## Abstract

We show that a strongly perturbed quantum system, being a semiclassical system characterized by the Wigner–Kirkwood expansion for the propagator, has the same expansion for the eigenvalues as for the Wentzel–Kramers–Brillouin series. The perturbation series is rederived by the duality principle in perturbation theory.

## 1. Introduction

Recently, we proposed a strongly coupled quantum field theory (Frasca 2006*a*,*b*). This approach is based on the duality principle in perturbation theory as exposed in Frasca (1998) and applied in quantum mechanics. In the latter paper, we showed that the dual perturbation series to the Dyson series, which applies to weak perturbations, is obtained through the adiabatic expansion when reinterpreted as done by Mostafazadeh (1997). This expansion grants a strong coupling expansion having a development parameter formally being the inverse of the parameter of the Dyson series and then holding just in the opposite limit of the perturbation going to infinity.

Owing to the relevance of this approach for quantum field theory, it is of paramount importance to show how this method is able to perform with an anharmonic oscillator, as this example has been set as a fundamental playground by Bender & Wu (1969, 1973) in their pioneering works. Besides, it would be helpful to trace the way the eigenvalues are obtained in this case in view of methods, such as the one of Janke & Kleinert (1995) and Kleinert (2004), providing excellent computational tools to this aim.

Indeed, in this paper we show that our method provides a sound solution to the strong coupling computation of the groundstate energy of an harmonic oscillator, this being given by the well-known semiclassical series for the eigenvalues in the Wentzel–Kramers–Brillouin (WKB) approximation. The relevance of the result relies on the fact that we obtain this result from the Wigner–Kirkwood (WK) expansion that our method yields. This provides an important conceptual result as, besides showing that the two approaches, WKB and WK, are equivalent in the strong coupling limit, we will be able to rederive the fact that (Simon 2005) *a strongly perturbed quantum system is a semiclassical system* from the duality principle in perturbation theory, demonstrating in this way the soundness of the approach. This fact may be interesting, for example, in the problem of the measure in quantum mechanics as we have already pointed out in Frasca (2003, 2004) for Quantum ElectroDynamics.

The result is obtained by exploiting the link between the WK expansion and the Thomas–Fermi approximation (Ring & Schuck 1980). Therefore, a semiclassical approximation relies on the foundation of a many-body approach and is equivalent to the WKB approximation in the strong coupling limit giving the same results for the eigenvalues. Indeed, the Thomas–Fermi approximation is also a semiclassical approximation and this fact is well known, having been proved by Lieb & Simon (1973, 1977). The interesting aspect here is that such a many-body approach gives back the Bohr–Sommerfeld quantization rule and we obtain also the higher-order corrections.

## 2. Method

In order to start our proof, we consider the simple case of a free particle undergoing the effect of a perturbation *V*(*x*) in a one-dimensional setting. Thus, the Schrödinger equation for the propagator *U* can be written as(2.1)where *λ* is an ordering parameter and *U*(*t*_{0}, *t*_{0})=*I*. The Dyson series for the propagator can straightforwardly be obtained when the limit *λ*→0 is considered, giving back the well-known solution series(2.2)*U*_{0} being the solution of the equation(2.3)given by(2.4)We recognize the interaction picture working here. At this point, it is interesting to note that the choice of a perturbation is completely arbitrary and one may ask what meaning could be attached to a series, with *λ*=1, where we take instead of *V*(*x*) as a perturbation giving the series(2.5)*K*_{0} being now the solution of the equation(2.6)being(2.7)The answer can be immediately obtained when, after reinserting the ordering parameter *λ*, we recognize that with a rescaling of time *t*→*λt* and taking the series(2.8)we recover the series (2.5), which is a strong coupling expansion. Setting *τ*=*λt*, we can write (2.5) as(2.9)We easily recognize that this series is dual to the Dyson series in the sense of Frasca (1998) having a development parameter that is formally the inverse with respect to that of the Dyson series. For this reason, we call this dual representation the free picture. This series is the 1+0-dimensional solution of the quantum field theory we presented in Frasca (2006*a*,*b*) and our aim is to get from this the groundstate energy showing that, in this case, we are working with a semiclassical expansion.

