Towards resurgence and trans-seriesAsymptotic analysis for mathematical functions and nonperturbative physics

Falk BruckmannU Regensburg, Summer term 2015

Version Monday 27th July, 2015, 11:13

content: from Taylor’s series to Ecale’s resurgence

via: Feynman’s diagrams, Airy’s function, Lefschetz’ thimbles, Borel’s resummation, Stirling’sformula, Stokes’ phenomenon, Laplace’s method

nonperturbative Physics: tunnelling and Planck’s constant, WKB and quantum mechanicalinstantons

more ‘practical’ examples than mathematical rigor

Reviews/Books/Lecture notes

[GD] Gerald Dunne, “Resurgence and Non-Perturbative Physics”, CERN Winter School onSupergravity, Strings and Gauge theory, February 2014,http://indico.cern.ch/getFile.py/access?resId=4&materialId=slides&confId=237741

[rather physical, but not very extensive mathematically]

[CMSO] Carl M. Bender, Steven A. Orszag, “Advanced Mathematical Methods for Scientistsand Engineers: Asymptotic Methods and Perturbation Theory”, Springer, 1999

[extensive for many mathematical functions incl. problems, definitions a bit hidden, WKBin detail]

[RD] R.B. Dingle, “Asymptotic Expansions: Their Derivation and Interpretation”, AcademicPress, 1973

[mathematical, and even philosophical sometimes]

[MRBS] M. Reed, B. Simon, “Methods of Modern Mathematical Physics, vol. IV: Analysisof operators”, Academic Press, 1978

[mathematically precise, mostly about operators and their eigenvalues]

[OC] Ovidiu Costin, “Asymptotics and Borel summability”, Chapman & Hall/CRC, 2009

[mathematically precise]

0-1

[JH] John K. Hunter, “Lecture Notes on Asymptotics”, University of California Davis, 2004,https://www.math.ucdavis.edu/∼hunter/asymptotics/asymptotics.html

[nice basics]

[JZJ] Jean Zinn-Justin, “Perturbation series at large orders in Quantum mechanics and fieldtheories: Application to the problem of resummation”, Phys. Rept. 70 (1981) 109

[the title says it all]

[DD] Daniele Dorigoni, “An Introduction to Resurgence, Trans-Series and Alien Calculus”,2014, http://arxiv.org/abs/1411.3585

[rather advanced, but also discussing a concret example]

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1 Introduction by examples

1.1 Taylor series

real functions for the moment

f(x) vs.∑n=0

cn(x− x0)n with nth derivative cn =f (n)(x0)

n!(1.1)

partial sum/upper bound = N : Taylor polynomial

full sum/upper bound = ∞: Taylor series

in general, Taylor series need not be convergent at all. And in fact the set of functionswith a convergent Taylor series is a meager set in the Frechet space of smooth functions.[wikipedia]

1.1.1 Eight examples

examples around x0 = 0:

ex. (0): f(x) = sin(x), only odd coeffs.: c2n+1 = (−1)n 1(2n+1)! (from exp-function)

converges to the function everywhere

ex. (1): f(x) = Ai(x), the Airy function

converges everywhere, too

interesting features: oscillating vs. exp. decay, asymptotic expansion (aroundinfinity) in Sec. 1.5.5

ex. (2): f(x) = arctan(x), only odd coeffs.: c2n+1 = (−1)n 12n+1 , convergent up to ±1

ex. (3): f(x) = 11+x2 = arctan′(x), only even coeffs.: c2n = (−1)n (two geometric series)

ex. (4): f(x) = 11±x , cn = (∓1)n, geometric series, both the same limited convergence

ex. (5): f(x) = Γ(x) around x0 = 1, convergence limited to |x− x0| < 1

here this is obviously due to a pole at x = 0

asymptotic expansion of log n! = log Γ(n+ 1) will be discussed in Sec. 1.3

ex. (6): f(x) = e−1/x, all coefficients vanish

e.g. f (1)(x) = −1x2 e−1/x x→0−→ 0 (since exponential is stronger in y2e−y

y→∞−→ 0)

Taylor series converges everywhere, but to another function! ‘nonper-turbative’

quantum mechanical tunnelling is of this form (in ~), see Sec. 1.2

related: y = f(x) solves the differential equation y′ − y/x2 = 0, plugging a Taylorseries ansatz for y into it yields y ≡ 0. “This is not progress.” [CMSO, p. 68]

1-1

ex. (7): f(x) = e−1/x2, same

note that around x0 = 1/2 the series converges between 0 and 1, ‘it feels thatsomething is going on at x = 0’

radius of convergence:

ratio test:∑

n bn converges, if limn→∞

∣∣∣ bn+1

bn

∣∣∣ < 1

radius of convergence: series converges for |x− x0| < r with r := limn→∞

∣∣∣ cncn+1

∣∣∣look back: (0): (2n+3)!

(2n+1)! = O(n2)→∞ = r X

(2): 2n+32n+1 = 1 +O(1/n)→ 1 = r X

(3,4): 1→ 1 = r X

(6,7): Taylor series ≡ 0: r =∞

these are examples for real functions that are smooth (= infinitely differentiable C∞),but not analytic (= locally given by a convergent power series)

in complex analysis such difference does not exist! (the converse always holds)

ex. (8): f(x) =∫∞

0 dt e−t

1+x2t= − e1/x

2Ei(−1/x2)x2 even has r = 0 [4, ex. (i)]

since c2n = (−1)nn! (f (n) even has one more factorial) and n!(n+1)! = 1

n+1 → 0 = r X

1.1.2 Going complex

some radii of convergence clearer from singularity structure in the complex plane

ex. (0): sin(z) has no poles (apart from ∞) ⇒ r =∞ X

an ‘entire’ function = holomorphic1 over the whole complex plane

ex. (3,4): 11+z2 and 1

1±z have poles at z = ±i resp. z = ∓1 ⇒ r = 1 X

ex. (2): arctan(z) = i2 log 1−iz

1+iz has cuts at z = ±i · [1,∞) (z = ±i are branch points)⇒ r = 1 X

ex. (7): e−1/z2has an infinite Laurent series: e(−1/z2) =

∑∞n=0

1n!

(− 1z2

)n ⇒ essential singularity

(alternative definition: if the singularity is neither a pole nor a removable singularity)

exp-function has an essential singularity at ∞ [a way to construct such functions]

different limits from different directions

limr→0 e− 1r2 = limy→∞ e

−y2= 0

limr→0 e− 1

(ir)2 = limy→∞ e+y2

=∞

1(once?) differentiable in its domain

1-2

limr→0 e− 1

(√ir)2 = limr→0

(cos( 1

r2 ) + i sin( 1r2 ))

not defined

oscillating between −1 and 1

Picard’s great theorem: in every neighborhood of an essential singularity, the functiontakes on every complex value, except possibly one, infinitely many times

outlook to trans-series: include functions of the form e−1/z2(as well as logarithms) to

define a more general series expansion

1.2 Tunnelling in Quantum mechanics

torwards example (6)

rectangular barrier V = V0 for x ∈ [−X/2, X/2], 0 elsewhere

wave functions ψ(x) are eigenfunctions of the hermitian Hamilton operator − ~2

2m∂2

∂x2 +V (x)· to eigenvalue (energy) E

take 0 < E < V0 and use a wave ansatz:

ψ(x) =

1 · eikx + αe−ikx for x < −X/2 k =

√2mE~ > 0

β+eκx + β−e

−κx for −X/2 < x < X/2 κ =

√2m(V0−E)

~ > 0

γeikx + 0 · e−ikx for x > X/2

(1.2)

‘1’ and ‘0’ are conventions and the other amplitudes are related to the reflection andtransmission coefficient:

R = |α|2 T = |γ|2 (1.3)

continuity and differentiability of ψ(x) at x = ±X/2 yields some T (E;V0, X,m), whichfor E V0 – i.e. large κ – reduces to

T ∝ e−2κX = e−positive

~ (1.4)

(for Physicists: in a slowly varying potential one has T ∝ e−2∫barrier κ(x)dx from WKB)

recall example (6): T has a vanishing Taylor series in small ~

⇒ if one views classical Physics as Quantum mechanics at vanishing Planck constant ~ = 0– where, e.g., position and momentum, whose commutator is i~, now commute – then thetunnelling effect cannot be obtained in a perturbative expansion in ~ around that limit

in the following we will typically expand around z =∞, so think of z = 1/~

1.3 Stirling’s formula (Gamma function)

consider x! = Γ(x + 1) with positive integer x for the moment (the perhaps more suitablevariable n will be our summation variable), needed for combinatorics

1-3

integral representation:

Γ(x) ≡∫ ∞

0dt e−t tx−1 (1.5)

1.3.1 Raw asymptotics from Laplace method

Stirling’s formula = approximation for factorials of large argument

x! ∼(xe

)x√2πx , log x! ∼ x log x− x+

1

2log(2πx) (1.6)

in the sense that limx→∞

lhsrhs = 1 (cf. the definitions in Footnote 26 later)

for later:

Γ(x) ∼(x− 1

e

)x−1√2π(x− 1) =

(x− 1

e

)xe

√2π

x− 1use x log(x− 1) ' x log x− 1

∼(xe

)x√2π

x(1.7)

derivation from the integral representation (1.5), in which we substitute t = sx (forx > 0), [CMSO, p. 275]

Γ(x) =

∫ ∞0dsx e

−sx sxxx

sx= xx

∫ ∞0ds e−x(s−log s) 1

s(1.8)

now use Laplace’s method (see Sec. 3.2 later)

for large x everything in the integrand away from the minimum of the ‘action’ (s− log s)is strongly suppressed

⇒ approximate the action up to second order around the minimum s = 1: s− log s ≈1 + (s−1)2

2 , which gives rise to a gaussian integral, when extending the s-integral range,the error of which is also small

⇒ take the slowly varying part of the integrand, 1s , at the minimum point s = 1,

Γ(x) ∼ xxe−x√

2π

xX (1.9)

(in path integrals, this is the semiclassical expansion)

1.3.2 Improved asymptotics and its limitations

define the remainder in the log-formula

Φ(x) = log x!−[x log x− x+

1

2log(2πx)

], lim

x→∞Φ(x) = 0 (1.10)

we will attempt to improve on it by inverse powers of x, by inspecting Fig. 1 one couldguess the first few terms

1-4

5 10 15 20

0.01

0.02

0.03

0.04 5 10 15 20

-0.0001

-0.00008

-0.00006

-0.00004

-0.00002

Fig. 1: Improving Stirling’s formula. Left: Φ(x) from (1.10) (dots) and 1/(12x), right: Φ(x)−1/(12x) and −1/(360x3). Note the different scales: approximation becomes better!

for Φ(x) one has the following asymptotic series (see App. 7.1)

Φ(x) =∞∑n=1

B2n

2n(2n− 1)x−2n+1 , with cut-off: ΦN (x) ≡

N∑n=1

B2n

2n(2n− 1)x−2n+1

(1.11)

where B.. are the Bernoulli numbers, the even ones (starting with zero index) are1, 1/6, −1/30, . . . , whereas there is only one nontrivial odd number, B1 = −1/2;relation to Riemann’s zeta function: Bn = −n ζ(1− n) for n ≥ 2

the Bernoulli numbers explode stronger than x2n and 2n(2n− 1) (similar to Γ)

B2n ∼ (−1)n+14√πn( nπe

)2n∼ (−1)n+12

(2n)!

(2π)2n(1.12)

e.g. B22 = 6192.12, and so the series is not converging

note that derivative or integral wrt. x in (1.11) does not cure such kind of divergence

this will be an example of an “asymptotic series”

thus one cannot get Φ(x) with arbitrary precision from the series, cf. Fig. 2

e.g. for x = 4 one can achieve an absolute precision of 9.5 · 10−13 with 13 terms and forx = 6 one can achieve 2.7 · 10−18 with 19 terms

we have expanded the Gamma function (its logarithm) around infinity, where it has anessential singularity, actually a limit of poles at nonpositive integers

can we estimate the maximal precision and at which term it is achieved?

minimum of term in the series wrt. the ordinal number n (extended to be noninteger):∂∂n |.. x

−2n+1|=0

using the asymptotics of the Bernoulli numbers, (1.12), plus the fact that√n and

2n(2n− 1) are subleading:

x∂

∂n

[( nπe

)2nx−2n

]= 0

∂

∂n

[2n log

( n

πex

) ]= 0 n = πx (1.13)

1-5

1 2 3 4 5

10-8

100

1012

1022

1032

5 10 15 20 25 30

10-18

10-16

10-14

10-12

10-10

10-8

10-6

Fig. 2: How the asymptotic series Φ(x), (1.11), starts to ‘fail’. Left: Φ(x)− ΦN=30(x) (dots)and the next term/correction with n = 31, which agree pretty well, but are not smallfor x ≤ 3. Right: the difference |Φ(x) − ΦN (x)| for fixed arguments x = 4, 6 (red,green) as a function of the cut-off N .

-30

-25

-20

-15

-10

-5

0 20 40 60 80 100

2

4

6

8

10

12

N

x

Fig. 3: Summary of the error (log10) of the asymptotic series for Φ (the region not plottedhas even larger errors). For a convergent series, the error should decrease with cut-offN , i.e. horizontally: NO! But the error decreases vertically, i.e. for x → ∞. If thishappens in a particular manner, one has an asymptotic series.

x-dependent, agrees with the numbers cited above, valley in Fig. 3 ‘looks linear’ indeed

value at this n:

B2n

2n(2n− 1)x−2n+1

∣∣∣n=πx

'√π

xe−2πx =

1 · 10−11 x = 4

3 · 10−17 x = 6(1.14)

roughly agrees with the errors cited above (now including the subleading terms)

exponentially small in x, 23.4.15

1-6

summary

log x! = log Γ(x+ 1) ∼ x log x− x+1

2log(2πx) + Φ(x) Φ(x) from (1.11) (1.15)

log Γ(x) ∼ x log x− x+1

2log(

2π

x) + Φ(x) (1.16)

where the second line [2] follows trivially from the first line and (F) below

Φ(x) can be represented as a series, Eq. (1.11), that does not converge, but when stoppedat its smallest term, produces results up to exponentially small errors for large and evenmoderate x, cf. Fig. 2 right panel and the error estimate above

recall that with a Taylor series around some finite x0 we have no chance of computingΓ(x) beyond x = 2x0: as we have visualized in Sec. 1.1, the Taylor series does notconverge anymore because of the pole of Γ at 0

even if the Taylor series converges – as is the case for the next two example functions– it is typically very expensive to obtain from that the function at large arguments tosome reasonable precision

1.3.3 Going complex: a puzzle

the Γ-function has poles at non-positive integers (incl. zero) and characteristic relations:

(F ) Γ(z + 1) = Γ(z) · z , log z = log Γ(z + 1)− log Γ(z) (1.17)

(R) Γ(1 + z) · Γ(1− z) =πz

sin(πz), log

πz

sin(πz)= log Γ(z + 1) + log Γ(−z + 1) (1.18)

which hold for complex argument z and follow from the factorial and Euler’s reflectionformula2

how does this compare to the ‘raw asymptotics’ log Γ(z + 1) = z log z − z + 12 log(2πz)

(without Φ for the moment), at imaginary z = iy?

(same argument in [GD, 49] for the derivative of log Γ called Ψ)

first relation

(F ) log(iy)?= log Γ(iy + 1)− log Γ(iy + 1)

∣∣y=y+iiy=iy−1

(1.19)

iπ

2+ log y

?= (iy) log(iy)− (iy) +

1

2log(2π(iy))

−[(iy − 1) log(iy − 1)− (iy − 1) +

1

2log(2π(iy − 1)

](1.20)

large y= i

π

2+ log y − 1

12y2+

i

12y3+

3

40y4+ . . . (1.21)

2Other variants of the reflection formula are equivalent: Γ(1− z)Γ(z) = π/ sin(πz) [wikipedia] and Γ(−z) =− πz sin(πz)

1Γ(z)

[2, typo in (3)!]. How to obtain: write the two Γs with (1.5) as integrals over t and s and

introduce new integration variables x = t + s and y = t/s, both range from 0 to ∞. The x-integral istrivial, the y-integral can be done [mathematica] and yields the rhs. for (R) for −1 < Re z < 1.

