Benjamin Bahr

On knottings in the physical Hilbert space of LQG as given by the EPRL model

We consider the EPRL spin foam amplitude for arbitrary embedded two-complexes. Choosing a definition of the face-and edge amplitudes which lead to spin foam amplitudes invariant under trivial subdivisions, we investigate invariance properties of the amplitude under consistent deformations, which are deformations of the embedded two-complex where faces are allowed to pass through each other in a controlled way. Using this surprising invariance, we are able to show that the physical Hilbert space, as defined by the sum over all spin foams, contains no information about knotting classes of graphs anymore.

Introduction

Loop Quantum Gravity is an approach to canonical quantum gravity (see [START_REF] Ashtekar | Quantization of diffeomorphism invariant theories of connections with local degrees of freedom[END_REF][START_REF] Rovelli | Quantum Gravity[END_REF][START_REF] Thiemann | Modern Canonical Quantum General Relativity (Cambridge Monographs on Mathematical Physics[END_REF] and references therein). The kinematical states ψ of the theory are given by elements in an auxiliary Hilbert space H kin , and are considered to correspond to quantized Cauchy data for the initial value problem of General Relativity. An orthonormal basis for H kin is given by the spin network functions, which carry a geometrical interpretation given by the geometric operators associated to areas and volumes. Quantum numbers associated to the states are graphs γ embedded in the Cauchy surface Σ, spins k (i.e. irreducible representations of SU(2)) along the edges of γ, together with SU (2)-intertwiners on its vertices (see figure 1).

Evolution of classical Cauchy data in General Relativity is subject to constraints, which ensure the four-diffeomorphism invariance of the theory. These constraints are divided into the (spatial) diffeomorphism constraints D a and the Hamiltonian constraint H. Correspondingly, the states ψ in H kin are not considered to be physical, but rather one is looking for states ψ phys which are solutions to the constraint equations Da ψ phys = Ĥψ phys = 0. In order to obtain the physical Hilbert space from the kinematical one, one usually employs a so-called "rigging map", which serves as a bona fide projector. Technically, this amounts to an anti-linear map η : D kin -→ D * kin (1.1) which maps elements of a dense subspace D kin of H kin (which is usually taken to be the finite linear span of the spin network functions) into its algebraic dual. Hence states are mapped into distributions, and one equips the range of η with an inner product via

η[φ] | η[ψ] phys := η[φ](ψ).
(1.2)

Generically the map η has a non-trivial kernel, and hence one has to divide the zero space out of the range of the inner product (1.2) in order to make it positive definite. As a result, several different kinematical states ψ will be mapped to the same physical state η[ψ] by (1.1). This is definitely a desired feature, since, in the case of systems with constraints, states that are e.g. related by a change of gauge differ kinematically, but should be the same physically.

In this article, we consider a specific proposal for the physical inner product (1.2) in Loop Quantum Gravity, which mimics a path integral formulation for GR in the sense that it includes a sum over histories of spin networks. This idea goes back to Rovelli and Reisenberger ([4,[START_REF] Baez | An Introduction to Spin Foam Models of Quantum Gravity and BF Theory[END_REF], see [START_REF] Perez | The spin-foam-representation of loop quantum gravity[END_REF] for a review), and lies at the foundation of spin foam models as understood in the context of Loop Quantum Gravity. 1Consider the 4d manifold M together with a foliation into 3d hypersurfaces M = Σ × [0, 1]. Embed two states ψ i and ψ f into the initial and final hypersurface respectively. Now consider a two-complex κ embedded in M that has ψ i and ψ f as boundary. This two-complex (which is interpreted as history of the spin network state evolving from ψ i to ψ f ) is assigned the so-called "spin foam amplitude" Z[κ], which is usually given in a local form

Z[κ] = f A f e A e v A v interior e B e v B v
boundary (1.3) where a particular spin foam model is chosen by a specification of the amplitudes A v , A e and A f associated to the interior vertices, edges and faces of the two-complex, as well as the amplitudes B v and B e associated to the vertices and edges of the boundary spin networks ψ i and ψ f .

The physical inner product, as given by the spin foam model, is then defined as the weighted sum over all two-complexes κ that have ψ i and ψ f as boundary (omitting the η from now on for reasons of brevity), i.e.

ψ f | ψ i phys = κ : ψ i κ -→ ψ f Z[κ] ( 1 . 4 )
This definition resembles a path-integral in the sense that it is a weighted sum over histories of spin networks, as proposed by Feynman. This particular form of the physical inner product as sum over embedded two-complexes has received increased interest recently (see in particular [START_REF] Rovelli | A new look at loop quantum gravity[END_REF]). Two of the spin foam models that are considered mostly today are the EPRL [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] and the FK model [START_REF] Freidel | A New Spin Foam Model for 4d Gravity[END_REF]. They have evolved out of a state-sum model constructed by Barrett and Crane [START_REF] Barrett | A Lorentzian Signature Model for Quantum General Relativity[END_REF], and amount to specific choices of A v . It has been shown [START_REF] Barett | Asymptotic analysis of the EPRL four-simplex amplitude[END_REF][START_REF] Conrady | On the semiclassical limit of 4d spin foam models[END_REF] that, in the limit of large quantum numbers, the amplitudes Z[κ], for κ being the complex dual to a four-simplex, are asymptotically equal to the cosine of the action of Regge-discretised gravity for that simplex, i.e. one has

Z[κ] ∼ e iS Regge + e -iS Regge , (1.5)
which supports the belief that the sum (1.4) is in some sense connected to a path integral for gravity. 2For all spin foam models that exist today however, the sum in (1.4) diverges, since there are infinitely many two-complexes, even if one sums only over diffeomorphism-equivalence classes of foams. 3 The task therefore still remains to make precise sense out of (1.4).

The scope of this article is to have a closer look at the symmetries of Z[κ], i.e. to investigate on which properties of κ the spin foam amplitude actually depends. The reason for this is two-fold. On the one hand, by finding a large symmetry group, one can restrict the sum over κ to a sum over equivalence classes [κ] under the symmetry, which makes it more likely to get hold of a mathematical meaningful concept of summation in (1.4). On the other hand, we will be able to make certain statements about the actual size of the physical Hilbert space H phys , i.e. to see which states in H kin are actually mapped to the same physical states. To be more specific, we will show that for the EPRL model, and any sensible definition of the sum (1.4), states ψ on graphs with the same combinatorics but different knotting classes will be mapped to the same physical state under (1.1). In other words, states in the physical Hilbert space will (at most) be labelled by combinatorial graphs, but will contain no knotting information.

The plan for this article is as follows: We start by recalling the definition of the Euclidean EPRL spin foam model for arbitrary two-complexes in section 2. We moreover discuss a specific choice for the edge-, face-and boundary amplitudes. In section 3 we define the consistent deformation of a spin foam κ, and show that the amplitude Z[κ] is invariant under it. In section 4 we describe a specific "unknotting" spin foam κ 0 , compute its amplitude, and -using the symmetry of the spin foam amplitudes demonstrated in the previous chapter -show that the physical Hilbert space contains no knotting information of graphs anymore.

The EPRL spin foam model a la KKL

Labels on two-complexes

In the following we recap in detail the definitions of the EPRL spin foam amplitudes, in order to make the reader familiar with our conventions, which keep close to those given in [START_REF] Kaminski | Spin-Foams for All Loop Quantum Gravity[END_REF]. The main difference to the more traditional ways of defining spin foams is that the two-complex in question is always regarded as embedded in a space-time manifold, and that at all times the orientation of the faces and edges of the two-complex are to be taken care of. 4For the rest of the article, fix a 4d manifold M ∼ Σ × [0, 1], where Σ i = Σ × {0} and Σ f = Σ × {1} are viewed as initial and final 3d hypersurface. By κ we always denote a piecewise analytic5 two-complex embedded in M, such that the intersections of κ with Σ i,f are piecewise analytic graphs embedded in either Σ i,f . The two-complex κ is endowed with an orientation, i.e. an orientation of all its 1-cells ("edges") and 2-cells ("faces") 6 . Furthermore, κ is equipped with irreducible Spin(4)-representations (j + f , j - f ) (where the j ± f ∈ 1 2 N are spins that arise from the equivalence Spin(4) SU(2) × SU(2))

along its faces f , and intertwiners ι e along its edges e. The convention is such that ι e is an intertwiner between the tensor product of representations belonging to faces of which e is a part of the boundary such that the orientation of e and the induced orientation of f coincide ("incoming faces"), and the tensor product of the faces that induce an orientation opposite to the one of e ("outgoing faces").

ι e : V j + f 1 ,j - f 1 -→ V j + f 2 ,j - f 2 ⊗ V j + f 3 ,j - f 3
and intertwines the action of Spin(4) SU(2) × SU(2) on both sides.

Each intertwiner map

ι e : f incoming V j + f ,j - f -→ f outgoing V j + f ,j - f (2.1)
is to commute with the group action on either side of (2.1), i.e.

ι e • f incoming ρ j + f ,j - f (g) = f outgoing ρ j + f ,j - f (g) • ι e (2.2)
for all g ∈ Spin(4), when

ρ j + f ,j - f is the irreducible representation on V j + f ,j - f
belonging to the face f .

