Hawking effect for a toy model of interacting fermions

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Hawking effect for a toy model of interacting fermions

Patrick Bouvier, Christian Gérard

To cite this version:

Patrick Bouvier, Christian Gérard. Hawking effect for a toy model of interacting fermions. 2013.

�hal-00869929�

OF INTERACTING FERMIONS

P. BOUVIER AND C. G ´ERARD

Abstract. We consider a toy model of interacting Dirac fermions in a 1 + 1 dimensional space time describing the exterior of a star collapsing to a black- hole. In this situation we give a rigorous proof of theHawking effect, namely that under the associated quantum evolution, an initial vacuum state will con- verge whent→+∞to a thermal state at Hawking temperature. We establish this result both for observables falling into the blackhole along null character- istics, and for static observables. We also consider the case of an interaction localized near the star boundary, obtaining similar results. We hence extend to an interacting model previous results of Bachelot and Melnyk, obtained for free Dirac fields.

1. Introduction

1.1. Introduction. The Hawking effect, see Hawking [Ha], predicts that in a space- time describing the collapse of a spherically symmetric star to a Schwarzschild black hole, an initial Boulware vacuum state will become an Unruh state at the future horizon: a static observer at infinity sees the Unruh state as a thermal state at Hawking temperature.

Despite the vast physical literature on the Hawking effect, there are few mathe- matically rigorous justifications of the Hawking effect. Dimock and Kay [DK1, DK2]

gave a construction of the Unruh state in the Schwarzschild space-time and on its Kruskal extension, using scattering theory for Klein-Gordon fields.

The first mathematical proof of the Hawking effect, in the original setting of Hawking, is due to Bachelot [Ba1]. Bachelot considered a linear Klein-Gordon field in the exterior of a spherically symmetric star, collapsing to a Schwarzschild black hole. This result was extended to linear Dirac fields in the same situation, first by Bachelot [Ba2], and then by Melnyk [Me]. The only proof to date in a non- spherically symmetric situation is due to H¨ afner [H], who gave a rigorous proof of the Hawking effect for Dirac fields for a star collapsing to a Kerr black hole.

The common theme of all the above mentioned results is that they deal with linear quantum fields: the time evolution of observables is implemented by a group of linear (symplectic or unitary) transformations on the phase space, and all the states are quasi-free.

This means that the problem can be reduced to a question about linear partial differential equations, with boundary conditions on the star boundary. The Hawk- ing effect emerges from the fact that the star boundary becomes asymptotically characteristic for large times. This leads to an exponentially fast concentration of Klein-Gordon or Dirac wave packets reflected by the star, which ultimately implies the Hawking effect.

Date: October 4, 2013.

2010Mathematics Subject Classification. 81T10, 81T20.

Key words and phrases. Hawking effect, interacting fermions, 1−dDirac equations.

1

In this paper we investigate the Hawking effect for a toy model of interact- ing Dirac fermions in 1 + 1 space-time dimensions. A mathematical discussion of interacting quantum fields is of course difficult, because there are few rigorous constructions of interacting quantum fields, even on Minkowski space.

For Klein-Gordon fields, there are the well-known constructions of the P(ϕ)

and ϕ

models due to Glimm and Jaffe, which were the main successes of the con- structive program from the seventies. We are not aware of any similar construction on a space-time which describes the exterior of a collapsing star, even when the interaction contains an ultraviolet and space cutoff.

For Dirac fields, the situation looks better, since fermionic fields are bounded, which in some situations allows to construct the interacting dynamics in a purely al- gebraic setting, independently of the choice of a representation. This is particularly convenient in the situation that we consider, since, even for free Dirac fields, two Fock representations in the exterior of the star at different times are inequivalent.

1.2. A toy model. To concentrate on the possibly new features introduced by the non-linear interactions and to keep the situation simple and manageable, we restrict ourselves to a toy model of Dirac fermions in 1 + 1 space-time dimensions:

we consider only 2 components spinors, and the effect of the metric is modeled by a vector potential. Note that if we forget about the non-linear interaction, our model is essentially identical to the one considered by Bachelot in [Ba2], after introduction of polar coordinates and suitable spin spherical harmonics.

Let us now briefly describe the model: the space-time is the region:

M = {(t, x) ∈ R

: x > z(t)},

where x = z(t) is the star boundary. We assume that z(t) ≡ z(0) for t ≤ 0, i.e.

the star is stationary in the past, the collapse starting at t = 0. As in [Ba2] we assume that z(t) ∼ −t − Ae

for t → +∞, i.e. the star boundary becomes asymptotically characteristic for large positive times.

The Dirac fields are two-components spinors ψ(t, x) ∈ C

, solving (in absence of interaction) the Dirac equation:

(1.1)

( ∂

ψ(t, x) + L∂

ψ(t, x) + iV (x)ψ(t, x) = 0, in z > z(t), ψ

(t, z(t)) = λ(t)ψ

(t, z(t)),

where L =

1 0 0 −1

and V (x) = V (x)

∈ M

( C ) is a matrix-valued potential representing the influence of the metric, with

V (x) → 0 at − ∞, V (x) → mΓ at + ∞, m > 0 is the mass of the field, and Γ ∈ M

( C ) satisfies

Γ = Γ

, Γ

= 1l, ΓL + LΓ = 0.

The reflection coefficient λ(t) equals

, so that the L

norm ˆ

z(t)

kψ(t, x)k

dx

is conserved. This implies that if h

:= L

(]z(t), +∞[; C

), the evolution group u

(s, t) : h

→ h

(see Subsect. 2.2) associated to (1.1) is unitary, and hence generates a fermionic dynamics

τ

(s, t) : CAR(h

) → CAR(h

),

where CAR(h) is the CAR C

−algebra associated to a Hilbert space h.

The self-interaction of the Dirac field is described by a perturbation of the form I = (ψ

(g)M ψ(g))

,

where n ≥ 2, M ∈ M

( C ) is a selfadjoint matrix, g ∈ L

( R ) is a compactly supported function. The associated interacting Dirac fields ψ

(t, x) formally solve the following non-linear Dirac equation:

(1.2)



 



 



∂

ψ

(t, x) + L∂

ψ

(t, x) + iV (x)ψ

(t, x)

− in(ψ

(t, g)|M ψ

(t, g))

C²

M ψ

(t, g)g(x) = 0, in x > z(t) ψ

(t, z(t)) = λ(t)ψ

(t, z(t)),

where ψ

(t, g) := ´

ψ

(t, x)g(x)dx ∈ C

. The properties of the interaction which are essential for our analysis are the following:

(1) I is bounded, which allows for a purely algebraic construction of the interacting dynamics τ

(s, t);

(2) I is even, which is the standard assumption needed to ensure locality, (3) I is localized in a (space) compact region.

