At the shoal of the resurgent depths
An introductory overview of Borel summation, transseries and resurgence,
applied to the Painlevé I equation and the quantum anharmonic oscillator
Bastiaan van Bloppoel
August 17, 2017
Supervisor: Dr. Marcel Vonk
Examinator: Dr. Alejandra Castro Anich
Abstract
An overview is provided of several subjects and techniques concerning asymptotic series. Borel summation is reviewed as a tool for calculating the sum of these series. The theory of transseries and resurgence is discussed briefly subsequently. As an example to show some of the essential concepts of these techniques in effect, the Painlevé I equation is discussed. Finally, the example of the quantum quartic anharmonic potential is treated. No new results are found, but a closer look is taken at existing literature to fit the main goal of this report, which is to provide an introductory overview of the treated subjects for the undergraduate physics student. The work brings the reader to the point where possible new research lies, and one such direction is outlined in some detail.
Contents
1 Preface
1
2 Introduction
1
3 Theory
3
3.1
Asymptotic series
. . . .
3
3.2
Borel resummation
. . . .
3
3.3
Transseries and Resurgence
. . . .
5
4 Painlevé I
7
4.1
Introduction
. . . .
7
4.2
Power series solution
. . . .
7
4.2.1
Borel resummation of the power series solution to
the PI equation
. . . .
8
4.3
Transseries solution
. . . .
9
4.3.1
Perturbative sector
. . . .
9
4.3.2
One-instanton sector
. . . .
10
4.4
Large-order relations
. . . .
11
5 Quantum quartic anharmonic oscillator
12
5.1
Introduction
. . . .
12
5.2
Ground-state energy perturbation series
. . . .
13
5.3
Large-order calculations for the instanton action
. . . .
15
6 Conclusion and Outlook
16
Appendices
17
1
Preface
The aim of this report is to provide a pedagogical introduction to the field of asymptotics, Borel summation and transseries and resurgence to the Bachelor student in physics, as these subjects are not treated in undergraduate physics courses. Because of the technical complexity of resurgence theory, we will only touch upon its most important results for this thesis.
No new results are provided by this work. Instead, we will work through a few examples to illustrate some of the basic concepts of these techniques.
The first of the greater examples will be the Painlevé I equation. For our purposes, this is a well tested arena to show how resurgence theory relates perturbative and nonperturbative physics. In the next and last example, we will look at the more physical problem of the quantum anharmonic oscillator. In this, we will calculate the ground state energies and by doing so we will be able to calculate its instanton action, the probability with which a particle in such a potential may tunnel out to infinity.
2
Introduction
In modern physics, it is very likely to encounter problems to which an exact solution can’t be found. When these problems are tackled, the physicist generally resorts to perturbation theory to find a solution. These perturbative methods produce power series in the variable that controls the strength of the perturbation. Take for instance a general observable φ and a perturbation parameter z, which we can tune to our liking. The perturbation series then looks like
φ(z) = φ0+ φ1z + φ2z2+ φ3z3+ ... (1)
Not only do we encounter such series when performing perturbative methods. Quite often, a dif-ferential equation will require us to plug in a power series solution which will take the same general form. Most of the series that are found in problems discussed in undergraduate physics courses are generally convergent. The idea then being that when the series is found, we add up all the corrections of the higher orders and slowly achieve a better approximation to the true answer. However, in reality we find that most of the series in modern physics problems tend to approach a value up to a certain order, after which the series diverges wildly. These types of series are typically formal asymptotic series, which will be discussed in section3.1. For now, it suffices to say that these series have coefficients which grow factorially and that the series has a zero radius of convergence. The origins of these characteristics can be understood intuitively by the following, slightly simplified arguments. For the factorial growth, we may turn to quantum field theory; when the interaction between particles is studied, this is described through Feynmann diagrams. The number of Feynmann diagrams grow factorially, so when these are summed for the partition function, there are factorial contributions to the coefficients of the power series. As for the radius of convergence, there exists an illuminating, simple argument provided by Suslov [9], which will be presented here to give an intuitive explanation of this property.
Consider a quantum particle in the following one-dimensional anharmonic potential, see figure
1:
U (x) = x2+ λx4 (2)
When the perturbation is turned off, that is when λ is zero, the well known harmonic oscillator is returned. When λ > 0, the system will have well-defined energy levels, but when λ < 0, the system will be metastable, as the particle can escape to infinity by the probability of tunneling through the potential barrier. This shows that the limits λ → 0+ and λ → 0− are not the same,
and so there must be a singularity at zero, which means that the radius of convergence is zero. Suslov even goes as far as to say that a zero radius of convergence is encountered in all fun-damental quantum field theories with a single coupling constant. The argument by Suslov is a simplification of an earlier argument by Freeman Dyson which shocked the scientific community in 1952, because nobody was ready to deal with the fact that all series in Quantum Electrodynamics would diverge. See [4] and [9].
-1.0 -0.5 0.5 1.0 0.5 1.0 1.5 2.0 (a) λ > 0 -1.0 -0.5 0.5 1.0 -0.2 -0.1 0.1 0.2 (b) λ < 0
Figure 1: Anharmonic potential for positive and negative λ
This calls for a mathematical tool to solve such divergent series. There exist various so called resummation techniques, and in this thesis, we will focus on the technique developed by Émile Borel and later greatly extended by Jean Écalle. Borel’s resummation method will be discussed in section 3.2. The largest part of the work on resurgence by Jean Écalle is beyond the scope of this thesis, though a few important objects and properties will be discussed here, as they are paramount to understanding that which goes beyond perturbative physics. Therefore, a rather simplified discussion of his work will be provided in section3.3.
