Wojciech Kamiński

Marcin Kisielowski

Jerzy Lewandowski

The EPRL intertwiners and corrected partition function

Keywords: numbers: 04, 60, Pp

Do the SU(2) intertwiners parametrize the space of the EPRL solutions to the simplicity constraint? What is a complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective in the general n-valent vertex case for the Barbero-Immirzi parameter less then 1. We find, however, that the EPRL map is not isometric. In the consequence, a partition function can be defined either using the EPRL intertwiners Hilbert product or the SU(2) intertwiners Hilbert product. We use the EPRL one and derive a new, complete formula for the partition function. Next, we view it in terms of the SU(2) intertwiners. The result, however, goes beyond the SU(2) spin-foam models framework and the original EPRL proposal.

I. INTRODUCTION

The main technical ingredient of the spin-foam models of 4-dimensional gravity is so called quantum simplicity constraint. Imposing suitably defined constraint on the domain of the (discrete) path integral turns the SU(2)×SU(2) (or SL(2,C)) BF theory into the spin-foam model of the Euclidean (respectively, Lorentzian) gravity [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF]. The formulation of the simplicity constraint believed to be correct, or at least fitting gravity the best among the known approaches [START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF], is the one derived by Engle, Pereira, Rovelli, Livine (EPRL) [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] (and independently derived by Freidel and Krasnov [START_REF] Freidel | A New Spin Foam Model for 4d Gravity[END_REF]). The solutions to the EPRL simplicity constraint are EPRL SU(2)×SU(2) intertwiners. They are defined by the EPRL transformation, which maps each SU [START_REF] Engle | Flipped spinfoam vertex and loop gravity[END_REF] intertwiner into a EPRL solution of the simplicity constraint. An attempt is made in the literature [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] to parametrize the space of the EPRL solutions by the SU(2) intertwiners. The vertex amplitude and the partition function of the EPRL model seem to written in in terms of that parametrization. The questions we raise and answer in this paper are:

• Is the EPRL map injective, doesn't it kill any SU(2) intertwiner?

• Is the EPRL map isometric, does it preserve the scalar product between the SU(2) intertwiners?

• If not, what is a form of a partition function derived from the SO(4) intertwiner Hilbert product written directly in terms of the SU(2) intertwiners, the preimages of the EPRL map?

We prove the EPRL map is injective in the general, n valent vertex case and for the Barbero-Immirzi parameter |γ| < 1. The proof in the |γ| > 1 has already been provided in [START_REF] Kamiński | Spin-Foams for All Loop Quantum Gravity[END_REF]. Hence, there are as many SU(2)×SU(2) EPRL intertwiners as there are the SU(2) intertwiners. Owing to this result the SU(2) intertwiners indeed can be used to parametrize the space of the EPRL SU(2)×SU(2)intertwiners. However, we find the EPRL map is not isometric. In consequence, there are two inequivalent definitions of the partition function. One possibility is to use a basis in the EPRL intertwiners space orthonormal with respect to the SO(4) representations. And this is what we do in this paper. A second possibility, is to use the basis obtained as the image of an orthonormal basis of the SU(2) intertwiners under the EPRL map. The partition function derived in [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] corresponds to the second choice, whereas the first one is ignored therein. The goal of this part of our paper is pointing out the first possibility and deriving the corresponding partition function. After the derivation, we compare our partition function with that of EPRL on a possibly simple example. We conjecture, that the difference converges to zero for large spins.

To make the paper intelligible we start presentation of the new results with the derivation of the partition function in Section II 4. The final formula for our proposal for the partition function Confidential: not for distribution. Submitted to IOP Publishing for peer review 31 May 2010 for the EPRL model is presented in Section II 5. The lack of the isometricity of the EPRL map is illustrated on specific examples in Section II 6. Finally, the proof of the injectivity of the EPRL map takes all the Section III B.

This work is written in terms of the EPRL framework [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] combined with our previous paper [START_REF] Kamiński | Spin-Foams for All Loop Quantum Gravity[END_REF] on the EPRL model.

II. OUR PROPOSAL FOR A PARTITION FUNCTION OF THE EPRL MODEL

Partition functions for the spin-foam models of 4-gravity: definition

Consider an oriented 2-complex κ whose faces (2-cells) are labeled by ρ with the irreducible representations of G =SU(2)×SU(2),

κ (2) f → ρ(f ),
and denote by H(f ) the corresponding Hilbert space. For every edge (1-cell) e we have the set/set of incoming/outgoing faces, that is the faces which contain e and whose orientation agrees/disagrees with the orientation of e. We use them to define the Hilbert space

H(e) = f incoming H(f ) ⊗ f outgoing H(f ) * . (2.1)
The extra data we use, is a subspace

H(e) SIMPLE ⊂ H(e) ( 2 . 2 )
defined by some constraints called the quantum simplicity constraints. In this paper, starting from section below, we will be considering the subspace proposed by Engle-Pereira-Rovelli-Livine.

For the time being H(e) SIMPLE is any subspace of the space of invariants of the representation

f incoming ρ(f ) ⊗ f outgoing ρ(f ) * , H(e) SIMPLE ⊂ Inv SU(2)×SU(2)   f incoming H(f ) ⊗ f outgoing H(f ) *   . (2.3)
(The subspace H SIMPLE e may be trivial for generic representations ρ(f ) and ρ(f ). Typically the simplicity constraints constrain also the representations themselves.) To every edge we assign the operator of the orthogonal projection onto H(e) SIMPLE ,

P SIMPLE e : f incoming H(f ) ⊗ f outgoing H(f ) * → f incoming H(f ) ⊗ f outgoing H(f ) * (2.4)
Our index notation is as follows (we drop 'SIMPLE' for simplicity)

(P e v) where the upper/lower indices of any vector v ∈ f incoming H(f ) ⊗ f outgoing H(f ) * correspond to incoming/outgoing faces. In the operator P e , for each face containing e, there are two indices, an upper and a lower one corresponding to the Hilbert space H(f ). If f is incoming (outgoing), then we assign the corresponding lower/upper index of P e to the beginning/end (end/beginning) of the edge. That rule is illustrated on Fig. 1. Now, for every pair of edges e and e , which belong to a same face f , and share a vertex v, there is defined the natural contraction at v of the corresponding vertex of P e with the corresponding vertex of P e . The contraction defines the following trace, e∈κ

P SIMPLE e → Tr   e∈κ (1)
P SIMPLE e   .

(2.6) Define partition function Z(κ, ρ) to be the following number:

Z(κ, ρ) := f ∈κ (2) d(f ) Tr   e∈κ (1)
P SIMPLE e   A(boundary), d(f ) := dimH(f ) (2.7)
where A(boundary) is a factor that depends only on the boundary of (κ, ρ), and we derive it elsewhere.

Partition functions for the spin-foam models of 4-gravity: the amplitude form

The partition function is usually rewritten in the spin-foam amplitude form [START_REF] Reisenberger | World sheet formulations of gauge theories and gravity[END_REF][START_REF] Baez | An introduction to Spinfoam Models of BF Theory and Quantum Gravity[END_REF][START_REF] Perez | Spinfoam models for Quantum Gravity[END_REF]. For that purpose one needs an orthonormal basis in each Hilbert space H(e) SIMPLE ; denote its elements by ι e,α ∈ H(e) SIMPLE , α = 1, 2, ..., n(e). Then

P SIMPLE e = ne α=1 ι e,α ⊗ ι † e,α (2.8)
where by ' † , for every Hilbert space H we denote the duality map Substituting the right hand side of (2.8) for each P SIMPLE e in (2.7), one writes the partition function in terms of the vertex amplitudes in the following way:

H v → v † ∈ H *
• For each edge of κ choose an element of the corresponding orthonormal basis; denote this assignment by

ι : e → ι e,αe (2.9)

• At each vertex v ∈ κ (0) :
take ι e1,αe 1 , ..., ι em,αe m where e 1 , ..., e m are the incoming edges

-take ι †
A v (ι) := Tr ι e1 ,α1 ⊗ ... ⊗ ι em,αm ⊗ ι † e 1 ,α 1 ⊗ ... ⊗ ι † e m ,α m (2.10)
where 'Tr' stands for the contraction (2.6) and can be defined by the evaluation of the spin-networks corresponding to the vertices (see [START_REF] Kamiński | Spin-Foams for All Loop Quantum Gravity[END_REF]).

• to each face f assign the face amplitude d(f ).