1.3. Results. Let us now describe the results of the paper.

The first step is to construct interacting Dirac fields, i.e. to quantize the non- linear Dirac equation (1.2).

Since we deal with fermions, the interaction term I above is bounded, and one can work in a purely algebraic setting: one can introduce C

−algebras A

= CAR(h

) of observables at time t, and it is easy to construct the interacting dynamics τ

(s, t) (see Sect. 4), which is a two parameter group of ∗−isomorphisms from A

to A

describing the time evolution.

We investigate the Hawking effect in three different situations.

1.3.1. Hawking effect I. In the first situation we take an observable at time t, lo- calized near the star boundary x = z(t), i.e. of the form α

(A) for some A ∈ A

, where α

is the group of left space translations. In terms of interacting space-time fields ψ

, a typical observable would be ψ

(t, x − t), i.e. a field falling into the black hole along null characteristics. This is the analog for interacting fields of the situation in [Ba2].

To evaluate the time-evolved state at time t acting on α

(A) we have to evolve α

(A) back to time 0, which yields

ω

(τ

(0, t) ◦ α

(A)),

where ω

is the vacuum state at time t = 0, τ

(t, 0) is the interacting dy- namics. Our goal is to compute the limit of the above quantity when t → +∞. We prove in Thm. 5.6 that the limit

(1.3) lim

t→+∞

ω

(τ

(0, t) ◦ α

(A)) = ω

(A) exists,

for any A in the C

−algebra A

. Let us describe the limiting state ω

, which is close to the one obtained by Bachelot in [Ba2]:

the algebra A

splits into the ( Z

−graded) tensor product A

⊗A b

(see Subsect.

A.2) of the left/right moving observables.

The limit state ω

acts on right moving observables as a vacuum state (com- posed with an appropriate wave morphism), while on left moving observables it acts as the thermal state ω

at inverse Hawking temperature β = 2πκ

, for the eternal black hole without interaction.

We also prove a similar result if the initial state ω

is replaced by another state

˜

ω which is even and belongs to the folium of ω

(see Corollary 5.8). As example

of such a state, one can choose an interacting vacuum state, whose existence is shown in Subsect. 5.5.

The first situation is graphically summarized in Figure 1 below: the grey re- gion is the support of the non-linear self-interaction. The curve x = z(t) is the star boundary. The dashed lines are the (backwards) characteristics for the Dirac equation, starting from the support of an observable at time T : left moving charac- teristics are reflected on the star boundary and asymptotically concentrated when T → +∞.

x

t t=T

x=z(t)

interaction region

Figure 1. Hawking effect I

1.3.2. Hawking effect II. In the second situation the observable A at time t is localized near the origin. In terms of space-time fields, a typical example would be simply ψ

(t, x). This is the analog for interacting fields of the situation considered by Melnik in [Me].

The situation is now more complicated: one has to be sure that the observable A, under backwards propagation, will split into left and right moving parts. One way to formulate this property is to introduce the (future) wave morphism γ

between the dynamics on the eternal black hole τ

and τ

(see Thm. 6.5). Then we have to require that A belongs to γ

A

. Observables outside this ∗−subalgebra will not see the Hawking effect.

It is easier to formulate our result if we assume the asymptotic completeness of γ

, i.e. that γ

A

= A

: then we prove in Thm. 6.18 that the limit

(1.4) lim

t→+∞

ω

(τ

(0, t)(A)) = ω

(A) exists, for A a local element of A

(i.e. A ∈ A

for some interval J b R ).

Without assuming asymptotic completeness, we have to restrict ourselves to observables A ∈ γ

A

. Such observables do not necessarily belong to A

for t large, i.e. the expression τ

(s, t)(A) may have no meaning. Therefore we replace A by E

A ∈ A

, where E

is the natural projection A

→ A

(see Remark 6.7).

Let us now describe the limiting state ω

. Again the algebra A

splits into a tensor product CAR(P

h

)b ⊗CAR(P

h

) of left/right moving observables (see Subsect. 6.1). In this case elements of CAR(P

h

) are left/right moving only asymptotically for large times.

On right moving observables the limit state ω

acts again as a vacuum state,

composed with a wave morphism. On left moving observables it acts as the thermal

state ω

. In contrast to case I, the potential term V is present in the thermal

state.

A similar result holds if we replace the initial state by another even, state ˜ ω be- longing to the folium of ω

, see Corollary 6.19. However we have now to assume that ˜ ω is invariant under the interacting stationary dynamics, τ

, describing the interacting Dirac field in the past.

Fig. 2 summarizes the second situation, with the same conventions as in Fig.

1: note that left moving characteristics starting at time T from close to the origin, reach the star boundary at time close to T /2: after time T /2 the situation for left moving observables is similar to case I.

x

t t=T

t=T/2 x=z(t) interaction region

Figure 2. Hawking effect II

1.3.3. Hawking effect III. In the two previous situations, the interaction region is far away from the star boundary: the effect of the self-interaction is decoupled from the effect of the asymptotically caracteristic boundary, which is essential in the Hawking effect.

For an initial observable starting at time T close to the star boundary z = z(T), the Hawking effect (in the free situation), is essentially due to what happens between the times T and T − 1, i.e. to the reflection on the asymptotically characteristic star boundary. Therefore we consider a third situation where the interaction is localized near the star boundary for times t ∈ [T − 1, T ]. We consider the following time-dependent interaction

I

(t) = 1l

(t)α

(I),

which is at time t localized near the star boundary x = z(t), and vanishes for t 6∈ [T − 1, T ]. We denote by ˜ τ

(s, t) the dynamics obtained as before by adding to the free dynamics τ

(s, t) the time-dependent interaction I

(t). We obtain a dynamics depending on the parameter T , which differs from the free dynamics τ

(s, t) only for T − 1 ≤ s ≤ t ≤ T . We show in Thm. 7.7 that the limit

T

lim

ω

(˜ τ

(0, T ) ◦ α

(A)) = ω

(A) exists

for A ∈ A

. The limiting state ω

is actually quite explicit, being the pullback

of the (free) limiting state ω

obtained by Bachelot in [Ba2] by a simple effective

interacting dynamics ˆ τ

(0, 1). The dynamics ˆ τ

(s, t) describes the combined

effect of interaction and reflection on the star boundary between times T + t and

T + s, in the limit T → +∞. The situation is summarized in Fig. 3 below.

x

t t=T

x=z(t)

interaction region

t=T-1

Figure 3. Hawking effect III

1.4. Plan of the paper. Let us now briefly describe the plan of our paper. In Sect. 2 we describe our geometrical setup and recall some results of [Ba2] about the linear case. The corresponding results for quantum dynamics are recalled in Sect. 3.