To provide an arena to test these techniques in a practical example, I have worked through the problem of solving the Painlevé I equation, which works well to illustrate many of the ideas in-volved. This discussion is presented in section4
Finally, the problem of the quantum anharmonic oscillator with a quartic potential is discussed as the main physical application of these techniques. This is the same problem which was presented as argument as to why many such series in physics diverge in the first place. We will calculate the ground-state energy of the potential. Combining these results with the large-order relation that will be discussed in section3.3, we will be able to calculate the instanton action of the potential, the probability with which a particle may tunnel out to infinity. This discussion is presented in section5
3
Theory
3.1
Asymptotic series
As described in the introduction, the perturbative expansions we encounter in many physical problems have a tendency of approaching a certain value up until order N in n, after which they diverge. Let’s say we have some perturbative expansion of a general quantity φ(z)
φ(z) =
∞
X
n=0
φnzn (3)
We call this series asymptotic to some function f (z), in the sense of Poincaré, if [6]
lim z→0 f (z) − N X n=0 φnzn z−N = 0 (4)
for all N. As we vary N, the partial sums of φ(z) will first approach the true value f (z), after which they will diverge. A logical next step in finding the value nearest to the true value would be to stop the series at the optimal N , since in this way, we can assign some value to the diverging series. This method is called optimal truncation. As shortly discussed in the introduction, we will often be looking at series with coefficients which grow factorially,
φn∼ A−nn! (5)
Often, the optimal value of N is there where the smallest term in the series is, so we want to find an N that minimizes |φnzn| and truncate the series at that point.
|φNzN| ∼ cN ! z A N (6) We then use Stirling’s approximation of the factorial to rewrite this expression,
∼ c expnNlog N − 1 − log A z o (7) At large N , the exponent has a saddle at
N∗= A z (8)
From this, we see that the method of optimal truncation indeed works best for small |z|, as the optimal value N∗ is larger here. We now estimate the error of the optimal truncation from the
next term in the series
(z) = φN∗+1z N∗+1 ∼ e −|A/z|, (9)
where we have used Stirling’s approximation again. The error we find is exponentially small and entirely nonperturbative in nature. So we see that the perturbative expansion alone is not enough to fully determine φ(z), and we need such additional, nonperturbative information to gain a more precise asymptotic approximation. To gain a more complete understanding of these nonperturbative contributions and their origins, we must dive into the field of Borel resummation.
3.2
Borel resummation
Borel’s resummation technique relies, in its basic principle, on a subtle, double transformation. For the rest of this chapter, we assume our asymptotic series φ(z) to be factorially divergent and to have a zero radius of convergence. Its Borel transform is then defined as
B[φ](ζ) = ∞ X n=0 φn n!ζ n, (10)
where, at first sight, it seems like we have only simply removed the factorial growth of the coeffi-cients. The transformation is more important than just this, for we have mapped the asymptotics onto the complex Borel plane, into a new series in the variable ζ. This new series now has a
finite radius of convergence and it is analytic in a disk around the origin. As we shall see, the Borel transform generally has singularities in the Borel plane, which are of great importance for the mathematical framework of resurgence. To return the Borel transform to the original series, we take the inverse Borel transform, which is a directional Laplace transform along a ray eiθ. To
make more explicit why this step works, the following identity is provided.
1 = R∞
0 dζe −ζζn
n! (11)
When we multiply our original divergent series φ(z) with this identity, we need to reverse order of summation and integration. This is actually an invalid step, but we find it gives a good approx-imation and as we shall see, the ambiguities involved later will justify this invalid step now. We receive
Sθ[φ](z) =
Z eiθ∞
0
dζB[φ](ζ)e−zζ (12)
This object is called the Borel resummation along θ, which has the same asymptotic expansion as φ(z). It is, in essence, an analytic continuation of φ(z) to a larger domain. Before we continue, let’s clarify this method with a simple example.
Example Consider the following series
f (z) =
∞
X
n=0
(−1)nzn (13)
This series converges for |z| < 1. We take its Borel transform
B[f ](ζ) = ∞ X n=0 (−1)nζn n! = e −ζ (14)
Now we take the inverse Borel transform to return our original function
Sθ[f ](z) = Z ∞ 0 dζe−ζe−zζ = Z ∞ 0 dζe−(1+z)ζ = e −(1+z)ζ −(1 + z) ∞ 0 = 1 1 + z (15) Where the function Sθ[f ](z) now converges for the much larger region Re(z) > −1, which can be
checked with an integral convergence test. This analytic continuation also allows us to calculate the interesting result of
∞ X 0 (−1)n =1 2 (16) when we plug in z = 1.