1-7

the algebraically small terms − 112y2 + i

12y3 + 340y4 + . . . have to be canceled from Φ(iy)−

Φ(iy) = Φ(iy)−Φ(iy− 1), which according to (1.15) has to be added on the right handside

this can be checked easily by expanding (iy − 1)−2n+1 in (1.11) (argument shift: finitenumber of summands there for each term here)

second relation

(R) logπ(iy)

sin(πiy)

?= log Γ(iy + 1) + log Γ(iy + 1)

∣∣y=−yiy=−iy

(1.22)

−πy + log(2πy)− log(1− e−2πy)?= −πy + log(2πy) (1.23)

the exponentially small terms − log(1−e−2πy) =∑∞

j=1e−2πjy

j have to come from Φ(iy)+Φ(iy) = Φ(iy) + Φ(−iy), which according to (1.15) has to be added on the right handside

however, Φ(iy) is formally an odd function, since its expansion (1.11) only has oddpowers; this implies Φ(iy) + Φ(−iy) = 0

‘something has to happen’ to the asymptotic expansion (1.15) in the complex plane [2]

for later keep in mind that along the real axis the series was alternating (since the B’sare), whereas at the imaginary axis we are looking at now the series has the same sign

1.4 The error function

start with real argument, needed for statistics

1.4.1 Definition and asymptotic series

incomplete Gaussian integration

erf(x) =2√π

∫ x

0dt e−t

2: −1→ 0→ 1 (odd) (1.24)

erfc(x) = 1− erf(x) =2√π

∫ ∞xdt e−t

2: 2→ 1→ 0 (1.25)

positive x: variable substitution s = t2 − x2 (not s2 = t2 − x2) [RD, first section]

now s ∈ [0,∞) ‘at the expense of’ introducing a non-rational integrand (from Jacobian)

erfc(x) =e−x

2

√π x

∫ ∞0ds e−s

1√1 + s

x2

(1.26)

expand the latter around the maximum of the integrand, i.e. for small s, “large s-contributions are cut-off by e−s anyway”

1√1 + s

x2

=1√π

∞∑n=0

(−1)nΓ(n+ 1/2)

Γ(n+ 1)

( sx2

)n(1.27)

1-8

now the s-integral cancels Γ(n+ 1) and one is left with (back to erf)

erf(x) = 1− e−x2

πx

∞∑n=0

(−1)n Γ(n+ 1/2)x−2n (1.28)

a series with exploding coefficients, similar to the Bernoulli numbers (1.12) in the pre-vious example, but alternating; limited precision, an error plot like Fig. 3 can easily beproduced (but does not look linear)

so what does ‘=’ actually mean? rhs. is not the definition, but the integral above; latera precise definition erf(x) ∼ rhs. ‘asymptotic to’ (cf. Eq. (2.5))

what have we done?

in going from (1.26) to (1.28), we have exchanged integration and limit (in the sum’scut-off)

this can be done if the sequence of functions fN (s) = e−s∑N

n=0 . . . sn converges uni-

formly3 to f(s) = e−s√

1 + s/x2 (for every fixed x), i.e. if (for all ε exists N0 such that)|fN (s)− f(s)| < ε is valid for all N > N0 and for all s

our sum (1.27) has radius of convergence 1 ← Γ(n+1/2)Γ(n+1)

/Γ(n+3/2)Γ(n+2) ∼

√n+ 1/

√n and

thus we can represent 1/√

1 + s/x2 by this sum only for s < x2, but we used it also for(analytically continued it to) s > x2 appearing in the integral; the factor e−s attenuatesthis problem, but ultimately the series diverges for all x [RD, 3]

let’s go on nonetheless

negative x similarly (odd function!)

erf(x) = −1− e−x2

πx

∞∑n=0

(−1)n Γ(n+ 1/2)x−2n (1.29)

two ‘parts’: the second part is of the same functional form as for positive x, since it ismanifestly odd (f(x2)/x)

the first term, an ‘odd constant’, should be read as sign x, which is non-continuous

for large argument the first term dominates, while the second term is exponentiallysmall

‘... exponentially small quantities, which are frequently negligible (and more frequentlyneglected)’ [1]

for real argument the two asymptotic regions are disconnected, so no problem with thediscontinuity in the leading sign(x), but think of connecting them on an arc in complexspace ...

( z|z| which on an arc takes values 1→ i→ −1→ . . . is not holomorphic because of z)

(cf. other functions that interpolate between −1 and +14)

3the interchangeability of integration and limit is then guaranteed for a compact s-range4 tanh(x): inf. many poles on the imaginary axis (2π-periodic); only exponential corrections for large x:

1-9

1.4.2 Going complex: the Stokes phenomenon

the integral definition of erf(x), (1.24), applies to all complex arguments z and resultsin an entire function (its Taylor series around z = 0 is absolutely convergent [RD, (2)])

as it turns out, the asymptotic behavior for erf(z), (1.28), is valid for all Re z > 0

likewise, (1.29) is valid for all | arg z| > π/2

for purely imaginary argument5, the following trick applies [RD], say positive imaginary

erf(iy) =iey

2

√π y

∫ y2

0ds e−s

1√1− s

y2

s > y2 : integrand imaginary (1.30)

=iey

2

√π y

Re

∫ ∞0ds e−s

1√1− s

y2

(1.26) above with x2 = −y2 (1.31)

=iey

2

π yRe

∞∑n=0

Γ(n+ 1/2) y−2n (for erf not erfc!) (1.32)

the function is thus exponentially growing and the corrections to it are given by anonalternating series

it does not contain a unity: it must be purely imaginary on general grounds

(should it have chosen +1 or −1?)

the result for negative imaginary argument is such that one can summarize for all z

erf(z) =

−1 Re z > 0

−0 Re z = 0

−1 Re z < 0

− e−z2

πz

∞∑n=0

(−1)n Γ(n+ 1/2) z−2n (1.33)

note that the leading factor (n = 0) is Γ(1/2) =√π, which some authors factor out

from one side of the imaginary axis to the opposite, the unity changes sign, but is atthe same time the subleading part there

a subleading term or ‘associated function’ [RD], here a constant, is ‘born’6 or changed atsuch a Stokes ray or Stokes line7, where the other function is maximally dominant(peak exponential dominant [RD])

here: imaginary axis

−1 + 21+e−2x = 1− 2e−2x +O(e−4x), seems to mean inf. many subleading terms

2π

arctan(x): cuts at 0 ± i[1,∞), where the real part jumps by 2, so this discontinuity is a feature of theoriginal function; expansion for large x: 1 + 2/π ·

∑l(−1)n 1

2n+11

x2n+1 , coefficients not exploding5yet another name: erfi(z) = −i erf(iz)6consider erfc where −2 is born7there are different conventions for Stokes and antiStokes lines, we follow [RD, GD, OC], being opposite to

[CMSO]

1-10

think of it as the ray = direction in which infinity is approached (infinity as a point onthe complexified sphere: rays meet), (anti-)‘Stokes lines are the asymptotes as z → z0

of the curves . . . ’ [CMSO, 116]

it can equally be characterized by the ‘late’ coefficients of the asymptotic series being‘of uniform sign’ [RD] yielding a nonalternating8 series [GD], cf. (1.32)

the coefficient in front of the leading part is analytic across the Stokes line

the other term that appears/jumps is (the asymptotic expansion of) the other linearlyindependent solution of a second order homogeneous differential equation, if the originalfunction obeys one

the error function solves (y′)′ = −2z(y′) (since y′ is a Gaussian) and so does a constant9

the real axis also constitutes a Stokes line, since now ±1 is maximally dominant

±1, however, is not an asymptotic expansion and it does not give birth to a change inthe associated function (exp(−z2)/z) across it

in between Stokes lines, the two terms must exchange the ‘leadingness’/dominance

this gives antiStokes lines

here: e−z2

= e−Re z2 ·phase|z|→∞≷≷ 1 for Re z2 ≶ 0, we have Re z2 = 0 at | arg z| = π4 ,

3π4

thus the diagonals | arg z| = π4 and | arg z| = 3π

4 are antiStokes lines10

one can write the function and the associated function – i.e. the two independent solu-tions of the corresponding differential equation – as f1(z) ∼ eS1(z) and f2(z) ∼ eS2(z) (atypical ansatz, cf. (1.78) below)

here S1(z) = 0 and S2(z) = −z2

the functions multiplying that behavior are slowly varying [3] (and not exponential ina large parameter)

then the following relations hold

Stokes line Im S1 = Im S2 (1.34)

antiStokes line Re S1 = Re S2 (1.35)

here 0 = Im (−(±iy)2) for real y and 0 = Re (−(e±iπ/4r)2) for real r

where is all that routed in?

8most of the time we deal with functions that are real for real argument, so the coefficients are real9and so does the term exp(−z2)/z·sum, e.g. expand it to some finite order in z, the differential equation is

fulfilled up to even higher order [mathematica]10open questions that should not have escaped the critical reader’s attention: beyond the antiStokes line, i.e.

for | arg z| > π/4, the term 1 is subleading to an infinite sum ez2

/z, ez2

/z3 etc., that is only asymptotic (incontrast to the leading ±1 at the real axis); so why keep the 1?? what would be worse (asymptotically!)without it? is it visible after having used the best approximation to the function we can get from theseries (the term with index n = z2 contributes the least)? likewise, why does the 1 disappear just at theimaginary axis,

1-11

[CMSO, 117]: “The Stokes phenomenon is not an intrinsic property of a function [likeAi(z)], but rather is a property of the functions that are used to approximate it. . . . Anasymptotic relation like Ai(z) ∼ Ai(z) (z → ∞) does not exhibit the Stokes phe-nomenon, . . . contains no useful information.”

well, the Stokes phenomenon only occurs at essential singularities of a function

now comes the clou: the asymptotic series – if worst, i.e. nonalternating and on the Stokesline – contains interesting analytical information

1.4.3 A magic computation

what is the formal series (1.33) teaching us?

“to make sense of this series”, we parametrise the Γ-growth of the coefficients in theseries by the integral representation of Γ

−e−z2

πz

∞∑n=0

(−1)n Γ(n+ 1/2) z−2n ≡ Φ(z) = −e−z2

πz

∞∑n=0

(−1)n∫ ∞

0dt e−t tn−1/2 1

(z2)n

(1.36)

price: new variable t including an integral: Borel transform, see Sec. 3

interchange integral and sum again (as when deriving the asymptotics of erf) and addup the geometric series

Φ(z) = −e−z2

πz

∫ ∞0

dt e−tt−1/2 1

1 + tz2

(1.37)

for real z = x (and in its vicinity) this integral is well-behaved11 and yields [mathemat-ica] −sign(x) + erf(x)

so we have made sense of this non-convergent series – via Borel resummation – andrecovered the original error function by virtue of (1.33)

imagine not this function, but only the series was given to us . . .

even more interesting: for z2 < 0, i.e. on the imaginary axis – where the series isnonalternating, so ‘worst’ – the integrand develops a pole at t = −z2

one distinguishes arguments left and right of the axis (similar to propagators)

z = ±ir(1 + iε/2) , z2 = −r2(1 + iε) , r > 0, ε > 0 (1.38)

for ε → 0 the pole in (1.37), at t = r2(1 + iε) walks through the positive real axis,therefore for ε ≷ 0 the values of Φ(z) differ by the residue

rest=r2

1

1 + t−r2

= −r2, total res = −e−z2

πz

∣∣∣z=±ir

× (e−tt−1/2)∣∣t=r2(−r2) =

1

π(∓i)

(1.39)

111/√t is harmless at t = 0

1-12

correct sign: for ε > 0 lift the contour above the pole, after which ε can be madenegative smoothly, and compensate by adding a small counterclock integral around thepole, which is given by 2πi · res, thus

limε→0

[Φ(±ir(1 + iε))− Φ(±ir(1− iε))

]= 2(±1) ε > 0 (1.40)

on the positive imaginary axis (upper sign), the function Φ left of the imag. axis is by2 larger than on the right of the imag. axis (on the negative imaginary axis the jumpis just the opposite)

the constant part in (1.33) is just the opposite, so that the resulting error function issmooth across the imaginary axis and given by the smooth part of Φ (the principalvalue of the integral [RD, 407, 413])

in other words, the asymptotic expansion of the exponentially dominating part aroundthe Stokes ray (ehere the imaginary axis) – where it dominates strongest – give usinformation on the subleading term/associated function

what is more, the leading term and its asymptotics are purely imaginary (1/z infront of the sum), whereas the term born is real, in general out of phase by π/2

and since the series (1.36) is around infinity, it is in powers z−2n, but it gives birth to12

ez2z, which cannot be obtained by a Taylor series expansion around infinity itself (like

e−1/x2around x = 0), it is ‘unnatural’ or nonperturbative

of course, all this rests on the fact that z = ∞ is an essential singularity of the errorfunction (since neither the limit of erf(z) nor of 1/erf(z) exists [wikipedia], cf. along pos.and neg. real axis) 30.4.15

‘The series is divergent; therefore we might be able to do something with it.’ O. Heavisideaccording to [GD]

1.5 The Airy function

example (1), start with real argument and a physical motivation from (wave) optics

1.5.1 Rainbows

come about by light being refracted [gebrochen] upon entering a water drop, then re-flected – say m times – within and finally refracted again upon leaving: see Fig. 4

this applies to every angle, but the relation between incoming and outgoing angle isnonlinear, we follow [5]

let n be the refraction index of water (n ≈ 4/3 in the visible), let the incident angle (tothe surface normal) have sin(i) = x, then the total angle θ, by which the light changes

12from e−tt−1/2r2 with t = −z2 = r2 (1.39), the whole associated function is constant due to the prefactor of

the series; note that in this complex direction ez2

= e−r2

is actually exp. small

1-13

Fig. 4: Formation of the (primary and secondary, m = 1, 2) rainbow according to R. Descartes[1637, wikipedia].

its direction is, cf. Fig. 5 left panel

θm = 2(

arcsin(x)− arcsin(x/n))

+m(π − 2 arcsin(x/n)

)(1.41)

if for several angles x (dx 6= 0) the same θ (dθ = 0), then the light is amplified: atextrema dθ/dx = 0

the minimum for the primary rainbow with m = 1 occurs at θ = 138, no angles θavailable below: caustic, Alexander’s dark space (seen from earth at 42)

the secondary rainbow with m = 2 has its extremum at smaller θ (seen in the sky abovethe primary one)

we will neglect the genuine beauty of rainbows: that the refraction index depends onthe color (= dispersion) and so do the rainbow angles

in wave optics, two or more rays in the same direction θ interfere and only amplify, iffthe difference in the accumulated phase is a multiple of 2π

; modulation on the ray optics amplification profile giving rise to superluminary[uberzahlig] rainbows

Huygens principle: the final amplitude of a wave can be obtained from superposingelementary waves incl. their phases and a decay with the inverse distance (mathemati-cally: Greens function as integral kernel etc.), we will consider the light waves as scalars(gauge potentials!)

example: 1d slit for observers far away at an angle α (Fraunhofer diffraction [Beugung])

amplitude ∝∫

slit dx eiαx×wave number k ∝ sin(α k

2×slit width)

αk ,

intensity ∝ . . .2: maxima and minima, decaying

1-14

Fig. 5: Some geometric details for the derivation of superluminary rainbows (left) and theintensity resulting from the Airy integral (1.43) (right), from [5].

here we have assumed that the slit is lit by light with a plane wave front

in general the diffraction pattern is the Fourier transform of the phase at the surface

for the rainbow: on a conveniently chosen surface (BB′ in Fig. 5 right panel) the phase

φ(x) = 4πradius

wave length

[1−

√1− x2 + 2

√n2 − x2

](1.42)

can be derived to be approximately cubic in the difference between x and x0 (recallthese are sines of the incoming rays), and eventually [Airy, 1836]

amplitude ∝∫ ∞−∞

dt ei(t3/3−ηt) (1.43)

where t ∝ (x−x0), η ∝ (θ−θ0) and we have extended the t integration range to infinity(which is beyond the validity range of our approximations)

result: maxima and minima in the ray optics-allowed region (inside the rainbow),cf. Fig. 5: superluminary rainbows, weakly decaying intensity

exp. decay into the so far empty region (above the rainbow), a typical wave phenomenonsmoothing the caustic

1.5.2 Definition and some basic properties

integral representation (in common with Gamma and error function):

Ai(x) :=1

2π

∫ ∞−∞

dt ei(t3

3+xt) (1.44)

integrand strongly oscillating13: Fig. 6

13still integrable, areas added become smaller in width, but wouldn’t be enough if of same sign, cf.∫| cos(..)|

1-15

-5 5

-0.15

-0.10

-0.05

0.05

0.10

0.15

-5 5

-0.15

-0.10

-0.05

0.05

0.10

0.15

Fig. 6: Real and imag. part of the Airy integrand exp(i(t3/3 + xt)) for x = 1/100. Note thatthe absolute value of the integrand is identical to 1/2π, ‘only’ the phase varies (cf. thehorizontal axis in Fig. 8), but very rapidly.

sign problem when trying to integrate

especially for large x, where the integral is actually exp. small (see below)

differential equation:

differentiate the integrand twice wrt. x, represent the occuring t2 as t-derivative of thefirst part, integration by parts on the second part pulls down an x

y′′ − xy = 0 (1.45)

important for the analysis of diff.eqns. like y′′+ g(x)y = 0 around a simple zero of g(x)

real function: one can also integrate over the even part cos( t3

3 + xt), which is real; thediff.eq. is real as well

-30 -25 -20 -15 -10 -5 5

-0.4

-0.2

0.2

0.4

Fig. 7: The Airy function Ai(x) for real arguments.

asymptotics, cf. Fig. 7

large x: exponentially decaying

1-16

small x: oscillating, i.e. two exp’s with imag. argument

Ai(x)→

exp

(− 2

3 x3/2)

x→∞

2 sin(

23(−x)3/2 + π

4

)x→ −∞

1

2√π|x|1/4

(1.46)

how to reconcile two vs. one exponential in different asymptotics?? [Stokes]

again not connected in real space, but we’ll go complex later

Quantum mechanics:

diff.eq. amounts to Schrodinger eq.