EPRL intertwiner

The EPRL model consists of a special choice of intertwiner (2.1). Equivalently, the model declares all amplitudes A e , A f , A v to be equal to zero whenever the corresponding representations j ± f and intertwiners ι e do not satisfy the following conditions for a fixed real γ = ±1, 0:

• For each face f there is a half-integer k f ∈ 1 2 N such that j ± f = |1 ± γ| 2 k f (2.3)
• For each edge e the associated intertwiner is the image of the following map φ, which maps SU(2) intertwiners to SU(2) × SU(2)-intertwiners: φ :

f incoming V k f ⊗ f outgoing V † k f -→ f incoming V j + f ,j - f ⊗ f outgoing V † j + f ,j - f φ(ι) m + 1 m - 1 ,...m + k m - k n + 1 n - 1 ,...n + l n - l = dh + dh -π j + in,1 (h + ) m + 1 m+ 1 . . . π j + in,k (h + ) m + k m+ k × π j - in,1 (h -) m - 1 m- 1 . . . π j - in,k (h -) m - k m- k × π j + out,1 ((h + ) -1 ) ñ+ 1 n + 1 . . . π j + out,k ((h + ) -1 ) ñ+ k n + k (2.4) × π j - out,l ((h -) -1 ) ñ- l n - l . . . π j - out,k ((h -) -1 ) ñ- k n - k × C k in,1 p 1 j + in,1 m+ 1 j - in,1 m- 1 • • • C k in,k p k j + in,k m+ k j - in,k m- k × C k out,1 q 1 j + out,1 ñ+ 1 j - out,1 ñ- 1 • • • C k out,l q l j + out,l ñ+ l j - out,l ñ- l × ιp 1 •••p k q 1 •••q l
I.e. ι e = φ(ι e ) for some SU (2)-intertwiner ιe . Here 2j ± x,i = |1±γ|k x,i for x = {in, out} are the SU(2) × SU(2) representations to the ingoing and outgoing faces. The C JM j 1 m 1 j 2 m 2 are the usual Clebsch-Gordon coefficients [START_REF] Edmonds | Angular Momentum in Quantum Mechanics[END_REF]. The complicated-looking formula (2.4) can be conveniently summarised with the help of the diagrammatic calculus given in figure 3 7 intertwiner. All face are outgoing, so there is no additional index out, in to distinguish them.

• • • • • • ι = φ(ι) ι k 1 k 2 k n j + 1 j + 1 j - 1 j - 1 j + 2 j + 2 j - 2 j - 2 j + n j + n j - n j - n
It should be noted that the map (2.4) is often referred to as "boost" in the literature. Since it is (whenever γ is such that (2.3) has solutions in the half-integers for all k f ) an isomorphism between the space of SU(2)-intertwiners and EPRL-intertwiners (i.e. those SU(2) × SU(2)-intertwiners which satisfy this particular quantization of the simplicity constraints [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF]), it provides an explicit one-to-one correspondence between data on the two-complex, and data associated to boundary states known from LQG. 8As indicated, the spin foam labels j ± f , ι e induce -via the above maps -spin network labels k e , ιv on the boundary graphs ψ i , ψ f , which can therefore be seen as elements in the kinematical Hilbert space H kin = L 2 (A, dµ AL ) of Loop Quantum Gravity. A spin foam κ with these labels is denoted as

ψ i κ -→ ψ f (2.5)
and is interpreted as the evolution of spin network functions starting as ψ i and ending as ψ f .

The vertex amplitude

In the following we present the definition of the vertex amplitude A v for a vertex v in the two-complex κ, which is a function of the representations j ± f and intertwiners ι e associated to faces and edges of κ. The data is supposed to satisfy the conditions (2.3) and (2.4).

Embed in M a small three-sphere, centered around the vertex v. The sphere S 3 should be placed such that it intersects each edge e which meets v at exactly one vertex v e , and each face f (which spans between two edges e 1 , e 2 ) meeting v at exactly one edge e f between the vertices v e 1 , v e 2 . Hence every vertex provides a dimensional reduction of the neighbourhood of v.

As a result, the intersection of the two-complex κ with the sphere results in a graph γ v , which consists of the edges and vertices e f , v e . Furthermore, the data j ± f , ι e determine data on the graph γ v in the following way: To every edge e f assign the orientation coinciding with the one of f , as well as the Spin(4)-representation j ± f . To the vertices v e assign the intertwiner ι e , when e is starting at the vertex v, and ι † e when e is ending at v. It is not hard to see that this results in a Spin(4)-network function T γv,j ± f ,ιe based on γ v . It is termed the vertex function, and the vertex amplitude is defined as the evaluation on the flat connection

A v := T γv,j ± f ,ιe (½) ( 2 . 6 ) = f δ a f b f e (ι e ) a 1 a 2 ... b 1 b 2 ...
Effectively the amplitude is computed by contracting the indices of the EPRL intertwiners ι e according to the combinatorics of the graph γ v .

j ± 1 j ± 1 j ± 2 j ± 2 j ± 3 j ± 3 ι e ι e S 3 γ v v κ
Figure 4: A vertex v in a spin foam κ, and the associated vertex function T γv ,j ± f ,ιe , which is a SU(2) × SU(2)-network function on the vertex graph γ v on a three sphere S 3 , which results as a dimensional reduction of the neighbourhood of v. The edges and vertices of γ v correspond to faces and edges in κ touching v.

Face-, edge-and boundary amplitudes

In this section we consider a specific choice for the amplitudes that are assigned to faces A f and edges A e within (1.3). We show how these amplitudes are derived by requiring the spin foam amplitude to be invariant under trivial subdivisions of the two-complex, which do not change its topology, as well as by requiring a functorial property of the spin foam amplitude. This particular choice of amplitudes has been considered before and is the most usual one (see e.g. [START_REF] Bojowald | Spin Foam Quantization and Anomalies[END_REF]). It is however not the only one, see e.g. [START_REF] Bianchi | Face amplitude of spinfoam quantum gravity[END_REF] for arguments for different amplitudes.

The spin foams described in this article are thought of as histories of spin networks of Loop Quantum Gravity. These spin networks themselves are invariant under trivial subdivisions, i.e. by dissecting an edge e of a graph γ with spin k into two edges e = e 1 •e 2 with spin k, and placing at the newly appeared vertex v the identity intertwiner ιv = ½ V k (See figure 6). This invariance suggests that one might desire a similar set of invariances to also hold for the spin foam amplitudes.

One way of subdividing a two-complex without changing its topology is by subdividing an edge e = e 1 • e 2 by adding another vertex v = e 1 ∩ e 2 .9 See figure 5.

The two new edges obtain the same orientation and intertwiner, i.e. ι e 1 = ι e 2 := ι e . Invariance of Z[κ] under this subdivision leads to the vertex-and edge amplitudes having to satisfy A e = 1

Av . The vertex amplitude A v can be evaluated using (2.6), realising

ι e ι e ι e v = A e ! = A 2 e A v ⇒ A e = 1 Av = 1 tr(ιeι † e )
Figure 5: Trivial subdivision of an edge with a vertex.

that γ v is an n-bridge graph 10 , where n is the number of faces meeting e. It is given by A v = tr(ι e ι † e ) (assuming that all faces are oriented along with e), leading to the edge amplitude

A e = 1 tr(ι e ι † e )
.

(2.7)

Another way of trivially subdividing the two-complex without changing its topology is by dissecting a face f with representation (j + f , j - f ) into two faces f 1 , f 2 , each with the same orientation and representations j ± 11 The intertwiner ι e on the newly appeared edge e = f 1 ∩ f 2 is chosen to be the identity intertwiner ι e = ½ V j + f ,jf (see figure 6).

f 1 = j ± f 2 = j ± f .
j f j f j f ι e = ι e = id, A f ! = A 2 f A e ⇒ A f = 1 Ae = (2j + f + 1)(2j - f + 1)
A f = 1 A e , (2.8)
where here e is supplied with the identity intertwiner. With (2.7) and (2.8) one arrives at an expression for the face amplitude

A f = (2j + f + 1)(2j - f + 1). (2.9)
It should be noted that, unlike the invariance under trivial subdivision of faces, the invariance of Z[κ] under subdivision of an edge does not follow from the invariance of the spin networks under subdivision of an edge. We feel, however, that it is a natural invariance to demand, not only because this way one is able to fix both face-and edgeamplitudes in terms of the vertex amplitude, but also because the history of the spin network is not changed by this move, because the topology of the two-complex remains the same. It therefore seems natural that the spin foam amplitude, which contains information about the dynamics, remains the same as well. It should also be noted that the two moves described above generate all subdivisions of the two-complex which do not change the history of the spin network, i.e. which are trivial subdivisions from the point of view of the dynamics.

