Christian Gérard
email: christian.gerard@math.u-psud.fr

Fumio Hiroshima
email: hiroshima@math.kyushu-u.ac.jp

Annalisa Panati
email: annalisa.panati@univ-tln.fr

Akito Suzuki

INFRARED PROBLEM FOR THE NELSON MODEL ON STATIC SPACE-TIMES

Keywords: Mathematics Subject Classification. 81T10, 81T20, 81Q10, 58C40 Quantum field theory, Nelson model, static space-times, ground state

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Infrared problem for the Nelson model on static space-times

Introduction

The study of Quantum Field Theory on curved space-times has seen important developments since the seventies. Probably the most spectacular prediction in this domain is the Hawking effect [Ha, FH, Ba], predicting that a star collapsing to a black hole asymptotically emits a thermal radiation. A related effect is the Unruh effect [Un, Un-W, dB-M], where an accelerating observer in Minkowski space-time sees the vacuum state as a thermal state.

Another important development is the use of microlocal analysis to study free or quasi-free states on globally hyperbolic space-times, which started with the seminal work by Radzikowski [START_REF] Radzikowski | Micro-local approach to the Hadamard condition in quantum field theory on curved space-time[END_REF][START_REF] Radzikowski | A local-to-global singularity theorem for quantum field theory on curved space-time[END_REF] , who proved that Hadamard states (the natural substitutes for vacuum states on curved space-times) can be characterized in terms of microlocal properties of their two-point functions. The use of microlocal analysis in this domain was further developed for example in [BFK], [Sa].

Most of these works deal with free or quasi-free states, because of the wellknown difficulty to construct an interacting, relativistic quantum field theory, even on Minkowski space-time.

However in recent years a lot of effort was devoted to the rigorous study of interacting non-relativistic models on Minkowski space-time, typically obtained by coupling a relativistic quantum field to non-relativistic particles. The two main examples are non-relativistic QED, where the quantized Maxwell field is minimally coupled to a non-relativistic particle and the Nelson model, where a scalar bosonic field is linearly coupled to a non-relativistic particle. For both models it is necessary to add an ultraviolet cutoff in the interaction term to rigorously construct the associated Hamiltonian.

In both cases the models can be constructed on a Fock space with relatively little efforts, and several properties of the quantum Hamiltonian H can be rigorously studied. One of them, which will also be our main interest in this paper, is the question of the existence of a ground state. Obviously the fact that H has a ground state is an important physical property of the Nelson model. For example a consequence of the existence of a ground state is that scattering states can quite easily be constructed. These states describe the ground state of H with a finite number of additional asymptotically free bosons.

When H has no ground state one usually speaks of the infrared problem or infrared divergence. The infrared problem arises when the emission probability of bosons becomes infinite with increasing wave length. If the infrared problem occurs, the scattering theory has to be modified: all scattering states contain an infinite number of low energy (soft) bosons (see eg [START_REF] Derezinski | Scattering theory of infrared divergent Pauli-Fierz Hamiltonians[END_REF]). Among many papers devoted to this question, let us mention [AHH, BFS, BHLMS, G, H, LMS, Sp] for the Nelson model, and [GLL] for non-relativistic QED.

Our goal in this paper is to study the existence of a ground state for the Nelson model on a static space-time, allowing also for a position-dependent mass. This model is obtained by linearly coupling the Lagrangians of a Klein-Gordon field and of a non-relativistic particle on a static space-time (see Subsect. 2.2). We believe that this model, although non-relativistic, is an interesting testing ground for the generalization of results for free or quasi-free models on curved space-times to some interacting situations. Let us also mention that for the Nelson model on Minkowski space-time the removal of the ultraviolet cutoff can be done by relatively easy arguments. After removal of the ultraviolet cutoff, the Nelson model becomes a local (although non-relativistic) QFT model. In a subsequent paper [START_REF] Gérard | Removal of UV cutoff for the Nelson model on static space-times[END_REF], we will show that the ultraviolet cutoff can be removed for the Nelson model on a static space-time.

Most of our discussion will be focused on the role of the variable mass term on the ground state existence. Note that when one considers a massive Klein-Gordon field in the Schwarzschild metric, the effective mass tends to 0 at the black hole horizon (see eg [Ba]). We believe that the study of the Nelson model with a variable mass vanishing at spatial infinity will be a first step towards the extension of the rigorous justification of the Hawking effect in [Ba] to some interacting models.

1.1. The Nelson model on Minkowski space-time. In this subsection we quickly describe the usual Nelson model on Minkowski space-time. The Nelson model describes a scalar bosonic field linearly coupled to a quantum mechanical particle. It is formally defined by the Hamiltonian

H = 1 2 p 2 + W (q) + 1 2 R 3 π 2 (x) + (∇ϕ(x)) 2 + m 2 ϕ 2 (x)dx + R 3 ϕ(x)ρ(x -q)dx,
where ρ denotes a cutoff function, p, q denote the position and momentum of the particle, W (q) is an external potential and ϕ(x), π(x) are the canonical field position and momentum. The Nelson model arises from the quantization of the following coupled Klein-Gordon and Newton system:

(1.1) (2 + m 2 )ϕ(t, x) = -ρ(x -q t ), qt = -∇ q W (q t ) -ϕ(t, x)∇ x ρ(x -q t )dx,
were 2 denotes the d'Alembertian on the Minkowski space-time R 1+3 . The cutoff function ρ plays the role of an ultraviolet cutoff and amounts to replacing the quantum mechanical point particle by a charge density.

To distinguish the Nelson model on Minkowski space-time from its generalizations that will be described later in the introduction, we will call it the usual (or constant coefficients) Nelson model.

For the usual Nelson model the situation is as follows: one assumes a stability condition (see Subsect. 4.5), implying that states with energy close to the bottom of the spectrum are localized in the particle position. Then if the bosons are massive i.e. if m > 0 H has a ground state (see eg [G]). On the contrary if m = 0 and ρ(x)dx = 0 then H has no ground state (see [START_REF] Derezinski | Scattering theory of infrared divergent Pauli-Fierz Hamiltonians[END_REF]).

1.2. The Nelson model with variable coefficients. We now describe a generalization of the usual Nelson model, obtained by replacing the free Laplacian -∆ x by a general second order differential operator and the constant mass term m by a function m(x). We set:

h := - 1≤j,k≤d c(x) -1 ∂ j a jk (x)∂ k c(x) -1 + m 2 (x),
for a Riemannian metric a jk dx j dx k and two functions c(x), m(x) > 0, and consider the generalization of (1.1):

(1.2)

∂ 2 t φ(t, x) + hφ(t, x) + ρ(x -q t ) = 0, qt = -∇ x W (q t ) -R 3 φ(t, x)∇ x ρ(x -q t )|g| 1 2 d 3 x.
Quantizing the field equations (1.2), we obtain a Hamiltonian H acting on the Hilbert space L 2 (R 3 )⊗Γ s (L 2 (R 3 )) (see Sect. 3), which we call a Nelson Hamiltonian with variable coefficients. Formally H is defined by the following expression:

H = 1 2 p 2 + W (q) (1.3) + 1 2 R 3 π 2 (x) + jk ∂ j c(x) -1 ϕ(x) a jk (x) ∂ k c(x) -1 ϕ(x) + m 2 (x)ϕ 2 (x)dx + R 3 ϕ(x)ρ(x -q)dx.
The main example of a variable coefficients Nelson model is obtained by replacing in the usual Nelson model the flat Minkowski metric on R 1+3 by a static Lorentzian metric, and by allowing also the mass m to be position dependent. Recall that a static metric on R 1+3 is of the form

g µν (x)dx µ dx ν = -λ(x)dtdt + λ(x) -1 h αβ (x)dx α dx β ,
where x = (t, x) ∈ R 1+3 , λ(x) > 0 is a smooth function, and h α,β (x) is a Riemannian metric on R 3 . We show in Subsect. 2.3 that the natural Lagrangian for a point particle coupled to a scalar field on (R 1+3 , g) leads (after a change of field variables) to the system (1.2).

1.3. The infrared problem. Assuming reasonable hypotheses on the matrix [a jk ](x) and the functions c(x), m(x) it is easy to see that the formal expression (1.3) can be rigorously defined as a bounded below selfadjoint operator H.

The question we address in this paper is the problem of existence of a ground state for H. Variable coefficients Nelson models are examples of an abstract class of QFT Hamiltonians called abstract Pauli-Fierz Hamiltonians (see eg [G], [BD] and Subsect. 4.1). If ω is the one-particle energy, the constant m := inf σ(ω) can be called the (rest) mass of the bosonic field, and abstract Pauli-Fierz Hamiltonians fall naturally into two classes: massive models if m > 0 and massless if m = 0.

For massive models, H typically has a ground state, if we assume either that the quantum particle is confined or a stability condition (see Subsect. 4.5). In this paper we concentrate on the massless case and hence our typical assumption will be that lim x→∞ m(x) = 0.

It follows that bosons of arbitrarily small energy may be present. The main result of this paper is that the existence or non-existence of a ground state for H depends on the rate of decay of the function m(x). In fact we show in Thm. 4.1 that if m(x) ≥ a x -1 , for some a > 0, and if the quantum particle is confined, then H has a ground state. In a subsequent paper [START_REF] Gérard | Absence of ground state for the Nelson model on static space-times[END_REF], we will show that if

0 ≤ m(x) ≤ C x -1-ǫ , for some ǫ > 0,
then H has no ground state. Therefore Thm. 4.1 is sharp with respect to the decay rate of the mass at infinity.

(If h = -∆ + λm 2 (x) for m(x) ∈ O( x -3/2
) and the coupling constant λ is sufficiently small the same result is shown in [START_REF] Gérard | Infrared Divergence of a Scalar Quantum Field Model on a Pseudo Riemannian Manifold[END_REF]).

1.4. Notation. We collect here some notation for the reader's convenience.

If

x ∈ R d , we set x = (1 + x 2 ) 1 2
. The domain of a linear operator A on some Hilbert space H will be denoted by DomA, and its spectrum by σ (A).

If h is a Hilbert space, the bosonic Fock space over h denoted by Γ s (h) is

Γ s (h) := ∞ n=0 ⊗ n s h.
We denote by a * (h), a(h) for h ∈ h the creation/annihilation operators acting on Γ s (h). The (Segal) field operators φ(h) are defined as φ(h) := 1 √ 2 (a * (h) + a(h)). If K is another Hilbert space and v ∈ B(K, K ⊗ h), then one defines the operators a * (v), a(v) as unbounded operators on K ⊗ Γ s (h) by:

a * (v) K⊗ n s h := √ n + 1 ½ K ⊗ S n+1 v ⊗ ½ n s h , a(v) := a * (v) * , φ(v) := 1 √ 2 (a(v) + a * (v). They satisfy the estimates (1.4) a ♯ (v)(N + 1) -1 2 ≤ v , where v is the norm of v in B(K, K ⊗ h).
If b is a selfadjoint operator on h its second quantization dΓ(b) is defined as:

dΓ(b) n s h := n j=1 ½ ⊗ • • • ⊗ ½ j-1 ⊗b ⊗ ½ ⊗ • • • ⊗ ½ n-j
.