With this data, with the vertex amplitudes and face amplitudes, the partition function takes the famous form

Z(κ, ρ) = f ∈κ (2) d(f ) ι v∈κ (0) A v (ι) A(boundary), (2.11)
The result is independent of the choice of the orthonormal basis of each H SIMPLE e .

The EPRL map

Now we turn to the EPRL intertwiners. For every edge e ∈ κ (1) H(e) SIMPLE = H(e) EPRL , P(e) SIMPLE = P (e) EPRL .

(2.12)

The definition of H(e) EPRL uses a fixed number γ ∈ R called Barbero-Immirzi parameter. The Hilbert space H(e) EPRL can be non-empty only if the 2-complex κ is labeled by EPRL representations. A representation ρ = (ρ j -, ρ j + ) of SU( 2)×SU( 2), where j ± ∈ 1 2 N define the SU(2) representations in the usual way, is an EPRL representation, provided there is k ∈ 1 2 N such that

j ± = |1 ± γ| 2 k. (2.13)
Therefore, we will be considering here labellings of the faces of the 2-complex κ with EPRL representations

f → ρ(f ) = (ρ j -(f ) , ρ j + (f ) ), (2.14)
each of which is defined in the Hilbert space

H(f ) = H j -(f ) ⊗ H j + (f ) .
Each labeling defines also a labeling with SU(2) representations given by (2.13),

f → ρ k(f ) , (2.15)
defined in the Hilbert space H k(f ) . Given an edge e and the corresponding Hilbert space

H(e) = f incoming H j -(f ) ⊗ H j + (f ) ⊗ f outgoing H * j -(f ) ⊗ H * j + (f ) , (2.16)
the natural isometric embeddings

C : H k → H j -⊗ H j + (2.17)
and the orthogonal projection operator

P : H(e) → H(e) ( 2 . 1 8 )
onto the subspace Inv SU(2)×SU(2) (H(e)) defines the natural map, the EPRL map:

ι EPRL : Inv SU(2) f incoming H k(f ) ⊗ f outgoing H * k(f ) → f incoming H j -(f ) ⊗ H j + (f ) ⊗ f outgoing H * j -(f ) ⊗ H * j + (f )
Its image is promoted to the Hilbert space (2.2),

H EPRL e := ι EPRL   Inv SU(2)   f incoming H k(f ) ⊗ f outgoing H * k(f )     . (2.19)

The problem with the EPRL intertwines

All the EPRL intertwiners can be constructed from the SU(2) intertwiners by using the EPRL map. The point is, that one has to be more careful while doing that. First, one has to make sure that the map ι EPRL is injective. If not, then Hilbert space of the SU(2)×SU(2) EPRL intertwiners is smaller then the corresponding space of the SU(2) intertwiners and we should know how big it is. For γ ≥ 1, the injectivity was proved in [START_REF] Kamiński | Spin-Foams for All Loop Quantum Gravity[END_REF]. In the next section we present a proof of the injectivity for |γ| < 1. Secondly, one should check whether or not the map ι EPRL is isometric. Given an orthonormal basis I e,1 , ..., I e,ne of the Hilbert space

Inv SU (2) f incoming H k(f ) ⊗ f outgoing H * k(f )
, we have a corresponding basis ι EPRL (I e,1 ), ..., ι EPRL (I e,ne ) of the corresponding Hilbert space H EPRL e . The question is, whether or not the latter basis is also orthonormal. We show in Section II 6, this is not the case. The direct procedure would be to orthonormalize the basis. We propose, however, a simpler solution.

A solution

An intelligent way, is to go back to the formula (2.

f → ρ(f ) = (ρ j - f , ρ j + f ).
(2.22)

The labeling is accompanied by the corresponding labeling with the SU(2) i

f → ρ k f , (2.23)
according to (2.13). For every edge e ∈ κ (1) , in addition to the Hilbert space H EPRL e ⊂ Inv SU(2)×SU( 2) (H e ), we also have its preimage, the Hilbert space

Inv   f incoming H (k f ) ⊗ f outgoing H * k f   .
(2.24)

Therein, we fix an orthonormal basis I e,a , a = 1, 2, ..., n e .

(2.25)

To define the partition function we proceed as follows:

• assign to every edge of κ a pair of elements of the basis, II : e → (I e,ae , I † e,be ), (2.26) more specifically, I e,ae is assigned to the end point and I † e,be to the beginning point of e, and we denote the assignment by the double symbol II;

• define for every edge an edge amplitude to be h e (II) := h be āe

• to every vertex v of κ assign the vertex amplitude with the trace defined by

Inv SU(2)×SU(2) f incoming H j -(f ) ⊗ H j + (f ) ⊗ f outgoing H * j -(f ) ⊗ H * j + (f ) . Then, we have h e,āb = c + ,c - f e c -c + a f e c -c + b .
(2.29)

We are in a position now, to compare our partition function with that of [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF]. The partition function of [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] is given by replacing the matrix h e,āb in (2.27) with δ ab . The example below gives quantitative idea about the difference between the two possible definitions of partition function for the EPRL model.

Example of the edge amplitude h ba showing that the EPRL map is not isometry

We will show in this section, that the EPRL map is not isometric. We calculate the edge amplitude defined in the previous section in a simple example, and see that its matrix is not proportional to the identity matrix, or even not diagonal. Consider an edge at which exactly four faces meet. Assume the orientation of each face is opposite to the orientation of the edge. We have an intertwiner I ∈ Inv SU(2) (H k1 ⊗ H k2 ⊗ H k3 ⊗ H k4 ) assigned to the end point of this edge. We choose a basis |k i m i (the eigenvector of the third component of angular momentum operator with eigenvalue m i ) in each space H ki , i ∈ {1, . . . , 4}. We choose a real basis of the space Inv SU(2) (H k1 ⊗ H k2 ⊗ H k3 ⊗ H k4 ) in the following form [START_REF] Alesci | Asymptotics of LQG fusion coefficients[END_REF]:

(I a ) k1m1k2m2k3m3k4m4 = √ 2a + 1 a m=-a a m =-a (-1) a+m k 1 k 2 a m 1 m 2 m δ m,-m k 3 k 4 a m 3 m 4 m
where j 1 j 2 j 3 m 1 m 2 m 3 is the Wigner 3j-Symbol, δ m,m is the Kronecker Delta.

Let ι a + ⊗ ι a -be the basis of the space Inv

H j + 1 ⊗ . . . ⊗ H j + 4 ⊗ Inv H j - 1 ⊗ . . . ⊗ H j -

4

. The intertwiner ι EPRL (I a ) expressed in this basis takes the following form (we skip in this section the subscript e indicating the dependence on edge):

ι EPRL (I a ) = f a + a - a ι a + ⊗ ι a -
where f a + a - a are real and are known as fusion coefficients [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF]. The tensor h āb could be expressed in terms of them:

h āb = (ι EPRL (I a )|ι EPRL (I b )) = a + a - f a + a - a f a + a - b
As an example we give the result of the calculation of the h a b matrix for γ = 1 2 , j 1 = 2, j 2 = 4, j 3 = 4, j 1 = 2; a, b ∈ {2, . . . , 6}):

          53723
         
We used the analytic expression for the fusion coefficient presented in [START_REF] Alesci | Asymptotics of LQG fusion coefficients[END_REF]. Clearly this matrix is nondiagonal. It shows that the EPRL map is not isometric. The edge amplitude h e bā is given by the inverse matrix: This part of the paper is devoted to the injectivity of EPRL intertwiner. More explicitly, we will prove the result stated in 1.

           46976713

A. Statement of the result

We assume that γ ∈ R and |γ| < 1. Suppose that

(k 1 , . . . , k n ) ∈ 1 2 N ∀ i j ± i = 1 ± γ 2 k i ∈ 1 2 N.
We consider the EPRL map

ι EPRL : Inv (H k1 ⊗ • • • ⊗ H kn ) → Inv H j + 1 ⊗ • • • ⊗ H j + n ⊗ Inv H j - 1 ⊗ • • • ⊗ H j - n (3.1) ι EPRL (I) j + 1 A1...j + n Anj - 1 B1...j - n Bn = I k1C1...knCn C k1C1 j + 1 D1j - 1 E1 • • • C knCn j + n Dnj - n En P j + 1 D1...j + n Dn j + 1 A1...j + n An P j - 1 E1...j - n En j - 1 B1...j - n Bn
with P standing for the orthogonal projections onto the subspaces of the SU(2) invariants of the Hilbert spaces

H j + 1 ⊗ • • • ⊗ H j + n , and respectively, H j - 1 ⊗ • • • ⊗ H j -
n . Now we can state our result. proves nontriviality. Now we extract conditions on k α from triangle inequalities (we assume for simplicity that k 1 ≥ k 2 )

k 1 + k 2 ≥ k α ≥ k 1 -k 2 i≥3 k i ≥ k α ≥ k i - j =i, j≥3 k j , i ≥ 3
For the existence of such k α we need only to show that

k 1 + k 2 ≥ k i - j =i, j≥3 k j , i ≥ 3 i≥3 k i ≥ k 1 -k 2
but these are exactly conditions for (k 1 , . . . , k n ) to satisfy triangle inequality.