Note finally that the invariance of the Spin foam amplitude under subdivision of an edge results in a formula for the amplitude which is homogenous in the scaling of the intertwiner of degree zero. The choice (2.7) is equivalent to a normalisation of the vertex amplitudes, and can therefore be absorbed into a redefinition of A v , see e.g. [START_REF] Barett | Asymptotic analysis of the EPRL four-simplex amplitude[END_REF] (which we, however, refrain from doing here). The choice of the face amplitude also seems natural from the point of view of BF theory. See however [START_REF] Bianchi | Face amplitude of spinfoam quantum gravity[END_REF] for a different amplitude, which in particular has the advantage of reducing the degree of divergence in the sum over spin foams (1.4).

Boundary amplitudes

In the following, we describe the boundary amplitudes B e , B v , which appear in the spin foam amplitude (1.3). The two-complex κ, which is embedded in M = Σ × [0, 1], induces boundary graphs γ i = κ ∩ Σ × {0} and γ f = κ ∩ Σ × {1}. The spin foams κ we are considering are all such that the initial and final dynamics of the spin network, which is described by the two-complex, is trivial. In more technical terms this means that there is an > 0 such that

κ ∩ Σ × [0, ] = γ i × [0, ] κ ∩ Σ × [1 -, 1] = γ f × [1 -, 1]
(2.10)

The equality signs in (2.10) are meant to denote diffeomorphic equivalence. This condition on the topology of the two-complex is not as restrictive as one might initially think. In fact it is generic and amounts to the condition that cutting of a two-complex is only allowed when not cutting directly through an inner vertex. The condition (2.10) also ensures that there is no nontrivial dynamics which is happening exactly at the initial or final boundary. In particular, let κ 1 and κ 2 be two twocomplexes with γ 1,f = γ 2,i = γ, then there is an obvious way to concatenate the two two-complexes by glueing them together at γ. Then (2.10) ensures that in this process no nontrivial vertices appear. The result of this glueing process is denoted by κ := κ 1 κ 2 .

Since the amplitude Z[κ] can be thought of exp[iS EH [g µν ]], where κ, j ± f , ι e is interpreted as an appropriate discretisation of g µν and the (Euclidean) Einstein-Hilbert is additive if the appropriate boundary terms are included, it appears natural that one should demand

Z[κ 1 κ 2 ] = Z[κ 1 ] Z[κ 2 ] ( 2 . 1 1 )
and this can be achieved, as we will show now, by an according choice for the boundary amplitudes B e , B v in (1.3).

There are several possibilities to choose the boundary amplitudes to ensure (2.11). One immediate choice is the following: For any vertex v in the boundary, denote the edge that ends at v by e v . Similarly, denote for every boundary edge e the face in κ that ends in it by f e . 12 Then, if one defines

B v := A ev -α B e := A fe α-1 (2.12)
for an α ∈ [0, 1], it is straightforward to verify (2.11). Note that for nonzero intertwiner ι e , the amplitudes A e , A f are always positive, hence (2.12) is well-defined.

For the rest of the article, we adopt the symmetric choice α = 1 2 , i.e.

B v := 1 A ev B e := 1 A fe (2.13)
which in particular leads to a more symmetric behaviour of the amplitudes under timereversal.

Consistent deformations

In this section we will demonstrate that the spin foam amplitudes Z[κ] defined by the EPRL model possess a large symmetry, if one adapts the choices for the amplitudes from the last chapter. This symmetry will help us in proving the core statements about the properties of the physical inner product (1.4) using the EPRL amplitude. The first, trivially notable symmetry is that of Diff(M) + , i.e. the orientation-preserving diffeomorphisms of M, since the actual value of the amplitude is only constructed using the combinatorial information of the embedding κ ⊂ M, e.g. which faces are connected to which edges. The change of the amplitude under an orientation-changing diffeomorphism is discussed in [START_REF] Kaminski | Spin-Foams for All Loop Quantum Gravity[END_REF].

In this chapter we will, however, turn to a larger invariance of Z[κ] than just diffeomorphisms. Provided we choose the amplitudes in (1.3) according to the last chapter, we will show that the spin foam amplitude is invariant under something which we term a consistent deformation, which does not only involve deformations of two-complexes which are induced by diffeomorphisms of the embedding manifold M, but in which e.g. some faces of κ are allowed to pass through each other in a controlled way. Since by letting the surfaces of κ intersect each other, one is, of course, creating new vertices, edges and faces. One therefore has to be careful what happens to e.g. the intertwiners in this case.

We will have a look at two different "moves" that incorporate non-diffeomorphic deformations of the spin foam, and which leave the spin foam amplitude invariant. We will subsequently discuss the kind of deformations that are generated by these moves and trivial subdivisions of the spin foam.

Pulling vertices apart

The first move we are going to consider will be able to subdivide an inner vertex into two vertices. Consider an inner vertex v of a spin foam that has the property that the corresponding vertex spin network γ v is disconnected in S 3 , i.e. γ v = γ 1 ∪ γ 2 . Let furthermore γ 1 and γ 2 not be linked, i.e. one can, by a diffeomorphism of S 3 , move e.g. γ 1 to the upper hemisphere and γ 2 to the lower hemisphere of S 3 . If this can be achieved, then one can place a three-dimensional hypersurface H in a neighbourhood of v ∈ M which intersects κ exactly at v (and the embedded sphere S 3 at the equator, see figure 7). It is clear that H divides a neighbourhood of v of κ into two parts which only intersect at v. The deformation we are defining consists of separating the two parts. 13 Of course, as soon as κ is separated that way, the combinatorics of the resulting κ is different: κ has two vertices v 1 , v 2 where κ had only one, namely v. The number of edges and faces under this deformation stay the same, as well as the labels on them. The amplitude Z[κ ] can hence be computed, and it is not hard to see that

v v 1 v 2 γ v γ v 1 γ v 2 ⇒
A v 1 A v 2 = A v (3.1)
since the evaluation of a disconnected spin network is just the product of the evaluation of each of its components. It follows immediately that

Z[κ] = Z[κ ]. (3.2)
This "pulling apart" of the vertex v was only possible since γ v was disconnected, so that its vertex amplitude A v factorises. Of course, the move can be reversed by choosing two vertices v 1 , v 2 in κ14 and moving them close to each other, so that they eventually overlap. This deformations, of course, needs to be done such that no other part of κ intersects with another part. The resulting vertex v has a vertex graph γ v which is obviously disconnected so that its vertex amplitude is the product of the amplitudes of v 1 and v 2 . Note that this move is possible independently of the labels on κ. Whether this move can be performed on a vertex or not depends only on the embedding and topology of the two-complex κ. Furthermore, it is noteworthy that the move can be generalised to vertices v whose vertex graph γ v have more than two connected components. Then v can, by successive application of the above move, be separated into as many vertices as there are connected components in γ v . The resulting spin foam κ has unchanged amplitude, and κ does not (modulo diffeomorphism) depend on the order in which the move above is applied.

Pulling edges apart

The second move (together with its inverse) which we are going to present is the higherdimensional equivalent of pulling the vertices v apart which have a factorising vertex amplitude. Consider an edge e in κ with ingoing faces f i,1 , . . . f i,n , and outgoing faces f o,1 , . . . , f o,m . Suppose that the associated intertwiner, which is an invariant map

ι e : n I=1 V j + i,I ,j - i,I -→ m J=1 V j + o,J ,j - o,J (3.3)
factorises in the following sense: The incoming and outgoing faces can each be separated into two sets such that there are EPRL intertwiners

ι 1 : k I=1 V j + i,I ,j - i,I -→ l J=1 V j + o,J ,j - o,J ι 2 : n I=k+1 V j + i,I ,j - i,I -→ m J=l+1 V j + o,J ,j - o,J such that ι e = ι 1 ⊗ ι 2 . (3.4)
Then it is possible to deform the two-complex by splitting the edge e into two edges e 1 and e 2 , each of which have the same starting-and ending vertices as e (see figure 8).

All the faces f i,I , f i,J with I = 1, . . . , k, J = 1, . . . l are attached to e 1 , and those with I = k + 1, . . . , n, J = l + 1, . . . m are attached to e 2 . Note that the resulting spin foam two complex has two edges where the original two-complex had only one, while the number of faces and vertices has not changed. We equip the edge e 1 with the intertwiner ι 1 , the edge e 2 with ι 2 , and call the resulting spin foam κ .

v i v i v f v f ι 1 ι 2 ι 1 ⊗ ι 2
ι e ι † e = (ι 1 ι † 1 ) ⊗ (ι 2 ι † 2 )
, we find that

A e 1 A e 2 = 1 tr(ι 1 ι † 1 ) 1 tr(ι 2 ι † 2 ) = 1 tr(ι e ι † e )
= A e (3.5)

If the beginning-and ending vertex of e are v i and v f respectively, then the vertex amplitudes A v i and A v f are a priori different, because the vertex graphs γ v i and γ v f have changed as well. In both graphs there is a vertex w i and w f respectively, which corresponds to the edge e. After the "splitting apart", in each vertex graph this vertex w i,f is replaced by two vertices w 1 i,f , w 2 i,f . Hence the "pulling apart" of the edge e in κ corresponds to pulling apart the two vertices w i and w f in the neighbouring graphs γ i and γ f .

w i w 1 i w 2 i S 3 S 3 γ v i γ v i
w f of ψ v f ).
Because of the definition of the vertex amplitude (2.6), the vertex amplitudes itself are invariant under this operation, e.g.

