- Email: [email protected]
Non-exponential decay for Polaron model
6 September 1999
Physics Letters A 260 Ž1999. 31–38 www.elsevier.nlrlocaterphysleta
Non-exponential decay for Polaron model L. Accardi a
a,)
, S.V. Kozyrev
b,1
, I.V. Volovich
c,2
Centro Vito Volterra UniÕersita di Roma Tor Vergata, Via di Tor Vergata, snc-00133 Roma, Italy b Institute of Chemical Physics, Kossygin Street 4, 117334 Moscow, Russia c StekloÕ Mathematical Institute, Gubkin Street 8, GSP-1, 117966 Moscow, Russia Received 26 April 1999; accepted 28 July 1999 Communicated by P.R. Holland
Abstract A model of a particle interacting with a quantum field is considered. The model includes as particular cases the polaron model and non-relativistic quantum electrodynamics. We compute matrix elements of the evolution operator in the stochastic approximation and show that, depending on the state of the particle, one can get non-exponential decay with the rate ty3r2 . In the process of computation a new algebra of commutation relations that can be considered as an operator deformation of the quantum Boltzmann commutation relations is used. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: Polaron; Stochastic limit; Relaxation
1. Introduction For many dissipative systems one has the exponential time decay of correlations. This result was established for various models and by using various approximations, see for example Ref. w1x. For certain models, in particular for the spin-boson Hamiltonian, also a regime with the oscillating behavior was found w2–4x. The presence of such a regime is very important in the investigation of quantum decoherence. The aim of this note is to show that for the model of particle interacting with quantum field, in particular for the polaron model, one can have not only the standard exponential decay but also the
) Corresponding author. E-mail: [email protected] 1 E-mail: [email protected] 2 E-mail: [email protected]
non-exponential decay Žas some powers of time. of correlations. We investigate the model describing interaction of non-relativistic particle with quantum field. This model is widely studied in elementary particle physics, solid state physics, quantum optics, see for example Refs. w5–8x. We consider the simplest case in which matter is represented by a single particle, say an electron, with position and momentum q s Ž q1 ,q2 ,q3 . and p s Ž p 1 , p 2 , p 3 . satisfying the commutation relations w q j , pn x s i d jn . The electromagnetic field is described by boson operators aŽ k . s Ž a1Ž k .,a 2 Ž k .,a 3 Ž k ..;a† Ž k . s Ž a†1 Ž k ., . . . ,a†3 Ž k .. satisfying the canonical commutation relations w a j Ž k .,a†nŽ kX .x s d jn d Ž k y kX .. The Hamiltonian of a free non relativistic atom interacting with a quantum electromagnetic field is H s H0 q l H I s v Ž k . a† Ž k . a Ž k . dk q 12 p 2 q l H I
H
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 1 0 - 1
Ž 1.
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
32
where l is a small constant, v Ž k . is the dispersion law of the field, H I s d 3 k Ž g Ž k . p P a† Ž k . eyi k q
H
qg Ž k . p P a Ž k . e i k q . q h.c. Ý 3js 1
Ž 2. 2
Ý 3js 1
Here p P aŽ k . s p j a j Ž k ., p s p j2 , 3 †Ž . Ž . 3 †Ž . Ž . a k a k s Ý js1 a j k a j k , kq s Ý js1 k j q j . For the polaron model the Hamiltonian has the form H s v Ž k . a† Ž k . a Ž k . dk q 12 p 2
H
q l d 3 k Ž g Ž k . a† Ž k . eyi k q q g Ž k . a Ž k . e i k q .
H
It is different from Ž1., Ž2. by a momentum p in the interaction Hamiltonian. For the analysis of this paper this difference is not important. In the present paper we will use the method for the approximation of the quantum mechanical evolution that is called the stochastic limit method, see for example w4,9–11x. The general idea of the stochastic limit is to make the time rescaling t ™ trl2 in the solution of the Schrodinger equation in interaction ¨ picture Ut Ž l . s e i t H 0 eyi t H , associated to the Hamiltonian H, i.e.
E
U Ž l . s yi l H I Ž t . Ut Ž l . Et t with H I Ž t . s e i t H 0 H I eyi t H 0 . We get the rescaled equation
E
Ut Žrll. 2 s y
i
2
Ut Žrll. 2
H Ž trl . Et l I and one wants to study the limits, in a topology to be specified, lim Ut Žrll. 2 s Ut ;
l™0
lim l ™0
1
l
HI
t
ž / l2
s Ht
Ž 3.
We will prove that Ut is the solution of the equation
E t Ut s yiHt Ut , U0 s 1
Ž 4.
The interest of this limit equation is in the fact that many problems become explicitly integrable. The stochastic limit of the model Ž1., Ž2. has been considered in w10–15x. After the rescaling t ™ trl2 we consider the simultaneous limit l ™ 0, t ™ ` under the condition
that l2 t tends to a constant Žinterpreted as a new slow scale time.. This limit captures the main contributions to the dynamics in a regime, of long times and small coupling arising from the cumulative effects, on a large time scale, of small interactions Ž l ™ 0.. The physical idea is that, looked from the slow time scale of the atom, the field looks like a very chaotic object: a quantum white noise, i.e. a d-correlated Žin time. quantum field b†j Ž t,k .,bj Ž t,k . also called a master field. If one introduces the dipole approximation the master field is the usual boson Fock white noise. Without the dipole approximation the master field is described by a new type of commutation relations of the following form w11x bj Ž t ,k . pn s Ž pn y k n . bj Ž t ,k .
Ž 5.
bj Ž t ,k . bn† Ž tX ,kX . s 2pd Ž t y tX . d Ž v Ž k . ykp q 12 k 2 . d Ž k y kX . d jn
Ž 6.
Such quantum white noises can be treated as an operator deformation of quantum Boltzmann commutation relations. Recalling that p is the particle momentum, we see that the relation Ž5. shows that the particle and the master field are not independent even at a kinematical level. This is what we call entanglement. The relation Ž6. is a generalization of the algebra of free creation–annihilation operators with commutation relations A i A†j s d i j and the corresponding statistics becomes a generalization of the Boltzmannian Žor Free. statistics. This generalization is due to the fact that the right hand side is not a scalar but an operator Ža function of the atomic momentum.. This means that the relations Ž5., Ž6. are module commutation relations. For any fixed value p of the atomic momentum we get a copy of the free Žor Boltzmannian. algebra. Given the relations Ž5., Ž6., the statistics of the master field is uniquely determined by the condition bj Ž t ,k . C s 0 where C is the vacuum of the master field, via a module generalization of the free Wick theorem, see w14x. In Section 2 the dynamically q-deformed commutation relations Ž7., Ž8., Ž14. are obtained and the
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
stochastic limit for collective operators is evaluated. In Section 3 the stochastic limit of the evolution equation is found. In Section 4 the non-exponential decay for vacuum vector in the polaron model is investigated.
