Multisymplectic formalism and the covariant phase space

Frédéric H élein * June 10, 2011

In most attempts for building the mathematical foundations of Quantum Fields Theory (QFT) two classical ways have been explored. The first one is often referred to as the Feynman integral or functional integral method. It is a generalization to fields of the path integral method for quantum mechanics and is heuristically based on computing integrals over the infinite dimensional set of all possible fields φ by using a kind of 'measure' -which should behave like the Lebesgue measure on the set of all possible fields φ -times e iL(φ)/ , where L is a Lagrangian functional (but attempts to define this 'measure' failed in most cases). The second one is referred to as the canonical quantization method and is based on the Hamiltonian formulation of the dynamics of classical fields, by following general axioms which were first proposed by Dirac and later refined. The Feynman approach has the advantage of being manifestly relativistic, i.e. it does not require the choice of a particular system of space-time coordinate, since the main ingredient is L(φ), which is an integral over all space-time. However the canonical approach, at least its classical formulation, seems to be based to the choice of a particular time coordinate which is needed to define the Hamiltonian function through an infinite dimensional Legendre transform.

However they are alternative formulations of the Hamilton structure of the dynamics of classical fields, which could be used as a starting point of a covariant canonical quantization 1 . We shall see two of them in this text: the covariant phase space and the multisymplectic formalism. The covariant phase space is based on the observation that the set of classical solutions to a variational problem, i.e. of critical points of some action functional L(φ), has an intrinsic canonical symplectic (or presymplectic) structure. The multisymplectic formalism is a generalization of the standard symplectic formalism, where the time in classical mechanics is replaced by the space-time: for instance if we start from a Lagrangian action L(φ) = X L(x, φ, dφ) we do not perform a Legendre transform with respect to a chosen time coordinates, but with respect to all space-time coordinates.

We expect that roughly speaking both the Feynman and the canonical approach should lead to equivalent theories. However if this fact is true, it should not be trivial for several reasons. A first obvious remark for that is that both theories are only heuristics and have no mathematical foundations, excepted in very simplified situations. A superficial difference between both approaches is the fact that one is based on the Lagrangian, the other one the Hamiltonian function. Moreover these two approaches answer to different questions, the Feynman offers a short and intuitive way to compute the quantities which can be measured in interaction processes between particles (although one cannot avoid the difficult step of regularizing and renormalizing the computed quantities). For the same task, the canonical approach seems to be more complicated, however it proposes a scheme to build mathematical objects (a complex Hilbert space of physical states and an algebra of self-adjoint operators acting on it, corresponding to observable quantities), the construction of which requires more effort by using the Feynman integral. But a deep difference between both methods is that the Feynman integral is a construction off shell, i.e. on the set of all possible fields, even those which are not solutions of the classical dynamical equations, whereas in many cases the canonical approach is a construction on shell, i.e. on the set of fields which are solutions of the dynamical equations (in particular in the covariant phase space method).

In this paper we shall present briefly the multisymplectic formalism and the covariant phase space and show the strong relation between both theories. To my knowledge this relation was discovered by J. Kijowski and W. Szczyrba in 1976 [START_REF] Kijowski | A canonical structure for classical field theories[END_REF], but their beautiful paper seems to have been ignored in the literature. We have included some historical comments. We shall conclude by presenting the geometric quantization scheme for linear field equations (i.e. free fields in the terminology of physicists) in the framework of multisymplectic geometry. The goal is to show how a canonical quantization could be performed in covariant way. [START_REF] Baez | Categorified Symplectic Geometry and the String Lie 2-algebra[END_REF] The multisymplectic formalism

Maps between vector spaces

We start with a simple variational problem: let X and Y be two vector space of dimension n and k respectively and assume that X is oriented, let U be an open subset of X and consider the set C ∞ (U, Y) of smooth maps u from U to Y. Let L : U × X × End(X, Y) -→ R be a Lagrangian density and consider the action functional on C ∞ (U, Y) defined by:

L[u] = U L(x, u(x), du x )β,
where β is a volume n-form on U. We use coordinates (

x 1 , • • • , x n ) on U s.t. β = dx 1 ∧ • • • ∧ dx n , coordinates (y 1 , • • • , y k ) on Y and v i µ on End(X, Y).
Then the critical points of L satisfy the Euler-Lagrange system of equations

∂ ∂x µ ∂L ∂v i µ (x, u(x), du x ) = ∂L ∂y i (x, u(x), du x ), ∀i s.t. 1 ≤ i ≤ k. (1)
We assume that the map

U × Y × End(X, Y) -→ U × Y × End(Y * , X * ) (x, y, v) -→ (x, y, ∂L ∂v (x, y, v))
, is a diffeomorphism: this is the analogue of the Legendre hypothesis used in Mechanics. We denote by p * = (p µ i ) µ,i the coordinates on End(Y * , X * ) and we define the Hamiltonian function

H on U × X × End(Y * , X * ) by H(x, y, p * ) := p µ i v i µ -L(x, y, v),
where we assume implicitely that v = (v i µ ) i,µ is the unique solution of ∂L ∂v (x, y, v) = p * . Then v i µ is actually equal to ∂H ∂p µ i . Moreover to any map u : U -→ Y we associate the map p * : U -→ End(Y * , X * ) s.t. p * (x) := ∂L ∂v (x, u(x), du x ), ∀x ∈ U. Then we can show [START_REF] Volterra | Sulle equazioni differenziali che provengono da questiono di calcolo delle variazioni[END_REF] that u is a solution of (1) iff (u, p * ) is a solution of the generalized Hamilton system:

       ∂u i ∂x µ (x) = ∂H ∂p µ i (x, u(x), p * (x)) ∂p µ i ∂x µ (x) = - ∂H ∂y i (x, u(x), p * (x)). (2)
System (2) can be translated as a geometric condition [START_REF] Kijowski | A finite dimensional canonical formalism in the classical field theory[END_REF] on the graph

Γ * := G(u, p * ) := {(x, u(x), p * (x))| x ∈ U} ⊂ U × Y × End(Y * , X * ).
Indeed consider a family of n vector fields

X 1 , • • • , X n : U -→ U ×Y×End(Y * , X * ) s.t. for any x ∈ U, (X 1 (x), • • • , X n (x)) is a basis of the tangent plane to G(u, p * ) at (x, u(x), p * (x)). Set β µ := ∂ ∂x µ β. Then (2) is equivalent to the condition that ∀ξ ∈ X × Y × End(Y * , X * ), dp µ i ∧ dy i ∧ β µ (ξ, X 1 , • • • , X n ) = dH(ξ)β(X 1 , • • • , X n ). (3)
In fact this can be easily checked by choosing

X µ := ∂ ∂x µ + ∂u i ∂x µ ∂ ∂y i + ∂p µ i ∂x µ ∂ ∂p µ i . More concisely we can introduce the n-multivector field X := X 1 ∧ • • • ∧ X n (so that X(x) ∈ Λ n T (x,u(x),p * (x)) Γ * , ∀x ∈ U). Then Equation (3) reads: ∀ξ ∈ X × Y × End(Y * , X * ), dp µ i ∧ dy i ∧ β µ (ξ, X) = dH(ξ)β(X). (4)
Equation ( 3) can be completed with the independence condition

β| Γ * = 0, (5)
where, if j Γ * : Γ * -→ U ×Y×End(Y * , X * ) denotes the inclusion map, β| Γ * := j * Γ * β. This condition garantees that locally Γ * is the graph of some map (u, p * ) over the 'space-time' X.

We will see now that the independence condition ( 5) can be further incorporated in a dynamical condition analogous to (3) by adding to the variables (x, y, p * ) a variable e dual to β. We define M := U × Y × R × End(Y * , X * ) with coordinates (x, y, e, p * ) = (x µ , y i , e, p µ i ) and the (n + 1)-form

ω := de ∧ β + dp µ i ∧ dy i ∧ β µ . (6)
We define H : M -→ R by H(x, y, e, p * ) := e + H(x, y, p * ). Then to any oriented n-dimensional submanifold Γ * = G(u, p * ) we associate the oriented n-dimensional submanifold Γ := {(x, u(x), e(x), p * (x))| x ∈ U} of M, where e is s.t. e(x) + H(x, u(x), p * (x)) = h, ∀x ∈ X, for some real constant2 h. Then Γ * is a solution of (3) and ( 5) iff Γ is a solution of:

∀ξ ∈ C ∞ (M, T M), ω(ξ, X) = dH(ξ)β(X), (7)
where C ∞ (M, T M) denotes the set of sections of T M over M, which can be iden-

tified with C ∞ (M, X × Y × R × End(Y * , X * )).
Note that conversely it is easy to check that any connected solution Γ of [START_REF] Donder | Introduction à la théorie des invariants intégraux[END_REF] is contained in a level set of H. We call a Hamiltonian n-curve any solution Γ of [START_REF] Donder | Introduction à la théorie des invariants intégraux[END_REF]. The (n + 1)-form ω is an example of a multisymplectic form and the pair (M, ω) is called a multisymplectic manifold. Using the notation ξ ω for the interior product of the vector ξ with the (n + 1)-form ω, we set:

Definition 1.1 Let M be a smooth manifold. A multisymplectic (n + 1)-form ω on M is a (n + 1)-form which is closed (i.e. dω = 0) and which is non degenerate (i.e. ∀m ∈ M, ∀ξ ∈ T m M, ξ ω = 0 =⇒ ξ = 0).

Higher order problems

The preceding can be extended to an action on maps u : U -→ Y of the form L[u] := U L(x, j r u(x))β, where j r u denotes the r-th order jet of u (i.e. all partial derivatives of u of order less than or equal to r). We denote by J r (U, Y) the r-th order jet space of maps from U to Y and we use the coordinates

x = (x µ ) µ and v = v i µ 1 •••µa i,µ 1 •••µa (for 1 ≤ i ≤ k, 0 ≤ a ≤ r and 1 ≤ µ b ≤ n) on J r (U, Y) s.t. v i µ 1 •••µa (j r u(x)) = ∂ a u i ∂x µ 1 • • • ∂x µa (x). It is convenient to introduce the multi-index notation M = µ 1 • • • µ a , where a ∈ N and ∀b ∈ [[1, a]], 1 ≤ µ b ≤ n and to set |M| = a. Then for |M| = r we note π M i (x, v) := ∂L ∂v i M (x, v). ( 8
)
The analogue of the Legendre hypothesis consists here in supposing that the map 8) is one to one. Next we define the vector space M with coordinates:

(v i M ) i,M ;|M |=r -→ (p M i ) i,M ;|M |=r defined by (
(x, v) : x µ v i v i µ • • • v i µ 1 •••µ r-1 p = (e, p * ) : e p µ i • • • p µ 1 •••µ r-1 i p µ 1 •••µr i for 1 ≤ µ, µ b ≤ n and 1 ≤ i ≤ k.
Clearly M contains J r-1 (U, Y) as a vector subspace. We also define recursively, for |M| ≤ r -1,

π M i (x, v) := ∂L ∂v i M (x, v) -D µ π M µ i (x, v M ),
where

D µ := ∂ ∂x µ + v i M µ ∂ ∂v i M .
Then we define a Hamiltonian function on M:

H(x, v, p * ) := p µ i v i µ + • • • + p µ 1 •••µr i v i µ 1 •••µr -L(x, v
), where we assume implicitely that, for

|M| = r, v i M = v i µ 1 •••µr is the solution of p M i = π M i (x, v), ∀M s.t. |M| = r, and we set p M i := π M i (x, v), ∀M s.t. |M| < r.
To any map u from U to Y we associate the map p * which is the image of j r u by the maps π M i . Then u is a critical point of L iff (j r-1 u, p * ) is a solution of the generalized Hamilton equations [START_REF] Donder | Théorie invariantive du calcul des variations[END_REF] 

      ∂u i µ 1 •••µa ∂x µ (x) = ∂H ∂p µ 1 •••µaµ i (x, u(x), p * (x)) for 0 ≤ a ≤ r -1 ∂p µ 1 •••µaµ i ∂x µ (x) = - ∂H ∂v i µ 1 •••µa (x, u(x), p * (x)) for 0 ≤ a ≤ r -1, (9)
Alternatively we can consider the map x -→ (x, j r-1 u(x), e(x), p * (x)), where e may be chosen so that e(x) + H(x, j r-1 u(x), p * (x)) = 0, ∀x and we can write [START_REF] Dolan | A co-variant approach to Ashtekar's canonical gravity[END_REF] in a way similar to (7) by using the Hamiltonian function

H(x, v, p) = e + H(x, v, p * ).
and the multisymplectic form

ω := de ∧ β + dp µ i ∧ dv i ∧ β µ + • • • + dp µ 1 •••µr i ∧ dv i µ 1 •••µ r-1 ∧ β µr .
An intrinsic geometrical multisymplectic formulation of these equations has been derived recently by L. Vitagliano [START_REF] Vitagliano | The Lagrangian-Hamiltonian Formalism for Higher Order Field Theories[END_REF].