A v i (j ± f , . . . , ι 1 ⊗ ι 2 , . . .) = A v i (j ± f , . . . , ι 1 , ι 2 , . . .) ( 3 . 6 )
because it is computed simply by contracting the indices of intertwiners on edges meeting in v i . Due to the orientation of the edge e, the intertwiner associated to

w f is ι † e = ι † 1 ⊗ ι † 2 ,
which is also of product form, and hence for A v f an equation similar to (3.6) holds. As a consequence, the whole spin foam amplitude is left invariant under the move described above:

Z[κ] = Z[κ ]. (3.7)
Note that, unlike in three dimensions (as e.g. figure 8 suggests), in four dimensions there may be several ways of pulling the edge apart into two edges. This can be seen most easily by realising that there are several ways of e.g. pulling apart the vertex w i in the vertex graph γ v i into w 1 i and w 2 i in figure 9: Instead of just separating the two vertices to produce a planar γ v i , one could have separated w i into two vertices w 1 i and w 2 i , producing a graph that has a nontrivial knotting class (see figure 10). If one chooses an according way of pulling the vertex w f apart, this defines a different way of pulling apart the edge e. The fact that there are different ways of pulling edges apart differing by knotting classes will play a paramount rôle later. Of course, we allow not only to "pull apart" some of the edges, but also allow for the the inverse move, which corresponds to "merging" two edges e 1 and e 2 in a spin foam κ which have the same beginning-and ending vertices. The resulting edge e then is equipped with the tensor product of the intertwiners ι e := ι 1 ⊗ ι 2 . The resulting spin foam κ then evidently has the same amplitude as κ, i.e.

w i w 1 i w 2 i S 3 S 3 γ v i γ v i
Z[κ ] = Z[κ].
It should be emphasized again that it is a question of the topology of the two-complex whether or not one can merge two edges into one, just as it is a question of topology whether or not one can merge or separate two vertices. In particular, the availability of this move does not depend on the spins j f or intertwiners ι 1 , ι 2 of κ. However, since by merging the resulting intertwiner ι e necessarily factorises ι e = ι 1 ⊗ ι 2 , and not every intertwiner is of this form, the inverse of this move, i.e. pulling edges apart, can only be performed if the intertwiner ι e factorises -unlike all the other moves it therefore depends on the actual data on κ, and not just on its topology.

Consistent deformations as equivalence relation

The two "moves" (and their inverses) that have been described in the last sections generate, together with trivial subdivisions of faces and edges (and their inverses), an equivalence relation of spin foams κ. They can be viewed as continuous deformations of the two-complex such that some bits of the two-complex are allowed to be put together or split apart. We will see later how one can use this in order to also allow faces to pass through each other in a controlled way. We term these "consistent deformations". Since the spin foam amplitude Z[κ] does not change under a consistent deformation, one can restrict the sum in the physical inner product (1.4) over the set of equivalence classes, where κ ∼ κ iff κ and κ are consistent deformations of each other:

ψ f |ψ i phys = κ : ψ i κ →ψ f Z[κ] = [κ] : ψ i κ →ψ f Z[κ]. (3.8)
It should be noted that inequivalent spin foams κ might have different "orbit size" under the action of consistent deformations, and one might define the inner product (3.8) with an additional factor F [κ] depending only on the equivalence class [κ], measuring the relative size of these orbits. Different choices for F all result in diffeomorphism-invariant physical inner products, so there is no a priori way of choosing one. 15 Since -even when restricting the sum to equivalence classes under consistent deformations -the sum (3.8) diverges anyway, we do not bother too much about the actual choice of F , set F [κ] ≡ 1 from now on, and remark that all the following statements in this article remain valid for "sensible" choices of F .

The physical Hilbert space

The "knotting spin foam"

In this section we will come to a crucial ingredient for our final result: We will consider two spin network functions ψ i , ψ f living on graphs γ i and γ f which have the same combinatorics (i.e. adjacency matrix), but different knotting classes. The goal in this section is to compute the spin foam amplitude Z[κ 0 ] for a spin-foam κ 0 : ψ i → ψ f which mediates between the two, and is essentially "unknotting" γ i into γ f . It is not hard to see that all these unknottings can be achieved by a successive application of the "move" depicted in figure 11.

Without loss of generality we assume that ψ i and ψ f are situated on graphs consisting of two loops, where in ψ f the two loops are linked, while in ψ i they are not linked. The transition between the two is given by the spin foam κ 0 , which can be described as follows:

The two loops start at ψ i as two linked loops, approach each other, meet at their respective vertex, and pass through each other, eventually ending linked as ψ f (see figure 12). The resulting spin foam κ 0 has one internal vertex v, which is the point in which the two loops meet. We now show that Z[κ 0 ] = 1, which will be the key point in our analysis. First we note that κ 0 has two internal faces, four internal edges (each going from one of the four boundary vertices to v) and one internal vertex. Using the definition (2.13), one easily sees that the face amplitudes and the boundary edge amplitudes cancel. Furthermore the boundary vertex amplitudes B v cancel two of the four edge amplitudes A e . If we denote the edges describing the history of the vertices of ψ i until they meet at v by e 1 and e 2 , we obtain

-→ k i k i k f k f ψ i ψ f
k k l l v ψ i ψ f
Z[κ 0 ] = A e 1 A e 2 A v = 1 (2j + k + 1)(2j - k + 1) 1 (2j + l + 1)(2j - l + 1) A v (4.1)
where 2j ± k = |1±γ|k and 2j ± l = |1±γ|l. Here we have used (2.7) and (2.13), remembering that the edges e 1 , e 2 carry the identity intertwiners. 16 16 To be precise, since the intertwiners on the boundary vertices carry the identity intertwiners, e.g. ι = id V k , the edges of

κ carry ι = φ(ι) = 2k+1 (2j + k +1)(2j - k +1) id V j + k ,j - k
, which is a nontrivial multiple of the identity intertwiner. However, due to the definition (2.7) of the edge amplitude, the spin foam is not sensitive to this factor.

All that remains to be done now is to compute the vertex amplitude for the one internal vertex v in κ 0 , which is the point where the two edges are crossing each other in order to "entangle the knot". In order to do so, one needs to intersect the spin foam with a small three-sphere around the vertex v, to obtain a graph γ v embedded in the three-sphere. It is not hard to see that γ v consists of two links, i.e. two loops, each with two vertices and two edges (see figure 13) The evaluation of this spin network function 17 can then be done, and one obtains

j ± k j ± l S 3 γ v
A v = (2j + k + 1)(2j - k + 1)(2j + l + 1)(2j - l + 1) (4.2)
which, with (4.1), results in

Z[κ 0 ] = 1 (4.3)

The spin foam sum

We have shown that the amplitude for the "unknotting" spin foam κ 0 :

ψ i → ψ f is Z[κ 0 ] = 1
, which is equal to the trivial amplitude. We now show that the spin foam κ 0 κ -1 0 : ψ i → ψ i is a consistent deformation of the identity foam id : ψ i → ψ i . This can be seen as follows:

Starting from the spin foam furthest to the left in figure 14, let the associated intertwiner to the edge between the two internal vertices be ι = ι 1 ⊗ ι 2 , where ι 1 is the identity map on V j ± k , and ι 2 the identity map on V j ± l . As discussed in the previous section, there are two ways of pulling the internal edge connecting the two internal vertices apart, differing by a knotting class of the resulting vertex amplitudes. One results immediately in κ 0 κ -1 0 : ψ i → ψ i , as one can readily see. The other one results in two "tubes" (the histories of the two loops) which are connected at the two internal vertices. Pulling the foam apart at these two vertices and then removing the two vertices (which have become trivial), one arrives at the identity foam id :

ψ i → ψ i
As a result, for any kinematical state ϕ we can write down a bijection Φ between the set of spin foams κ : ϕ → ψ i modulo consistent deformations and spin foams κ :

ϕ → ψ f ⇔ ⇔ κ 0 κ -1 0 id ⇔ ⇔ Figure 14:
The spin foams id and κ 0 κ -1 0 can be consistently deformed into each other by application of the pulling moves described above (and their inverses), and removing/adding trivial vertices. modulo consistent deformations by

Φ : [κ] -→ [κκ 0 ] Φ -1 : [κ ] -→ [κ κ -1 0 ] ( 4 . 4 )
That Φ is not only well-defined but also a bijection follows from κ 0 κ -1 0 ∼ id, i.e. the two spin foams are consistent deformations of each other. With the functorial property (2.11) and (4.3) one finds that

Z[κ] = Z[Φκ] ( 4 . 5 )
Therefore, for each (equivalence class of a) spin foam κ : ϕ → ψ f there is exactly one (equivalence class of a) spin foam κ : ϕ → ψ i , and the spin foam amplitude is the same for each of the two. It follows that

ψ i |ϕ phys = [κ]:ϕ→ψ i Z[κ] = Φ[κ]:ϕ→ψ f Z[κ] = Φ[κ]:ϕ→ψ f Z[Φκ] = κ :ϕ→ψ f Z[κ ] ( 4 . 6 ) = ψ f |ϕ phys
It should be noted that this property follows entirely from the symmetries of the amplitude Z[κ], and is independent of the actual way in which the vastly sum κ over spin foams is realised, as long as it is realised such that the symmetries of Z[κ] are respected.