The Nelson model on static space-times

In this section we discuss the Nelson model on static space-times, which is the main example of Hamiltonians that will be studied in the rest of the paper. It is convenient to start with the Lagrangian framework.

2.1. Klein-Gordon equation on static space-times. Let g µν (x) be a Lorentzian metric of signature

(-, +, +, +) on R 1+3 . Set |g| = det[g µν ], [g µν ] = [g µν ] -1 . Consider the Lagrangian L free (φ)(x) = 1 2 ∂ µ φ(x)g µν (x)∂ ν φ(x) + 1 2 m 2 (x)φ 2 (x),
for a function m : R 4 → R + and the associated action:

S field (φ) = R 4 L free (φ)(x)|g| 1 2 (x)d 4 x,
where φ : R 4 → R. The Euler-Lagrange equations yield the Klein-Gordon equation:

2 g φ + m 2 (x)φ = 0, for 2 g = -|g| -1 2 ∂ µ |g| 1 2 g µν ∂ ν . Usually one has 1 2 m 2 (x) = 1 2 (m 2 + θR(x)),
where m ≥ 0 is the mass and R(x) is the scalar curvature of the metric g µν , (assuming of course that the function on the right is positive). In particular if m = 0 and θ = 1 6 one obtains the so-called conformal wave equation. We set x = (t, x) ∈ R 1+3 . The metric g µν is static if:

g µν (x)dx µ dx ν = -λ(x)dtdt + λ(x) -1 h αβ (x)dx α dx β ,
where λ(x) > 0 is a smooth function and h αβ is a Riemannian metric on R 3 . We assume also that m 2 (x) = m 2 (x) is independent on t.

Setting φ(t, x) = λ|h| -1/4 φ(t, x), we obtain that φ(t, x) satisfies the equation:

∂ 2 t φ -λ|h| -1/4 ∂ α |h| 1 2 h αβ ∂ β |h| -1/4 λ φ + m 2 λ φ = 0. We note that |h| -1/4 ∂ α |h| 1 2 h αβ ∂ β |h| -1/4 is (formally) self-adjoint on L 2 (R 3 , dx
) and is the Laplace-Beltrami operator ∆ h associated to the Riemannian metric h αβ (after the usual density change u → |h| 1/4 u to work on the Hilbert space L 2 (R 3 , dx)).

2.2.

Klein-Gordon field coupled to a non-relativistic particle. We now couple the Klein-Gordon field to a non-relativistic particle. We fix a mass M > 0, a charge density ρ : R 3 → R + with q = R 3 ρ(y)d 3 y = 0 and a real potential W : R 3 → R. The action for the coupled system is

S = S part + S field + S int , for S part = R M 2 | ẋ(t)| 2 -W (x(t))dt, S int = R 4 φ(t, x)ρ(x -x(t))|g| 1 2 (x)d 4 x.
The Euler-Lagrange equations are:

2 g φ(t, x) + m 2 (t, x)φ(t, x) + ρ(x -x(t)) = 0, M ẍ(t) = -∇ x W (x(t)) -R 3 φ(t, x)∇ x ρ(x -x(t))|g| 1 2 d 3 x.
Doing the same change of field variables as in Subsect. 2.1 and deleting the tildes, we obtain the system:

(2.1)

∂ 2 t φ -λ∆ h λφ + m 2 λφ + ρ(x -x(t)) = 0, M ẍ(t) = -∇W (x(t)) -R 3 φ(t, x)∇ρ(x -x(t))d 3 x. 2.3.
H part (x, ξ) = 1 2M ξ 2 + W (x), H field (ϕ, π) = 1 2 R 3 π 2 (x) -ϕ(x)λ(x)∆ h λ(x)ϕ(x) + m 2 (x)λ(x)ϕ 2 (x)dx, H int (x, ξ, ϕ, π) = R 3 ρ(y -x)ϕ(y)dy.
The classical phase space is as usual

R 3 ×R 3 ×L 2 R (R 3 )×L 2 R (R 3 ), with the symplectic form (x, ξ, ϕ, π)ω(x ′ , ξ ′ , ϕ ′ , π ′ ) = x • ξ ′ -x ′ • ξ + R 3 ϕ(x)π ′ (x) -π(x)ϕ ′ (x)dx.
The usual quantization scheme leads to the Hilbert space:

L 2 (R 3 , dy) ⊗ Γ s (L 2 (R 3 , dx)),
where Γ s (h) is the bosonic Fock space over the one-particle space h, and to the quantum Hamiltonian:

H = (- 1 2 ∆ y + W (y)) ⊗ ½ + ½ ⊗ dΓ(ω) + 1 √ 2 a * (ω -1 2 ρ(• -y) + a(ω -1 2 ρ(• -y) ,
where

ω = (-λ∆ h λ + m 2 λ) 1 2 ,
dΓ(ω) is the usual second quantization of ω and a * (f ), a(f ) are the creation/annihilation operators on Γ s (L 2 (R 3 , dx)).

The Nelson Hamiltonian with variable coefficients

In this section we define the Nelson model with variable coefficients that will be studied in the rest of the paper. We will deviate slightly from the notation in Sect. 2 by denoting by x ∈ R 3 (resp. X ∈ R 3 ) the boson (resp. electron) position. As usual we set

D x = i -1 ∇ x , D X = i -1 ∇ X .
3.1. Electron Hamiltonian. We define the electron Hamiltonian as:

K := K 0 + W (X),
where

K 0 = 1≤j,k≤3 D Xj A jk (X)D X k , acting on K := L 2 (R 3 , dX), where: (E1) C 0 ½ ≤ [A jk (X)] ≤ C 1 ½, C 0 > 0.
We assume that W (X) is a real potential such that K 0 + W is essentially selfadjoint and bounded below. We denote by K the closure of K 0 + W . Later we will assume the following confinement condition :

(E2) W (X) ≥ C 0 X 2δ -C 1 , for some δ > 0.
Physically this condition means that the electron is confined. As is well known (see eg [GLL]) for the question of existence of a ground state , this condition can be replaced by a stability condition, meaning that states near the bottom of the spectrum of the Hamiltonian are confined in the electronic variables by energy conservation.

We will discuss the extension of our results when one assume the stability condition in Subsect. 4.5.

Field Hamiltonian. Let:

h 0 := - 1≤j,k≤d c(x) -1 ∂ j a jk (x)∂ k c(x) -1 , h := h 0 + m 2 (x),
with a jk , c, m are real functions and:

(B1) C 0 ½ ≤ [a jk (x)] ≤ C 1 ½, C 0 ≤ c(x) ≤ C 1 , C 0 > 0, ∂ α x a jk (x) ∈ O( x -1 ), |α| ≤ 1, ∂ α x c(x) ∈ O(1), |α| ≤ 2, ∂ α x m(x) ∈ O(1), |α| ≤ 1. Clearly h is selfadjoint on H 2 (R 3
) and h ≥ 0. The one-particle space and oneparticle energy are:

h := L 2 (R 3 , dx), ω := h 1 2 . The constant:
inf σ(ω) =: m ≥ 0, can be viewed as the mass of the scalar bosons.

The following lemma is easy;

Lemma 3.1. (1) One has Kerω = {0}, (2) Assume in addition to (B1) that lim x→∞ m(x) = 0. Then inf σ(ω) = 0.
Proof. It follows from (B1) that

(u|hu) ≤ C 1 (c -1 u| -∆c -1 u) + (c -1 u|c -1 m 2 u), u ∈ H 2 (R 3 ). Therefore if hu = 0 u is constant. It follows also from (B1) that c(x) -1 preserves H 2 (R 3 ).
Therefore by the variational principle

m 2 = inf σ(h) ≤ C 1 inf σ(-∆ + c -2 (x)m 2 (x)) = 0.

This proves (2). 2

The Nelson Hamiltonian defined below will be called massive (resp. massless) if m > 0 (resp. m = 0.) The field Hamiltonian is dΓ(ω), acting on the bosonic Fock space Γ s (h).

Nelson Hamiltonian.

Let ρ ∈ S(R 3 ), with ρ ≥ 0, q = R 3 ρ(y)dy = 0. We set:

ρ X (x) = ρ(x -X) and define the UV cutoff fields as:

(3.1) ϕ ρ (X) := φ(ω -1 2 ρ X ), where for f ∈ h, φ(f ) is the Segal field operator: φ(f ) := 1 √ 2 (a * (f ) + a(f )) . Note that setting ϕ(X) := φ(ω -1 2 δ X ), one has ϕ ρ (X) = ϕ(X -Y )ρ(Y )dY .
Remark 3.2. One can think of another definition of UV cutoff fields, namely:

φχ (X) := φ(ω -1 2 χ(ω)δ X ), for χ ∈ S(R), χ(0) = 1.
In the constant coefficients case where h = -∆ both definitions are equivalent. In the variable coefficients case the natural definition (3.1) is much more convenient.

The Nelson Hamiltonian is:

(3.2)

H := K ⊗ ½ + ½ ⊗ dΓ(ω) + ϕ ρ (X),
acting on

H = K ⊗ Γ s (h).
Set also:

H 0 := K ⊗ ½ + ½ ⊗ dΓ(ω),
which is selfadjoint on its natural domain. The following lemma is standard.

Lemma 3.3. Assume hypotheses (E1), (B1). Then H is selfadjoint and bounded below on D(H 0 ).

Proof. it suffices to apply results on abstract Pauli-Fierz Hamiltonians (see eg [START_REF] Georgescu | Spectral theory of massless Nelson models[END_REF]Sect.4]). H is an abstract Pauli-Fierz Hamiltonian with coupling operator v ∈ B(K, K ⊗ h) equal to:

L 2 (R 3 , dX) ∋ u → ω -1 2 ρ(x -X)u(X) ∈ L 2 (R 3 , dX) ⊗ L 2 (R 3 , dx)
Applying [START_REF] Georgescu | Spectral theory of massless Nelson models[END_REF]Corr. 4.4], it suffices to check that ω

-1 2 v ∈ B(K, K ⊗ h). Now ω -1 2 v B(K,K⊗h) = ( sup X∈R 3 ω -1 ρ X 2 ) 1 2
Using that h ≥ CD 2

x and the Kato-Heinz inequality, we obtain that ω -2 ≤ C|D x | -2 , hence it suffices to check that the map

L 2 (R 3 , dX) ∋ u → |D x | -1 ρ(x -X)u(X) ∈ L 2 (R 3 , dX) ⊗ L 2 (R 3 , dx)
is bounded, which is well known. 2

Existence of a ground state

In this section we will prove our main result about the existence of a ground state for variable coefficients Nelson Hamiltonians. This result will be deduced from an abstract existence result extending the one in [BD], whose proof is outlined in Subsects. 4.1, 4.2 and 4.3.

Theorem 4.1. Assume hypotheses (E1), (B1). Assume in addition that: m(x) ≥ a x -1 , for some a > 0, and (E2) for some δ > 3 2 . Then inf σ(H) is an eigenvalue.