More general multisymplectic manifolds

Assume that we start with an action L which is an integral of a Lagrangian density which depends on the first order derivatives of the field. This may be for instance a variational problem on maps u between two manifolds X and Y with a functional L[u] := X L(x, u(x), du x )β or a variational problem on sections of a fiber bundle π : Z -→ X . Then a natural multisymplectic manifold is the vector bundle Λ n T * (X × Y) in the first case or Λ n T * Z in the second case. Both manifolds are indeed endowed with a canonical (n + 1)-form ω which is the straightforward analogue of the canonical symplectic form on any cotangent bundle [START_REF] Dedecker | On the generalization of symplectic geometry to multiple integrals in the calculus of variations[END_REF][START_REF] Kijowski | Multiphase spaces and gauge in the calculus of variations[END_REF][START_REF] Hélein | Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl[END_REF]. We may call this manifold the universal multisymplectic manifold associated with the Lagrangian problem. Although this construction seems to be similar to the symplectic one for Hamiltonian mechanics, it is different because, say for maps between two manifolds X and Y of dimensions n and k respectively and a first order variational problem, on the one hand the Lagrangian density depends on n + k + nk variables (in other words the analogue of the product of the time real line and of the tangent bundle in mechanics has dimension n + k + nk), whereas on the other hand the analogue of the cotangent bundle is Λ n T * (X × Y) and has dimension n + k + (n+k)! n!k! . This means that we have much more choices in the Legendre transform, which is not a map in general but a correspondence, as soon as n ≥ 2 and k ≥ 2. This is why it is possible and often simpler to impose extra constraints in the Legendre transform, which means that we replace the universal multisymplectic manifold Λ n T * Z (whatever Z is: a Cartesian product X × Y or the total space of a bundle) by some submanifold of it. Most Authors prefer to use the affine multisymplectic submanifold

Λ n 2 T * Z: if Z = X × Y, it is the subbundle of Λ n T * (X × Y) over X × Y, the fiber over the point (x, y) ∈ X × Y of which is the subspace of n-forms p ∈ Λ n T * (x,y) (X × Y) s.t. ∀η 1 , η 2 ∈ T y Y, (0, η 1 ) ∧ (0, η 2 ) p = 0.
It Z is a fiber bundle over X , Λ n 2 T * Z, which is the subbundle over Z, the fiber over z ∈ Z of which is the subspace of n-forms p ∈ Λ n T * z Z s.t. for any pair of vertical vectors η 1 , η 2 ∈ T z Z, η 1 ∧ η 2 p = 0 (by 'vertical' we mean that η 1 and η 2 are maps to 0 ∈ T π(z) X by the differential of π). In both cases an n-form reads p = eβ + p µ i dy i ∧ β µ in local coordinates and the latter theory is actually the right generalization of ( 6). This theory is usually refered to as the De Donder-Weyl theory although it was discovered by V. Volterra (see §1.8). Note that Λ n 2 T * Z can alternatively be defined as being the affine dual of the first jet bundle of sections of Z over X [START_REF] Gotay | Momentum maps and classical relativistic fields, Part I/ covariant field theory[END_REF].

Premultisymplectic manifolds

A variant consists in manifolds equipped with a closed (n + 1)-form but without assuming a non-degeneracy condition, as for instance in [START_REF] Kijowski | A canonical structure for classical field theories[END_REF]:

Definition 1.2 We call a triple (M, ω, β) an n-phase space if M is a manifold, ω is a closed (n + 1)-form, called a premultisymplectic form and β is a non vanishing n-form.
Examples of premultisymplectic manifolds can be built easily by starting from a multisymplectic manifold (M, ω) with a Hamiltonian function H on it which has no critical points (as for instance H(x, y, e, p * ) = e + H(x, y, p * ) for the previous theory). Then we let η be a vector field on M s.t. dH(η) = 1 everywhere and we set β := η ω. For any h ∈ R the level set [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF]. In particular ω| M h is obsviously closed but may be degenerate in general: indeed if Γ is a Hamiltonian ncurve contained in M h then any vector tangent to Γ is in the kernel of ξ -→ ξ ω.

M h := H -1 (h) is a submanifold. Then (M h , ω| M h , β| M h ) is a premultisymplectic manifold
In fact an n-phase space (M, ω) carries an intrinsic dynamical structure: we say that an n-dimensional submanifold Γ of M is a Hamiltonian n-curve if:

∀v ∈ C ∞ (M, T m M), (v ω) | Γ = 0 and β| Γ = 0. ( 10
)
This definition is motivated by the fact that (if Γ is connected) Γ is a solution of ( 7) iff there exists some h ∈ R s.t. Γ is contained in M h and Γ is a Hamiltonian n-curve in the n-phase space (M h , ω| M h , β| M h ) (see [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF]). However there are examples of premultisymplectic manifolds which do no arise from this construction as for instance the example in [START_REF] Hélein | The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables[END_REF][START_REF] Rovelli | A note on the foundation of relativistic mechanics -II: Covariant Hamiltonian general relativity[END_REF] obtained by starting from the Palatini formulation of gravity.

Action principle

We assume here that we are working in an premultisymplectic manifold (M, ω, β) and that the form ω is exact, i.e. is of the form ω = dθ, where θ is an n-form on M.

This is true e.g. in a submanifold of Λ n T * Z, where ω is precisely defined as the differential of a canonical 'Poincaré-Cartan' form θ. To any oriented n-dimensional submanifold Γ we associate the action

A[Γ] := Γ θ. (11)
One can then show that any n-dimensional submanifold Γ on which β does not vanish is a critical point of A iff it is a Hamiltonian n-curve, i.e. a solution of [START_REF] Forger | Covariant Poisson bracket in geometric field theory[END_REF] (see [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF]). Actually if Γ is the image of a given configuration by some Legendre transform, then A[Γ] coincides with the Lagrangian action of the configuration we started with [START_REF] Hélein | Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl[END_REF]. Note that in the case where ω is not exact one could define a similar action on a homology class of n-dimensional submanifolds by replacing Γ by ∆ ω, where ∆ is a (n + 1)-chain connecting Γ with a particular n-dimensional submanifold which generates the homology class.

Observable functionals

An observable functional is a functional on the 'space' of all solutions: this notion will be central in the next section concerning the covariant phase space. A particular class of such functionals arise in the context of multisymplectic manifolds or premultisymplectic manifolds as follows. In the following we denote by F the set of n-dimensional oriented submanifolds (fields) and M and by E the subset of F composed of Hamiltonian n-curves.

In a multisymplectic manifold (M, ω)

We define an infinitesimal symplectomorphism of (M, ω) to be a vector field ξ ∈ C ∞ (M, T M) s.t. L ξ ω = 0 (i.e. the Lie derivative of ω by ξ vanishes). Note that since ω is closed, this relation is equivalent to d(ξ ω) = 0. An important case occurs when ξ ω is exact: then there exists an (n -1)-form F s.t.

dF + ξ ω = 0. ( 12)

Any (n -1)-form F on M s.t. there exists a vector field ξ satisfying ( 12) is called an observable (n -1)-form. In the case where n = 1 then F is a function and in fact any function on M is an observable 0-form because the symplectic form is non degenerate. However if n ≥ 2 then an arbitrary (n -1)-form on M is not observable in general, but if it is so then the vector field ξ s.t. [START_REF] García | Geometría simplética en la teoria de campos[END_REF] holds is unique: we shall denote it by ξ F . Observable (n -1)-forms can be integrated over hypersurfaces in an n-curve to produce observable functionals. For that purpose, given some Hamiltonian function H on M we define a slice Σ to be codimension one submanifold of M s.t. for any Hamiltonian n-curve Γ the intersection of Σ with Γ is transverse. We also assume that Σ is co-oriented, which means that ∀m ∈ Σ the 1-dimensional quotient space T m M/T m Γ is oriented. Then we can endow Σ ∩ Γ with an orientation and define

Σ F : F -→ R Γ -→ Σ∩Γ F
Then one can recover two important notions in the semi-classical theory of fields.

First one can define a bracket between observable (n -1)-forms F and G by the formula {F,

G} := ξ F ∧ ξ G ω = ξ F dG = -ξ G dF.
Obviously {F, G} is also an (n -1)-form. Moreover one can prove that it is also observable and that ξ {F,G} = [ξ F , ξ G ] [START_REF] Kijowski | A finite dimensional canonical formalism in the classical field theory[END_REF][START_REF] Hélein | The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables[END_REF]. Then the set of observable (n -1)forms equipped with this 'Poisson bracket' becomes almost a Lie algebra (it satisfies the antisymmetry relation {F, G} + {G, F } = 0, but not the Jacobi identity; we have instead {{G, H},

F } + {{H, F }, G} + {{F, G}, H} = d(ξ F ∧ ξ G ∧ ξ H ω),
which, in the case where n = 2 can be understood as a Lie 2-algebra structure [START_REF] Baez | Categorified Symplectic Geometry and the String Lie 2-algebra[END_REF]). However we can define the bracket

Σ F, Σ G := Σ {F, G} (13)
which coincides with the Poisson bracket on functionals on fields used by physicists. We will also meet an interpretation of this bracket in the next Section.

A second important notion is the relation between observable forms and the dynamics. Indeed if Γ is a Hamiltonian n-curve and if F is an observable (n -1)form then one can use the dynamical equation [START_REF] Donder | Introduction à la théorie des invariants intégraux[END_REF] with the vector field ξ

F . It gives us, ∀m ∈ Γ, ∀X ∈ Λ n T m Γ, dF (X) = -ω(ξ F , X) = -dH(ξ F )β(X). ( 14
)
Hence we see that if dH(ξ F ) vanishes, then dF | Γ vanishes. This implies by using Stokes theorem that the restriction of the functional Σ F to the set E of Hamiltonian n-curves does not depend on Σ but on its homology class. For that reason we say that an observable

(n -1)-form F is dynamical if dH(ξ F ) = 0.
In a premultisymplectic manifold (M, ω, β)

The definition of an observable (n -1)-form F , of the bracket and of the observable functionals Σ F can be adapted mutatis mutandis to the case of an premultisymplectic manifold (M, ω, β). The difference is that in such a space the dynamical condition dH(ξ) = 0 is empty (think that M is the level set of some Hamiltonian function H on a multisymplectic manifold, then the fact that ξ is tangent to this level set forces it to be in the kernel of dH). Hence any observable (n -1)-form is a dynamical one.

