Summation of divergent series: Order-dependent mapping

Summation of divergent series: Order-dependent mapping

Jean Zinn-Justin CEA, IRFU and Institut de Physique Théorique, Centre de Saclay, F91191 Gif-sur-Yvette Cedex, France

Summation methods play a very important role in quantum field theory because all perturbation series are divergent and the expansion parameter is not always small. A number of methods have been tried in this context, most notably Padé approximants, Borel–Padé summation, Borel transformation with mapping, which we briefly describe and one on which we concentrate here, Order-Dependent Mapping (ODM). We recall the basis of the method, for a class of series we give intuitive arguments to explain its convergence and illustrate its properties by several simple examples. Since the method was proposed, some rigorous convergence proofs were given. The method has also found a number of applications and we shall list a few.

Divergent series; Summation methods; Borel transformation; Quantum mechanics.
journal: Applied Numerical Mathematics

1 The initial motivation: Perturbative quantum field theory

In quantum field theory, the main analytic calculation tool is the perturbative expansion. As an illustration, we consider the important example of the \phi^{4} field theory ZJbook. In the statistical formulation, one considers the Euclidean (or imaginary time) action \mathcal{S}, local functional of the field \phi(x), x\in\mathbb{R}^{d},

\mathcal{S}(\phi)=\int{\rm d}^{d}x\,\left[{\textstyle{\frac{1}{2}}}\sum_{\mu=1% }^{d}\left[\partial_{\mu}\phi(x)\right]^{2}+{\textstyle{\frac{1}{2}}}r\phi^{2}% (x)+\frac{g}{4!}\phi^{4}(x)\right], (1)

where r and g are two parameters. To this action is associated a functional measure \mathop{\rm e}\nolimits^{-\mathcal{S}(\phi)}/\mathcal{Z}, where \mathcal{Z} is the partition function given by the field integral

\mathcal{Z}=\int[{\rm d}\phi]\mathop{\rm e}\nolimits^{-\mathcal{S}(\phi)}. (2)

The limit d=0 corresponds to a simple integral.

The case d=1 corresponds to the quantum quartic anharmonic oscillator.

Dimensions d>1 correspond to quantum field theory and the expression (1) is then somewhat symbolic since the theory has to be modified at short distance to regularize UV divergences and renormalized to cancel them.

In particular, the dimensions d=2,3 are especially relevant to classical statistical physics and the theory of phase transitions. Finally, d=4 is relevant to the theory of fundamental interactions at the microscopic scale. The corresponding relativistic quantum field theory is part of the so-called Higgs mechanism.

For the field theory (1), the perturbative expansion amounts to an expansion in powers of the positive parameter g.

For d>1, the difficulty of evaluating the successive perturbative terms increases very rapidly. Moreover, questions like regularization and renormalization arise. Therefore, the calculation of renormalization group functions in the d=3 (\boldsymbol{\phi}^{2})^{2} field theory up order g^{7} rBNGMo is a remarkable achievement.

1.1 Large order behaviour of perturbative series

In the \phi^{4} field theory (1), g=0 corresponds to a singularity since the integral (2) is not defined for g<0. The perturbative series is divergent. For d<4, the large order behaviour can be inferred from a steepest descent calculation of the field integral (2) rLOBLip rLOBgen. For the quartic anharmonic oscillator (d=1) the result was derived earlier from the Schrödinger equation rCBTTW. For any physical observable f, the results have the general structure

f_{k}\mathop{\propto}_{k\to\infty}(-1)^{k}k^{b}a^{k}k!\,, (3)

where a depends only on d and b is a half-integer that depends on the observable. The coefficient A=1/a has the value

\displaystyle d=0: \displaystyle A \displaystyle=3/2\,, (4)
\displaystyle d=1: \displaystyle A \displaystyle=8\quad\,, (5)
\displaystyle d=2: \displaystyle A \displaystyle=35.10268957367896(1)\quad\textrm{(Zinn-Justin)}, (6)
\displaystyle d=3: \displaystyle A \displaystyle=113.38350781527714(1)\quad\textrm{(Zinn-Justin)}. (7)

For d=4, to the contribution coming from the steepest descent calculation, a contribution due to the large momentum singularities of Feynman diagrams has in general to be added.

Finally, notice that for d<4, Borel summability has been proved.

Similar results can be obtained for a number of quantum field theories. When the formal expansion parameter is Planck’s constant, a divergence of the form (3) is in general found (except for some fermion theories), but the parameter a may be complex. For an early review, see ZJLOreport.

It follows from the large order behaviour analysis that, when the expansion parameter is not small, a summation of the perturbative expansion is indispensable.

1.2 Series summation

In the study of the fundamental interactions at the microscopic scale, it was realized that in the case of the strong nuclear force, unlike QED, the expansion parameter was large and, therefore, perturbation theory useless, leading many physicists even to reject quantum field theory as a framework to describe such phenomena.

Before the large order behaviour was even known, in rDBMP it was proposed, instead, to sum the perturbative expansion, using Padé approximants and the idea was applied to a phenomenological model, the \phi^{4} field theory in d=4 dimensions. Since only two or three terms could be calculated, the possible convergence of the Padé summation could not be checked very well. However, the results obtained in this way made much better physical sense than those of plain perturbation theory. For a review see ZJthesis.

In the seventies, one outstanding problem for which summation methods was required, is the determination of critical exponents and other critical quantities in the theory of second order phase transitions. Following Wilson, for a whole class of physical systems, these quantities can be obtained from the ({\boldsymbol{\phi}}^{2})^{2} field theory in d=3 dimensions. One verifies immediately that the expansion parameter, the renormalized interaction g_{\rm r}, is of order 1 and a series summation is required (we do not discuss here the \varepsilon=4-d expansion, but the problem is analogous).

To deal with the practical problem of series summation, a method was proposed based on Borel–Padé approximants rBNGMo. With the knowledge of the large order behaviour, a more efficient method could be developed, combining a Borel transformation (actually Borel–Leroy) and a conformal mapping rLGZJ, rRGZJ, which we briefly present in next section. However, another method based only on the analytic properties of the series, the order-dependent mapping was also investigated, which we describe in more detail in section LABEL:ssODMdef (a general reference is ZJbook).

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