In Sect. 4 we construct the interacting dynamics in the algebraic, i.e. represen- tation independent setting, by adapting standard perturbation arguments.

Sect. 5 resp. Sect. 6, Sect. 7 are devoted to the proof of the Hawking effect in the first, resp. second and third setup. In Appendix A we recall some stan- dard facts about CAR algebras, the fermionic exponential law and perturbations of C

−dynamics.

1.5. Notations. If h

are Hilbert spaces i = 1, 2 we write T : h

→ ˜ h

if T ∈ B (h

, h

) is bijective with bounded inverse. We will use the same notation if A

are C

−algebras and T : A

→ A

is a ∗−isomorphism.

Various objects in the text, like Hilbert spaces, selfadjoint operators, C

−algebras,

∗−morphisms or states, are decorated with sub- and supercripts. As a rule sub- scripts are used to label a time or a time interval, while superscripts are used to label the various interaction terms, like 0 for no interaction, V for interaction po- tential, or int for the non-linear interaction. Superscripts l/r are also used to denote left/right moving observables. Subscripts vac and β in states are used to denote vacuum or thermal states, at temperature β

.

2. Classical free dynamics

In this section we describe our setup and recall some results of [Ba2] about the free classical dynamics. We also collect some additional results which will be important in later sections.

2.1. Notations and hypotheses.

2.1.1. Collapsing star. We first recall the framework of Bachelot [Ba2], describing a star collapsing to a black hole, in a 1 + 1 dimensional space-time.

The space-time is

M := {(t, x) ∈ R

: x > z(t)}

where the star boundary is x = z(t) with:

z ∈ C

( R ), z(t) = z(0), t ≤ 0,

z(t) = −t − Ae

+ ζ(t), t ≥ 0, (2.1)

− 1 ≤ z(t) ˙ ≤ 0, t ≥ 0, for A, κ > 0 and

(2.2) |ζ(t)| + | ζ(t)| ≤ ˙ Ce

, t ∈ R , C > 0.

The reflection coefficient on the star boundary is:

λ(t) = 1 + ˙ z(t) 1 − z(t) ˙

.

Without loss of generality we can assume that z(0) = 0. The second condition in (2.1) means that the collapse start at t = 0, the star being stationary in the past.

2.1.2. Dirac operators. We now define various one dimensional Dirac operators.

We set

h

:= L

(]z(t), +∞[, C

), t ∈ R , h

:= L

( R , C

),

h

:= L

(J, C

), J b R interval.

We set

L :=

1 0 0 −1

and fix a matrix-valued potential (representing the influence of the metric):

R 3 x 7→ V (x) ∈ M

( C ), with V = V

, V ∈ C

( R ), and:

(2.3)

|V (x) − V (∞)| + hxi|V

(x)| ∈ O(hxi

), x → +∞,

|V (x)| + hxi|V

(x)| ∈ O(hxi

), x → −∞, for some > 0. We assume that

V

= mΓ, Γ ∈ M

( C ), where m > 0 is the mass of the field and

Γ = Γ

, Γ

= 1l, ΓL + LΓ = 0.

Let us now introduce Dirac operators; We set:

(2.4) b

:= iL∂

− V (x) acting on h

, with domain

Dom b

= {u ∈ H

(]z(t), +∞[, C

) : u

(z(t)) = λ(t)u

(z(t))}, and:

(2.5) b

:= iL∂

− V (x) acting on h

with domain

Dom b

= H

( R , C

).

2.2. Classical free dynamics. The classical free dynamics is generated by the following Dirac equation:

(2.6)



 



 



∂

ψ(s, x) + L∂

ψ(s, x) + iV (x)ψ(s, x) = 0, in x > z(s), s ∈ R , ψ

(s, z(s)) = λ(s)ψ

(s, z(s)), s ∈ R ,

ψ(t, x) = ψ(x), in x > z(t).

In this subsection we recall some results of [Ba2], about the existence and properties of solutions of (2.6).

Definition 2.1. A {u(s, t)}

with values in B(h

, h

) is called a (two-parameter) propagator if:

i) u(s, t) ∈ U(h

, h

), ii) u(t, t) = 1l, t ∈ R ,

iii) u(s, t

)u(t

, t) = u(s, t), s, t

, t ∈ R ,

iv) ∀ (s

, t

) ∈ R

, ∀J b ]z(t

), +∞[ ∀f ∈ h

the map (s, t) 7→ u(s, t)f ∈ h

is continuous at (s

, t

).

In the above definition we denoted by U (h

, h

) the group of unitary operators from h

to h

.

Note that condition iv) is the appropriate replacement for the strong continuity of (s, t) 7→ u(s, t) in the case h

≡ h.

The following result can be found in [Ba2].

Theorem 2.2. Assume the hypotheses in Subsect. 2.1. Then there exists a unique propagator u

(s, t) ∈ B (h

, h

) such that:

u

(s, t) : Dom b

→ Dom b

, s, t ∈ R ,

∂

u

(s, t) = ib

u

(s, t) on Dom b

,

∂

u

(s, t) = −iu

(s, t)b

on Dom b

.

It follows that if ψ ∈ Dom b

, then ψ(s, x) = u

(s, t)ψ(x) solves (2.6) in the strong sense. For the Dirac equation without boundary condition we will set ac- cordingly:

u

(s, t) := e

∈ U (h

, h

).

2.3. Additional results. In this subsection we collect some known results from Bachelot [Ba2] about the classical dynamics u

(s, t). For free Dirac fields outside of a collapsing star, they are sufficient to obtain a proof of the Hawking effect, as done in [Ba2]. In the toy model of interacting Dirac fields that we consider, they will also be important.

We first define the left translations:

Definition 2.3. If f ∈ h

, we set f

(·) := f (· + t) ∈ h

.

2.3.1. Finite propagation speed. We first collect some properties of finite propaga- tion speed for u

(s, t) and u

(s, t).