Moreover if ω is exact, i.e. ω = dθ, we know that Hamiltonian n-curve are critical points of the action [START_REF] Forger | Multisymplectic and polysymplectic structures on fiber bundles[END_REF]. We can thus see that dynamical observable (n-1)forms correspond to symmetries of the variational problems and the conservation law dF | Γ = 0 for a Hamiltonian n-curve Γ is nothing but Noether's first theorem [START_REF] Noether | Invariante Variationsprobleme[END_REF][START_REF] Kosmann-Schwarzbach | Les théorèmes de Noether -Invariance et lois de conservation au XXe siècle[END_REF]. Indeed for any observable (n -1)-form F ,

L ξ F θ = d(ξ F θ) + ξ F dθ = d(ξ F θ) + ξ F ω = d(ξ F θ -F ).
Hence L ξ F θ is exact, so that ξ F is a symmetry of the action Γ θ up to a divergence term. The conserved current is just F | Γ .

Hamilton-Jacobi equations

The Hamilton-Jacobi equation for a Hamiltonian function H on a multisymplectic manifold of the form Λ n T * Z (or on a submanifold of it) is the following equation on an (n -1)-form S on Z (i.e. a section of the vector bundle Λ n-1 T * Z -→ Z):

H(z, dS z ) = 0. ( 15
)
Alternatively the unknown may be chosen to be λ := dS: we then require that λ is a closed n-form on Z (or a section λ of Λ n T * Z -→ Z s.t. λ * ω = 0) which is a solution of λ * H = 0. Then if for instance Z = X × Y, and if we denote by π the projection from X × Y to X , λ := π * dS provides us with a null Lagrangian functional X λ on X (i.e. a Lagrangian density which satisfies the Euler-Lagrange equation for any map). In constrast with non relativistic quantum Mechanics, the usefulness of this equations in the quantization of fields is not clear for the moment. One of the interests of the Hamilton-Jacobi equation is that it allows in principle to prove under some circumstances that some solutions of the Euler-Lagrange system of equations are glogal minimizers, by following a classical strategy designed by K. Weierstrass and D. Hilbert (see [START_REF] Weyl | Geodesic fields in the calculus of variation for multiple integrals[END_REF][START_REF] Dedecker | On the generalization of symplectic geometry to multiple integrals in the calculus of variations[END_REF][START_REF] Rund | The Hamilton-Jacobi theory in the calculus of variations, its role in mathematics and physics[END_REF]). This strategy is the exact analogue in the general theory of calculus of variations of the theory of calibrations used in minimal surfaces.

Note that one could impose extra conditions such as requiring that λ = ds 1 ∧ • • • ∧ ds n , where s 1 , • • • , s n are functions on Z plus the fact that the graph of λ is foliated by solutions to the Hamilton equations (this provides then a generalization of the picture built by Hamilton in order to conciliate the Fermat principle with the Huygens principle): this was achieved by Carathéodory [START_REF] Carathéodory | Variationsrechnung und partielle Differentialgleichungen erster Ordnung[END_REF] in his theory (see §1.8).

Some historical remarks

The generalization of the Hamilton equations to variational problems with several variables developped first along two directions. One of these is the question of deciding whether a given solution to a variational problem is a minimum of the action functional. This question was answered locally for 1-dimensional variational problems by C.G.J. Jacobi (by following a remark of Legendre) in 1837 [START_REF] Jacobi | Zur Theorie des Variations-Rechnung und des Differential-Gleichungen[END_REF] by founding a method to check that the second variation is nonnegative which is based on solutions to the so-called Jacobi equation. Note that this method was extended to several variables by A. Clebsch [START_REF] Clebsch | Ueber die zweite Variation vielfache Integralen[END_REF] in 1859. Later on a global, nonlinear version of these ideas was developped by K. Weierstrass and D. Hilbert to prove the minimality of some solutions. This theory is connected with another famous work of Jacobi of the same year (1837), who obtained the Hamilton-Jacobi equation [START_REF] Jacobi | Ueber die Reduction der Integration des partiellen Differentialgleichungen erster Ornung zwischen irgend einer Zahl Variabeln auf die Integration eines einzigen Systemes gewöhnlicher Differentialgleichungen[END_REF] by generalizing the work of Hamilton relating the Fermat principle to the Huygens principle. In 1890 V. Volterra wrote two papers [START_REF] Volterra | Sulle equazioni differenziali che provengono da questiono di calcolo delle variazioni[END_REF][START_REF] Volterra | Sopra una estensione della teoria Jacobi-Hamilton del calcolo delle varizioni[END_REF] where, to my knowledge for the first time 3 , two different generalizations of the Hamilton system of equations to variational problems with several variables were proposed. In [START_REF] Volterra | Sopra una estensione della teoria Jacobi-Hamilton del calcolo delle varizioni[END_REF] Volterra extended the Weierstrass-Hilbert theory to variational problems with several variables. This theory was further developped by G. Prange in 1915 [START_REF] Prange | Die Hamilton-Jacobische Theorie für Doppelintegrale[END_REF] and by C. Carathéodory in his book in 1929 [START_REF] Carathéodory | Variationsrechnung und partielle Differentialgleichungen erster Ordnung[END_REF] and is called today the Carathéodory theory. In 1934 H. Weyl [START_REF] Weyl | Geodesic fields in the calculus of variation for multiple integrals[END_REF], inspired by Carathéodory's theory, proposed a variant of it which is based on the same theory as the one proposed by Volterra in [START_REF] Volterra | Sulle equazioni differenziali che provengono da questiono di calcolo delle variazioni[END_REF] and that we described in §1.1. Today this theory is called the De Donder-Weyl one by many Authors4 . Its geometrical framework is the affine multisymplectic manifold Λ n 2 T * Z. A second direction was the notion of invariant integrals due to H. Poincaré [START_REF] Poincaré | Les méthodes nouvelles de la mécanique céleste[END_REF] and further developped by E. Cartan [START_REF] Cartan | Leçons sur les invariants intégraux[END_REF] in 1922: here one emphasizes the relationship of Hamilton equations with the search of invariants which may be functions or differential forms. This point of view is strongly related with the covariant phase space theory (see §2.1 below). This theory was developped in full generality by T. De Donder [START_REF] Donder | Théorie invariantive du calcul des variations[END_REF] in 1935 and his main contribution was to deduce the extension of the affine ('De Donder-Weyl') theory to Lagrangian densities depending on an arbitrary number of derivatives, i.e. to the theory expounded in §1.2. Hence although Weyl's and De Donder's contributions are almost simultaneous they are independant in their inspiration: Weyl starting point was the so-called Carathéodory theory, motivated by the search for generalizations to several variables of the Hamilton-Jacobi equation, whereas De Donder's starting point was the theory of integral invariants.

The fact that a continuum of different theories may exists for a given variational problem was first understood by T. Lepage [START_REF] Lepage | Sur les champs géodésiques du calcul des variations[END_REF] in 1936 and completely described by P. Dedecker in 1953 [START_REF] Dedecker | On the generalization of symplectic geometry to multiple integrals in the calculus of variations[END_REF]. Today we can picture these various theories as submanifolds of the universal multisymplectic manifold Λ n T * Z introduced by J. Kijowski [START_REF] Kijowski | Multiphase spaces and gauge in the calculus of variations[END_REF] in 1974.

Recently the so-called De Donder-Weyl theory (but that we should call the 'first Volterra theory') has beed studied by many Authors starting with the important work by the Polish school around 1970, i.e. by W. Tulczjew, J. Kijowski, W. Szczyrba and later on in many papers which are refered to in e.g. [START_REF] Hélein | Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory, in Noncompact problems at the intersection of geometry, analysis, and topology[END_REF][START_REF] Forger | Multisymplectic and polysymplectic structures on fiber bundles[END_REF]. However the Lepage-Dedecker theory has received much less attention (to my knowledge it was only considered by J. Kijowski [START_REF] Kijowski | Multiphase spaces and gauge in the calculus of variations[END_REF], F. Hélein, J. Kouneiher [START_REF] Hélein | Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl[END_REF][START_REF] Hélein | The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables[END_REF][START_REF] Hélein | Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory, in Noncompact problems at the intersection of geometry, analysis, and topology[END_REF] and M. Forger, S. V. Romero [START_REF] Forger | Covariant Poisson bracket in geometric field theory[END_REF]), probably because of its complexity. The latter theory leads however to interesting phenomena, particularly for gauge theories [START_REF] Hélein | Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl[END_REF][START_REF] Hélein | Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory, in Noncompact problems at the intersection of geometry, analysis, and topology[END_REF], since first class Dirac constraints simply disappear there.

The modern formulation using the multisymplectic (n + 1)-form as the key of the structure of the theory seems to start with the papers of J. Kijowski [START_REF] Kijowski | A finite dimensional canonical formalism in the classical field theory[END_REF], H. Goldschmidt and P. Sternberg [START_REF] Goldschmidt | The Hamilton-Cartan formalism in the calculus of variations[END_REF] in 1973 and the introduction of observable (n -1)-forms apparentely goes back to the work of K. Gawédski [START_REF] Gawȩdski | On the generalization of the canonical formalism in the classical field theory[END_REF] in 1972.

An example

Let X be the n-dimensional Minkowski space-time with coordinates x = (x 0 , x 1 , • • • , x n-1 ) and consider the linear Klein-Gordon equation on X :

ϕ + m 2 ϕ := ∂ 2 ϕ ∂t 2 -∆ϕ + m 2 ϕ = 0, (16)
where t = x 0 and ∆ :

= n-1 i=1 ∂ 2 (∂x i ) 2 .
We use the notations x := (x 1 , • • • , x n-1 ) ∈ R n-1 and x = (x 0 , x) ∈ R n and we define the Euclidean scalar product x • y :=

x 1 y 1 + • • • + x n-1 y n-1 on R n-1 and the Minkowski product x • y = η µν x µ y ν = x 0 y 0 -x 1 y 1 -• • • -x n-1 y n-1 = x 0 y 0 -x • y,
on X . The multisymplectic formulation of ( 16) takes place in M := Λ n T * (X × R), equipped with the multisymplectic form

ω := de ∧ β + dp µ ∧ dϕ ∧ β µ .
Note that ω = dθ λ , where

θ λ := eβ + λp µ dϕ ∧ β µ -(1 -λ)ϕdp µ ∧ β µ ,
where λ ∈ R is a parameter to fixed later. The Hamiltonian function on M corresponding to solutions of ( 16) is

H(x, ϕ, e, p) := e + 1 2 η µν p µ p ν + 1 2 m 2 ϕ 2 .
To a solution ϕ of ( 16)we associate a Hamiltonian n-curve Γ = {(x, ϕ(x), e(x), p(x))

| x ∈ X } which satisfies      p µ (x) = η µν ∂ϕ ∂x ν (x) e(x) = - 1 2 η µν ∂ϕ ∂x µ (x) ∂ϕ ∂x ν (x) - 1 2 m 2 ϕ(x) 2 . ( 17
)
We define E to be the set of Hamiltonian n-curves s.t. for all time t, x -→ ϕ(t, x) is rapidly decreasing at infinity. We denote by P n-1 H M the set of dynamical observable (n -1)-forms F and sp H M := {ξ | L ξ ω = 0, dH(ξ) = 0}. Note that (n -1)-forms F in P n-1 H M are found by looking at vector fields ξ in sp H M and by solving ξ ω + dF = 0. They are of the form

F = ζ θ + F Φ ,
where ζ is a vector field on the Minkowski space-time X which is a generator of the action of the Poincaré group and

