Path integral representation of fractional harmonic oscillator

ARTICLE IN PRESS

Physica A 371 (2006) 303–316 www.elsevier.com/locate/physa

Path integral representation of fractional harmonic oscillator Chai Hok Eaba, S.C. Limb, a

Department of Chemistry, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand b Faculty of Engineering, Multimedia University, Cyberjaya 63100, Selangor DE, Malaysia Received 6 December 2005; received in revised form 17 January 2006 Available online 2 May 2006

Abstract Fractional oscillator process can be obtained as the solution to the fractional Langevin equation. There exist two types of fractional oscillator processes, based on the choice of fractional integro-differential operators (namely Weyl and Riemann-Liouville). An operator identity for the fractional differential operators associated with the fractional oscillators is derived; it allows the solution of fractional Langevin equations to be obtained by simple inversion. The relationship between these two fractional oscillator processes is studied. The operator identity also plays an important role in the derivation of the path integral representation of the fractional oscillator processes. Relevant quantities such as two-point and n-point functions can be calculated from the generating functions. r 2006 Elsevier B.V. All rights reserved. Keywords: Fractional oscillator process; Path integral; Partition function

1. Introduction During the past two decades we witness an increasing use of differential equations of fractional order in condensed matter physics and physics of mesoscopic systems. For examples, anomalous diffusion [1–3] that occurs in complex heterogeneous and disordered media can be described by fractional diffusion equations; non-Debye dielectric relaxation phenomena [3,4] and complex viscoelasticity [3,5] are modeled by fractional differential equations. The general trend seems to indicate that many transport phenomena in media with fractal characteristic can be described by fractional differential equations. One way to obtain concrete realization of a particular fractional model is to associate it with a fractional generalization of an ordinary stochastic process. The most well-known among these fractional stochastic processes include fractional Brownian motion [6,7] and fractional Levy motion [8,9]. In many applications in physics, in particular in condensed matter physics and quantum theory, the path integral approach provides a convenient and powerful tool for studying various physical systems [10,11]. In the case of fractional Brownian motion, its path integral representation has been studied by Sebastian [12]. Laskin [13] has considered quantum mechanical path integral over Levy paths.

Corresponding author.

E-mail addresses: [email protected] (C.H. Eab), [email protected] (S.C. Lim). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.03.029

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In this paper, we study another type of fractional process, namely the fractional Ornstein-Uhlenbeck process or the fractional oscillator process. In view of the fact that harmonic oscillator is one of the simplest non-trivial examples, and in addition it can be regarded as one-dimensional free Euclidean massive scalar field, therefore its fractional generalization is expected to be of interest. The main aim of this paper is to obtain the path integral representation of fractional oscillator, which so far has not been studied. It is hoped that our results may provide insights for the generalization of path integral approach to fractional derivative quantum fields [14,15]. The plan of this paper is as follows. In the next section we obtain two types of fractional oscillator processes from the fractional Langevin equation. We then proceed to derive the path integral representations of fractional oscillator. Path integral defined on various intervals representing fractional oscillator processes of Riemman-Liouville and Weyl types, and fractional oscillator process at positive temperature. We show that the propagator (correlation function) and the partition function of the fractional oscillator can be obtained by using the path integral method. Finally, we consider briefly the computation of the n-point functions of the fractional oscillator process. 2. Fractional oscillator processes The fractional derivative of order a40 denoted by da =dta
a Dat can be defined in terms of its inverse the fractional integral, a I at
a Da t : Z t 1 a a D f ðtÞ
I f ðtÞ ¼ ðt  uÞa1 f ðuÞ du; tXa. (1) a t a t GðaÞ a A fractional derivative of arbitrary order a Dat , with n  1paon, can be defined through fractional integration of order na and successive ordinary derivative of order n:
n d a an D f ðtÞ ¼ f ðtÞ. (2) a t a Dt dt For 0oao1, or n ¼ 1, one gets Z 1 d t a D f ðtÞ ¼ ðt  uÞa1 f ðuÞ du: a t GðaÞ dt a

(3)

(1) and (2) are known as the Riemann–Liouville (RL) and Weyl integral and derivative if a ¼ 0 and a ¼ 1, respectively [16–19]. We first recall that Brownian motion B(t) and white noise Z(t) are related through the equation dB
Dt B ¼ ZðtÞ, (4) dt where the correlation function   Brownian motion process B(t) is a zero mean Gaussian process with   BðtÞBðsÞ ¼ t ^ s ¼ minðt; sÞ, and the white noise Z(t) is defined by /Z(t)S ¼ 0, ZðtÞZðsÞ ¼ dðt  sÞ. Note that (4) is to be understood in the sense of distribution or generalized functions [20]. Note that t denotes the Euclidean time in all our discussion in this paper. Fractional generalization of (4) leads to fractional Brownian motion (fBm) X(t) with a a Dt X ðtÞ

¼ ZðtÞ.