For our aims, we need to exploit this strong coupling series to unveil its nature. We already see from the leading-order solution (2.7) that it coincides with the leading order of the well-known semiclassical WK series (Ring & Schuck 1980; Schulman 1981; Janke & Kleinert 1995; Kleinert 2004) as should be expected (Simon 2005). We need to prove this also for higher orders. Indeed, the computation is straightforward and gives, at least for the first correction,(2.10)that is, what one should expect for the WK expansion. Thus, we have obtained the semiclassical WK series out of a dual strong coupling expansion for a quantum mechanical system by the duality principle in perturbation theory. This in turn means, as already known (Simon 2005), that a strongly perturbed quantum system is a semiclassical system. This is in agreement with the fact that a large mass expansion in quantum mechanics gives rise to a WK expansion out of the WKB expansion (Osborn & Molzahn 1986). We just point out that a wide application of the WK expansion is seen in statistical mechanics (Schulman 1981) and in this case is obtained with the standard substitution , which we will use in the following.

The WK series appears rather singular depending on ascending power of *τ* and having terms proportional to gradients of the potential making the dependence on *λ* anomalous at best for such an expansion. But as we will see below, the series for the groundstate energy is well defined and is in agreement with the corresponding expression for a WKB series in spite of the very singular nature of this expansion. This is easy to prove using techniques of many-body physics that will permit us to show that the leading order of the WK expansion is the well-known Thomas–Fermi approximation.

The next step is to recognize that we can resum all the terms with *p*^{2}/2*m* giving finally, after projecting on the momentum eigenstates with a Wigner transformation of the propagator,(2.11)Now being(2.12)The density matrix can be obtained by inverse Laplace transforming the series (2.11) divided by *β* (Ring & Schuck 1980), i.e.(2.13)with *c*>0, giving the expression(2.14)and the leading order is just the Thomas–Fermi approximation. This series does not have a clear dependence on *λ*, but this is a strong coupling series that has meaning in the sense of distributions.

We now impose the normalization condition(2.15)where *n*=0, 1, 2, …, and this gives the WKB energy levels and their higher-order corrections as we will see in a while. Indeed, we can substitute equation (2.14) into the above normalization condition giving the series (Bender *et al*. 1977)(2.16)*x*_{T1} and *x*_{T2} being the solutions of the equation *E*=*λV*(*x*) that determine the region where the integral is meaningful. Use has been made of the fact that . We recognize here the WKB quantization condition and its higher-order corrections as promised.

Our aim now is to apply the above expansion to the computation of the groundstate energy of a pure quartic oscillator *H*=*p*^{2}/2*m*+*λx*^{4}/4. From Janke & Kleinert (1995) and Kleinert (2004), we know that at the leading order one should have *E*=0.667986259155777…(*λ*/4)^{1/3}. We know also from (Bender *et al*. 1977) that this value can be obtained by pushing the series (2.16) to the higher orders. In order to have an idea of what we get, for our example we get the series(2.17)where(2.18)*K* and *E* being the elliptic integrals of first and second kind, respectively, and . The numerical solution is rather satisfactory as we get from the Bohr–Sommerfeld term that , which, as is well known, has an error of about 20%, while with the first-order correction one has , which improves to 10% already at this order. But, as already shown by Bender *et al*. (1977), we know that the semiclassical approach is able to produce the exact value by going to higher orders.

## 3. Conclusion

Thus, we can conclude that the WK series produces eigenvalues through the Bohr–Sommerfeld rule and its higher-order corrections. Besides, our result has been obtained by considering the duality principle in perturbation theory and applying it to the Dyson series, rederiving by this means the equivalence between strong coupling perturbation theory and semiclassical expansion, making sound the main result of this paper.

## Acknowledgments

I would like to thank Hagen Kleinert for inviting me to Berlin to discuss with him these questions.

## Footnotes

- Received January 4, 2007.
- Accepted May 24, 2007.

- © 2007 The Royal Society