F Φ := p µ Φ(x) -ϕη µν ∂Φ ∂x ν (x) β µ ,
where Φ is a solution of [START_REF] Goldschmidt | The Hamilton-Cartan formalism in the calculus of variations[END_REF]. Note that moreover

ξ Φ := ξ F Φ = Φ(x) ∂ ∂ϕ + η µν ∂Φ ∂x ν (x) ∂ ∂p µ -m 2 ϕΦ(x) + p µ ∂Φ ∂x µ (x) ∂ ∂e .
In the following we shall denote by

P (λ) µ := ∂ ∂x µ θ λ
and we observe that since L ∂ ∂x µ θ λ = 0, we have dP

(λ) µ + ∂ ∂x µ ω = 0. Hence ξ P (λ) µ = ∂ ∂x µ .
The brackets of two dynamical observable forms F, G ∈ P n-1 H M are given as follows: for any pair Φ, Ψ of solutions of ( 16),

{F Φ , F Ψ } = η µν ∂Φ ∂x ν (x)Ψ(x) -Φ(x) ∂Ψ ∂x ν (x) β µ . (18)
We observe that d{F Φ , F Ψ } = 0. Hence P n-1 H M, {•, •} can be understood as a kind of central extension of the Lie algebra (sp H M, [•, •]) and the Lie sub algebra spanned by forms F Φ as an infinite dimensional analogue of the Heisenberg algebra with central charges given by [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF]. Lastly

P (λ) µ , F Φ = L ∂ ∂x µ F Φ -d ∂ ∂x µ F Φ = F ∂Φ ∂x µ -d ∂ ∂x µ F Φ (19)
and

P (λ) µ , P (λ) ν = 0.
For the purpose of quantization we look at functionals of the form F = Σ F Φ which are simultaneously eigenvectors of the linear operators

F -→ Σ P (λ) µ , F , for µ = 0, • • • , n -1.
We find by using relation [START_REF] Hélein | Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage-Dedecker versus De Donder-Weyl[END_REF] that the eigenvector equation reduces to ∂Φ ∂x λ = cΦ. This implies (by using the eigenvalue equation for µ = 0, • • • , n -1) that Φ(x) = αe ik•x . But because Φ should also be a solution of ( 16) we must then have

η µν k µ k ν = (k 0 ) 2 -| k| 2 = m 2 . ( 20
)
(We remark that the maps { Σ P (λ) λ , •} play the role of the generators of a Cartan subalgebra.) Let us denote by C the mass shell, i.e. the set of all k = (k 0 , k) ∈ R 4 which are solutions of [START_REF] Hélein | The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables[END_REF]. This set actually splits into two connected components according to the sign of k 0 : we let

C + := {k ∈ C | k 0 > 0}. For any k ∈ C + we define α k := F ie ik•x / √ 2π 3 = i √ 2π 3 e ik•x (p µ -iϕk µ ) β µ α * k := F -ie -ik•x / √ 2π 3 = -i √ 2π 3 e -ik•x (p µ + iϕk µ ) β µ .
The vector fields associated to these observable forms are:

ξ k := ξ α k = e ik•x √ 2π 3 i ∂ ∂ϕ -k µ ∂ ∂p µ + η µν p µ k ν -im 2 ϕ ∂ ∂e , ξ * k := ξ α * k = e -ik•x √ 2π 3 -i ∂ ∂ϕ -k µ ∂ ∂p µ + η µν p µ k ν + im 2 ϕ ∂ ∂e .
We then define the observable functionals

a k := Σ α k and a * k := Σ α * k .
As the notations suggest these functionals are the classical analogues of respectively the annihilation and the creation operators. The advantage however is that our functionals a k and a * k are independant of the coordinate system. We can choose Σ to be the hyperplane x 0 = t = 0 and, for any function f , denote by f | 0 the restriction of f to Σ. Then, for any Γ ∈ E we have

a k (Γ) = i √ 2π 3 R 3 ∂ϕ ∂t | 0 ( x) -ik 0 ϕ| 0 ( x) e -i k• x d x = i ∂ϕ ∂t | 0 ( k) + k 0 ϕ| 0 ( k),
where, for all function ψ on R 3 ,

ψ( k) := 1 √ 2π 3 R 3 ψ( x)e -i k• x d x.
Similarly we have:

a * k (Γ) = -i √ 2π 3 R 3 ∂ϕ ∂t | 0 ( x) + ik 0 ϕ| 0 ( x) e i k• x d x = -i ∂ϕ ∂t | 0 (-k) + k 0 ϕ| 0 (-k).
Hence we deduce that

ϕ| 0 ( k) = 1 2k 0 a k (Γ) + a * k (Γ) and ∂ϕ ∂t | 0 ( k) = 1 2i a k (Γ) -a * k (Γ) ,
where k = (k 0 , k) := (k 0 , -k). Thus denoting dµ(k)

= 1 2k 0 dk 1 dk 2 dk 3 = 1 2k 0 d k, we have ϕ(0, x) = 1 √ 2π 3 R 3 1 2k 0 d k e i k• x a k (Γ) + a * k (Γ) = 1 √ 2π 3 C + dµ(k) a k (Γ)e -ik•x + a * k (Γ)e ik•x and ∂ϕ ∂t (0, x) = -i √ 2π 3 R 3 1 2 d k e i k• x a k (Γ) -a * k (Γ) = -i √ 2π 3 C + dµ(k)k 0 a k (Γ)e -ik•x -a * k (Γ)e ik•x .
Recall that these integrals can be interpreted as integrals over C + through the parametrization R 3 ∋ k -→ (k 0 , k) ∈ C + and that dµ is a measure on C + invariant by the action of the Lorentz group. Note also that in order to agree with some textbooks one should add an extra factor √ k 0 inside the integrals. By using the relation [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF] we obtain, ∀k, ℓ ∈ C + ,

{α k , α ℓ } = -ie i(k+ℓ)•x (2π) 3 (k µ -ℓ µ )β µ , {α * k , α * ℓ } = ie -i(k+ℓ)•x (2π) 3 (k µ -ℓ µ )β µ , {α k , α * ℓ } = ie i(k-ℓ)•x (2π) 3 (k µ + ℓ µ )β µ .
These brackets cannot be integrated over the slice Σ := {x 0 = 0} in the measure theoretical sense 5 , but one can make sense of their integrals as distributions over the variables k ± ℓ ∈ R 3 :

{a k , a ℓ } = {a * k , a * ℓ } = 0, ∀k, ℓ ∈ C + , and {a k , a * ℓ } = i2k 0 δ( ℓ -k).
A way to regularize these operators and their brackets is, by using functions f, g ∈ L 2 (C + ), to define

a f := C + dµ(k)f (k)a k , and a * g := C + dµ(k)g(k)a * k . Then {a f , a * g } = i C + dµ(k)f (k)g(k).
2 The covariant phase space

A short historical review

The simplest version of the covariant phase space is the set E of solutions of a Hamiltonian time evolution problem. In this case the Cauchy problem consists in choosing some point m 0 in the ordinary phase space (classically positions and momenta) and some time t 0 and in looking for solutions of the Hamiltonian vector flow which coincide with m 0 at time t 0 . This problem has an unique solution in all regular cases and this means that E is in one to one correspondence with the set of initial data {m 0 }. In other words to each time t 0 it corresponds a natural 'Cauchy coordinates system' on E, which is just the set of initial conditions {m 0 }.

The key property is that the Hamiltonian flow preserves the symplectic structure: this means all the symplectic structures induced by these 'Cauchy coordinates systems' on E coincides. Hence this defines a canonical symplectic structure on E.

The substitution of the ordinary phase space by the space of solutions is a classical analogue of the transition from the Schrödinger picture to the Heisenberg picture in quantum Mechanics: in the Schrödinger picture the dynamics of a particle is described by the evolution of some time dependant 'state' which is represented by a complex line in some complex Hilbert space (the quantum analogue of a point in the ordinary phase space), whereas in the Heisenberg picture the state (still a complex line in a complex Hilbert space) does not evolve with time so that it may be interpreted as a quantum analogue of a solution of the dynamical equations, i.e. of a point in E (actually more precisely on a Lagrangian submanifold in the phase sapce, according to A. Weinstein).

In Mechanics this concept is relatively old: the idea of contempling the space of solutions of a mechanical problem itself has probably his roots in the method of the variation de la constante of J.L. Lagrange and the notion of 'Lagrange bracket' is very close to the symplectic structure on the phase space. The observation that this space carries an intrinsic symplectic structure was clearly formulated by H. Poincaré [START_REF] Poincaré | Les méthodes nouvelles de la mécanique céleste[END_REF] in his theory of invariant integrals (invariants intégraux ) and later further developped by E. Cartan [START_REF] Cartan | Leçons sur les invariants intégraux[END_REF] and fully recognized by J.M. Souriau [START_REF] Souriau | Structure des systèmes dynamiques[END_REF]. T. De Donder [START_REF] Donder | Introduction à la théorie des invariants intégraux[END_REF] extended the notion of integral invariant to variational problems with several variables, being hence very close from the notion of covariant phase space in this context, although it is not clear that he realized it. Actually it seems difficult to decide when the concept of covariant phase space in fields theory merged out. My own guess is that such an idea could also have been inspired by quantum fields theory, since it may be thought as the classical analogue of the Heisenberg picture in quantum fields theory. First known works in this direction are the R.E. Peierls bracket in 1952 [START_REF] Peierls | The commutation laws of relativistic field theory[END_REF], followed by the paper of I. Segal [START_REF] Segal | Quantization of nonlinear systems[END_REF] in 1960. Peierls defined a bracket on the set of solutions to a relativic hyperbolic wave equation which can be understood now as the restriction of the Poisson bracket associated to the covariant phase structure on a certain class of functionals on the phase space. Segal proved that the set of solutions of a non linear field relativistic wave equation precisely carries a symplectic structure and proposes to use this symplectic structure to quantize fields (and his paper is among the ones at the origin of the geometric quantization method). This idea was later developped in a more and more general framework by P. L. García [START_REF] García | Geometría simplética en la teoria de campos[END_REF] in 1968, García and A. Pérez-Rendón [START_REF] García | Symplectic approach to the theory of quantized fields, I[END_REF] in 1971, H. Goldschmidt, S. Sternberg [START_REF] Goldschmidt | The Hamilton-Cartan formalism in the calculus of variations[END_REF] in 1973. To my opinion the more achieved presentation is the one by J. Kijowski and W. Szczyrba [START_REF] Kijowski | A canonical structure for classical field theories[END_REF] in 1976, which gives the first elementary but general presentation of this structure, by using the multisymplectic formalism.

A more recent apparition of this idea can be found in the papers by C. Crnkovic and E. Witten [START_REF] Crnkovic | Covariant description of canonical formalism in geometrical theories[END_REF] and by G. Zuckerman [START_REF] Zuckerman | Action functional and global geometry[END_REF], where the Authors apparentely ignored the previous references and have rediscovered this principle, being guided by the concept of the variational bicomplex by F. Takens [START_REF] Takens | A global formulation of the inverse problem of the calculus of variations[END_REF] and from the work of A.M. Vinogradov [START_REF] Vinogradov | The C-spectral sequence, Lagrangian formalism, and conservations laws, I and II[END_REF]. This was followed by several developments in the physical (e.g. [START_REF] Dolan | A co-variant approach to Ashtekar's canonical gravity[END_REF]) and the mathematical literature, where this principle is often refered to as the Witten covariant phase space. A general presentation in the framework of the secondary calculus of Vinogradov was done by E. Reyes [START_REF] Reyes | On covariant phase space and the variational bicomplex[END_REF] and L. Vitagliano in [START_REF] Vitagliano | Secondary calculus and the covariant phase space[END_REF] and in relation to multisymplectic geometry (as in the present paper) by M. Forger and S.V. Romero in [START_REF] Forger | Covariant Poisson bracket in geometric field theory[END_REF].