What we have seen until now is that the Borel resummation provides us with an analytic con-tinuation of the original perturbative series. In some cases, this gives an answer to our original question; how do we associate a value to a divergent series? However, we will often find that when we try to perform Borel resummation along a given ray θ, we will encounter singularities along this path and thus cannot calculate the integral directly. Instead, we must dodge these singularities by defining lateral Borel resummations Sθ± which pass the singularities slightly above
or below, see Figure 2 for a schematic representation of such deviations of the contour integral around the singularity. This freedom of choice gives rise to ambiguities, ones which, in this case, we call nonperturbative ambiguities. They are called such, because they are completely nonper-turbative in nature, our pernonper-turbative expansion would never have noticed the singularities in the Borel plane, let alone the subtlety of a choice of path around them. It is also good to point out that when we encounter perturbative expansions which have such singularities, we call them non-Borel summable, because we can’t directly find an answer to the original question regarding the value of the divergent sum. Now, the important subtle point of these singularities and ambiguities is that the different possible integration paths in the lateral Borel summations produce functions with the same asymptotic behaviour, but differing by exponentially small terms. To illuminate
Figure 2: Schematic representation of two possible lateral Borel summation contours. (From [6])
this property, consider a Borel transform which has one singularity, a simple pole, at a distance A from the origin along a direction θ. The difference between the two lateral Borel summations is
Sθ+− Sθ− ∝ I A dζ e −ζz ζ − A, (17)
which amounts to a residue integral around the pole, so then we are left with I A dζ e −ζz ζ − A = 2πiResζ=A e−ζz ζ − A = 2πie −Az ∝ e−Az (18)
This last, exponentially small term is very important within the context of the resurgence frame-work, which will become more explicit in the next section. It is an entirely nonperturbative contribution. We call A the instanton action, because the first time these contributions were found in a physical context, they described the probability of a particle tunneling through a potential barrier, as we shall see in section5. By now, we still call this the instanton action, even when the physical interpretation of the problem at hand is not such. This is the same A which we introduced in (5) for the growth of the coefficients. The fact that it returns here is no coincidence. As we shall see, this connection between the singularity structure of the Borel plane and the divergent growth of the coefficients is a very important and interesting phenomenon in the framework of resurgence. To summarize, we have taken an asymptotically divergent series on which we performed the Borel transformation and the inverse Borel transformation and we have found that, due to singularities in the complex Borel plane, there are certain ambiguities as to our choice in resummation path which give rise to nonperturbative contributions in the problem. At this point, we could look at the practical side of things and develop a machine for producing answers in numbers to our divergent series. To do such, we would perform what is called Borel-Padé summation. In this, we take a Padé approximant of the Borel transform, which essentially turns the polynomial into a rational function, mimicking the singularity structure of the Borel transform. We then take the Laplace transform to perform the Borel resummation and end up with a number which would be a good approximation to the true answer to the original problem. I will not say more about this procedure in this section, for it is more illuminating to see it in action whilst treating an example, as we will do in section4.2.1. Now we will turn to the framework of transseries and resurgence, as this will allow us to understand these nonperturbative results much better.
3.3
Transseries and Resurgence
In the previous section, we have found that when we apply perturbative methods on problems which provide us with divergent series, there exists a connection between the singularity structure of the Borel plane, as represented in the exponential instanton factors (18), and the factorial growth of the coefficients of the series φn ∼ n!A−n, where A is exactly that connecting element.
This relation conceals important information about the nonperturbative physics that lies behind it. If we wish to fully understand the complete picture, we must go beyond the perturbative expansion and work with an object that incorporates both perturbative and nonperturbative contributions.
The transseries (see, for instance, [1,6]) is such a mathematical object. It is a double expansion in powers of both the perturbation parameter z and the previously found nonperturbative exponential e−Az φ(z, σ) = ∞ X n=0 σne−nA/zznβ ∞ X g=0 φ(n)g zg, (19) .
What we see here is an object with various sectors. If we take n = 0, we return our original perturbative expansion. For all the n > 0, we have an infinite set of nonperturbative sectors which have their own perturbative expansion at each given n. For the same historical reason as why we call the A in the exponent the instanton action, the different sectors of the transseries are called instanton sectors, numbered accordingly by their respective level in n. So the perturbative sector is called the zero-instanton sector, after which we have the one-instanton sector, and so forth. Concerning the clarification of some parameters, we have a σ, which is called the transseries parameter. It serves to control the behaviour of the transseries in different instanton sectors of the Borel plane. It is determined by some physical condition of the problem at hand. The znβ is a factor included for generality, where β is an exponent which allows the sum to start at some non-zero power of z and varies per considered example.
The most important element of the transseries is the instanton action A. To illuminate its im-portance, I will briefly introduce some essential results of Jean Écalle’s resurgence theory. A full treatment of resurgence theory would be beyond the scope of this report, though very interesting for a possible more advanced follow-up project.
The etymology of the name, and with that the most essential feature of the theory, comes from the fact that the coefficients of the various instanton sectors are connected by a web of large-order relations, which are approximations valid for g → ∞, hence large-order. The coefficients of the one sector resurge in the other. The derivation of the large-order relations is a technical and tedious one, involving Jean Écalle’s alien calculus and bridge equation and the Stokes automorphism. I refer the reader to [1] for a detailed treatment of this derivation. For now, we will take a look at the large-order relation as a given and tested result of resurgence theory.
φ(0)g ' +∞ X k=1 Sk 1 2πi Γ(g − kβ) (kA)g−kβ +∞ X h=1 Γ(g − kβ − h + 1) Γ(g − kβ) φ (k) h (kA) h−1. (20)
In its closed form, the formula for the large-order relations is slightly daunting and hard to decipher, so let’s write out a few terms for it to be more insightful.
φ(0)g ' S1 2πi Γ(g − β) Ag−β φ(1)1 + A g − β − 1φ (1) 2 + ... + + S 2 1 2πi Γ(g − 2β (2A)g−2β φ(2)1 + 2A g − 2β − 1φ (2) 2 + ... + + S 3 1 2πi Γ(g − 3β) (3A)g−3β φ(3)1 + 3A g − 3β − 1φ (3) 2 + ... + ... (21)
From this formula we can see clearly at first sight that we have a large-order factorial growth in which the instanton action A plays an important role. We also see that the coefficients in different instanton sectors are all interrelated, that is, they resurge in the different sectors. The most fas-cinating results perhaps, is that our perturbative expansion knew all about the nonperturbative physics and the resurgence of the different instanton sectors all along. We only needed to dig it out with the right tools.
After having introduced a lot of new concepts it will be beneficial to look at an example in which we apply these techniques. The next section will serve to do so. We will discuss the Painlevé I equation as our vehicle to test these techniques and relations.
