- Email: [email protected]
The mean energy for the anharmonic oscillator in the strong-coupling regime
ARTICLE IN PRESS
Physica A 368 (2006) 111–118 www.elsevier.com/locate/physa
The mean energy for the anharmonic oscillator in the strong-coupling regime N.F. Svaiter Centro Brasileiro de Pesquisas Fisicas-CBPF, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ 22290-180, Brazil Received 13 September 2005; received in revised form 13 October 2005 Available online 24 February 2006
Abstract We consider a single anharmonic oscillator with frequency o and coupling constant l, respectively, in the strongcoupling regime. We are assuming that the system is in thermal equilibrium with a reservoir at temperature b1 . Using the strong-coupling perturbative expansion, we obtain the mean energy for the oscillator in the regime lbo, up to order pffiffiffi 1= l. r 2006 Elsevier B.V. All rights reserved. Keywords: Anharmonic oscillator; Strong-coupling; Mean energy
1. Introduction The strong-coupling regime in quantum field theory is one of the unsolved problems of theoretical physics of the last century. There are many situations where one has to account for non-perturbative coupling regions, and also discuss the physics of strongly coupled systems. Actually, the use of the perturbative expansion outside the weak-coupling regime is fundamental for our understanding of the whole perturbative renormalization program in quantum field theory. In this paper we present a method for calculating the mean energy pfor ffiffiffi a single oscillator with the anharmonic contribution in the strong-coupling regime up to the order 1= l. The anharmonic oscillator is described by the Hamiltonian H¼
p2 1 l þ mo2 x2 þ x4 4! 2m 2
(1)
and for simplicity we are assuming that our system is one-dimensional and is in thermal equilibrium with a reservoir at temperature b1 . We are working in the imaginary time formalism and making use of the Kubo–Martin–Schwinger (KMS) condition [1,2]. To find the partition function and the mean energy, our approach consists in the combination of two techniques used currently in the literature: the strong-coupling expansion [3–6], and the zeta-function method [7,8]. E-mail address: [email protected]. 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.12.067
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
112
The anharmonic oscillator, with a ðl=4!Þx4 ðtÞ term is formally very similar to the field theory describing a scalar field with a quartic self-interaction. Therefore, let us briefly discuss the strong-coupling expansion in Euclidean field theory at zero temperature. The basic idea of the approach is the following: in a formal representation for the generating functional of complete Schwinger functions of the theory ZðhÞ, we treat the Gaussian part of the action as a perturbation with respect to the remaining terms of the functional integral, i.e., in the case for the ðlj4 Þd theory, the local self-interacting part, in the functional integral. The main difference between the strong-coupling expansion from the standard perturbative expansion is that, in the former, we have an expansion of the generating functional of complete Schwinger functions in inverse powers of the coupling constant. We are developing our perturbative expansion around the independent-value generating functional Q0 ðhÞ, where different points of the Euclidean space are decoupled, since the gradient terms are dropped [9–13]. The fundamental problem of the strong-coupling expansion is how to give meaning to the independentvalue generating functional and to the non-conventional representation for the Schwinger functional. A solution to this problem was presented a long time ago [9,14], where the independent-value generating functional describing Euclidean free and also self-interacting scalar fields were presented. We would like to stress that there are two different ways to analyze the independent-value generating functional. The first is a lattice regularization. Meanwhile a naive use of a continuum limit of the lattice regularization for the independent-value generating functional leads to a Gaussian theory. To obtain this result, note that in the absence of the kinetic term, a model described by the independent-value generating functional exhibits a statistically independent behavior. If we construct a one-dimensional lattice, there is no correlations between xi at one lattice site and xj at another different lattice site. But the central limit theorem states that the probability distribution of a sum of N independent random variables becomes Gaussian when N goes to infinity. Therefore in the continuum limit the independent-value generating functional becomes Gaussian. A way to avoid this problem is to use a non-translational functional measure which leads us to a useful representation for the independent-value generating functional. In this paper we show how it is possible to compute the partition p function and the mean energy of the ffiffiffi anharmonic oscillator in the strong-coupling regime, up to the order 1= l. The paper is organized as follows: in Section 2, the strong-coupling expansion for a single anharmonic oscillator is presented. In Section 3 we calculate the partition function and the mean energy of the system. Finally, Section 4 contains our conclusions. We are using the natural units, i.e., _ ¼ c ¼ kB ¼ 1. 2. The strong-coupling expansion in one-dimensional quantum mechanical system Let us consider a one-dimensional quantum mechanical system. The partition function for the system assuming that it is in thermal equilibrium with a reservoir at temperature b1 is given by " Z !# 2 Z b 1 dx ½dxðtÞ exp dt m þ V ðxðtÞÞ , (2) ZðbÞ ¼ 2 dt xð0Þ¼xðbÞ 0 where in the functional integral we require that xðtÞ is periodic with period b, i.e., xðtÞ ¼ xðt þ bÞ. There are many different physical situations that can be analyzed starting from the partition function. We would like to discuss the case of a single anharmonic oscillator, where, for simplicity, we are choosing m2 ¼ 1, and therefore the contribution of V ðxðtÞÞ is given by V 1 ðxÞ ¼
1 2 2 l 4 o x þ x . 2 4!