33
exist. Then it is natural to conjecture that its algebra shall be obtained as the stochastic limit Ž l ™ 0. of the algebra Ž7., Ž8.. Notice that the factor qlŽ t y tX , x . is an oscillating exponent and one easily sees that lim ql Ž t , x . s 0 , l™0
2. Deformed commutation relations
l™0
In this section we reproduce in the brief form the notations and the main results of the work w14x. In order to determine the limit Ž3. one rewrites the rescaled interaction Hamiltonian in terms of some rescaled fields al, j Ž t,k .: 1
l
HI
t
ž / l2
H
l 1
s
l
ž ž
exp i
t
/
exp y i
ž
H0 e i k q a j Ž k . exp y i
l2
t
l2
bj Ž t ,k . bn† Ž tX ,kX . s 2pd Ž t y tX . d Ž v Ž k . ykp q 12 k 2 . d Ž k y kX . d jn
X X s al , n Ž tX ,kX . al , j Ž t ,k . qy1 l Ž t y t ,kk .
where 1
and the limit of Ž7. gives the module free relation
Ž 13 .
Operators al, j Ž t,k . also obey the relation al , j Ž t ,k . al , n Ž tX ,kX .
s d 3 kp Ž g Ž k . al Ž t ,k . qg Ž k . a†l Ž t ,k . . q h.c.
al , j Ž t ,k . [
1
q Ž t , x . s 2pd Ž t . d Ž x . Ž 11 . l2 l Thus it is natural to expect that the limit of Ž8. is bj Ž t ,k . pn s Ž pn y k n . bj Ž t ,k . Ž 12 . lim
Ž v Ž k . y kpq 12 k 2 .
t
l2
/
H0
/
ei k q aj Ž k .
It is now easy to prove that operators al, j Ž t,k . satisfy the following q–deformed module relations,
Ž 14 .
In what follows we will not write indexes j, n explicitly. The relation Ž14. should disappear after the limit, see w14x. In fact, if the relation Ž14. would survive in the limit then, because of Ž11., it should give bŽ t,k . bŽ tX ,kX . s 0, hence also b† Ž t,k . b† Ž tX ,kX . s 0, so all the n–particle vectors with n G 2 would be zero.
al , j Ž t ,k . a†l , n Ž tX ,kX . 3. Evolution equation
s a†l , n Ž tX ,kX . al , j Ž t ,k . ql Ž t y tX ,kkX . 1 q
X
l2
1 2
ql Ž t y t , v Ž k . y kp q k
= d Ž k y kX . d jn al , j Ž t ,k . pn s Ž pn y k n . al , j Ž t ,k .
2
. Ž 7. Ž 8.
where ql Ž t y tX , x . s eyi
tytX x l2
Ž 9.
is an oscillating exponent. This shows that the module q–deformation of the commutation relations arise here as a result of the dynamics and are not put artificially ab initio. For a discussion of q-deformed commutation relations see for example w16x. Now let us suppose that the master field bj Ž t ,k . s lim al , j Ž t ,k . l™0
Ž 10 .
Let us find stochastic differential equation for the model we consider. In the introduction we claimed that the stochastic limit for the Schrodinger equation ¨ in interaction picture will have the form Ž4.: E t Ut s yiHt Ut . But in this equation both Ht and Ut are distributions. We need to regularize this product of distributions. In the present section we will make the following regularization: roughly speaking we replace Ht by Htq0 q const. We investigate the evolution operator in interaction picture Ut Ž l.. We start with the equation tqdt
Ž l. Utqd t s 1 q Ž yi l .
ž
q Ž yi l . =
t1
Ht dt
2
2
Ht
tqdt
Ht
H I Ž t 1 . dt 1
dt 1
H I Ž t 1 . H I Ž t 2 . q . . . U tŽl .
/
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
34
Ž l. Ž l. where dt ) 0. We get for dUt Ž l. s Utqd t y Ut
dUt Ž l. s Ž yi l .
ž
tqdt
Ht
t1
=
Ht dt
H I Ž t 1 . dt 1 q Ž yi l .
2
tqdt
Ht
dt 1
H I Ž t 1 . H I Ž t 2 . q . . . U tŽl .
/
2
Let us make the rescaling t ™ trl2 in this perturbation theory series. We get tqdt
dUt Žrll. 2 s Ž yi .
ž
Ht
q Ž yi . =
1
2
HI
l
dt 1
tqdt
Ht
t2
1
l
dt 1
HI
l2
1
t1
2
l
HI
Here ² P : is the stochastic limit of the vacuum expectation of boson field Žthat acts as a conditional expectation on momentum of quantum particle p.. We will prove this result by analizing of the perturbation theory series. We have
t1
ž / l2
q . . . Ut Žrll. 2
ž / / l2
Ž 15 .
To find the stochastic differential equation we need to collect all the terms of order dt in the perturbation theory series Ž15.. Terms of order dt are contained only in the first two terms of these series. Let us investigate the first two terms. For the first term of the perturbation theory we get tqdt
Ht
s
dt1
1
l
tqdt
Ht
HI
² X dB Ž t ,k . Ut : s 0 ; X .
t1
ž /
Ht dt
In the case of Boltzmannian statistics we get the same relation: the Žfree. independence means that roughly speaking dB Ž t,k . kills all creations in Ut . We have the following Lemma. The stochastic differential dB(t,k) and the eÕolution operator Ut are free independent. This means that for an arbitrary obserÕable X,
² X dB Ž t ,k . Ut : s lim
q Žy i .
=
ž /
H
qg Ž k . a†l Ž t 1 ,k . Ž 2 p q k . .
Ž 16 .
In the stochastic limit l ™ 0 this term gives us
Hdk Ž g Ž k . Ž 2 p q k . dB Ž t ,k . where the stochastic differential dB Ž t,k . is the stochastic limit of the field alŽ t,k . in the time interval Ž t,t q dt .: dB Ž t ,k . s lim
tqdt
H l™0 t
dt al Ž t ,k . s
tqdt
Ht
qg Ž k . dB Ž t ,k . Ž 2 p q k . .
tqdt
Ht
dt b Ž t ,k .
We will prove that the stochastic differential dB Ž t,k . and the evolution operator Ut are free independent. In the bosonic case independence would result in the relation w dB Ž t,k .,Ut x s 0. From this relation follows that ² X dB Ž t,k .Ut : s 0 for arbitrary observable X.
1
l
dt al Ž t ,k .
HI
n
1
H0tdt1 l HI
H0tdt1 . . . H0t tn
Ny 1
t1
ž / l2
dt N
1
l
q...
HI
t1
ž / l2
...
ž / /; l2
q...
where l1 H I Ž t krl2 . is given by the formula Ž16.. Here lim l™ 0 Xl s X. Let us analyze the N-th term of perturbation theory. The N-th term of perturbation theory is the linear combination of the following terms Žwe omit integration over k, k n . ² Xl
†
t q dt
t
ž
2
dt 1 dk Ž g Ž k . Ž 2 p q k . al Ž t 1 ,k .