4
Painlevé I
4.1
Introduction
The approach of this section will be the following. We will take a look at the Painlevé I equation to test the previously introduced techniques and to show that the large-order relations as predicted by resurgence theory hold. The Painlevé I equation provides us with a good arena to do so, because in this case we can find the instanton action both through analytical and numerical methods, whereas in other examples it may be that only the latter is possible. First, we will approach our problem in the ’naive’ way, by trying a power series solution. In this, we can confirm the divergence of the coefficients of the resulting power series and develop a machine for assigning values to the diverging sum by using Borel’s resummation method. Then we will seek a more complete answer with our knowledge of transseries to find an analytical approach to the instanton action and the coefficients of the instanton sectors. At last we will take only our perturbative coefficients and use the large-order relations to find the instanton action and the coefficients of the instanton sector using numerical methods.
The Painlevé I equation is a second order, non-linear differential equation. This equation ap-pears in many contexts in physics. An important application of the Painlevé I is, for instance, in two-dimensional quantum gravity, where it provides the all-genus solution [6]. The equation is as follows.
φ(z)2−1
6φ”(z) = z (22)
4.2
Power series solution
To obtain the power series solution we insert the following ansatz into the Painlevé I equation (henceforth it will be referred to as the PI equation). The ansatz can be justified by means of the method of dominant balance, among other techniques. For now, we will assume it correct.
φ(z) = z1/2
∞
X
n=0
φnz−5n/2 (23)
After calculating derivatives and plugging it into the PI equation, we obtain z ∞ X n=0 φnz−5n/2 X∞ m=0 φmz−5m/2 −1 6 ∞ X n=0 φn 1 − 5n 2 −1 − 5n 2 z(−3−5n)/2= z (24) Which becomes z ∞ X n=0 Xn m=0 φmφn−m z−5n/2−1 6 ∞ X n=0 φn 1 − 5n 2 −1 − 5n 2 z(−3−5n)/2= z (25)
Notice that, in order to equate the power in z1 on the l.h.s and the r.h.s, we need φ2
0 = 1, which
implies φ0= ±1. We are free to choose φ0= 1, because equation (23) is a formal solution, so the
sign can be absorbed in the factor z1/2. All other powers of z must cancel. We shift the second term in (25) and get
∞ X n=0 φmφn−m= 1 6φn−1 1 − 5(n − 1) 2 −1 − 5(n − 1) 2 n−1 X m=1 φmφn−m+ 2φ0φn= φn−1 24 (25(n − 1) 2 − 1) φn= φn−1 48 (25(n − 1) 2− 1) −1 2 n−1 X m=1 φmφn−m (26)
Here we have the recursion formula for our perturbative coefficients. This can be plugged in to Mathematica to calculate them and put them in a list. A few of these are listed in Table1. Figure
φ0 φ1 φ2 φ3 φ4 1 −1 48 − 49 4608 − 1225 55296 − 4412401 42467328
Table 1: The first five perturbation coefficients of the Painlevé I power series solution
Had we not known about Borel resummation, this series would have been useless, because we wouldn’t have known what to do with divergent series. Let us now see how the technique of Borel resummation can indeed provide us with finite answers for given z.
n φn
Figure 3: A plot showing the absolute value of the perturbative coefficients of the power series solution to the Painlevé I equation
4.2.1 Borel resummation of the power series solution to the PI equation
We will briefly go through the steps involved in calculating the sum of the Painlevé I power series solution for given z. We have our series (23) with coefficients (26). The next step is to take its Borel transform. B[φ](ζ) = ζ1/2 ∞ X n=0 φn n!ζ −5n/2. (27)
Now we wish to take its inverse Borel transform to assign a value to the original sum for given z. Before we can do so, there is a problem we must overcome. Since we don’t have a closed form for the Borel transform, the procedure of taking the inverse Borel transform can only be performed term by term, which will only deliver us our original asymptotic series. No progression at all. So we need to approximate the Borel transform with a Padé approximant, which will give us converging answers for large-order N in n. The Padé approximant works as follows. Given a polynomial of degree N , where N is even, the Padé aproximant expands this function as a ratio of two power series in degrees N/2 such that the first N + 1 derivatives remain the same.
B[N ][φ](ζ) ' P[N/2][φ](ζ)
Q[N/2][φ](ζ) . (28)
Intuitively, one can understand the value of this process by considering the growth of the polyno-mial B[N ][φ](ζ) as ζ → ∞ and comparing this to the growth of the rational function on the r.h.s.
of (28) in the same limit. The polynomial will grow quickly, but the ratio of the two polynomials of the same degree will remain about constant, which is much better for integration. Mathematica has a built-in function for taking Padé approximants of polynomials, called PadeApproximant.
After choosing a suitable order N in n and taking the Padé approximant, we are ready to in-tegrate along a given path. We choose a value for z in our original series of which we would like to know the sum and plug it into the Mathematica function NIntegrate. Since the PI equation has singularities on the positive real axis, we would like to bypass these. This can be done by first taking an integral from 0 to · i and then one form · i to ∞. I have done so for z = 10, N = 100 and = 1 and the result is.
φ(10) ' Z C B[100][φ](10) ' Z C P[50][φ](10) Q[50][φ](10) = 0.316207 − 4.16334 × 10 −16i (29)
To illuminate the existence of the ambiguity of choosing a path around the singularities, we take a look at the same integral, only now taken along the path with = −1 and the result is 0.316207 + 4.16334 × 10−16i, as expected. The difference is non-perturbative and it is the analogue of the 2πie−Az term in (18). We won’t further discuss these result, as this section was merely to show how one can use the Borel transform to assign values to asymptotic series.
4.3
Transseries solution
Now that we have seen what the power series solution and its resulting perturbative series can do for us, we will turn to the transseries solution. The aim of this subsection is to calculate an exact expression for the instanton action and to find the coefficients of the one-instanton sector of the transseries.