Important maps

Every I ∈ Inv (H k1 ⊗ • • • ⊗ H kn ) may be uniquely written as

I k1A1k2A2...knAn = kα C kαAα k1A1k2A2 I kα kαAαk3A3...knAn ,
where

I kα ∈ Inv (H kα ⊗ H k3 ⊗ • • • ⊗ H kn ). Summation is taken over such k α that (k α , k 1 , k 2 ) and (k α , k 3 , . . . , k n ) satisfy triangle inequality. This gives us decomposition of Inv (H k1 ⊗ • • • ⊗ H kn ) into orthogonal subspaces ⊕ α H α ,
where each H α is isomorphic to Inv (H kα ⊗ H k3 ⊗ • • • ⊗ H kn ). Let us define maps which assign these partial isometries

Q kα : Inv (H k1 ⊗ • • • ⊗ H kn ) → Inv (H kα ⊗ H k3 ⊗ • • • ⊗ H kn ) , Q kα I = I kα
Adjoints to them are embeddings Q * kα .

Q * kα I kα k1A1k2A2...knAn = C kαAα k1A1k2A2 I kα kαAαk3A3...knAn .

These maps are also well defined in a case that α does not occur in the decomposition ⊕H α but (k α , k 1 , k 2 ) satisfies triangle inequalities. Then the space Inv (H kα ⊗ H k3 ⊗ • • • ⊗ H kn ) is trivial and the maps Q kα and Q * kα too. Let us fix (k 1 , . . . , k n ) and (j ± 1 , . . . , j ± n ) satisfying triangle inequalities and such that j

+ i +j - i = k i . Lemma 7. Suppose k α , j ± α are such that j + α + j - α = k α and (k α , k 1 , k 2 ) and (j ± α , j ± 1 , j ± 2
) satisfy triangle inequalities. Then there exists an operator

G kαj + α j - α : Inv H j + α ⊗ H j + 3 ⊗ • • • ⊗ H j + n ⊗ Inv H j - α ⊗ H j - 3 • • • ⊗ H j - n → → Inv H j + 1 ⊗ • • • ⊗ H j + n ⊗ Inv H j - 1 ⊗ • • • ⊗ H j - n such that for all I ∈ Inv H k β ⊗ H k3 ⊗ • • • ⊗ H kn and φ ∈ Inv H j + α ⊗ H j + 3 ⊗ • • • ⊗ H j + n ⊗ Inv H j - α ⊗ H j - 3 • • • ⊗ H j - n ι k1...kn Q * k β I, G kαj + α j - α φ =    ι kαk3...kn I, φ , k β = k α 0, k β > k α * , k β < k α Proof. We define G kαj ± α as G kαj ± α (φ) j + 1 A1...j + n Anj - 1 B1...j - n ...Bn = βC j + α Aα j + 1 A1j + 2 A2 C j - α Bα j - 1 A1j - 2 A2 φ j + α Aαj + 3 A3...j - α Bαj - 3 B3... ,
with β nonzero constant to be defined later. Let us compute

ι k1 ...kn Q * k β I, G kαj ± α φ
In the definition of ι one can skip projection because both φ and G kαj ± α φ are invariants. Let us write explicitly ι k1...kn Q * k β I, G kαj ± α φ . We have

I k β A...knAn C k β A k1A1k2A2 C k1A1 j + 1 B1j - 1 C1 • • • C knAn j + n Bnj - n Cn βC j + 1 B1j + 2 B2 j + α B C j - 1 C1j - 2 C2 j - α C φ j + α Bj + 3 B3...j - α Cj - 3 C3... = = βC k β A k1A1k2A2 C k1A1 j + 1 B1j - 1 C1 C k2A2 j + 2 B2j - 2 C2 C j + 1 B1j + 2 B2 j + α B C j - 1 C1j - 2 C2 j - α C I k β A...knAn C k3A3 j + 3 B3j - 1 C3 • • • C knAn j + n Bnj - n Cn φ j + α Bj + 3 B3...j - α Cj - 3 C3... .
We need only to show that βC

k β A k1 A1k2A2 C k1A1 j + 1 B1j - 1 C1 C k2A2 j + 2 B2j - 2 C2 C j + 1 B1j + 2 B2 j + α B C j - 1 C1j - 2 C2 j - α C = C k β A j + α Bj - α C , k β = j + α + j - α 0 k β > j + α + j - α .
The second equality is obvious because there exists no intertwiner if k β > j + α + j - α . The first will be proved in the next subsection III B 41 .

Relation among intertwiners

We know that C

k β A k1A1k2A2 C k1A1 j + 1 B1j - 1 C1 C k2A2 j + 2 B2j - 2 C2 C j + 1 B1j + 2 B2 j + α B C j - 1 C1j - 2 C2 j - α C is proportional to C k β A j + α Bj - α C .
In order to prove that the factor of proportionality is nonzero we will show that

C kαA k1A1k2A2 C k1A1 j + 1 B1j - 1 C1 C k2A2 j + 2 B2j - 2 C2 C j + 1 B1j + 2 B2 j + α B C j - 1 C1j - 2 C2 j - α C C j + α Bj - α C kαA = 0
and that would be β -1 . In fact it is enough to show that the intertwiner

C k1A1 j + 1 B1j - 1 C1 C k2A2 j + 2 B2j - 2 C2 C j + 1 B1j + 2 B2 j + α B C j - 1 C1j - 2 C2 j - α C C j + α Bj - α C kαA
is not equal zero or equivalently the same for

C k1A1 j + 1 B1j - 1 C1 C j + 2 B2j - 2 C2 k2A2 C j + 1 B1 j + 2 B2j + α B C j - 1 C1 j - 2 C2j - α C C j + α Bj - α C kαA
We only sketch the proof. First of all, we remind some facts about intertwiners and diagrammatic notation.

Let P k stands for projection onto symmetric subspace in H ⊗2k 1/2 equivalent to H k (k is a half natural number).

P k : H ⊗2k 1/2 → H ⊗2k 1/2 .
In this subsection we regard H k as this subspace of H The intertwiner

C k1k2k3 : C → H k1 ⊗ H k2 ⊗ H k3 is proportional to P k1 ⊗ P k2 ⊗ P k3 ⊗2k1+2k2+2k3 .
In the diagrammatic language this can be depicted as on figure 2. We skip the index k in P k on the diagrams for notations' brevity. The line with symbol k denotes H ⊗2k 1/2 . We have to notice important properties,that in diagrammatic language is shown on figures 3 and 4.

Our intertwiner can be written as shown on the figure 5. Now using properties mentioned earlier we see that one can merge j + ij with j - ij into k ij and that intertwiner is equal to intertwiner shown on the figure 2 and is nonzero.

Inductive steps

Induction starts with n = 1. In this case k 1 = 0 and so also j ± 1 = 0. The map ι 0 = C 0 00 : C → C ⊗ C is the identity.

Suppose now, that we have just proved Hyp n -1.

In the decomposition of given I ∈ Inv (H k1 ⊗ • • • ⊗ H kn ) into subspaces H α we choose minimal k α such that Q kα I is nonzero. We know, by lemmas 4 and 5 that for either

C k 1 A 1 j + 1 B 1 j - 1 C 1 C j + 2 B 2 j - 2 C 2 k 2 A 2 C j + 1 B 1 j + 2 B 2 j + α B C j - 1 C 1 j - 2 C 2 j - α C C j + α Bj - α C kαA .
or for

(j + α , j - α ) = 1 + γ 2 k α - , 1 -γ 2 k α +
all necessary assumptions of lemma 7 are satisfied. From the Hyp n -1 for the sequences (k α , k 3 , . . . k n ), (j ± α , j ± 3 , . . . j ± n ) we know that there exists This finishes inductive step and the proof.