(5)

In the case where a Dat is the fractional derivative of order a of RL type with a ¼ 0, then for X(0) ¼ 0, (5) can be inverted to give a X ðtÞ ¼ 0 Da t ZðtÞ
0 I t ZðtÞ Z t 1 ¼ ðt  uÞa1 ZðuÞ du. GðaÞ 0

ð6Þ

The fractional Brownian motion defined by (6) is called fBm of Riemann-Liouville type [21]. If the fractional derivative in (5) is of Weyl type, its direct inversion is not possible since it is divergent; and it needs to be

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modified as done by Mandelbrot and van Ness in order to obtain the usual fBm [6]. It can be shown that the RL fBm approaches the standard fBm of Mandelbrot and van Ness when t ! 1 [21]. Next we consider the Ornstein-Uhlenbeck process (or oscillator process) Y(t), which is given by the solution to the Langevin equation ðDt þ oÞY ðtÞ ¼ ZðtÞ.

(7)   Solutions for a processes begins at time   t ¼ 0 and t ¼ 1 are Gaussian processes with Y ðtÞY ðsÞ ¼ eoðtþsÞ ½e2oðt^sÞ  1ð2oÞ1 and Y ðtÞY ðsÞ ¼ eojtsj ð2oÞ1 as their respective correlation functions. If we allow the definitions of RL and Weyl fractional derivatives to include the a ¼ 1 case, then these OU processes can respectively be regarded as the non-stationary ‘‘RL OU process’’ and the stationary ‘‘Weyl OU process’’. Basically there exist two ways to ‘‘fractionalize’’ the Langevin Eq. (7). One way to generalize the OU process Y(t) to a fractional OU process Q(t) is to consider the following fractional Langevin equation [19,22,23] ð0 Dat þ oÞQðtÞ ¼ ZðtÞ;

a40.

(8)

Deterministic version of such a fractional oscillator equation has been considered by several authors [24,25] with the white noise replaced by a deterministic force. (8) can be solved by using Laplace transform. For simplicity, we consider the case ap1, and Q(0) ¼ 0. Then the solution of (8) is given by Z t QðtÞ ¼ ðt  uÞa1 E a;a ðoðt  uÞa ÞZðuÞ du (9) 0

This is a non-stationary process and its correlation function has a rather complicated form [19,22,23]: 1   X QðsÞQðtÞ ¼

ðoÞjþk2 saj tak1 2 F 1 ð1; 1  ak; 1 þ aj; s=tÞ, Gðaj þ 1ÞGðakÞ j;k¼1

(10)

where we have assumed sot. On the other hand, if one replace RL fractional derivative by the Weyl derivative, the solution contains integral which is not convergent just like in the case of fBm. In order to obtain the standard fBm, the Weyl fractional integral needs to be modified in order to make the integral convergent [6]. Another way to introduce fractional OU process is to use the following fractional Langevin equation: ðDt þ oÞa QðtÞ ¼ ZðtÞ.

(11)

We shall show below that the fractional OU process or fractional harmonic oscillator given by (11) seems to be the ‘‘correct’’ generalization of ordinary OU or oscillator process. (11) can be solved by using Fourier or Laplace transform. However, for our purpose of deriving the path integral representation for the fractional OU process, we solve (11) by inverting the fractional differential operator ða Dt þ oÞa . Before we do that, we give some remarks about (11). Q(t) has to be considered as a generalized random process [20], and the elliptic pseudo-differential operator ðD2t þ o2 Þa : SðRÞ ! SðRÞis continuous and essentially self-adjoint in L2 ðRÞ with SðRÞas a core [26]. In order to invert (11), we need the following results: the identity below holds for both the Weyl and Riemann-Liouville fractional derivatives: ða Dt þ oÞa
eot a Dat eot .

(12)

To verify this we note that the fractional operator on the left-hand side of (12) can be formally expanded as [17] 1
 X a a ok a Dak (13a) ða Dt þ oÞ ¼ t k k¼0 ¼

1
 X a k¼0

k

oak a Dkt ,

(13b)

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where a j

! ¼

Gða þ 1Þ . Gðj þ 1ÞGða  j þ 1Þ

(14)

Consider first the case of Weyl fractional derivative1 Dat . Let f(t) be an arbitrary analytic function, then using Leibniz rule one gets ! 1 X a    k ot ak f ðtÞ eot 1 Dat ½eot f ðtÞ ¼ eot 1 Dt e 1 Dt k¼0 k ! 1 X a ot f ðtÞ, ð15aÞ ¼e ok eot 1 Dak t k k¼0

e

ot

a ot 1 Dt ½e f ðtÞ

¼e ¼e

ot

ot

! 1 X a  k¼0

k

1 X

a

k¼0

k

ak ot e 1 Dt



k 1 Dt f ðtÞ



! oak eot 1 Dkt f ðtÞ.