Remark 4.2. The condition δ > 3 2 in Thm. 4.1 comes from the operator bound ω -3 ≤ C x 3+ǫ , ∀ ǫ > 0 proved in Thm. A.8. Remark 4.3. From Lemma 3.1 we know that inf σ(ω) = 0 if lim x→∞ m(x) = 0. Therefore the Nelson Hamiltonian can be massless using the terminology of Subsect.

3.2.

Remark 4.4. In a subsequent paper [START_REF] Gérard | Absence of ground state for the Nelson model on static space-times[END_REF] we will show that if 0 ≤ m(x) ≤ C x -1-ǫ , for some ǫ > 0, then H has no ground state. Therefore the result of Thm. 4.1 is sharp with respect to the decay rate of the mass at infinity. 4.1. Abstract Pauli-Fierz Hamiltonians. In [BD], Bruneau and Dereziński study the spectral theory of abstract Pauli-Fierz Hamiltonians of the form

H = K ⊗ ½ + ½ ⊗ dΓ(ω) + φ(v),
acting on the Hilbert space H = K ⊗ Γ s (h), where K is the Hilbert space for the small system and h the one-particle space for the bosonic field. The Hamiltonian H is called massive (resp. massless) if inf σ(ω) > 0 (resp. inf σ(ω) = 0). Among other results they prove the existence of a ground state for H if v is infrared regular.

Although most of their hypotheses are natural and essentially optimal, we cannot directly apply their abstract results to our situation. In fact they assume (see [BD, Assumption E]) that the one-particle space h equals L 2 (R d , dk) and the one-particle energy ω is the multiplication operator by a function ω(k) which is positive, with ∇ω bounded, and lim k→∞ ω(k) = +∞. This assumption on the one-particle energy is only needed to prove an HVZ theorem for massive (or massless with an infrared cutoff) Pauli-Fierz Hamiltonians.

In our case this assumption could be deduced (modulo unitary equivalence) from the spectral theory of h. For example it would suffices to know that h is unitarily equivalent to -∆. This last property would follow from the absence of eigenvalues for h and from the scattering theory for the pair (h, -∆) and require additional decay properties of the [a ij ](x), m(x) and of some of their derivatives.

We will replace it by more geometric assumptions on ω (see hypothesis (4.4) below), similar to those introduce in [GP], where abstract bosonic QFT Hamiltonians were considered. Since we do not aim for generality, our hypotheses on the coupling operator v are stronger than necessary, but lead to simpler proofs. Also most of the proofs will be only sketched.

Let h, K two Hilbert spaces and set H = K ⊗ Γ s (h).

We fix selfadjoint operators K ≥ 0 on K and ω ≥ 0 on h. We set inf σ(ω) =: m ≥ 0.

If m = 0 one has to assume additionally that Kerω = {0} (see Remark 4.5 for some explanation of this fact).

Remark 4.5. It X is a real Hilbert space and ω is a selfadjoint operator on X , the condition Kerω = {0} is well known to be necessary to have a stable quantization of the abstract Klein-Gordon equation ∂ 2 t φ(t) + ω 2 φ(t) = 0 where φ(t) : R → X . If Kerω = {0} the phase space Y = X ⊕ X for the Klein-Gordon equation splits into the symplectic direct sum Y reg ⊕ Y sing , for Y reg = Kerω ⊥ ⊕ Kerω ⊥ , Y sing = Kerω ⊕ Kerω, both symplectic spaces being invariant under the symplectic evolution associated to the Klein-Gordon equation. On Y reg one can perform the stable quantization. On Y sing ,if for example Kerω is d-dimensional, the quantization leads to the Hamiltonian -∆ on L 2 (R d ). Clearly any perturbation of the form φ(f ) for ½ {0} (ω)f = 0 will make the Hamiltonian unbounded from below. So we will always assume that (4.1) ω ≥ 0, Kerω = {0}.

Let H 0 = K ⊗ ½ + ½ ⊗ dΓ(ω).
We fix also a coupling operator v such that:

(4.2) v ∈ B(K, K ⊗ h).
The quadratic form φ(v) = a(v) + a * (v) is well defined for example on K ⊗ DomN 1 2 . We will also assume that:

(4.3) ω -1 2 v(K + 1) -1 2 is compact.
Proposition 4.6 ([BD] Thm. 2.2). Assume (4.1), (4.3). Then H = H 0 + φ(v) is well defined as a form sum and yields a bounded below selfadjoint operator with Dom|H|

1 2 = Dom|H 0 | 1 2 .
The operator H defined as above is called an abstract Pauli-Fierz Hamiltonian.

4.2. Existence of a ground state for cutoff Hamiltonians. We introduce as in [BD] the infrared-cutoff objects

v σ = F (ω ≥ σ)v, H σ = K ⊗ ½ + ½ ⊗ dΓ(ω) + φ(v σ ), σ > 0,
where F (λ ≥ σ) denotes as usual a function of the form χ(σ

-1 λ), where χ ∈ C ∞ (R), χ(λ) ≡ 0 for λ ≤ 1, χ(λ) ≡ 1 for λ ≥ 2.
An important step to prove that H has a ground state is to prove that H σ has a ground state. The usual trick is to consider

Hσ = K ⊗ ½ + ½ ⊗ dΓ(ω σ ) + φ(v σ ),
where:

ω σ := F (ω ≤ σ)σ + (1 -F (ω ≤ σ))ω = ω + (σ -ω)F (ω ≤ σ).
Note that since ω σ ≥ σ > 0, Hσ is a massive Pauli-Fierz Hamiltonian. Moreover it is well known (see eg [G], [BD]) H σ has a ground state iff Hσ does. The fact that Hσ has a ground state follows from an estimate on its essential spectrum (HVZ theorem). In [BD] this is shown using the condition that h = L 2 (R d , dk) and ω = ω(k). Here we will replace this condition by the following more abstract condition, formulated using an additional selfadjoint operator r on h. Similar abstract conditions were introduced in [GP].

We will assume that there exists an selfadjoint operator r ≥ 1 on h such that the following conditions hold for all σ > 0:

(4.4) (i) (z -r) -1 : Domω σ → Domω σ , ∀ z ∈ C\R,
(ii) [r, ω σ ] defined as a quadratic form on Domr ∩ Domω is bounded, (iii) r -ǫ (ω σ + 1) -ǫ is compact on h for some 0 < ǫ < 1 2 . The operator r, called a gauge, is used to localize particles in h.

We assume also as in [BD]:

(4.5) (K + 1) -1 2 is compact.
This assumption means that the small system is confined.

Proposition 4.7. Assume (4.1), (4.2), ( 4.3), (4.4),(4.5) . Then

σ ess ( Hσ ) ⊂ [inf σ( Hσ ) + σ, +∞[.
It follows that Hσ (and hence H σ ) has a ground state for all σ > 0.

Proof. By (4.3), φ(v σ ) is form bounded with respect to H 0 (and to K ⊗ ½ + ½ ⊗ dΓ(ω σ )) with the infinitesimal bound, hence H σ , Hσ are well defined as bounded below selfadjoint Hamiltonians. We can follow the proof of [START_REF] Derezinski | Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians[END_REF]Thm. 4.1] or [START_REF] Gérard | Spectral and scattering theory for some abstract QFT Hamiltonians[END_REF]Thm. 7.1] for its abstract version. For ease of notation we denote simply Hσ by H, ω σ by ω and v σ by v.

The key estimate is the fact that for χ ∈ C ∞ 0 (R) one has

(4.6) χ(H ext )I * (j R ) -I * (j R )χ(H) ∈ o(1), when R → ∞.
(The extended operator H ext and identification operator I(j R ) are defined for example in [START_REF] Gérard | Spectral and scattering theory for some abstract QFT Hamiltonians[END_REF]Sect.2.4]). The two main ingredients of the proof of (4.6) are the estimates:

(4.7) [F ( r R ), ω σ ] ∈ O(R -1 ), F ∈ C ∞ 0 (R), and
(4.8) ω -1 2 σ F ( r R ≥ 1)v σ (K + 1) -1 2 ∈ o(R 0 ). Now (4.8) follows from the fact that v σ (K + 1) -1 2 is compact (note that ω -1 2 σ is bounded since ω σ ≥ σ)
, and (4.7) follows from Lemma 4.8. The estimate (4.6) can then be proved exactly as in [START_REF] Gérard | Spectral and scattering theory for some abstract QFT Hamiltonians[END_REF]Lemma 6.3]. Note that here we prove only the ⊂ part of the HVZ theorem, which is sufficient for our purposes. The details are left to the reader. 2 Lemma 4.8. Assume conditions (i), (ii) of (4.4). Then for all F ∈ C ∞ 0 (R) one has:

F (r) : Domω σ → Domω σ , [F ( r R ), ω σ ] ∈ O(R -1
). Proof. The proof of the lemma is easy, using almost analytic extensions, as for example in [GP]. The details are left to the interested reader. 2 4.3. Existence of a ground state for massless models. Let us introduce the following hypothesis on the coupling operator ([BD, Hyp. F]):

(4.9) ω -1 v(K + 1) -1 2 is compact.
Theorem 4.9. Assume (4.1), (4.2), (4.3), (4.4), (4.5) and (4.9). Then H has a ground state.

Proof. we can follow the proof in [START_REF] Bruneau | Pauli-Fierz Hamiltonians defined as quadratic forms[END_REF]Sect. 4]. The existence of ground state for H σ ( [START_REF] Bruneau | Pauli-Fierz Hamiltonians defined as quadratic forms[END_REF]Prop. 4.5]) is shown in Prop. 4.7. The arguments in [START_REF] Bruneau | Pauli-Fierz Hamiltonians defined as quadratic forms[END_REF]Sects 4.2,4.3] based on the pull-through and double pull-through formulas are abstract and valid for any one particle operator ω. The only place where the fact that h = L 2 (R d , dk) and ω = ω(k) appears is in [START_REF] Bruneau | Pauli-Fierz Hamiltonians defined as quadratic forms[END_REF]Prop. 4.7] where the operator |x| = |i∇ k | enters.

In our situation it suffices to replace it by our gauge operator r. The rest of the proof is unchanged. 2 4.4. Proof of Thm. 4.1. We now complete the proof of Thm. 4.1, by verifying the hypotheses of Thm. 4.9. We recall that h = L 2 (R d dx), ω = h 1 2 and we will take r = x = (1 + x 2 ) 1 2 . Proof of Thm. 4.1. We saw in the proof of Lemma 3.3 that v, ω -1 2 v are bounded, hence in particular (4.2) is satisfied. By hypothesis (E2), (K + 1) -1 2 is compact, which implies that conditions (4.3) and (4.5) are satisfied.