Xl
= 1q Ž y i .
t1
l
¦H
l™ 0
=
dt al Ž t ,k .
t Ny1
H0
t
H0 dt
1
...
dt N al´ 1 Ž t 1 ,k 1 . . . . al´ N Ž t N ,k N . :
Let us shift alŽt ,k . to the right using dynamically q-deformed relations. In the following we will use notions of the work w14x. Let us enumerate annihilators in the product al´ 1 Ž t 1 ,k 1 . . . . al´ N Ž t N ,k N . as alŽ t m j ,k m j ., j s 1, . . . J, and enumerate creators as a†lŽ t mXj ,k mXj ., j s 1, . . . I, I q J s N. This means that if ´ m s 0 then al´ m Ž t m ,k m . s alŽ t m j ,k m j . for m s m j Žand the analogous condition for ´ m s 1..
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
We will use the following recurrent relation for correlator Žanalogous formula was proved in the work w14x.: tqdt
² Xl
Ht
dt
t
H0 dt
1
...
t Ny1
H0
dt N
=al Ž t ,k . al´ 1 Ž t 1 ,k 1 . . . . al´ N Ž t N ,k N . : I
s
tqdt
Ý d Ž k y k m . ² XlH X j
dt
t
js1
t
H0 dt
1
...
t Ny1
H0
dt N
= al´ 1 Ž t 1 ,k 1 . . . . aˆ†l Ž t mXj ,k mXj . . . . al´ N Ž t N ,k N . : = =
ž
Ł
qy1 l Ž t y t mXj ,kk m i .
Ł
ql Ž t y t mXj ,kk mXi .
Ł
X
qy1 l
Ł
X
ql Ž t y t mXi ,kk mXi .
X X m i)m j
=
m i-m j
=
1 2
ql t y t mXj , v Ž k . y kp q k
l2
X m i)m j
=
X
m i-m j
2
Ht
/
s
Ž t y t m ,kk m . i
I
t
js1
Ž 17 .
t Ny1
0
0
=
l
2
dt N
ql yt mXj , v Ž k . y kp q 12 k 2
ž
Ł
X
qy1 l Ž yt mXj ,kk m i .
Ł
X
ql Ž yt mXj ,kk mXi .
X
m i)m j
=
Ł X
X
m i-m j
=
tqdt
Ht
=
l
1
2
l
Ł
X
Ł
X m i-m j
t2
l
HI
t1
Ht dt Hdk < g Ž k . <
l2
2
2
ž
exp yi
2
t1
ž / ž / t1 y t 2
Ž2 pqk .
tqdt
Ht
s
dt 1
tqdt
Ht
1
t1
Ht dt dt 1
2
l
2
eyi
0
t 1 yt 2
0
it x
it x
2
1
s x y i0
spd Ž x . y i PP
1 x
we get for the second term
/
1
2
v Ž k . y kp q 12 k 2 y i0
Let us denote
Ž g < g . y Ž p . s yiHdk < g Ž k . < 2 Ž 2 p q k .
Ł
X
m i-m j
qy1 l Ž yt m i ,kk m i . =
qy1 l Ž t ,kk m i .
X
X
m i)m j
Ł
X X m i-m j
ql Ž t ,kk mXi . ql Ž t ,kk mXi .
2
1
v Ž k . y kp q 12 k 2 y i0
s dk < g Ž k . < 2 Ž 2 p q k .
Ł
/
x
l2
Hyt rl dt e Hy`dte
ql Ž yt mXi ,kk mXi .
qy1 l Ž t ,kk m i .
2
Ž v Ž k . y kp q 12 k 2 .
l2
H
dt ql Ž t , v Ž k . y kp q 12 k 2 .
m i)m j
=
1
dt 1
l
HI
yidt dk < g Ž k . < 2 Ž 2 p q k .
m i)m j
=
tqdt
Ht
2
yi
= al´ 1 Ž t 1 ,k 1 . . . . aˆ†l Ž t mXj ,k mXj . . . . al´ N Ž t N ,k N . : =
1
t1
Ht dt
due to q–module relations on alŽ t,k ., p. Performing integration over t 1 , t 2 and using the formulas
i
Ý d Ž k y k m . ² XlH dt1 . . . H X j
dt 1
=
Here the notion aˆ†l means that we omit the operator a†l in this product. The right hand side of the Eq. Ž17. is equal to
1
Žwe use that ql is an exponent.. The first three lines of this formula do not depend on t and the last two lines do not depend on t 1 , . . . ,t N . Therefore the stochastic limits for these values can be made independently Žthe limit of product is equal to the product of limits.. It is easy to see that the stochastic limit for the multiplier that depends on t Žof the last two lines. is equal to zero. This finishes the proof of the Lemma. The second term of the perturbation theory series is equal Žup to terms of order Ž dt . 2 . tqdt
1
35
H
2
ž
= pd Ž v Ž k . y kp q 12 k 2 . yi PP
1
v Ž k . y kp q 12 k 2
/
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
36
Combining all the terms of order dt we get the following result: Theorem. The stochastic differential equation for Ut s lim l ™ 0 Ut(rll) 2 haÕe a form
ž
dUt s yi dk Ž g Ž k . Ž 2 p q k . dB Ž t ,k .
H
/
Ž 18 .
Eq. Ž18. can be rewritten in the language of distributions as dUt dt
ž
² X < Ut < X : s dp < f Ž p . < 2 eytŽ g < g .yŽ p.
s yi dk Ž g Ž k . Ž 2 p q k . b Ž t ,k .
H
qg Ž k . b† Ž t ,k . Ž 2 p q k . . y Ž g < g . y Ž p . Ut
/
Ž 19 . Here we understand the singular product of distributions bŽ t,k .Ut in the sense that Ž19. is equivalent to Ž 18 . . We have to stress that dB Ž t, k . s Ht tqd t dt bŽt ,k . / bŽ t,k . dt and we can not obtain Ž19. dividing Ž18. by dt.
v Ž k . y kp q 12 k 2 s 1 y 12 p 2 q 12 Ž k y p .
Ž g < g . y Ž p . s yiHdk < g Ž k . < 2 Ž 2 p q k . =
Let us investigate the behavior of ²Ut : using stochastic differential equation Ž18.. We get ² dUt : s ² Ž yi . dkg Ž k . Ž 2 p q k . dB Ž t ,k .
H
/
ydt Ž g < g . y Ž p . Ut : Using the free independence of dB Ž t,k . and Ut we get d dt
²Ut : s ²
d dt
Because U0 s 1, we have the solution ²Ut : s eyt Ž g < g .yŽ p. In this section we calculate the matrix element ² X < Ut < X : where X s f Ž p . m F in the momentum
2
1 1 2
1 y p q 12 Ž k y p . 2
2
s y2 i dk < g Ž k . < 2 y i Ž I1 q I2 . ,
H
I1 s Ž y2 q 10 p 2 . dk < g Ž k . < 2
H
=
1 1 2
1 y p 2 q 12 Ž k y p .