ð15bÞ

For the RL fractional derivative, we can verify (15a) in the same way. For the verification of (15b), one can either use 0 Dat f ðtÞ ¼ 1 Dat ðyðtÞf ðtÞÞ or apply the generalized Leibniz rule [19]: a 0 Dt ½f ðtÞgðtÞ

¼

1
 X a k¼0

k

ak f ðtÞ 0 Dt



k 0 Dt gðtÞ



¼

1
 X a k¼0

k

k 0 Dt f ðtÞ



ak gðtÞ 0 Dt



,

(16)

for the analytic functions f ðtÞ ¼ eot and g(t). We remark that the operator identity (12) can also be verified by considering its Fourier transform in the case of Weyl fractional derivative, and using the Laplace transform for the Riemann–Liouville fractional derivative. By using the operator identity (12), we can now invert (11): QðtÞ ¼ ðeot a Dat eot Þ1 ZðtÞ ¼ ðeot a I at eot ÞZðtÞ.

ð17Þ

In contrast to fBm, both the Weyl and RL Weyl fractional oscillators can be obtained by inversion (17). The Weyl oscillator process Q(t) turns out to be a stationary centred or zero mean Gaussian process; and its correlation function for t4s is given by     QðtÞQðsÞ ¼ ðeot 1 I at eot ÞZðtÞðeos 1 I as eos ÞZðsÞ Z s Z   eoðtþsÞ t ¼ dv dueoðuþvÞ ðt  vÞa1 ðs  uÞa1 ZðvÞZðuÞ 2 ½GðaÞ 1 1 Z eoðtþsÞ s ¼ due2ou ½ðt  uÞðs  uÞa1 ½GðaÞ2 1 Z 1 eoðtþsÞ 2a1 ¼ ðt  sÞ due2oðtsÞu ½uð1  uÞa1 ½GðaÞ2 0 t  s a1=2 1 ¼ pffiffiffi K a1=2 ðoðt  sÞÞ, ð18Þ pGðaÞ 2o where Kv(Z) is the modified Bessel function of second kind or MacDonald function [27].

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On the other hand, the RL fractional oscillator is a non-stationary centred Gaussian process with the following correlation function (for sot):     Q0 ðtÞQ0 ðsÞ ¼ ½eot 0 I at eot ZðtÞ½eos 0 I as eos ZðsÞ Z eoðtþsÞ s due2ou ½ðt  uÞðs  uÞa1 ð19aÞ ¼ ½GðaÞ2 0 Z  eoðsþtÞ sa ta1 1 h s ia1 2ou ¼ ðuÞ du ð1  uÞ 1  e t ½GðaÞ2 0  eoðsþtÞ sa ta1 s ; 2os , ¼ Bð1; aÞF 1; 1  a; 1 þ a; 1 t ½GðaÞ2

ð19bÞ

where B(a, b) is the beta function, F1(a, b, c, x, y) is the confluent hypergeometric function in two variables [27]. By using the series expansion of F1, (19) can be written as [28]:   eoðsþtÞ sa ta1 X ð1Þmþn ð1  aÞn s m Q0 ðtÞQ0 ðsÞ ¼ ð2osÞn , a½GðaÞ2 m;n ð1 þ aÞmþn m!n! t

(20)

where ðaÞn ¼ Gða þ nÞ=GðaÞ is the Pochhammer’s symbol. (20) can be transformed to become the sum of Gauss hypergeometric functions [28]: 1  X s ð1Þn ð2osÞn X ð1 þ nÞm ð1  aÞm s m F1 1; 1  a; 1 þ a; ; 2os ¼ t ð1 þ aÞn n! ð1 þ a þ nÞm m! t n¼0

¼

1 X ð2osÞn 2 F 1 ð1 þ n; 1  a:1 þ a þ n; s=tÞ. ð1 þ aÞn n¼0

ð21Þ

Next, we show that RL fractional oscillator asymptotically approaches the Weyl fractional oscillator as t ! 1. From (19a) one gets Z   eoðstÞ t Q0 ðtÞQ0 ðsÞ ¼ du½uðu  ðt  sÞÞa1 e2ou . (22) ½GðaÞ2 ts For finite (ts) and t ! 1, (22) becomes Z   eoðstÞ 1 Q0 ðtÞQ0 ðsÞ ¼ du½uðu  ðt  sÞÞa1 e2ou ½GðaÞ2 ts
 21=2a jt  sj a1=2 ¼ pffiffiffi K a1=2 ðojt  sjÞ, o pGðaÞ

ð23Þ

which is identical to (18). Since both Q(t) and Q0(t) are centred Gaussian processes one can conclude that limt!1 Q0 ðtÞ ¼ QðtÞ, where the equality is in the sense of finite distribution. We can extend the fractional oscillator to positive temperature. For an oscillator in Euclidean time at positive temperature T ¼ 1=b40; one imposes the periodic time condition QðtÞ ¼ Qðt þ bÞ which is just the Kubo-Martin-Schwinger condition [29,30]. The usual definitions of RL integro-differential operators are not suitable for dealing with periodic functions since they do not preserve periodicity. In order to deal with periodic functions, it is necessary to modify the definitions of fractional integro-differential operators such that they transform a periodic function into another periodic one. Here we shall follow the definitions of the Weyl fractional integro-differential operators for the periodic functions given by Samko et al. [17]. Consider a b-periodic function f(x) on R. Its Fourier series is given by sffiffiffi 1 1 X f ðxÞ ¼ cn eion x , (24a) b n¼1

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sffiffiffi Z 1 b ion x cn ¼ e f ðxÞ dx; b 0

(24b)

with on ¼ 2np=b. It is assumed that the functions to be considered have zero mean value: Z b bc0 ¼ f ðxÞ dx ¼ 0.