(3)
Let us suppose an anharmonic oscillator in the strong-coupling regime. To generate the correlation functions by functional differentiations, we couple linearly the anharmonic oscillator to a t-dependent external source. It is convenient to consider hðtÞ to be complex. Consequently, hðtÞ ¼ ReðhÞ þ i ImðhÞ. In this paper we are concerned with the case ReðhÞ ¼ 0. Therefore the generating functional at finite temperature
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
113
Zðb; hÞ is given by Z Zðb; hÞ ¼
Z ½dxðtÞ exp
Z
b
b
1 xðtÞKxðt0 Þ 2 0 0 xð0Þ¼xðbÞ Z b 1 l 2 2 4 þ dt so x ðtÞ x ðtÞ þ hðtÞxðtÞ , 2 4! 0 dt
dt0
where we have also integrated over all periodic paths and K Kðo; s; t t0 Þ is defined by d2 Kðo; s; t t0 Þ ¼ 2 þ ð1 sÞo2 dðt t0 Þ, dt
ð4Þ
(5)
where s is a complex parameter defined in the region 0pReðsÞo1. Note that we are considering a modification of the strong-coupling expansion. We split the quadratic part in the functional integral, which is proportional to the frequency squared, into two parts; one contributes together with the derivative term in the action as the perturbation, and the other appears in the independentvalue generating functional. In general, we must assume ReðsÞa1, because ReðsÞ ¼ 1 introduces infrared divergences in the calculations. The zero frequency case ðs ¼ 1Þ in the modified kernel can only be assumed in some very special situations, as for example to calculate the renormalized vacuum energy of a scalar field in the presence of boundaries, where Dirichlet boundary conditions are assumed [6]. As we will see, the choice of a suitable s will simplify our calculations in the situation which we are interested. This modification will be clarified in the next section. When s is a real parameter defined in the region ½0; 1Þ the kernel Kðo; s; t t0 Þ is associated with a positive definite elliptic operator. Functional differentiation gives the thermal average of a time-ordered of position operators, i.e., the correlation functions for a stochastic process. For sake of completeness, we would like to present the simple result for the partition function of the anharmonic oscillator in the regime l5o, in first order in l. One finds [15] 1 Z bo 3l b 1 2 bo 2 coth ZðbÞ ¼ 2 sinh 1 dt (6) þ Oðl Þ . 2 4! 0 4 2 To find the partition function for the anharmonic oscillator in the strong-coupling regime it is natural to use an alternative perturbation expansion, that is, the strong-coupling perturbative expansion. The idea is to treat the Gaussian part of the action as a perturbation with respect to the non-Gaussian terms in the functional integral. We get the following formal representation for the generating functional at finite temperature Zðb; hÞ: Z Z b 1 b d d Zðb; hÞ ¼ exp Kðo; s; t t0 Þ dt dt0 Qðb; s; hÞ, (7) 2 0 dhðtÞ dhðt0 Þ 0 where Qðb; s; hÞ, the new independent-value functional integral, is given by Z b Z 1 l 4 2 2 ½dxðtÞ exp dt so x ðtÞ x ðtÞ þ hðtÞxðtÞ , Qðb; s; hÞ ¼ N 2 4! 0 xð0Þ¼xðbÞ
(8)
and the modified kernel Kðo; s; t t0 Þ was defined by Eq. (5). The factor N is a normalization that can be found using that Qðb; s; hÞjh¼0 ¼ 1. The main difference from the standard representation for the generating functional is that we have an expansion of the generating functional in inverse powers of the coupling constant. We are developing our perturbative expansion around the independent-value generating functional Qðb; s; hÞ. One way to proceed is to neglect high-orders terms in the perturbative expansion. Therefore in the leading order, we have that Zðb; hÞ can be written as Z Z b 1 b d d Kðo; s; t t0 Þ Zðb; hÞ ¼ 1 dt dt0 Qðb; s; hÞ. (9) 2 0 dhðtÞ dhðt0 Þ 0
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
114
Since we are mainly interested in presenting the partition function, we can also assume that the external source is constant i.e., hðtÞ ¼ h. As we will see, this assumption will lead us to redefine the original representation for the independent-value generating functional obtained in Ref. [9]. Of course, the above assumption greatly simplifies our problem, but we still have some work to do. To evaluate ln Zðb; hÞ, note that we have two steps to follow. The first one is to give meaning to the independent-value generating functional, and the second one is to regularize and renormalize the kernel Kðo; s; t t0 Þ integrated over the volume ½0; b. We would like to stress that the parameter s was introduced only to simplify our calculations in some situations. Therefore s can be complex if we are able to work in all orders of perturbation theory. The generating functional does not depends on the value for s. Since we concentrate in the leading order, some care has to be taken to prevent a complex generating functional. A simple way to avoid the problem is assume that the parameter s is real. Therefore we will impose that ImðsÞ ¼ 0. In this situation the independent-value functional Qðb; s; hÞ should be a normalized, positive definite functional. In the next section, we will use a combination of the original representation for the independent-value generating functional, and also use the generalized zeta-function method to regularize and renormalize the kernel Kðo; s; t t0 Þ integrated over the Euclidean time. 3. The mean energy for the anharmonic oscillator in the strong-coupling regime In the present section we study the single anharmonic oscillator in the strong-coupling regime. Let us proceed in deriving the mean energy. The mean energy can be obtained from ln Zðb; hÞjh¼0 , i.e., EðlÞ ¼ ðq=qbÞ ln Zðb; hÞjh¼0 . To have a well-defined meaning to the mean energy that can be obtained from Eq. (9) we may proceed as follows. First, we use the formal definition of the independent-value generating functional Qðb; s; hÞ. Second, we have to regularize and renormalize the kernel Kðo; s; t t0 Þ integrated over the Euclidean time. The success of our method, depends critically on the possibility to handle the independent-value generating functional and the kernel integrated on the Euclidean time ½0; b. Since we are concerned with the strong-coupling regime, to evaluate ln ZðbÞ let us use the leading term. Using the cumulant expansion idea, which relates the mean of a exponential to the exponential of means after some simple calculations we obtain 1 q2 Qðb; s; hÞ a 1 d zðsÞ ln Zðb; hÞ ¼ þ , (10) 2 Qðb; s; hÞ 2 2 ds qh s¼0 where zðsÞ is the global generalized zeta-function associated with the operator ððd2 =dt2 Þ þ ð1 sÞo2 Þ and a is a infinite constant. There are some issues that we would like to discuss. Note also that, we assume thermal equilibrium and since we are working in the Euclidean formalism, the spectrum of the operator D ððd2 =dt2 Þ þ ð1 sÞo2 Þ has a denumerable contribution. Remind that s is a complex parameter defined in the region 0pReðsÞo1. At this point, let us impose that ImðsÞ ¼ 0. With this choice, the operator D is a positive definite elliptic operator acting on xn ðtÞ defined in a compact manifold. This operator has a complete set of orthonormal eigenfunctions xn ðtÞ and associated eigenvalues an . We have d2 (11) 2 þ ð1 sÞo2 xn ðtÞ ¼ an xn ðtÞ dt Rb with the boundary conditions xn ð0Þ ¼ xn ðbÞ. Note that we have 0 dtxn ðtÞxn0 ðtÞ ¼ dnn0 . The generalized zetafunction associated with the operator ððd2 =dt2 Þ þ ð1 sÞo2 Þ, is defined by zðd2 =dt2 Þþð1sÞo2 ðsÞ ¼
1 X
as n ,
(12)
n¼1
where the spectrum is given by " # 2pn 2 2 an ¼ þ ð1 sÞo ; b
nZ.