I2 s 6 dk < g Ž k . < 2 p Ž k y p .
H
Ut : s y Ž g < g . y Ž p . ²Ut :
2
One can expect non-exponential relaxation when supp f Ž p . ; < p < - '2 4 . In this case ReŽ g < g .y Ž p . s 0 and there is no dumping. All decay in this case is due to interference. We will use the approximation diam supp g Ž k . 4 diam supp f Ž p .. Physically this means that the particle is more localized in momentum representation than the field. This assumption seems natural because the field’s degrees of freedom are fast and the particles one are slow. Under this assumption we can estimate the matrix element Ž20.. We will prove that in this case there will be polynomial decay. For < p < - '2 we get
4. Non-exponential decay
ž
Ž 20 .
H
We investigate the polaron model when v Ž k . s 1. For this choice of v Ž k . we get
qg Ž k . dB † Ž t ,k . Ž 2 p q k . . ydt Ž g < g . y Ž p . Ut
representation and F is the vacuum vector for the master field. This matrix element is equal to
2
, 1 1 2
1 y p q 12 Ž k y p . 2
2
Here only I1 and I2 depend on p and therefore can interfere. Let us find the asymptotics of Ž g < g .y Ž p . on p Žwe investigate the case when p is a small parameter.. We will use the following assumption on g Ž k .: let g Ž k . be a very smooth function. This means that
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
< g Ž k .< 2 s l F Ž l k ., F Ž k . ) 0 is compactly supported smooth function, l is a small parameter. Let us consider the Taylor expansion on the small parameter p
We get for X Ž t . s ² X < Ut < X : X Ž t . s dp < f Ž p . < 2 eytŽ g < g .yŽ p.
H
l F Ž l k . s l F Ž lŽ k y p . .
ž žH
s exp it 2
E q l 2 Ý pi i
E ki
37
F Ž lŽ k y p . . q . . . .
dk < g Ž k . < 2
y dk < g Ž k . < 2
H
We get that l F Ž lŽ k y p .. is a leading term with respect to l. Taking l ™ 0 we get that we can use < g Ž k y p .< 2 instead of < g Ž k .< 2 in the formulas for I1 and I2 for sufficiently smooth g Ž k .. Let us calculate I1 a d I2 . Using assumptions considered above we get I1 s Ž y2 q 10 p 2 . dk < g Ž k y p . < 2
1 1 q 12 k 2
= dp < f Ž p . < 2 e i t ŽA p
H
`
H0
1 y 12 p 2 q 12 Ž k y p .
2
s Ž y2 q 10 p 2 . dk < g Ž k . < 2
H
y p 2 dk < g Ž k . < 2
H
Q s 6 dk < g Ž k . < 2 k
H
1 q 12 k 2
Ž 1 q 12 k 2 .
I s pQ, 2 2
1 1 q 12 k 2
We get
Ž g < g . y Ž p . s y 2 iHdk < g Ž k . < 2 q 2 i dk < g Ž k . < 2
H
A s 10 dk < g Ž k . < 2
H
1 1 q 12 k 2
Ž 1 q 12 k 2 .
2
3
p
ž
B y iAt
/
2
Acknowledgements
1 1 q 12 k 2
H
1
2
y iAp 2 y ipQ,
y dk < g Ž k . < 2 =
2
dp p 2 eyB p e i A t p s
We get that for large t the decay of the matrix element X Ž t . s ² X < Ut < X : is proportional to Ž At . y3 r2 where A is the functional of the cut-off function given by Ž21.. To conclude, in this paper we obtain that in the polaron model for some Žsymmetric and very smooth. cut-off functions we have the polynomial relaxation, the matrix element being proportional to ty3 r2 . The dependence on the parameter B that corresponds to the size of the support of the smearing function f Ž p . of quantum particle in the momentum space for large t is not important. We can say that particles with large momentum decay exponentially and the particles with small momentum decay as ty3 r2 and the decay for large t does not depend on the smearing function f Ž p ..
1
1
qp Q . 2
4p
1
2
Let us estimate this integral for f Ž p . s eyB p , B 4 1. Let us consider for simplicity the case Q s 0 Žfor example g Ž k . is spherically symmetric.. In this case the integral is equal to
H
=
//
Ž 21 .
S. Kozyrev and I. Volovich are grateful to Centro Vito Volterra where this work was started for kind hospitality. This work was partially supported by INTAS 96-0698 and RFFI-9600312 grants. I.V. Volovich is supported in part by a fellowship of the Italian Ministry of Foreign Affairs organized by the Landau Network-Centro Volta.
38
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
References w1x W.H. Louisell, Statistical properties of Radiation, Wiley, New York, 1973. w2x A.O. Caldeira, A.J. Leggett, Phys. Rev. Lett. 46 Ž1981., p.211. w3x A.J. Leggett, S. Chakravarty, A. Garg, L.D. Chang, Rev. Mod. Phys. 59 Ž1987. N1. w4x L. Accardi, S.V. Kozyrev, I.V. Volovich, Phys. Rev. A 57 Ž1997. N3, quant-phr9706021. w5x N.N. Bogoliubov, Ukr. Mat. J. 2 Ž1950. N2, 3–24. w6x H. Frohlich, Adv. Phys. 3 Ž1954. 325. w7x S.S. Schweber, An introduction to relativistic quantum field theory, Row, Petersen &., Elmsford, NY, 1961. w8x R.P. Feynman, Statistical mechanics, W.A. Benjamin Inc., Advanced Book Program, Reading, Massachusetts, 1972. w9x Van L. Hove, 1955, Physica, 21, 617.
w10x L. Accardi, Y.G. Lu, The Wigner Semi–circle Law in Quantum Electro Dynamics, Commun. Math. Phys. 180 Ž1996. 605, Volterra preprint no. 126 Ž1992. w11x L. Accardi, Y.G. Lu, I. Volovich, Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations, publications of IIAS ŽKyoto. Ž1997. w12x J. Gough, Infinite Analysis, Quantum Probab. Rel. Top., 1 no. 3 Ž1998. 439. w13x M. Skeide, Hilbert modules in quantum electro dynamics and quantum probability. Volterra preprint no. 257 Ž1996., Comm. Math. Phys. Ž1998. w14x L. Accardi, S.V. Kozyrev, I.V. Volovich, Dynamical q-deformation in quantum theory and the stochastic limit, q-algr9807137, J. Phys. A Žto be published. w15x L. Accardi, S.V. Kozyrev, I.V. Volovich, Teor. Mat. Fiz. 116 Ž1998. 401-416. w16x I.Ya. Aref’eva, I.V. Volovich, Phys. Lett. 268 Ž1991. 179.