Our perturbative expansion was a power series in z−5/2. For reasons that have to do with the physical interpretation of the PI equation in string theory however (see section 5.2 of [1]), the nonperturbative expansion should be in powers of z−5/4. So we have for our transseries ansatz
φ(z) = z1/2 ∞ X n=0 σ1ne−nAz 5/4 z−5nβ4 ∞ X g=0 φ(n)g z−5g/4 (30)
We calculate its derivative to z twice, plug it into the PI equation and rescale it to x = z−5/4 to clean things up. Then we obtain
x−4/5 ∞ X n=0 n X m=0 σ1ne−nA/xxnβ ∞ X g=0 g X k=0 φ(m)k φ(n−m)g−k xg − 1 96 ∞ X n,g
σ1nφ(n)g e−nA/xxnβ+g−2/5h25(nA)2x−2/5− 5nAx3/5− 10nA(−5nβ − 5g + 2)x3/5
+ (−5nβ − 5g + 2)(−5nβ − 5g − 2)x8/5i= x−4/5 (31)
4.3.1 Perturbative sector
To check our transseries ansatz, let’s see if our perturbative coefficients return. Taking n = 0 and g = 0, we find that φ(0)0 = 1 and we have
x−4/5 ∞ X g=0 ∞ X k=0 φ(0)k φ(0)g−kxg− 1 96 X g=0 φ(0)g xg+6/5(−5g + 2)(−5g − 2) = x−4/5 g → g − 2 g X k=0 φ(0)k φ(0)g−k= 1 96φ (0) g−2(−5(g − 2) + 2)(−5(g − 2) − 2) φ(0)g = 1 192φ (0) g−2(25(g − 2) 2− 4) − 1 2 g−1 X k=1 φ(0)k φ(0)g−k
Here, we recognize our recursion relation for the perturbative coefficients and we find that
φ(0)g = (
φg/2, even g
This is simply due to the fact that we have rescaled our perturbation parameter from z−5/2 to z−5/4.
4.3.2 One-instanton sector
Now we turn to the one-instanton sector of the transseries solution to find an exact expression for the instanton action and a recursion relation for the one-instanton coefficients. Taking n = 1, we have 2σ1e−A/xxβ−4/5 ∞ X g=0 g X k=0 φ(0)k φ(1)g−kxg = 1 96 ∞ X g=0 σ1φ(1)g e
−A/xxβ+g−2/5h25A2x−2/5− 5Ax3/5− 10A(−5β − 5g + 2)x3/5
+ (−5β − 5g + 2)(−5β − 5g − 2)x8/5i (33)
Equating powers of xβ−4/5, observe that
2 = 25 96A 2 =⇒ A = ±8 √ 3 5 (34)
This is our exact expression for the instanton action. Since exponential suppression is desired in the transseries, we only take the positive solution. The solution for negative A, however, is also a valid solution when we wish to calculate the two-parameters transseries. We will discuss this further in section6. To determine β we equate powers of xβ+1/5, obtaining
2φ(1)1 −25 96A 2φ(1) 1 = 25A 96 1 − 2βφ(1)0 , (35) which sets β = 1/2.
Now that A and β are known, the expression can be reduced to a recursion relation for the coefficients of the one-instanton sector.
2σ1e−A/xx−3/10 ∞ X g=0 g X k=0 φ(0)k φ(1)g−kxg = 1 96 ∞ X g=0 σ1φ(1)g e −A/xx1/10xgh192x−4/10+ 80√3gx6/10+ (25g2+ 25g + g 4)x 16/10i (36)
Now we skip a few steps for brevity, notifying that the steps involved in this calculation are analogous to those used in deriving the recursion relation for the perturbative sector. The result is φ(1)2g+1= −1 320√3(g + 1) (100g2+ 100g + g)φ(1)g − 768 g+2 X k=2 φ(0)k φ(1)g+2−k (37) In conclusion of this subsection the first two sectors of the transseries solution of the PI equation are presented. φ(0) = 1 − 1 48x 2− 49 4608x 4− 1225 55296x 6− ... φ(1) = 1 − 5 64√3x + 75 8192x 2− 341329 23592960√3x 3+ ... (38)
We have seen how the transseries ansatz has allowed us to calculate the instanton action and various instanton sectors of the solution. We will now use these results to see how the large-order relation as predicted by resurgence theory can be studied and tested to high precision.
4.4
Large-order relations
In general, it is not always possible to find expressions for the instanton action and the various instanton sectors through analytic calculations. One must trust in the large-order relation pro-vided by resurgence theory instead. The aim of this subsection is to justify this trust by checking whether the results provided by the large-order relation match our analytical results of the previous subsection. We will discuss what this ’trust’ really means a bit more in section6.
First, let’s take our large-order relation (21) adapted to the PI equation.
φ(0)2g ' S1 2πi Γ(2g − 1/2) A2g−1/2 φ(1)0 + A 2g − 3/2φ (1) 1 + ... + S 2 1 2πi Γ(2g − 1) (2A)2g−1 φ(2)0 + 2A 2g − 2φ (2) 1 + ... + O(3−g) (39)
We see here that our perturbative coefficients are related to the quantities of our interest. It is possible to extract the instanton action and the coefficients of the various instanton sectors from this relation. This is the gist of using the large-order relation to find the wanted quantities. The idea behind the extraction of the various elements found in this equation lies in the following. We define a new sequence from the known sequence of the perturbative coefficients as follows
χg=
φ(0)2(g+1) 4g2φ(0) 2g
. (40)
This allows for the extraction of A, because we have χg= 1 A2 1 + 1 g + O 1 g2 . (41)
So the process comes down to looking at the convergence of χ−1/2g as g → ∞. This convergence
is generally very slow. There exist various tricks from numerical analysis to speed up the conver-gence of a series. In the present, we will be using the Richardson transform. I refer the reader to AppendixAfor an explanation on this technique.