(25)

0

In other words, when dealing with of periodic functions, we shall discard the constants. P fractionaln integrals ikn x Since the nth derivative f n ðxÞ ¼ 1 ðio Þ c e , the fractional integration can then be defined as n n n¼1 a 1 I t f ¼

1 X

cn ion x ; a e ðio Þ n n¼1

a tI 1f ¼

1 X

cn ion x . a e ðio Þ n n¼1

(26)

The fractional differentiation can be defined in a similar way as a 1 Dt f ¼

1 X

ðion Þa cn eion x ;

n¼1

a t D1 f ¼

1 X

ðion Þa cn eion x .

(27)

n¼1

In this way, the fractional integral and derivative of a b-periodic function is again a b-periodic function. By adopting the definitions of the Weyl fractional integro-differential operators for periodic functions [17], we can now consider the finite temperature fractional oscillator process Qb(t), which satisfies the following fractional Langevin equation ð1 Dt þ oÞa Qb ðtÞ ¼ Zb ðtÞ.

(28)

with the periodic boundary condition Qb ðt þ bÞ ¼ Qb ðtÞ, and periodic white noise Zb ðt þ bÞ ¼ Zb ðtÞ,   Zb ðtÞ ¼ 0;

1   1 X Zb ðtÞZb ðsÞ ¼ eion ðtsÞ b n¼1

Inversion of (28) gives     Qb ðtÞQb ðsÞ ¼ ðeot 1 I at eot ÞZb ðtÞðeos 1 I as eos ÞZb ðsÞ 1 eoðtþsÞ X ð1 I at eðoþion Þt Þð1 I as eðoion Þs Þ ¼ b n¼1 1 1 X eion ðtsÞ ¼ . b n¼1 ðo2n þ o2 Þa

(29)

ð30Þ

When a ¼ 1, the covariance reduces to that of ordinary harmonic oscillator at finite temperature. 3. Path-integral representation of fractional oscillator In this section we consider the path integral of fractional harmonic oscillator. For this purpose we need to know the action of the fractional oscillator and how to evaluate the resulting path integral. As we shall see below, there seem to exist two possible actions for the fractional oscillator, and they give equivalent Gaussian path integrals under certain mild conditions. Since they are nonlocal actions the usual techniques of path integrals need to be modified before they are applied. Our main aim is to evaluate the path integral in order to obtain correlation function (or propagator) and partition function for some boundary conditions using an adapted techniques of Gaussian functional integral and Fourier diagonalization. We would like to stress that such calculations is made possible because of the representation of the operator ðDt þ oÞa in the form given by (9). This operator identity enables the classical equation of motion, which is a fractional differential equation, becomes solvable. We also give a brief discussion on the calculation of n-point functions, in view of the fact that the fractional oscillator process is non-Markovian and the two-point function (propagator) cannot be used to generate n-point functions.

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Since the fractional oscillator process Q(t) from the fractional Langevin Eq. (8) is linear with respect to the white noise [30], we can intuitively write down the propagator of Q(t) as Z Qb   D½Q exp½SL ½Q, (31) Z Qb ; bjQa ; a ¼ Qa

where D½Q ¼ limn!1 1=N n dQðt1 Þ dQðt2 Þ dQðt3 Þ    dQðtn Þ, with tjþ1  tj ¼ e ¼ 1=n. Note that Nn involves e which is a rather complicated; its explicit value is not necessary for our subsequent discussion. We remark that the Jacobian of transformation in the path integral can be shown to be given by the functional determinant det½ðD2t þ o2 Þa , its evaluation requires regularization techniques such as the generalized zeta function method [Ref. 31]. The action of the fractional oscillator is given by Z 1 b SL ½Q ¼ dt½ðDt þ oÞa QðtÞ2 . (32) 2 a The subscript L denotes the action is associated with the fractional Langevin equation (11). When