Proposition 2.4. (1) if suppf ⊂ [R, +∞[ then suppu

(s, t)f ⊂ [R +|t−s|, +∞[;

(2) if suppf ⊂ [a, b] then suppu

(s, t)f ⊂ [a − |t − s|, b + |t − s|];

(3) if suppf ⊂ [0, R] then suppu

(s, t)f

⊂ [z(s), R − s] for all s ≤ t.

Proof. the proof of (1) can be found in [Ba2]. (2) follows from classical arguments,

see e.g. [CP]. (3) is shown in [Ba2, Proof of Thm. VI.5]. 2

Statement (2) of Prop. 2.4 and the uniqueness in Thm. 2.2 imply the following fact:

Proposition 2.5. Let J b R an interval. Then there exists c ≥ 0 such that u

(s, t)f = u

(s, t)f, ∀ f ∈ h

, c + t/2 ≤ s ≤ t.

2.3.2. Scattering results. One can split h

as direct sum:

h

= h

⊕ h

, for

(2.7) h

:= {f = (f

, f

) ∈ h

: f

= 0}, h

:= {f = (f

, f

) ∈ h

: f

= 0}.

If f ∈ h

, we denote by f

its orthogonal projection on h

. If V ≡ 0 we easily see that:

(2.8) u

(0, t)f = f

, f ∈ h

, u

(t, 0)f = f

, f ∈ h

. Proposition 2.6. The strong limit

w

:= s− lim

t→+∞

u

(0, t)u

(t, 0) exists on h

.

Proof. See [Ba2, Prop. VI.4]. 2 Proposition 2.7.

w− lim

t→+∞

u

(0, t)f

= 0, ∀ f ∈ h

.

Proof. We follow some arguments in [Ba2]. By density we can assume that f ∈ h

is compactly supported. We write for 0 ≤ T ≤ t:

ku

(T, t)f

− u

(T, t)f

k = ku

(t, T )u

(T, t)f

− f

k

= k ´

T

u

(t, σ)V u

(σ, t)f

dσk ≤ ´

T

kV u

(σ, t)f

kdσ.

By Prop. 2.4 (3) we know that suppu

(σ, t)f

⊂ [z(σ), R − σ] for some R ≥ 0, hence by hypothesis (2.3) we have kV u

(σ, t)f

k ∈ O(hσi

). It follows that

(2.9) lim

T→+∞

sup

T≤t

ku

(T, t)f

− u

(T, t)f

k = 0.

Next we write

u

(0, t)f

= u

(0, T )u

(T, t)f

+ u

(0, T ) u

(T, t)f

− U

(T, t)f

= u

(0, T )u

(T, 0)u

(0, t)f

+ u

(0, T ) u

(T, t)f

− U

(T, t)f

. We know from [Ba2, Lemma VI.8] that w− lim

u

(0, t)f

= 0. Using (2.9) and an /2 argument we obtain the proposition. 2

2.3.3. Limits of quasi-free states. The following theorem is the key result of [Ba2].

Theorem 2.8. For f ∈ h

one has:

t→+∞

lim (u

(0, t)f

|1l

(b

)u

(0, t)f

) = (f |(1l + e

)

f ).

The analogous result for f ∈ h

follows immediately from Prop. 2.6 and (2.8).

Proposition 2.9. For f ∈ h

one has:

t→+∞

lim (u

(0, t)f

|1l

(b

)u

(0, t)f

) = (w

f |1l

(b

)w

f ).

We recall that (f |1l

(b

)f ) is the covariance of the quasi-free vacuum state for

the Dirac field in the exterior of the star at t = 0, while (f|(1l + e

)

f ) is

the covariance of the thermal state at Hawking temperature κ/2π near the black

hole horizon.

3. Free quantum dynamics

In this section we define the free quantum dynamics corresponding to the classical dynamics constructed in Subsect. 2.2.

Let us first introduce some notation. For t ≥ 0 we set A

:= CAR(h

), A

:=

CAR(h

) and for an interval J b R , A

:= CAR(h

) (see Subsect. A.1). Note that A

, A

⊂ A

isometrically.

We start by a definition analogous to Def. 2.1.

Definition 3.1. A family {τ(s, t)}

is a (two-parameter) quantum dynamics if:

i) τ(s, t) : A

→ ˜ A

, ii) τ(t, t) = 1l, t ∈ R ,

iii) τ(s, t

)τ(t

, t) = τ(s, t), s, t

, t ∈ R ,

iv) ∀ (s

, t

) ∈ R

, ∀J b ]z(t

), +∞[ ∀A ∈ A

the map (s, t) 7→ τ (s, t)A ∈ A

is continuous at (s

, t

).

Since u

(t, s) is a propagator, it generates a (free) quantum dynamics τ

(s, t).

Definition 3.2. We denote by τ

(s, t) the quantum dynamics defined by:

τ

(s, t)(ψ

(f )) := ψ

(u

(s, t)f ), f ∈ h

.

Similarly we define the quantum dynamics τ

(s, t), τ

(s, t) associated to u

(s, t) and u

(s, t).

Note that τ

(s, t) is a stationary quantum dynamics on A

, i.e. τ

(s, t) = τ

(s + t

, t + t

), for all s, t

, t ∈ R .

We also define the (one-parameter) dynamics α

on A

defined by (3.1) α

(ψ

(f )) := ψ

(f

), f ∈ h

.

The properties of propagators recalled in Subsect. 2.3 immediately carry over to quantum dynamics. For example the following fact follows from Prop. 2.5.

Lemma 3.3. Let J b R an interval. Then there exists c ≥ 0 such that τ

(s, t)A = τ

(s, t)A, ∀ A ∈ A

, c + t/2 ≤ s ≤ t.

4. Interacting quantum dynamics

In this section we construct the interacting dynamics τ

(s, t) that we will consider in the sequel. It will be obtained by perturbing the free dynamics τ

(s, t) by a bounded interaction term I localized in a bounded region of space. As usual, since we consider fermionic fields , interacting dynamics can be constructed at the algebraic level.

Formally the construction of the interacting dynamics τ

(s, t) defined in Def.