Physics Letters A 260 Ž1999. 31–38 www.elsevier.nlrlocaterphysleta
Non-exponential decay for Polaron model L. Accardi a
a,)
, S.V. Kozyrev
b,1
, I.V. Volovich
c,2
Centro Vito Volterra UniÕersita di Roma Tor Vergata, Via di Tor Vergata, snc-00133 Roma, Italy b Institute of Chemical Physics, Kossygin Street 4, 117334 Moscow, Russia c StekloÕ Mathematical Institute, Gubkin Street 8, GSP-1, 117966 Moscow, Russia Received 26 April 1999; accepted 28 July 1999 Communicated by P.R. Holland
Abstract A model of a particle interacting with a quantum field is considered. The model includes as particular cases the polaron model and non-relativistic quantum electrodynamics. We compute matrix elements of the evolution operator in the stochastic approximation and show that, depending on the state of the particle, one can get non-exponential decay with the rate ty3r2 . In the process of computation a new algebra of commutation relations that can be considered as an operator deformation of the quantum Boltzmann commutation relations is used. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: Polaron; Stochastic limit; Relaxation
1. Introduction For many dissipative systems one has the exponential time decay of correlations. This result was established for various models and by using various approximations, see for example Ref. w1x. For certain models, in particular for the spin-boson Hamiltonian, also a regime with the oscillating behavior was found w2–4x. The presence of such a regime is very important in the investigation of quantum decoherence. The aim of this note is to show that for the model of particle interacting with quantum field, in particular for the polaron model, one can have not only the standard exponential decay but also the
) Corresponding author. E-mail: [email protected] 1 E-mail: [email protected] 2 E-mail: [email protected]
non-exponential decay Žas some powers of time. of correlations. We investigate the model describing interaction of non-relativistic particle with quantum field. This model is widely studied in elementary particle physics, solid state physics, quantum optics, see for example Refs. w5–8x. We consider the simplest case in which matter is represented by a single particle, say an electron, with position and momentum q s Ž q1 ,q2 ,q3 . and p s Ž p 1 , p 2 , p 3 . satisfying the commutation relations w q j , pn x s i d jn . The electromagnetic field is described by boson operators aŽ k . s Ž a1Ž k .,a 2 Ž k .,a 3 Ž k ..;a† Ž k . s Ž a†1 Ž k ., . . . ,a†3 Ž k .. satisfying the canonical commutation relations w a j Ž k .,a†nŽ kX .x s d jn d Ž k y kX .. The Hamiltonian of a free non relativistic atom interacting with a quantum electromagnetic field is H s H0 q l H I s v Ž k . a† Ž k . a Ž k . dk q 12 p 2 q l H I
H
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 1 0 - 1
Ž 1.
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
32
where l is a small constant, v Ž k . is the dispersion law of the field, H I s d 3 k Ž g Ž k . p P a† Ž k . eyi k q
H
qg Ž k . p P a Ž k . e i k q . q h.c. Ý 3js 1
Ž 2. 2
Ý 3js 1
Here p P aŽ k . s p j a j Ž k ., p s p j2 , 3 †Ž . Ž . 3 †Ž . Ž . a k a k s Ý js1 a j k a j k , kq s Ý js1 k j q j . For the polaron model the Hamiltonian has the form H s v Ž k . a† Ž k . a Ž k . dk q 12 p 2
H
q l d 3 k Ž g Ž k . a† Ž k . eyi k q q g Ž k . a Ž k . e i k q .
H
It is different from Ž1., Ž2. by a momentum p in the interaction Hamiltonian. For the analysis of this paper this difference is not important. In the present paper we will use the method for the approximation of the quantum mechanical evolution that is called the stochastic limit method, see for example w4,9–11x. The general idea of the stochastic limit is to make the time rescaling t ™ trl2 in the solution of the Schrodinger equation in interaction ¨ picture Ut Ž l . s e i t H 0 eyi t H , associated to the Hamiltonian H, i.e.
E
U Ž l . s yi l H I Ž t . Ut Ž l . Et t with H I Ž t . s e i t H 0 H I eyi t H 0 . We get the rescaled equation
E
Ut Žrll. 2 s y
i
2
Ut Žrll. 2
H Ž trl . Et l I and one wants to study the limits, in a topology to be specified, lim Ut Žrll. 2 s Ut ;
l™0
lim l ™0
1
l
HI
t
ž / l2
s Ht
Ž 3.
We will prove that Ut is the solution of the equation
E t Ut s yiHt Ut , U0 s 1
Ž 4.
The interest of this limit equation is in the fact that many problems become explicitly integrable. The stochastic limit of the model Ž1., Ž2. has been considered in w10–15x. After the rescaling t ™ trl2 we consider the simultaneous limit l ™ 0, t ™ ` under the condition
that l2 t tends to a constant Žinterpreted as a new slow scale time.. This limit captures the main contributions to the dynamics in a regime, of long times and small coupling arising from the cumulative effects, on a large time scale, of small interactions Ž l ™ 0.. The physical idea is that, looked from the slow time scale of the atom, the field looks like a very chaotic object: a quantum white noise, i.e. a d-correlated Žin time. quantum field b†j Ž t,k .,bj Ž t,k . also called a master field. If one introduces the dipole approximation the master field is the usual boson Fock white noise. Without the dipole approximation the master field is described by a new type of commutation relations of the following form w11x bj Ž t ,k . pn s Ž pn y k n . bj Ž t ,k .
Ž 5.
bj Ž t ,k . bn† Ž tX ,kX . s 2pd Ž t y tX . d Ž v Ž k . ykp q 12 k 2 . d Ž k y kX . d jn
Ž 6.
Such quantum white noises can be treated as an operator deformation of quantum Boltzmann commutation relations. Recalling that p is the particle momentum, we see that the relation Ž5. shows that the particle and the master field are not independent even at a kinematical level. This is what we call entanglement. The relation Ž6. is a generalization of the algebra of free creation–annihilation operators with commutation relations A i A†j s d i j and the corresponding statistics becomes a generalization of the Boltzmannian Žor Free. statistics. This generalization is due to the fact that the right hand side is not a scalar but an operator Ža function of the atomic momentum.. This means that the relations Ž5., Ž6. are module commutation relations. For any fixed value p of the atomic momentum we get a copy of the free Žor Boltzmannian. algebra. Given the relations Ž5., Ž6., the statistics of the master field is uniquely determined by the condition bj Ž t ,k . C s 0 where C is the vacuum of the master field, via a module generalization of the free Wick theorem, see w14x. In Section 2 the dynamically q-deformed commutation relations Ž7., Ž8., Ž14. are obtained and the
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
stochastic limit for collective operators is evaluated. In Section 3 the stochastic limit of the evolution equation is found. In Section 4 the non-exponential decay for vacuum vector in the polaron model is investigated.