The basic principle

δΓ = Γ ξ = Γ ξ + ζ.
Moreover the condition on δΓ of being tangent to E forces ξ to be a solution of the Jacobi equation:

∀v ∈ C ∞ (M, T m M), (v L ξ ω) | Γ = 0. ( 21
)
Note that, although ξ is not a vector field defined on M (neither on a neighbourhood of Γ in M) but only on Γ, one can make sense of L ξ ω| Γ because Γ is a solution of [START_REF] Forger | Covariant Poisson bracket in geometric field theory[END_REF].

Then for any slice Σ (see §1.6), any Γ ∈ E and δΓ ∈ T Γ E, we define

Θ Σ Γ (δΓ) := Σ∩Γ ξ θ,
where ξ is a section of j * Γ T M over Γ s.t. δΓ = ξ and ξ θ is the interior product of θ by ξ. This hence define a 1-form Θ Σ on E 2.2.1 The dependance of Θ Σ on Σ This is the first natural question. For that purpose let us consider a smooth 1-parameter family of slices (Σ t ) t and compute the derivative:

d dt Θ Σt Γ (δΓ) = d dt Σt∩Γ ξ θ = Σt∩Γ L ∂ ∂t (ξ θ) = Σt∩Γ ∂ ∂t d(ξ θ) + d ξ ∧ ∂ ∂t θ . But d(ξ θ) = L ξ θ -ξ dθ and thus ∂ ∂t d(ξ θ) = ∂ ∂t (L ξ θ) - ∂ ∂t ξ ω.
However we can assume w.l.g. (see [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF]) that the vector fields ∂ ∂t and ξ admit extensions s.t. ξ, ∂ ∂t = 0. Then the preceding relation gives us

∂ ∂t d(ξ θ) = L ξ ∂ ∂t θ -ξ ∧ ∂ ∂t ω.
Hence

d dt Θ Σt Γ (δΓ) = Σt∩Γ L ξ ∂ ∂t θ - Σt∩Γ ξ ∧ ∂ ∂t ω + Σt∩Γ d ξ ∧ ∂ ∂t θ . (22)
First let us consider a smooth curve s -→ Γ s ∈ E s.t. Γ 0 = Γ and dΓs dt = δΓ. Then the first term in the r.h.s. of ( 22) is equal to

Σt∩Γ L ξ ∂ ∂t θ = d ds Σt∩Γs ∂ ∂t θ s=0 = δS dΣ t dt Γ (δΓ),
where we have posed:

S dΣ t dt (Γ) := Σt∩Γ ∂ ∂t θ.
Second we can assume w.l.g. (see [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF]) that we can choose ∂ ∂t in such a way that it is tangent to Γ. Let (X 2 , • • • , X n ) be a system of tangent vectors on Γ s.t. ∀t, ∀m ∈

Σ t ∩Γ, (X 2 (m), • • • , X n (m)) is a basis of T m (Σ t ∩Γ) and ( ∂ ∂t (m), X 2 (m), • • • , X n (m)) is a basis of T m Γ. Then if ψ is a n-volume form on Γ s.t. ψ( ∂ ∂t , X 2 , • • • , X n ) = 1
, the second term in the r.h.s. of [START_REF] Jacobi | Zur Theorie des Variations-Rechnung und des Differential-Gleichungen[END_REF] reads

- Σt∩Γ ξ ∧ ∂ ∂t ω = - Σt∩Γ ω ξ, ∂ ∂t , X 2 , • • • , X n ψ
and vanishes because of the Hamilton equations [START_REF] Forger | Covariant Poisson bracket in geometric field theory[END_REF]. Lastly we assume that the restriction of ξ to Σ t ∩ Γ has compact support or is rapidly decreasing: this occurs for instance if the Hamilton system encodes hyperbolic wave equations, if Σ is a level hypersurface of some time coordinate and if we impose that the Hamiltonian n-curves in E have a prescribed behaviour at infinity in space for all time. Then the last term in the r.h.s. of ( 22) vanishes. Then Relation ( 22) can be rewritten

d dt Θ Σt Γ (δΓ) = δS dΣ t dt Γ (δΓ), ∀δΓ ∈ T Γ E or d dt Θ Σt = δS dΣ t dt . (23)
We can also define the functional

S Σ 2 Σ 1 (Γ) := Γ∩{t 1 ≤t≤t 2 } θ,
which represents the 'action' between the slices Σ 1 :

= {t = t 1 } an Σ 2 := {t = t 2 }. Then S Σ 2 Σ 1 (Γ) = t 2 t 1 S dΣ t
dt (Γ)dt and thus we deduce by integrating [START_REF] Kijowski | A finite dimensional canonical formalism in the classical field theory[END_REF]

over [t 1 , t 2 ] that Θ Σ 2 -Θ Σ 1 = δS Σ 2 Σ 1 . (24)

The symplectic form

In view of the preceding we are led to the conclusion that, although the 1-form Θ Σ depends on Σ, its differential δΘ Σ does not depend on Σ since [START_REF] Kijowski | Multiphase spaces and gauge in the calculus of variations[END_REF] tells us that Θ Σ 2 -Θ Σ 1 is an exact form. Of course one should be careful in using the identity δ • δ = 0 since E is not a smooth manifold (see [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF] for a rigorous proof that δΘ Σ does not depend on Σ). All that motivates the definition of the following 2-form on E: Ω := δΘ Σ .

We will prove that Ω has the following expression:

∀δ 1 Γ, δ 2 Γ ∈ T Γ E, Ω Γ (δ 1 Γ, δ 2 Γ) = Σ∩Γ ξ 1 ∧ ξ 2 ω, (25)
where ξ 1 , ξ 2 are sections over Γ of j * Γ T M s.t. δ 1 Γ = Γ ξ 1 and δ 2 Γ = Γ ξ 2 . To prove [START_REF] Kijowski | A canonical structure for classical field theories[END_REF] we need to compute δΘ Σ Γ (δ 1 Γ, δ 2 Γ). For that purpose we first assume that we can extend the two tangent vectors δ 1 Γ and δ 2 Γ to commuting vector fields on E around Γ (actually we can assume that [ξ 1 , ξ 2 ] = 0). Then

δΘ Σ Γ (δ 1 Γ, δ 2 Γ) = δ 1 Γ • Θ Σ Γ (δ 2 Γ) -δ 2 Γ • Θ Σ Γ (δ 1 Γ) -Θ Σ Γ ([δ 1 Γ, δ 2 Γ]) = δ 1 Γ • Σ∩Γ ξ 2 θ -δ 2 Γ • Σ∩Γ ξ 1 θ . Thus δΘ Σ Γ (δ 1 Γ, δ 2 Γ) = Σ∩Γ L ξ 1 (ξ 2 θ) - Σ∩Γ L ξ 2 (ξ 1 θ) .
We use then the following identity (see [START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF]): for any pair of vector fields X 1 and X 2 and for any p-form β,

L X 1 (X 2 β) -L X 2 (X 1 β) = X 1 ∧ X 2 dβ + [X 1 , X 2 ] β + d(X 1 ∧ X 2 β).
Setting X 1 = ξ 1 , X 2 = ξ 2 and β = θ, we obtain using [ξ 1 , ξ 2 ] = 0 and dθ = ω that

L ξ 1 (ξ 2 θ) -L ξ 2 (ξ 1 θ) = ξ 1 ∧ ξ 2 ω -d(ξ 1 ∧ ξ 2 θ). Thus δΘ Σ Γ (δ 1 Γ, δ 2 Γ) = Σ∩Γ ξ 1 ∧ ξ 2 ω -d(ξ 1 ∧ ξ 2 θ). (26)
Hence if we assume that the restriction of ξ 1 and ξ 2 to Σ t ∩ Γ has compact support or is rapidly decreasing (as in the preceding paragraph) we obtain [START_REF] Kijowski | A canonical structure for classical field theories[END_REF]. Hence we conclude that, under some hypotheses, one can endow the set E of solutions to the Hamilton equations with a symplectic form Ω given by [START_REF] Kijowski | A canonical structure for classical field theories[END_REF]. This form does depend not on Σ under the condition that the boundary terms [START_REF] Jacobi | Zur Theorie des Variations-Rechnung und des Differential-Gleichungen[END_REF] and -Σ∩Γ d(ξ 1 ∧ ξ 2 θ) in [START_REF] Kirillov | Geometric quantization[END_REF] vanish. This means that, on each slice Σ, the Jacobi vector fields ξ, ξ 1 , ξ 2 decreases sufficiently rapidly at infinity. Such a condition is true if, for instance, the manifold X is a Lorentzien manifold, the slice Σ is (a lift of) a spacelike hypersurface of X and we impose in the definition of E that all Hamiltonian n-curves Γ in E are asymptotic to a given 'ground state' Hamiltonian n-curve Γ 0 at infinity on each slice Σ.

Σt∩Γ d ξ ∧ ∂ ∂t θ in
With such a symplectic structure Ω on E we can define a Poisson bracket on real-valued functionals on E, which is nothing but [START_REF] García | Symplectic approach to the theory of quantized fields, I[END_REF].

Γ Γ Γ - + 0 ' 2 Γ 0 1 Σ Σ σ 2 1 σ Figure 1: A geometric comparison of Θ Σ1 (δΓ) with Θ Σ2 (δΓ)
purpose imagine that our problem models a hyperbolic time evolution problem and that there are well-defined notions of time and space coordinates on M (as it is the case for any wave equation on a curved space-time).

Consider a Hamiltonian n-curve Γ and let Γ ′ be another Hamiltonian n-curve, which we suppose to be close to Γ. More precisely we assume that Γ ′ = Γ + εδΓ + o(ε), where ε > 0 is a small parameter : by this condition we mean that there exists a vector field ξ ∈ C ∞ (M, T M) s.t. δΓ = Γ ξ and Γ ′ is the image of Γ by the flow map e εξ . We also assume that, for all 'time', Γ ′ is asymptotic to Γ at infinity in space. Let Σ 1 and Σ 2 be two slices, assume that these slices are space-like hypersurfaces and, in order to fix ideas, we suppose that Σ 2 is in the future of Σ 1 . These slices cross transversally Γ and Γ ′ and we denote by σ 1 (resp. σ 2 ) the piece of Σ 1 (resp. Σ 2 ) which is enclosed by the intersections with Γ and Γ ′ (see the picture). We also denote by Γ -the part of Γ which is in the past of Σ 1 , by Γ + the part of Γ which is in the future of Σ 2 and by Γ ′ 0 the part of Γ ′ which is between Σ 1 and Σ 2 (see again the picture). Lastly we consider the (not necessarily Hamiltonian) n-curve Γ ε , which is the union of Γ -, σ 1 , Γ ′ 0 , σ 2 and Γ + . Of course Γ ε is not smooth, but it can be approached by a sequence of smooth n-curves, so that the following makes sense. We also endow Γ ε with the orientation which agrees with that of Γ on Γ -∪ Γ + and with that Γ ′ on Γ ′ 0 . Let us use that fact that Γ is a Hamiltonian n-curve, hence a critical point of [START_REF] Forger | Multisymplectic and polysymplectic structures on fiber bundles[END_REF]. It implies that

Γε θ = Γ θ + o(ε). (27)
However the l.h.s. of ( 27) can be decomposed as

Γε θ = Γ - θ + σ 1 θ + Γ ′ 0 θ + σ 2 θ + Γ + θ,
wheras its r.h.s. is

Γ θ + o(ε) = Γ - θ + Γ 0 θ + Γ + θ + o(ε),
where Γ 0 is the part of Γ between Σ 1 and Σ 2 . Hence [START_REF] Koenigsberger | Die Prinzipien der Mechanik für mehrere unavhängige Variable[END_REF] reduces to

σ 1 θ + Γ ′ 0 θ - Γ 0 θ + σ 2 θ = o(ε). (28)
We now recognize that, on the one hand,

σ 1 θ = εΘ Σ 1 (δΓ) + o(ε), σ 2 θ = -εΘ Σ 2 (δΓ) + o(ε)
(the sign being due to the orientation of σ 2 ). On the other hand

Γ 0 θ = S Σ 2 Σ 1 (Γ 0 ) and Γ ′ 0 θ = S Σ 2 Σ 1 (Γ ′ 0 ) = S Σ 2 Σ 1 (Γ 0 + εδΓ 0 ) + o(ε). Hence (28) gives us εΘ Σ 1 (δΓ) + ε δS Σ 2 Σ 1 Γ (δΓ) -εΘ Σ 2 (δΓ) = o(ε).
Thus by dividing by ε and letting ε tend to 0, we recover [START_REF] Forger | Multisymplectic and polysymplectic structures on fiber bundles[END_REF].