4.4 corresponds to the quantization of the following non-linear Dirac equation:

(4.1)



 

 

 

 

 

 

∂

ψ(s, x) + L∂

ψ(s, x) + iV (x)ψ(s, x)

− in(ψ(s, g)|M ψ(s, g))

C²

M ψ(s, g)g(x) = 0, ψ

(s, z(s)) = λ(s)ψ

(s, z(s)), s ∈ R , ψ(t, x) = ψ(x), in x > z(t),

where ψ(s, g) := ´

ψ(s, x)g(x)dx ∈ C

, M ∈ M

( C ) is a selfadjoint matrix and

g ∈ L

(J ) for some J b R is a compactly supported function.

4.1. Construction of the interacting dynamics.

Definition 4.1. Let M ∈ M

( C ) with M = M

. We set for g ∈ L

( R ):

ψ

(g)M ψ(g) :=

2

X

i=1

ψ

(g ⊗ e

)λ

ψ(g ⊗ e

) ∈ A

, where M = P

i=1

λ

|e

)(e

| (i.e. (e

, e

) is a basis of eigenvectors of M ).

We fix g ∈ L

(J ) for an interval J b ]z(0), +∞[ , 2 ≤ n ∈ N and set:

(4.2) I := (ψ

(g)M ψ(g))

∈ CAR

(h

).

The interaction term I represent a localized, even, self-interaction of the Dirac field in M.

Remark 4.2. All the results below extend immediately to the case when I is replaced by a finite sum of I

, associated to matrices M

and compactly supported space- cutoffs g

. The only important properties of I is that it should be even and localized.

For later use we state the following fact, which follows immediately from the CAR and the fact that I ∈ CAR

(h

).

Lemma 4.3. Let B = Q

i=1

ψ

(f

). Then there exists C

such that

(4.3) k[I, B]k ≤ C

n

Y

i=1

kf

k

n

X

i=1

|(g|f

)|.

Using the results of Subsect. A.6 we can now construct the interacting dynamics {τ

(s, t)}

.

Definition 4.4. Let I be as in (4.2) , I(s, t) := τ

(s, t)I and R(s, t) := R

(s, t) ∈ U (A

) be obtained as in Prop. A.11. We set:

(4.4) τ

(s, t)(A) := R(s, t)τ

(s, t)(A)R(s, t)

, A ∈ A

, s, t ∈ R

Then by Prop. A.11 {τ

(s, t)}

is a dynamics, called the interacting quantum dynamics.

For the convenience of the reader, we recall that R(s, t) ∈ A

solves:

(4.5)

( ∂

R(s, σ) = −iR(s, σ)I(s, σ), R(s, s) = 1l.

We can also define the corresponding interacting dynamics without boundary conditions, acting on A

. We set I

(s, t) := τ

(s, t)(I) and define R

(s, t) ∈ U (A

) as above and:

(4.6) τ

(s, t)(A) := R

(s, t)τ

(s, t)(A)R

(s, t)

, A ∈ A

, s, t ∈ R . Again τ

(s, t) is stationary.

Remark 4.5. Let us faithfully represent A

= CAR(h

) in the fermionic Fock

space Γ

(h

) (see Subsect. A.1) by the Fock representation π

. Then τ

(s, t) is

implemented in the Fock representation by the unitary group e

, where H

=

dΓ(b

) is the second quantization of b

. The dynamics τ

(s, t) is implemented

by e

for H

= H

+ π

(I).

4.2. Properties of τ

(s, t).

Lemma 4.6. There exists c ≥ 0 such that:

R

(s, t) = R(s, t), c + t/2 ≤ s ≤ t.

Proof. The interaction I defined in (4.2) belongs to A

for some interval J b R . We apply then Lemma 3.3 to each term in the series defining R(s, t), see Lemma A.10. 2

Lemma 4.7. Let J b R an interval. Then there exists c ≥ 0 such that τ

(s, t)(A) = τ

(s, t)(A), ∀ A ∈ A

, c + t/2 ≤ s ≤ t.

Proof. It suffices to apply Lemmas 4.6 and 3.3 to the definition of τ

, τ

. 2 5. Hawking effect I

In this section we study the Hawking effect in the situation referred to as case I in the introduction (see Subsect. 1.3). For t ∈ R we set A

:= CAR(h

) ⊂ A

, called the left/right moving observables.

The algebra A

splits into a twisted tensor product of the left/right moving CAR algebras A

. The first step consists in studying the evolution τ

(0, t) ◦ α

on left/right moving observables.

5.1. Left propagation.

Proposition 5.1. Let A ∈ A

. Then

t→+∞

lim τ

(0, t) ◦ α

(A) − τ

(0, t) ◦ α

(A) = 0.

To prove Prop. 5.1, we will need the following lemma.

Lemma 5.2. For any > 0 and A ∈ A

, there exists T such that sup

t≥T

kτ

(T, t) ◦ α

(A) − τ

(T, t) ◦ α

(A)k ≤ .

Proof. Let us set A(s, t) = τ

(s, t) ◦ α

(A) to simplify notation. We first claim that

(5.1) kτ

(s, t) ◦ α

(A) − A(s, t)k ≤ ˆ

s

k[I, A(σ, t)]kdσ.

Let us prove (5.1). By Def. 4.4 we have:

τ

(s, t) ◦ α

(A) − A(s, t) = R(s, t)A(s, t)R(s, t)

− A(s, t)

= [R(s, t), A(s, t)]R

(s, t), using that R(s, t) is unitary. Set :

F

(σ) := [R(s, σ), A(s, t)], G

(σ) := [I(s, σ), A(s, t)].

We note first that

G

(σ) = [τ

(s, σ)(I), τ

(s, t) ◦ α

(A)] = τ

(s, σ)([I, A(σ, t)]), using that τ

is an homomorphism. Since τ

is isometric, we have (5.2) kG

(σ)k = k[I, A(σ, t)]k.

Recalling that R(s, σ) solves

( ∂

R(s, σ) = −iR(s, σ)I(s, σ),

R(s, s) = 1l,

we see next that F

(·) solves the equation:

(5.3)

( ∂

F

(σ) = −iF

(σ)I(s, σ) − iR(s, σ)G

(σ), F

(s) = 0,

which clearly has a unique solution. We look for F

(σ) of the form F

(σ) = H

(σ)R(s, σ). We obtain the equation:

(5.4)

( ∂

H

(σ) = −iR(s, σ)G

(σ)R(s, σ)

, H

(s) = 0.

Since R(s, σ) is unitary we obtain

kτ

(s, t) ◦ α

(A) − A(s, t)k = kF

(t)k = kH

(t)k

≤ ´

s

kG

(σ)kdσ = ´

s

k[I, A(σ, t)]kdσ, which proves (5.1).