33
exist. Then it is natural to conjecture that its algebra shall be obtained as the stochastic limit Ž l ™ 0. of the algebra Ž7., Ž8.. Notice that the factor qlŽ t y tX , x . is an oscillating exponent and one easily sees that lim ql Ž t , x . s 0 , l™0
2. Deformed commutation relations
l™0
In this section we reproduce in the brief form the notations and the main results of the work w14x. In order to determine the limit Ž3. one rewrites the rescaled interaction Hamiltonian in terms of some rescaled fields al, j Ž t,k .: 1
l
HI
t
ž / l2
H
l 1
s
l
ž ž
exp i
t
/
exp y i
ž
H0 e i k q a j Ž k . exp y i
l2
t
l2
bj Ž t ,k . bn† Ž tX ,kX . s 2pd Ž t y tX . d Ž v Ž k . ykp q 12 k 2 . d Ž k y kX . d jn
X X s al , n Ž tX ,kX . al , j Ž t ,k . qy1 l Ž t y t ,kk .
where 1
and the limit of Ž7. gives the module free relation
Ž 13 .
Operators al, j Ž t,k . also obey the relation al , j Ž t ,k . al , n Ž tX ,kX .
s d 3 kp Ž g Ž k . al Ž t ,k . qg Ž k . a†l Ž t ,k . . q h.c.
al , j Ž t ,k . [
1
q Ž t , x . s 2pd Ž t . d Ž x . Ž 11 . l2 l Thus it is natural to expect that the limit of Ž8. is bj Ž t ,k . pn s Ž pn y k n . bj Ž t ,k . Ž 12 . lim
Ž v Ž k . y kpq 12 k 2 .
t
l2
/
H0
/
ei k q aj Ž k .
It is now easy to prove that operators al, j Ž t,k . satisfy the following q–deformed module relations,
Ž 14 .
In what follows we will not write indexes j, n explicitly. The relation Ž14. should disappear after the limit, see w14x. In fact, if the relation Ž14. would survive in the limit then, because of Ž11., it should give bŽ t,k . bŽ tX ,kX . s 0, hence also b† Ž t,k . b† Ž tX ,kX . s 0, so all the n–particle vectors with n G 2 would be zero.
al , j Ž t ,k . a†l , n Ž tX ,kX . 3. Evolution equation
s a†l , n Ž tX ,kX . al , j Ž t ,k . ql Ž t y tX ,kkX . 1 q
X
l2
1 2
ql Ž t y t , v Ž k . y kp q k
= d Ž k y kX . d jn al , j Ž t ,k . pn s Ž pn y k n . al , j Ž t ,k .
2
. Ž 7. Ž 8.
where ql Ž t y tX , x . s eyi
tytX x l2
Ž 9.
is an oscillating exponent. This shows that the module q–deformation of the commutation relations arise here as a result of the dynamics and are not put artificially ab initio. For a discussion of q-deformed commutation relations see for example w16x. Now let us suppose that the master field bj Ž t ,k . s lim al , j Ž t ,k . l™0
Ž 10 .
Let us find stochastic differential equation for the model we consider. In the introduction we claimed that the stochastic limit for the Schrodinger equation ¨ in interaction picture will have the form Ž4.: E t Ut s yiHt Ut . But in this equation both Ht and Ut are distributions. We need to regularize this product of distributions. In the present section we will make the following regularization: roughly speaking we replace Ht by Htq0 q const. We investigate the evolution operator in interaction picture Ut Ž l.. We start with the equation tqdt
Ž l. Utqd t s 1 q Ž yi l .
ž
q Ž yi l . =
t1
Ht dt
2
2
Ht
tqdt
Ht
H I Ž t 1 . dt 1
dt 1
H I Ž t 1 . H I Ž t 2 . q . . . U tŽl .
/
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
34
Ž l. Ž l. where dt ) 0. We get for dUt Ž l. s Utqd t y Ut
dUt Ž l. s Ž yi l .
ž
tqdt
Ht
t1
=
Ht dt
H I Ž t 1 . dt 1 q Ž yi l .
2
tqdt
Ht
dt 1
H I Ž t 1 . H I Ž t 2 . q . . . U tŽl .
/
2
Let us make the rescaling t ™ trl2 in this perturbation theory series. We get tqdt
dUt Žrll. 2 s Ž yi .
ž
Ht
q Ž yi . =
1
2
HI
l
dt 1
tqdt
Ht
t2
1
l
dt 1
HI
l2
1
t1
2
l
HI
Here ² P : is the stochastic limit of the vacuum expectation of boson field Žthat acts as a conditional expectation on momentum of quantum particle p.. We will prove this result by analizing of the perturbation theory series. We have
t1
ž / l2
q . . . Ut Žrll. 2
ž / / l2
Ž 15 .
To find the stochastic differential equation we need to collect all the terms of order dt in the perturbation theory series Ž15.. Terms of order dt are contained only in the first two terms of these series. Let us investigate the first two terms. For the first term of the perturbation theory we get tqdt
Ht
s
dt1
1
l
tqdt
Ht
HI
² X dB Ž t ,k . Ut : s 0 ; X .
t1
ž /
Ht dt
In the case of Boltzmannian statistics we get the same relation: the Žfree. independence means that roughly speaking dB Ž t,k . kills all creations in Ut . We have the following Lemma. The stochastic differential dB(t,k) and the eÕolution operator Ut are free independent. This means that for an arbitrary obserÕable X,
² X dB Ž t ,k . Ut : s lim
q Žy i .
=
ž /
H
qg Ž k . a†l Ž t 1 ,k . Ž 2 p q k . .
Ž 16 .
In the stochastic limit l ™ 0 this term gives us
Hdk Ž g Ž k . Ž 2 p q k . dB Ž t ,k . where the stochastic differential dB Ž t,k . is the stochastic limit of the field alŽ t,k . in the time interval Ž t,t q dt .: dB Ž t ,k . s lim
tqdt
H l™0 t
dt al Ž t ,k . s
tqdt
Ht
qg Ž k . dB Ž t ,k . Ž 2 p q k . .
tqdt
Ht
dt b Ž t ,k .
We will prove that the stochastic differential dB Ž t,k . and the evolution operator Ut are free independent. In the bosonic case independence would result in the relation w dB Ž t,k .,Ut x s 0. From this relation follows that ² X dB Ž t,k .Ut : s 0 for arbitrary observable X.
1
l
dt al Ž t ,k .
HI
n
1
H0tdt1 l HI
H0tdt1 . . . H0t tn
Ny 1
t1
ž / l2
dt N
1
l
q...
HI
t1
ž / l2
...
ž / /; l2
q...
where l1 H I Ž t krl2 . is given by the formula Ž16.. Here lim l™ 0 Xl s X. Let us analyze the N-th term of perturbation theory. The N-th term of perturbation theory is the linear combination of the following terms Žwe omit integration over k, k n . ² Xl
†
t q dt
t
ž
2
dt 1 dk Ž g Ž k . Ž 2 p q k . al Ž t 1 ,k .