a ¼b1, this action does not reduce to the harmonic oscillator; it differs from it by the boundary terms 1=2o QðtÞ2 a . It is then obvious that this provides a way to ‘‘fractionalize’’ the path integration when Qb ¼ Qa, that is for the case with such a periodic boundary condition. By employing fractional integration by parts [16,17], one can show that there exists another possible action given by Z 1 b Sf ½Q ¼ dtQðtÞðD2t þ o2 Þa QðtÞ. (33) 2 a We note that (33) differs from SL[Q] in boundary terms which contain Q and its derivatives of all order. If one considers the case a ¼ 1, then Sf differs from the action for the ordinary harmonic oscillator by the boundary term 1=2½QðtÞ dQðtÞ=dtba . However, Sf determines the equation of motion and the classical solution S f ½Qcl  ¼ 0. For this reason, it is clear that SL and Sf are identical on the space of piecewise continuous functions Q : ½a; b ! R, which satisfy the boundary conditions QðaÞ ¼ QðbÞ ¼ 0. Now we explore the path integration with the action SL given by (32), which can also be expressed by using (9) in the following form Z 1 b SL ½Q ¼ dt½eot 1 Dat eot QðtÞ2 . (34) 2 a Before we proceed to obtain the path integral representation of fractional oscillator process, let us introduce   Rb some notations. We denote inner product and norm are respectively by f ; g ¼ a dtf ðtÞgðtÞ and ffiffiffiffiffiffiffiffiffiffiffiffi ffi q   f ¼ f ; f . We shall use the following compact notations for operators related to the fractional oscillator: K ¼ ða Dt þ oÞa ;

K ¼ ða Dt þ oÞa .

(35)

The right inverse of each of these operators is defined as K G ¼ 1,

KG ¼ 1; 

(36)



Note that K; K ; G and G : SðRÞ ! SðRÞ, where SðRÞ is Schwartz space of test functions. We have Kf ðtÞ ¼ eot 1 Dat eot f ðtÞ;

K f ðtÞ ¼ eot t Da1 eot f ðtÞ,

(37)

Gf ðtÞ ¼ eot 1 I at eot f ðtÞ;

G f ðtÞ ¼ eot t I a1 eot f ðtÞ.

(38)

The generating function for the corresponding path integration can be defined by introducing a source term. The initial and end points are not explicitly specified, and they are to be understood within the context: Z

  ZðJÞ ¼ D½Q exp S L ½Q þ J; Q (39)

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We want to obtain the classical path for ap1. Consider the following variation of the function Q(t) around x(t) classical path by q(t): QðtÞ ¼ xðtÞ þ qðtÞ

(40)

where the classical path begins at x(a) ¼ Xa and ends at x(b) ¼ Sb xðbÞ ¼ X b , while qðaÞ ¼ 0; qðbÞ ¼ 0. The condition for the action with a source term to be a minimum is dSL ½Q  d J; Q ¼ 0. Thus equation of motion is given by K fKx ¼ J.

(41)

with fðtÞ ¼ yðb  tÞyðt  aÞ. The solution is then given by x ¼ x0 þ GG fJ,

(42)

where x0(t) is the solution of homogeneous equation with J ¼ 0. Using the solution in the action and separate the classical and fluctuation part, we get   2 1 Z½J  ¼ exp hJ; x0 i þ G fJ Z ½0, 2

(43)

where   Z½0 ¼ Z Qb ; bjQa ; a ¼ expfS L ½x0 gZ 0 ,

(44)

Z Z0 ¼ Zð0; bj0; aÞ ¼

D½q expfSL ½qg.

(45)

Then the mean and correlation function are given by   QðtÞ ¼ lim

d

J!0 dJðtÞ

ln Z½J ¼ x0 ðtÞ,

(46)

     Cðt; sÞ ¼ QðtÞQðsÞ  QðtÞ QðsÞ   d2 ln Z½J ¼ G fdt ; G fds . J!0 dJðtÞdJðsÞ

¼ lim

ð47Þ

As we are dealing with the Gaussian measure, suffice to restrict our consideration explicitly for x0(t) and C(t, s) in the following cases. 3.1. Finite interval ½0; b First we consider the finite interval ½0; b case with the boundary conditions QðbÞ ¼ X b andQð0Þ ¼ X 0 . The characteristic function is fb0 ðtÞ ¼ yðb  tÞyðtÞ. Here, we would like to solve the homogeneous equation of motion K fKx ¼ 0 on a finite interval ½0; b. We have Kx0 ¼ Aðb  tÞa1 eot ,

(48)

where A is an arbitrary constant. If we write x0 ðtÞ ¼ x¯ 0 ðtÞfb0 ðtÞ þ X 0 , with x¯ 0 ðtÞ ¼ x0 ðtÞ  X 0 such that x¯ 0 ð0Þ ¼ 0 and

(49) fb0 ðtÞ

¼ yðb  tÞyðtÞ, then we have

Kx0 ¼ Kfx¯ 0 þ KX 0 .

(50)

The solution is x¯ 0 ðtÞ ¼ x0  X 0 ¼ oa X 0 Gf1 þ AGfðb  tÞa1 eot þ Bta1 eot .