φ ∈ Inv H j + α ⊗ • • • ⊗ H j + n ⊗ Inv H j - α ⊗ • • • ⊗ H j -

IV. SHORT DISCUSSION

We studied in this paper properties of the solutions to the EPRL simplicity constraints which were derived in [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF]. We also pointed out two different possibilities of defining the partition function out of them. Our definition is (2.7). It uses only the subspace of the SO(4) intertwiners which solve the EPRL simplicity constraint. The comparison and contrast between our definition and that of [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] is provided by (2.27) and the comments which follow that equality. The difference follows from the fact proven in Section II 6 above, that the EPRL map that is not isometric. The example considered in that section gives also quantitative idea of the difference. The question of which definition of the partition function is correct can not be answered at this stage. Finally, we studied the "size of the space of the EPRL solutions". We have shown that the EPRL map does not kill any SU(2) intertwiner. The proof is presented in detail in Section III.

FIG. 1 :

 1 FIG. 1: According to this rule, given an edge e ( e ) contained in incoming (outgoing) face f , the indices of Pe ( P e ) corresponding to H(f ) are assigned to the beginning and, respectively, to the end of the edge. The oriented arc only marks the orientation of the polygonal face f .

  defined by the Hilbert scalar product. In the Dirac notation ι e,α = |e, α , and ι † e,α = e, α|.

e 1 ,α e 1 ,

 11 ..., ι † e m ,α e m where e 1 , ..., e m are the outgoing edges -define the vertex amplitude

  7) for the partition function and repeat the step leading to (2.11) with each projection P SIMPLE e = P EPRL e written in terms of the corresponding basis ι EPRL (I e,1 ), ..., ι EPRL (I e,ne ). The suitable formula reads e,a ) ⊗ ι EPRL (I e,b ) † (2.20) where h e a b, a, b = 1, ..., n e define the inverse matrix to the matrix h e, ba := (ι EPRL (I e,b )|ι EPRL (I e,a )) (2.21) given by the Hilbert product (•|•) in the Hilbert space (2.16). (In the Dirac notation, P EPRL e = h e a b|e, a e, b|.) Now, we are in a position to write the resulting spin-foam amplitude formula for the partition function. It is assigned to a fixed 2-complex κ and a fixed labeling of the faces by the EPRL representations

Fig. 1 ,ι•

 1 EPRL (I e,ae ) ⊗ e outgoing ι EPRL (I e ,b e ) †   to every face f assign the amplitude d f Finally, the spin-foam amplitude formula for the partition function reads Z(κ, ρ) = f d f II e h e (II) v A v (II)A(boundary). (2.27) The matrix (2.21) can be written in terms of the EPRL fusion coefficients, ι EPRL (I e,a ) =: f e c -c + a ι e,c -⊗ ι e,c + , (2.28) defined by the decomposition into an orthonormal basis ι e,c -⊗ ι e,c + in

  OF THE MAP I → ι EPRL (I)

FIG. 4 :

 4 FIG. 4: An equalityP k 1 +k 2 +k 3 • P k 1 +k 2 ⊗ I = P k 1 +k 2 +k 3 .

FIG. 5 :

 5 FIG. 5: Intertwiner proportional toC k 1 A 1

ι

  kαk3...kn Q kα I, φ = 0.We haveι k1...kn I, G kαj ± α φ = k β ≥kα ι k1...kn Q * k β Q k β I, G kαj ± α φ = ι kαk3...kn Q kα I, φ = 0.

  ⊗2k 1/2 . Let us also denote the canonical map: C → H 1/2 ⊗ H 1/2 . An intertwiner proportional to C k 1 k 2 k 3 , k12 = k1 + k2 -k3 and etc.

		k1
		P
	k12	k23	k13
	P			P
	k2			k3
	FIG. 2: k1	k2	= k1 + k2
	FIG. 3: An equality between H 2k 1 1/2 ⊗ H 2k 2 1/2 and H 2k 1 +2k 2

Although it seems to be standard, we include it for a sake of completeness.

Acknowledgments

We would like to thank John Barrett, Jonathan Engle for coming to Warsaw and delivering lectures on the SFM. JL acknowledges the conversations with Laurent Freidel, and exchange of e-mails with John Baez, Carlo Rovelli, Roberto Pereira and Etera Livine. MK would like to thank Jonathan Engle from Albert Einstein Institute in Potsdam and Carlo Rovelli, Matteo Smerlak from Centre de Physique Theorique de Luminy for discussions and hospitality during his visits at their institutes. The work was partially supported by the Polish Ministerstwo Nauki i Szkolnictwa Wyzszego grants 182/NQGG/ 2008/0, 2007-2010 research project N202 n081 32/1844, the National Science Foundation (NSF) grant PHY-0456913, by the Foundation for Polish Science grant Master and a Travel Grant from the QG research networking programme of the European Science Foundation.

Theorem 1. For any sequence (k 1 , . . . , k n ) ∈ 1 2 N such that Inv (H k1 ⊗ • • • ⊗ H kn ) is nontrivial the map

B. Proof of the theorem

In order to make the proof transparent, we divide it into subsections. In subsection III B 1 some auxiliary definitions are introduced. We state also an inductive hypothesis, that will be proved in subsection III B 5. The injectivity of EPRL map follows from that result. The main technical tool of the proof is placed in subsection III B 3, where the map 7 is defined.

Auxiliary definitions

Let us introduce some notations Definition 2. For x ∈ R we define

One can define map ι under condition

Con n: Sequences of half natural numbers (k 1 , . . . , k n ) and (j ± 1 , . . . , j ± n ) are such that

with P standing for projections onto invariant subspaces. We will use the letter ι k1...kn for all sequences (k 1 , . . . , k n ), (j ± 1 , . . . , j ± n ) if it do not cause any misunderstanding. We will base our prove on the following inductive hypothesis:

Hyp n: Suppose that (k 1 , . . . , k n ) and (j ± 1 , . . . , j ± n ) satisfy condition Con n and that

This in fact proves the injectivity.

Useful inequalities

Both [x] ± are increasing functions and satisfy (x, y ∈ R, j ∈

In order to prove the last point, we notice that

Lemma 4. Suppose that (k, l, j) satisfies triangle inequality and that 1+γ 2 k ∈ 1 2 N then

Proof. In the first case suppose that k + l = j holds, then 1+γ 2 k + 1+γ 2 l ± = 1+γ 2 j ± that proves triangle inequality.

In the second case, we restrict our attention to 1+γ 2 k, 1+γ 2 l + , 1+γ 2 j -. We have

• 1+γ 2 k ≤ 1+γ 2 l + + 1+γ 2 j -from the property d) listed above.

The case of 1+γ 2 k, 1+γ 2 l -, 1+γ 2 j + is analogous.

Lemma 5. Suppose that (k 1 , . . . , k n ) satisfies triangle inequality and that 1+γ 2 k i ∈ 1 2 N for i = 2, . . . , n, then 1+γ 2 k 1 ± , 1+γ 2 k 2 , . . . , 1+γ 2 k n also satisfy triangle inequalities.

Proof. That follows from the monotonicity of functions [x] ± and the fact that in the inequality

all terms but one are half-integer.

Lemma 6. Suppose that (k 1 , . . . , k n ) satisfies triangle inequality then

is nontrivial.

Proof. We will find an k α such that both (k α , k 1 , k 2 ) and (k α , k 3 , . . . , k n ) satisfy triangle inequalities. By induction there would be

and then 0 = C kαA k1A1k2A2 φ kαAk3A3...knAn


Alessandro Pesci
email: pesci@bo.infn.it

A proof of the Bekenstein bound for any strength of gravity through holography

Keywords: numbers: 04.20.Cv, 04.40.-b, 04.62.+v, 04.70.Dy, 05.20.-y, 05.30.-d, 05.70.-a, 12.38.Mh, 25.75.Nq, 51.20.+d

come

A proof of the Bekenstein bound for any strength of gravity through holography

I. MOTIVATION

In the description of thermodynamic systems, recent works [1,2] have shown that the generalized covariant entropy bound (GCEB) [3], which can be considered as the most general formulation of the holographic principle for semiclassical circumstances, is universally satisfied if and only if the statistical-mechanical description is characterized by a lower-limiting spatial scale l * , determined by the assigned local thermodynamic conditions. 1 In [5] the consequence has been drawn that l * entails also a lower-limit to the temporal scale, and this supports, both for its existence and value, the recently proposed universal bound [6] to relaxation times of perturbed thermodynamic systems, bound attained by black holes. 2 All this amounts to say that the GCEB is satisfied if and only if a fundamental discreteness is present in the spatio-temporal description of statistical-mechanical systems, with a value changing from point to point being determined by local thermodynamic conditions.