We now check condition (4.4). Note that

ω σ = f (h) where f ∈ C ∞ (R) with f (λ) = λ 1 2 for λ ≥ 2. Clearly Domω σ = H 1 (R d ) which is preserved by (z -x ) -1
, so (i) of (4.4) is satisfied. Condition (iii) is also obviously satisfied. It remains to check condition (ii). To this end we write

ω σ = f (h) = (h + 1)g(h) where g ∈ C ∞ (R) satisfies g (n) (λ) ∈ O( λ -1 2 -n ), n ∈ N, and hence (4.10) [ x , ω σ ] = [ x , h]g(h) + (h + 1)[ x , g(h)].
Since ∇a jk (x), ∇c(x), ∇m(x) are bounded and Domh = H 2 (R d ) we see that

(4.11) [ x , h](h + 1) -1 2 , [[ x , h], h](h + 1) -1 are bounded.
In particular the first term in the r.h.s. of (4.10) is bounded. To estimate the second term, we use an almost analytic extension of g satisfying:

(4.12) g|R = g, | ∂g ∂z (z)| ≤ C N z -3/2-N |Imz| N , N ∈ N, suppg ⊂ {z ∈ C||Imz| ≤ c(1 + |Rez|)}, (see eg [DG1, Prop. C.2.2]),

and write

g(h) = i 2π C ∂g ∂ z (z)(z -h) -1 dz ∧ dz.
We perform a commutator expansion to obtain that:

[ x , g(h)] = g ′ (h)[ x , h] + R 2 , for R 2 = i 2π C ∂g ∂ z (z)(z -h) -2 [[ x , h]h](z -h) -1 dz ∧ dz. Since |g ′ (λ)| ≤ C λ -3/2 , (h + 1)g ′ (h)[ x , h]
is bounded. To estimate the term (h + 1)R 2 , we use again (4.11) and the bound

(h + 1) α (z -h) -1 ≤ C z α |Imz| -1 , α = 1 2 , 1.
We obtain that

(h + 1)R 2 ≤ C [[ x , h]h](h + 1) -1 C | ∂g ∂ z (z)| z 2 |Imz| -3 dzdz.
This integral is convergent using the estimate (4.12). This completes the proof of (4.4). It remains to check condition (4.9), i.e. the fact that the interaction is infrared regular. This is the only place where the lower bound on m(x) enters. By Thm. A.8 we obtain that ω -3/2 x -3/2-ǫ is bounded for all ǫ > 0. By condition (E2), we obtain that X 3/2+ǫ (K + 1) -1 2 is bounded for all ǫ > 0 small enough. Therefore to check (4.9) it suffices to prove that the map

L 2 (R 3 , dX) ∋ u → x 3/2+ǫ ρ(x -X) X -3/2-ǫ u(X) ∈ L 2 (R 3 , dX) ⊗ L 2 (R 3 , dx)
is bounded, which is immediate since ρ ∈ S(R 3 ). This completes the proof of Thm. 4.1. 2 4.5. Existence of a ground state for non confined Hamiltonians. In this subsection we state the results on existence of a ground state if the electronic potential is not confining. As explained in the beginning of this section, one has to assume a stability condition, meaning that states near the bottom of the spectrum of H are confined in electronic variables from energy conservation arguments. Definition 4.10. Let H be a Nelson Hamiltonian satisfying (E1), (B1). We assume for simplicity that the electronic potential W (X) is bounded. Set for R ≥ 1:

D R = {u ∈ DomH |½ {|X|≤R} u = 0}. The ionization threshold of H is Σ(H) := lim R→+∞ inf u∈DR, u =1 (u|Hu).
The following theorem can easily be obtained by adapting the arguments in this section.

Theorem 4.11. Assume hypotheses (E1), (B1), W ∈ L ∞ (R 3 ) and m(x) ≥ a x -1 for some a > 0. Then if the following stability condition is satisfied:

Σ(H) > inf σ(H),
H has a ground state.

Sketch of proof.

Assuming the stability condition one can prove using Agmontype estimates as in [Gr] (see [P] for the case of the Nelson model H)[ then e β|X| χ(H σ ) is bounded uniformly in 0 < σ ≤ σ 0 for σ 0 small enough. From this fact one deduces by the usual argument that H σ has a ground state ψ σ and that (4.13) sup σ>0

) that if χ ∈ C ∞ 0 (] -∞, Σ (
X N ψ σ < ∞.
One can then follow the proof in [START_REF] Panati | Existence and nonexistence of a ground state for the massless Nelson model under binding condition[END_REF]Thm. 1.2]. The key infrared regularity property replacing (4.9) is now

sup σ>0 ω -1 vψ σ H⊗h < ∞.
This estimate follows as in the proof of (4.9) from Thm. A.8 and the bound (4.13).

The details are left to the reader. 2

Appendix A. Lower bounds for second order differential operators

In this section we prove various lower bounds for second order differential operators. These bounds are the key ingredient in the proof of the existence of a ground state for the Nelson model.

A.1. Second order differential operators. Let us introduce the class of second order differential operators that will be studied in this section. Let:

h 0 = 1≤j,k≤d c(x) -1 D j a jk (x)D k c(x) -1 , h = h 0 + v(x),
with a jk , c, v real functions and:

(A.1) C 0 ½ ≤ [a jk (x)] ≤ C 1 ½, C 0 ≤ c(x) ≤ C 1 , C 0 > 0, ∂ α x a jk (x) ∈ O( x -1 ), |α| ≤ 1, ∂ α x c(x) ∈ O(1), |α| ≤ 2, (A.2) v ∈ L ∞ (R d ), v ≥ 0.
Clearly h 0 and h are selfadjoint and positive with domain H 2 (R d ). We will always assume that d ≥ 3.

A.2. Upper bounds on heat kernels. If K is a bounded operator on L 2 (R d , c 2 dx) we will denote by K(x, y) ∈ D ′ (R 2d ) its distribution kernel. In this subsection we will prove the following theorem. We set:

ψ α (t, x) := x 2 x 2 + t α , α > 0.
Theorem A.1. Assume in addition to (A.1), (A.2) that: v(x) ≥ a x -2 , a > 0, then there exists C, c, α > 0 such that: (A.3) e -th (x, y) ≤ Cψ α (t, x)ψ α (t, y)e ct∆ (x, y), ∀ t > 0, x, y ∈ R d .

If c(x) ≡ 1 or if h 0 is the Laplace-Beltrami operator for a Riemannian metric on R d , then Thm. A.1 is due to Zhang [Zh].

Remark A.2. Conjugating by the unitary

U : L 2 (R d , dx) → L 2 (R d , c 2 (x)dx), u → c(x) -1 u, we obtain h0 := U h 0 U -1 = c(x) -2 1≤j,k≤d D j a jk (x)D k , h := U hU -1 = h0 + v(x),
which are selfadjoint with domain H 2 (R d ). Let e -t h(x, y) for t > 0 the integral kernel of e -t h i.e. such that e -t hu(x) = R d e -th (x, y)u(y)c 2 (y)dy, t > 0.

Then since e -th (x, y) = c(x)e -t h(x, y)c(y), it suffices to prove Thm. A.1 for e -t h.

By the above remark, we will consider the operator h0 (resp. h) and denote it again by h 0 (resp. h). We note that they are associated with the closed quadratic forms:

Q 0 (f ) = R d j,k ∂ j f a jk ∂ k f dx, Q(f ) = Q 0 (f ) + R d |f | 2 c 2 v dx, with domain H 1 (R d ).
Let us consider the semi-group {e -th } t≥0 generated by h. Since DomQ 0 = H 1 (R d ), we can apply [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF]Thms. 1.3.2,1.3.3] to obtain that e -th is positivity preserving and extends as a semi-group of contractions on L p (R d , c 2 dx) for 1 ≤ d ≤ ∞, strongly continuous on L p (R d , c 2 dx) if p < ∞. In other words {e -th } t≥0 is a Markov symmetric semigroup.

We first recall two results, taken from [PE] and [D].

Lemma A.3. Assume (A.1), (A.2). Then there exist c, C > 0 such that:

0 ≤ e -th (x, y) ≤ Ce ct∆ (x, y), ∀ 0 < t, x, y ∈ R d .

Proof. Since v(x) ≥ 0 it follows from the Trotter-Kato formula that 0 ≤ e -th (x, y) ≤ e -th0 (x, y), a.e. x, y.

The stated upper bound on e -th0 (x, y) is shown in [START_REF] Porper | Two sided estimates of fundamental solutions of second order parabolic equations and some applications[END_REF]Thm. 3.4]. 2

The following lemma is an extension of [START_REF] Davies | Heat Kernels and Spectral Theory[END_REF]Lemma 2.1.2] where the case c(x) ≡ 1 is considered. Lemma A.4. Assume (A.1), (A.2). Then:

(1) e -th is ultracontractive, i.e. e -th is bounded from L 2 to L ∞ for all t > 0, and

c t := e -th L 2 →L ∞ = sup f ∈L 2 e -th f ∞ f 2 ≤ ct -d/4
with some constant c > 0.

(2) e -th is bounded from L 1 to L ∞ for all t > 0 and

e -th L 1 →L ∞ ≤ c 2 t/2 .
(3) The kernel e -th (x, y) satisfies:

0 ≤ e -th (x, y) ≤ c 2 t/2 . Proof. From Lemma A.3 we obtain that e -th f ∞ ≤ C e ct∆ |f | ∞ ≤ C ′ t -d/4 f 2 ,
using the explicit form of the heat kernel of the Laplacian. This proves (1).

Taking adjoints we see that e -th is also bounded from

L 1 to L 2 with e -th L 1 →L 2 ≤ c t . It follows that e -th L 1 →L ∞ ≤ e -th/2 L 2 →L ∞ e -th/2 L 1 →L 2 ≤ c 2 t/2 , which proves (2). Statement (3) is shown in [D, Lemma 2.1.2]. 2
We will deduce Thm. A.1 from the following result. e -th (x, y) = e -th (x, y) ǫ e -th (x, y)

1-ǫ ≤ Ct -ǫd/2 e -ǫ(x-y) 2 /2t t -(1-ǫ)d/2 ψ α (t, x) 1-ǫ ψ α (t, y) 1-ǫ ≤ C ′ t -d/2 e -c(x-y) 2 /2t ψ β (t, x)ψ β (t, y), for β = (1 -ǫ)α.
This completes the proof of Thm. A.1. 2 It remains to prove Theorem A.5. To this end, we employ the following abstract result. Lemma A.6. ([MS,Theorem B]) Let (M, dµ) be a locally compact measurable space with σ-finite measure µ and let A be a non-negative self-adjoint operator on L 2 (M, dµ) such that

(i) e -tA1 := (e -tA | L 1 ∩L 2 ) clos L 1 →L 1 , t ≥ 0 is a C 0 -semi-group of bounded opera- tors, i.e., e -tA1 L 1 →L 1 ≤ c 1 , t ≥ 0. (ii) e -tA is bounded from L 1 to L ∞ with: e -tA1
L 1 →L ∞ ≤ c 2 t -j , t > 0, for some j > 1. Assume moreover that there exists a family of weights ψ(s, x) (s > 0) such that:

(B1) ψ(s, x), ψ(s, x) -1 ∈ L 2 loc (M \ N, dµ) for all s > 0, where N is a closed null set. (B2) There is a constant c independent of s such that, for all t ≤ s, ψ(s, •)e -tA ψ(s, •) -1 f 1 ≤ c f 1 , f ∈ D s , where D s := ψ(s, •)L ∞ c (M \ N, dµ) (
B3) There exists 0 < ǫ < 1 and constants ĉi > 0, i = 1, 2 such that for any s > 0 there exists a measurable set

Ω s ⊂ M with (a) |ψ(s, x)| -ǫ ≤ ĉ1 for all x ∈ M \ Ω s , (b) |ψ(s, x)| -ǫ ∈ L q (Ω s ) and |ψ(s, •)| -ǫ
L q (Ω s ) ≤ ĉ2 s j/q with q = 2/(1ǫ) and j > 1 is the exponent in condition (ii).