(51)

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From the boundary condition at t ¼ 0, we have x¯ 0 ð0Þ ¼ 0, which implies B ¼ 0. Thus x¯ 0 ðtÞ ¼ x0  X 0 ¼ oa X 0 Gf1 þ AGfðb  tÞa1 eot ,

(52)

with Gf1 ¼

eot GðaÞ

Z

t

duðt  uÞa1 eou ¼

0

oa gða; otÞ GðaÞ

and g(a, x) is the incomplete gamma function. Let Z t a1 ot ot 1 M ðtÞ ¼ Gfðb  tÞ e ¼ e dsðt  sÞa1 ðb  sÞa1 e2os . GðaÞ 0

(53)

(54)

In order to evaluate A, we only need to know the values of M(0) and M(b). It is obvious that M(0) ¼ 0. By a straight-forward calculation, we get MðbÞ ¼

eob gð2a  1; 2obÞ . ð2oÞ2a1 GðaÞ

(55)

Thus, 

 X 0 gða; obÞ ð2oÞ2a1 GðaÞ X 0 gða; otÞ xðtÞ . M ðtÞ  ¯ ¼ Xb  X0 þ GðaÞ eob gð2a  1; 2obÞ GðaÞ With the help of the identity Gða; xÞ ¼ GðaÞ  gða; xÞ for incomplete gamma function [23], we obtain   X 0 Gða; obÞ ð2oÞ2a1 GðaÞ X 0 gða; otÞ . x0 ðtÞ ¼ X 0 þ X b  MðtÞ  ob GðaÞ e gð2a  1; 2obÞ GðaÞ By substituting (57) into (34), we obtain the classical action as   1 X 0 Gða; obÞ 2 ð2oÞ2a1 GðaÞ2 SL ½x0  ¼ X b  2 GðaÞ gð2a  1; 2obÞ and the propagator (31) can now be written as ( )     1 X 0 Gða; obÞ 2 ð2oÞ2a1 GðaÞ2 Z½0 ¼ Z X b ; bjX 0 ; 0 ¼ exp  X b  Z0 . 2 GðaÞ gð2a  1; 2obÞ

(56)

(57)

(58)

(59)

The direct evaluation of Z0 as a Dirichlet boundary condition is difficult as it cannot be diagonalized by Fourier sine series with the fundamental period (half wavelength) b, as in the case of ordinary harmonic oscillator. However, Z0 is a normalization factor, its value can be determined indirectly by imposing the following normalization condition: Z 1 Z 1   1¼ dX b Z X b ; bjX 0 ; 0 ¼ Z ð0; bj0; 0Þ dX b expfS L ½x0 g 1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pgð2a  1; 2obÞ ¼ Zð0; bj0; 0Þ , ð60Þ ð2oÞ2a1 GðaÞ2 which implies sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2oÞ2a1 GðaÞ2 . Zð0; bj0; 0Þ ¼ 2pgð2a  1; 2obÞ

(61)

It can be easily verified that the results obtained for the fractional oscillator in the limit o-0, coincide with those obtained by Sebastian for the fractional Brownian motion of Riemann–Liouville type [12]. Now, the generating function 2 can be evaluated explicitly by substituting (57), (59) into (43) and (44), and express the term G fJ corresponding to finite interval ½0; b. The mean is just the classical

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  path (57), QðtÞ ¼ x0 ðtÞ. The correlation function (47) associated to the finite interval t; s 2 ½0; b is then given by Z b Z 1 1 ox dxe duðu  xÞa1 yðb  uÞyðu  xÞeou dðu  tÞ Cðt; sÞ ¼ 2 ½GðaÞ 0 1 Z 1  eox dvðv  xÞa1 yðb  vÞyðv  xÞeov dðv  tÞ 1 Z eoðtsÞ t^s ¼ dxðt  xÞa1 ðs  xÞa1 e2ox , ð62Þ ½GðaÞ2 0 which is exactly the same as (19a) obtained earlier for the Riemann–Liouville fractional oscillator process. 3.2. Lower semi infinite interval ð1; b For the range of integration ð1; b, we have from SL[Q] the equation of motion given by (42)), except that f is to be replaced by y which denotes the step function y(bt), that is K yKx ¼ 0. Then the corresponding solution similar to (51) is given by Z 1 ot t e x0 ðtÞ ¼ A dsðt  sÞa1 ðb  tÞa1 e2os GðaÞ 1 Z 1 ot 1 e ¼A dssa1 ðb  t þ sÞa1 e2os : ð63Þ GðaÞ 0 By using the integral identity ([23, p. 319, 3.383.8]), we get
 1 ðb  tÞ a1=2 bo e K a1=2 ððb  tÞoÞ. x0 ðtÞ ¼ A pffiffiffi 2o p

(64)

The value of A can be obtained by taking the limit t-b in (63), which gives for a41/2 A ¼ Xb

eob ð2oÞ2a1 GðaÞ . Gð2a  1Þ

(65)

Thus we get GðaÞ ð2ðb  tÞoÞa1=2 K a1=2 ððb  tÞoÞ. x0 ðtÞ ¼ X b pffiffiffi pGð2a  1Þ

(66)

For ap1/2, the integral diverges, which implies A must be zero. Thus in this case the only possible solution is with the boundary condition x0 ðbÞ ¼ X b ¼ 0, i.e. x0 ðtÞ ¼ 0

for all t 2 ð1; b and for

ap12.