The exact value of l * is set from the GCEB [1,2]. This does not necessarily mean however that l * has to depend on gravity. In effect, the derivation of l * from the GCEB is through Raychaudhuri equation so that gravity cancels out [2]. Indeed, this scale appears intrinsically unrelated to gravity (in particular l * must not be thought as something around the Planck scale; it is in general much larger than this) since, as we will discuss later, it can be fully described simply as a consequence or expression of the flat-space quantum description of matter [2].

Among the various entropy bounds, the GCEB appears the most general one, subsuming in some way all other previous bounds at the conditions in which they are supposed to hold. The Bekenstein universal entropy bound (UEB) [START_REF] Bekenstein | [END_REF], the first proposed among them, has been shown to be implied by GCEB for weak field conditions (a strengthened version of it, actually, in case of non-spherical systems) [11]. This bound sets an upper limit to the S/E ratio (S is entropy and E is total mass-energy) for a given arbitrary system, in terms of its size. The original motivation was a gedanken experiment, in the context of black hole physics, in which the variations of total entropy are considered when objects with negligible self-gravity are deposited at the horizon. In spite of how it was derived, it has been clear since the beginning that gravity should not play any role in establishing the UEB [START_REF] Bekenstein | [END_REF]. Since, besides the systems with negligible self-gravity, also the objects with the strongest gravitational effects, namely the black holes, appear to satisfy the bound (actually they attain it), the original proposal stressed moreover the UEB should hold true for whatever strength of the gravitational interaction (assuming the conditions be such that the terms entering the bound can be appropriately defined). A strong support, if not a proof, of this statement under safe conditions for the definition of system's "energy" and "size", has been given in [12], through the consideration of self-gravitating radiation confined in a spherical box; in particular, some key elements have been pointed out there, apparently crucial in determining the validity of the UEB under strong self-gravity.

What is left is the intriguing situation of a bound belonging in principle to pure flat-spacetime physics, which however should hold true, exactly with the same form, also under conditions in which gravity is strong. The main goal of present work is just trying to face again this issue, this time from a holographic standpoint. This is done giving a general argument supposed to prove the UEB for any strength of gravity (at suitable conditions, i.e. with all the terms involved in the bound unequivocally defined) as a result of the existence and value of the limiting scale l * , that is from holography in its semiclassical formulation. We then explore what this holographic perspective says about the connection of the UEB with the η/s Kovtun-Son-Starinets bound (KSS) [13] and with the proliferation of species problem.

II. THE LIMITING LENGTH

Given a spacelike 2-surface B the covariant entropy bound [14,15] is a conjecture that connects the entropy content of adjacent regions to the area A of B. It states that S(L) ≤ A/4 (in Planck units, the units we will use throughout this paper) where L is a lightsheet of A (a 3d null hypersurface generated by not expanding light rays orthogonal to B and followed until a caustic or a singularity is reached) and S(L) is the entropy on it. The generalization in [3], i.e. the GCEB, is in that the lightsheet L is allowed to terminate on a second spacelike 2-surface with area A , before the caustic or singularity are reached, with the bound becoming S(L) ≤ (A-A )/4.

The proof of the GCEB for perfect fluids given in [1] (other proofs, in terms of conditions which are proven to suffice for the validity of the bound, are in [3,16]) has allowed to identify a condition not only sufficient but also necessary for the validity of the GCEB [1,2]. This condition, which turns out to be in relation with the condition (1.9) in [3], has been expressed [2] in terms of a lower-limiting spatial length l * to the size of any system, for which an assigned statistical-mechanical description is given. This amounts to put everywhere a local limit to the size l of the disjoint sub-systems of which a large system can be thought as composed of. The condition is [2]

l ≥ l * , (1)
with, for a perfect fluid,

l * = 1 π s ρ + p = 1 πT 1 - µn ρ + p , (2)
being for thin slices the GCEB saturated when their thickness is as small as l * (when feasible). Here s, ρ, n, p, T , µ are respectively local entropy, mass-energy and number densities, local pressure, temperature and chemical potential (with, in this latter, possible rest-mass energy included).

The need for such limiting scale l * can be recognized through consideration of thin plane slices. The variation of the cross-sectional area of the lightsheet we trivially can construct on them is quadratic in the thickness l of the layer, whereas its entropy content is (obviously) linear in l. Hence, if a lower limit is not envisaged for l, for thin enough (terminated) lightsheets the GCEB would be definitely violated.

Looking at a system consisting of a single thin slice of thickness l and area A, condition (1) amounts to say that, for l < l * , a statistical-mechanical description according to which the system has a pressure p and a proper energy E and entropy S (to give ρ = E/Al and s = S/Al) matching the assigned values which define l * , cannot be given, that is, the size of the layer somehow forbids that the thermodynamic potentials have the values assigned. One important consequence of this is that the scale associated to local thermal equilibrium at any point cannot be lower than l * there. The concept of local thermal equilibrium scale l eq appears, in fact, such that if slices of thickness l eq are physically cutted, the intensive thermodynamic potentials in them have to remain unchanged (that is they would have negligible boundary effects), and this if l eq < l * would imply a violation of the above. Indeed, in general the scale of thermal equilibrium can be expected to be much larger than l * . This is implied by the notion itself of l * : for macroscopic systems at global equilibrium, for example, the scale of thermal equilibrium is given by system's size, which can be large at will, whereas l * remains fixed, as determined by the assigned local thermodynamic parameters.

Also, we have no reason to expect that the unallowance of an assigned statistical-mechanical description should set in exactly at the scale l * ; we should expect instead that in general there will be a gap between l * and the lowest scale l alw allowed by the material medium, being the gap dependent on the properties of the latter (after all, we know that there are fluids much more entropic than others so that, for these latter, both the GCEB for any configuration and thus also the inequality (1), have to be satisfied to spare). This implies that a mechanism must be at work, which, even putting l * and the GCEB aside completely, fixes l alw .

This mechanism is quite naturally provided by the uncertainty principle. If we ideally slice up a macroscopic system with finer and finer spacing l, we can expect to obtain thinner and thinner slices inside a region of thermal equilibrium, each of them with always a same statistical-mechanical description. For point-like constituents (i.e. particles or molecules for which the intrinsic spatial quantum uncertainty is larger than their size as composite objects), this will remain true down to a certain limiting thickness of the slices l qm , driven by quantum mechanics and roughly corresponding to the representative de Broglie wavelength λ of the constituent particles, at which scale the statistical-mechanical properties of the material medium (such as temperature and energy density for example) begin to be affected by slicing, if physical. It is not possible, for example, consider systems consisting of a slice of a photon gas at temperature T with thickness l < l qm = 1 πT [2]. Put another way, a statistical-mechanical description (with boundary effects included) for 3d containers, appears quantum-mechanically untenable when one of the dimensions becomes l λ, if λ is the pretended particle's wavelength.

For a system consisting of particles with a same wavelength λ, this suggests l qm = λ. In the general case we still will write l qm = λ with λ representing a typical or characteristic wavelength of constituent particles, meaning that below λ the quantum uncertainty from the boundary starts to affect the intrinsic thermal distribution of the particles in the system.

For 'virtual' or 'mathematical' slicing, for l < l qm particles 'in' a slice affect also adjacent slices, so that extensivity is lost in the sense that entropy or energy cannot be recovered as sum on slices of thickness l < l qm . The entropies or energies of these slices must instead be combined in a more complicated manner, to give however always those same values on scales l > l qm . The net effect is thus anyhow that total entropy or energy are determined as sum on slices, each not thinner than l qm = λ.

In this sense we thus can never go below λ and, obviously, below the size l c of the constituent particles due to compositeness, if dominant. Moreover, for point-like constituents the limiting scale λ can always be attained. In case of composite constituents we should stop at l c ≥ λ, but we can always imagine an equivalent system composed of particles with mass and other properties equal to those in the original system but point-like, so that the limiting scale λ can still be reached.