− . . . ψ′′ + (V − E)ψ = 0 with V − E ∝ x (1.47)

linear(ly approximated) potential shoulder = constant force, ‘crossed’ by energy

exp. decay and oscillating behavior known from potential step at some x0, where:

ψ(x) =

eikx + αe−ikx x < x0

γe−κx x > x0

(1.48)

similar to near barrier, Sec. 1.2

1.5.3 Saddle points and steepest descent integration

can we find a better integral representation for Ai(x), in complex t-space?

varying the contour in the bulk . . . : allowed since no poles (exp of polynomial)

. . . and at infinity: contribution vanishes, iff integrand becomes small there

xt always negligible compared to t3/3

eit3/3 is exponentially small, iff

Re (ie3i arg t/3) < 0 sin(3 arg t) > 0 arg t ∈ [0,π

3] ∪ [

2π

3, π] ∪ [

4π

3,5π

3] (1.49)

3 ‘good’ wedges/sectors in complex t-space touching the real line used so far

alternative contour:

can we avoid cancellation in the integrand by demanding a stationary phase on thenew contour or since Ai(x) as integral is real and positive for x > 0, can we make theintegrand real and positive as well? (*)

x > 0, along some t-contour want vanishing phase:

Re( t3

3+ xt

)= 0 (Im t)2 =

(Re t)2

3+ x or Re t = 0 (1.50)

1-17

two hyperbolas through the points 0 ± i√x, cf. Fig. 8, and purely imaginary t’s, but

these are not connected to the original contour

indeed the upper hyperbola starts and ends in one of the ‘good’ wedges from above14

for any phase close to zero the curve runs into ‘bad’ wedges, see Fig. 8; so do contoursof phase ±2π, ±4π etc.

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

Fig. 8: The integrand of the Airy function for x = 1. Black: absolute value (logarithmically)with two saddle points, lightblue: the ‘good’ wedges of (1.49) where the integrand isasymptotically small, red: contour lines with phases Re (t3/3 + xt) = 0, ±2π, ±4π,light red: two phases near 0. Can you read off which saddle is higher? (seen from theupper saddle the value at the origin is large, from the lower saddle it is small⇒ uppersaddle is smaller [although it is in the large wedge!])

saddles

(upper) hyperbola contour passes through the extremal point 0 + i√x of the abs. value

why is this a saddle? |eig(t)| = e−Im g(t), the imaginary (and real) part of an analyticfunction is harmonic, i.e. ∂2

aIm g(a+ib) = −∂2b Im g(a+ib) (and ∂a,bIm g(a+ib) vanishes

at the saddle), convex vs. concave ⇒ saddle

even more: constant phase contours perpendicular to constant absolute value contours15

and therefore follow steepest descent (and steepest ascent); at the saddles two phasecontours meet and along ‘the correctly chosen contour’ the absolute value decreases

some saddles are called ‘unstable’ (here integrating along the imaginary line) or ‘notcontributing’ (saddle at 0− i

√x)

the latter is relevant for complex arguments of the Airy function, see Sec. 1.5.4 below,

14its asymptotics is Im t∝|Re t|/√

3, while the wedges are bounded by the angle π/3 with slope tan(π/3)=√

315from Cauchy-Riemann: ∇tIm g ⊥ ∇tRe g and both are perpendicular to the .. constant contours

1-18

and the other solution Bi of the diff. eq., see Sec. 1.5.6 below

explicitly: if parametrized by s = Re t:

Ai(x)x>0=

1

2π

∫ ∞−∞

ds(1 +

i2s/3

2√s2/3 + x

) exp(− 2

9(4s2 + 3x)

√s2/3 + x

)(1.51)

Gaussian-like and thus amenable to numerics: ‘sign problem’ solved

not found in integral databases (!), but after relating the Airy function to the modifiedBessel function16:

Ai(x) =1

π

√x

3K1/3

(2

3· x3/2

)(1.52)

I found that integral representation in wikipedia

Jacobian: dtds = dRe t

ds + idIm tds

can change the complex integral away from the phase defining the contour (!), so theintegral is not purely positive (∗), a sign problem remains!?

here: 1 + i odd(s), does not contribute to the integral (we know the result is real)

from that integral one can obtain the asymptotics for large pos. x, similar to the Laplacemethod17:

Ai(x)x>0=

1

π

∫ ∞0ds exp

(− 2

3x3/2 (1 +

4s2

3x)

√1 +

s2

3x︸ ︷︷ ︸min. at s = 0 (x > 0)

)

=1

π

∫ ∞0ds exp

(− 2

3x3/2

(1 +

3s2

2x+O(s4/x2)

))→ 1

πexp

(− 2

3x3/2

) ∫ ∞0ds e−

√xs2 = exp

(− 2

3x3/2

) 1

2√π x1/4

(1.53)

keeping the next term 5s4/24x2 in the exponent, the integral gets multiplied by 1− 548x3/2 ,

which we will recover later in Sec. 1.5.5, but the next-to-next term −7s6/432x3 isnegative, such that the s-integral does not converge (!)

saddle points for x < 0? for that we are now . . .

1.5.4 Going complex

Airy-argument x replaced by z: entire function

essential singularity at ∞ (different limits when going there in different directions, cf.(1.46) and Fig. 10), ∞ is an ‘irregular singular point’ of the differential equation (1.45)[CMSO, 62-64]

16actually this identification holds for | arg x| < 2π/3, across that line K is not smooth, but Ai is17the Laplace method itself does not work, since (1 + 4s2/3x)

√1 + s2/3x is minimal at s2/x = −9/4, thus

outside of the s-integration range (x > 0)

1-19

integral definition: choose a contour close to the real axis18, see App. 7.2, i.e. arg t = εand arg t = π − ε ⇒ integral converges for all z

same wedges (zt negligible wrt. t3/3)

watch how the saddle points and contours change with z

let’s take z = reiφ with φ ∈ [0, π], don’t mix up with the phase of integrand ; contours

the saddle points rotate counterclockwise – just like z – with half the velocity of z

they are at ±i eiφ/2√r = ±

√−z, value at the saddle points:

exp(∓ 2

3r3/2 cos

3φ

2

)(1.54)

; value and potential dominance of saddle point integration

phase contours change as well

phase at the saddle points: ∓ 23 r

3/2 sin 3φ2

for r = 1 see Fig. 9 7.5.15

before φ = 2π/3

contours deformed, phases at saddles: 0→ − 23r

3/2 (green in the figure (r = 1)) → 0 X

Ai as the integral is complex, its value starts to grow exponentially for φ > π/3, cf. (1.54)and (1.55) and Fig. 10 below

at φ = 2π/3

the landscape is simply rotated19 from φ = 0, but the wedges did not rotate ⇒ choosea different contour

on vanishing phase: from the left wegde to first saddle, then from the first to the secondsaddle and then from the second saddle to the right wedge (small saddle 1 saddle2 further at the perp. contour)

again, all other constant phase contours start/end in ‘bad’ wedges

for the first two parts we integrate along t = e−iπ/6s and Jacobian: dtds = e−iπ/6; second

part exp. small (cf. the value at its saddle): Jacobian of the first part gives the phaseof Ai for large |z|

plausible asymptotics: Ai(re2πi/3)r→∞−→ e−iπ/6 exp(

2

3r3/2)︸ ︷︷ ︸

saddle value

1

2√πr1/4︸ ︷︷ ︸

cf. Gaussin int. (1.53)

(see later)

after φ = 2π/3

the two wedges are disconnected by the phase 0 contour (red) in the middle: no contourdirectly from left to right wedge

18without ε: t real and ei(t3/3+Re zt)e−Im z·t grows for t→ −∞ or t→∞ as soon as Im z 6= 0

19since t3

3+ re2πi/3 · t = (e2πi/3t)3

3+ r · (e2πi/3t)

1-20

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

Fig. 9: Contours of absolute value and phase (green: −2/3 (not −2π/3), red: 0, magenta: 2/3)of the integrand exp(i(t3/3 + zt)) in the t-plane for Ai(z) with |z| = 1 and varyingphase, from left to right (then top to bottom): arg z = φ ∈ π 1

10 ,13 ,

23−0.1, 2

3 ,23+0.1, 1.

1-21

but we can use the down wedge to go there from the left wedge and to leave from thereto the right wedge: two contours with phases between 0 and 2/3 (red and magenta inthe figure) and between 0 and −2/3 (red and green), plus Jacobians (!)

qualitatively different contours, one of them visiting the other – lower right – saddle,which according to the lower sign of (1.54) is exp. suppressed20 ⇒ associated functionborn, arg z = 2π/3 is a Stokes line

summary of the Airy function, see Fig. 10

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

Fig. 10: Airy function Ai with complex argument. Left: logarithm of the absolute value, right:phases (arg; red for 0, yellow for π, green for −π/6). Lines at angles π/3, 2π/3.

kept using the dominant saddle up to φ < π ⇒ (makes plausible) asymptotics for φ 6= π:

Ai(z)|z|→∞−→

exp(−23z

3/2)

2√πz1/4

=exp(−2

3 cos(3φ2 ) r3/2)

2√πr1/4

exp(i[

sin(3φ

2)− φ

4

]) (1.55)

exponentially decaying for φ ∈ [−π/3, π/3], otherwise exponentially growing, cf. Fig. 10

φ = π/3: saddles at same abs-contour in Fig. 9 upper right (and cross the wedge edge)

exchange dominance ⇒ antiStokes

φ = 2π/3: phase −π/6, cf. Fig. 10, cos(3φ2 ) = −1: maximal growth

φ = π: both saddles contribute equally ⇒ sine, see the second line of (1.46)

1.5.5 Asymptotic series expansion

already several times:

Φ(z) :=Ai(z)

exp(− 2

3z3/2)

2√πz1/4

z→∞−→ 1 (1.56)

20recall that the saddle we always visit has become exp. large beyond π/3

1-22

improve Φ by inverse powers of z, need z− half integer, actually multiples of three:

Φ(z) =

∞∑n=0

cnz−3n/2 cn =

1

2πβn

≈(n!)2 (see below)︷ ︸︸ ︷Γ(n+

1

6)Γ(n+

5

6)

n!β = − 3

4(1.57)

as usual, e.g. in Quantum mechanics, by recursion relation from the differential equation

the coefficient alternate, they first decrease, but then explode ⇒ asymptotic series:

cn = 1,− 5

48,

385

4608,− 85085

663552, . . .

(Γ(

1

6)Γ(

5

6) = 2π

)(1.58)

= 1,−0.10, 0.084,−0.13, 0.29,−0.88, 4.5,−15, 480,−3300, . . . (1.59)

asymptotics, generalizing Stirling’s formula (1.7) to

Γ(an+ b) = nan(ae

)an(an)b−1/2

√2π(1 +O(1/n)

)(1.60)

one finds Γ(n+ 16)Γ(n+ 5

6) = n!2

n

(1 +O(1/n)

)and thus

cnn→∞−→ 1

2π

1

nβnn! (1.61)

= ?,−0.12, 0.090,−0.13, 0.30,−0.91, 3.4,−15,−480, 3300, . . . (1.62)

similar to the series for the error function, (1.36), but additional n in the denominator(could be eliminated by a β- or g-derivative) → later branch cut instead of pole

get rid of the factorial growth similar to the calculation in Sec. 1.4.3

divide coefficients by n! and reinstate integral representation of it:

Φ(z) =∞∑n=0

cnn!

∫ ∞0dt e−t(tg)n g := z−3/2 (1.63)

=

∫ ∞0dt e−t

∞∑n=0

cnn!

(tg)n (1.64)

note that the sum can be expressed in the hypergeometric function (expand the latter!)

∞∑n=0

cnn!

(tg)n =∞∑n=0

1

2πβn

Γ(n+ 16)Γ(n+ 5

6)

n!n!(tg)n = 2F1

(1

6,5

6, 1∣∣βtg) (1.65)

coefficients nonalternating for g < 0 (gn compensates βn in cn) meaning arg z = ±2π3

expect Stokes phenomenon = where the second saddle comes into play

1-23

2F1(..|βgt) has a branch cut for βgt > 1 (g < 0), the imaginary part jumps by−2F1(..|1− βgt):

limε→0

Im[

2F1

(1

6,5

6, 1∣∣βt(g + iε

)− 2F1

(1

6,5

6, 1∣∣βt(g − iε)] ε > 0 (1.66)

= limε′→0

Im[

2F1

(1

6,5

6, 1∣∣βtg(1− iε′

)− 2F1

(1

6,5

6, 1∣∣βtg(1 + iε′

)]ε′ =

ε

−g> 0 (1.67)

= −2F1

(1

6,5

6, 1∣∣1− βtg)= −1 for βtg = 1

→ − Γ(2/3)Γ(5/6)2

(βtg)1/6

for βtg →∞(1.68)

similar if asymptotic form (1.61) of cn used, but start at n = 1

∞∑n=1

1

2π

1

nβnn!

n!(tg)n = − 1

2πlog(1− βtg) (1.69)

− 12π log(1− βtg) has a branch cut for βgt > 1, the imaginary part jumps by −1

limε→0

Im[− 1

2πlog(1− βt(g + iε)) +

1

2πlog(1− βt(g − iε))

]= −1 (1.70)

which agrees with the exact jump for βtg = 1

let’s again define arguments after and before one of these Stokes lines:

z± = re2πi/3e±iε ε > 0 (1.71)

as shown, the function Φ jumps for g below vs. above the negative axis by the t-integralof that jump

defining

g± = (z±)−3/2 = r−3/2︸ ︷︷ ︸|g|>0

(−1)e∓iε′

= −|g|(1∓ iε′) ε′ =2

3ε > 0 (1.72)

we have thus obtained in (1.67)

limε′→0

(Φ(z)

∣∣g=g+

− Φ(z)∣∣g=g−

)(1.73)

= − i∫ ∞−1/β|g|︸ ︷︷ ︸

>0

dt e−t 2F1

(1

6,5

6, 1∣∣1 +β|g|︸︷︷︸

<0

t) as

= −i∫ ∞−1/β|g|

dt e−t · 1 (1.74)

= − i 2√πe− 2

3|g|Ai(|g|−2/3)

|g|1/6as= −i e−

43|g| (1.75)

as ‘as’ we have included the expressions using only the asymptotics (1.69) with the log’sunity jump (generically it is more likely to be able to sum up the approximated lateterms only21), indeed ‘as’ follows from Ai by using the asymptotics of the latter, (1.55)

21we were lucky since (1.75) is one of the view known integrals for 2F1, note that it yields Ai again, but at adifferent (positive) argument

1-24

note that the original series (1.64) is in small g, real, but nonconverging; this jump isimaginary and of the famous nonperturbative form e−1/|g|

recall that Ai has an essential singularity at ∞

going back to r = |z|

∆Φ(re2πi/3) ≡ limε→0

(Φ(re2πi/3(1 + iε))− Φ(re2πi/3(1− iε))

)ε > 0

= −i 2√π e−

23r3/2

r1/4Ai(r)as= −i e−

43r3/2

(1.76)

again, the first expression asymptotes to the second one by virtue of (1.55) (for largepositive r)

and, including the prefactor of the series, (1.56)

∆Ai(re2πi/3) =exp(−2

3(re2πi/3)3/2)

2√π(re2πi/3)1/4

∆Φ(re2πi/3) =exp(2

3r3/2)

2√πr1/4

e−πi/6∆Φ(re2πi/3)

= eπi/3Ai(r)as= eπi/3

exp(−23r

3/2)

2√π r1/4

(1.77)

again: Ai(re2πi/3) is exponentially large and has phase e−πi/6 (below (1.55) and Fig. 10),whereas its change is exponentially small and related by a relative factor of i

1.5.6 The big picture

coming back to the original question of one vs. two exponentials for z = x ≷ 0

Ai solves a second order diff. eq. y′′ = zy, see (1.45), and another independent solutionexists, called Bi(z)

integral def.: others contours, namely t ∈ (−∞,−i∞) and t ∈ (+∞,−i∞) [CMSO, p.314] ending in other wedges(cf. App. 7.2); Bi is real for real argument

two different asymptotics: ansatz f(z) = exp(azb)zc ⇒ a = ±2/3, b = 3/2, c = 1/422

f±(z) =exp(±2

3 z3/2)

z1/4(1.78)

with corrections ⇒ asymptotic series (recurrence relation from diff. eq.):

y±(z) =exp(±2

3 z3/2)

2√π z1/4

∞∑n=0

(±1)n(3/4)nΓ(n+ 1/6)Γ(n+ 5/6)

2πn!z−3n/2 (1.79)

so far we used y− (y± = z−1/4E± from [RD, 22])

22demand y′′ =[(abzb−1 − cz−1

)2+ ab(b− 1)zb−2 + cz−2

]y

!= z1y; b < 0: all z-powers in [..] negative; b > 0:

z2b−2 is the leading term generating z1 ⇒ b = 3/2; coefficient in front must be 1 ⇒ a = ±2/3; subleadingterm z−1/2 must have vanishing prefactor ⇒ c = 1/4

1-25

not smooth due to the fractional powers of z

going around: f±(e2πiz) = −if∓(z) (from e2πi·3/2 = −1, e2πi·1/4 = i)

likewise: y±(e2πiz) = −iy∓(z) (from e2πi·(−)3n/2 = (−1)n)

to describe entire functions Ai and Bi, the coefficients in front of y± must jump ⇒Stokes phenomenon

dominance: f±(z) ∝ exp(± 2

3 |z|3/2[cos(3

2 arg z) + i sin(. . .)])

f+ and thus y+ dominates for | arg z| < π/3

f− and thus y− dominates in the complement π/3 < | arg z| < π

x > 0: Ai fine-tuned to be exponentially small only, Bi necessarily contains an exp.large part

same asymptotics of f+ and f− at arg z = ±π/3, (±)π, since cos(±π2 ) = 0 (*)

both are O(1)

exchange dominance ⇒ antiStokes lines

indeed (*) is the requirement that the real parts of the exponentials agree

Stokes lines

arg z = 0 (pos. real axis): f+ maximally dominant, non-alternating

arg z = ± 2π3 : f− maximally dominant, non-alternating

where we computed the Stokes phenomenon in the previous section

indeed the imaginary parts of the exponentials agree, sin(± 32 arg z) = 0, for these arg z

answer to original question: the two parts of the real line are antiStokes and Stokes (!)