Xl
= 1q Ž y i .
t1
l
¦H
l™ 0
=
dt al Ž t ,k .
t Ny1
H0
t
H0 dt
1
...
dt N al´ 1 Ž t 1 ,k 1 . . . . al´ N Ž t N ,k N . :
Let us shift alŽt ,k . to the right using dynamically q-deformed relations. In the following we will use notions of the work w14x. Let us enumerate annihilators in the product al´ 1 Ž t 1 ,k 1 . . . . al´ N Ž t N ,k N . as alŽ t m j ,k m j ., j s 1, . . . J, and enumerate creators as a†lŽ t mXj ,k mXj ., j s 1, . . . I, I q J s N. This means that if ´ m s 0 then al´ m Ž t m ,k m . s alŽ t m j ,k m j . for m s m j Žand the analogous condition for ´ m s 1..
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
We will use the following recurrent relation for correlator Žanalogous formula was proved in the work w14x.: tqdt
² Xl
Ht
dt
t
H0 dt
1
...
t Ny1
H0
dt N
=al Ž t ,k . al´ 1 Ž t 1 ,k 1 . . . . al´ N Ž t N ,k N . : I
s
tqdt
Ý d Ž k y k m . ² XlH X j
dt
t
js1
t
H0 dt
1
...
t Ny1
H0
dt N
= al´ 1 Ž t 1 ,k 1 . . . . aˆ†l Ž t mXj ,k mXj . . . . al´ N Ž t N ,k N . : = =
ž
Ł
qy1 l Ž t y t mXj ,kk m i .
Ł
ql Ž t y t mXj ,kk mXi .
Ł
X
qy1 l
Ł
X
ql Ž t y t mXi ,kk mXi .
X X m i)m j
=
m i-m j
=
1 2
ql t y t mXj , v Ž k . y kp q k
l2
X m i)m j
=
X
m i-m j
2
Ht
/
s
Ž t y t m ,kk m . i
I
t
js1
Ž 17 .
t Ny1
0
0
=
l
2
dt N
ql yt mXj , v Ž k . y kp q 12 k 2
ž
Ł
X
qy1 l Ž yt mXj ,kk m i .
Ł
X
ql Ž yt mXj ,kk mXi .
X
m i)m j
=
Ł X
X
m i-m j
=
tqdt
Ht
=
l
1
2
l
Ł
X
Ł
X m i-m j
t2
l
HI
t1
Ht dt Hdk < g Ž k . <
l2
2
2
ž
exp yi
2
t1
ž / ž / t1 y t 2
Ž2 pqk .
tqdt
Ht
s
dt 1
tqdt
Ht
1
t1
Ht dt dt 1
2
l
2
eyi
0
t 1 yt 2
0
it x
it x
2
1
s x y i0
spd Ž x . y i PP
1 x
we get for the second term
/
1
2
v Ž k . y kp q 12 k 2 y i0
Let us denote
Ž g < g . y Ž p . s yiHdk < g Ž k . < 2 Ž 2 p q k .
Ł
X
m i-m j
qy1 l Ž yt m i ,kk m i . =
qy1 l Ž t ,kk m i .
X
X
m i)m j
Ł
X X m i-m j
ql Ž t ,kk mXi . ql Ž t ,kk mXi .
2
1
v Ž k . y kp q 12 k 2 y i0
s dk < g Ž k . < 2 Ž 2 p q k .
Ł
/
x
l2
Hyt rl dt e Hy`dte
ql Ž yt mXi ,kk mXi .
qy1 l Ž t ,kk m i .
2
Ž v Ž k . y kp q 12 k 2 .
l2
H
dt ql Ž t , v Ž k . y kp q 12 k 2 .
m i)m j
=
1
dt 1
l
HI
yidt dk < g Ž k . < 2 Ž 2 p q k .
m i)m j
=
tqdt
Ht
2
yi
= al´ 1 Ž t 1 ,k 1 . . . . aˆ†l Ž t mXj ,k mXj . . . . al´ N Ž t N ,k N . : =
1
t1
Ht dt
due to q–module relations on alŽ t,k ., p. Performing integration over t 1 , t 2 and using the formulas
i
Ý d Ž k y k m . ² XlH dt1 . . . H X j
dt 1
=
Here the notion aˆ†l means that we omit the operator a†l in this product. The right hand side of the Eq. Ž17. is equal to
1
Žwe use that ql is an exponent.. The first three lines of this formula do not depend on t and the last two lines do not depend on t 1 , . . . ,t N . Therefore the stochastic limits for these values can be made independently Žthe limit of product is equal to the product of limits.. It is easy to see that the stochastic limit for the multiplier that depends on t Žof the last two lines. is equal to zero. This finishes the proof of the Lemma. The second term of the perturbation theory series is equal Žup to terms of order Ž dt . 2 . tqdt
1
35
H
2
ž
= pd Ž v Ž k . y kp q 12 k 2 . yi PP
1
v Ž k . y kp q 12 k 2
/
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
36
Combining all the terms of order dt we get the following result: Theorem. The stochastic differential equation for Ut s lim l ™ 0 Ut(rll) 2 haÕe a form
ž
dUt s yi dk Ž g Ž k . Ž 2 p q k . dB Ž t ,k .
H
/
Ž 18 .
Eq. Ž18. can be rewritten in the language of distributions as dUt dt
ž
² X < Ut < X : s dp < f Ž p . < 2 eytŽ g < g .yŽ p.
s yi dk Ž g Ž k . Ž 2 p q k . b Ž t ,k .
H
qg Ž k . b† Ž t ,k . Ž 2 p q k . . y Ž g < g . y Ž p . Ut
/
Ž 19 . Here we understand the singular product of distributions bŽ t,k .Ut in the sense that Ž19. is equivalent to Ž 18 . . We have to stress that dB Ž t, k . s Ht tqd t dt bŽt ,k . / bŽ t,k . dt and we can not obtain Ž19. dividing Ž18. by dt.
v Ž k . y kp q 12 k 2 s 1 y 12 p 2 q 12 Ž k y p .
Ž g < g . y Ž p . s yiHdk < g Ž k . < 2 Ž 2 p q k . =
Let us investigate the behavior of ²Ut : using stochastic differential equation Ž18.. We get ² dUt : s ² Ž yi . dkg Ž k . Ž 2 p q k . dB Ž t ,k .
H
/
ydt Ž g < g . y Ž p . Ut : Using the free independence of dB Ž t,k . and Ut we get d dt
²Ut : s ²
d dt
Because U0 s 1, we have the solution ²Ut : s eyt Ž g < g .yŽ p. In this section we calculate the matrix element ² X < Ut < X : where X s f Ž p . m F in the momentum
2
1 1 2
1 y p q 12 Ž k y p . 2
2
s y2 i dk < g Ž k . < 2 y i Ž I1 q I2 . ,
H
I1 s Ž y2 q 10 p 2 . dk < g Ž k . < 2
H
=
1 1 2
1 y p 2 q 12 Ž k y p .