(13)
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
115
Using the definition for the global generalized zeta-function and the spectrum of the operator given by Eq. (13) we have the generalized zeta-function given by " #s 1 X 2pn 2 2 zðd2 =dt2 Þþð1sÞo2 ðsÞ ¼ þ ð1 sÞo . (14) b n¼1 Here, it is useful to define the modified Epstein zeta-function in the complex plane s, i.e., the function zðs; nÞ by zðs; nÞ ¼
1 X
ðn2 þ n2 Þs ;
n2 40.
(15)
n¼1
The series defined by Eq. (15) converges absolutely and defines in the complex s plane an analytic function for ReðsÞ412. It is possible to analytically extend the modified Epstein zeta-function where the integral representation is valid for ReðsÞo1 [16–18]. It is not difficult to write the generalized zeta-function in terms of the modified Epstein zeta-function. We have 2s pffiffiffiffiffiffiffiffiffiffiffi ob b zðd2 =dt2 Þþð1sÞo2 ðsÞ ¼ z s; 1 s , (16) 2p 2p where zðs; nÞ, is the modified Epstein zeta-function. As we discussed, the series representation for zðs; nÞ converges for ReðzÞ412 and its analytic continuation defines a meromorphic function of s which is analytic at s ¼ 0. The modified Epstein zeta-function has poles at s ¼ 12; 12, etc. Using the analytic extension of the modified Epstein zeta-function it is not difficult to show that the values for the modified Epstein zeta-function zðs; nÞ, at s ¼ 0 and ðq=qsÞzðs; nÞjs¼0 are given by zðs; nÞjs¼0 ¼ 0
(17)
and also q zðs; nÞ ¼ 2 lnð2 sinh pnÞ. qs s¼0
(18)
Since we are interested in calculating the derivative of the generalized zeta-function at the origin of the complex s plane, we have ! 1 q 1 d b 2s 1 b 2s q zðd2 =dt2 Þþð1sÞo2 ðsÞ ¼ zðs; nÞ zðs; nÞ . þ (19) 2 qs 2 ds 2p 2 2p qs s¼0 s¼0
Choosing s ¼ 0, and using Eqs. (17) and (18) in Eq. (19) we obtain the well-known result in the literature. For the general case ðsa0Þ we have 1 q ob z 2 2 ¼ ln 2 sinh ð1 sÞ . (20) 2 ðsÞ 2 qs ðd =dt Þþð1sÞo s¼0 2 The need to introduce a mass scale m is irrelevant for our problem. To show this it is sufficient to note that to take into account the scaling properties we have to make the change q q 1 zðs; nÞ ! zðs; nÞ ln (21) zðs; nÞ qs qs 4pm s¼0 s¼0 s¼0 and since zðs; nÞjs¼0 ¼ 0, the scaling behavior does not appear in our problem. We would like to point out that in the original derivation for the independent-value model for scalar fields in a d-dimensional Euclidean space, a result was obtained which is well defined for all functions which are square integrable in Rn i.e., hðxÞ 2 L2 ðRn Þ. This observation allow us to conclude that we need also to use a normalization in the situation that we are investigating. It is possible to show that the independent-value
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
116
generating function can be written as Z b Z 1 1 du 1 l ð1 cosðhuÞÞ exp so2 u2 u4 . Qðb; s; hÞ ¼ exp dt 2b 0 2 4! 1 juj
(22)
There is no need to go into details of this derivation. The reader can find it in Refs. [9,13,14]. It is important to stress that the independent-value generating function defined by Eq. (22) does not reduce to the conventional free independent-value model, if we choose l ¼ 0. The non-Gaussian contribution represents a discontinuous perturbation of the free theory, that is, a pseudo-free theory. Actually, this is the key point of a program to investigate non-renormalizable models in field theory. In order to study Qðb; s; hÞ, let us define Rðo; s; l; hÞ given by Z 1 du 1 l ð1 cosðhuÞÞ exp so2 u2 u4 . Rðo; s; l; hÞ ¼ (23) 2 4! 1 juj Using a series representation for cos x, it is not difficult to show that Z 1 X ð1Þk 2k 1 1 l 4 2k1 2 2 h du u exp so u u . Rðo; s; l; hÞ ¼ 2 2 4! ð2kÞ! 0 k¼1
(24)
Now let use the fact that the s parameter can be chosen in such a way that the calculations becomes tractable. Analyzing only the independent-value generating functional it is not possible to write Qðb; s; hÞ in a closed form even in the case of constant external source. One way to obtain a closed expression is to choose s ¼ 0. Therefore we have Rðo; s; l; hÞjs¼0
Z 1 X ð1Þk 2k 1 l 4 2k1 h ¼2 du u exp u . 4! ð2kÞ! 0 k¼1
At this point let us use the following integral representation for the Gamma function [19]: Z 1 1 n=p n n1 p dx x expðmx Þ ¼ m G ; ReðmÞ40; ReðnÞ40; p40. p p 0
(25)
(26)
It is clear that the ðlxp Þ theory, for even p44, can also easily handle applying our method. Using the result given by Eq. (26) in Eq. (25) we have Rðo; s; l; hÞjs¼0 ¼
1 X k¼1
gðkÞ
h2k lk=2
,
(27)
where the coefficients gðkÞ are given by gðkÞ ¼
1 ð1Þk k ð4!Þk=2 G . 2 ð2kÞ! 2
(28)
Substituting Eqs. (27) and (28) in Eq. (22), we obtain that the independent-value generating function Qðb; s; hÞjs¼0 can be written as " # Z b 1 X 1 h2k Qðb; s; hÞjs¼0 ¼ exp dt gðkÞ k=2 . (29) 2b 0 l k¼1 It is easy to calculate the second derivative for the independent-value generating function with respect to h. Note that Qðb; s; hÞjh¼s¼0 ¼ 1. Thus we have ! ! 1 1 q2 Qðb; s; hÞ 1X h2k2 1X h2k js¼0 ¼ gðkÞð2kÞð2k 1Þ k=2 exp gðkÞ k=2 þ GðhÞ, (30) 2 k¼1 2 k¼1 qh2 l l
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
117
where GðhÞ is given by GðhÞ ¼
1 X k;q¼1
gðk; qÞ
h2kþ2q2 lðkþqÞ=2
!