in total three of each such lines

cf. error function: prefactor is e−z2

(vs. e0), (1.33), only two of each, no multivaluedness

Ai again:

Ai(z) ∼ y−(z) for | arg z| < 2π

3(1.80)

no admixture of y+ by convention at z = x > 0: one exponential

no possibility for the prefactor of y+ to jump away from 0 unless Stokes line at ±2π/3

y− changes on the way from small to large (at the antiStokes line ±π/3), cf. Fig. 10 left

from the jump of the last section (second line of (1.77)):

Ai(z) ∼ y−(z) + eπi/3y−(ze−2πi/3) for arg z >2π

3(1.81)

note that for argZ = 2π/3 both arguments are on Stokes lines, now extrapolate toreal negative argument z = reiπ, means that going π/3 further: both arguments on

1-26

antiStokes lines, where y± = O(1)

y−(reiπ) =exp(−2

3r3/2(−i))

2√π r1/4

e−iπ/4∑n

(−1)n . . . r−3n/2(−i)n (1.82)

eπi/3y−(reiπ/3) =exp(−2

3r3/2i)

2√π r1/4

eπi/3e−iπ/12︸ ︷︷ ︸eπi/4

∑n

(−1)n . . . r−3n/2(i)n (1.83)

the second contribution is exactly the complex conjugate of the first one ⇒ Ai(−r) real

both O(1): two exponentials in form of a sine or cosine

r = −x:

Ai(x)→2

sin( 23

(−x)3/2+π4

)︷ ︸︸ ︷cos(

2

3(−x)3/2 − π

4)

2√π |x|1/4

(1.84)

cf. the lower line of (1.46) (the next term is prop. to sin(23(−x)3/2 − π

4 ), all even andodd terms are of these two forms, but the series is only asymptotic anyway)

Bi again: homework . . .

follow the saddles it visits as z rotates

produce and understand its characteristics as in Fig. 10

Stokes phenomenon as in the previous section

consistency with Bi being a particular linear combination of Ai(ωz) and Ai(ω2z), withω = e−2πi/3 a third root of unity, which are solutions of the diff. eq. as well

a small check: which of the integrands’ portraits in Fig. 11 below – for Ai(z) at differentzs – is connected23 to the Stokes and antiStokes line?24

23with a single z one cannot get access to those lines as they only define directions asymptotically24solution: in the right plot, the two saddles have the same value (black contours!) and the two potential

asymptotics f±(z) are about to interchange their dominance: antiStokes (while the integration contourkeeps using only one of the saddles and is blind to that event), to make the connection rigorous, thisbehavior should remain for |z| → ∞; in the left plot two saddles are used: Stokes line, to make thatrigorous one should show, that this is not the case away from the plot and thus a subleading term (thesaddles are of different heights: black contours, for the order see the caption of Fig. 8) is born: after asmall deformation the red lines will generically leave the saddles, i.e. open up there, for that there are 2×2possibilities, 2 of them are shown in Fig. 9 and these are related by continuity: indeed Stokes, in the othertwo possibilities basically the purple and green line take over the shape from the red, they are again relatedby continuity, now nothing but phases change, the Stokes phenomenon could be absent . . .

1-27

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

Fig. 11: Like Fig. 9: contours of absolute value (black) and phase (colored) of the integrand ofthe Airy function Ai(z) with z = exp(2πi/3) and z = exp(πi/3). Which is connectedto the Stokes line, which to the antiStokes line?

1-28

2 Asymptotic series

for the moment real arguments x around some x0

for complex argument recollect how z0 = ∞ was approached in the examples: |z| → ∞ issimilar to a series with real argument x = |z| when the phase arg z is specified and fixed (butthe coefficients are complex), arg z may be viewed as some parameter (which can change theproperties of the series substantially: alternating vs. non-alternating)

2.1 Remainder and definitions

given a series for a function (see Sec. 1.1), it is all about the remainder/rest term(Restglied)

f(x)−N∑n=0

cn(x− x0)n ≡ RN (x) (2.1)

[this differs from [GD] because the latter is incorrect]

for a Taylor series cn = f (n)(x0)n! and the remainder can be written in Lagrange form,

valid for N + 1 times differentiable functions f :

RN (x) =f (N+1)(ξ)

(N + 1)!(x− x0)N+1 for some ξ between x and x0 (2.2)

as if the next term in the series, but at a different argument

series is convergent, if

|RN (x)| N→∞−→ 0 for x fixed (2.3)

two examples of convergent Taylor series:

ex. (0), f(x) = sinx: f (N+1)(ξ) = ± sincos

(ξ)∈ [−1, 1] and xN+1

(N+1)!

N→∞−→ 0 indeed

for all fixed x fixed

ex. (4), f(x) = 11−x : f (N+1)(ξ) = (N+1)!

(1−ξ)N+2 , RN (x) = 11−ξ

(x

1−ξ

)N+1

let x ∈ [0, 1/2), then the same for ξ and (1− ξ) ∈ (1/2, 1] and∣∣∣ x

1−ξ

∣∣∣ < 1

RN ∝ (# < 1)N+1 N→∞−→ 0: series converges X(proven for restricted x)

but we know that this need not be the case

idea

converging series: sitting at a fixed x, the error reduces as N gets large, i.e. with betterapproximating functions (x− x0)N

2-29

asymptotic series: sitting at any fixed approximation N , the error reduces when movingtowards x0

that alone would be too simple: getting the first coefficient right, c0 = f(x0), then

f(x)− f(x0)− c1(x− x0)1 − c2(x− x0)2 − . . . x→x0−→ 0 trivially, since all functions are ofthe form (x− x0)fixed, positive

example: want to describe f(x) = x at x0 = 0 by αx, the remainder R1 = (1 − α)x1

vanishes as x → 0 independent of α; but we want a condition that enforces α = 1, sothe remainder should not be of the last retained order x1, but much smaller, in general:RN (x) |x− x0|N

some formal definitions first [JH, 19]

‘much smaller than’, ‘dominated by’: [CMSO, p. 78]

f(x) g(x) (x→ x0) iff limx→x0

f(x)

g(x)= 0 (2.4)

for that to make sense, we need g(x) 6= 0 in some vicinity of x0

the most general math. definition is:

iff ∀ε > 0 ∃δ such that |f(x)| ≤ ε|g(x)| follows from |x− x0| < δ

if x0 =∞: ∀ε∃∆ such that . . . follows from x > ∆

alternative notion: f = o(g) “little-o”25

funny example [CMSO, 79]: x −10 for x→ 0, even for x > 0

asymptotic power26 series 21.5.15

f(x) ∼∞∑n=0

cn(x− x0)n as x→ x0 iff RN (x) |x− x0|N ∀N (2.5)

f ‘is asymptotic to’ the series on the right hand side [CMSO, p.89]

with the definitions (2.1), (2.4) one can equally write [MRBS, 26]

f(x)−N∑n=0

cn(x− x0)n |x− x0|N ∀N (2.6)

limx→x0

f(x)−∑N

n=0 cn(x− x0)n

(x− x0)N= 0 ∀N (2.7)

25 cf. big-O = ‘bounded by’, ‘at most of order’ : iff limx→x0

sup∣∣∣ f(x)g(x)

∣∣∣ <∞ or there exists α > 0∃δ or ∃∆ such

that |f(x)| ≤ α|g(x)| follows from |x− x0| < δ or from x > ∆ [CMSO, 318] ; a weaker condition of course,e.g. f = O(f), but f 6= o(f)

26 One can define more general asymptotic series∑∞n=0 φn(x−x0) by virtue of ‘asymptotic sequences’, i.e. a set

of functions φn which by themselves become ever smaller, φn+1(x−x0) φn(x−x0) [OC, 3] (coefficients?sometimes in the literature); here (x− x0)n+1 (x− x0)n.One can also define ‘asymptotic to’ between functions [CMSO, 79]: f ∼ g iff f−g g (or limx→x0 f/g = 1),then also g ∼ f since g − f f (transitive relation a s well).

2-30

analogously for x0 =∞: [RD, 16]

f(x) ∼∞∑n=0

cnx−n iff lim

x→∞xN(f(x)−

N∑n=0

cnx−n) = 0 ∀N (2.8)

nonintegral powers possible: (x − x0)αn (or x−αn when x0 = ∞) with α > 0 [CMSO,89], cf. α = 3/2 in the Airy example (1.57)

note that a series is asymptotic only relative to a function, whereas convergence is anabsolute concept, that can be read off from the (decay of) the coefficients alone [CMSO,119] (the convergent series can be used to define the function)

a series might be asymptotic without being convergent, since we have another tool,x→ x0, but:

‘∑∞

n=0 cn(x − x0)n is just a formal symbol. It does not have any meaning as a series.’[4]

‘saying that∑∞

n=0 cn(x − x0)n is asymptotic to f(x) gives one no information aboutf(x) at fixed x 6= x0.’ [MRBS, 39, with modified symbols]

is the series meaningless in this case (zero radius of convergence)?

we know now how to use the symbol ∼, but is it as symmetric as it looks?

2.2 Some properties

uniqueness of the series/its coefficients

any function has at most one asymptotic series [MRBS, 26] (around a given point)

proof: assume two series and take their difference ⇒ same coefficients

if the Taylor series exists – i.e. if the function is smooth in some real interval aroundx0 – then it is asymptotic to the function and thus it is its asymptotic series that weseeked [MRBS, 27]

proof: use Lagrange remainder (2.2), take the supremum of f (N+1)(ξ) over ξ ∈ [x, x0]

once again, the Taylor series need not converge or may converge to another function

in other words, if a function can be expanded in an asymptotic series, then the coeffi-cients are unique, from (2.7) we get recursively: [CMSO, 90]

at N = 0: limx→x0

f(x)−c01 = 0 ⇒ c0 = lim

x→x0

f(x)

at N = 1: limx→x0

f(x)−c0−c1(x−x0)(x−x0) = 0 ⇒ c1 = lim

x→x0

f(x)−c0x−x0

etc.

if all these limits exist27, these are basically the derivatives = back to Taylor

27e.g. the function need not be defined at x0, but its limit for x→ x0 must exist

2-31

in turn, functions cannot be expanded around their poles, since c0 = ∞ (e.g. 1/x nearx0 = 0 or ex near x0 =∞ [CMSO, 89])

within the framework of asymptotic series, can we say that for examples (6,7)

e−1/x(2) ∼ 0 (x→ 0) and e−x(2) ∼ 0 (x→∞) (2.9)

yes, from definition (2.7) we just need that exp. is stronger than every power, expectedfrom Taylor series

but then 0 is the unique asympt. series for these functions and cannot be ‘improved’among asymptotic series

it also hints at the . . . [CMSO, 123]

existence, but non-uniqueness of the function

for any sequence cn there exists a smooth function for which f(x) ∼∑

n cn(x−x0)n

(Borel-Ritt Theorem [JH, 22])

but it is not unique [RD, 16][CMSO, 123]

‘without implying any belittlement of Poincare’s contribution [to propose a definitionof asymptotic series] it is imperative to ferret out its shortcomings’ [RD, 16]

two functions with the same asymptotic series may differ by a function with zero asymp-totic series; if they differ by a smooth function, that function has zero Taylor series

e.g. for x0 = 0: f(x) and f(x) + e−1/x(2)have the same asymptotic series28, cf. (2.9)

even if a series converges, like 0 in example (2.9), reading it as an asymptotic series, thefunction asymptotic to it is not unique

however, the quality of the best asymptotic approximation to the function might beinfluenced [CMSO, 123]: comparing two functions to the same series (convergent or atits lowest contribution), the remainder will generically be different

when trying to enforce uniqueness, one must remove those kind of functions

arithmetical operations [CMSO, 125]

asympt. series can be added and multiplied by numbers X

and multiplied and divided by each other X(denominator should not vanish [OC, 7])

integration X

if f is integrable near x0 [CMSO, 126])

term-by-term: the x-powers in the asymptotic series are all integrable (unless x0 =∞,where the first two terms c0 and c1/x are not integrable and have to be moved from theseries to the function) and we do not care how the decay of the coefficients is modified,since we don’t need convergence

28of course do not multiply f(x) · e−1/x(2) , since the latter kills the asympt. series of the former

2-32

definite integration∫ xx0dy f(y), so no undetermined constant (series starts with (x−x0)1)

differentiation not in general (!)

differentiating our standard examples e−1/x(2)at x0 = 0 term-by-term is fine

but differentiaton can reveal highly-oscillatory terms: [JH, 21], [CMSO, 125]

(x→ 0) f(x) = e−1/x2sin(e1/x2

) ∼ 0 (old derivation, sine bounded) (2.10)

f ′(x) = − 2

x3cos(e1/x2

) +2

x3e−1/x2

sin(e1/x2)︸ ︷︷ ︸

∼0

(2.11)

both highly oscillatory, f squeezed by envelop ±e−1/x2, f ′ liberated since envelop ±2/x3

f ′ has no asymptotic series at x0 = 0, since the limit of cos(e1/x2)/x3 for x→ 0 to deter-

mine c0 is not defined (and so are the limits for higher coefficients); this actually meansthat f ′ is not defined at x = 0 and thus f is not differentiable there, but continuous29

and it does possess an asymptotic series

a convergent power series expansion of an analytic function can be differentiated term-by-term [JH, 21],

for a sufficient condition for the derivative to be defined (in complex space) see [OC, 8]

2.3 An integral counting Feynman diagrams

we will analyse the following model for a field theory [JZJ]

Z(g) =1√2π

∫ ∞−∞

dt e−12t2− g

4t4 (g > 0) (2.12)

g is the coupling of a scalar φ4 ‘field’ theory in zero dimensions/anharmonic oscillator, Z isthe path integral/the partition function, normalized such that Z(0) = 1

Z can be expressed in closed form in terms of a Bessel function

Z(g) =1

2√π

1√ge1/8gK1/4

( 1

8g

)(2.13)

which we can always use to cross-check some calculations30, otherwise forget it imme-diately since we are typically not in such a comfortable situation

rather the first thing one tries is perturbation theory in g meaning expanding theinteraction part

Z(g) =1√2π

∫ ∞−∞

dt e−12t2∞∑n=0

(− g

4 t4)n

n!(2.14)

29so is e.g. x sin(1/x)30note that expanding K1/4

(18g

)for small positive g gives

√g e−1/8g times a polynomial in g, so the other

g-dependent factors are canceled; Z(g) decays like 1/g1/4 for large g

2-33

exchanging sum and integral and integrating it term by term

Z(g)“ = ”1√2π

∞∑n=0

1

n!(−1

4)n∫ ∞−∞

dt e−12t2t4n︸ ︷︷ ︸√

2 4nΓ(2n+1/2)

gn (2.15)

Z(g) ∼ 1√π

∞∑n=0

(−1)nΓ(2n+ 1/2)

n!gn (g → 0) (2.16)

this model can be used to count the multiplicity of Feynman diagrams = integrandsto a given order in g; here they are ‘all the same’ and finite, in most field theories theintegrands after expanding the interaction term are highly nontrivial and infinite (need‘renormalization’), these aspects are not reflected by our model31

counting: order gn contribution to Z in momentum space has n vertices and thus 4nlegs Φ(p1) . . .Φ(p4n), which need to be paired up otherwise the integral over that p (withGaussian) vanishes; how many possibilities to built (unordered) pairs of 4n elements?(4n− 1) · (4n− 3) · . . . · 7 · 5 · 3 · 1 = 4nΓ(2n+ 1/2)/

√π, this many Feynman diagrams

contribute at (g/4)n X

so far just a formal manipulation (interchanging sum and integral), how to justify “∼”?

need to analyze limg→0Z(g)−

∑Nn=0 cng

n

gN, cf. (2.7); up to a mult. constant the numerator is∫∞

0 dt e−t2/2(e−y−

∑Nn=0

(−1)n

n! yn) with positive y := g4 t

4 (let g > 0) since this is how weobtained the series (exchanging integral and finite sum is ok); invoke the Lagrange form(. . .) = (−1)N+1e−ξ/(N + 1)! · yN+1 for some ξ ∈ [0, y]; it follows that |(. . .)| ≤ e0/(N +

1)! · yN+1 and similarly for the integral |∫∞

0 dt e−t2/2(e−y−

∑Nn=0

(−1)n

n! yn)| ≤ #NgN+1.

qed.

the series is alternating (should always help convergence), but the coefficients explode,since from (1.60)

Γ(an+ b)

n!≈ n(a−1)n n→∞−→

0 a < 1

∞ a < 1(2.17)

we have

Z(g) ∼∞∑n=0

cngn (g → 0), cn

n→∞−→ (−1)n1√πnn(4

e

)n 1√n

(2.18)

and so this is just an asymptotic series (the same applies to the Taylor series from theexact result (2.13), the g-polynomial mentioned in Footnote 30)

once again, we can only get a limited precision from this perturbation theory for everyfixed g

note that this is not so, if we perturb in the quadratic (mass) term

31sometimes one can establish bounds for the individual diagrams [6], which attributes the remaining singu-larities to the number of diagrams and thus strengthens the analogy to our model

2-34

first introduce it as

Z(g,m) :=m√2π

∫ ∞−∞

dt e−m2

2t2− g

4t4 = Z(g/m4) (m > 0, g > 0) (2.19)

the latter can be shown by a change of the integration variable; again normalized suchthat Z(0,m) = 1

expansion:

Z(g,m) =m√2π

∞∑n=0

1

n!(−1

2)n

∫ ∞−∞

dt e−g4t4t2n︸ ︷︷ ︸

1√2g1/4

(2√g

)nΓ(n/2+1/4)

m2n

=1

2√π

m

g1/4

∞∑n=0

(−1)nΓ(n/2 + 1/4)

n!