Then there is a constant C such that |e -tA (x, y)| ≤ Ct -j |ψ(t, x)ψ(t, y)|, ∀ t > 0, a.e. x, y ∈ M.

To verify condition (B2) of Lemma A.6, we will use the following lemma. Lemma A.7. ([MS,Criterion 2]) Let e -tA be a C 0 -semi-group on L 2 (M, dµ). Denote by •, • the scalar product on L 2 (M, dµ). Then:

e -tA f L ∞ ≤ f L ∞ , f ∈ L 2 ∩ L ∞ , t > 0,
if and only if:

(A.4) Re f -f ∧ , Af ≥ 0, f ∈ D(A)
,

where f ∧ = (|f | ∧ 1)sgnf with sgnf (x) := f (x)/|f |(x) if |f |(x) = 0 and sgnf (x) = 0 if f (x) = 0.
Proof of Thm. A.5: We will prove that there exists α > 0 such that the hypotheses of Lemma A.6 are satisfied for (M, dµ) = (R d , c 2 (x) dx), A = h and ψ(s, x) = ψ α (s, x). For ease of notation we will often denote ψ α simply by ψ.

From the discussion before Lemma A.4, we know that e -th extends as a C 0 -semigroup of contractions of L 1 (R d , c 2 dx), which implies that hypothesis (i) holds with c 1 = 1. Hypothesis (ii) with j = d/2 follows from (2) of Lemma A.4. Note that d/2 > 1 since d ≥ 3.

We now check that conditions (B) are satisfied by ψ α provided we choose α = α 0 a 1 2 for some constant α 0 . Since ψ, ψ -1 are bounded, condition (B1) is satisfied for all α > 0.

Set Ω s := {x ∈ R d | x 2 ≤ s}. Then ψ(x) -ǫ = x 2 + s x 2 αǫ ≤ 2 αǫ , ∀ x ∈ Ω s ,
which proves the bound (a) of (B3) for all α > 0. Take now 0 < ǫ < d d+4α so that we see that d -2αǫq > 0 for q = 2/(1ǫ). If 0 ≤ s < 1 Ω s = ∅ and (b) of (B3) is satisfied. If s ≥ 1 we have: ψ -ǫ q L q (Ω s ) = Hence (b) is satisfied for j = d/2. It remains to check (B2). To avoid confusion, we denote by g, f the scalar product in L 2 (R d , c 2 (x)dx) and by (g|f ) the usual scalar product in L 2 (R d , dx).

Since ψ, ψ -1 are C ∞ and bounded with all derivatives, we see that {ψe -th ψ -1 } t≥0 is a C 0 -semi-group on L 2 (R d , c 2 dx), with generator

h ψ := ψhψ -1 , Domh ψ = H 2 (R d ).
We claim that there exists α > 0 such that (A.5) e -th ψ L 1 →L 1 ≤ C, uniformly for 0 ≤ t ≤ s.

By duality, (A.5) will follow from (A.6):

(A.6) e -th * ψ L ∞ →L ∞ ≤ C, uniformly for 0 ≤ t ≤ s.

To prove (A.6), we will apply Lemma A.7. To avoid confusion, ∂ j f (x) will denote a partial derivative of the function f , while ∇ j f (x) denote the product of the operator ∇ j and the operator of multiplication by the function f .

Setting b i = ψ -1 ∂ i ψ, we have:

h * ψ = ψ -1 hψ = -c(x) -2 j,k ∇ j a jk (x)∇ kj,k c -2 (x)b j (x)a jk (x)∇ k c -2 (x)∇ j a jk (x)b k (x) + v(x)c -2 (x) j,k b j (x)a jk (x)b k (x) = -c(x) -2 j,k ∇ j a jk (x)∇ k -2c(x) -2 j,k b j (x)a jk (x)∇ k + w(x), where:

w(x) = v(x)c(x) -2 j,k b j (x)a jk (x)b k (x) c(x) -2 j,k a jk (x)∂ j b k (x)c(x) -2 j,k (∂ j a jk )(x)b k (x). Clearly Domh * ψ = H 2 (R d ). To simplify notation, we set A(x) = [a jk (x)], F (x) = (b 1 (x), . . . , b d (x)). The identity above becomes:

(A.7) h * ψ = -c -2 ∇ x A∇ x -c -2 F A∇ x -c -2 ∇ x AF + v -c -2 F AF, = -c -2 ∇ x A∇ x -2c -2 F A∇ x + w.
We note that b j (x) = αsx j x -2 ( x 2 + s) -1 , which implies that:

|b j (x)| ≤ Cα x -1 , |∇ x b j (x)| ≤ Cα x -2
, for some C > 0.

Since v(x) ≥ a x -2 , this implies using also (A.1) that:

(A.8) v(x)c(x) -2 F AF (x) ≥ 0, w(x) ≥ 0, for α > 0 small enough. This implies that

(A.9) Re f, h * ψ f = -(∇ x f |A∇ x f ) + (f |(c 2 v -F AF )f ) ≥ 0, for f ∈ H 1 (R d ).
It follows that h * ψ is maximal accretive, hence e -th * ψ is a C 0 -semi-group of contractions by the Hille-Yosida theorem.

To check condition (A.4) in Lemma A.7 we follow [MS], with some easy modifications. We write We have:

f -f Λ , h * ψ f = (∇(f -f Λ )|A∇f ) -2(F (f -f Λ )|A∇f ) + ((f -f Λ )|c 2 wf ) =: C 1 (f ) + C 2 (f ) + C 3 (f ).
Using (A.10), we have:

C 1 (f ) = (∇(f -f Λ )|A∇f ) = (∇f | χ |f | A∇f ) -(∇|f ||f χ |f | 2 A∇f ) + (∇χ| f |f | A∇f ) =: B 1 (f ) + B 2 (f ) + B 3 (f ).
Clearly B 1 (f ) is real valued. Next: (A.11) ReB

2 (f ) = - 1 2 (∇|f || χ |f | 2 A(f ∇f + f ∇f )) = -(∇|f || χ |f | A∇|f |),
using (A.10). Similarly:

(A.12) ReB 3 (f ) = 1 2 (∇χ| 1 |f | A(f ∇f + f ∇f )) = (∇χ|A∇χ),

Theorem A. 5 .

 5 Assume the hypotheses of Thm. A.1. Then there exists C, α > 0 such that: e -th (x, y) ≤ Ct -d/2 ψ α (t, x)ψ α (t, y). Proof of Theorem A.1: Combining Lemma A.3 with Thm. A.5 we get:

  2αǫq-1 dr = C ′ s d/2 .

f

  f Λ = sgnf χ, χ := ½ {|f |≥1} (|f | -1), and note that if f ∈ Domh * ψ ⊂ H 1 (R d ) then |f |, sgnf, χ ∈ H 1 (R d ) with (A.10) ∇sgnf = ∇f |f | f ∇f |f | 2 , ∇χ = ½ {|f |≥1} ∇|f |, ∇|f | = 1 2|f | (f ∇f + f ∇f ).

using again (A.10). We estimate now ReC 2 (f ). We have: (A.13)

We estimate now ReC 3 (f ). We have: (A.14) ReC 3 (f ) = Re(ff Λ |c 2 wf ) = Re(χ|c 2 w|f |) = (χ|c 2 w|f |) = (χ|c 2 wχ) + (χ|c 2 w).

Collecting (A.11) to (A.13), we obtain that:

+(χ|c 2 w).

We use now the point-wise identity:

Hence the first line in the rhs of (A.15) is positive. Concerning the third line, we recall that (A.8) implies that w ≥ 0 if α = α 0 a. Since χ ≥ 0 the third line is also positive. Therefore: (A.7) and the fact that χ is real. Using (A.9) we obtain condition (A.4). This completes the proof of Thm. A.5. 2 A.3. Lower bounds for differential operators. We now deduce lower bounds for powers of h from the heat kernel bounds in Subsect. A.2.

Theorem A.8. Assume hypotheses (A.1), (A.2) and

We start by an easy consequence of Sobolev inequality.

Lemma A.9. On L 2 (R d ) the following inequality holds:

Proof. We have

By the Sobolev inequality ([RS2, Equ. IX.19]):

for r = 2d/(d + 2γ). We write then f = x -α x α f and use Hölder inequality to get:

We choose q = 2, p = d/γ. The function x -α belongs to L d/γ if α > γ. This implies the lemma. 2 Proof of Thm. A.8. We first recall the formula:

In the estimates below, various quantities like (f |h -δ f ) appear. To avoid domain questions, it suffices to replace h by h + m, m > 0, obtaining estimates uniform in m and letting m → 0 at the end of the proof. We will hence prove the bounds (A.17)

Moreover we note that it suffices to prove (A.17) for f ≥ 0. In fact it follows from (A.16) that (h + m) -β has a positive kernel. Therefore

and (A.17) extends to all f ∈ C ∞ 0 (R d ). We will use the bound (A.3) in Thm. A.1, noting that if (A.3) holds for some α 0 > 0 it holds also for all 0 < α ≤ α 0 . We use the inequality

as long as β > α, using again (A.16). Integrating this point-wise inequality, we get that (f |h -2β f ) ≤ C(f | x 2α (-∆) -2(β-α) x 2α f ). We can apply Lemma A.9 as long as 2(βα) < d/2, and obtain

if α < β < α + d/4. Since α can be taken arbitrarily close to 0, this completes the proof of the theorem. 2


Jérôme Denis Ltci

David Pontille
email: pontille@ehess.fr

Placing Subway Signs: Practical Properties of Signs at Work

Keywords: Ecology, Ethnography, Placement Work, Pragmatics, Subway Signs

This article complements previous studies stressing the importance of the emplacement of signs in public spaces by focusing on the placement process itself. The authors' ethnographic study of the Paris subway shows that the way-finding system operates in a process of graphical ordering within which the standardized shape, content and emplacement of each sign are crucial. However, the placement work does not simply consist of following instructions. The authors show that, in order to find the proper place for boards, stickers and posters, workers have to explore the environment and reconcile the sign placement policy with the ecology of the local sites. In doing so, workers do not just discover an emplacement designated for a particular sign but enact an available place, mobilizing numerous embodied practices. During this process, signs themselves are not treated as stabilized entities; they oscillate between what the authors term ʻpractical properties' that arise from the combination of objects, bodies and environment.

Introduction

Various studies have shown that the understanding of our ʻvisual culture' (Jenks, 1995;Van Leeuwen and Jewitt, 2001) and the role of signs in our world raise the question of the relationships that signs have with their surroundings. This issue highlights a particular semiotic property: emplacement. We summarize three recent approaches to illustrate the interest of such a focus.