Thus, λ (or the maximum between λ and l c ) could possibly play the role of l alw above. However, how confronts λ with l * as given by ( 2) ? After all, if it could be λ < l * for some system with point-like constituents, the GCEB could be violated (choosing slices with thickness as small as λ). The consideration of various statistical-mechanical systems with point-like constituents hints that

λ ≥ l * , (3)
with in general λ l * , being the bound attained for ultra-relativistic systems (with µ = 0) [1,2]. Uncertainty principle seems to conspire in order that the limit in (3) be strictly attained in the most challenging cases and, as a consequence, be satisfied by far by more ordinary systems. According to this, the conjecture can be made that inequality (3) is guaranteed and exactly required by quantum mechanics. In this perspective inequality (1), namely the GCEB when the inequality is taken in a general-relativistic context, precisely predicts and is predicted by quantum mechanics. 3 In any case, even with this conjecture aside, GCEB + quantum mechanics means l * ≤ λ ≤ l alw universally.

All the above says that an upper-bound to the quantity s ρ+p given by

s ρ + p = πl * ≤ πλ (4)
has to be introduced if the GCEB is to be saved, apparently being however this exactly provided simply by quantum mechanics. This inequality expresses the fact that the requirement of a lower bound l * to λ translates into an upper bound πλ to s ρ+p . Whenever an hypothetical configuration with assigned s, p, ρ is considered, if it presents in some region s ρ+p values exceeding πλ, it would challenge the GCEB there, but also, according to the conjecture above, it would be there simply unphysical in the sense of quantum-mechanically untenable.

Put another way, any statistical-mechanical description, consistent as such with quantum mechanics, appears to intrinsically require locally an upper limit πλ to s ρ+p . In [18], a connection between entropy and energy densities (like the ones proven to be sufficient to derive the GCEB) has been assumed, and has been proposed to be chosen as a constitutive principle for the construction of a theory of quantum gravity. The perspective emerging from this section is that, at our level of description (i.e. a statistical-mechanical description for matter (with entropy, energy and other quantities defined in the volumes under consideration) and a classical description for gravity), the relation between entropy and energy density to which the GCEB is crucially tied (relation (1)), is seemingly a straight consequence of quantum mechanics (through (3) or ( 4)). It can anyway be considered as a new assumption as far as the starting point is chosen to be a theory in which quantum mechanics is not already in. In this case, relations (3) or (4) would predict or bring in, it.

III. THE BEKENSTEIN BOUND DERIVED

Having l * , we will prove here the UEB in a full-gravity setting. From the above this can also be interpreted as implying in particular that the UEB can actually be derived from the GCEB, i.e. from holography, without any restriction on the strength of gravity.

The UEB says that for a physical system which fits into a sphere of radius R in asymptotically flat 4d spacetime

S E ≤ 2πR, (5)
where S and E are its entropy and total mass-energy [START_REF] Bekenstein | [END_REF]. Many discussions arose in the past, regarding the validity of the bound or its precise meaning even for the case of weakly gravitating systems (see [19,20] and references therein). No controversy seems to survive today for the validity of ( 5) if applied to complete, weakly self-gravitating, isolated systems [19]. However, a completely satisfactory proof of the bound even at these conditions, within the realm of full quantum field theory, is still lacking, not least due to the need to address the entropy of vacuum fluctuations (see [21] and references therein). If, to gain a first approach to the bound, this is intended as applied first only to the fields, superimposed to vacuum, which make up the system under consideration (so that for example E is the energy above the vacuum state), still a general proof is lacking. To our knowledge, the most general proof to date is, in fact, given in terms of free fields [22]. 4 If already in Minkowski spacetime some ambiguities in the definition of R and E have been pointed out [25], for curved spacetime it is far from clear even the meaning they can have in the general case. At least for spherically-symmetric asymptotically-flat spacetimes however, most of the inadequacies disappear and these are the conditions assumed here. Let us proceed to derive what the notion of l * implies for the UEB under the stated circumstances, first with gravity off. Even if in these circumstances the GCEB, and thus holography, lose any meaning, we know that conditions (1) and (3) have to hold true, and l * definition through (2) remains meaningful.

For any given macroscopic system, which we assume at local thermodynamic equilibrium, we first show that the bound

S E + pdV ≤ πR (6)
holds true, being dV system's volume element and R the circumscribing radius. The system, of course, in addition of being radially inhomogeneus, can be also in general non-spherically-symmetric, as it is clear that for gravity off the metric, being constant, is anyhow 'spherically symmetric' around every point. It can be easily shown, however, that every non-spherically-symmetric system can always be reduced to a spherically-symmetric one with a same 'circumscribing' radius and a larger ratio in equation ( 6).

Let us consider thus an isolated spherically-symmetric system with radius R and imagine to ideally perform a partition of it through a number of concentric spherical shells, each with thickness l i (with thickness l 0 of the central sphere we intend its diameter 2r 0 ) small enough that the shell can be considered at thermal equilibrium with statistical-mechanical local parameters constant on it5 (actually l 0 can be also as large as 2R in the homogeneous case). We have

S E + pdV = S i (E i + p i V i ) ≤ S i E i + p i V i = πl * i ≤ πl i = πR + πr 0 . (7)
In the expression above, l * i is l * at the local conditions of layer i. In [START_REF] Landau | Statistical Physics[END_REF] the first inequality comes from ai bi ≤ ai bi if a i , b i ≥ 0; the second from (1). If we allow the spherical shells to be as thin as possible (l i approaching λ i , where λ i is the typical wavelength (spelled above) on layer i of point-like, or point-like reduced, constituents), the quantities S i , E i and p i V i in (7) still represent, evidently, total entropy S, total energy E and pdV respectively. As long as R max{λ i } as must be always true for macroscopic systems, we have in (7) r 0 R so that (7) reduces to (6).

When the assumption is made p ≤ ρ (p = ρ (stiff matter) the stiffest equation of state compatible with causality [26]), we have S/(2E) =

S i /(2 E i ) ≤ S i / (E i + p i V i )
, so that from (6), S/E ≤ 2πR, that is the UEB ( 5) is obtained.

From the first inequality in [START_REF] Landau | Statistical Physics[END_REF] we see thus that when R max{λ i }, the UEB ( 5) is satisfied with orders of magnitude to spare (in the homogeneous case, for example, the first inequality in [START_REF] Landau | Statistical Physics[END_REF] introduces a factor ∼ R λ ). In general moreover λ (or l alw ) l * and this contributes further in obtaining S E 2πR. These results show that to challenge the UEB (5) systems with size R ≈ λ must be considered. The importance of being

R λ (8)
for a meaningful analysis of UEB, has been pointed out in early papers [START_REF] Bekenstein | [END_REF]23]. Moreover already [START_REF] Bekenstein | [END_REF] argued that a system consisting of a photon gas of limiting size R ≈ λ comes close to challenge the UEB (5), in agreement with what we find. What is stressed here however is that condition (8) on system's size is actually all what is needed to imply the bound: once this condition is fullfilled, automatically the UEB is satisfied and nothing is left to be proven. This relies on inequality (3), which is required by the GCEB, but, if we trust the conjecture introduced above, is also a direct expression of quantum mechanics alone. The linking between the UEB and the uncertainty principle, contained all the arguments above for the bound, seems to be expressed as straight as possible by inequality (3), by which the entire UEB is summarized: nothing more appears to be needed.

We proceed now to inspect what happens with gravity. As stated above, in search for conditions at which E and R can meaningfully be defined, we limit to spherically-symmetric asymptotically-flat spacetimes. We proceed thus to consider a spherically-symmetric asymptotically-flat spacetime from a spherically-symmetric system with radius R, assuming to subdivide the system in N concentric spherical shells each thin enough to be considered at thermal equilibrium, and we do not pose any constraint on the strength of gravity.

Following [12], and in the sense spelled out there, we restrict our consideration to spacetimes which admit configurations with a moment of time symmetry (in short, these spacetimes do admit a certain given "instant of time" configuration in which the extrinsic curvature of the corresponding spacelike hypersurface vanishes). The arguments given for this in [12] (strictly, for radiation) can be summarized in the quest for local extrema of entropy for given energy. Here we argue also that apparently only configurations belonging to these spacetimes allow to be completely probed through not expanding lightsheets.