1 1X h2k gðkÞ k=2 exp 2 k¼1 l
! (31)
and gðk; qÞ ¼ kqgðkÞgðqÞ. We are interested in the case h ¼ 0, therefore the double series does not contribute to Eq. (30), since limh!0 GðhÞ ¼ 0. Using the fact that we are interested in the case h ¼ 0, we have the simple result that in Eq. (30) only term k ¼ 1 contributes. We get rffiffiffiffiffiffi q2 Qðb; s; hÞ 3p ¼ . 2 8l qh h¼s¼0
(32)
Substituting the result obtained from the generalized zeta-function method given by Eqs. (20) (choosing s ¼ 0) and (32) in Eq. (10) we have that ln ZðbÞ is given by rffiffiffiffiffiffi 3p a ob þ ln 2 sinh ln ZðbÞ ¼ . 8l 2 2
(33)
Note that in the free energy we will have a infinite contribution, but since we are interested in the mean energy, this contribution can be discarded. Using the definition for the mean energy, we get rffiffiffiffiffiffih 3p o o i þ ob EðlÞ ¼ . 8l 2 e 1
(34)
At this point, some comments are in order. Although we are obtaining the mean energy EðlÞ in the regime lbo, our result is valid in the regime of large fluctuations. In this case there is no constrain over l and ever for small coupling constant our results can be used. In this situation we see that the mean energy of the system in the complex coupling constant plane present a branch point at l ¼ 0. This situation also appears in the zerodimensional lj4 theory [20]. 4. Conclusions In this article we studied the strong-coupling regime in one-dimensional models, after analytic continuation to imaginary time. One-dimensional models are very simple system for which we can apply our method in obtaining thermodynamics quantities in the leading order in the inverse of coupling constant. We calculate the partition function and the mean energy for the anharmonic oscillator, using the strong-coupling perturbative expansion and the generalized zeta-function analytic regularization. The picture that emerges from our method is the following: in the strong-coupling perturbative expansion we may split the problem of defining the generating functional into two parts. The first is how to define precisely the independent-value generating functional. The second part is to go beyond the independent-value approximation and take into account the contributions from the perturbative term. Concerning the first problem, we would like to stress that a naive use of a continuum limit of the lattice regularization for the independent-value generating functional leads to a Gaussian theory, where we simple make use of the central limit theorem. The fundamental modification which allow us to avoid the central limit theorem is a change in the measure in the functional integral. The second problem can be controlled using an analytic regularization. Since we assume that the external source is constant i.e., hðtÞ ¼ h, the strong-coupling perturbative expansion, in combination with an analytic regularization procedure, is a useful method to compute global quantities, as the mean energy. To obtain local quantities, as the Green’s functions of the model, we have to use Eq. (9) without assuming that the external source is constant. It is well known that if someone try to find the energy levels of the anharmonic oscillator using a weakcoupling perturbative expansion, the perturbative series does not converge no matter how small the coupling constant is. Therefore the ground state energy E 0 ðlÞ of the anharmonic oscillator is analytic in a cut-plane and
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
118
using perturbation theory we can shown that [21,22] E 0 ðlÞ ¼
1 X
lk f k .
(35)
k¼0
The small l behavior of ImðE 0 ðlÞÞ is related to the behavior of the coefficient f k of the above equation, when the order k becomes large. For large l it is possible to show that [15] E 0 ðlÞ ¼ l1=3
1 X
an l2n=3 .
(36)
n¼0
If we compare the result of the ground state energy E 0 ðlÞ of the anharmonic oscillator given by Eq. (36) with Eq. (34) we note a discrepancy between our results. The result given by Eq. (35) was obtained in a smallcoupling expansion framework and the second one using, not only the strong-coupling perturbative expansion but also a non-translational functional measure. These facts can explain the discrepancy between our results and the results presented in Ref. [15]. There are several directions for investigations. It should be possible to apply the method to more realistic theories. To mention a few: since scalar fields play a fundamental role in the standard model, the study of the strongly coupled ðlj4 Þd theory at finite temperature and also the renormalized vacuum energy of scalar quantum fields in the presence of macroscopic structures deserves future investigations. Acknowledgments I would like to thank, G. F. Hidalgo and S. Joffily for enlightening discussions, and also L. A. Oliveira for comments on the manuscript. This paper was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnolo´gico do Brazil (CNPq). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570. P.C. Martin, J. Schwinger, Phys. Rev. 115 (1959) 1342. S. Kovesi-Domokos, Il Nuovo Cimento 33A (1976) 769. C.M. Bender, F. Cooper, G.S. Guralnik, D.H. Sharp, Phys. Rev. D 19 (1979) 1865. N.F. Svaiter, The strong-coupling expansion and the singularities of the perturbative expansion, Proceedings of the ‘‘X Brazilian School of Cosmology and Gravitation’’, Rio de Janeiro, Brazil, AIP, 2003. N.F. Svaiter, Physica A 345 (2004) 517. R.T. Seeley, Am. Math. Proc. Symp. Pure Math. 10 (1967) 288. S.W. Hawking, Commun. Math. Phys. 55 (1977) 133. J.R. Klauder, Acta Phys. Austriaca 41 (1975) 237. J.R. Klauder, Phys. Rev. D 14 (1976) 1952. R. Menikoff, D.H. Sharp, J. Math. Phys. 19 (1978) 135. R.J. Rivers, Path Integral Methods in Quantum Field Theory, Cambridge University Press, Cambridge, 1987. J.R. Klauder, Beyond Conventional Quantization, Cambridge University Press, Cambridge, 2000. J.R. Klauder, Ann. Phys. 117 (1979) 19. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, NY, 1996, pp. 930–931. L.H. Ford, Phys. Rev. 21 (1980) 933. L.H. Ford, N.F. Svaiter, Phys. Rev. 51 (1995) 6981. E. Elizalde, A. Romeo, J. Math. Phys. 30 (1989) 1133. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980. A.P.C. Malbouisson, R. Portugal, N.F. Svaiter, Physica A 292 (2001) 485. G. Parisi, Phys. Lett. 66B (1977) 167. E. Brezin, G. Parisi, J. Stat. Phys. 19 (1978) 269.