(m2

√g

)n(2.20)

in the late coefficients, the ratio of Γs approaches 1/nn/2 (instead of nn in (2.18); a = 1/2in (2.17)) decaying fast enough to yield an infinite radius of convergence for this series

a function of g/m4 indeed, thus small m can be read as large g: no problem

the reason for the series to be asymptotic only is that the integral over e−1/2·u2−g/4·u4

is divergent for g < 0 since u4 is the leading power (but not so for m2 < 0 since thisis the coefficient of the subleading power)

nevertheless, this is the perturbative expansion in field theory, since what can be solvedexactly as the starting point is the interaction-free (massive) theory being Gaussian

outlook:

a similar accuracy limitation applies to perturbative Quantum Electrodynamics and isrooted in a similar (Dyson) instability, see below

from our experience Borel resummation should be applicable since in |cn/n!| → 4n/n,i.e. the leading order nn has canceled, Borel unambiguous for the alternating series atg > 0

for the nonalternating series at g < 0 an imaginary nonperturbative jump shall occur,whose details are related to high order perturbation theory

2.4 Particle in magnetic field: Zeta function regularization and asymptotic series

aim: for the partition function of a particle with charge q in a constant magnetic field Bz = B,show that the expansion in B is only asymptotic and give a physical reason: a virtual (Dyson)instability

maybe one of the simplest QFT computations

on the way we will learn an interesting technique

2-35

2.4.1 Landau levels

first heuristic, then detailed

if the reader is familiar with Eq. (2.30), then this subsection can be skipped

nonrelativistic and classical:

circle perpendicular to B (mass spectrometer, including electric fields, measure m/q)

& constant velocity vz longitudinal to B ⇒ helix

radius, angular velocity, energy and angular momentum:

r =mv

qB, ω =

v

r=qB

m, E =

m

2v2 +

m

2v2z =

ω

2L+

p2z

2mL = mωr2 (2.21)

quantum via Bohr-Sommerfeld:

angular momentum is quantized (~ = 1), thus

E = kqB

2m+

p2z

2m, k ∈ Z (2.22)

relativistic (c = 1):

E ≈ m+ Enonrel, E =√m2 + kqB + p2

z (2.23)

finally correct from Dirac equation:

first massless

/Dψ = iλψ, /D = γµ(∂µ + iqAµ), γµ, γν = 2δµν (2.24)

where we have changed to Euclidean space32, used that /D is antihermitian there (γs canbe chosen hermitian, iAµ are antihermitian and so are ∂µ (momenta ∂µ/i hermitian))

the gauge fields have to generate the magnetic field

∂µAν − ∂νAµ = Fµν = B(δµ,xδν,y − δµ,yδν,x), e.g. Ay = Bx, Arest = 0 (2.25)

the eigenvalues come in pairs due to chiral symmetry

γµ, γ5 = 0 ⇒ /D, γ5 = 0 ⇒ /D(γ5ψ) = −iλ(γ5ψ) (2.26)

such that λ2 – the eigenvalues of − /D2– are two-fold degenerate33

32motivation: better tractable (technical), thermodynamics33note that ‖γ5ψ‖2 = ‖ψ‖2, so γ5ψ is normalized, iff ψ is, as eigenmodes to different λ they are also linear

independent; the degeneracy also applies to zero modes, on which /D also commutes with γ5, such that thelatter can be measured and γ5ψ ∝ ψ does not give an independent state; nonetheless, for log(iλ + m) welater compute, the zero modes give logm = 1

2logm2 with matches with (2.31).

2-36

this operator can be written as

/D2

=1

4(γµγν + γνγµ)(DµDν +DνDµ) +

1

4(γµγν − γνγµ)(DµDν −DνDµ)

= D2µ +

i

4[γµ, γν ]qFµν (2.27)

with the representation γ1,2 =

(0 iσ1,2

−iσ1,2 0

)in terms of Pauli matrices, in which

γ5 = diag(1, 1,−1,−1) we obtain i2 [γ1, γ2]qF12 =

(−σ3 0

0 −σ3

)qB and thus

− /D2= −D2

µ + qB diag(1,−1, 1,−1) (2.28)

eigenvalues

parametrize diagonal entries by σ ∈ −1, 1 (twice the spin), both two-fold degenerate

remember that a four-spinor contains both orientations of spin-1/2 and particle/antiparticle

the temporal34 and (to B) longitudinal part: −D2t − D2

z = −∂2t − ∂2

z give rise to theusual momenta p2

t + p2z, over which one integrates with prefactor Lt/(2π) · Lz/(2π)

L.. are the extensions of these direction, taken to be large (discrete→ continuous evals)

perpendicular part turns into a 1D harmonic oscillator, since ∂y commutes with /D

−D2x −D2

y → −∂2x − (ipy + iqBx)2 = −∂2

X + (qB)2X2 x+pyqB

= X, ∂x = ∂X

→ 2|qB|(n+ 1/2), n = 0, 1, . . . (2.29)

both momenta px,y disappeared and a new quantum number n appeared

from the shift x → X on can deduce the degeneracy to be LxLy|qB|/2π i.e. prop. tothe magnetic flux (instead of LxLy/(2π)2, which would apply for a neutral particle;moreover |qB|

∑n has the same dimension as

∫dpxdpy)

altogether we have the following eigenvalue and trace:

λ2 = p2t + p2

z + |qB|(2n+ 1) + qBσ,∑λ

= 2V4|qB|(2π)3

∫ ∞−∞dpt

∫ ∞−∞dpz

∞∑n=0

∑σ=±1

(2.30)

where V4 = LtLxLyLz

the energies (still massless) E2 = p2z + |qB|

(2n + 1 + sign(qB)σ

)are indeed of the

expected form (2.23) since (..) is integer 28.5.15

34which can be generalized to finite temperature (particle in a potential well of finite size Lt = 1/T ) withantiperiodic boundary conditions: −∂2

0 → (2π(k + 1/2)T )2, k ∈ Z (no degeneracy); this quantization isremoved for the T → 0 limit that we analyze here

2-37

remark about finite temperature

replace pt by 2πkT+πT with k ∈ Z, where the second summand comes from antiperiodicboundary conditions (wave ansatz and exp(πi) = −1); in the trace replace 1/(2π) ·

∫dpt

by∑

k∈Z

2.4.2 Zeta function, heat kernel, Mellin transform, Schwinger proper time

the partition function of this system35 is given by det( /D +m), the vacuum energy36 density(free energy density at zero temperature) is − 1

V4log det( /D+m) etc., but all this is obviously

divergent:

log det( /D +m) =∑λ

log(iλ+m) =1

2

∑λ

log(λ2 +m2) =1

2tr log(− /D2

+m2) (2.31)

idea of zeta function regularization:

use

log y = −∂s=0y−s ∂s=0 ≡

∂

∂s

∣∣∣s=0

(2.32)

after interchanging ∂s and∑

λ one has to sum up (λ2+m2)−s which for large enough s(real part of it) converges; then analytically continue to s = 0

in this (particular) way one gets rid of an infinity, the remaining finite part depends onm2, B etc.

Riemann zeta function (‘the archetypal example’ [wikipedia]):

ζR(s) =

∞∑n=1

1

nsRe s > 1 (2.33)

as is turns out, one can analytically continue to s = 0, only at s = 1 a pole appears, so

ζR(0) = 1 + 1 + 1 + . . . = −1/2 (2.34)

ζR(−1) = 1 + 2 + 3 + . . . = −1/12 (2.35)

ζR(−2) = 1 + 4 + 9 + . . . = 0 (2.36)

this function is very interesting for number theory and statistics, it has trivial zerosat s = −2,−4, . . . and (infinitely many) nontrivial zeros in the strip 0 < Re s < 1,conjectured37 to be at Re s = 1/2 and at ‘kind of random’ Im s etc.

35path integral over ψ and ψ with integrand exp(ψ( /D+m)ψ) giving det( /D+m) due to the Grassmann natureof ψ and ψ

36with energies write logZ = 1/2 ·Lt/2π∫dpt

∑E log(p2

t +E2(α)), where Lt = β = 1/T and we introduced anexternal parameter α (e.g. B) entering only through the energies; then do the following trick: ∂ logZ/∂α =1/2 · β/2π ·

∑E

∫dpt2E/(p

2t +E2) · ∂E/∂α = 1/2 · β

∑E ∂E/∂α and logZ = β

∑E E/2 +α−indep., which

is the sum of vacuum energies (the free energy F = −T logZ = −∑E E/2 has an unsual sign due to

fermions: determinant in numerator)37B. Riemann (1859): “Hiervon ware allerdings ein strenger Beweis zu wunschen; ich habe indess die Auf-

suchung desselben nach einigen fluchtigen vergeblichen Versuchen vorlaufig bei Seite gelassen, da er fur dennachsten Zweck meiner Untersuchung entbehrlich schien.” checked numerically for 1010 solutions, unsolvedClay Millenium Problem

2-38

how to compute? integral representation related to the one of Γ, (1.5)

y−s =1

Γ(s)

∫ ∞0dt e−tyts−1 Re y,Re s > 0 (2.37)

which can be read as the Mellin transform (of e−ty)

perform the sum (y = n in (2.33)):∑∞

n=1 e−tn = 1/(et − 1) and write that in terms of

another sum t/(et − 1) =∑∞

k=0Bktk/k!

for small t (∫∞

1 is ok) integrate∫ 1

0 dt ts−2+k = 1/(s + k − 1), which is still valid for Re

s > 1 (or better Re s > 1− k)

this expression, however, can be continued to smaller s easily, with the poles compen-sated by the prefactor 1/Γ(s) for all s but s = 1

the integral∫∞

1 remains together with the k-sum [7, (3.4)], which converges, cf. (1.12)

zeta function for an operator A

with eigenvalues ηn:

ζA(s) ≡∑n

(ηn)−s = trA−s (2.38)

from which

ζ ′A(s) =∑n

− log ηn · (ηn)−s = −tr[

logA ·A−s]

(2.39)

for the determinant assume that A has dimension energy2 such that we can regularizeit with a scale µ of dimension energy and (2.32) as follows:

log det( Aµ21

)= tr log

( Aµ21

):= −∂s=0 tr

( Aµ21

)−s= tr

[log

A

µ21

( Aµ21

)−s]s=0

= −ζ ′A(0)− logµ2 ζA(0) (2.40)

later A = − /D2+m2 and we need 1

2 log det . . .

note that in this regularization/definition det(AB) 6= det(A) det(B) ‘multiplicative anomaly’

compute again with (2.37)

ζA(s) =1

Γ(s)

∫ ∞0dt ts−1 tr e−tA (2.41)

the latter is also called the heat38 trace

its asymptotic (!) expansion reads

tr e−tA ∼ 1

(4πt)dim/2

∞∑n=0

an(A)tn (t→ 0) (2.42)

with an the Seeley-de Witt coefficients depending on the operator and space-time39

38heat equation (diffusion): ∂tψ = ∆ψ, similar to a Schrodinger equation, from which we know the timeevolution operator: U = et∆

39e.g. on its curvature, if A is the (covariant) Laplace-Beltrami operator

2-39

2.4.3 Result and interpretation

heat trace

tr e−t(− /D2+m2) =

∑λ

e−(λ2+m2)t (2.43)

= 2V4|qB|(2π)3

e−m2t(∫ ∞−∞

dp e−p2t)2

︸ ︷︷ ︸π/t

∞∑n=0

e−2|qB|t(n+1/2)

︸ ︷︷ ︸1

e|qB|t−e−|qB|t

∑σ=±1

e−qBtσ︸ ︷︷ ︸eqBt + e−qBt

the last two terms combine with the degeneracy |qB| to

|qB| coth(|qB|t) = qB coth(qBt) =1

t+

(qB)2

3t+O((qB)4t3) (2.44)

thus, the expansion near t = 0 has 1/t2, t0 . . . and thus a0,2,... with a2 ∝ (qB)2

the integration variable t is actually a proper time [Eigenzeit; Schwinger [10]] and hasdimension length2

the heat trace is also extensive in the volume (at constant B; like a free energy)

zeta function

the zeta function of our operator is related to the Hurwitz zeta function, since∑n

∫d2p

(p2 +m2 + qBσ + 2|qB|n

)−s ∝∑n

(m2 + qBσ + 2|qB|n

)1−s ∝∑n

(x+ n)1−s

(2.45)

from the heat trace we have

ζ(s) =V4

(2π)2

1

Γ(s)

∫ ∞0dt e−m

2tts−2qB coth(qBt) (2.46)

due to a potential nonintegrable singularity at t → 0 this integral is well-defined forRe s > 2 (coth provides another t−1) and we will have to analytically continue it tos ≈ 0

separate these terms:

ζ(s) =V4

(2π)2

1

Γ(s)

∫ ∞0dt e−m

2tts−2( 1

t︸︷︷︸→ζ0(s)

+(qB)2

3t︸ ︷︷ ︸

→ζ2(s)

+[qB coth(qBt)− 1

t− (qB)2

3t]

︸ ︷︷ ︸→ζ(s)

)(2.47)

the integrals (with prefactor etc.) define three functions of s and qB, that are analyticalfor Re s > 2, Re s > 0 and Re s > −2, respectively

2-40

the first two terms can be continued easily:

ζ0(s) =V4

(2π)2

1

Γ(s)

∫ ∞0dt e−m

2tts−3 =V4

(2π)2m4 Γ(s− 2)

m2sΓ(s)(2.48)

ζ0(0) =V4

(2π)2

m4

2, ζ ′0(0) =

V4

(2π)2m4(

3

4− logm) (2.49)

ζ2(s) =V4

(2π)2

1

Γ(s)

(qB)2

3

∫ ∞0dt e−m

2tts−1 =V4

(2π)2

(qB)2

3

1

m2s(2.50)

ζ2(0) =V4

(2π)2

(qB)2

3· 1, ζ ′2(0) =

V4

(2π)2

(qB)2

3(−2 logm) (2.51)

note that the B-independent part ζ0 can be obtained directly from the B = 0-eigenvaluesλ2 = p2

t + p2z + p2

x + p2y, their degeneracy V4/(2π)4

∫d4p · 4 (from Dirac space) and the

Gaussian heat trace∫d4p exp(−p2t) = π2t−2

for the remainder

ζ(s) =V4

(2π)2

1

Γ(s)

∫ ∞0dt e−m

2tts−2[qB coth(qBt)− 1

t− (qB)2

3t]

(2.52)

(recall Re s > −2 here) one can go to s ≈ 0, via ts/Γ(s) = s+O(s2) and gets:

ζ(0) = 0, ζ ′(0) =V4

(2π)2

∫ ∞0dt e−m

2tt−2[qB coth(qBt)− 1

t− (qB)2

3t]

(2.53)

one is tempted to do a perturbative expansion in B (or the charge q/coupling e); ex-changing integral and sum we will end up with an asymptotic series, keep in mind thatthe integral above defines ζ ′(0) uniquely (no nonintegrability around t = 0: t−2+3,5,...)

expansion in B:

from

qB coth(qBt)− 1

t− (qB)2

3t =

∞∑n=2

bn(qB)2nt2n−1, bn =4nB2n

(2n)!(2.54)

one gets

ζ ′(0) ∼ V4

(2π)2

∞∑n=2

bn(qB)2n

∫ ∞0dt e−m

2tt2n−3︸ ︷︷ ︸m4Γ(2n−2)/(m2)2n

(2.55)

∼ V4

(2π)2m4

∞∑n=2

Γ(2n− 2)

(2n)!B2n

(2qB

m2

)2n(B → 0) (2.56)

now, while Γ(2n − 2)/Γ(2n + 1) decays like 1/n3, cf. (1.60), the Bernoulli numbersexplode like n2n, cf. (1.12)

⇒ the series does not converge, but is asymptotic only

2-41

it alternates (from B2n)

note that the integrand [..] in (2.53) is negative, and so is the first term in the seriesabove, since B4 = −1/30

what is so special about B = 0 (essential singularity)? see Sec. 2.5.3

the final result is, incl. a factor 1/2 from (2.31)

−ζ ′(0)− logµ2ζ(0)

2V4=

1

2(2π)2

−m4

[34− 1

2log

m2

µ2

]+

(qB)2

3log

m2

µ2

− ζ ′(0)

2V4(2.57)

note that only a finite number of terms depends on the scale µ40

the µ-terms come from ζ(0) and from the factor 1/Γ(s) in it, (2.41), ζ(0) vanishes unlessthe remaining integral is infinite

in both the expansion in qB or Seeley-de Witt coefficients (which are related by theargument qBt in coth), we see that all terms from some order on have high powers of tthat render the integrand integrable at t = 0

what is the meaning of these first terms?