On one hand, in fact, time-symmetric configurations with R < 2M (M is the mass of the system) when evolved back into the past inevitably lead to a white hole [12]. Assuming white holes cannot exist (as it is likely to be [START_REF] Penrose | Quantum Gravity 2: A second Oxford symposium[END_REF]), the instant-of-time-symmetry constraint above implies the piece of spacetime with R < 2M should be excluded. On the other hand, spherically-symmetric systems inside their own Schwarzschild radius are not completely covered by the future-directed ingoing lightsheets (the only left with non-positive expansion), the part covered always satisfying the Bousso bound [14]. Now, the recipe of relating, as in the GCEB, the area of spacelike 2-surfaces with the entropy content of adiacent null hypersurfaces appears to be the most convenient approach in view of covariance, and of most general applicability, to the entropy bounds. If, according to this, we assume that the UEB (in which S is the entropy of a given system) refers to systems completely coverable by not expanding lightsheets, 6 we are led, too, to exclude configurations with R < 2M .

Let us assume R > 2M (with also r > 2m(r) for any r < R as well, being m(r) the mass of the part of the system enclosed inside the area radius r at a given coordinate time). Using comoving coordinates the metric can be written

ds 2 = -e 2Φ dt 2 + e 2Λ da 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) ( 9 )
with Φ and Λ depending each on a and t. In this expression, the function Λ is related to m [START_REF] Misner | [END_REF] by

e Λ = 1 + U 2 - 2m r -1 2 ∂r ∂a , (10)
with U = e -Φ ∂r ∂t the comoving proper-time derivative of r and

m(r) = r 0 4πr 2 ρdr = a 0 ρ 1 + U 2 - 2m(r ) r 1/2 dV, (11)
where dV = 4πr 2 e Λ da is proper volume. For any given system (thus for assigned local thermodynamic quantities), and for each a, entropy is assigned irrespective of U . From first equality in (11) with r = R, we see M grows with R so that the configurations which maximize S M R are those with minimum M . Since any U = 0 implies an increase of M , these configurations correspond to U = 0 ∀a, namely they are time-symmetric, as it could have been expected from what mentioned above. This means that in our search for a proof of the UEB we can restrict the consideration to the U = 0 case. We have

S i M i = s i ρ i l i r i -r i-1 = s i ρ i 1 α i ≤ s i ρ i + p i 2 α i = πl * i 2 α i ≤ πl * i 8m i l i ≤ 8πm i , (12)
where the index i labels the quantites for the shell i as before and

α i ≡ 1 -2mi ri 1/2
. Here very small l i are assumed. The first inequality comes from causality (according to [26]); the second from

1 αi ≤ 1 α * i
, where α * i denotes the α at a proper distance l i from the horizon of a Schwarzschild black hole with mass m i (and

1 α * i = 4mi
li , see e.g. [START_REF] Misner | Gravitation[END_REF]); the third corresponds to (1). Using (12) we get

S M = S i M = 1 M S i M i M i ≤ 1 M 8πm i M i = 1 M M 0 8πmdm = 4πM ≤ 2πR, (13)
that is the UEB ( 5) with E = M . In (12) the second inequality turns into an equality when 2m i and r i are just close enough that, if m i is put into the sphere "i -1", a black hole is formed. This happens when l i α i = r i -2m i , or l i = r i α i . If r i becomes closer to 2m i the inequality would be violated for the assigned l i , but we can choose a smaller l i , such that the equality holds true. Continuing this process we reach a situation which demands for l i = λ i or smaller. We meet here a crucial point. We have to face how the notion of black hole should be modified in a context in which the quantum nature of matter is taken into account. A macroscopic state with r i α i < λ i seems impossible to be conceived due to the quantum uncertainty in the position of the constituent particles. In such a state in fact, as a fraction of the wavelength of the most external particles extends to r < 2m i , there would be a non-vanishing probability to have an horizon. As a consequence, other particles, also due to their own random motion, could find themselves inside the just considered horizon, which would have to be larger, and so on, so that the (logical) process would end only with all particles inside. This is a quantum-mechanical argument regarding the macroscopic state, irrespective of the motion of the particles in the gravitational field. This suggests we have to jump directly from the macroscopic state with r i α i = λ i to the black hole. According to this, the second inequality in ( 12) is always satisfied and is just attained when a black hole is formed.

Relation λ i ≤ r i α i , indeed, turns out to be the kind of inequality found in [12] (cf. eqs. ( 2) and (52) there) considering the physical feasibility of configurations arbitrarily close to the Schwarzschild radius, given the intrinsic quantum-mechanical size of constituent particles. 7 Looking at (12) the attaining of the UEB limit by not collapsed bodies is possible only with stiff matter. 8 We notice a benefit from gravity. In flat spacetime, we have seen, the configurations able in principle to attain the UEB ( 5) are from a statistical-mechanical point of view rather peculiar, as they correspond to systems with R ≈ λ. Thanks to gravity, we can consider equivalent systems (that is with a same S/E R just attaining the UEB) at any radius, with sound statistical-mechanical meaning.

If we consider a plane layer thin enough to allow for statistical-mechanical quantities to be considered constant in it, from (1)-( 2) we have S ≤ π(ρ + p)Al • l, with A the surface area of the layer and l the proper thickness. For weak-field conditions, this expression becomes

S ≤ π (E + pAl) • l, (14)
with E the total mass-energy of the layer of matter. When p ρ inequality ( 14) can be written as

S ≤ πEl, (15)
which is the expression of the "generalized" Bekenstein bound given in [11] (eq. ( 15) there; with l intended as the smallest size of the body) when applied to this layer, and which is tighter, for non-spherically-symmetric bodies as in the present case, than the original one of Bekenstein. For non-negligible p, the general bound to the entropy on the layer can be expressed as

S ≤ 2πEl, (16)
since p ≤ ρ if we require causality [26]. For spherical shells, inequalities ( 14), ( 15) and ( 16) remain unaffected, with l still denoting layer's thickness (and thus inequality (15) becomes here much stronger than what apparently implied by [11], which refers to the smallest size of the body, that is here the diameter 2R of the shell). In the limit of l approaching the radius R, inequality (16) becomes the UEB (5).

Let us consider a plane layer of matter just attaining the bound in inequality ( 16) (thus with l = λ = l * and ρ = p) with assigned area. Imagine to increase the energy per particle, at the same time reducing the thickness of the layer to coincide, still, with particle's new λ, so that the entropy in the layer is always at the limit. Let us imagine also, we do this while lowering energy density of the amount needed to leave total energy constant. No end can be, seemingly, envisaged for such a procedure from a quantum-mechanical standpoint, so that inequality (16) can hold true without any limit to the smallness of l. One additional information, not coming from quantum mechanics, must be introduced, namely the concept of gravitational collapse, to understand that beyond a certain limiting energy per particle (the Planck scale) particles behave as, let say, black holes, and any further increase of particle's energy necessarily brings to an increase in the thickness of the layer instead of a decrease (reflecting this, a modification of Heisenberg's relations at Planck scale) if the layer has not to become thinner than particle's size. And in this new scenario inequality (16) will hold true anyway.

We notice that a decreasing λ implies total entropy S ∝ l = λ decreasing as λ. Entropy per unit area σ in the region where the particle is localized, goes, however, as S λ 2 and thus as 1 λ , that is it increases when λ decreases. When λ approaches the Planck scale, σ reaches its maximum value which, as the 2-surface near the particle can no longer be considered plane, is σ max = S A bh = 1 4 ; any further increase in particle's energy cannot give any variation to this value of σ. The maximum entropy per unit area of a layer with thickness equal to constituent particle's size is thus given by σ max = 1 4 1 l 2 p , being l p the Planck length. From inequality (16) taken alone, σ could have grown instead without any limit (σ ∼ 1 λ ). We see that if quantum mechanics means setting a limit 7 We find that the violation (at ∼ Planck scale) of the Bousso bound, and of the UEB, found in [START_REF] Geršl | [END_REF], appears to be accompanied by a violation of this condition in that at the circumstances considered in [START_REF] Geršl | [END_REF] as challenging the bound, the radius of the system turns out to be smaller than λ/ 1 -2M R . 8 For gravitating systems, the peculiarity of stiff matter can be seen also in the approximate area scaling of entropy of stiff-matter balls (beyond a certain radius depending on central pressure) [31] (see also [32]).

on specific entropy (entropy per unit energy), gravity does not produce any change to this limit but amounts to set, from it, a limit on absolute entropy per unit area.

In conclusion, in this section we have given a proof of the UEB for any strength of the gravitational interaction using holography in its most general semiclassical formulation as given by the GCEB. This has been done using the l * concept. In the following section the connection of the UEB with the η/s bound and with the proliferation of species problem is considered, still from l * standpoint.