We can now complete the proof of the lemma. Assume first that A belongs to CAR

(h

) for some interval J ⊂ [0, R] (recall that z(0) = 0). By linearity we may assume that A = Q

i=1

ψ

(f

) with suppf

⊂ [0, R]. By Prop. 2.4 (3) we know that suppu

(σ, t)f

⊂ [z(σ), −σ + R] hence for σ ≥ σ(J ) we have [I, A(σ, t)] = 0 by Lemma 4.3, hence

(5.5) τ

(s, t)(A) − A(s, t) = 0, σ(J) ≤ s ≤ t, A ∈ CAR

(h

).

Let now A ∈ A

and > 0. By density we can choose J as above and ˜ A ∈ CAR

(h

) such that kA − Ak ≤ ˜ /2. Applying (5.5) to ˜ A we obtain T = σ(J ) such that

sup

T≤t

kτ

(T, t) ◦ α

(A) − A(T, t)k ≤ . This completes the proof of the lemma. 2

Proof of Prop. 5.1. Let A ∈ A

. Again let us set A(s, t) = τ

(s, t) ◦ α

(A), so that we need to show that

t→+∞

lim τ

(0, t) ◦ α

(A) − A(0, t) = 0.

We fix > 0 and T as in Lemma 5.2. We have:

τ

(0, t) ◦ α

(A) = τ

(0, T ) ◦ τ

(T, t) ◦ α

(A)

= R(0, T )τ

(0, T ) ◦ τ

(T, t) ◦ α

(A)R(0, T )

= R(0, T )A(0, t)R(0, T ) + O(), by Lemma 5.2. By (4.5) we have:

∂

(R(0, σ)BR(0, σ)

) = −iR(0, σ)[I(0, σ), B]R(0, σ)

, B ∈ A

hence:

R(0, T )A(t)R(0, T )

− A(0, t) = −i ˆ

0

R(0, σ)[I(0, σ), A(0, t)]R(0, σ)

dσ.

By (5.2) for s = 0 we have k[I(0, σ), A(0, t)]k = k[I, A(σ, t)]k. To complete the proof of the proposition, it suffices to show that

(5.6) lim

t→+∞

[I, A(σ, t)] = 0, ∀ σ ≥ 0.

Since kA(σ, t)k = kAk, it suffices by density and linearity to prove (5.6) if A = Q

i=1

ψ

(f

) for f

∈ h

with compact support. By Lemma 4.3 it suffices hence to prove that

w− lim

t→+∞

u

(0, t)f

= 0.

But this follows from Prop. 2.7. This completes the proof of the proposition. 2 5.2. Right propagation.

Proposition 5.3. The strong limit s− lim

t→+∞

τ

(0, t) ◦ α

=: γ

exists on A

.

Before proving the proposition, let us note that γ

is an even homomorphism (see Subsect. A.1).

Lemma 5.4. The homomorphism γ

is even i.e. P ◦ γ

= γ

◦ P.

Proof. α

is even, so it suffices to prove that τ

(0, t) is even. This follows if we prove that R(s, t) ∈ CAR

(h

). We note that R(s, σ) and P R(s, σ) solve the same differential equation, using that I is even. 2

Proof of Prop. 5.3. Let A ∈ A

. By (2.8) we have α

= τ

(t, 0) on A

. Therefore we will be able to prove the proposition by the Cook argument. We will first prove that

(5.7) lim

t→+∞

τ

(0, t) ◦ τ

(t, 0)(A) =: γ

(A), A ∈ A

. exists, and then that

(5.8) s− lim

t→+∞

τ

(0, t) ◦ τ

(t, 0)(A), A ∈ γ

A

.

exists. Let us first prove (5.7). Since τ

(0, t) and τ

(t, 0) are free dynamics, this follows from Prop. 2.6 which states that:

(5.9) lim

t→+∞

u

(0, t)u

(t, 0)f = w

f exists for f ∈ h

. It follows that

γ

(ψ

(f)) = ψ

(w

f ), f ∈ h

.

To prove (5.8) we will need some estimates on the speed of convergence in (5.9), for well chosen initial data.

Assume that f ∈ h

is smooth with compact support. Then u

(t, 0)f = f

≡ 0 near x = z(t) hence u

(t, 0)f ∈ Dom b

. It follows that:

∂

u

(0, t)u

(t, 0)f = iu

(0, t)(b

− b

)u

(t, 0)f = iu

(0, t)V f

.

From hypothesis (2.3) we obtain that kV f

k ∈ O(t

) hence by integrating from t to +∞, we obtain:

(5.10) w

f − u

(0, t)u

(t, 0)f ∈ O(t

).

Let us now prove (5.8). By linearity, density and using that τ

(0, t) and τ

(t, 0) are isomorphisms, we can assume that A = ψ

(w

f ) for f ∈ h

smooth with compact support. We have

τ

(0, t) ◦ τ

(t, 0)(A) = R(0, t)τ

(0, t) ◦ τ

(t, 0)(A)R(0, t)

= R(0, t)AR(0, t)

.

We apply once more the Cook argument and compute

∂

R(0, t)γ

(A)R(0, t)

= −iR(0, t)[I(0, t), A]R(0, t)

. As before

k[I(0, t), A]k = k[I, τ

(t, 0)(A)]k = k[I, ψ

(u

(t, 0)w

f )]k

= k[I, ψ

(u

(t, 0)f ]k + O(t

),

by (5.10). Since f has compact support, and u

(t, 0)f = f

, we obtain that [I, u

(t, 0)f ] = 0 for t large enough. Therefore k∂

R(0, t)γ

(A)R(0, t)

k ∈ O(t

), which proves (5.8) by the Cook argument. 2

5.3. Hawking effect I.

5.3.1. The limit state. In the rest of the paper we denote by β = 2πκ

the inverse Hawking temperature.

Let us denote by ω

the gauge-invariant quasi-free thermal state on A

with covariance:

ω

(ψ

(f )ψ(g)) = (f |(1l + e

)

g), f, g ∈ h

. This state restricts to a quasi-free state on A

, still denoted by ω

.

We denote by ω

the gauge-invariant quasi-free vacuum state on A

with covariance:

ω

(ψ

(f )ψ(g)) = (f|1l

(b

)g), f, g ∈ h

.

The state ω

◦ γ

is a gauge-invariant state on A

, which is even by Lemma 5.4.