O(B0)

proportional to m4 for dimensional reasons

such terms contribute to the cosmological constant, if constant external magnetic fieldsmake sense in this context

O(B2) 11.6.15

note that the Maxwell action contains a term F 2µν/4 = B2/2 times V4, too, but without

electric charge (field energy of photons, independent of the charge they could couple to)

so far B was the bare field entering the Lagrangian, the physical renormalized field Brcan be defined by demanding the Maxwell form of the quadratic term even afterthe fermionic contribution (in the ‘effective action’):

B2

2− 1

2(2π)2

(qB)2

3log

m2

µ2=B2r

2(2.58)

where the minus sign is from det = exp(log det) vs. exp(−SMaxwell)

then, however, Br depends on the scale µ, that we introduced in the regularisation andno physical quantity should depend on it

in fact, µ plays the role of a cut-off, see below (not the renorm. scale as many authorsnote in passing)

once again: zeta function regularization defines log det to be finite, (2.40), so no infinitiesappear anymore, but µ should be removed; do this by saying that B is not physical,but only a ‘bare’ parameter in the Lagrangian (not infinite here)

40does that reflect the renormalizability of QED? the arguments below seem to hold for any zeta functionregularized theory = all determinants . . .

2-42

this removed divergence will give the renormalization of B or the electric charge, sinceqB = qrBr is an invariant (protected by gauge symmetry), i.e. independent of therenormalization scale µR:

∂µR(qrBr) = 0 ⇒ − qrBrµR∂µRBr = µR∂µRqr =: β (2.59)

this way of determining the running of the coupling encoded in the beta function is called‘background field method’; it even works for nonabelian gauge theory backgrounds toone-loop [7], where ghost contributions turn the beta function negative, which is thefamous asymptotic freedom

since µ is like a cut-off, we turn to another definition of the beta function:

β = µ∂µq = − qBµ∂µB = − q

2B2µ∂µB

2 (2.60)

we compute it by replacing qB → qrBr in our result and bringing it to the other side:

B2 = Z3B2r , Z3 = 1 +

q2r

12π2log

m2

µ2(2.61)

where Z3 is called renormalization factor (for the gauge field, the inverse of it or thecharge)

now since qr and Br depend on the renorm. scale µR, but not on the cut-off µ, we get

β = − q

2B2B2r

q2r

12π2µ−2

µ=

q3r

12π2(2.62)

where we equated the bare and renormalized quantities on the right hand side, theirdifferences being of higher order

thus, from the positive sign of β and (2.59): |qr| increases with increasing scale µR, i.e.with better resolution in the UV = screening (and not asymptotic freedom): from farthe charge is screened by virtual particle-antiparticle pairs to be smaller than nearby

meaning of µ? compare to cut-off regularization for O((qB)2)

only in pt,z, i.e. replace (in (2.44)):∫ ∞−∞

d2p e−p2t =

π

t→ 2π

∫ Λ

0dp p e−p

2t =π

t(1− e−Λ2t) (2.63)

indeed becomes 1/t for large Λ, but for every finite Λ the small-t behavior is improvedto O(t0)

hence in ζ2(s) ∝ (qB)2 we have to consider a factor (1− e−Λ2t) = O(t) in the t-integral;in (2.47) we have noted that this integal converged for Re s > 0, which now becomesRe s > −1; thus we can go to s ≈ 0 immediately, again via ts/Γ(s) = s + O(s2); thenew ζ-function will thus vanish at s = 0, but have a nonvanishing derivative there:

ζΛ,2(s) =V4

(2π)2s

(qB)2

3

∫ ∞0dt e−m

2tt−1(1− e−Λ2t) (2.64)

ζΛ,2(0) = 0 ζ ′Λ,2(0) = V4(qB)2

12π2log(1 +

Λ2

m2

)(2.65)

2-43

now compare the two expressions (at this order):

−ζ ′2(0)− logµ2ζ2(0)

2V4=

(qB)2

2(12π2)log

m2

µ2(2.66)

−ζ ′Λ,2(0)− logµ2ζΛ,2(0)

2V4= − (qB)2

2(12π2)log

Λ2

m2(Λ m) (2.67)

thus µ has the role of a UV cut-off Λ (solely from ζ ′Λ); indeed the bare and renormalizedB’s should be related by a cut-off

accuracy of the perturbative series in qB

like in for the Gamma funtion asymptotics in (1.13) we assume the index n to becontinuous and consider the minimum of the terms cnx

2n, x = 2qB/m2 (x→ 0) in thesum (2.56), for large n:

log |cnx2n| ∝ n log(nx) (2.68)

which is minimal at n ≈ 1/(ex), for consistency ex must be smallX[e is Euler’s constant,not fundamental charge]

(note that the expansion of (1.13) was for x→∞ and n ∝ x)

the contribution there is x1/x = exp(log x·1/x) which is similar to the famous exp(−1/x)(since log x < 0) and nonperturbative

some realistic numbers

strongest continuous magnetic field in the lab: B = 45T = 45kg/C/s [wikipedia]

with elementary charge |eB| ≈ 10−18kg/s ≈ 300(eV )2 and electron mass x ≈ 2 · 10−9

⇒ one can add up 108 terms before noticing the nonconvergent character of the series

will be different in magnetized neutron stars, where the magnetic field can be a factor108 larger (but constant?!)

2.5 Asymptotic series in QED

2.5.1 Good news: the anomalous magnetic dipole moment of the muon (‘g-2’)

one of the best showcases for agreement between theory and experiment, we follow [8] ([GD]differs in the last digits)

magnetic moment ~µ along the spin ~s:

~µ = g|e|2m

~s, g = 2︸ ︷︷ ︸Dirac theory

(1 + a), a :=g − 2

2(2.69)

as an expansion in the fine structure constant

α =e2

4π(2.70)

2-44

(since vertices with factors e come in pairs)

one can compute a in QED (= virtual photons, electrons, muons and tau-leptons) up tofive loops, which needs 12 000 Feynman diagrams [9]; these diagrams have two muon legsand one external photon leg (the (experimental) magnetic field the magnetic momentcouples to) and thus are vertex corrections

the values known up to now are [9]:

aQED =5∑

n=1

cn(απ

)nc1 = 1/2 [Schwinger ’48,’49]

c2 = 0.76586.., c3 = 24.0505.., c4 = 130.88, c5 = 753 (2.71)

one can watch the coefficients exploding

with α = 1/137.036 (at the electron mass), however, each coefficient comes with onemore power of the suppression factor 137π = 430 and therefore, the summands stilldecrease,

1011 · aQED = 116 140 973 + . . .+ 5 = 116 584 718 (2.72)

with an error of 0.08 (!!)

the contributions from electroweak and strong interactions to 1011aQED are much smaller,154(1) and 6900(40), but sizeable in the agreement between theory and experiment; thelatter extends over 6 digits (!) ‘only’ and is of order 300

deviations in such high precision experiments are a chance to see effects of new particles(that couple to the muon)

therefore, one needs to know the strong contribution to better accuracy, partiallythrough lattice simulations . . .

2.5.2 Bad news: the Dyson instability

instability in a system of N charged bosons (no Pauli blocking here) with equal charge e [11,1952!], we follow [6], N is large

mean kinetic energy T and mean potential energy V – from pairwise repulsive Coulombicinteraction – gives a total energy of

E ≈ NT +N2

2e2V (2.73)

however, if the Coulomb potential was attracting between like charges, i.e.e2 < 0, the system would be unstable

beyond a critical number of particles

Nc =1

−e2

T

V(2.74)

the energy of the ground state decreases like −N2

2-45

relativistic: particle creation possible41

= barrier penetration

large numbers of particles can always be created (after a sufficient time42)

the Hamiltonian cannot be bounded from below ⇒ e2 = 0 is a singularity

perturbative expansion of any physical quantity (must be even in e)

F (e2) =∑n

cn(e2)n (2.75)

sensitive to the above instability ‘at order Nc when the diagrams corresponding to thecreation of Nc particles appear’ [6]

terms in the series decrease/increase before/after that and around Nc are O(1):

cNce2Nc

cNc+1e2(Nc+1)≈ 1 ⇒ cNc+1

cNc≈ 1

e2∝ Nc ⇒ cn ∝ n! (2.76)

having generalized to all late coefficients cn (justified?) we have obtained the typicalfactorial growth of the asymptotic series reflecting the singularity

2.5.3 Good news again: Euler-Heisenberg effective action and pair creation

[12, 1936!], for the history and more details see [13, 14]

integrating out the electrons in QED gives an effective action for the electromagnetic fields~E and ~B, back in Minkowski space and restricting to constant (uniform) fields

in modern language:

Γ( ~E, ~B) =~E2 − ~B2

2−i log det ′(i /D +m)︸ ︷︷ ︸

ΓD

(2.77)

in the Dirac determinant det′ the quadratic (and constant) terms are removed, sincethey have been used to renormalize E and B (see the end of Sec. 2.4.3) and so ΓD startsquartic

it ‘polarizes the vacuum’ like a medium ~D = ~E +O((E,B)3) etc.

light-by-light scattering becomes possible due to the ‘four-vertex’ at O((E,B)4); ofcourse looking at it with a UV magnifying glass, the fermion loop connecting the photonlegs is recovered

ΓD is a function of e ~E and e ~B and the only possible Lorentz tensors are ~E2 − ~B2 =:a2 − b2 and ~E · ~B =: ab

41these pairs have opposite charges, but in this fictitious world they would separate (unlike charges repel!)into two clusters of like charges, for which the argument applies

42would not play a role, if this time is larger than the age of the universe

2-46

the result by E&H is [14, (1.2)]

ΓD = − V4

8π2

∫ ∞0dt e−m

2t 1

t

[ea cot(eat)eb coth(ebt)− 1

t2− −(ea)2 + (eb)2

3︸ ︷︷ ︸first two terms

](2.78)

the square bracket is O(t2) and thus the integral is fine

this is very similar to the Schwinger time representiation we have already derived inEuclidean space, now q = e

indeed, it agrees exactly with the one from Euclidean space, (2.57) and (2.53), for purelymagnetic fields, because

(i) in this case ~E = 0, a = 0, b = | ~B| = B, ea cot(eat)→ 1/t

(ii) magnetic fields are not effected by the Wick rotation t→ iτ to move from Minkowskito Euclidean space

it means that from the same calculation we can get information about the effect ofpurely electric fields, because

(i) in this case ~B = 0, a = | ~E| = E, b = 0, eb coth(ebt)→ 1/t

(ii) electric fields – like time – differ by a factor i between Minkowski and Euclideanspace

note that y cot(y) = (iy) coth(iy) and thus (‘E’ always stands for Minkowskian)

ΓD(E) = − V4

8π2

∫ ∞0dt e−m

2t 1

t2[eB coth(eBt)− 1

t2− (eB)2

3

]B=iE

(2.79)

for such a value of B, many mathematical bugs/features occur signalling new Physicsfrom electric fields:

the integral is not well-defined, since cot has infinitely many poles

the corresponding asympotic series

ΓD(E) ∼∞∑n=2

Γ(2n− 2)

(2n)!

[(−1)nB2n

](2eE

m2

)2n(E → 0) (2.80)

is not alternating anymore, but of same sign

we can compute the integral from residues provided the integration contour in t isdeformed (assume eE > 0 for simplicity):

limε→0

( ∫ ∞+iε

0−∫ ∞−iε

0

)dt e−m

2t 1

t2[eE cot(eEt)− 1

t2+

(eE)2

3

]= −2πi

∞∑k=1

rest=kπ/(eE)e−m2t 1

t2[eE cot(eEt)− 1

t2+

(eE)2

3

]= −2πi

∞∑k=1

e−m2kπ/(eE) 1

(kπ/(eE))2

[1− 0 + 0

](2.81)

2-47

(minus from orientation of curves, k = 0 excluded since the integrand is constructed tohave no pole there), such that

∆ΓD(E) = iV4

4π3(eE)2

∞∑k=1

e−m2πeE

k

k2(2.82)

but this is imaginary

resolution:

an imaginary part in the effective action, i.e. in the amplitude between in- and out-vacuum (= persistence) means an instability of the vacuum (= nonpersistence)

〈0; out|0; in〉 = eiΓ ⇒ |〈0; out|0; in〉|2 = e−2 Im Γ (2.83)

here wrt. the creation of particle-antiparticle pairs: after the creation of the pair, theopposite charges are torn apart by the electric field and give rise to a different quantumstate

(not so for the magnetic field with the circular motion in it: annihilation and return tothe old state)

the imag. part of Γ thus has the physical interpretation of a pair production rate

like in the Dyson argument, this instability is the rationale why the series in small realmagnetic fields is only asymptotic, Sec. 2.4.3

a careful analysis of the Wick rotation shows that one has to approach the real axisin complex t-space from a particular imaginary part and only half of the residues areneeded, such that [12, 10], cf. [GD, 7]:

Im ΓDV4

=1

8π3(eE)2

∞∑k=1

exp(−m2πeE k)

k2(2.84)

which is extensive in the volume Xand nonperturbative in E (!)

again realistic numbers

it is clear that the electric fields have to be of the order of the electron mass squared,which is huge: Ecrit = 1018Volt/m (otherwise the rate is exponentially suppressed),similar for light-by-light scattering

modern laser experiments with non-constant fields come closer to this value 18.6.15

2-48

3 Borel resummation and transform

3.1 Idea and nomenclature

idea: how to resum a non-convergent series with coefficients cn “to a function”, provided theseries with coefficients cn

n! converges [Borel, 1899]

simple version [GD], [CMSO, 382]

use g as small coupling in which the system is expanded

give the series names:

f(g) =

∞∑n=0

cngn (g → 0, formal) , B[f ](t) =

∞∑n=0

cnn!tn . . .Borel transform (3.1)

then the integral

1

g

∫ ∞0dt e−t/g B[f ](t) = Sf(g) =

∫ ∞0dt e−t B[f ](tg) . . .Borel resummation (3.2)

– where we gave two equivalent notions related by variable subsitution – gives the desiredfunction

Sf(g) “ = “

∞∑n=0

cngn = f(g) (small g) (3.3)

as it will turn out, the relation between the two sides is just ∼, i.e. an asymptotic series

the naive proof exchanges integral and sum (we have done this already in Sec. 1.4.3 andespecially Sec. 1.5.5)

Sf(g) =

∫ ∞0dt e−t

∞∑n=0

cnn!gntn “ = “

∞∑n=0

cngn 1

n!

∫ ∞0dt′ e−ttn︸ ︷︷ ︸n!

(3.4)

the mathematical proof uses the same Γ-type integrals and writes down conditions onthe series/functions to make that work

look for a function f with the given series as its asymptotics; recall that there alwaysexist a smooth such function, actually infinitely many differing by subdominant func-tions

sometimes a different series is considered [DD]

f(x) =

∞∑n=0

cn x−n−1 (x→∞) (3.5)

x = 1/g up to another power of x, will also become complex; same B[f ] as before, butthe integral for Sf reads

Sf(x) =

∫ ∞0dt e−txB[f ](t) (3.6)

3-49

3.2 Watson’s lemma and Laplace’s method

Lemma: suppose a function φ (that will be B[f ] later) obeys [CMSO, 263]

(i) φ(t) continuous on t ∈ (0,∞) (3.7)

(ii) φ(t) = O(eCt) as t→∞ for some C > 0 (3.8)

(iii) φ(t) ∼ tA∞∑n=0

antBn as t→ 0+ with A > −1, B > 0 (3.9)

then one can compute the following x-parametric integral ove φ and gets an asymptoticseries in x,∫ ∞

0dt e−xtφ(t) ∼

∞∑n=0

anΓ(A+Bn+ 1) · x−(A+Bn+1) as x→ +∞ (3.10)

(ii) and (iii) required such that this integral exists for t→∞ and t→ 0, respectively

there exist other versions, e.g. where the upper limit of the integral is finite [OC, 31] orwhere the powers of t are more general than B×integer etc.

proof by Laplace’s method, cf. Sec. 1.3: [CMSO, 261-4]

(1) restrict the integration range to around the maximum of the exponential function(the ‘action’), here t = 0, the error is exponentially small in x:∫ ∞

0dt e−xtφ(t) =

∫ ε

0dt e−xtφ(t) +O(e−x∆, e−xε, e−(x−C)∆) ∀ε (3.11)

here we have made use of the property (ii), from which the existence of some positive

∆ follows (cf. Footnote 25), thus split∫∞ε =

∫ ∆ε +

∫∞∆ and that φ(t) is bounded in the

interval t ∈ (ε,∆) (from (i))

x must be large enough for the last term to be small, ε can be chosen arbitrarily small

(2) expand the non-exponential function around that maximum using the asymptoticseries (iii) (keeping the exponential function intact for the last step) and let’s be pedan-tic:

∀# > 0 ∃δ : t < δ ⇒∣∣φ(t)− tA

N∑n=0

antBn∣∣ ≤ # tA+BN (3.12)

integrating this with e−xt up to that δ gives:∫ δ

0dt e−xt

∣∣φ(t)−N∑n=0

antA+Bn

∣∣ ≤ #

∫ δ

0dt e−xttA+BN (3.13)

(since integrand positive) < #

∫ ∞0dt e−xttA+BN = #

Γ(A+BN + 1)

xA+BN+1

3-50

we need the integral up to ε, the remainder is again exponentially suppressed for largex; moreover we use Cauchy-Schwarz to write

∣∣∣ ∫ ε

0dt e−xtφ(t)−

∫ ε

0dt e−xt

N∑n=0

antA+Bn

∣∣∣ (3.14)

≤∫ ε

0dt e−xt

∣∣φ(t)−N∑n=0

antA+Bn

∣∣ (3.15)