IV. SOME RELATED ISSUES

In [4], the UEB ( 5) is used to put a local constraint on s/ρ values for a fluid,

s ρ ≤ 2πL, ( 17
)
in terms of the correlation length L of the material medium, arguing that this is the minimum size for a parcel of fluid to have a consistent fluid description (i.e. with the usual equations of dissipative hydrodynamics). In the perspective we have presented here, we obtain (see ( 16))

s ρ ≤ 2πl * ≤ 2πλ, (18)
with l * given by ( 2). Here the first inequality comes simply from p ≤ ρ. Bound ( 18) is, in general, much stronger than bound (17). If we think for example to an ordinary gas, L is given by the mean free path, and is much larger than the quantum uncertainty λ on the positions of the molecules. The fact that ( 3) is in general a strong inequality (whichever is the state of the material medium, i.e. if gaseous, liquid or solid) gives anyway that bound (18) is in general satisfied to spare. For an assigned material medium a bound of the kind ( 17) cannot be the final word on s/ρ. L, in fact, changes in general with s and ρ, even when intensive thermodynamic parameters and s/ρ ratio are held fixed. Assuming thus that, while holding the ratio s/ρ fixed, conditions can always be envisaged for which the correlation length is reduced to the particle's λ corresponding to the assigned intensive parameters (below λ we cannot go, due to quantum correlation), starting from (17) we are brought eventually to (18). Some consequences can be drawn from this, concerning the viscosity to entropy ratio η/s. For nonrelativistic systems, assuming η = 1 3 Lρa (a is thermal velocity) [START_REF] Huang | Statistical Mechanics[END_REF], we get

η s = 1 3
Lρa s

1 3 L ρ + p s a = 1 3π L l * a, (19)
where last equality comes from (2). We can consider now another configuration of the same system which has minimum η/s for the assigned intensive parameters. To this aim we act as described above on ρ, s and p, while keeping the intensive variables unchanged (and ρ+p s and λ), till obtaining L approaches its quantummechanically limiting value λ, if feasible. We have

η s min = 1 3π λ l * a. (20)
Now if, to try to understand what is going on, we consider a Boltzmann gas, we have (from [2]) λ l * ∝ m T ∝ 1/a (being 1 3 ma 2 = T [START_REF] Huang | Statistical Mechanics[END_REF]), with m the mass of constituent particles, and this means that in the expression (20) no dependence on a is left. Using the expression for λ l * given in [2] (with the parameter χ 1 there, since we are assuming L, the mean free path, is approaching λ) we obtain η s min ≈ 0.7 (for g = 2, being g the number of degrees of freedom per particle). Thus we notice that: i) the KSS bound [13] η/s ≥ 1/4π seems satisfied even if a 1 (see also [4]); ii) considering (along the lines of [4]) the KSS bound also as an entropy bound (s ≤ 4πη), (nonrelativistic) systems which are far away, by many orders of magnitude, from the attaining of (18) (s 2πρλ), can anyway be near to saturate the KSS bound, so that the latter appears for such systems much tighter than (18) intended as an entropy bound.

For ultrarelativistic systems, if we refer to a fluid consisting of particles whose statistical equilibrium is determined completely by collisions through radiation quanta (being, these, photons or neutrinos, or gluons), assuming that η ≈ 1 3 τρ γ [START_REF] Misner | [END_REF], where τ is the average time for a quantum to collide and ρ γ is the energy density of radiation, and that the contribution to total entropy from radiation is dominant, we get

η s ≈ 1 3 τ ρ γ s γ = 1 4 τ ρ γ + p γ s γ = 1 4π τ l * γ = 1 4π L l * γ (21)
where the quantities with subscript γ refer to radiation and as correlation length L we intend the average path for a quantum to collide. At conditions, if allowed, such that L has its limiting value, namely the wavelength itself λ γ of the quantum of radiation, the ratio η s has its minimum given by

η s min ≈ 1 4π λ γ l * γ = 1 4π , (22)
which just corresponds to attain, through conventional quantum mechanical arguments, the string theoretical KSS bound. Here, last equality follows from the saturation of the bound (3) for ultrarelativistic fluids (with µ = 0) [2]. As far as the mentioned conditions can be considered conceivable in the quark-gluon plasma, what we have just obtained could be considered also as a back-of-the-envelope description of the very low values of η s [35,36] experimentally found for it (see also [4]). 9 We see that the viscosity bound, as the UEB, can be thought as coming from l * concept (inequality (3)). Some consequences of the results we have presented, can also be drawn concerning the so-called proliferation of species problem. The problem [3,12,20,38] (see also [15] and references therein) is in that if one allows for an unlimited proliferation of particle species, one should expect the entropy in a box at fixed energy to challenge the entropy limit envisaged by the UEB. Indeed, also the statistical-mechanical main proof of the UEB as given in [22], refers to a limited number (a very liberal limit, actually) of non-interacting fields.

Many ways to face this problem have been proposed in the years ( [19] and [15], and references therein), and we refer the reader to [19] for a quite recent review on this. N could be intrinsically required to be limited for physics to be consistent (see [39], concerning the stability of the quantum vacuum itself). However, even if N could be allowed to be arbitrary, something which is assumed neglibible in present arguments in favour of the UEB, and in its counter-arguments, could turn out however to be essential when N is large, so that the UEB could still be satisfied [19]. As a general point we could say that the UEB could be true whatever the consistency of physics demands to N , i.e. whether it must be limited or can be arbitrary (see also [21]).

This is also what is obtained in our approach. Condition (3) translates in this case to the request that some representative value of the collection of λ's (corresponding to the different species) be not smaller than l * , as defined by (2). This looks quite insensitive to the number of species. If for example we consider N copies of the electromagnetic field at a given T , with N arbitrary, denoting with ρ N , s N , ... the quantities referring to the system with N copies and with ρ 1 , s 1 , ... the quantities for the case of one copy, we get ρ N = NbT 4 and s N = 4 3 NbT 3 with b a constant, so that l * N = l * 1 , and condition (3), which is in this case λ ≥ l * N , is satisfied, attained to be more precise, as in the case with only one copy. For the chosen circumstances however also the ratio S/E is insensitive to N .

If the N copies have to be taken with total energy fixed, in the N -copy case the temperature must be different, T → T , with T satisfying NT 4 = T 4 . In this case ρ N = ρ 1 and s N = 4 3 NbT 3 = N 1/4 s 1 , so that l * N = N 1/4 l * 1 . Also photon wavelengths however must be different, λ = 1 πT → λ = 1 πT and thus, still, condition (3), namely λ ≥ l * N if we have N copies, is satisfied (attained) as for the one-copy case. The proliferation of species, thus, could be not a problem for condition (3), too. It would hold true regardless of the properties of N , being it bounded or arbitrary.

In conclusion, in this paper a proof of the UEB, the universal entropy bound, for any strength of gravity has been given, using holography in its most general semiclassical formulation as expressed by the GCEB, the generalized covariant entropy bound. A modified formulation and a proof of the UEB for plane layers and for spherical shells has been also proposed. What has been used in the proofs is only the l * concept (equation (2) and inequality (1) or (3)), a concept essential for the GCEB but fully meaningful also in a world without gravity. In such a world the GCEB would lose any meaning, contrary to the UEB. The emerging perspective is thus that both the GCEB and the UEB arise from the l * concept, being the former obtained when gravity is turned on and the latter straight from l * without any reference to gravity. Some consequences from l * standpoint as for the η/s bound and the proliferation of species problem have also been drawn.

NOTE ADDED IN PROOF

When the present manuscript was (almost) completed new results have been published dealing with the relation between the viscosity bound and the generalized second law of thermodynamics [40]. The KSS bound is derived in them using an approach completely different from that presented here, but still relying on the existence and value of the minimum scale l * .

Through the consideration of single-particle systems, in[11,17] the role, under weak field conditions, of the UEB (or, at these conditions, of the GCEB) in predicting the uncertainty principle has been stressed.

Other statistical-mechanical arguments which give support to (5) can be found in[23,24], in addition to the original argument[START_REF] Bekenstein | [END_REF].

A partition like this is supposed to be always feasible, on the basis of the notion itself of local equilibrium.

An extension of the UEB could however be envisaged, in which S and E are the entropy and energy not of the whole system but instead of the part on the future-directed ingoing lightsheet from the boundary. In these circumstances the UEB would appear to coincide with the covariant entropy bound.

Regarding this point of the role that ordinary quantum mechanics can play in providing a very low η/s, see also[START_REF] Gyulassy | NATO/ASI: Structure and dynamics of elementary matter[END_REF].