I2 s 6 dk < g Ž k . < 2 p Ž k y p .
H
Ut : s y Ž g < g . y Ž p . ²Ut :
2
One can expect non-exponential relaxation when supp f Ž p . ; < p < - '2 4 . In this case ReŽ g < g .y Ž p . s 0 and there is no dumping. All decay in this case is due to interference. We will use the approximation diam supp g Ž k . 4 diam supp f Ž p .. Physically this means that the particle is more localized in momentum representation than the field. This assumption seems natural because the field’s degrees of freedom are fast and the particles one are slow. Under this assumption we can estimate the matrix element Ž20.. We will prove that in this case there will be polynomial decay. For < p < - '2 we get
4. Non-exponential decay
ž
Ž 20 .
H
We investigate the polaron model when v Ž k . s 1. For this choice of v Ž k . we get
qg Ž k . dB † Ž t ,k . Ž 2 p q k . . ydt Ž g < g . y Ž p . Ut
representation and F is the vacuum vector for the master field. This matrix element is equal to
2
, 1 1 2
1 y p q 12 Ž k y p . 2
2
Here only I1 and I2 depend on p and therefore can interfere. Let us find the asymptotics of Ž g < g .y Ž p . on p Žwe investigate the case when p is a small parameter.. We will use the following assumption on g Ž k .: let g Ž k . be a very smooth function. This means that
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
< g Ž k .< 2 s l F Ž l k ., F Ž k . ) 0 is compactly supported smooth function, l is a small parameter. Let us consider the Taylor expansion on the small parameter p
We get for X Ž t . s ² X < Ut < X : X Ž t . s dp < f Ž p . < 2 eytŽ g < g .yŽ p.
H
l F Ž l k . s l F Ž lŽ k y p . .
ž žH
s exp it 2
E q l 2 Ý pi i
E ki
37
F Ž lŽ k y p . . q . . . .
dk < g Ž k . < 2
y dk < g Ž k . < 2
H
We get that l F Ž lŽ k y p .. is a leading term with respect to l. Taking l ™ 0 we get that we can use < g Ž k y p .< 2 instead of < g Ž k .< 2 in the formulas for I1 and I2 for sufficiently smooth g Ž k .. Let us calculate I1 a d I2 . Using assumptions considered above we get I1 s Ž y2 q 10 p 2 . dk < g Ž k y p . < 2
1 1 q 12 k 2
= dp < f Ž p . < 2 e i t ŽA p
H
`
H0
1 y 12 p 2 q 12 Ž k y p .
2
s Ž y2 q 10 p 2 . dk < g Ž k . < 2
H
y p 2 dk < g Ž k . < 2
H
Q s 6 dk < g Ž k . < 2 k
H
1 q 12 k 2
Ž 1 q 12 k 2 .
I s pQ, 2 2
1 1 q 12 k 2
We get
Ž g < g . y Ž p . s y 2 iHdk < g Ž k . < 2 q 2 i dk < g Ž k . < 2
H
A s 10 dk < g Ž k . < 2
H
1 1 q 12 k 2
Ž 1 q 12 k 2 .
2
3
p
ž
B y iAt
/
2
Acknowledgements
1 1 q 12 k 2
H
1
2
y iAp 2 y ipQ,
y dk < g Ž k . < 2 =
2
dp p 2 eyB p e i A t p s
We get that for large t the decay of the matrix element X Ž t . s ² X < Ut < X : is proportional to Ž At . y3 r2 where A is the functional of the cut-off function given by Ž21.. To conclude, in this paper we obtain that in the polaron model for some Žsymmetric and very smooth. cut-off functions we have the polynomial relaxation, the matrix element being proportional to ty3 r2 . The dependence on the parameter B that corresponds to the size of the support of the smearing function f Ž p . of quantum particle in the momentum space for large t is not important. We can say that particles with large momentum decay exponentially and the particles with small momentum decay as ty3 r2 and the decay for large t does not depend on the smearing function f Ž p ..
1
1
qp Q . 2
4p
1
2
Let us estimate this integral for f Ž p . s eyB p , B 4 1. Let us consider for simplicity the case Q s 0 Žfor example g Ž k . is spherically symmetric.. In this case the integral is equal to
H
=
//
Ž 21 .
S. Kozyrev and I. Volovich are grateful to Centro Vito Volterra where this work was started for kind hospitality. This work was partially supported by INTAS 96-0698 and RFFI-9600312 grants. I.V. Volovich is supported in part by a fellowship of the Italian Ministry of Foreign Affairs organized by the Landau Network-Centro Volta.
38
L. Accardi et al.r Physics Letters A 260 (1999) 31–38
References w1x W.H. Louisell, Statistical properties of Radiation, Wiley, New York, 1973. w2x A.O. Caldeira, A.J. Leggett, Phys. Rev. Lett. 46 Ž1981., p.211. w3x A.J. Leggett, S. Chakravarty, A. Garg, L.D. Chang, Rev. Mod. Phys. 59 Ž1987. N1. w4x L. Accardi, S.V. Kozyrev, I.V. Volovich, Phys. Rev. A 57 Ž1997. N3, quant-phr9706021. w5x N.N. Bogoliubov, Ukr. Mat. J. 2 Ž1950. N2, 3–24. w6x H. Frohlich, Adv. Phys. 3 Ž1954. 325. w7x S.S. Schweber, An introduction to relativistic quantum field theory, Row, Petersen &., Elmsford, NY, 1961. w8x R.P. Feynman, Statistical mechanics, W.A. Benjamin Inc., Advanced Book Program, Reading, Massachusetts, 1972. w9x Van L. Hove, 1955, Physica, 21, 617.
w10x L. Accardi, Y.G. Lu, The Wigner Semi–circle Law in Quantum Electro Dynamics, Commun. Math. Phys. 180 Ž1996. 605, Volterra preprint no. 126 Ž1992. w11x L. Accardi, Y.G. Lu, I. Volovich, Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations, publications of IIAS ŽKyoto. Ž1997. w12x J. Gough, Infinite Analysis, Quantum Probab. Rel. Top., 1 no. 3 Ž1998. 439. w13x M. Skeide, Hilbert modules in quantum electro dynamics and quantum probability. Volterra preprint no. 257 Ž1996., Comm. Math. Phys. Ž1998. w14x L. Accardi, S.V. Kozyrev, I.V. Volovich, Dynamical q-deformation in quantum theory and the stochastic limit, q-algr9807137, J. Phys. A Žto be published. w15x L. Accardi, S.V. Kozyrev, I.V. Volovich, Teor. Mat. Fiz. 116 Ž1998. 401-416. w16x I.Ya. Aref’eva, I.V. Volovich, Phys. Lett. 268 Ž1991. 179.