Since ϕ was arbitrary, from (4.6) it follows immediately that the difference ψ i -ψ f has zero physical norm, i.e. that ψ i and ψ f are mapped to the same physical state in the sense of the rigging map procedure (1.1).

The calculations presented here were for the case of two loops interlinking each other. It should, however, have become clear that the calculations would have lead to the same result if the two edges passing through each other would not have been closed to form loops, but would have been edges contained in a larger graph, as long as the rest of the graph had 'stayed constant' throughout the evolution. In this case, where κ 0 would have described the evolution of a large graph, where two edges gradually pass through each other, on the right hand side of (4.1) the amplitudes of the remaining bits of the twocomplex would have been present, which would however have combined to 1 (since the trivial transition has unit amplitude, due to the choice of B v , B e ), leading to Z[κ 0 ] = 1 as well.

As a result, since any transition between two graphs γ i , γ f with the same combinatorics but different knotting classes can be performed by subsequent crossing of edges (where all one needs to do is trivially subdividing each pair of edges that shall "pass through" each other during the evolution described by the spin foam with a trivial vertex each), and because of (2.11), formula (4.6) also holds for states ψ i , ψ f situated on γ i , γ f . Hence, the states in the physical Hilbert space H phys do not contain any knotting classes of graphs.

Summary and Discussion

In this article we have investigated properties of the physical Hilbert space H phys as defined by the sum-over-spin foams procedure using the Euclidean EPRL amplitude. The result was that H phys does not contain any knotting information of the graphs in the following sense: The bona fide projector η : H kin → H phys maps states that have the same combinatorics but different knotting classes to the same physical state.

This result depends on several assumptions. The most visible assumption is that of the face-and edge amplitudes (2.7) and (2.8), as well as the boundary amplitudes (2.13). This choice was a natural consequence of demanding the invariance of the spin foam amplitude Z[κ] under trivial subdivisions (figures 5 and 6), and the product property (2.11). The invariance of Z[κ] under subdividing an edge with a vertex (without conditions on the vertex, since the vertices do not carry any data) does, strictly speaking, not follow from the canonical framework. However, it appears to be natural in the light of interpreting a spin foam as a history of a spin network. If one believes that the whole dynamical information is captured by the evolution of the graph and the representation data distributed on it, then that dynamical information is not changed under subdividing an edge with a vertex. Hence the amplitude Z[κ] should not be sensitive to this change.

The result we have obtained rested on a special property of the EPRL amplitude, which leads to an invariance of the spin foam amplitude Z[κ] under a "consistent deformation" of κ. It should be noted that we do not propose such a consistent deformation to correspond to a physical symmetry, or be related to diffeomorphisms (other than diffeomorphisms being a subset of the consistent deformations). In particular, the consistent deformations do not form a group acting on the set of spin foams in a straightforward way 18 . Nevertheless, the invariance is a fact, and it allows to severely reduce the summation over all spin foams to all spin foams modulo consistent deformations. Hence, as a convenient tool it allows to prove statements about the physical Hilbert space.

The central reason for the independence of the physical Hilbert space of knottings is that the "unknotting" spin foam κ 0 has unit amplitude, i.e. Z[κ 0 ] = 1, and the ultimate reason for this is the definition of the vertex amplitude itself. The value of A v depends only on the evaluation of a spin network embedded in S 3 , and this evaluation is independent of the knotting of the embedded graph. It is just sensitive to its combinatorics. It should be noted that this depends of the generalisation to the EPRL amplitude given in [START_REF] Kaminski | Spin-Foams for All Loop Quantum Gravity[END_REF], which was until then only defined for the case in which the corresponding graph in S 3 had a trivial knotting anyway. It appears to be possible to change the generalisation of the amplitude from [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] to general spin foams by changing its sensitivity to the knotting of the vertex graph, e.g. by multiplying the vertex amplitude by a nontrivial function of knotting invariants, or by equipping the vertices of the spin foam itself with "knotting charges", which, for the sake of consistency, should be trivial in the case of the four-simplex amplitude.

Although the physical Hilbert space does not contain any knotting information of the graphs, it should be emphasized that this does not mean that the theory is insensitive to knotting within the space-time four-manifold M = Σ × [0, 1]! The latter is a question concerning the expectation values of some (four-dimensional) observables which are computed in the theory. The shape of the boundary Hilbert space is not necessarily making any statements of what possible expectation values arise in the theory, and whether they differ for different observables (e.g. corresponding to differently knotted surfaces in M) or not. A prominent example which illustrates this is three-dimensional gravity as described by the Ponzano Regge model. The boundary Hilbert space, in the Loop formulation, is given by planar spin networks embedded in the two-dimensional boundary manifold [START_REF] Noui | Three dimensional loop quantum gravity: physical scalar product and spin foam models[END_REF], and therefore contains no knotting information, not even on the kinematical level. The expectation value of observables corresponding to curves embedded in the three manifold (e.g. Wilson lines) however do depend on the knotting of the embedding [START_REF] Barrett | The Ponzano-Regge model[END_REF], being related to well-known knot invariants. It is very well possible that this also happens in the fourdimensional theory. Before one has made a more precise sense of the sum-over-spin foams and has a good idea of what observables in the theory should be, it will nevertheless be hard to make any certain statements about this issue.

1 Confidential 10 Figure 1 :

 1101 Figure 1: A spin network function consists of an embedded graph γ ⊂ Σ, spins k e along its edges and interwiners ι v along its vertices.

3 Figure 2 :

 32 Figure 2: An edge e with bordered by three faces. The intertwiner ι e associated to it maps ι e : V j +

Figure 3 :

 3 Figure 3: Diagrammatic representation of the map φ. The lines carry SU (2) representations, the box denotes integration over SU(2) × SU(2), and each vertex carries an SU(2)intertwiner. All face are outgoing, so there is no additional index out, in to distinguish them.

Figure 6 :

 6 Figure 6: Trivial subdivision of a face with an edge

Figure 7 :

 7 Figure 7: A vertex v in a spin foam which has a disconnected vertex graph γ v can be pulled apart into two vertices v 1 and v 2

Figure 8 :

 8 Figure 8: Pulling the edge e with ι e = ι 1 ⊗ ι 2 apart into two edges e 1 , e 2 with ι 1 on one and ι 2 on the other.

Figure 9 :

 9 Figure 9: Pulling the edge e apart corresponds to pulling apart the corresponding vertex w i of the vertex function ψ v i (and w f of ψ v f ).

Figure 10 :

 10 Figure 10: Different way of pulling the vertex w i apart into two vertices w 1 i , w 2 i , resulting in a vertex graph with different knotting class in S 3 .

Figure 11 :

 11 Figure 11: Graphs with different knotting classes can be related to each other by subsequently crossing lines.

Figure 12 :

 12 Figure 12: We consider the example of ψ i consisting of two loops (with spins k and l), and ψ f consisting of the same loops which are linked with each other.

Figure 13 :

 13 Figure 13: The neighbouring spin network, situated on the graph γ v = S 3 ∩ κ 0 .

The mathematical concept as abstract state sum model is much older (see e.g. literature in[START_REF] Barrett | A Lorentzian Signature Model for Quantum General Relativity[END_REF][START_REF] Barrett | The Ponzano-Regge model[END_REF]), but differs slightly from the way spin foams are understood in this article.

It should furthermore be mentioned that there is a vast arsenal of technical tools for performing calculations in spin foam models built upon the definition of certain coherent states given in[START_REF] Livine | Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity[END_REF], which also allows for an interpretation of the intertwiners in terms of four-dimensional geometry, see e.g.[START_REF] Freidel | Twisted geometries: A geometric parametrisation of SU(2) phase space[END_REF][START_REF] Freidel | U(N) Coherent States for Loop Quantum Gravity[END_REF].

Even for a fixed two-complex the sum over representation labels diverges. See e.g.[START_REF] Bojowald | Spin Foam Quantization and Anomalies[END_REF].

The orientation of the individual cells provides a natural pairing between the different tensors associated to the cells. In the numerous figures we often refrain from adding arrows indicating orientations whenever possible in order not to overburden the reader's eye.

[START_REF] Baez | An Introduction to Spin Foam Models of Quantum Gravity and BF Theory[END_REF] It is, up to this point, not clear which is the most general category of two-complexes one can work with. It is sure that piecewise-analytic works, just as piecewise analytic graphs in LQG are a convenient choice. Whereas LQG can also be defined with smooth graphs (which is much more involved, however, and spin network functions do not form a basis for 'smooth LQG'), it is not known if the EPRL model can be defined for, say, CW

complexes.[START_REF] Perez | The spin-foam-representation of loop quantum gravity[END_REF] The 0-cells ("vertices") are equipped with the +-orientation by default.