As an epigraphist, [START_REF] Petrucci | Public Lettering. Script, Power, and Culture[END_REF] reconstructs the history of ʻpublic lettering' and traces its social functions from the 11th century through to the modern period. He notably sheds light on the relations between the functions and styles of letters and their location of appearance. Public lettering is here described as an instrument of public power: dominant groups managed the emplacement of inscriptions in urban spaces, such as monuments, façades or central squares, to enhance their visibility. From this perspective, public lettering and their locations are part of a programme: they perform a political public space.

From a semiotic standpoint, [START_REF] Scollon | Discourses in Place:. Language in the Material World[END_REF] make a plea for studying ʻsigns in place'. They suggest investigating the issue of indexicality (relationships that signs have with their surroundings) with the concept of ʻemplacement'. They advocate a ʻgeosemiotics' that offers many resources for deviating from traditional semiotics that generally interprets isolated signs. The main idea is that signs fully acquire their meaning only from the place they occupy in a particular setting.

Finally, from an actor-network perspective, Latour and Hermant (1998: 27-35) explore the invisible infrastructures that support Paris as a city. They show that street name plates have to be at a ʻproper' place, which they analyse as the outcome of a large network of inscriptions and administrative departments. Here, sign emplacement is the result of the alignment of heterogeneous elements, each raising political, moral and technical issues. The sign in place is therefore an element of a broad assemblage of devices that perform the city.

Even though these studies vary in their approaches and objects, they all emphasize the place of signs as a crucial condition of felicity but hardly tackle the placement process. [START_REF] Petrucci | Public Lettering. Script, Power, and Culture[END_REF] studies designers and typographs but gives very little attention to the workers who engrave, stick bills or put public inscriptions in place. Focusing on indexicality, [START_REF] Scollon | Discourses in Place:. Language in the Material World[END_REF] mention that indications on a board are useless when the sign is still in the hands of its manufacturer, but do not study the activities of those involved in its installation. Finally, Latour and Hermant (1998: 27-35) are less interested in the placement process itself than in the stream of successive transformations in which a sign goes from one city department to another, one map to one delivery slip, one plate to one street.

Yet, in order to be in place, signs have to be put in a particular site, and this operation remains largely undocumented. The placement of signs is a full-time activity that requires specific operations, skills and know-how. These operations play a great part in the setting of visual environments. We think that an analysis of such operations, through an ethnographic study of placement, is useful for understanding signs and visual communication from a pragmatic point of view: neither per se, nor by observing their uses, but by questioning the practical conditions of their configuration.

From this perspective, we assume that the ʻproper' place is neither a potentiality inscribed in the sign, nor an available spot ready to be filled within the environment. Environmental properties of public inscriptions are supposed to be literally highlighted. Beyond the sole problem of indexicality, it is the practical abilities of signs that are at stake in the installation process. Therefore, semiotic features are never taken for granted during the course of placement in workers' operations. We will show that placement work requires signs to be treated in heterogeneous ways. For workers, focusing on linguistic, material or spatial features is a practical issue that has to be handled according to each situation.

Subway signs in Paris

In this article, we will focus on the case of subway signs, using the results of an ethnographical fieldwork within the Régie Autonome des Transports Parisiens (RATP). In Paris, subway signs became a central issue at the beginning of the 1990s. At this time, a team of specialists (designers, cartographers and architects) was brought together with the aim of reconsidering the whole organization of subway signs. Surveys, field studies and experiments were conducted. This resulted in the creation of a complete way-finding system (a ʻsignage', signalétique in French) and the writing of a particularly ambitious policy. For the first time, the content, shape, size, colours and emplacement of subway signs were standardized and detailed in extremely precise guidelines. In doing so, the RATP established a real ʻwriting policy' [START_REF] Foucault | Discipline and Punish[END_REF] that deeply transformed the subway spaces. Because it notably engendered the multiplication of signs within halls, corridors, staircases and platforms, it thus performed a hybrid setting in which ʻthe distinction between the building and its signs, between the text and the territory, becomes indistinct' (Fuller, 2002: 236). This policy, which was specific to the Parisian subway system, transformed signage into a sociopolitical apparatus that was supposed to make every kind of passenger in every situation aware of the signs' emplacement. It set the basis for a politics of attention in public spaces in which everyone was supposed to be aware of graphical objects that had been placed in the environment (Denis and Pontille, forthcoming).

The placement process was crucial for enacting such an ambitious policy. Since its implantation, a new department, that also deals with the signs' maintenance and repair, has been devoted to this activity. The team is composed of five workers whose previous domains of intervention varied from electricity to mechanics. At the time of our study, they had been in the department for five to eight years. They were not aware of semiotics or any kind of sign theories, but they became real subway sign experts on the job. Because they continually consulted the guidelines on a day-to-day basis, they had a sound knowledge of the details of the policy. Moreover, after walking miles and miles within the Parisian subway system, they acquired a rich experience of sites and situations that was invaluable during the placement work.

Ethnography of signs placement

To grasp the pragmatics of placement, we carried out situated observations of workers who daily put up Paris subway signage1 . We followed the workers, who worked in pairs, during their rounds in subway stations. Our purpose was to reconstitute the spectrum of their activities, to understand the constraints they have to deal with in everyday work and to identify the key resources they use for the signs' placement.

Besides field notes, we took systematic photographs during workers' daily interventions to give an account of different aspects of their work conditions: their main operations, their gestures and the tools they used, the ways they dealt with their surroundings, and so on. Photographs are particularly useful for this kind of study that deals with visible materials [START_REF] Wagner | ʻVisible Materials, Visualised Theory and Images of Social Research'[END_REF]; in our case, that meant not only signs but also the handling process of visual artifacts. Taking pictures is a way of precisely documenting the semiotic work that takes place within subway stations, and of emphasizing features that can easily be overlooked when one studies signs, such as the role of materiality or seemingly insignificant gestures [START_REF] Piette | Le mode mineur de la réalité[END_REF].

We also paid particular attention to the vocabulary the workers used during their hesitations and discussions about signs and their placement. Such a focus is important in two regards. First, it shows that workers are highly reflexive and that they continuously talk to each other in order to achieve their tasks. Second, it emphasizes the judgement processes that placement work requires. As we will see, to put a board in place cannot be reduced to a mere implementation of placement rules: it is always a decision process with varying degrees of difficulty.

These investigations were helpful in identifying the variety of situations in which workers daily act and in understanding that their work involves heterogeneous ways of treating signs. Furthermore, such an ethnography allows us to emphasize broader political issues. In modern societies, which set up ʻinformation' as a main component of citizenship [START_REF] Barry | Political Machines: Governing a Technological Society[END_REF], urban spaces are generally looked upon as functional and inhabitable in themselves. The mobility and orientation of passersby should emerge naturally from architectural buildings, design objects and influx (auto)management. Studying placement processes of signs provides an occasion to ʻsurface the invisible work' [START_REF] Star | ʻThe Ethnography of Infrastructure'[END_REF] that enables visual environments to achieve permanency and efficiency. It is a way for us to shed more light on the material dimensions of the so-called information society and its infrastructures. When signs are essential to the everyday running of subway systems, airports and cities, we assume that the activities of sign workers can be seen as a keystone of public places performation.

Placement process and the emergence of the properties of signs

Our first observations clearly showed that the workers cannot simply follow directions when they put signs in place. They must make judgements to accomplish the placement process. Of course, in their day-to-day activity, guidelines are useful. They frequently consult them either to confirm a decision or to pick up precise instructions about the way boards have to be ʻofficially' placed. But these documents all remain on their desks. They do not accompany workers all the way to the sites of their interventions.

When they begin a sequence of work, operators only have at their disposal a work order (WO) that provides a short description of what they have to do: which board needs attention, its location in the station, and the expected operations to restore the situation. We will see that workers never follow these directions automatically: they have to reconcile work orders and the signage policy with the ecology of the local sites. They must find their own way of identifying the proper place for signs within each particular setting. During such an inquiry, they are confronted with numerous dilemmas that involve signs in very distinctive ways.

We will show that to place the subway signs properly, the workers deal with heterogeneous features that emerge in the course of their activities. In doing so, they face what we term practical properties of signs. These properties are practical in the sense that they are not present within signs, but occur during the encounter of objects, bodies and environment [START_REF] Law | ʻObject Lessons'[END_REF]. Throughout the placement process, signs are not stabilized entities with univocal features: according to each situation, they oscillate from one quality to another. We will show that, in the case of the Parisian subway signage, signs appear as texts, connected entities, deictics, landmarks, or material objects. Each of these practical properties is connected to specific operations by which workers can achieve placement.

Textuality

In the course of placement work, subway signs are regularly handled as texts. This is the case, for example, when workers have to replace boards for several reasons: a board whose information is considered as erroneous, a station name incorrectly spelt, information that must be updated, and so on. In such situations, the focus of work is the discursive content of each board: workers look upon signs as intended to be read and the right placement is guided by the need for boards to be legible and relevant. This textual dimension is an important feature of the placement work since the discursive content does not stop at boards themselves: it is reproduced in the documents that are associated with boards. Each WO contains details about the board to be placed or replaced, and generally indicates its informational content. Partly abbreviated and coded, these descriptions are supposed to help workers to know which board and what place their intervention concerns. Textual dimension of boards is therefore a powerful means of coordination in the actors' network that takes part in the signs placement process. Nevertheless, what is indicated in the document may not be sufficiently explicit, may be only partly reliable in relation to the complete content of the board, or indeed false. Placement work then becomes more difficult, as the following scene shows.

Before entering the station, Robert and Herbert read the WO. It says ʻmissing board'. A deeper reading of the WO comments section allows Robert to specify the task to be done. In more detail, it says: ʻtwo boards to be removed from the customer service booth window'. The information is completely contradictory: there is no missing board; on the contrary, they have to remove two boards. Once on the scene, Herbert and Robert look at the boards and consult each other.

Herbert:

So, what do we have to take off, finally?

Robert: Two boards have to be removed, so ... this one (pointing out a little red one on the window) and this one (a pink one on the wall, on the right).

Herbert: Are you sure? Because the WO says ʻafter-sales service' and ʻcustomer service information line 4'.

The text on boards and the text on the WO do not match and Robert and Herbert are overcome by doubt. To clear up the uncertainty, they seek help from the information officer in the customer service booth, who indicates the boards that have to be taken away: the pink one on the wall and another, identical one, on the window. Robert and Herbert remove the two boards.

Even if they seem transparent, textual properties of signs can lose some of their signification in situation. Occasionally, the interaction with another interlocutor can be helpful, such as the verbal precisions from the information officer in this example. The use of oral interrogation is sometimes essential to ʻclarify' and ʻcontextualize' writing, especially in workplaces [START_REF] Grosjean | ʻL'oral et l'écrit dans les communications de travail ou les illusions du "tout écrit"'[END_REF]. But in most cases, this help is not available. Generally, workers have no other interlocutors and the only way to deal with their doubts is to follow their own points of view.

In these sequences of placement work, the textual properties emerge as the result of the active involvement of workers: they read both the boards and the WO seeking an alignment that has not already been made. They have to find a way of matching information and identifying their precise tasks. Here, since the discursive content is almost exclusively what matters, signs are considered as just vehicles for information. Thus, the purpose of the placement work in this case is to guarantee a proper display for ʻmessages' that appear on boards.