Physica A 368 (2006) 111–118 www.elsevier.com/locate/physa
The mean energy for the anharmonic oscillator in the strong-coupling regime N.F. Svaiter Centro Brasileiro de Pesquisas Fisicas-CBPF, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ 22290-180, Brazil Received 13 September 2005; received in revised form 13 October 2005 Available online 24 February 2006
Abstract We consider a single anharmonic oscillator with frequency o and coupling constant l, respectively, in the strongcoupling regime. We are assuming that the system is in thermal equilibrium with a reservoir at temperature b1 . Using the strong-coupling perturbative expansion, we obtain the mean energy for the oscillator in the regime lbo, up to order pffiffiffi 1= l. r 2006 Elsevier B.V. All rights reserved. Keywords: Anharmonic oscillator; Strong-coupling; Mean energy
1. Introduction The strong-coupling regime in quantum field theory is one of the unsolved problems of theoretical physics of the last century. There are many situations where one has to account for non-perturbative coupling regions, and also discuss the physics of strongly coupled systems. Actually, the use of the perturbative expansion outside the weak-coupling regime is fundamental for our understanding of the whole perturbative renormalization program in quantum field theory. In this paper we present a method for calculating the mean energy pfor ffiffiffi a single oscillator with the anharmonic contribution in the strong-coupling regime up to the order 1= l. The anharmonic oscillator is described by the Hamiltonian H¼
p2 1 l þ mo2 x2 þ x4 4! 2m 2
(1)
and for simplicity we are assuming that our system is one-dimensional and is in thermal equilibrium with a reservoir at temperature b1 . We are working in the imaginary time formalism and making use of the Kubo–Martin–Schwinger (KMS) condition [1,2]. To find the partition function and the mean energy, our approach consists in the combination of two techniques used currently in the literature: the strong-coupling expansion [3–6], and the zeta-function method [7,8]. E-mail address: [email protected]. 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.12.067
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
112
The anharmonic oscillator, with a ðl=4!Þx4 ðtÞ term is formally very similar to the field theory describing a scalar field with a quartic self-interaction. Therefore, let us briefly discuss the strong-coupling expansion in Euclidean field theory at zero temperature. The basic idea of the approach is the following: in a formal representation for the generating functional of complete Schwinger functions of the theory ZðhÞ, we treat the Gaussian part of the action as a perturbation with respect to the remaining terms of the functional integral, i.e., in the case for the ðlj4 Þd theory, the local self-interacting part, in the functional integral. The main difference between the strong-coupling expansion from the standard perturbative expansion is that, in the former, we have an expansion of the generating functional of complete Schwinger functions in inverse powers of the coupling constant. We are developing our perturbative expansion around the independent-value generating functional Q0 ðhÞ, where different points of the Euclidean space are decoupled, since the gradient terms are dropped [9–13]. The fundamental problem of the strong-coupling expansion is how to give meaning to the independentvalue generating functional and to the non-conventional representation for the Schwinger functional. A solution to this problem was presented a long time ago [9,14], where the independent-value generating functional describing Euclidean free and also self-interacting scalar fields were presented. We would like to stress that there are two different ways to analyze the independent-value generating functional. The first is a lattice regularization. Meanwhile a naive use of a continuum limit of the lattice regularization for the independent-value generating functional leads to a Gaussian theory. To obtain this result, note that in the absence of the kinetic term, a model described by the independent-value generating functional exhibits a statistically independent behavior. If we construct a one-dimensional lattice, there is no correlations between xi at one lattice site and xj at another different lattice site. But the central limit theorem states that the probability distribution of a sum of N independent random variables becomes Gaussian when N goes to infinity. Therefore in the continuum limit the independent-value generating functional becomes Gaussian. A way to avoid this problem is to use a non-translational functional measure which leads us to a useful representation for the independent-value generating functional. In this paper we show how it is possible to compute the partition p function and the mean energy of the ffiffiffi anharmonic oscillator in the strong-coupling regime, up to the order 1= l. The paper is organized as follows: in Section 2, the strong-coupling expansion for a single anharmonic oscillator is presented. In Section 3 we calculate the partition function and the mean energy of the system. Finally, Section 4 contains our conclusions. We are using the natural units, i.e., _ ¼ c ¼ kB ¼ 1. 2. The strong-coupling expansion in one-dimensional quantum mechanical system Let us consider a one-dimensional quantum mechanical system. The partition function for the system assuming that it is in thermal equilibrium with a reservoir at temperature b1 is given by " Z !# 2 Z b 1 dx ½dxðtÞ exp dt m þ V ðxðtÞÞ , (2) ZðbÞ ¼ 2 dt xð0Þ¼xðbÞ 0 where in the functional integral we require that xðtÞ is periodic with period b, i.e., xðtÞ ¼ xðt þ bÞ. There are many different physical situations that can be analyzed starting from the partition function. We would like to discuss the case of a single anharmonic oscillator, where, for simplicity, we are choosing m2 ¼ 1, and therefore the contribution of V ðxðtÞÞ is given by V 1 ðxÞ ¼
1 2 2 l 4 o x þ x . 2 4!