Geometric quantization

We address here the question of building a geometric quantization scheme, or at least a prequantization scheme for fields based on the covariant phase space structure. This was more or less the programm envisioned by G. Segal in 1960 [START_REF] Segal | Quantization of nonlinear systems[END_REF]. We present here some attempt of that by using the multisymplectic theory on a very elementary example, which is the one presented in §1.9.

Canonical vector fields on the set of solutions E

We can associate to each F ∈ P n-1 H M a tangent vector field Ξ F on E wich is given by

∀Γ ∈ E, Ξ F (Γ) := Γ ξ F .
In the case of the Klein-Gordon equation ( 16) it is interesting to represent solutions in E by local coordinates. The more convenient way is based on the Fourier transform: any Hamiltonian n-curve Γ is characterized by a solution ϕ to [START_REF] Goldschmidt | The Hamilton-Cartan formalism in the calculus of variations[END_REF] and by writing

ϕ(x) = 1 √ 2π 3 C + dµ(k) u k e -ik•x + u * k e ik•x , (29)
we get formally a map

E -→ C C + × C C + ϕ -→ (u k , u * k ) k∈C + .
Note that the image of E is characterized by the reality condition u k = u * k , ∀k ∈ C + . We can obviously extend this map to the complexification E C of E and then this map is an isomorphism.

The creation and annihilation canonical transformations

Now given some function f ∈ L 2 (C + ) consider

α f := C + dµ(k)f (k)α k = i √ 2π 3 C + dµ(k)e ik•x f (k)(p µ -iϕk µ )β µ .
(Note that the observable functional a f defined in §1.9 is obtained by integration of α f over a slice.) Then

ξ f := ξ α f = i √ 2π 3 C + dµ(k)e ik•x f (k) ik µ ∂ ∂p µ -(m 2 ϕ + iη µν p µ k ν ) ∂ ∂e + ∂ ∂ϕ
is completely characterized by the fact that it preserves ω and dH and through its action on ϕ:

dϕ (ξ f ) = i √ 2π 3 C + dµ(k)e ik•x f (k).
We can easily integrate ξ f on M and its action on F :

U(s, ϕ) = ϕ + s i √ 2π 3 C + dµ(k)e ik•x f (k).
In terms of the coordinates (u k , u * k ) k∈C + it gives:

U(s, u k , u * k ) = (u k , u * k + isf (k)).
Hence we can symbolically denote

Ξ f := Ξ α f = i C + f (k) ∂ ∂u * k .
There is no integration measure used here, the sign stands uniquely for summing linearly independant vectors: the meaning is that

Ξ f 1 √ 2π 3 C + d k 2k 0 u k e -ik•x + u * k e ik•x = 1 √ 2π 3 C + dµ(k)if (k)e ik•x .
A completely analogue computation can be done for

α * g := C + dµ(k)g(k)α * k = -i √ 2π 3 C + dµ(k)e -ik•x g(k)(p µ + iϕk µ )β µ ,
where g ∈ L 2 (C + ). Denoting ξ * g := ξ α * g we have

dϕ ξ * g = -i √ 2π 3 C + dµ(k)e -ik•x g(k).
Hence Ξ * g := Ξ α * g is given by

Ξ * g = -i C + g(k) ∂ ∂u k .

Spacetime translations

We now look at the canonical vector fields on E associated with spacetime translations P ζ , where ζ is constant vector field on X . We recall that ξ P

(x)) | x ∈ X } ⊂ E by U(x, •) is Γ s = {(x, ϕ(x -sζ), e s (x), p s (x)) | x ∈ X },
where the value of e s (x) and p s (x) is completely determined by the constraint that Γ s ⊂ E and by the knowledge of ϕ(x -sζ). This can be proved by a simple change of variable. Similarly we determine the action of Ξ ζ on the coordinates (u k , u * k ) k∈C + by computing its action on ϕ:

(Ξ ζ ϕ) (x) = d ds 1 √ 2π 3 C + dµ(k) u k e -ik•(x-sζ) + u * k e ik•(x-sζ) |s=0 = 1 √ 2π 3 C + dµ(k) ik • ζu k e -ik•(x-sζ) -ik • ζu * k e ik•(x-sζ) = i C + k • ζ u k ∂ ∂u k -u * k ∂ ∂u * k ϕ (x). Hence Ξ ζ = i C + k • ζ u k ∂ ∂u k -u * k ∂ ∂u * k .

Geometric prequantization

We recall very briefly the prequantization scheme due to B. Kostant and J.-M. Souriau (generalizing previous constructions by B.O. Koopman, L. Van Hove and I. Segal, see [START_REF] Kirillov | Geometric quantization[END_REF][START_REF] Śnyatycki | Geometric quantization and quantum Mechanics[END_REF]). We let (M, ω) be a simply connected symplectic manifold and we assume for simplicity that there exists a 1-form θ with ω = dθ. We consider the trivial bundle L := M × C and denote by Γ(M, L) the set of square integrable sections of L. Using θ we can define a Hermitian connection ∇ acting on Γ(M, L) by

∀ξ ∈ Γ(M, T M), ∀ψ ∈ Γ(M, L), ∇ ξ ψ = ξ • ψ - i θ(ξ)ψ.
Then to each function F ∈ C ∞ (M, R) we associate the operator F acting on Γ(M, L)

F ψ = F ψ + i ∇ ξ F ψ = (F -θ(ξ F )) ψ + i ξ F • ψ,
where dF + ξ F ω = 0. This construction is called the prequantization of (M, ω).

For instance if (M, ω) = (R 2n , dp i ∧ dq i ), then ω = dθ, with θ = p i dq i and q i = q i + i ∂ ∂p i and p i = -i ∂ ∂q i . Of course one needs further restrictions in order to recover an irreducible representation of the Heisenberg algebra (and hence the standard quantization): this will be the purpose of introducing a polarization and a tensorization of the line bundle L with the bundle of half volume forms transversal to the leaves of the polarization (see [START_REF] Kirillov | Geometric quantization[END_REF][START_REF] Śnyatycki | Geometric quantization and quantum Mechanics[END_REF]).

We will propose an extension of this procedure to our setting, concerned with the quantization of fields. We consider the trivial bundle L := E C × C over E C , where E C is the complexification of the set of solutions to the Klein-Gordon equation [START_REF] Goldschmidt | The Hamilton-Cartan formalism in the calculus of variations[END_REF] as before. On the set Γ(E C , L) of smooth sections of L (we are here relatively vague about the meaning of "smooth") we define a notion of covariant derivative along any vector field of the type Ξ F , where F ∈ Γ(E C , L) by

∀ψ ∈ Γ(E C , L), ∇ Ξ F ψ := Ξ F • ψ - i Σ ξ F θ ψ,
where θ = θ λ . Then we define the prequantization of F ∈ P n-1 H M to be the operator acting on Γ(E C , L) by:

F ψ := Σ F ψ + i ∇ Ξ F ψ = i Ξ F • ψ + Σ F -ξ F θ ψ.

Prequantization of the creation and annihilation observables

We look here for the expressions of the prequantization of a f and a * g given in §1.9. We first set the fact that if ϕ is given in terms of (u k , u * k ) k∈C + by ( 29), then 1 √ 2π where k = (k 0 , -k). We deduce the following

Σ∩Γ α f = 1 √ 2π 3 R n d x C + d k 2k 0 e -i k• x f (k) ϕ(0, x)k 0 + ip 0 (0, x) = C + d k 2k 0 f (k) u k + u * k 2 + u k -u * k 2 = C + d k 2k 0 f (k)u k . Similarly Σ∩Γ α * g = C + d k 2k 0 g(k)u * k .
We moreover observe that Using the previous results we can now express, for ψ ∈ Γ(E C , L),

ξ f θ = 1 √ 2π 3 C +
∇ Ξ f ψ := Ξ f • ψ - i Σ ξ f θ ψ = i C + f (k) ∂ψ ∂u * k - i C + d k 2k 0 f (k) 2 u k ψ and ∇ Ξ * g ψ := Ξ * g • ψ - i Σ ξ * g θ ψ = -i C + g(k) ∂ψ ∂u k - i C + d k 2k 0 g(k) 2 u * k ψ.
For the prequantizations we obtain:

a f ψ = C + f (k) ∂ψ ∂u * k + C + d k 2k 0 f (k) 2 u k ψ,
P (λ) ζ ψ = i Ξ ζ • ψ = C + k • ζ u k ∂ ∂u k -u * k ∂ ∂u * k ψ.
Note that if we need to compute Σ P (λ)

ζ , it is more suitable to set λ = 1, since it gives then the standard expression for the stress-energy tensor. For instance, if

ζ = ∂ ∂x 0 - Σ∩Γ P (1) 0 = R n-1 d x (p 0 ) 2 2 + 3 i=1 (p i ) 2 2 + m 2 ϕ 2 2
gives the total energy in the frame associated with the coordinates x µ .

Introducing a polarization

We choose to impose the extra condition ∇ Ξ * g ψ = 0, ∀g (covariantly antiholomorphic sections), which gives us:

ψ(u k , u * k ) = h(u * k ) exp - 1 2 C + d k 2k 0 u k u * k = h(u * k )|0 .
The advantage of this choice is that all observables functional (creation and annihilation, energy and momentum) of are at most linear in the variables (u k , u * k ), hence we do not need to use the Blattner-Kostant-Sternberg correction for these operators [START_REF] Kirillov | Geometric quantization[END_REF][START_REF] Śnyatycki | Geometric quantization and quantum Mechanics[END_REF]. As a result P |0 = 0, so that the energy of the vacuum vanishes without requiring normal ordering. However we did not take into account the metaplectic correction, which requires a slight change of the connection: if we would do that we would find that the vacuum as an infinite energy (as in the standard quantization scheme), which can be removed by a normal ordering procedure. The mysterious thing here (as was already observed) is that by ignoring the metaplectic correction (which however is fundamental for many reasons) we do not need the normal ordering correction.