(67)

The correlation differs considerably from the finite interval case. From (47), we get the correlation function: C b ðt; sÞ ¼ yðb  maxðt; sÞÞCðt; sÞ.

(68)

For bXt, s, we have Z 1 Cðt; sÞ ¼ dxeox x I ab eox dðx  tÞeox x I ab eox dðx  sÞ 1

¼

1 eoðtþsÞ ½GðaÞ2

Z

minðt;sÞ

dxðt  xÞa1 ðs  xÞa1 e2ox .

ð69Þ

1

Let t4s, then change variable x-sx, (69) becomes Z 1 1 oðtsÞ Cðt; sÞ ¼ e dxxa1 ðt  s þ xÞa1 e2ox . ½GðaÞ2 0

(70)

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By using the integral (69) one gets
 1 ðt  sÞ a1=2 Cðt; sÞ ¼ pffiffiffi K a1=2 ððt  sÞoÞ, 2o pGðaÞ

313

(71)

which agrees with (18), the correlation function for the Weyl fractional oscillator process. We remark that the correlation function Cb(t, s) depends on the b in the sense that b4t4s. 3.3. Periodic boundary condition (finite temperature case) Similar to the Harmonic oscillator problem, we consider an oscillator on a heat bath at temperature T, the partition function then can be written as path integral on the periodic boundary condition on ½0; b, where b ¼ 1=kT . The periodic boundary condition is essential for thermodynamic equilibrium (KMS condition) [29,30]. The action satisfying periodic boundary condition is given by  1 2 1 SL ½Q ¼ KQ ¼ S f ½Q ¼ Q; K KQ (72) 2 2 with discrete Q(t) that can be expressed in terms of the complete Fourier set feion t ; n ¼ 0; 1; 2; . . .g, on ¼ 2np=b: sffiffiffi 1 1 X QðtÞ ¼ (73) q~ eion t , b n¼1 n and q~ n ¼ q~ n . Using the fact that Keiok t ¼ ðo  iok Þa eiok t we can write   Z n o 1 dReðq  2 ~ Þ dImðq~ n Þ dq~  1 2a  2 ~ pffiffiffi exp ðo2 þ o2k Þa q~ n  þ J~ n q~ n Z½J ¼ pffiffiffiffiffiffi exp  o q~  þ J  q~  P pffiffiffi n n¼1 2 p p 2p ( ) 1 jJ n j2 1 X . ¼ Z½0 exp 2 n¼1 ðo2 þ o2k Þa Using the fact that sffiffiffi sffiffiffi 1 1 d 1 X 1 X @ ion t @ ¼ e ¼ eion t  ~ dJðtÞ b n¼1 b n¼1 @J n @J~ n

(74a)

(74b)

(75)

the correlation can be thus obtained as Cðt; sÞ ¼

1 d d 1 X 1 ln Z½J ¼ eion ðtsÞ . dJðtÞ dJðsÞ b n¼1 ðo2 þ o2n Þa

(76)

Denoted (74a) for J ¼ 0 by Z ao and evaluation of the integral gives: Z ao ¼

1 1 1 P 2 . a o n¼1 ðon þ o2 Þa

(77) can be related to the partition function Z 1o of harmonic oscillator as

a

 a . Z ao ¼ Z1o ¼ 2 sin h ob=2

(77)

(78)

Note that the partition function for the fractional harmonic oscillator can also be calculated by using the thermal zeta function technique [31]. 3.4. Infinite interval Finally we consider the case with the action defined on R, with the boundary condition such that the classical solution gives limt!1 xðtÞ ¼ 0. The equation of motion is then given by (41) with f

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replaced by 1 or K KxðtÞ ¼ JðtÞ.

(79)

The homogeneous solution has no other solution than the trivial solution x0(t) ¼ 0, which implies SL ½x0  ¼ 0. The solution of (79) is then given by xðtÞ ¼ GG JðtÞ.

(80)   Thus, we can calculate Z½J; QðtÞ in a similar way as carried out for the previous cases. As for the correlation function, from (68), and (71), we have Cðt; sÞ ¼ lim C b ðt; sÞ. b!1

(81)

Alternatively, we can use (47) for direct calculation based on the generating function which gives Z 1 dxeox x I a1 dðx  tÞeox x I a1 dðx  sÞ Cðt; sÞ ¼ 1