(a) 2.7704 < (χg)−1/2< 2.7718 (b) 2.771280(< χ−1/2) g < 2.771283
Figure 4: The fifth (blue), tenth (red) and twenty fifth (green) Richardson transforms of the sequence χ−1/2g are displayed. Two levels of zoom are taken to show how the higher orders of the
Richardson transform are important for greater accuracy.
Figure4 is presented to show how increasing numbers of Richardson transforms on the sequence χg can dramatically increase its convergence. The first coefficient in the twenty fifth Richardson
transform χ[25]1
−1/2
already has an error of merely 1.646 × 10−7 from the exact answer 8
√ 3 5 .
great improvement. Even greater precision can be achieved by looking at the hundredth element of the twenty fifth Richardson transformχ[25]100
−1/2
, with an error of just 1.553 × 10−31. This ac-curacy is very important in using the large-order relation to determine further quantities, because in every subsequent approximation, previous errors are carried along.
Now that we have approximated the instanton action to great precision, let’s look at the next quantity of interest. This would be the first one-instanton coefficient
2πiA2g−1/2 Γ(2g − 1/2)φ (0) 2g ∼ S1φ (1) 0 , (42)
but since we are free to choose φ(1)0 = 1, because we can adapt our transseries parameter σ1
accordingly, and the first Stokes factor is known from literature to be S1 = −i3
1/4
2√π (both [1]), we
will move to the next coefficient. We now look at the sequence 2g
A
2πiA2g−1/2 S1Γ(2g − 1/2)
φ(0)2g − φ(1)0 ∼ φ(1)1 (43)
In Figure 5 we see the convergence of the above towards φ(1)1 = 5
64√3 = 0.045105... The recipe
for all the following coefficients remains the same. We approximate a term, subtract it from the left-hand side, approximate the next, and so forth. This way, we can calculate as many instanton coefficients as we wish.
(a) Original sequence and the two transforms (b) Zoomed in to the two transforms
Figure 5: The original sequence 2gA 2πiA2g−1/2 S1Γ(2g−1/2)φ (0) 2g − φ (1) 0
in blue, with its tenth (red) and twenty fifth (green) Richardson transforms are displayed. Once again, it is zoomed to show accuracy.
We have seen how we can use the large-order relation to calculate the instanton action and all the coefficients of the various instanton sectors. Moreover, it can all be done by just looking at the perturbative coefficients of an asymptotic series. I emphasize the fact that we have a powerful tool in hand. To present this in the framework of a more intuitive physical problem, we will now turn to the quantum quartic anharmonic oscillator.
5
Quantum quartic anharmonic oscillator
5.1
Introduction
Now that we have been able to test the large-order relation in an environment where we were able to check its validity, we will apply it in a familiar and frequently discussed physical example. The quantum quartic anharmonic oscillator is a problem that has been treated by many physicists. We will be following the outline of the second section of the landmark article by Bender and Wu [3]. Interestingly enough, that paper was written before Jean Écalle had developed his resurgence theory in the 80’s, so what we will be doing in this section is new compared to what Bender and
Wu do in that article, in the sense that we will be using the large-order relation to approximate the instanton action simply from the perturbative coefficients that Bender and Wu discovered back in 1969. The problem at hand is one of the simplest one-dimensional physical examples of which the power series solution diverges. In this case, the instanton action has an intuitive physical meaning. The name even derives from examples like these. It describes the probability with which a particle in the potential may tunnel through the potential barrier, out to infinity. This tunneling happens at an instance, hence the name.
This section will obey the following outline: First, we will follow section II of [3] for the cal-culation of the perturbative coefficients of the ground-state energy of the anharmonic oscillator. Since the derivation is presented compactly there, we will expand a few steps to benefit the in-tended audience of this report. After that, we will use these perturbative coefficients along with the large-order relation to approximate the instanton action of the potential.
The quartic anharmonic oscillator is described by the time-independent Schrödinger equation − ¯h 2 2m d2 dx2 + V (x) Φ(x) = EΦ(x) (44)
with the quartic anharmonic potential V (x) = 1
2mω
2x2+1
4λx
4, (45)
where λ is our perturbation parameter. When we turn off the perturbation, the familiar harmonic oscillator is returned, of which we know the solution to the ground-state wave function and energy:
Φ0(x) = e− mω 2¯hx 2 , E0= ¯ hω 2 (46)
5.2
Ground-state energy perturbation series
Henceforth, we will be working with natural length and energy scales and set constants to 1 to bypass clutter in calculations, apart from m = 12, which makes things neater. When we turn on the perturbation, our Schrödinger equation becomes the differential equation
(d 2 dx2 + x2 4 + λ x4 4 )Φ(x) = E(λ)Φ(x) , (47)
where we have expanded the ground state energy in the perturbation parameter λ
E(λ) = 1 2+ ∞ X n=1 φnλn. (48)
We have extracted the first element from the sum, φ0= 12, for reasons that become clear later, in
calculations. The wave function Φ(x) is subject to the boundary condition lim
x→±∞Φ(x) = 0 . (49)
The combination of the boundary condition and the need to expand the wavefunction in both x and λ lead us to the following ansatz for the wavefunction
Φ(x) =
∞
X
n=0
λne−x2/4Bn(x) . (50)
Here, the Bn(x) are polynomials in x to be determined, with B0= 1. They act as our expansion
in x for Φ(x). Let’s take the derivatives to x
Φ0(x) = ∞ X n=0 λne−x2/4−x 2Bn(x) + B 0 n(x) Φ”(x) = ∞ X n=0 λne−x2/4−1 2Bn(x) + x2 4 Bn(x) − xB 0 n(x) + Bn”(x) . (51)
Then we plug it into the differential equation (47) − d 2 dx2 + x2 4 + λ x4 4 Φ(x) = 1 2Φ(x) − x2 4 Φ(x) + x2 4 Φ(x) + ∞ X n=0 λne−x2/4xBn0(x) − Bn”(x) + λ x4 4 Bn(x) =1 2+ ∞ X n=1 λnφn Φ(x) , (52)
where we see how the extracted φ0 = 12 helps us clean things up by canceling the 12Φ(x) terms.