(3)
Let us suppose an anharmonic oscillator in the strong-coupling regime. To generate the correlation functions by functional differentiations, we couple linearly the anharmonic oscillator to a t-dependent external source. It is convenient to consider hðtÞ to be complex. Consequently, hðtÞ ¼ ReðhÞ þ i ImðhÞ. In this paper we are concerned with the case ReðhÞ ¼ 0. Therefore the generating functional at finite temperature
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
113
Zðb; hÞ is given by Z Zðb; hÞ ¼
Z ½dxðtÞ exp
Z
b
b
1 xðtÞKxðt0 Þ 2 0 0 xð0Þ¼xðbÞ Z b 1 l 2 2 4 þ dt so x ðtÞ x ðtÞ þ hðtÞxðtÞ , 2 4! 0 dt
dt0
where we have also integrated over all periodic paths and K Kðo; s; t t0 Þ is defined by d2 Kðo; s; t t0 Þ ¼ 2 þ ð1 sÞo2 dðt t0 Þ, dt
ð4Þ
(5)
where s is a complex parameter defined in the region 0pReðsÞo1. Note that we are considering a modification of the strong-coupling expansion. We split the quadratic part in the functional integral, which is proportional to the frequency squared, into two parts; one contributes together with the derivative term in the action as the perturbation, and the other appears in the independentvalue generating functional. In general, we must assume ReðsÞa1, because ReðsÞ ¼ 1 introduces infrared divergences in the calculations. The zero frequency case ðs ¼ 1Þ in the modified kernel can only be assumed in some very special situations, as for example to calculate the renormalized vacuum energy of a scalar field in the presence of boundaries, where Dirichlet boundary conditions are assumed [6]. As we will see, the choice of a suitable s will simplify our calculations in the situation which we are interested. This modification will be clarified in the next section. When s is a real parameter defined in the region ½0; 1Þ the kernel Kðo; s; t t0 Þ is associated with a positive definite elliptic operator. Functional differentiation gives the thermal average of a time-ordered of position operators, i.e., the correlation functions for a stochastic process. For sake of completeness, we would like to present the simple result for the partition function of the anharmonic oscillator in the regime l5o, in first order in l. One finds [15] 1 Z bo 3l b 1 2 bo 2 coth ZðbÞ ¼ 2 sinh 1 dt (6) þ Oðl Þ . 2 4! 0 4 2 To find the partition function for the anharmonic oscillator in the strong-coupling regime it is natural to use an alternative perturbation expansion, that is, the strong-coupling perturbative expansion. The idea is to treat the Gaussian part of the action as a perturbation with respect to the non-Gaussian terms in the functional integral. We get the following formal representation for the generating functional at finite temperature Zðb; hÞ: Z Z b 1 b d d Zðb; hÞ ¼ exp Kðo; s; t t0 Þ dt dt0 Qðb; s; hÞ, (7) 2 0 dhðtÞ dhðt0 Þ 0 where Qðb; s; hÞ, the new independent-value functional integral, is given by Z b Z 1 l 4 2 2 ½dxðtÞ exp dt so x ðtÞ x ðtÞ þ hðtÞxðtÞ , Qðb; s; hÞ ¼ N 2 4! 0 xð0Þ¼xðbÞ
(8)
and the modified kernel Kðo; s; t t0 Þ was defined by Eq. (5). The factor N is a normalization that can be found using that Qðb; s; hÞjh¼0 ¼ 1. The main difference from the standard representation for the generating functional is that we have an expansion of the generating functional in inverse powers of the coupling constant. We are developing our perturbative expansion around the independent-value generating functional Qðb; s; hÞ. One way to proceed is to neglect high-orders terms in the perturbative expansion. Therefore in the leading order, we have that Zðb; hÞ can be written as Z Z b 1 b d d Kðo; s; t t0 Þ Zðb; hÞ ¼ 1 dt dt0 Qðb; s; hÞ. (9) 2 0 dhðtÞ dhðt0 Þ 0
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
114
Since we are mainly interested in presenting the partition function, we can also assume that the external source is constant i.e., hðtÞ ¼ h. As we will see, this assumption will lead us to redefine the original representation for the independent-value generating functional obtained in Ref. [9]. Of course, the above assumption greatly simplifies our problem, but we still have some work to do. To evaluate ln Zðb; hÞ, note that we have two steps to follow. The first one is to give meaning to the independent-value generating functional, and the second one is to regularize and renormalize the kernel Kðo; s; t t0 Þ integrated over the volume ½0; b. We would like to stress that the parameter s was introduced only to simplify our calculations in some situations. Therefore s can be complex if we are able to work in all orders of perturbation theory. The generating functional does not depends on the value for s. Since we concentrate in the leading order, some care has to be taken to prevent a complex generating functional. A simple way to avoid the problem is assume that the parameter s is real. Therefore we will impose that ImðsÞ ¼ 0. In this situation the independent-value functional Qðb; s; hÞ should be a normalized, positive definite functional. In the next section, we will use a combination of the original representation for the independent-value generating functional, and also use the generalized zeta-function method to regularize and renormalize the kernel Kðo; s; t t0 Þ integrated over the Euclidean time. 3. The mean energy for the anharmonic oscillator in the strong-coupling regime In the present section we study the single anharmonic oscillator in the strong-coupling regime. Let us proceed in deriving the mean energy. The mean energy can be obtained from ln Zðb; hÞjh¼0 , i.e., EðlÞ ¼ ðq=qbÞ ln Zðb; hÞjh¼0 . To have a well-defined meaning to the mean energy that can be obtained from Eq. (9) we may proceed as follows. First, we use the formal definition of the independent-value generating functional Qðb; s; hÞ. Second, we have to regularize and renormalize the kernel Kðo; s; t t0 Þ integrated over the Euclidean time. The success of our method, depends critically on the possibility to handle the independent-value generating functional and the kernel integrated on the Euclidean time ½0; b. Since we are concerned with the strong-coupling regime, to evaluate ln ZðbÞ let us use the leading term. Using the cumulant expansion idea, which relates the mean of a exponential to the exponential of means after some simple calculations we obtain 1 q2 Qðb; s; hÞ a 1 d zðsÞ ln Zðb; hÞ ¼ þ , (10) 2 Qðb; s; hÞ 2 2 ds qh s¼0 where zðsÞ is the global generalized zeta-function associated with the operator ððd2 =dt2 Þ þ ð1 sÞo2 Þ and a is a infinite constant. There are some issues that we would like to discuss. Note also that, we assume thermal equilibrium and since we are working in the Euclidean formalism, the spectrum of the operator D ððd2 =dt2 Þ þ ð1 sÞo2 Þ has a denumerable contribution. Remind that s is a complex parameter defined in the region 0pReðsÞo1. At this point, let us impose that ImðsÞ ¼ 0. With this choice, the operator D is a positive definite elliptic operator acting on xn ðtÞ defined in a compact manifold. This operator has a complete set of orthonormal eigenfunctions xn ðtÞ and associated eigenvalues an . We have d2 (11) 2 þ ð1 sÞo2 xn ðtÞ ¼ an xn ðtÞ dt Rb with the boundary conditions xn ð0Þ ¼ xn ðbÞ. Note that we have 0 dtxn ðtÞxn0 ðtÞ ¼ dnn0 . The generalized zetafunction associated with the operator ððd2 =dt2 Þ þ ð1 sÞo2 Þ, is defined by zðd2 =dt2 Þþð1sÞo2 ðsÞ ¼
1 X
as n ,
(12)
n¼1
where the spectrum is given by " # 2pn 2 2 an ¼ þ ð1 sÞo ; b
nZ.