  We must understand the induced vector field Ξ ζ on F . Let U(s, •) be the flow mapping of the vector field ζ: U(s, x, ϕ, e, p) = (s, x + ζ, ϕ, e, p). Then the image of Γ = {(x, ϕ(x), e(x), p

e

  -i k• x p 0 (0, x)d x = u k -u * k 2i ,

d k 2k 0 2 ,

 02 e ik•x f (k) 2 (ϕk µ + ip µ ) β µ = α f

  We observe that we have formally [ a f , af ′ ] = [ a * g , a * g ′ ] = 0 and [ a f , a * g ] = C + d k 2k 0 f (k)g(k).Prequantization of the stress-energy tensorIt relies on finding the prequantization of P (λ) ζ = ζ µ P (λ) µ . In principle one should compute the functionals of Σ P (λ) ζ and Σ ξ P (λ) ζ θ λ . But as observed in the previous section we have P (λ) ζ = ζ θ λ = ξ P (λ) ζ θ λ because Lie ζ θ λ = 0. Hence Σ P (λ) ζ -ξ P (λ) ζ θ λ = 0 and so the prequantization of P (λ) ζ is just

The word 'covariant ' refers here to a construction which does rely on the choice of a particular system of coordinates on space-time and hence which respects the basic principles of Relativity.

W.l.g. we can assume that the constant h is zero, so that Γ is included in H -1 (0).

This was followed by a work by L. Koenigsberger[START_REF] Koenigsberger | Die Prinzipien der Mechanik für mehrere unavhängige Variable[END_REF] in 1901, quoted by T. De Donder in[START_REF] Donder | Théorie invariantive du calcul des variations[END_REF], which unfortunately I have difficulties to understand.

including, in previous papers, the Author of this note, who ignored until recently the work of Volterra

in contrast with the integrals Σ∩Γ α k and Σ∩Γ α * k which exists if the restrictions to Σ of ϕ and of its time derivative are Lebesgue integrable

We expound here briefly the principle of the covariant phase space using the multsymplectic formalism. Our presentation will be heuristic and we refer to[START_REF] Kijowski | A canonical structure for classical field theories[END_REF][START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF] for details. We assume that we are given a premultisymplectic manifold (M, ω, β) (see §1.4) and, as in §1.5, that ω is exact, i.e. ω = dθ, for some n-form θ. We note E the set of Hamiltonian n-curves in (M, ω, β), i.e. the set of oriented ndimensional submanifolds Γ ⊂ M which satisfy[START_REF] Forger | Covariant Poisson bracket in geometric field theory[END_REF]. Given some Γ ∈ E, the tangent space 6 to E at Γ represents the set of infinitesimal deformations δΓ of Γ which preserves the equation[START_REF] Forger | Covariant Poisson bracket in geometric field theory[END_REF]. Such a deformation δΓ can be represented by a vector field ξ tangent to M defined along Γ, i.e. a section over Γ of j * Γ T M, which is the pull-back image of the tangent bundle T M by the embedding map j Γ : Γ -→ M. Given δΓ, the vector field ξ is of course not unique, since for any tangent vector field ζ on Γ (i.e. a section of the subbundle T Γ ⊂ j * Γ T M), ξ + ζ represents also δΓ. If so we write:

Note that since E may not be a manifold in general, the usual definition of a tangent space should be replaced by a suitable notion, see[START_REF] Kijowski | A canonical structure for classical field theories[END_REF][START_REF] Hélein | The use of the covariant phase space on non nonlinear fields[END_REF]

2.3 A geometric view of the proofWe can give an alternative proof of Relation[START_REF] Kijowski | Multiphase spaces and gauge in the calculus of variations[END_REF] with a more geometric flavor. We will be even more heuristic, however the validity of our argument is strongly based on the fact that the Lagrangian action can be represented by[START_REF] Forger | Multisymplectic and polysymplectic structures on fiber bundles[END_REF]. For that


Nathalie Lazaric
email: nathalie.lazaric@gredeg.cnrs.fr

Organizational Routines and Cognition: Introduction to the Special Issue on Routines

This article introduces this special issue on routines. It offers some suggestions as to why the concept of routines is considered central in methodological considerations of capabilities and organizational evolution. The contributors to this special issue propose various analytical tools, and provide some missing pieces from the puzzle related to the prominent role of routines. Issues discussed in the papers include methodological individualism. Routines lie between the individual and the firm levels of analysis because they are enacted by individuals in a social context. It is also suggested that a multilevel research agenda provides a finer grained analysis because organizational routines are not isolated units but are entangled among the various organizational layers.

In everyday language, a routine is regarded as automatic behavior, in contrast to designed and implemented strategic plans [START_REF] Cohen | What's different is routine[END_REF][START_REF] Cohen | Reading Dewey: Reflections on the Study of Routine[END_REF][START_REF] Lazaric | Routines and routinization: an exploration of some micro-cognitive foundations[END_REF]. Routines are sometimes seen as unconscious acts, while decision-making is deliberate and intentional. This interpretation, which is still shared by some researchers, is unlikely to progress the scientific debate and is out of line with current research on routines. At least since [START_REF] Nelson | An Evolutionary Theory of Economic Change[END_REF], routines are seen as pertaining to organized groups, rather than individuals. Contemporary studies in the cognitive, social and organizational sciences are further refining our use of this concept, involving the notion of procedural memory [START_REF] Cohen | Organizational Routines are Stored as Procedural Memory: Evidence from a Laboratory Study[END_REF], or recurrent interaction patterns [START_REF] Cohen | Routines and Other Recurring Action Patterns of Organizations: Contemporary Research Issues[END_REF] and involving change driven by individuals [START_REF] Feldman | Organizational Routines as a Source of Continuous Change[END_REF]. These complementary views acknowledge the role of individuals and organizations in building new routines.

The concept of routines is at the center of methodological considerations of capabilities and organizational evolution according to [START_REF] Salvato | Beyond collective entities: Multilevel research on organizational routines and capabilities[END_REF] and [START_REF] Rerup | Routines as a source of change in organizational schema: The role of trialand-error learning[END_REF]. These authors suggest unpacking the organizational 'black box' in order to grasp the organizational complexity of the notion of routines. Along these lines, there needs to be an easier connection between the micro and macro levels of this concept. Contributors to this special issue propose various analytical tools, but are in consensus about the need to make progress in this direction. Their diverse insights provide some of the missing pieces in the puzzle and suggest a revised research agenda for observing the prominent role of routines in everyday life. Every 'recurrent interaction pattern' in an organization may be hiding a potential routine. However, not every interaction constitutes a routine, which means that some clarification is required to avoid inaccurate labeling. The definition provided by [START_REF] Pentland | The foundation is solid, if you know where to look: Comment on Felin and Foss[END_REF] who emphasizes that organizational routines are 'a repetitive recognizable patterns of interdependent actions, carried out by multiple actors' (Feldman and Pentland 2003: 93), is a good starting point. This should not prevent us from continuing the debate since many components of routines are previously-invisible sub-patterns that emerge through the repetition of sequences of actions, but were not designed intentionally by the actors involved. The question should perhaps be why do routines emerge, and in what context, and how are these actions intermingled in the organizational layers that support and enable their creation? The contributors to this special issue propose different solutions to these questions. We can investigate these frameworks without any pretensions to classifying the appropriateness or accuracy of the arguments.

Most of the contributors support the idea that routines emerge to coordinate actions into the division of labor within organizations. But is there an 'invisible hand' working to aggregate these various actions?

Management and authority appear to be good candidates for the formulation of routines, but were they the only triggers this would mean that organizations were purely bureaucratic processes responding to simplistic feedbacks. The emergence of organizational principles can be interpreted in various ways, some of which we investigate here. We are interested also in 'where to look' to paraphrase [START_REF] Pentland | The foundation is solid, if you know where to look: Comment on Felin and Foss[END_REF].

Micro-foundations and methodological individualism

For some authors, methodological individualism is a pre-requisite for understanding the intraorganizational dimension of routines and their motivational components [START_REF] Abell | Building micro foundations for the routines, capabilities, and performance links[END_REF]. [START_REF] Felin | The Endogenous Origins of Experience, Routines and Organizational Capabilities: The Poverty of Stimulus[END_REF] claim that individuals are a central and crucial piece in the puzzle. That said, methodological individualism is prone to ambiguity and is used in different ways. For instance, it is not always made clear whether it is being proposed that social phenomena should be explained in terms of individuals or social phenomena should be explained by individuals plus relation between individuals (Hodgson, 2007: 97). As Hodgson notes, the second option paves the way for including social structures as an explanatory variable and thus to re-frame the debate. For escaping this dichotomy on methodological individualism versus methodological collectivism, Giddens (1982) and others see agents and structures as inclusive mechanisms, i.e. a source of a permanent creation with enabling and constraining forces.

At first glance, there would seem to be simple disagreement between the proponents of methodological individualism and the defenders of the multilevel perspective. However, the controversy goes deeper, since the authors bring their own epistemological and ontological visions of routines and organizations. This is not new [START_REF] Lazaric | The role of routines, rules and habits in collective learning: some epistemological and ontological considerations[END_REF], but since these discussions have not been resolved, they are re-debated from the perspective of the organizational structure and its foundations.

For instance, Teppo and Felin (2011) consider these issues from the point of view of individuals.

Individualism and other causal mechanisms appear to be critical for understanding organizational management, notably how intentional human action and diverse interactions produce certain strategic phenomena (Abell et al., 2008: 492). Starting from [START_REF] Coleman | Foundations of Social Theory[END_REF] insights, Abell et al. try to build the micro-foundations of management, which they maintain are required to capture firm outcomes and to avoid a too simplistic vision of their micro complexities. Micro-foundations are important because 'the individual level considerations have been consistently black boxed in the received capabilities view' (Abell et al., 2008: 494). In this latter perspective, in order to get a better understanding of the routinization process, the formalization of routines should integrate key insights from the economic theory of the firm as depicted by such authors as [START_REF] Alchian | Production, information costs, and economic organization[END_REF] and [START_REF] Holmström | Moral Hazard in Teams[END_REF].

In sum:

How things are done in organizational settings, both in terms of structure and overall efficiency or creativeness, is a function of who is doing. Even in highly routinized environments, the origins of heterogeneous routines are fundamentally at the individual level. (Felin and Foss: 2005: 450).

For this reason, 'collective structures are dependent on the individuals who make up the organization' (ibid: 450). [START_REF] Felin | The Endogenous Origins of Experience, Routines and Organizational Capabilities: The Poverty of Stimulus[END_REF] go a step further, claiming that the behavioral roots of this debate impede organizations from fully integrating feedbacks in a manner suitable to deal with novelties.

According to them the 'poverty of stimulus' makes organizations more receptive to past experience than to becoming inventors of new inputs supported by individuals. In this debate, [START_REF] Hodgson | Poverty of stimulus and absence of cause: Some question for Felin and Foss[END_REF] while endorsing the role of human agency, argue that the term 'methodological individualism' is misleading. Opposing individuals to social structures creates a false dichotomy. Indeed human capacities remain caused and its source of change has multiple ranges of explanation going beyond the solely individual arena.