≤ #Γ(A+BN + 1)x−(A+BN+1) +

∫ ε

δdt e−xt

∣∣φ(t)−N∑n=0

antA+Bn

∣∣︸ ︷︷ ︸

O(e−xδ, e−xε)

(3.16)

together with the ε→∞ equivalence of (1) we have almost got an asymptotic series forthe integral we want; just need the integrals in the sum

(3) ‘the most convenient way to evaluate’ them [CMSO] is to extend the integrationrange back to infinity, again the error is exponentially small (and exchange allowedwith finite sum)

N∑n=0

an

∫ ε

0dt e−xttA+Bn =

N∑n=0

an

[ ∫ ∞0−∫ ∞ε

]dt e−xttA+Bn (3.17)

=N∑n=0

anΓ(A+Bn+ 1)x−(A+Bn+1) +O(e−xε) (3.18)

and we get (∀# ∃∆ such that from x > ∆ (to make exp. terms arb. small) follows . . . )∫ ∞0dt e−xtφ(t) ∼

∞∑n=0

anΓ(A+Bn+ 1)x−(A+Bn+1) (x→ +∞) qed. (3.19)

nomenclature following [OC, 27]:

integrals of the form

limx→∞

∫dt exη(t)φ(t) (3.20)

with an ‘action’ η and an ‘observable’ φ is treated by the Laplace method, if everythingis real, by the stationary phase method, if everything is real except η that is purelyimaginary, and by the steepest descent or saddle point method, if η and φ are analytic

brought to some particular form, Watson’s lemma can apply to all of them

3-51

3.3 Some properties and examples

Watsons’ lemma for A = 0, B = 1, φ = B[f ] and an = cn/Γ(n+ 1) yields:

if: B[f ](t) ∼∞∑n=0

cnn!tn (t→ 0+)

then:

∫ ∞0dt e−xtB[f ](t) = Sf(x) ∼

∞∑n=0

cnx−n−1 (x→ +∞)

provided B[f ] is continuous and bounded by an exponential

(3.21)

(3.22)

‘Borel resummation is the inverse of Watson’s lemma’ [CMSO, 382]

note the similarity of the inverse Borel transform B[f ] → Sf (3.22) to the Laplacetransform φ→ L[φ]

L[φ](x) :=

∫ ∞0dt e−xtφ(t) (3.23)

hence Sf = L[B[f ]]; ‘the theory of Borel summability is nothing other than the theoryof Laplace transforms, written in slightly different variables’ [15]

from that we can take over various properties, e.g. [DD, 6]

af + bg aB[f ] + bB[g] . . . linearity (3.24)

∂xf(x) −tB[f ](t) (differentiate integral or redefine cns) (3.25)

f(x+ 1) e−t B[f ](t) (integral or from previous relation) (3.26)

f(λx) 1

λB[f ]

( tλ

)(variable substitution in integral or redefine cns) (3.27)

f(x) · g(x) (B[f ] ∗ B[g])(t) . . . convolution:(φ ∗ η)(t) =

∫ t

0ds φ(s)η(t− s) (3.28)

lhs: ‘multiplicative’ model, rhs: ‘convolutive’ model

what is the precise meaning of this?

→ (Borel transform): from f to B[f ] through dividing coefficients by n!

← (inverse Borel transform): from B[f ] to Sf via integral

exponentially growing coefficients = not too bad = geometric series

alternating or not: encoded in β

cn = βn (3.29)

f(x) =

∞∑n=0

βnx−n−1 (∗)=

1

1− β/x1

x=

1

x− β(3.30)

B[f ](t) =

∞∑n=0

βn

n!tn

(∗∗)= eβt ∼ t0

∞∑n=0

ant1n (t→ 0) indeed (3.31)

Sf(x) =

∫ ∞0dt e−t(x−β) (∗∗∗)

=1

x− β∼ f(x) (x→ 0) indeed (3.32)

3-52

(∗): the sum converges for∣∣βx

∣∣ < 1, i.e. |x| > |β| (from ratio test/radius of convergence),

but can of course be analytically continued to all x but β

(∗∗): Watson’s (ii) obeyed for all β, negative β = altern. series converges best for t→∞

(∗ ∗ ∗): the integral exists for Re (x− β) > 0, i.e. Re x > Re β (from t→∞)

which is a different range [4, 8] . . .

factorially growing coefficients = worse, but this is what Borel resummation was inventedfor

cn = βnn! (3.33)

f(x) =1

x

∞∑n=0

n!(β/x)n formal series only (3.34)

B[f ](t) =

∞∑n=0

(βt)n =1

1− βt(3.35)

the Borel transform is now a geometric series

fine around t→ 0 and t→∞, but continuity = Watson’s (i) becomes crucial

B[f ](t) has a pole for t ∈ (0,∞) just for real positive β = series of coherent sign

otherwise we can perform the inverse/resummation:

x > 0, β < 0 : Sf(x) =

∫ ∞0dt e−xt

1

1− βt(−β)t=s

=1

−β

∫ ∞0ds ex/β·s

1

1 + s

=1

βe−x/βEi(x/β) (3.36)

this resembles example (8) (zero radius of convergence of the Taylor series for y2 = −βx )

and the Stieltjes integral∫∞

0 dt e−t/(1 + xt) [CMSO, 120]

recall that β is negative, for comparison to [DD] use that for y > 0: Ei(−y) = −Γ(0, y)

asymptotic series∑∞

n=0 βnn!x−n−1 as x→∞ indeed

differential equation

by a shift of the summation index one can show that f formally solves (‘Euler’s equation’[DD]):

df

dx+

1

βf =

1

βx(3.37)

the general solution of which is

f = Sf + c e−x/β Sf from (3.36) (3.38)

for β < 0 the homogeneous solution e−x/β diverges as x→ +∞ and thus has no asymp-totic expansion around it; the Borel resummation finds the only particular solution thatconverges

3-53

radius of convergence for the Borel transform

is finite, if and only if the formal series f is of Gevrey-1 type, meaning coefficients growfactorially: |cn| = O(βnn!) [DD]

that does not buy us too much, since the radius of convergence could be finite endedby a pole on the positive t-axis, as for positive β above

if the series f(x) already converges, say for |x−1| < r, then its Borel transform is anentire function with an exponential bound er|t| [DD, 6] (like for exp. growing coefficients)

conversely, if B[f ] has only a finite radius of convergence, then the radius of convergenceof the original series is 0 (like for fact. growing coefficients)

stronger factorially growing coefficients: cnn→∞−→ βnΓ(γn+ δ) [GD]

dividing by n! = Γ(n+ 1) is not enough for γ > 1, since Γ(γn+#)Γ(n+#′) ∝ n

(γ−1)n cf. (1.60)

define more general Borel transforms by making use of Watson’s lemma for arbitrary(A,B), so far (0, 1)

keep the formal series f(x), use an = cn/Γ(A + Bn + 1), other t-powers in the Boreltransform to match Watson’s lemma and x→ xB to match the original series

B(A,B)[f ](t) ∼∞∑n=0

cnΓ(A+Bn+ 1)

tA+Bn (t→ 0+)

x−1+(1+A)/B

∫ ∞0dt e−x

1/BtB(A,B)[f ](t) ≡ S(A,B)f(x)

∼∞∑n=0

cnx−n−1 (x→ +∞)

provided A > −1, B > 0 and B[f ] is continuous and bounded by an exponential

let’s now treat the stronger factorially growing case, let γ, δ > 0 such that we can use(A,B) = (δ − 1, γ)

f(x) =∞∑n=0

βnΓ(γn+ δ)x−n−1 (formal) (3.39)

B(δ−1,γ)[f ](t) =∞∑n=0

βntγn+δ−1 =1

1− βtγtδ−1 (3.40)

S(δ−1,γ)f(x) = x−1+δ/γ

∫ ∞0dt e−x

1/γ t tδ−1

1− βtγ(3.41)

again, this is well-defined unless β is positive, the nonalternating case

for β < 0 we can substitute (−β)tγ = s and get

S(δ−1,γ)f(x) =1

γx−1

∫ ∞0

ds

s

1

1 + s(xs

−β)δ/γ exp

(− (

xs

−β)1/γ

)(3.42)

which agrees with [GD] up to some notions

3-54

the Feynman diagram example:

let’s start from the asymptotic series, (2.16)

Z(g) ∼ 1√π

∞∑n=0

(−1)nΓ(2n+ 1/2)

n!gn =

1√πgf(

1

g) (g → 0+) (3.43)

f(x) =∞∑n=0

(−1)nΓ(2n+ 1/2)

n!x−n−1 (x→ +∞) (3.44)

and see what the Borel resummations give us

the conventional one is fine since the ratio of Γ’s in cn can be tamed by 1/n!, the Boreltransform

B[f ](t) =∞∑n=0

(−1)nΓ(2n+ 1/2)

n!n!tn =

2√π

K(12 −

1√1+4t

)

(1 + 4t)1/4(3.45)

(with K the complete elliptic integral of the first kind) must then be integrated over t(with e−xt) and this becomes dirty

but we can also use (A,B) = (−1/2, 2) (which obey the inequalities) to exactly cancelthe Γ-factor in cn

B(−1/2,2)[f ](t) =∞∑n=0

(−1)n(((((

(Γ(2n+ 1/2)

n!

1

((((((Γ(2n+ 1/2)

t2n−1/2 =e−t

2

√t

(3.46)

this is simple and leads to the following integral for the inverse Borel transform

S(−1/2,2)f(x) = x−3/4

∫ ∞0

dt√te−√x t−t2 t=u2/

√4x

=1√2x

∫ ∞0du e−

u2

2−u

4

4x (3.47)

and, approaching the partition function like in (3.43)

1√πgS(−1/2,2)f(

1

g) =

1√2π

∫ ∞0du e−

u2

2− g

4u4

(3.48)

which is nothing but the original integral definition of Z(g)

hence the Borel resummation has recovered not just any of the functions being asymp-totic to the perturbative expansion of Z(g), but Z(g) itself(!); does this also hold forthe conventional Sf sketched above?

we have to understand what characterises Z(g) among these functions, such that it isobtained from the Borel formalism 25.6.15

the magnetic example, (2.56): the zeta function

(2π)2ζ ′(0)

V4m4=

1

g2

∞∑n=2

Γ(2n− 2)B2n

(2n)!g2n+2 (g =

2qB

m2→ 0) (3.49)

3-55

gives rise to the following series f(x):

f(x) =

∞∑n=2

Γ(2n− 2)B2n

(2n)!x−n−1 (x =

1

g2→∞) (3.50)

which (A,B) to take here such that the sum in B[f ] is a known one? B2n/(2n)! can besummed up, cf. (2.54); so let’s cancel Γ(2n − 2) meaning (A,B) = (−3, 2); A violatesits condition, but it works because the sum starts at n = 2 only

B(−3,2)[f ](t) =∞∑n=2

B2n

(2n)!t2n−3 (3.51)

in fact, the lowest order is t1, which could have even been t−1+ε; the sum is

B(−3,2)[f ](t) =t−2

2

[coth

( t2

)− 2

t− t

6

](3.52)

together with e−√x t in the integral for the inverse transform, S(−3,2)f(t) is similar to

(should agree exactly with) the integral representation (2.53) of ζ ′(0)

we again recover the integral representation of the function we knew the asymptoticexpansion of [14, (1.22)], but here we biased ourselves to it; basically we have undonethe t-integral (2.55) that introduced Γ(2n− 2)

factorial growth even stronger [CMSO, 382]

naively, if

f(x) =∞∑n=0

βn(n!)2x−n−1 (x→ +∞) (3.53)

then we tame it by two integrals

f(x) =

∞∑n=0

(β/x)n( ∫ ∞

0dt e−xttn

)2 1

x“ = “

∫ ∞0dt

∫ ∞0ds e−x(t+s) 1

x− βts(3.54)

for β not just positive real – e.g. in the alternating case – no pole appears in theintegration range of t and s, and thus “ = “ should become an asymptotic series ∼

3.4 The Watson-Nevanlinna-Sokal Theorem

want to characterize the function that agrees with its Borel resummed asymptotic series ,uniqueness of functions to the same asymptotic series as well

funny enough the math literature discusses this in terms of the small expansion parameter g(coupling; not the large x) and it will be complexified

the original theorem by Watson [1912], cf. [MRBS, 40], used non-optimal (too strong) as-sumptions (on the region Ω below)

later improved by Nevanlinna [1918] and Sokal [15]

3-56

definition: an analytic function f(g), i.e. a function given by a convergent power series,possesses a uniform remainder in some region Ω ∈ C, if

|RN (g)| ≤ # · σN+1 · (N + 1)! · |g|N+1 (g → 0) (3.55)

for all N and g in Ω [was RN (g) |g|N ∀N , (2.5)]

(this is also called strong asymptotic series [MRBS, 40] and the uniqueness of a functionto a given asympotic series follows (but they only consider a particular Ω))

Theorem [15], cf. [OC, 115]

if f(g) has a uniform remainder in the circular region Ω = g |Re g−1 > 1/R, i.e.where the distance of g to R/2 + 0i is smaller than R/2, then the Borel transformB[f ](t) =

∑∞n=0

cnn! t

n

(a) converges for |t| < 1/σ and

(b) has an analytic continuation to the striplike region |g − R+| < 1/σ satifying anexponential bound |B[f ](t)| ≤ #O(e|t|/R) (for every strip with 1/σ′ < 1/σ)

(c) the Borel resummation Sf(g) is an absolutely convergent integral43and agrees withf(g) in Ω, namely:

1

g

∫ ∞0dt e−t/gB[f ](t) = f(g) ∀g ∈ Ω (3.56)

note that this is the first version of the Borel resummation from (2.2); for generalcomplex (non-positive) g’s the two versions differ: the second occurs from the first by avariable substitution tII = tI/g, i.e. the integration range in the second case is not thepositive real axis, but (0, e−i arg g · ∞); then (b) guarantees this integral only for smallarg g

the restriction Re g−1 > 1/R follows naturally from the exponential bound in (b)

when bounded as assumed the Borel sum∑

n(σt)n indeed converges (at least) for|t| < 1/σ

what about the generic situation? so far the function f(g) had a uniform remainder ina ball Ω right of the point g = 0 around which we expanded it, whereas the strip in theBorel transform was around the positive real axis

the relation of the ball to the point can easily be changed by considering the rotationgeiγ which rotates t in the same way (linearity, transfer it via redefinition of cn)

if the expansion works for balls in all directions, then the function f has a finite radius ofconvergence and the Borel transform B[f ] should be entire with an exponential boundin all directions, this is precisely guaranteed by the union of strips in all directions,which cover the complex space (the coefficient in the exponent is the minimum of allradii, which indeed determines the radius of convergence of f etc.)

43even the integral with absolute value of the integrand exists

3-57

3.4.1 Illustration by examples

ex. (7) what about the famous e−1/g(2), that can be added to a function without changing its

asymptotic expansions?

if the theorem applied to say just e−1/g(2), the Borel transform would be the same

as without it, so zero, and so is the Borel resummation, thus f = 0; so one of theassumptions must be violated (in complex g-space, which we already used in Sec. 1.1.2to reveal the ‘problematic’ behavior of these functions)

for e−1/g2the limit g → 0 depends on the direction, since along the imaginary axis

the function diverges, cf. Sec. 1.1.2; on the other hand, if a function has an asymptoticseries as g → 0, then c0 is this limit, cf. Sec 2.2

this actually applies to e−1/gα for α > 1, since along arg g = φ the absolute value isexp(−1/rα · cos(φα)) and the argument of the cosine can be negative for φ large, i.e.close to π/2 (this argument seems not to apply to α = 1)

furthermore, the functions e−1/gα for 0 < α < 1 also dominate all powers and so leadto the same ambiguity in the asymptotic series for g → 0

the limit argument above does not apply here, but the uniform bound is not obeyed

the asymptotic series of these functions are identical to zero, so the remainder RN (g) isthe function itself (and thus N -independent); what is the maximum of |e−1/gα/gN | overg? the denominator wants to make this expression diverge near zero, but it is kept finiteby the exponential function; nevertheless, the maximum grows with N , but how? thederivative vanishes at g = (α/N)1/α, which is real and approaches 0 as N → ∞, so isin Ω; the value at this g is ( Neα)N/α and should be bounded by #σN N ! = #′(σNe )N

√N

which can obviously not be the case for 1/α > 1

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

Fig. 12: Contour lines of the absolute value of exp(−1/g2) (left) and exp(−1/√g) (right) in

the complex g-plane.

exp. growing coefficients: cn = βn giving f(g) = 1/(1 − βg) (not f(x)) and B[f ](t) =exp(βt)

on the g-side we have to check for what β the pole g∗ = 1/β is in Ω; it is outside of Ω,iff Re g−1

∗ = Re β < 1/R

on the t-side the absolute value |eβt| = eRe(βt) should be bound by the exponential e|t|/R,

3-58

which gives the same condition Re β ≤ 1/R

for σ look at the remainder, which is a geometric series, too (def.: (2.1)) RN (g) =∑∞N+1(βg)n = (βg)N+1

1−βg and inspect whether |RN | ≤ #(σ|g|)N+1(N + 1)! in Ω, thatshould be the case for all σ due to the factorial growth

indeed, B[f ](t) converges for all t, i.e. for all strip widths 1/σ

fact. growing coefficients cn = βnn!, where f(g) has zero radius of convergence andB[f ](t) = 1/(1− βt)

B[f ] has a strip up to its pole t = β−1 (which should not be close to the real axis, soβ should not give rise to nonalternating series), but there is no exponential bound onB[f ]

can we find B[f ] inbetween, that does have an exponential bound in the strip and thestrip being limited by a pole on say the negative t-axis

construct a Bose-Einstein like Borel transform

B[f ](t) =1

−α+ eβt0 < α < 1, β > 0 (3.57)

the second inequality guarantees exponential decay for t near the real axis, the firstinequality means that the pole t∗ = 1

β logα appears on the negative t-axis, i.e. 1/σ = t∗

the finite radius of convergence in B[f ] already tells us that the radius of convergenceof the (formal) series f(g) must be zero

in fact the inverse Borel (Laplace) transform is

Sf(g) =1

1 + βg1F2

(1, 1 +

1

βg; 2 +

1

βg

∣∣α) Re1

g> −β (3.58)

this hypergeom. function has a branch cut at α > 1, which is excluded here, but fromthe third argument poles appear at 2 + 1

βg = 0,−1,−2, . . .; together with the prefactor

this gives poles at g∗ = − 1β (1, 1

213 , . . .) which accumulate near zero and prevent a finite

radius of convergence

these zeros come from the left and are not in the ball Ω; actually the condition Re 1/g >−β means g outside of the ball of radius 1/2β around −1/2β + 0i, which excludesprecisely those poles

in Ω the theorem should give the function f(g); actually Sf(0) = 1/(1−α) (as the limitwhen approaching g = 0 not from negative g)

to check the expectation σ = β/ logα (from the Borel side) one has to analyse theuniform remainder of the hypergeometric function in the g-ball Ω

what about the radius R? the hypergeometric function seems to have no poles in theright half-plane Re g > 0, the exponential decay exp(−βt) with positive β is limited byexp(t/R) for all positive R; thus one can consider arbitrarily large balls Ω in g whichindeed cover the right half-plane

3-59

4 An(other) example towards resurgence

4.1 Introduction

Sine-Gordon model:

LEucl. =1

2

( ∂φ∂xµ

)2+ V (φ) V (φ) =

m4

g

(1− cos

(√gmφ))

(4.1)

degenerate minima of the potential ⇒ tunneling events = nonperturbative objects

in 1+1d: static kinks, breathers etc.