A network of signs

In a way-finding system, no sign can stand alone, whether it is in halls, corridors, or on platforms. Each board is designed and arranged in relationship to others. Therefore, for the workers, environment is never solely architectural. During the placement process, they are constantly aware of the numerous signs that are already in place. The network of enamelled boards, stickers and posters constitute a graphical setting they literally inhabit. Both their sound knowledge of the signage policy and the skills they acquired during their continuous displacements within the whole subway system make them experts of such a hybrid environment. This expertise sometimes places them in a position where they go beyond the placement or replacement orders that appear on WOs. Because they never look just at the board that needs attention, but also at the network in which it stands, they may come to take initiatives for displacements, indeed removals, of certain boards.

Nathan has to replace a provisory board in a corner. After finding the sign in question, he hesitates, looking at both left and right corridors and the boards already in place. The WO specifies he has to change the board indicating ʻPlace de la Chapelle' for a new one because the old one is worn. But after another look at the two boards that frame this one, he decides to remove it instead of placing a new one. He explains aloud: ʻThis is a provisional board. It's at the proper place but the two others at its left and right are new ones: they are the definitive signs, enamelled. So this one has no more reason to be here. I don't have to replace it'. In saying this, he pulls off the provisional sign (Figure 1). A particular place in a subway station is therefore more than just a micro-local occurrence of isolated symbols. This is particularly visible in the following situation where workers initiate a real dynamic relation with the network of signs.

Nathan and Stephan are in a huge station. They try to find the location of a worn board they have to replace. This station consists of three subway lines with a connection to a railway station. There are many platforms and countless corridors. Stephan has the new board in his hand and starts to read the signs he encounters. He tries to find boards that relate to the one he holds. Nathan also looks for clues in the environment (Figure 2). Stephan finds a track in a corridor. They go down to a platform where signs indicate the same directions as the one they have to put up. Stephan points out a particular one that carries some information in common with the one he holds. He says: ʻwe just have to follow the others and we will find where one is missing.' After a while, Nathan finds the empty place: across the rails, he points to a board that is exactly the same as Stephan's, but with reversed arrows. The board to be changed must be placed on the other side of this corridor. As they reach the place, Stephan confirms Nathan's intuition.

Networks of signs can be very complex. In large stations, all signs do not refer to each other and a glance is not enough to determine the position of the board to be placed. Workers themselves sometimes have to pace around platforms and corridors in order to follow the trail of signs. In such situations, the environment has to be explored as a graphical ecology where species of signs are tied up with each other and where the equilibrium of places has to be maintained.

Deictics

During the placement process, signs are also mobilized according to their deictic abilities, that is their ʻproperties … of pointing to the world outside of language' (Scollon and Scollon, 2003: 209). Boards are seen by workers as deictics in two different ways. First, deictic properties are constituted as resources for action. In relying on them, workers can identify the part of the station where they have to put up a board. The site that the sign points to is thus as much a space for passengers' movements as a workplace for operators.

Nathan and Stephan are on their way to replace a board in a very large station that has to be placed at the top of the entrance stairs. They drive around the square above the station, looking for the right entrance. There are a lot of entrances and they stop the van each time to check if the entrance has the same identification code as the one that figures on the WO. But after they have driven around the entire square and looked at six entrances, they still have not found the right one. Stephan decides to change their tactics. He goes to the back of the van and looks at the board itself. He reads its information and the ones on the WO in order to understand where they have to put up the board (Figure 3). The first lines indicate that this access to the station is only for passengers who already have tickets, which means that it is an entrance that provides direct access to the platforms without encountering any ticket booths. The next few lines specify that the nearest ticket booth is located at the main entrance of the inner square. Therefore, there are only two possible solutions for finding the correct place: it has to be one of the entrances on the left but to the right of the main entrance on the inner square.

Nathan parks the van in front of the first one and Stephan confirms that the board has to be placed right here.

In this case, deictic properties of signs are considered as stable and easily available. This is why they are clearly a resource for workers' action. But certain situations show that such qualities sometimes have to be enacted. Indexicality is then not present at the beginning of the placement work, but emerges as its result. This is what happens when a fresh site has to be indicated and brand-new boards have to be placed. Deictic properties of the future board are never self-evident, nor do they rely on the informational content or directional symbols such as arrows. Workers have to enact them by choosing the position of the board very precisely. This choice sometimes requires the reconstruction of a physical link between the site that has to be indicated and the future board location.

A new museum opened recently in the east of Paris. Its president asked the RATP for its name to be indicated on exit boards on the platforms and in the hall of the nearest station. Paul is sent over there to identify the right place for these boards and to decide which street name will be associated with that of the museum.

In order to make his decision, he first looks around the outside of the station and tries to identify the right exit. On his way to one of the three possible exits, he mobilizes several elements to confirm his choice. First, he looks for the museum building. Once it is in sight, he looks again at the other exits. His choice seems to be made: one of the exits is clearly further away and the other one would involve crossing a wide avenue.

Then he goes down to the station and walks back along the way that will be indicated by the exit boards: from the hall to the platform. In doing so, he can identify the boards to be replaced by the new ones that will include the museum name.

From his observations, and then through his own movements, Paul experiences the potential deictic properties of the future boards. He has put to the test the deictic relation in covering the space between the indicated site and the future deictics' location. But this first move is not sufficient for his choice to be definitive. Paul is still wondering if the name of the street he chose is the right one. To strengthen his choice, he mobilized other resources.

In the station, Paul carefully reads the street names on existing boards. He pays particular attention to the ones that point to the exit he chose. The boards that will carry the name of the new museum must refer to a sole street. To check this, he walks again as far the exit. Outside, another device allows him to confirm that the places for the future boards and street name are the right ones: the neighbourhood map. He reads it carefully, spots the museum on it and the junction where his chosen exit is sited (Figure 4). Finally, he tries to read the street name plates on the other side of the avenue to be sure there is no possible confusion. Once all crosschecking has been done, he confirms in his notebook the location of each future board and the street name to be associated with that of the museum.

The other deictics such as the neighborhood map offer important resources for corroborating the emplacement choice. Actually, this case clearly shows that they also constitute specific constraints. For the deictic properties of a future board to be enacted, workers cannot simply place the sign in the right place and ignore other way-finding artifacts. This could create situations of cognitive conflicts where indications on different signs are not adjusted to each other, and may be completely contradictory. Here, the awareness of the connection between signs goes beyond halls, corridors, stations and their signage. A larger network has to be performed: heterogeneous deictics have to be aligned with others2 .

These cases show that the capacity for signs to point to specific sites is the result of embodied practices. Deictic properties have to be enacted. Movement here is essential. The stability of a board, its installation in a fixed place, is the outcome of the circulation of Paul's body. He builds the deictic thread from the signs within the station to the site of the museum with his feet, by pacing up and down spaces. Thus, it is by his own movements that he can successfully form a network from a set of heterogeneous signs (various indexes of the station, the neighborhood map, street nameplates, etc.). In doing so, we could say that Paul carries out the correspondence between the world and the inscriptions that represent it, just as some scientists do [START_REF] Latour | ʻThe "Pedofil" of Boa-Vista: A Photo-Philosophical Montage'[END_REF]. However, as [START_REF] Fuller | ʻThe Arrow-Directional Semiotics: Wayfinding in Transit'[END_REF] put it, signage is not a matter of representation: it deals with the ordering of the environment and its users. By walking back and forth, Paul put to the test, step by step, the link that names and arrows on the board forge between places.

Landmarks

A fourth way consists of the consideration of the signs as direct supporting resources for action. In this case, emplacement assembles both visual and pragmatic properties. The ʻpush' sticker that has to be stuck on certain doors (Figure 5) is a typical example of such a situation. Admittedly, it has to be seen but, most of all, its emplacement has to precisely indicate where one has to put one's hand to open the door properly. Here, the sign is conceived of as a landmark. From this point of view, the crucial point in the placement work is its ability to become a resource for action and the availability of the sign matters much more than its legibility. It has to be salient within the environment in order to carry an almost transparent perception/action process, in the same way that some cognitive systems rely on perceptive tasks [START_REF] Hutchins | ʻHow a Cockpit Remembers Its Speeds'[END_REF].

To enact this property is not too difficult for the workers when it is about signs that involve the hand's movement. It is more complex to assess the way more distant boards can catch the eye.

Herbert and Leonard have just finished putting up a board that indicates the name of the station on a platform wall. Leonard steps back, and looks at the result of their work. He glances at the entire platform, estimating its length, and says, doubtfully: ʻwe should have provided two boards! The passengers in the last carriage cannot see anything. If I was there I could definitely not see it [the board].' Here, the emplacement of the board they just put up is not at stake. It is the need for an additional one that is questioned. In order to check its relevance, they decide to wait for the next train. When it comes, they observe the passengers' faces and glances. Some of them are obviously looking for the name of the station. They do not immediately find the board. After this test, Herbert and Leonard agree to order another board for the end of the platform.

When they enact signs as landmarks, placement workers deal with action. And since the experiment is carried out on the spot, they have to make their own way of putting the board's availability to the test. First, they take the place of passengers who get out of the carriage. In so doing, they assess the qualities of the board with their own eyes: they perform an ʻembodied representation' of users [START_REF] Barrey | ʻDesigner, Packager, Merchandiser: Trois professionnels pour une même scène marchande'[END_REF][START_REF] Denis | ʻProjeter le marché. Les saisies du public dans un service de production télévisuelle'[END_REF], grounded in their ordinary competencies as passengers. Second, they take advantage of the direct presence of users within their workplace: to complete the first experiment, they carry out a quick observation of passengers' attitudes towards the signs. Users are within easy reach, moving in their ʻnatural' environment. By examining their movements, workers can assess the landmarks-to-be in their own environment.

Materiality

Finally, a simple look at the subway signs during work processes allows the understanding of a very simple fact: signs that may be considered as texts, connected entities, deictics or landmarks are also objects. Installation is made by handling activities: workers have to carry boards, put them up on walls, take them down, and so on (Figure 6). Dealing with materiality is thus a crucial way for workers to treat subway signs. In carrying out all these operations, workers do not pay any attention to the discursive content of the board, nor to its graphical composition. All that matters is the efficiency of hanging, the straight alignment of the board with the tiles and its cleanness. When workers treat the subway signs as objects, their activity is oriented towards a unique purpose: to stabilize boards in the physical environment.

Nathan and Stephan have to put up a new signboard to replace the older one that broke when it fell off the wall. They first have to put up new metal brackets, for which they drilled holes beforehand. Then they measure both board and brackets, and scribble a few marks on the wall. They use the tiles as a guide to make sure the new board will be positioned straight. For these operations they use two different drills, glue cement, a pen, a measuring tape, a screwdriver, plugs and screws (Figure 7). Once the board is put up on the wall, Stephan carefully wipes it and says: ʻHere it is, a brand new sign.' Material properties of signs thus raise particular issues that we call ʻecological': dealing with it implies a whole infrastructure of tools, devices, materials (concrete, plaster, tiles) and energy supply. For instance, during this sequence of activity, Nathan and Stephan had to pause to recharge the battery of the second drill. Operators are then confronted with the interdependence of technical networks. Therefore, during placement work, as in repair and maintenance situations [START_REF] Graham | ʻOut of Order: Understanding Repair and Maintenance'[END_REF], workers have to deal with the strengths and limitations that make up the material order of inherited infrastructures (such as the electricity network).