Which is, strictly speaking, only true if all faces are outgoing. See e.g.[START_REF] Barett | Asymptotic analysis of the EPRL four-simplex amplitude[END_REF] for an introduction to the diagrammatic calculus, which allows to avoid writing down expressions such as (2.4).

It has been shown[START_REF] Kaminski | The EPRL intertwiners and corrected partition function[END_REF] that φ is no isometry, which has lead to discussion about the correct way in which to sum over the intertwiner data in(1.4). Since we perform calculations with single amplitudes only and leave the explicit form of the sum over spin foams in(1.4) open, the results of this article are independent of this question.

Note that one doesn't have to subdivide the faces touching e as well.

By n-bridge graph we mean a graph consisting of two vertices v 1 , v 2 , and n edges each of which starts at v 1 and ends at v 2 .

See also[START_REF] Bojowald | Spin Foam Quantization and Anomalies[END_REF] for a more refined argument for this kind of invariance, considering the anomalies in the sum(1.4).

Note that this is only well-defined if the topology of the two-complex satisfies (2.10).

This separation can be achieved by e.g. defining a homotopy of the two-complex in the piecewise analytic category.

Which can, in M, be moved close to each other by a diffeomorphism of M.

In defining the spatially-diffeo invariant Hilbert space of canonical LQG, a similar freedom of choosing such a factor exists, see e.g.[START_REF] Sahlmann | Exploring the diffeomorphism invariant Hilbert space of a scalar field[END_REF] for a discussion.

Which does not notice the fact that the two loops form a nontrivial knot -which is the central reason for the results of this article.

This is because e.g. "splitting a vertex" is not applicable to all spin foams, but only to those which have a topology that allows for this move to occur, i.e. which contain vertices v with disconnected vertex graphs γ v . However, it is possible to construct a groupoid with the spin foams as objects, and arrows between spin foams whenever there is a consistent deformation between them.

Acknowledgements

It is a pleasure to thank Jurek Lewandowski for invitations to Warsaw, and him, Wojtiech Kaminski and Frank Hellmann for enlightening discussions. The author furthermore would like to thank Bianca Dittrich for urging him to write this article.


Cédric Montero
email: cedric.montero@univ-montp2.fr

Joseph Gril

Clémentine Legeas

David G Hunt

Bruno Clair

Hygromechanical history and mechanosorptive creep

Keywords: clear wood, creep, mechanosorption, moisture content

An experiment was performed on Norway spruce to simplify our understanding of mechanosorptive creep. Two well-matched sets of specimens were subjected to the same humidity history but loaded in four-point bending at different times. One set was loaded dry, and thus experienced a considerable creep at first adsorption (the so-called ++ effect). The other set was then loaded wet and only exhibited, together with the dry-loaded set, the usual pattern of creep increase during drying (-effect) followed by a partial recovery during remoistening (+ effect). Both sets converged to the same response after a few cycles. The results confirm that a single type of mechanosorption, combined with strain-dependent hygroexpansion and humidity-dependent viscoelascity, is sufficient to account for all observed features of longitudinal creep of wood at ambient temperature.

Introduction

During the lifetime of a timber structure, wood is subjected to continuous changes of humidity. Like other polymeric and hygroscopic materials, loaded wood creeps with time and the amount of creep is higher under the more humid conditions. Moreover, as evidenced by [START_REF] Armstrong | Effect of moisture changes on creep in wood[END_REF], the adsorption-desorption process, associated with local variations of moisture content (MC) may advance the creep considerably. This particular feature of wood material, a coupling between mechanical response of wood and adsorption-desorption processes [START_REF] Armstrong | Influence of moisture changes on deformation of wood under stress[END_REF][START_REF] Armstrong | The effect of moisture content changes on the deformation of wood under stress[END_REF] was named mechanosorptive effect by [START_REF] Grossman | Requirements for a model that exhibits mechano-sorptive behaviour[END_REF]. It was given various physical interpretations [START_REF] Boyd | An anatomical explanation for visco-elastic and mechano-sorptive creep in wood, and effects of loading rate on strength[END_REF][START_REF] Back | Transient effects of moisture sorption on the strength properties of paper and wood-based materials[END_REF][START_REF] Mukudai | Evaluation on non-linear viscoelastic bending deflection of wood[END_REF][START_REF] Bazant | Constitutive equation of wood at variable humidity and temperature[END_REF][START_REF] Hoffmeyer | Mechano-sorptive creep mechanism of wood in compressive and bending[END_REF][START_REF] Navi | Micromechanics of creep and relaxation[END_REF] although to date no clear consensus was reached.

It was found that during a bending test in changing humidity, the first increase of MC and any decrease cause an increase in deflection, whereas subsequent adsorption in the same MC range produces little creep or some recovery. According to these statements, the first adsorption of water has a special influence on the deflection. A first approach to the quantification of mechanosorptive creep was made by [START_REF] Ranta-Maunus | The viscoelasticity of wood at varying moisture content[END_REF] based on plywood creep data. This author defined "hygroviscoelastic constants" to model the evolution of elastic compliance J induced by change of MC. Three notations were proposed to designate the derivative of J with respect to MC during first adsorption to a higher MC than has been experienced since the start of loading a ++ , any desorption a -, or subsequent adsorption within the MC range already experienced a + (Figure 1a). Later, [START_REF] Hunt | Creep trajectories for beech during moisture changes under load[END_REF] observed that the values of these constants depend on strain and he proposed a limit to the amount of mechanosorptive creep, approached asymptotically by repeated humidity cycles. [START_REF] Hunt | Stable-state creep limit of softwood[END_REF] proposed a simple procedure, based on overloading followed by one humidity cycle, to obtain rapidly the creep limit. The authors showed evidence of an asymptotic trend of decreasing compliance with increasing MC, and described the compliance increase during drying as "pseudo-creep" and the decrease during humidifying as "pseudorecovery" (Figure 1b). They interpreted this phenomenon as a stress-dependent (or straindependent) coefficient of moisture expansion. A practical consequence of this finding was to allow desorption (-effect) and adsorption (+ effect) to be considered as equally contributing to mechanosorption, which simplified greatly the numerical models that were proposed later [START_REF] Martensson | Tensile behaviour of hardboard under combined mechanical and moisture loading[END_REF][START_REF] Salin | Numerical prediction of checking during timber drying and a new mechano-sorptive creep model[END_REF][START_REF] Mohager | Long term bending creep of wood in cycling relative humidity[END_REF][START_REF] Hanhijärvi | Deformation kinetics based rheological model for the time-dependent and moisture induced deformation of wood[END_REF]. According to these models, the mechanosorptive strain depends on the sum of the absolute value of MC increments, the "cumulative moisture content change". Experimental results in longitudinal tension [START_REF] Muszyński | Hygro-mechanical behavior of red spruce in tension parallel to the grain[END_REF]) confirmed the applicability of this assumption to describe the creep response to a succession of humidity cycles under a constant tensile load, thanks to the fitting of the mechanosorptive strains to a single exponential curve. As the test started with a humidification period, these data suggested that the first wetting did not contribute differently from subsequent swelling, which thus contradicts the existence of a ++ effect. Already in 1984, Hunt suggested that a part of the strong increase of creep when "the maximum humidity since loading" is exceeded was caused by the normal increase of the viscoelastic compliance with humidity. Such analysis relies strongly on the possibility to separate a mechanosorptive contribution from the total strain. [START_REF] Hanhijärvi | Experimental indication of interaction between viscoelastic and mechano-sorptive creep[END_REF] tried to elucidate the interaction between viscoelastic creep (at constant humidity) and mechanosorptive creep (resulting from MC changes). Although they did observe an interaction based on the analysis of the kinetics, their procedure did not allow to conclude about the existence of the ++ effect because they loaded their specimen in the wet state so they only involved the "usual" type of mechanosorption.

The results presented in this paper are aiming at answering the question of the existence or non-existence of the ++ effect when wood creeps along the fibres. For that purpose, it is necessary to compare the mechanosorptive creep of wood submitted to humidity increase after being loaded dry, with that of wood directly loaded wet. In the first case, ++ effect is expected to occur while in the second, it is not. After this initial difference, both specimens should experience the same hygromechanical loading so that their final response can be compared. Bending was preferred to tensile load in this work in order to ensure no experimental artefacts even if the interpretation of a bending test is tricky especially in transient adsorption-desorption phases, as pointed out by [START_REF] Muszyński | Hygro-mechanical behavior of red spruce in tension parallel to the grain[END_REF], and so special precautions were taken.

Materials and method

Wood material

Clear Norway spruce specimens were selected. Specimen size: 100 mm (long.) x 10 mm (rad.) x 2 mm (tang.) (Figure 2a). The ring width was 2.3 mm, so that the cross-section contained about 4 rings loaded in parallel. Specimen selection A total of 18 specimens paired in two sets were required for this test. As the objective of the study was to compare 2 groups, a careful matching was necessary. 80 specimens were cut and their structural, hygroscopic and mechanical properties were measured by various non-destructive methods (Table 1). Average microfibril angle (MFA) and grain angle at 2 positions per specimen were determined by X-ray diffraction. The MFA was calculated according to the calibration of [START_REF] Yamamoto | Method of determining the mean microfibril angle of wood over a wide-range by the improved Cave's method[END_REF] and the grain angle with the methodology described in [START_REF] Brémaud | Acoustical properties of wood in string instruments soundboards and tuned idiophones: biological and cultural diversity[END_REF].