We absorb the λ in the Bn(x) in the quartic term and divide common factors to obtain ∞ X n=0 λnxBn0(x) − Bn”(x) + x4 4 Bn−1(x) = ∞ X n=1 λnφn X∞ m=0 λmBm(x) (53) Now we need to expand the Bn(x). The value of the following substitution becomes clearer as we
continue to calculate. We let x =√2y and Bn(y) =
2n
X
j=1
y2jBn,j(−1)n. (54)
We take its derivatives, note that dxd becomes √1 2 d dy B0n(y) = √1 2 2n X j=1 2jy2j−1Bn,j(−1)n Bn”(y) = 1 2 2n X j=1 2j(2j − 1)y2j−2Bn,j(−1)n (55)
Now we plug these into equation (53) and shift some indices to equate powers of y
∞ X n=0 λn 2n X j=1 2jy2jBn,j(−1)n− 2n X j=0 (j + 1)(2j + 1)y2jBn,j+1(−1)n+ 2n X j=3 y2jBn−1,j−2(−1)n−1 = ∞ X n=1 λnφn X∞ m=0 λm 2m X j=1 y2jBm,j(−1)m . (56)
It can be shown that [3], from this relation, φn= (−1)n−1Bn,1. This reduces equation (55) to the
following recursion relation
2jBn,j = (j + 1)(2j + 1)Bn,j+1+ Bn−1,j−2− n−1
X
p=1
Bn−p,1Bp,j (57)
Now we can use equations (54) and (57) to solve for all Bi,j, allowing us to subsequently extract
all the φnwe need for our perturbative coefficients. The method is as follows. We look at equation
(54) for chosen n to find the polynomials in Bn and their boundary conditions, after which we
take equation (57) to determine the value of the coefficients. We write out two sets of equations to illustrate the method.
n = 0; B0= 1 B0,0 = 1
n = 1; B1= −B1,1y2− B1,2y4 B1,j = 0 ∀ j ≥ 3
n = 2; B2= B2,1y2+ B2,2y4+ B2,3y6+ B2,4y8 B2,j = 0 ∀ j ≥ 5 (58)
We detect a pattern for the boundary conditions, namely every Bn,2n+i = 0 ∀ i ≥ 1. This is the
key to solving the recursion relation (57). Let’s write the j belonging to the n to show the main procedure. n = 1; j = 1; 2B1,1= 6B1,2 j = 2; 4B1,2= 1 n = 2; j = 1; 2B2,1= 6B2,2 j = 2; 4B2,2= 15B2,3 j = 3; 6B2,3= 28B2,4+ B1,1. (59)
Now we see how the recursion relation can be solved step by step manually. As this becomes very tedious work very quickly, we plug it into Mathematica to do the manual labour for us. We end up with a list for our perturbative coefficients φn. The first five are listed in Table2.
φ1 φ2 φ3 φ4 φ5 3 4 − 21 8 333 16 − 30885 128 916731 256
Table 2: The first five perturbative coefficients of the ground state energy of the anharmonic oscillator
Now that we can calculate a large number of coefficients, we can turn to the large-order relation once again to find an accurate approximation of the instanton action of this system.
5.3
Large-order calculations for the instanton action
Using the same procedure as we did for the calculation of the large-order approximation to the instanton action of the PI equation, we shall now calculate the instanton action of the anharmonic oscillator. We adapt our large-order relation (21) for the anharmonic oscillator and only pay attention to that which provides us with the instanton action, as we are not interested in the instanton coefficients right now.
φ(0)0 ' S1 2πi Γ(n − β) An−β φ(1)1 + A g − β − 1φ (1) 2 + ... . (60)
Now we consider the sequence χn= 1 n φ(0)n+1 φ(0)n ' 1 A 1 −β n+ O 1 n2 (61) to extract the instanton action from our perturbative coefficients by looking at the convergence of χ−1n as n → ∞. Again, we have a slow convergence, so we take Richardson transforms to speed things up (see AppendixA). Figure6is provided to show how the series converge. We find that for the twentyfifth Richardson transform at n = 30, χ[25]30 = 0.333333989870... . So we might suspect that A = 13. This result is consistent with tunneling calculations. Take for instance [8], where the result A = 1
12 is off our A = 1
3 by exactly the difference in normalization in the potential, 1 4 (see
equation (19) of [8]). We could now continue to calculate all the instanton coefficients, but this lies beyond the aim of this report. It has now been shown how the large-order relation can be used to find an approximation to the instanton action of a physical problem numerically without a priori knowledge of its value through analytic calculations involving transseries solutions. Our ’trust’ in the large-order relation has been justified and we now might suspect that quantum mechanical problems in general lead to resurgent functions. Also, it would be interesting to work through this problem by calculating the two-parameters transseries solution. We will briefly discuss these last two points in section6.