(13)
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
115
Using the definition for the global generalized zeta-function and the spectrum of the operator given by Eq. (13) we have the generalized zeta-function given by " #s 1 X 2pn 2 2 zðd2 =dt2 Þþð1sÞo2 ðsÞ ¼ þ ð1 sÞo . (14) b n¼1 Here, it is useful to define the modified Epstein zeta-function in the complex plane s, i.e., the function zðs; nÞ by zðs; nÞ ¼
1 X
ðn2 þ n2 Þs ;
n2 40.
(15)
n¼1
The series defined by Eq. (15) converges absolutely and defines in the complex s plane an analytic function for ReðsÞ412. It is possible to analytically extend the modified Epstein zeta-function where the integral representation is valid for ReðsÞo1 [16–18]. It is not difficult to write the generalized zeta-function in terms of the modified Epstein zeta-function. We have 2s pffiffiffiffiffiffiffiffiffiffiffi ob b zðd2 =dt2 Þþð1sÞo2 ðsÞ ¼ z s; 1 s , (16) 2p 2p where zðs; nÞ, is the modified Epstein zeta-function. As we discussed, the series representation for zðs; nÞ converges for ReðzÞ412 and its analytic continuation defines a meromorphic function of s which is analytic at s ¼ 0. The modified Epstein zeta-function has poles at s ¼ 12; 12, etc. Using the analytic extension of the modified Epstein zeta-function it is not difficult to show that the values for the modified Epstein zeta-function zðs; nÞ, at s ¼ 0 and ðq=qsÞzðs; nÞjs¼0 are given by zðs; nÞjs¼0 ¼ 0
(17)
and also q zðs; nÞ ¼ 2 lnð2 sinh pnÞ. qs s¼0
(18)
Since we are interested in calculating the derivative of the generalized zeta-function at the origin of the complex s plane, we have ! 1 q 1 d b 2s 1 b 2s q zðd2 =dt2 Þþð1sÞo2 ðsÞ ¼ zðs; nÞ zðs; nÞ . þ (19) 2 qs 2 ds 2p 2 2p qs s¼0 s¼0
Choosing s ¼ 0, and using Eqs. (17) and (18) in Eq. (19) we obtain the well-known result in the literature. For the general case ðsa0Þ we have 1 q ob z 2 2 ¼ ln 2 sinh ð1 sÞ . (20) 2 ðsÞ 2 qs ðd =dt Þþð1sÞo s¼0 2 The need to introduce a mass scale m is irrelevant for our problem. To show this it is sufficient to note that to take into account the scaling properties we have to make the change q q 1 zðs; nÞ ! zðs; nÞ ln (21) zðs; nÞ qs qs 4pm s¼0 s¼0 s¼0 and since zðs; nÞjs¼0 ¼ 0, the scaling behavior does not appear in our problem. We would like to point out that in the original derivation for the independent-value model for scalar fields in a d-dimensional Euclidean space, a result was obtained which is well defined for all functions which are square integrable in Rn i.e., hðxÞ 2 L2 ðRn Þ. This observation allow us to conclude that we need also to use a normalization in the situation that we are investigating. It is possible to show that the independent-value
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
116
generating function can be written as Z b Z 1 1 du 1 l ð1 cosðhuÞÞ exp so2 u2 u4 . Qðb; s; hÞ ¼ exp dt 2b 0 2 4! 1 juj
(22)
There is no need to go into details of this derivation. The reader can find it in Refs. [9,13,14]. It is important to stress that the independent-value generating function defined by Eq. (22) does not reduce to the conventional free independent-value model, if we choose l ¼ 0. The non-Gaussian contribution represents a discontinuous perturbation of the free theory, that is, a pseudo-free theory. Actually, this is the key point of a program to investigate non-renormalizable models in field theory. In order to study Qðb; s; hÞ, let us define Rðo; s; l; hÞ given by Z 1 du 1 l ð1 cosðhuÞÞ exp so2 u2 u4 . Rðo; s; l; hÞ ¼ (23) 2 4! 1 juj Using a series representation for cos x, it is not difficult to show that Z 1 X ð1Þk 2k 1 1 l 4 2k1 2 2 h du u exp so u u . Rðo; s; l; hÞ ¼ 2 2 4! ð2kÞ! 0 k¼1
(24)
Now let use the fact that the s parameter can be chosen in such a way that the calculations becomes tractable. Analyzing only the independent-value generating functional it is not possible to write Qðb; s; hÞ in a closed form even in the case of constant external source. One way to obtain a closed expression is to choose s ¼ 0. Therefore we have Rðo; s; l; hÞjs¼0
Z 1 X ð1Þk 2k 1 l 4 2k1 h ¼2 du u exp u . 4! ð2kÞ! 0 k¼1
At this point let us use the following integral representation for the Gamma function [19]: Z 1 1 n=p n n1 p dx x expðmx Þ ¼ m G ; ReðmÞ40; ReðnÞ40; p40. p p 0
(25)
(26)
It is clear that the ðlxp Þ theory, for even p44, can also easily handle applying our method. Using the result given by Eq. (26) in Eq. (25) we have Rðo; s; l; hÞjs¼0 ¼
1 X k¼1
gðkÞ
h2k lk=2
,
(27)
where the coefficients gðkÞ are given by gðkÞ ¼
1 ð1Þk k ð4!Þk=2 G . 2 ð2kÞ! 2
(28)
Substituting Eqs. (27) and (28) in Eq. (22), we obtain that the independent-value generating function Qðb; s; hÞjs¼0 can be written as " # Z b 1 X 1 h2k Qðb; s; hÞjs¼0 ¼ exp dt gðkÞ k=2 . (29) 2b 0 l k¼1 It is easy to calculate the second derivative for the independent-value generating function with respect to h. Note that Qðb; s; hÞjh¼s¼0 ¼ 1. Thus we have ! ! 1 1 q2 Qðb; s; hÞ 1X h2k2 1X h2k js¼0 ¼ gðkÞð2kÞð2k 1Þ k=2 exp gðkÞ k=2 þ GðhÞ, (30) 2 k¼1 2 k¼1 qh2 l l
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
117
where GðhÞ is given by GðhÞ ¼
1 X k;q¼1
gðk; qÞ
h2kþ2q2 lðkþqÞ=2
!