t  s a1=2 1 K a1=2 ðoðt  sÞÞ, ¼ pffiffiffi pGðaÞ 2o

ð82Þ

which is just the correlation of the Weyl fractional oscillator process. This result is just a consequence of the fact that the RL fractional oscillator approaches the Weyl fractional oscillator as time becomes very large. 3.5. n-Point functions Due to the fact that fractional oscillator processes do not satisfy the Markov property, its two-point function or propagator cannot be used to calculate the n-point functions since the Chapmann–Komogorov condition fails. In this subsection we show that the three-point function (and hence n-point functions) of the fractional oscillator can be explicitly calculated. For aot1 ot2 ot3 o    otn ob, where a and b are as defined in the previous section, the n-points distribution is the integral of the from: Z n h i Y   1 ðnÞ n D½Q r fX ; tgj¼1 ¼ d Qðtj Þ  X j eS½Q Z j¼1 Z 1 Z Pn 1 1 i p ðQðt ÞX ÞS ½Q ¼ dp dp    dpn D½Q e j¼1 j j j Z ð2pÞn 1 1 2 Z 1 Pn 1 1 i pX j¼1 j j Z ½J  ¼ dp dp    dp e ð83Þ n Z ð2pÞn 1 1 2 P where JðtÞ ¼ i nj¼1 pj dðt  tj Þ. We can write explicitly (81) as Z 1 Pn h i 1 i pX j¼1 j j dp dp    dp e rðnÞ fX ; tgnj¼1 ¼ n ð2pÞn 1 1 2 2 2 3 X n n X  exp4i pj x0 ðtj Þ  12 pj G fdtj 5 j¼1 j¼1 Z 1 Pn 1Pn 1 i p ðx ðt ÞX Þ p Cðt ;t Þp ¼ dp1 dp2    dpn e j¼1 j 0 j j 2 i;j¼1 i i j j . ð84Þ n ð2pÞ 1 The verification of (84) is directly consequence of (47). It is obvious that the integral is Gaussian. Thus we can integrate and get h i 1 1 0y 1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi e1=2X ½Cn  X , rðnÞ fX ; tgnj¼1 ¼ ð2pÞn=2 det Cn

(85)

where Cn are the n  n matrices, with element ½Cn ij ¼ Cðti ; tj Þ, and X 0y ¼ ðX 1  x0 ðt1 Þ; X 2  x0 ðt2 Þ; . . . ; X n  x0 ðtn ÞÞ.

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We shall consider the case of finite interval ½0; b as an illustration. In this case, we must consider the two end points as conditional. Let us begin with the one point conditional distribution:

 rð1Þ X ; tjX b ; b; X 0 ; 0 Z   1  D½QeS½Q d QðbÞ  X b dðQðtÞ  X ÞdðQð0Þ  X 0 Þ ¼

Z X b ; b; X 0 ; 0 2 1 ð86Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi e1=2C 11 jX x0 ðtÞj . 2pC 11 However, we can calculate it as a special three-points distribution where the two end points are considered: Z  

 1 D½QeS½Q d QðbÞ  X b dðQðtÞ  X ÞdðQð0Þ  X 0 Þ rð3Þ X b ; b; X ; t; X 0 ; 0 ¼ Z



 Z X b ; b; X 0 ; 0 ð1Þ ¼ r X ; tjX b ; b; X 0 ; 0

Z  1 2C1 jX x0 ðtÞj2 Z X b ; b; X 0 ; 0 , ð87Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi e 11 Z 2pC 11 where Z

Z

1

Z¼

1

dX 0 1

 dX b Z X b ; b; X 0 ; 0 .

(88)

1

Since the integration by X0 or Xb, say Xb, will lead to the function that independent of the other X0. Thus we R V =2 have to take the cut-off for X0 with V =2 dX 0 ¼ V , we get in the un-normalized form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pgð2a  1; 2obÞ (89) Z ¼ VZ ½0; b; 0; 0 ð2oÞ2a1 GðaÞ2 and from (59), thus we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (  )

  Z X b ; b; X 0 ; 0 1 ð2oÞ2a1 GðaÞ2 X 0 Gða; obÞ 2 ð2oÞ2a1 GðaÞ2 1 exp 2 X b  ¼ . V 2pgð2a  1; 2obÞ GðaÞ gð2a  1; 2obÞ Z

 Since we know C 11 ¼ Cðt; tÞ and ½C 1 1 11 ¼ 1=Cðt; tÞ, thus for finite interval, we get Z eoðtþsÞ t^s dxðt  xÞa1 ðs  xÞa1 e2ob Cðt; tÞ ¼ lim s!t GðaÞ2 0 gð2a  1; 2otÞ ¼ . ð2oÞ2a1 GðaÞ2

(90)

ð91Þ

We can thus re-express the 3-point distribution as 2

  1 1 1  1 ½X x0 ðtÞ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2Cðb;bÞ rð3Þ X b ; b; X ; t; X 0 ; 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2Cðt;tÞ V 2pCðb; bÞ 2pCðt; tÞ

h

X 0 Gða;obÞ GðaÞ

X b

i2 .

(92)

From Eq. (57) we can see that expression of x0(t) is linear combination of X0 and Xb. Thus, the integration of the X will allow one to recover Z½X b ; b; X 0 ; 0=Z, while the integration of any of the two end points will not lead to Z½X b ; b; X ; t=Z or Z½X ; t; X 0 ; 0=Z. Acknowledgements S.C. Lim would like to thank the Malaysian Ministry of Science, Technology and Innovation for Grant IRPA 09-99-01-0095 EA093.

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