(a) Original sequence and the two transforms (b) Zoomed in to the two transforms
Figure 6: The original sequence χ−1n in red, with its tenth (blue) and twenty fifth (green) Richardson
6
Conclusion and Outlook
The purpose of this paper was to contribute a review of the existing literature on subjects con-cerning asymptotic series. We have discussed Borel summation as a tool for calculating the sums of such series. Whilst treating this subject, we found that there exist certain ambiguities concern-ing the sconcern-ingularities in the Borel plane. This lead us to the discovery that the structure of the singularities and the factorial growth of the original asymptotic series are linked by the instanton action. In order to better understand the complete picture of asymptotics, we turned to transseries and resurgence. Transseries offered us a mathematical object that incorporates both perturbative and nonperturbative sectors, opening the door to nonperturbative physics. Resurgence theory bestowed the large-order relation on us, revealing the fact that all the different instanton sectors of the transseries are connected by a web of large-order relations and thereby accentuating the connection between the singularities in the Borel plane and the factorial growth of the coefficients. To illuminate these objects and properties, we reviewed the Painlevé I equation by studying the asymptotics of its instantons. At last, we calculated the instanton action of the quantum quartic anharmonic oscillator with these newfound tools to concretize the somewhat abstract mathemati-cal phenomena in a physimathemati-cal example.
Where has this journey along the shoals of the resurgent depths brought us? What are possi-ble next ventures in future research? We briefly touched upon whether or not to ’trust’ the large order relations in section 4.4. This trust becomes truth for all functions which fall in the class of resurgent functions. The resurgent functions can be defined as a class of formal series eφ such that the analytic continuation of the formal Borel transform eφ satisfies a certain condition re-garding the possible singularities [7]. This is a rather vague definition and for practical reasons it would be interesting to prove whether a certain class of functions can fall into this definition all together, because then we would know to what physical problems the large-order relations apply. Understandably, this is a rather ambitious task. Closer to the scope of this report is perhaps the following. As mentioned in section4.3.2, there are such things as two-parameters transseries. In the case where the instanton actions are ±A, like we found for the Painlevé I equation, the two-parameters transseries ansatz is
φ(z, σ1, σ2) = ∞ X n=0 ∞ X m=0 σn1σm2 z−βnme−(n−m)A/z ∞ X g=0 φ(n|m)g zg. (62)
One would expect this ansatz to work, but research has found [1, 5] that when these are studied in the case of Painlevé I, all terms with n ∧ m 6= 0 need logarithmic factors for the solution to be correct. Interestingly, there is as of yet no physical interpretation for these logarithms. For the case of the anharmonic oscillator, one might question what happens when we construct a potential which has instanton actions A1= −A2. Do we indeed need a two-parameters transseries ansatz of
the form φ(z, σ1, σ2) = ∞ X n=0 ∞ X m=0 σ1nσ m 2 e−(n−m)A/zΦ(n|m)(z) , (63) with Φ(n|m)(z) = min(n,m) X k=0 logkz ∞ X g=0 φ(n|m)[k]g zg, (64)
to find correct solutions? We might expect it to be so, because it appears to be a necessity for the Painlevé I problem. But what if we construct a potential such that A1 6= −A2? If we find that
logarithms aren’t needed in this case, then isn’t A1= −A2 just a special case of this? It would be
interesting to know.
Hopefully this report has emphasized the importance and beauty of nonperturbative physics to the undergraduate student. It is an invaluable field of knowledge to the modern theoretical physicist.
Appendices
A
Richardson transform
The Richardson transform is a handy, simple numerical tool to speed up the convergence of a sequence. Its basic principle lies in the following (here, I follow the explanation of [10]).
Consider the sequence
χg= a0+ a1 g + a2 g2 + a3 g3 + O 1 g4 . (65)
In many cases of interest, we can reduce our problem to such a general form. Take for instance our sequence for the extraction of the instanton action in the Painlevé I example (41). We are then interested in extracting the number a0, which we do by taking the N -th Richardson transform of
the sequence χg(see [2] for a derivation of this result).
χ[N ]g ≡ N X k=0 (−1)N −k (g + k) N k!(N − k)!χg+k (66)
The way the Richardson transform works is by removing the next-to leading tail in (65). This takes us closer to the limit because it increases the negative power in g
χ[N ]g = a0+ aN +1 gN +1+ O 1 gN +2 . (67)
There is a limit to how many Richardson transforms can be taken, because if our original list {χg}
References
[1] Inês Aniceto, Ricardo Schiappa, and Marcel Vonk. The resurgence of instantons in string theory. arXiv preprint arXiv:1106.5922, 2011.
[2] Carl M Bender and Steven A Orszag. Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media, 2013.
[3] Carl M Bender and Tai Tsun Wu. Anharmonic Oscillator. Phys. Rev., 184(5):1231–1260, 8 1969. doi: 10.1103/PhysRev.184.1231. URLhttps://link.aps.org/doi/10.1103/PhysRev. 184.1231.
[4] Freeman J Dyson. Divergence of perturbation theory in quantum electrodynamics. Physical Review, 85(4):631, 1952.
[5] Stavros Garoufalidis, Alexander Its, Andrei Kapaev, and Marcos Marino. Asymptotics of the instantons of Painlev{é} I. International Mathematics Research Notices, 2012(3):561–606, 2011.
[6] Marcos Marino. Lectures on non-perturbative effects in large N gauge theories, matrix models and strings. Fortschritte der Physik, 62(5-6):455–540, 2014.
[7] David Sauzin. Resurgent functions and splitting problems. arXiv preprint arXiv:0706.0137, 2007.
[8] Mikhail A Shifman. ITEP lectures in particle physics and field theory, volume 62. World Scientific, 1998.
[9] I M Suslov. Divergent perturbation series. Journal of Experimental and Theoretical Physics, 100(6):1188–1233, 2005.