Since h

= h

⊕ h

, we can by Def. A.7 define the following state on A

: Definition 5.5. We set

ω

:= ω

⊗ b (ω

◦ γ

), which is a state on A

.

5.3.2. Main result I. The following theorem is the main result of this section.

Theorem 5.6.

t→+∞

lim ω

◦ τ

(0, t) ◦ α

(A) = ω

(A), A ∈ A

.

Proof. By linearity and density we may assume that A = A

× A

, A

∈ A

, A

∈ A

. By Prop. 5.3 we have τ

◦ α

(A

) = γ

(A

) + o(t

). Applying Lemma 5.7 we obtain that

t→+∞

lim ω

(A

A

) = ω

⊗ b ω

◦ γ

(A

A

), which completes the proof of the theorem. 2

Lemma 5.7. Let A

∈ A

, A

∈ A

. Then

t→+∞

lim ω

(τ

(0, t) ◦ α

(A

)A

) = ω

⊗ b ω

(A

A

), . Proof. By linearity and density we can assume that

A

=

n₁

Y

i=1

ψ

(f

)

p₁

Y

i=1

ψ(g

), A

=

n₂

Y

i=1

ψ

(f

)

p₂

Y

i=1

ψ(g

), where

f

, g

∈ h

, for 1 ≤ i ≤ n

, 1 ≤ j ≤ p

, f

, g

∈ h

, for 1 ≤ i ≤ n

, 1 ≤ j ≤ p

. To simplify notation we set

γ

:= τ

(0, t) ◦ α

, γ

:= τ

(0, t) ◦ α

,

so that from Prop. 5.1 we have γ

(A

) = γ

(A

) + o(t

). It follows that γ

(A

) =

n1

Y

i=1

ψ

(u

(0, t)f

)

p₁

Y

i=1

ψ(g

) + o(t

).

Using the CAR and Prop. 2.7, we obtain that:

γ

(A

)A

= (−1)

Q

i=1

ψ

(f

) Q

i=1

ψ

(u

(0, t)f

) Q

i=1

ψ(g

) Q

i=1

ψ(g

) + o(t

).

Since ω

is a gauge invariant quasi-free state (see Subsect. A.3), we see that ω

(γ

(A

)A

) = o(t

) if n

+ n

6= p

+ p

, and if n

+ n

= p

+ p

= n we have:

(5.11)

ω

(γ

(A

)A

)

= (−1)

P

σ∈Sn

(σ) Q

k=1

ω

(ψ

(F

)ψ(G

)) + o(t

), where:

F

=

u

(0, t)f

for 1 ≤ k ≤ n

,

f

for n

+ 1 ≤ k ≤ n, , G

=

u

(0, t)g

for 1 ≤ k ≤ p

, g

for p

+ 1 ≤ k ≤ n.

Recall that ω

(ψ(f )ψ(g)) = (f |1l

(b

)g)

and w− lim u

(0, t)f

= 0 for f ∈ h

by Prop. 2.7. We see the sum on the r.h.s. is o(t

) unless n

= p

and n

= p

. If this is the case the only permutations σ contributing to the sum are of the form σ

× σ

where σ

∈ S

. Collecting these terms we obtain that:

ω

(γ

(A

)A

) = ω

(γ

(A

))ω

(A

) + o(t

).

By the result of Bachelot [Ba2] recalled in Thm. 2.8 we know that

t→+∞

lim ω

(γ

(A

)) = ω

(A

).

Now we use Remark A.4 and the definition of the Z

−graded tensor product of two states (see Lemma A.6) to see that

t→+∞

lim ω

(γ

(A

)A

) = ω

⊗ω e

(A

A

), which completes the proof of the lemma. 2

5.4. Change of initial state. We assumed in Subsect. 2.1 that the star was stationary in t ≤ 0. It is hence natural to take as dynamics in the past the station- ary interacting dynamics τ

(s, t) on A

defined as follows: we first define the stationary analog of τ

(s, t), acting on A

by

τ

(s, t)ψ

(f ) := ψ

(e

f ), f ∈ h

.

We can then define the stationary interacting dynamics τ

(s, t) associated to I in Def. 4.1. It suffices to repeat the construction in Subsect. 4.1 with τ

(s, t) instead of τ

(s, t).

An adapted choice of the initial state in Thm. 5.6 would be an even state ˜ ω on A

, invariant under τ

(s, t). The following easy result shows that Thm. 5.6 will extend to ˜ ω, provided that ˜ ω belongs to the folium of ω

, i.e. is represented by a density matrix in the GNS representation of ω

. Recall that such states are physically interpreted as local perturbations of ω

.

Corollary 5.8. Let ω ˜ a state on A

which is even and belongs to the folium of ω

. Then

t+∞

lim ω ˜ ◦ τ

(0, t) ◦ α

(A) = ˜ ω

(A), A ∈ A

, where:

˜

ω

= ω

⊗ b (˜ ω ◦ γ

).

Proof. Since ˜ ω belongs to the folium of ω

, we are, by linearity and density, reduced to compute the limit:

t→+∞

lim ω

(P

(ψ, ψ

)γ

(A)

A

P(ψ, ψ

)),

where A

∈ A

, A

∈ A

and P(ψ, ψ

) is a polynomial in CAR

(h

). Moreover since ˜ ω is even, we see that P (ψ, ψ

) ∈ CAR

(h

). By the same argument as in the proof of Lemma 5.7, we see that

P

(ψ, ψ

)γ

(A)

A

P (ψ, ψ

) = γ

(A)

P

(ψ, ψ

)A

P (ψ, ψ

) + o(t

), hence as in Lemma 5.7 we have:

lim

ω

(P

(ψ, ψ

)γ

(A)

A

P(ψ, ψ

))

= ω

(A

)ω

(P

(ψ, ψ

)A

P(ψ, ψ

)) = ω

(A

)˜ ω(A

).

We can then complete the proof as in Thm. 5.6. 2

5.5. Existence of interacting initial vacua. It remains to construct even states

˜

ω which belong to the folium of ω

and are invariant under τ

(s, t). To do this it is convenient to work in the GNS representation of the vacuum state ω

, i.e. the Fock representation. We refer the reader to Subsect. A.4.

Recall that b

is defined in Subsect. 2.1.2. It is easy to show that (5.12) σ

(b

) =] − ∞, −m] ∪ [m, +∞[.