Material issues occur in another way during placement work: not as an outcome of work but as a constraint. In these cases, boards are grasped as objects ʻpresent-at-hand' whose material properties create difficulties.

Nathan and Stephan have to change a board in order to update its information. The amount of handling they have to do for this seemingly simple operation is surprising. Sixteen small screws have to be removed from the metal frame in order to take down the broken board (Figure 8). Once it is done, they put up the new plaque very carefully: a false move could damage it. Then they screw back the frame before hanging it up on the ceiling. Workers' practices depend largely on designers and manufacturers who do not necessarily take their activities into account when they make brackets or frames. When they consider boards as objects, workers are confronted with a material assemblage that is the result of activities prior to their own. In these cases, they handle signs as stabilized entities whose materiality is a particularly resistant feature in the completion of their work.

Therefore, there are two ways that material properties emerge during placement work. On the one hand, materiality is an outcome of placement work process, it appears as the stabilized result of the workers' distributed activity. On the other hand, materiality is a preexisting component of the situation, the sign that there is already a stabilized object that may be particularly resistant to the workers' interventions. The main challenge for workers when making placements is to face up to the constraints of the material features assembled in a sign in order to enact the ones required by its proper placement.

The articulation of the properties of signs

For our purposes, we have successively isolated several ways of treating signs. Each of them was illustrated by different micro-local empirical scenes, where specific qualities emerge. Such a perspective is particularly helpful for shedding light on the multiplicity of signs' practical properties that occur throughout placement work. However, it does not reflect the entire process that encompasses the placement of subway signs. In most cases, the same sign can oscillate from one quality to another. The main issue for workers is then to articulate these various properties during the same intervention. In order to understand this articulation process, it is necessary to follow workers in a longer sequence of action. This is the aim of this last section.

During our inquiry, a new directive was issued by the security department: from that time on, emergency exits had to be indicated by a set of new signs and to be marked out from platforms to streets. Herbert and Leonard were sent to a station after the security staff had pointed out different emplacements for these signs. Their task consisted of confirming or rejecting these choices.

For the observer, the situation seems clear. From a geosemiotic perspective, emergency signs are the archetype of deictics [START_REF] Scollon | Discourses in Place:. Language in the Material World[END_REF]. When validating or rejecting the security staff's propositions, there is no doubt that workers will mainly assess emplacements according to signs' deictic properties. The right place will be the one that allows the sign to effectively point to an exit. However, as soon as Herbert and Leonard go down to the station's hall and check for the first proposition, things get more complicated.

The security department suggested putting up an emergency sign on a large existing board. Straightaway Leonard rejects this choice. He explains that the board already displays the names of two exits, several figures and the names of lines that point to connections to subway lines and bus routes. Here, Leonard's refusal is based on the discursive content of the sign: in his opinion, the board will display too much information if one adds an emergency sticker. Furthermore, he underlines the possible contradiction of the texts: each word, figure and icon on this board is devoted to the day-to-day normal flux, an emergency sign would strongly interfere with them.

But the presence of emergency signs in the station is compulsory. Herbert and Leonard then have to come up with another solution to finding a place for this first sign. In other words, they have to find a position for it within the internal network of signs, without weakening its consistency. Their first movement is to walk through the hall and to act as passengers. In so doing, they treat the future signs as landmarks. They adopt different points of view in the corridor; they walk all around the entrance hall and try to discover which place would make the signs most salient for passengers.

For instance, in front of the revolving doors, Leonard says: ʻHere riders can't see the stairs that lead to the exit. So we could put stickers on the doors ...' Herbert does not say anything, but he seems doubtful. Leonard continues: ʻOr we could put a sticker on the window of the customer service booth. It is not great, but that is an option.'

Here, workers rely on passengers' ordinary competencies to judge where pertinent and visible information should be placed. However, some new difficulties make them adopt another viewpoint. As they consider the possibility of putting up a new board dedicated to the emergency sign, materiality suddenly appears as a crucial issue.

Herbert and Leonard are back in front of the existing board. Leonard looks up: ʻThe concrete ceiling can withstand another board hanging on a chain. It could be a solution. The board will be visible whatever the circumstances since the light is sufficiently intense.' But Herbert notices: ʻThe gap between the ceiling and passengers' heads is too short. Some tall passengers may hurt themselves.' Leonard turns around, looks up again and adds right away: ʻand the board would be hidden by this video camera.

Besides it may hide information given on the existing board' (Figure 9). As soon as workers take into account the material property of signs, they have to deal with several other elements that are present in the environment. No piece of information must weaken, disrupt or refute another one, nor can a sticker or a board hide an existing one. Furthermore, in this example, a whole ecology of entities emerges as pertinent: lights, video camera, pillars and passengers themselves play a role in the decision process.

This case shows how workers can find themselves in a particularly challenging situation during the placement process. They try a lot of various options, assess and discuss potential means, and hesitate for a long time before coming to a final decision. The latter requires the articulation of all signs' properties that occurred during their inquiry. In this example, Herbert and Leonard finally choose to put up stickers on the revolving doors. Such a solution sheds a new light on the placement process: its result represents the identification of an available emplacement. And availability is a very complex thing. First, the place for the future sign has to be graphically available: it should not already display any other inscriptions. By choosing the back of revolving doors for ʻexit' stickers, workers avoid any risk of disrupting the surrounding graphical order. Second, the place must be materially available: the stickers will not hamper passengers during their displacements. Third, the place must be pragmatically available. Ordinarily, the back of revolving doors is inoperative. Exit stickers will not come into conflict with other forms of action by pointing to a new possible exit in case of emergency. In such an operation, the doors are themselves transformed. Initially devoted to prevent passengers from entering without a ticket, the back of the doors is now used as a security apparatus.

Conclusion

The placement process relies on workers' activities that cannot simply consist of following directions. Almost each intervention brings them to the same crucial question: where is the right place for this sign? To answer this question, workers carry out a meticulous exploration of the environment in which they strive to reconcile the signage policy with the ecology of the local sites. In doing so, we showed that they are not discovering an emplacement that was waiting for a sign: they enact an available place, mobilizing numerous embodied practices (such as walking around the station or adopting passengers' positions). During such an inquiry, signs themselves are not treated as stabilized entities. They oscillate from one quality to another as the placement process goes on. Thus, in enacting the right place, operators face the multiplicity of what we termed the practical properties of signs. We showed that, according to the situation, the subway signs occurred as texts, connected entities, deictics, landmarks or material objects. These properties are not already there, within signs, but arise out of the action and have to be taken into account to achieve the placement work. To put subway signs in the ʻright' place, these emerging properties have to be prioritized and articulated. Thus, this is not only the environment, but also signs themselves that are enacted and stabilized during the placement process.

To conclude, we would like to tackle the question of the possible generalization of this ethnography of placement work. How could such research be relevant for studying other subway systems or even the placement of other kinds of graphical objects in public spaces? Since it offers a way to focus on the backstage view of ʻsigns in place', we think that the analysis of the practical properties of signs that occurs during the placement work could be useful, to a larger extent, in geosemiotics studies. However, such research has to be mobilized with certain precautions.

First, one has to keep in mind that placement work is not a generic activity. Placing signs is always accomplished in a specific environment and within a larger sociopolitical setting. For instance, we saw that in Paris, the subway signage policy is a very ambitious one. It was conceived as an important component of a strong process of ordering, for the sake of which the amount of signs within the environment is extremely large and the whole way-finding system is highly standardized. One cannot understand the complexity of the placement work that we described without taking into account this policy: it is present within workers' activity through the guidelines that lie on their desks and the work orders that they hold in their hands when entering a station.

In other terms, the generalization of our placement work study could not just rely on the collection of observations of micro-tasks in other settings. It would require a broader view of workers' activity that would allow researchers to be aware of the political dimension that accompanies the placement process in the form of normative documents and situated directions. The organization of work itself is also important: placement does not always appear as an official dedicated activity. In a comparative setting, such a perspective would offer a useful means for a full understanding of the culturally specific features of each situation, without referring to mysterious impalpable social forces.

Second, one has to be aware that the analysis of placement work may lead to an inadequate impression of stability. Once an available place has been identified and the sign has been fixed to a wall, one could easily think that the process of ordering is achieved, and then begin to focus on other parts of it (for instance, the situated uses of boards and stickers). As demonstrated by some of the situations described in this article, such a view would be a complete misunderstanding. Social order is a matter of continuous performation, and in respect of its other domains [START_REF] Graham | ʻOut of Order: Understanding Repair and Maintenance'[END_REF], graphical order has to be maintained from day to day. A sign is never placed once and for all. It may be damaged, contain obsolete information, or simply get worn. The sites that it points to may also change. The activity of placement is thus always tightly linked to the activities of repair and maintenance. The study of the latter would provide a decisive complement for the foray we made into the backstage view of ʻsigns in place'.

Figure 1 .

 1 Figure 1. Removing a provisional board. © Photo: J. Denis and D. Pontille.

Figure 2 .

 2 Figure 2. Reading and observing the surrounding signs © Photo: J. Denis and D. Pontille.

Figure 3 .

 3 Figure 3. Checking out a board to deduce its location. © Photo: J. Denis and D. Pontille.

Figure 4 .

 4 Figure 4. Reading a neighborhood map to confirm a choice. © Photo: J. Denis and D. Pontille.

Figure 5 .

 5 Figure 5. Putting up a sticker. © Photo: J. Denis and D. Pontille.

Figure 6 .

 6 Figure 6. Carrying a board. © Photo: J. Denis and D. Pontille.

Figure 7 .

 7 Figure 7. Drilling. © Photo: J. Denis and D. Pontille.

Figure 8 .

 8 Figure 8. Removing screws. © Photo: J. Denis and D. Pontille.

Figure 9 .

 9 Figure 9. The board, among other objects. © Photo: J. Denis and D. Pontille.

This study is part of a larger ethnographic fieldwork about Paris subway signage: conception and design, production and installation, repair and maintenance of signs in the environment. It is also based on interviews of different stakeholders and on the analysis of several documents such as guidelines and reports.

The performation of such a large network is important because subway systems are not "nonplaces"[START_REF] Augé | Non-Places: Introduction to an Anthropology of Supermodernity[END_REF]: they are tightly tied to the city.

Acknowledgements

This research is part of a collective project entitled ʻEcologies and Politics of Writing' (Département Sciences Économiques et Sociales, LTCI (UMR 5141) CNRS, TELECOM ParisTech and Anthropology of Writing Team, IIAC (UMR 8177) CNRS-EHESS) and supported by the Agence Nationale de la Recherche, Grant n°NT05-1_42082. We would like to acknowledge the staff of the Régie Autonome des Transports Parisiens (RATP) for their active cooperation in this study. We are also grateful to Delphine Burali for her support in translation, and to two anonymous referees for their helpful comments on a previous version of this paper.