to ∩r>0(Fr)∗ and that we have h0|0i = 1. More generally ifubelongs to∩ℓ=0,1Cℓ((t, t), Hs+1−ℓ(Rn)) then fort∈(t, t) we can consider ϕ[u]t
the (unique) element ofE0s+1 such that [ϕ[u]t]t= [u]t (i.e. ϕ[u]t is the solution of the linear Klein–Gordon equation (1) with the same Cauchy data at time t as [u]t). Then since
||ϕ[u]t||E0s+1 = ||[u]t||s+1, we can consider |fi(ϕ[u]t) to be the evaluation of the functional
|fionϕ[u]t, for allr >0 such that r >||[u]t||s+1 and, for all|fi ∈Fr. This define a linear formh[u]t| over Fr :
∀r >||[u]t||s+1; h[u]t|:Fr ∋ |fi 7−→ h[u]t|fi:=|fi(ϕ[u]t).
Then h[u]t| belongs to∩r>||[u]t||s+1(Fr)∗.
Definition 2.3 For any r0 ∈(0,∞] and any k, ℓ∈N a continuous operator T from F(k)(0,r
0) to F(ℓ)(0,r
0) is a family (Tr)0<r<r0, where, for any r ∈ (0, r0), Tr : F(k)r −→F(ℓ)r is a continuous linear operator with norm ||Tr|| and such that, ∀r, r′ ∈(0, r0), if r < r′, then the restriction of Tr to F(k)r′ coincides with Tr′. Moreover, if X : (0, r0) −→ (0,∞) is a locally bounded function, we say thatthe norm ofT is controlled by X if,∀r ∈(0, r0),
||Tr|| ≤X(r).
For simplicity we systematically denote each operator Tr by T in the following. In the case where r0 =∞, we will just write that T is a continuous operator from F(k) to F(ℓ).
The following result concerns an example of a continuous operator from F(1) to Fwith a norm controlled by the constant function||ψ||E0s+1.
Lemma 2.1 Let s ∈ R and ψ ∈ E0s+1. Then for any r ∈ (0,∞) and |fi ∈ F(1)r , the functional δδψ|fi defined by
∀ϕ ∈BEs+1
0 (0, r), δ|fi δψ (ϕ) :=
X∞ p=1
p|fpi(ψ⊗ϕ⊗p−1) belongs to Fr and
Nr
δ|fi δψ
≤ ||ψ||E0s+1Nr(1)(|fi). (20) Proof — For any ϕ ∈ E0s+1 we have
∀p∈N, p|fpi ψ ⊗ϕ⊗p−1≤p⌈fp⌋||ψ||E0s+1||ϕ||pE−s+11
0 . Hence if we denote |gi:= δδψ|fi we deduce that ⌈gp−1⌋ ≤p||ψ||E0s+1⌈fp⌋and so
Nr
δ|fi δψ
= X∞
p=1
⌈gp−1⌋rp−1 ≤ X∞
p=1
p||ψ||E0s+1⌈fp⌋rp−1 =||ψ||E0s+1Nr(1)(|fi).