We assume that Kerb

= {0} and equip h

with the complex structure j :=

i sgn(b

), and denote by Z the associated one-particle space. If π

is the cor- responding Fock representation we have:

ω

(A) = (Ω|π

(A)Ω),

where Ω ∈ Γ

(Z) is the vacuum vector. In other words (Γ

(Z), π

, Ω) is the GNS triple associated to ω

.

From Subsect. A.4 we know that:

π

(τ

(s, t)A) = e

π

(A)e

, A ∈ A

, for H

= dΓ(|b

|), and if Q = dΓ(sgn(b

)), then:

ψ

(e

f ) = e

ψ

(f)e

, f ∈ h

, θ ∈ R . It is also well known that if

H := H

+ π

(I), then

π

(τ

(s, t)(A)) = e

π

(A)e

, A ∈ A

.

Since τ

(s, t) is implemented by e

in the Fock representation, eigenvectors of H will yield invariant states for τ

(s, t), which obviously belong to the folium of ω

.

Existence of eigenvectors is ensured by the following theorem, whose proof follows by adapting arguments in [A, DG].

Theorem 5.9. On has

σ

(H) = [inf σ(H) + m, +∞[.

Therefore inf σ(H) is an eigenvalue of H.

To be able to apply Corollary 5.8, we need however the existence of an even eigenstate ψ of H , i.e. such that Qψ = 2nψ for some n ∈ Z . Note that since I is even we have [H, Q] = 0, which does not imply the existence of even eigenstates of H . However this is clearly true for small interactions. In fact setting H (λ) = H

+λI and E(λ) = inf σ(H (λ)) we have

Lemma 5.10. Assume |λ| is small enough.Then 1l

(H (λ)) is rank one and Q1l

(H (λ)) = 0.

Therefore for λ small enough, H (λ) has a unique ground state Ω(λ) of zero charge and the associated state satisfies the hypotheses of Corollary 5.8.

6. Hawking effect II

In this section we study the Hawking effect in case II (see Subsect. 1.3). Com- pared to Sect. 5, the observable is not translated to the left, therefore the influence of the potential V and of the non-linear self-interaction has to be taken into ac- count. To this end we use tools from scattering theory, both for classical and quantum dynamics.

6.1. Asymptotic velocity for Dirac equations. In this subsection we state some results of Daud´ e [Da] on the existence of the asymptotic velocity observable for stationary Dirac equations. The asymptotic velocity provides a convenient way to separate left and right propagating initial states. More details can be found in [Da].

Theorem 6.1. Let χ ∈ C

( R , C

). Then (1) the limits

χ

:= s− lim

t→±∞

e

χ( x

t )e

exist.

(2) there exist bounded selfadjoint operators P

on h

such that χ

= χ(P

) for χ ∈ C

( R , C

).

(3) one has

[P

, b

] = 0, σ(P

) = [−1, 1], 1l

(P

) = 1l

(b

).

Remark 6.2. Since it is known (see e.g. [Ba2, Lemma III.1] that b

has no eigenvalues, we have actually 1l

(P

) = 0, i.e. any initial state has a non- vanishing asymptotic velocity.

We will only use the future asymptotic velocity P

which we will denote simply by P . Moreover we will set

P

:= 1l

(P), so that by Remark 6.2 we have

P

+ P

= 1l.

We set now V

:= lim

V (x), so that V

= 0, V

= V

, see Subsect. 2.1. We set also

b

:= LD

+ V

, with domain H

( R , C

), acting on h

.

From Thm. 6.1 and the short-range nature of the interaction V (see (2.3)), we obtain by standard arguments the existence of wave operators:

Proposition 6.3. The limit s− lim

t→+∞

e

e

=: w

exists on P

h

.

Prop. 6.3 yields the following result for free quantum dynamics.

Proposition 6.4. The limit γ

:= s− lim

t→+∞

τ

(t, 0) ◦ τ

(0, t) exists on CAR(P

h

) The map γ

: CAR(P

h

) → CAR(h

) = A

are ∗−morphisms with

γ

(ψ

(f)) = ψ

(w

f), f ∈ P

h

.

The similar limit γ

with V

replaced by V

and P

replaced by P

exists on CAR(P

h

) also exists, but will play no role in the sequel. The ’right’ analog of γ

is the wave morphism γ

introduced below in Prop. 6.13.

6.2. Wave morphisms. We now prove an analog of Prop. 6.4 for interacting dynamics.

Theorem 6.5. The limit γ

:= s− lim

t→+∞

τ

(t, 0) ◦ τ

(0, t) exists on A

and is a ∗−morphism of A

.

The morphism γ

is an example of a wave morphism.

Proof of Thm. 6.5. The proof relies once again on the Cook argument, combined with minimal velocity estimates for the Dirac equation. For more details on minimal velocity estimates see [Da]. Since b

has no point spectrum we see that the space D of vectors in ∩

Dom hxi

such that f = χ(b

)f for some χ ∈ C

( R \[−m, m]) is dense in h

. The strong minimal velocity estimates (see [Da]) give

(6.1) ∀f ∈ D, ∃ 0 < c

< 1 such that k1l

( |x|

t )e

f k ∈ O(t

), ∀ N ∈ N . We can now argue as in the proof of Prop. 5.3, since by (6.1) t 7→ (g|e

f) is integrable for f ∈ D. 2

We denote by E

: A

→ A

the ∗−homomorphism defined in A.1.3, associated to the inclusion h

⊂ h

. Since ∪

h

is dense in h

, we have:

(6.2) s− lim

t→+∞

E

= 1l, in A

.

We now combine Thm. 6.5 with Lemma 4.7 to obtain the following result.

Proposition 6.6. Let A = γ

(B) ∈ A

. Then for any > 0 there exist C

, T

> 0 such that

sup

t/2+C≤s≤t/2+2C, t≥T

kτ

(s, t) ◦ E

(A) − τ

(s, t)(B)k ≤ .

Remark 6.7. We do not know if A ∈ γ

A

belongs to A

for all t large enough, so a priori τ

(s, t)(A) does not makes sense. Replacing A by E

(A) ∈ A

fixes this problem, at the price of an error kA − E

(A)k which is o(t

).

Proof. Since A = γ

(B) we have:

τ

(0, t)(B) = τ

(0, t)(A) + o(t

), t → +∞.

Since τ

and τ

are stationary dynamics, this implies that for any c > 0:

(6.3) lim

t→+∞

sup

0≤s≤t/2+c

kτ

(s, t)(B) − τ

(s, t)(A)k = 0.