Behaviorist roots for the foundation of routines

Some misunderstandings about the notion of routines emerge from readings of [START_REF] Nelson | An Evolutionary Theory of Economic Change[END_REF] book. They do not define this concept as behavioral "lock in" or the product of past behavior, but as a source of knowledge that needs constant renewal and new understanding in order to make sense within the firm and to achieve appropriate coordination. As [START_REF] Winter | Problems at the Foundation? Comments on Felin and Foss[END_REF] explains in his contribution here, the reformulation of this notion based on the heritage of Michael Polanyi transforms the 'state of the art' through the integration of two important concepts: 'personal knowledge' and 'tacit knowledge' . This integration represents a departure from the traditional Simonian vision of routines as programs, an idea diffused by the Carnegie School (see [START_REF] Feldman | Organizational Routines as a Source of Continuous Change[END_REF] for a discussion). Also, the notion of 'personal knowledge' implicitly transforms the traditional vision of organizations by emphasizing the difficulties involved in articulating and transferring knowledge inside firms and enabling the emergence of organizational knowledge not based on initially individual skills. The capacity for firms to survive depends on the transformations that are put in place promoted either by the permanent reconfiguration of routines or their new combination. Some transformations are motivated by the individual and organizational knowledge present in the firm. The literature tends to focus not on these sources of change but on the changes promoted by exogenous shocks. This dynamics is based on Schumpeterian thinking in which firms are continuously motivated to innovate in order to survive in a non-stationary environment, and where innovation itself promotes further transformations. This implies a double transformation motivated by internal and external forces, whose real origins are sometimes difficult to identify [START_REF] Winter | Toward a neo-Schumpeterian theory of the firm[END_REF]. In this context, the notion of organizational routines, a pillar of the evolutionary edifice, is a source of both stability and change in the organization. The idea of change is a difficult one, and many talk of routines and innovation as oppositional. However, this hides the underlying organizational reality. Routines evolve according to various internal or external pressures and create focal points around which organization members agree to organize their work or activities. The outside observer might see only the external forces pushing for a modification of routines, but not the internal 'governance' pushing for their transformation. There is a view that changes to routines are more likely to be determined by changes in the environment (Cohen et al., 1996: 683;Andersen, 1994). However, Schumpeterian theory reveals two sources of renewal. First, the 'combinatorics of routines', which occurs through a combination of sub-elements. Although relevant, this approach is undeveloped (Becker, Knudsen and March, 2006: 362). Second, the unreliable process of replication of internal elements. Indeed, the firm's ability to copy and/or to extend routines is a source of competitive advantage. This mechanism may be imperfect and costly, but can also be a valuable source of evolution and change. Winter (1996) develops this in trying to explain the extension and renewal of knowledge bases in a given competitive environment [START_REF] Becker | The influence of Knowledge in the replication of routines[END_REF]. Based on the reasons put forward above, [START_REF] Winter | Problems at the Foundation? Comments on Felin and Foss[END_REF] explains in his contribution the objections to imputing Skinner-type 'behaviorist' arguments to his previous work with Nelson.

Organizations and motivational feedbacks

In their essay [START_REF] Felin | The Endogenous Origins of Experience, Routines and Organizational Capabilities: The Poverty of Stimulus[END_REF] underline the excessive emphasis placed on experience and repetition as the organizational antecedents to routines and capabilities. According to them, lessons learned by the organizations are 'black boxed', impeding the integration of endogenous sources of novelty. The 'poverty of stimulus' illustrates this process where internal elements interfering with recurrent internal patterns of interactions are not fully understood. This is a 'chicken and egg' type problem in the sense that whether and to what extent we are able to differentiate the initial cause from the final cause of this organizational process. To escape this circularity, we need to identify the emergence of the organizational principles, their origin, legitimacy and potential efficiency. In this direction, [START_REF] Salvato | Beyond collective entities: Multilevel research on organizational routines and capabilities[END_REF], advocate for multilevel research on routines and capabilities.

They suggest that routines should be broken down into sequences of individual actions to understand their evolution and effectiveness while 'breaking organization-level routines across hierarchical levels to understand the role of different individual rationalities in routines performance' (Rerup and Salvato,ibid: 14). Observation of individuals in organizations performing different activities and encompassing different understandings of routine would seem appropriate. In this context, the problem of the 'poverty of stimulus' can be transformed in feedback opacity inside and outside the organization.

Individuals have various goals and various motivations which may be in accordance or conflict with the organization's goals. Organizational layers create opacity between individual behavior and organizational principles.

For [START_REF] Greve | Organizational routines and performance feedback[END_REF], this opacity stems from the fact that feedbacks in organizations are not automatic, and within organizations, multilevel organizational goals tend to make feedback more difficult because: the organization level sets the goal variables for lower levels ( …) but does not directly influence lower level -routine modification. First often the individual behavior is part of multi person routines ( ...) as the result the person and the decision are not tightly associated. Second the goal is often assigned higher level the specific routines (…)[Consequently] the goal and the routine are not tightly associated. In such cases, is often not clear whether the individual will feel responsible for making any specific change, and that change will be a modification of a routines (Greve, 2008: 199).

In contexts where the feedbacks between hierarchy and individuals are confused, motivational issues may be a way of reducing this gap. [START_REF] Witt | Emergence and functionality of organizational routines: An individualistic approach[END_REF] claims that we should look more closely at various sources of motivation, i.e. intrinsic and extrinsic ones. Indeed, the expression of individual knowledge inside organizations presupposes sufficient mutual fits in cognitive frames (proposition 2). If individuals are sharing the same cognitive framework, identification with organizational goals will be easier. Thus, shared cognitive frameworks need to be established among individuals based on common knowledge and common goals. This may allow organizations to identify common values reducing the cognitive distance between individuals and instilling some coordinating principles. For Witt (2011, proposition 4) the sharing of cognitive frameworks and social values enables the identification of intrinsic motivations and the building of patterns of recurrent interactions. The self determination theory describes motivation as an autonomous variable which may sustain some degree of regulation and control [START_REF] Deci | Intrinsic motivation and self-determination in human behavior[END_REF][START_REF] Deci | A meta analytic review of experiments examining the effects of extrinsic rewards on intrinsic motivation[END_REF]. Starting from this premise, intrinsic motivation refers to performance of an activity because it is interesting and provides rewards in terms of the basic psychological need for autonomy, competence and relatedness. This source of motivation involves a conscious act and the commitment of individuals to take part in a new learning process. Extrinsic motivation, on the other hand, refers to the performance of an activity because of its instrumental or separable external outcomes, e.g. monetary reward [START_REF] Deci | A self determination theory view of Rewards, Motivation and Organizations[END_REF].

For [START_REF] Postrel | Incentives, Routines, and Self-Command[END_REF], going beyond individual impulsiveness and executing new routines requires new values and learning of new habits. Consequently, incentive systems on their own are not sufficient to change routines because they often conflict with employees' intrinsic motivations (fear of instrumentalism and loss of autonomy). As [START_REF] Leibenstein | Inside the Firm: the Inefficiencies of Hierarchy[END_REF][START_REF] Leibenstein | A branch of economics is missing: micro-micro theory[END_REF] explains, authority and hierarchy cannot substitute for this deliberate effort and cannot control the daily routines of members because they are able to exercise discretionary power. This means that the origin of the effort and, more specifically, the conditions under which individuals can reconcile their personal interests with those of the organization, become critical. Building intrinsic motivation is difficult because 'individual and organizational rationality are engineered "outcomes", not natural endowments' (Postrel and Rumelt, 1992:432). However, a closer look at this issue might help to clarify some of the discrepancy observed between 'ostensive' and 'performative' routines, and illustrate the difficulties of trying to control routines in the face of individuals' free will [START_REF] Feldman | Reconceptualizing Organizational Routines as a Source of Flexibility and Change[END_REF].

As [START_REF] Feldman | Organizational Routines as a Source of Continuous Change[END_REF] suggests, routines are 'emergent accomplishments' that need to be made sense of in order to be deployed by workers in the right way (i.e. requiring minimal supervision of their application). Social mechanisms are required to support the organization's values and social goals. In this perspective, an 'individualist' view of routines may provide a better understanding of the social underpinnings that become taken for granted when routines are activated (see also [START_REF] Lazaric | Knowledge, hierarchy and the selection of routines: an interpretative model with group interactions[END_REF][START_REF] Zbaracki | When Truces Collapse: A longitudinal study of price adjustment routines[END_REF][START_REF] Witt | Emergence and functionality of organizational routines: An individualistic approach[END_REF].

Routines as multilevel mechanisms

Organizational routines, rather than being isolated units may appear to be intertwined with various feedbacks [START_REF] Salvato | Beyond collective entities: Multilevel research on organizational routines and capabilities[END_REF]. The emergence of mechanisms of change stemming from individuals, but which have an impact on the organization ('upward causation'), and the changes within organizations that radically affect individuals ('reconstitutive downward causation ') (Hodgson, 2007:108) are worthy of investigation. Routines clearly lie between the individual and firm levels of analysis because they are enacted by individuals, but within a social context. In his contribution [START_REF] Vromen | Routines as multilevel mechanisms[END_REF] claims that we should conceive routines as multilevel mechanisms, i.e. patterns of behaviors that generate firm behaviors. This proposal helps to broaden our view of routines by encompassing others' interaction mechanisms and by decomposing emergent outcomes and the links within and among them. More precisely ' it helps in getting a clearer picture of how skills and routines are ontologically (rather than metaphorically) related to each other'. Vromen does not claim that his standpoint solves all the problems, but that it can be seen as a counter proposal to avoid a takeover by reductionist arguments. [START_REF] Pentland | The foundation is solid, if you know where to look: Comment on Felin and Foss[END_REF] extends the debate and invites us to consider the granularity of the routines for appreciating their 'practical consequences'. For him, the intersection between action and experience matters especially in the context of ecologies of routines that emerge in networks such as the 'purchase-to-pay cycle'. He encourages a focus on real routines in order to avoid confusion generated 'by disconnecting words from their meaning in practice'. Based on the empirical evidence, former deliberating about foundations should be directed to the tools and methods required to analyze them (a similar judgment is found in [START_REF] Becker | Applying routines in analysing organizations[END_REF].

Technological artefacts as mediators of activities

A valuable contribution to this debate in describing how organizational routines co-evolve with various technological artefacts and more precisely about how 'artefacts and materiality [are brought] to the center of the routines and the Routines Theory', is discussed by D'Adderio (2011) in her contribution. From this perspective, technological artefacts are sources of new forms of procedural memory, but also generate potential rigidity [START_REF] Cyert | Configuring Software, Reconfiguring Memories: the Influence of Integrated Systems on Reproduction of Knowledge and Routines[END_REF][START_REF] D'adderio | The performativity of routines: Theorising the influence of artefacts and distributed agencies on routines dynamics[END_REF]. Artefacts then are 'mediators of human cognition and activity', and are at the interface between ostensive visions of routines and their performance. As a result, they capture, encode and select among performances while having an impact on performative routines -and vice versa [START_REF] D'adderio | Artefacts at the centre of routines: performing the material turn in routines theory[END_REF]. This finer grained analysis provides a means of grounding and situating routines and empirical research. Pentland recommends some other guidelines to enable a better understanding of the link between 'stability and change in routines', the links between 'routines and capabilities' and the need to investigate more fully the 'ecologies of routines' (for complementary discussions see [START_REF] Pentland | Issues in empirical field of Organizational routines[END_REF]Becker and Lazaric, 2009). To sum up and to paraphrase Pentland, in this debate, we should always be connected to our object of observation. Routines are often considered to be mundane, ordinary 'objects' performed in daily life. Thus, in exploring them as 'scientific objects' we may be tempted increase their complexity as our analytical representations hide our normative vision of what organizations are (or what they should be) in some specific context. I invite readers of this special issue to join in this discussion in their future research.

  Nelson and Winter for many years worked with the Rand Corporation, which oversaw several military and civilian technological programs, and they worked closely also with the Carnegie School. Based on this experience Nelson and Winter became thoroughly familiar with the radical technical uncertainty characterizing innovation. Publication of March and Simon's (1963) Organizations had a decisive impact on the whole scientific community (Nelson, 2006). Nelson and Winter were influenced by this book; this led to their consideration of dysfunction in major military organizations and the role of institutions in launching technological innovations. They saw the firm as bearing the mark of these large structures whose decisions, far from responding to market signals, relied on specific rules. Their original Schumpeterian vision of the firm was modulated by their observation of these dysfunctions. Instead of the idea of routines evoked by Schumpeter, or Simon's concept of a program, Nelson and Winter reformulated the notion. Their definition of routines is based on several essential questions related to where individual skills and organizational knowledge reside, as the basis for a successful production process.