1 1X h2k gðkÞ k=2 exp 2 k¼1 l
! (31)
and gðk; qÞ ¼ kqgðkÞgðqÞ. We are interested in the case h ¼ 0, therefore the double series does not contribute to Eq. (30), since limh!0 GðhÞ ¼ 0. Using the fact that we are interested in the case h ¼ 0, we have the simple result that in Eq. (30) only term k ¼ 1 contributes. We get rffiffiffiffiffiffi q2 Qðb; s; hÞ 3p ¼ . 2 8l qh h¼s¼0
(32)
Substituting the result obtained from the generalized zeta-function method given by Eqs. (20) (choosing s ¼ 0) and (32) in Eq. (10) we have that ln ZðbÞ is given by rffiffiffiffiffiffi 3p a ob þ ln 2 sinh ln ZðbÞ ¼ . 8l 2 2
(33)
Note that in the free energy we will have a infinite contribution, but since we are interested in the mean energy, this contribution can be discarded. Using the definition for the mean energy, we get rffiffiffiffiffiffih 3p o o i þ ob EðlÞ ¼ . 8l 2 e 1
(34)
At this point, some comments are in order. Although we are obtaining the mean energy EðlÞ in the regime lbo, our result is valid in the regime of large fluctuations. In this case there is no constrain over l and ever for small coupling constant our results can be used. In this situation we see that the mean energy of the system in the complex coupling constant plane present a branch point at l ¼ 0. This situation also appears in the zerodimensional lj4 theory [20]. 4. Conclusions In this article we studied the strong-coupling regime in one-dimensional models, after analytic continuation to imaginary time. One-dimensional models are very simple system for which we can apply our method in obtaining thermodynamics quantities in the leading order in the inverse of coupling constant. We calculate the partition function and the mean energy for the anharmonic oscillator, using the strong-coupling perturbative expansion and the generalized zeta-function analytic regularization. The picture that emerges from our method is the following: in the strong-coupling perturbative expansion we may split the problem of defining the generating functional into two parts. The first is how to define precisely the independent-value generating functional. The second part is to go beyond the independent-value approximation and take into account the contributions from the perturbative term. Concerning the first problem, we would like to stress that a naive use of a continuum limit of the lattice regularization for the independent-value generating functional leads to a Gaussian theory, where we simple make use of the central limit theorem. The fundamental modification which allow us to avoid the central limit theorem is a change in the measure in the functional integral. The second problem can be controlled using an analytic regularization. Since we assume that the external source is constant i.e., hðtÞ ¼ h, the strong-coupling perturbative expansion, in combination with an analytic regularization procedure, is a useful method to compute global quantities, as the mean energy. To obtain local quantities, as the Green’s functions of the model, we have to use Eq. (9) without assuming that the external source is constant. It is well known that if someone try to find the energy levels of the anharmonic oscillator using a weakcoupling perturbative expansion, the perturbative series does not converge no matter how small the coupling constant is. Therefore the ground state energy E 0 ðlÞ of the anharmonic oscillator is analytic in a cut-plane and
ARTICLE IN PRESS N.F. Svaiter / Physica A 368 (2006) 111–118
118
using perturbation theory we can shown that [21,22] E 0 ðlÞ ¼
1 X
lk f k .
(35)
k¼0
The small l behavior of ImðE 0 ðlÞÞ is related to the behavior of the coefficient f k of the above equation, when the order k becomes large. For large l it is possible to show that [15] E 0 ðlÞ ¼ l1=3
1 X
an l2n=3 .
(36)
n¼0
If we compare the result of the ground state energy E 0 ðlÞ of the anharmonic oscillator given by Eq. (36) with Eq. (34) we note a discrepancy between our results. The result given by Eq. (35) was obtained in a smallcoupling expansion framework and the second one using, not only the strong-coupling perturbative expansion but also a non-translational functional measure. These facts can explain the discrepancy between our results and the results presented in Ref. [15]. There are several directions for investigations. It should be possible to apply the method to more realistic theories. To mention a few: since scalar fields play a fundamental role in the standard model, the study of the strongly coupled ðlj4 Þd theory at finite temperature and also the renormalized vacuum energy of scalar quantum fields in the presence of macroscopic structures deserves future investigations. Acknowledgments I would like to thank, G. F. Hidalgo and S. Joffily for enlightening discussions, and also L. A. Oliveira for comments on the manuscript. This paper was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnolo´gico do Brazil (CNPq). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570. P.C. Martin, J. Schwinger, Phys. Rev. 115 (1959) 1342. S. Kovesi-Domokos, Il Nuovo Cimento 33A (1976) 769. C.M. Bender, F. Cooper, G.S. Guralnik, D.H. Sharp, Phys. Rev. D 19 (1979) 1865. N.F. Svaiter, The strong-coupling expansion and the singularities of the perturbative expansion, Proceedings of the ‘‘X Brazilian School of Cosmology and Gravitation’’, Rio de Janeiro, Brazil, AIP, 2003. N.F. Svaiter, Physica A 345 (2004) 517. R.T. Seeley, Am. Math. Proc. Symp. Pure Math. 10 (1967) 288. S.W. Hawking, Commun. Math. Phys. 55 (1977) 133. J.R. Klauder, Acta Phys. Austriaca 41 (1975) 237. J.R. Klauder, Phys. Rev. D 14 (1976) 1952. R. Menikoff, D.H. Sharp, J. Math. Phys. 19 (1978) 135. R.J. Rivers, Path Integral Methods in Quantum Field Theory, Cambridge University Press, Cambridge, 1987. J.R. Klauder, Beyond Conventional Quantization, Cambridge University Press, Cambridge, 2000. J.R. Klauder, Ann. Phys. 117 (1979) 19. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press, NY, 1996, pp. 930–931. L.H. Ford, Phys. Rev. 21 (1980) 933. L.H. Ford, N.F. Svaiter, Phys. Rev. 51 (1995) 6981. E. Elizalde, A. Romeo, J. Math. Phys. 30 (1989) 1133. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980. A.P.C. Malbouisson, R. Portugal, N.F. Svaiter, Physica A 292 (2001) 485. G. Parisi, Phys. Lett. 66B (1977) 167. E. Brezin, G. Parisi, J. Stat. Phys. 19 (1978) 269.