The specific modulus and damping coefficient were measured under free-free vibration conditions [START_REF] Brémaud | Diversité des bois utilisés ou utilisables en facture d'instruments de musique[END_REF]. Dimensional changes from 70 to 92% RH and from 92 to 50% RH were measured in the tangential direction with a calliper. (1) (°) 0.0 2.4 0.6 (0.5) Density

(2) (g cm -3 ) 0.397 0.532 0.429 (0.016) Specific modulus

(2) (GPa) 24.9 31.0 28.1 (1.6) tan δ L

(2) (‰) 5.6 8.8 6.7 (0.6) α T 70-92%RH

(3) (%/%) 0.330 0.472 0.387 (0.032) α T 92-50%RH

(3) (%/%) 0.289 0.516 0.410 (0.042)

(1) X-ray diffraction (2) estimation from vibrational properties at T=22°C, 44%RH, (3) hygroscopic coefficient in tangential direction for swelling (from 70 to 92% RH, T=20°C) and shrinkage (from 92 to 50% RH, T=20°C).

Previous experiments have shown that a close positioning of the specimen within the original log is not sufficient to ensure correct matching. The MFA is presented for information but was not used to select specimens because the measurement performed was clearly affected by the position of the X-ray beam, both in earlywood and latewood, where MFA differs [START_REF] Herman | Growth rate effects on intra-ring and interring trajectories of microfibril angle in Norway spruce (Picea abies)[END_REF]. To guarantee a good matching between the two groups, a selection procedure by specimen pairing was developed. A first selection of 55 specimens was done to remove specimens with local disturbances of grain angle or inappropriate dimensions. The selection was made in order to minimize, for each property, the difference of average and standard deviation (StD) between both sets. The values of each set for each measured property are listed in Table 2. The small difference in average and StD of specimen properties between each group attests the quality of the matching.

Creep equipment and measurements

Four-point bending devices with 40 mm inner span and 80 mm outer span were used. A serial arrangement with a pair of 2 specimens was designed to increase the repeatability in the same environmental conditions (Figure 2b). Two groups of 6 specimens (labelled "dry-loaded set" and "wet-loaded set") and one group of 2 specimens per set (labelled "unloaded set") were placed in a controlled environment produced by a regulated climatic chamber (T ± 0.35°C, RH ± 2%). On each of these specimens, the strain ε + was measured on the tensile face and ε -on the compression face by means of strain gages with 3 wires and 5 mm grid length (KYOWA KFG-5-120-C1-11-L1M3R), fixed by gluing (Superglue Loctite Cyanoacrylate), and plugged into a data logger (two TML TDS-102-2) (figure 2c). One gage-less specimen per set was weighed occasionally to check any balance drift and to measure their transverse dimensions for the calculation of stress corrections. These operations were performed through a 5 cm diameter side opening and thus induced negligible perturbation of the climate of the chamber. The dimensions of the gauge support was 9.4 x 2.8 mm, so that in early transient phases of humidity changes, the perturbation due to the humidity barrier imposed by the gauge occurred at most on 1/3 of the specimen width.

Hygromechanical history

All creep results presented were measured in the longitudinal direction, and the experiments were performed at constant temperature T = 23°C ± 0.35 °C. The total duration of the test was 72 days. A preliminary acclimatizing phase of around 215 h without loading of the specimens was applied to release macroscopic residual strains. After this phase, the specimens of the dry-loaded set were loaded in the dry state (~10.4% MC). After a 5 h period of dry creep, the specimens were subjected to a humidity increase up to the maximum value allowed by the climatic chamber (92% RH) to experience the "++" situation. This increase has been performed with slow RH variation (from 40% RH to 92% RH in 12 h, ie 4% RH/h) to avoid a high MC gradient within the specimen. After this increase, the dry-loaded set was subjected to wet creep (92% RH) during 7 h. The second group of specimens from the wet-loaded set, located in the same chamber, was then loaded wet (~19% MC) and subjected during 11 days to viscoelastic creep only. From that time, both specimens underwent creep at high humidity (~19% MC) punctuated by drying and moistening cycles of total duration of 1 day each (at the rate of 4% RH/h). After 63 days of loading, both groups were unloaded and the recovery in the wet state was recorded.

The force was applied manually within 30 s with a lead mass designed to produce within the inner span a maximum stress σ 0 of about 5 MPa under wet conditions (Table 2). Bending tests up to failure performed under similar conditions in an air dry environment with two matched specimens, allowed evaluation of the stress level as 6% of the failing load, well below the observed elastic limit of 60%, although the non-linearity of mechanosorptive behaviour in bending was found to start at about 10-20% of the ultimate stress [START_REF] Hunt | Linearity and non-linearity in mechano-sorptive creep of softwood in compression and bending[END_REF].

Mechanical analysis

The interpretation of bending tests requires special precautions especially during transient adsorption-desorption conditions where the non-homogeneity of stress distribution interacts in a complex manner with that of moisture. The use of a pair of strain gages instead of a measure of deflection allows, to some extent, to counterbalance the drawbacks of bending as a method for observing the intrinsic behaviour of the material. As the specimens were loaded at low stress level, we assume the applicability of the Bernouilli hypothesis. The strain field through the thickness can be calculated by linear interpolation between ε + and ε -:

ε(y) = ε + Δε.y (1)
where ε = (ε + +ε -)/2 and Δε=(ε + -ε -)/2 are the mean and differential strain, respectively, and y is the position along thickness relative to neutral axis (Figure 2d). The mean strain ε reveals stress independent phenomena occurring during adsorption-desorption such as the hygrothermal expansion (swelling and shrinkage), but also allows to check the linearity of the mechanical response, especially differences between compression and tension. The differential strain Δε, half of the difference of strain between the two faces, is proportional to sample curvature. It mostly results from the action of the bending moment but can be also produced by differences of hygrothermal expansion between the tensile and compressive faces (causing pseudo creep/recovery, Figure 1b).

The stress is estimated by assuming a linear stress distribution through the thickness and refers to the maximum stress within the inner span. This is only a rough estimate during transient adsorption-desorption conditions, where the rigidity of the material and the strain distribution is heterogeneous. This difficulty exists for any loading mode but is exacerbated in bending due to the dominant contribution of the upper and lower faces. In equilibrium conditions, the validity of this estimate relies on the linearity of the mechanical response.

The effect of the change of transverse dimensions according to MC was considered in the stress determination, by means of the following equation:

( ) 0 0 1 T w w σ σ α = + - (2)
where α T is the moisture expansion coefficient measured on an unloaded specimen (α T = 0.375 %/%), w is the mean MC of the specimen and σ 0 , w 0 refer to the air-dry condition (w 0 =11% MC). This corrected stress can be referred to as the "nominal stress", by analogy with the situation of a tensile mechanosorptive creep test, where a constant load is applied on a varying cross-section. In such case, no correction would be required as the water causing the transverse expansion is not supposed to contribute to the rigidity. Equation ( 2) takes into account the additional lever arm effect resulting from the application of a bending moment.

Moisture content estimation

The moisture content evolution was estimated based on the average of mean strain ε of loaded specimens, revealing the swelling and shrinkage of specimen. For that purpose, two specimens per set were placed in similar creep devices to those of the loaded specimens but were not loaded during the test and one specimen per set was gauge-less and weighed occasionally. Loading at this stress level did not influence significantly the hygroscopic behaviour of the specimens (Figure 3a). Due to the slow RH changes imposed on the wood, the MC did not deviate substantially from an equilibrium value. The evolution of MC was described by a third order polynomial function between the average of mean strain of all loaded specimens and occasional MC measurements (Figure 3b). The rather high values of MC in the driest condition (40% R.H.) can be explained by the fact that the wood was never oven dried before the test [START_REF] Kollmann | Principles of wood science and technology I[END_REF].

Results

Experimental and analysed results

Figure 4 gives an overview of the experimental data of this study. The complete hygrothermal history is shown in Figure 4a. A constant force was applied first to the dry-loaded set at t = 215 h and then to the wet-loaded set at t = 239 h (Figure 4b). As the dry-loaded set was loaded earlier, its value of differential strain Δε(t) remains above that of the wet-loaded set (Figure 4c) even if a part of the gap was already visible before loading, due to a small humidity-induced curvature of the specimens ( Δε dry 0 = -2µε and Δε wet 0 = -16µε ). This artefact can be explained by minor misalignments of strain gages and/or of fibres for each specimen. The relevant parameter for the analysis of mechanosorption is the MC of the specimens. Figure 5a shows the MC estimate based on the correlation shown in Fig. 3b. The slight drop observed during the constant RH phases can be explained by the transient expansion observed during changes of relative humidity [START_REF] Hunt | Longitudinal shrinkage-moisture relations in softwood[END_REF]. Figure 5b illustrates that the nominal stress, calculated according to equation (2), drops by 4% from the dry to wet condition. Higher levels of nominal stress are retained temporarily during the humidity cycles. Thus such a test cannot be strictly considered as creep, which theoretically requires a constant stress. The variation is, however, small enough to allow the calculation of a compliance by dividing the differential strain by the nominal stress. The time-dependent compliance J for each loaded specimen is finally calculated as:

J = Δε -Δε 0 σ ( 3
)
where σ is calculated from equation ( 2), Δε the differential strain induced by the stress and Δε 0 the average of differential strain during the 10 min preceding loading at time t 0 . Figure 5c shows that, as expected, the initial compliance is much higher for the wet-loaded set than for the dry-loaded set.

During the first wetting, the dry-loaded set creeps markedly and remains above that of the wetloaded set during the entire test. As shown in Figure 5c, after the last cycle, both curves are almost superimposed. This result suggests that the response of wood to repeated moisture cycles does not depend, after a sufficient number of cycles, on the humidity condition at initial loading.

Within each set, the specimens gave very close results: the StD of the difference in compliance, during the whole loaded period, was 0.005 GPa -1 for each sets.

Discussion

The construction of creep trajectories was not straightforward in this study because the estimation of MC was based on the average of the mean strain of loaded specimens. This probably accounts, in Figure 6, for the loopy shape of cycle portions and overlapping of successive cycles. However, a precise MC estimation is not required to compare the creep of the two sets.

The first question to address is whether the compliance increase of the dry-loaded set during the first humidifying resulted from viscoelasticity only or involved mechanosorption. In the high humidity region, the mechanosorptive effect is not expected to dominate the "pure" creep that would have been observed at constant humidity. Indeed, through 260 h of creep in the wet condition, the compliance of the wet-loaded set increased from 0.1135 to 0.1386 GPa -1 , to compare to the level of 0.1487 GPa -1 already attained by the loaded-dry set. A study of the kinetics is required to evaluate how much additional creep the loaded-wet set would have experienced if it had been loaded as long as the loaded-dry set. Based on Hanhijarvi's method (Hanhijarvi and Hunt 1998), a linear extrapolation in logarithm of time since mechanical loading on the last 100 h shows an estimation of the level of wet-loaded set compliance of 0.1392 GPa -1 which leaves for the dryloaded set at least 0.0085 GPa -1 of compliance that can be only attributed to mechanosorption. Our results show that adsorption did induce mechanosorptive creep during the first adsorption of the dry-loaded set.

The second question concerns the nature of this mechanosorptive response. During the first wetting, the dry-loaded set was subjected to a typical ++ situation. If the concept of ++ effect applied, an additional creep should have been observed for that set, compared to the wet-loaded set that only experienced "subsequent" adsorption or desorption in the later cycles. The closeness of the response of both sets during the final humidity cycle, demonstrated clearly in Figure 7, suggests very strongly that the marked compliance increase of the dry-loaded set during the first wetting did not correspond to such an additional mechanosorption. It clearly demonstrates that the concept of ++ effect is non-relevant for wood: whatever creep occurred during the first wetting, it was not of a different nature from what happened during later cycles.

These conclusions rely strongly on the quality of the matching. Thanks to the combination method, we have sufficiently reduced the effect of wood variability. Since the possibility of a ++ effect has been ruled out by the preceding argument, the creep of the dry-loaded set during first adsorption can be only be interpreted as + effect. During a moisture cycle under load, as shown in Fig. 1b, pseudo-creep occurs during desorption and pseudo-recovery during adsorption. In addition to this reversible response, which has nothing to do with creep, a net increase of compliance is generally observed between successive cycles. It is usually difficult to separate the respective contributions of desorption and adsorption, as well as that of non-mechanosorptive creep, as we were able to do at the beginning of our creep test. The presented results support the hypothesis that adsorption contributes to mechanosorption, just like desorption. They do not provide guidance on whether both sorption directions contribute equally, as is considered in most models, or by a different amount. However, considering the thermodynamics of the sorption process, even a quantitative difference between the observed effects of adsorption and desorption would not necessarily mean that the mechanisms are intrinsically different.

Conclusions

Two sets of matched specimens loaded at different humidity levels exhibited the same total response after several humidity cycles under load. The results provided evidence that the first humidifying stage, in a creep test, does not involve the particular phenomenon designated as "++" effect. On the other hand, the marked creep increase during first wetting was partly explained by the increase of viscoelastic compliance, but required also the contribution of mechanosorption.

These findings give strong arguments to simplify our understanding of creep under changing humidity, which could now be reduced to three effects: (1) a single process of mechanosorptive creep, regardless of whether it is adsorption or desorption; (2) an added effect of pseudo-creep, based on the differential swelling/shrinkage rates which depend on strain level, and is therefore not creep at all; (3) viscoelastic creep, whose rate increases strongly at high humidities.

These three effects apply in combination, according to humidity and load history. This comparative approach has been supported by this and previous experiments and still needs to be confirmed by further experiment and modelling. The creep data obtained in this study will be also useful for the estimation of long-term wood creep, that involves the complex combination of pure creep and mechanosorption, as well as the quantification of pseudo-creep and progress towards a mechanosorptive creep limit.

Figure 1 :

 1 Figure 1: Hypothetical plots of compliance against moisture content (a) with constant values of a ++ , a + and a -(humidifying ---, dehumidifying -) (from Hunt 1984) and (b) toward a creep limit with strain dependency of hydroviscoelastic constant (from Hunt 1989)

Figure 2 :

 2 Figure 2: Log extraction and specimens orientation within the log (a). Experimental equipment of 4 point bending in serial arrangement (b) and gage dimensions (c). Scheme of strain field along specimen thickness (d) with indication of strain gauges measurements ε + and ε -, the mean strain ε and the differential strain Δε

Figure 3 :

 3 Figure 3: (a) Comparison of mean strain of loaded and unloaded specimens. (b) Determination of specimens MC. based on mean strain of loaded specimens at occasional MC measurements x with the relation w=aε 3 + bε 2 +cε+d (with a=38.22, b=10.68, c=8.45, d=10.42 and ε in ‰)

Fig 4 :

 4 Fig 4 : Overview of the experimental data: (a) hygrothermal environment ( ….. RH [%],# Temperature [°C]), (b): force [N], (c): differential strain [µε] vs. elapsed time [h]. Means of six specimens for each set dry-loaded set ---and wet-loaded set ---in (b) and (c) representations. Indications of differential strain induced by mechanical loading δε(t)=Δε(t)-Δε(t 0 -).

Figure 5 :

 5 Figure 5: Time evolution of adapted data: (a) moisture content calculated from mean strain of specimens;

Table 1 :

 1 Variability of specimen properties characterized using various non destructive methods

	Property		Min	Max	Mean (±St.D)
	Mean fibril angle	(1) (°)	9	17	13 (2)
	Grain angle

Table 2 :

 2 Variability of each set of specimens using results of non destructive methods after specimen selection (9 specimens per set). For explanations see also Table1.

	Property		Min	Dry-loaded set Max Mean (±StD)	Min	Wet-loaded set Max Mean (±StD)
	Density			0.395 0.450	0.421 (0.020)	0.394	0.441	0.423 (0.015)
	Specific modulus	(2) (GPa)	27.3	29.8	28.8 (1.0)	27.4	31.1	29.4 (1.3)
	(2) (‰) α T 70-92%RH tan δ L (3) (%/%) α T 92-50%RH (3) (%/%)	6.1 0.363 0.455 7.1 0.336 0.405	6.4 (0.4) 0.406 (0.036) 0.372 (0.025)	5.9 0.362 0.330	6.7 0.452 0.401	6.2 (0.3) 0.407 (0.034) 0.370 (0.024)
	Stress σ	0 (MPa)*		4.85	5.14	4.98 (0.13)	4.81	5.15	5.01 (0.12)
	* applied with lead masses

Acknowledgements

The authors gratefully acknowledge the financial support of the Austrian Research Promotion Agency (FFG, project number 815234/12791) and the wood industry partnership Building with Wood within CEI-Bois for supporting this research work within project "MechWood". The "MechWood" research partners are thanked for their cooperation and collaboration in this project. The authors would also like to thank Mr G. Camp (LMGC) and Mr. P. Valorge (LMGC) for technical achievements, Mr G. Calchera (CIRAD) for rupture experiments, and Dr A. Van der Lee (IEM) for X-ray measurements of microfibrillar angle.

(b) nominal stress σ; (c) compliance J (# dry-loaded set, # wet-loaded set)

Creep trajectories relative to estimated moisture content

Figure 6 shows the compliance as a function of the estimated MC. These creep trajectories, which eliminate time, allow discussion of the applicability of the hygroviscoelastic constant concept. The wet-loaded set exhibits a rather typical mechanosorptive response. After the initial wet creep, each drying-wetting cycle induced a significant compliance increase. The approach toward a creep limit is clearly visible.

In the dry-loaded set, the initial wetting period has produced a considerable creep, and the first cycle has a significant effect although smaller than for the wet-loaded set. Then the contributions of subsequent cycles are almost identical in both sets. Obviously, the wet-loaded set tends to catch up with the dry-loaded set. This is clearly demonstrated in Figure 7, presenting the relation between the two sets.