V is minimal at φ = 0 with the expansion m2

2 φ2 − g4! φ

4 + #φ6 + . . . meaning infinitely manyinteractions and m and g are mass and coupling constant

goal: asymptotic behavior of perturbation series and what to learn from it [21, DD, 35]

0 2 4 6 8 10g

0.6

0.8

1.0

1.2

Z(g)

Fig. 13: The partition function Z(g) for the Sine-Gordon reduced model (4.3) for positive g,for large g it decays to zero.

we perform the following reduction: no kinetic term (0d), m = 1, restrict φ to oneperiod:

Z(g) = c

∫ π/√g

−π/√gdφ exp

(1

g

[cos(√g φ)− 1)

])=

c√g

∫ π

−πdu exp

(1

g

[cosu− 1)

])(4.2)

the constant c could be conveniently fixed such that Z(g = 0) = 1 (see below); to makecontact to [DD] use g = 4λ and a different c

initially g > 0, but we will quickly consider negative and even complex g

[DD] also considers operator insertions of the form ei integer u, which yield very similarproperties

like for the integral in Sec. 2.3, there exists an exact solution44:

Z(g) =

√2π

ge−1/gI0(

1

g) (4.3)

44use for instance exp(1/g · cosu) = I0(1/g) +∑∞k=1 .. cos(ku), [CMSO, 571]

4-60

with the modified Bessel function, we have fixed : c = 1/√

2π

one can see that this is a well behaved function for g ∈ (0,∞), cf. Fig. 13

one can even write down the expansion for small g45, but we again pretend not to knowthis solution and approach the problem perturbatively

4.2 Perturbative expansion(s)

the integrand

consider the u-integrand in (4.2) without the prefactor 1/√g, cf. Fig. 14

-3 -2 -1 1 2 3u

0.2

0.4

0.6

0.8

1.0

exp([cos u -1]/g)

Fig. 14: The u-integrand in (4.2) without the prefactor 1/√g for g = 1, 1/2, 1/4 (black, blue,

red) and in the approximation cosu− 1 ≈ −u2/2 (lighter colors)

for small g it is well-approximated by exp(−u2/(2g)) which has width√g

together with the prefactor 1/√g the integral approaches 1 for g → 0 as expected46

this behavior also suggests how to obtain the perturbative series in small g: expand thedeviation from the Gaussian

yet another integral representation:

Z =1√

2πg1/4

∫ π/g1/4

−π/g1/4

dv exp(1

g

[cos(g1/4v)− 1)

])(4.4)

where we expand:

exp(1

g

[cos(g1/4v)− 1)

])= e− v2

2√g exp

( ∞∑k=2

(−1)k

(2k)!gk/2−1v2k

)︸ ︷︷ ︸a2(v)+a3(v)g1/2+a4(v)g...

(4.5)

45mathematica gives∑n=0 Γ(n+ 1/2)2/π/n! · (2g)n + ie−2/g∑(−1)n.. same ..

46in the opposite limit g →∞ the integrand becomes a constant and the prefactor makes it decay like 1/√g

4-61

as g → 0 the first factor becomes sharply peaked around v = 0 (like for u in Fig. 14),whereas the remainder is smooth there ⇒ Laplace method

the second factor can be cast into a series in g

exp(a2 + a3g1/2 + . . .) = 1 + (a2 + a3g

1/2 + . . .)1 +1

2(a2 + a3g

1/2 + . . .)2 + . . . (4.6)

where for a given order in g we only need a finite number of terms to write

exp(. . .)

=∞∑n=0

cn(v) gn/2 (4.7)

moreover, the ai are integer powers of v and so the cn’s are polynomials

we can extend the integration range from (−π/g1/4, π/g1/4) to (−∞,∞) with onlyexponentially small errors at every order47, which do not contribute to the asymptoticseries in g (a typical step in the Laplace method)

finally, one interchanges those integrals and the infinite g-sum to get [DD, (7.12) withq = 0 and 2λ = g/2]

Z(g) ∼ 1

π

∞∑n=0

Γ(n+ 1/2)2

n!

(g2

)n(g → 0+) (4.8)

. . . an asymptotic series, that diverges and is not even alternating

better derivation [CMSO, 265,269]

yet another integral representation starting with the right hand side of (4.2)

Z(g) =1√2πg

2

∫ π

0du exp

(− 2

gsin2 u

2

)=

1√πg

∫ 2

0dt

e−t/g√1− t/2

√t

(4.9)

(cross-check for the Jacobian: needs to diverge at both int. limits) with the expansion

t−1/2 1√1− t/2

= t−1/2∞∑n=0

Γ(n+ 1/2)√π n! 2n

tn (4.10)

and g = 1/x one can now apply Watson’s lemma for a finite interval (just preposition(ii) is not needed anymore) with A = −1/2, B = 1 arriving at (4.8)

actually the upper limit of the integral does not matter in the perturbative expansiong → 0

proofs: in the derivation above, the upper limit in the Watson lemma integral des notappear at all in the series expansion of it

alternatively, in the u-integral one can bound sin2(u/2) from above for any inter-val [u, π], hence the absolute value of the integral over that interval is bounded byexp(−that bound/g) and has zero asymptotic series

4-62

this raises the uniqueness question again: if we could Borel resum the perturbativeseries, which of these integrals (with varying upper bound) would we get

this nonconvergence of the series is typically rooted in some pole/brunch cut of thefunction, when the argument g is generalized to be complex

let’s analyse negative g’s (from our experience turning the above series alternating is‘the place to look for’)

the u-integrand as shown in Fig. 15 is now largest at the endpoints u = ±π, where itgrows like exp(−2/g) = exp(2/|g|), the width decreases like

√g as for positive g and is

again met by the prefactor

⇒ Z(g = 0) diverges when approached from negative g’s

moreover, Z(g) is purely imaginary for neg. g from the prefactor 1/√g

the expansion around u = π for negative g differs from the expansion around u = 0 forpositive g basically only by this factor since

cos(u− π + ε)− 1

g=

cos(u+ ε)− 1

−g− 2

g(g < 0) (4.11)

and the extension to the infinite integral (after expanding the smooth part) is the same;hence [DD, (7.14)]

Z(g) ∼ e−2/g

√−1

1

π

∞∑n=0

Γ(n+ 1/2)2

n!

(−g2

)n(g → 0−) (4.12)

where the imaginary factor emerges from the prefactor

471/g1/4 ·∫ π/g1/4

0dv exp(− v2

2√g) =

√π/2 erf(π/

√2g) = O(e−#/g) +

√π/2, where the latter is the infinite

integral

-3 -2 -1 0 1 2 3u

0.1

1

10

100

1000

104exp([cos u -1]/g)

Fig. 15: The u-integrand in (4.2) without the prefactor 1/√g – like in Fig. 14 – now on a

logarithmic scale and for negative g = −1,−1/2,−1/4 (black, blue, red) and in theapproximation cosu− 1 ≈ −2 + (u− π)2/2 (lighter colors).

4-63

for negative g nonalternating, too

note that the expansion around the vacuum u = 0(= v = φ) would become alternatingfor negative g, where the vacuum actually represents a maximum of the action/minimumof the integrand, around which the expansion makes no sense

likewise, the expansion around the saddle u = π makes no sense for positive g, but thereit would be alternating

4.3 Borel resummation

first attempt

large argument x if needed:

Z(g) =2

gf(2

g

), f(x) ∼ 1

π

∞∑n=0

Γ(n+ 1/2)2

n!x−n−1 (x→∞) (4.13)

Γ(n + 1/2) asymptotes like n! (Gevrey-1), so the conventional Borel transform withanother n! should suffice

actually Γ(n+ 1/2)2/n!2n→∞−→ 1/n like for the Airy function in Sec. 1.5.5, and indeed a

hypergeometric function appears

B[f ](t) =1

π

∞∑n=0

Γ(n+ 1/2)2

n!2tn = 2F1

(1

2,1

2, 1|t

)(4.14)

which also equals 2K(t)/π with K the complete elliptic integral of the first kind

2F1

(12 ,

12 , 1|t

)has a branch cut for t > 148, exactly on the positive axis, where we want

to integrate t

the jump means that an imaginary (nonperturbative) part is born = Stokes line

we can compute it first defining lateral Borel resummations (along different t-rays)

Sθf =1

g

∫ eiθ∞

0dt e−t/gB[f ](t) (4.15)

here 2.7.15

limε→0

[Sε − S−ε

]f(g) =

1

g

∫ ∞0dt e−t/g lim

ε→0

[1F2

(1

2,1

2, 1|t+ iε

)− 1F2

(. . . , |t− iε

)]=

1

g

∫ ∞1dt e−t/g

2πiΓ(1)

Γ(1/2)2 1F2

(1

2,1

2, 1|1− t

)[DD, (7.18)]

=2πi

gπe−1/g

∫ ∞0ds e−s/g 1F2

(1

2,1

2, 1| − s

)(4.16)

this ambiguity is, up to some imaginary and nonperturbative prefactor, exactly theBorel resummation of 1F2

(12 ,

12 , 1| − t

), i.e. (4.14) with alternating coefficients and this

series would appear from the expansion around the maximum (at g > 0), cf. (4.12)

48still B[f ] has a finite radius of convergence as expected from the original series being Gevrey-1

4-64

some formalization:

saddles: ‘vacuum’ at u = 0: I maximum at u = π: II

action: S := −1g

[cosu− 1

]= +2

g sin2 u2 SI = 0 SII = 2

g

partition function: Z = 1√2πg

π∫−πdu e−S = 1√

πg

2∫0

dt e−t/g 1√1−t/2

√t

(*)

series (formal): ΦI,II(g) = 1π

∑∞n=0

Γ(n+1/2)2

n!

(± g

2

)n a more convenient Borel resummation: A = −1/2, B = 1 (not writing them), ‘from

field to action variable’ [DD]:

B[ΦI,II ](t) ≡1

π

∞∑n=0

Γ(n+ 1/2)1

n!

(± t

2

)nt−1/2 =

1√π

1√1∓ t/2

√t

(4.17)

SΦI,II(g) =1√g

∫ ∞0dt e−t/gB[ΦI,II ](t) (4.18)

cf. (4.10), lateral Borel resummations SθΦ accordingly

the second case is Borel resummable for real g (and around), since the integrand is ok;a variable shift t→ t+ 2 yields

SΦII(g) = e2/g 1√πg

∫ ∞2dt

1√t/2− 1

√t

(4.19)

which is purely real

comparing to Z from (*) (watch the integration limits) one finally gets

Z(g) = Re SΦI(g) (4.20)

where the real part of SΦI(g) is unambiguous

the ambiguity in it is purely imaginary (well, Z is real by construction) and just pro-jecting onto the real part is also called ‘median summation’

this imaginary part occurs if the integral is extended to t ∈ [0,∞); the denominator√1− t/2 is certainly purely imaginary for t > 2, but depends on how one approaches

it (lateral sums!); thus the ambiguity is a sign whereas the value is given by the othersaddle

Im SΦI(g) = ±e−SII︷︸︸︷2/g SΦII(g) (4.21)

thus just Borel resumming the series around the vacuum gives rise to animaginary ambiguity, that is nonperturbative in the coupling and requiresthe nontrivial saddle to cancel it

this agreement/cancellation is for a whole series term-by-term

but remember that relative to the original integral interval [−π, π] the saddle at u± πis a maximum, this will be clarified in the next section

4-65

the semiclassical expansion for Z can in general be written as a trans-series∑A

σAe−SAΦA(g) (g → 0) (4.22)

where SA = #A/g and here σ1 = 1, σII = ±i for arg g = 0∓

4.4 Going complex: saddles and Lefschetz thimbles

as for the Airy function in Fig. 9, we complexify the coupling and the integration variable

want to see where asymptotic wedges are ‘good’, how lines (of constant absolute value and)of constant phase – also called Lefschetz thimbles49 – cover the complex u-space and whichones could be used for an alternative representation of the original integral: Fig. 16

let JI,II(θ) be the thimbles connected to the two saddles and asymptoting into goodwedges, depending on θ ≡ arg g

near50 real positive g: the contour JI (red) changes qualitatively from NW-SE for θ = 0−

(upper left panel) to SW-NE for θ = 0+ (middle left)

orientation: fixed to agree with the integral orientation of the interval

to be equivalent to the integration over the interval [0, π], one has to add the thimbleJII (green in upper left, blue in middle left) with a definite sign

define JII(θ ≈ 0) to be directed as S-N, which by periodicity of the integrand can beput to u = 0 or u = π

orientation: this way JII does not change abruptly51 for θ ≈ 0

in general, an integration cycle Σ, depending on θ again, can be written as a sum overthimbles

Σ(θ) =∑

thimbles A

nAJA(θ) (4.23)

with integer coefficients (intersection numbers) n, here we have

Σ(0−) = JI(0−) + JII(0

−) (4.24)

Σ(0+) = JI(0+)− JII(0+) (4.25)

the the jump in the coefficient (in sign) compensates for the different imaginary tails ofthe JI ’s

49for fields φ(x) they are defined by flow equations ∂φ(x;t)∂t

= − δS[φ]δφ(x;t)

and the initial condition that at large t

the function φ(x; t) reaches a saddle point/classical configuration φcl(x) defined by δS[φ]/δφ(x; t)∣∣φ=φcl

=

0; for n-dimensional integrals these are n-dimensional manifolds (varieties) (counting: complexified, butconstrained); this is related to quantization via complex Langevin equations . . .

50directly at real positive g, i.e. the Stokes line JI = JI(0) also touches the saddle II (upper right)51near real negative g, i.e. the second Stokes line, the role of I and II is interchanged (lower panels); verify

that purely imaginary g’s constitute an anti-Stokes line (middle right panel)

4-66

“So far we have made no approximations in our analysis. If one could evaluate theintegrals along the cycles exactly, one would obtain an exact result for Z.” [21]

how to understand the series ΦI,II and their Borel resummations in this context?

ΦI(g) is the perturbative expansion of the action integral along JI(g)

it came about by the Laplace method over u ∈ [−π, π] expanding the integrand nearu ≈ 0, where the action is minimal (and the integral weight is maximal), but still it isnot Borel resummable for real positive g

ΦII(g) is the perturbative expansion of the action integral52 along JI(g) and is real andresummable

formally for the resummed expansions [Unsal @ Lattice2016, Pham ’83!?]

S0±ΦI(g) =

∫JI(0±)

du e−S(u) (4.26)

(presumable for all θ and A)

just using JI and Borel resumming for θ = 0± one does not see the necessary contribu-tions from the thimble II, which here are purely imaginary and cancel the ambiguity

the coefficients nA (from geometry) agree with the coefficients σA in (4.22) (from trans-series)

52details: for u ∈ [π − i∞, π + i∞], the action becomes S = 2g

cosh2 y2

and indeed y = 0 has minimal action;one obtains (4.12), where the imaginary prefactor arises from the Jacobian and exp(−2/g) because thevariable subtitution must be 2 cosh2 y

2= 2 + t

4-67

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Fig. 16: Integrand exp(−2g sin2 u

2 ), (4.9) for g ∈ exp(−i/10), 1, exp(i/10), i,− exp(−i/10),−1(left to right) in the complex u-plane. Grey: contour lines of the absolute value(spaced according to logarithm) with the ‘empty regions’ having small values =‘good wedges’. Colors = phases of the integrand, red: 0, green and blue: slightlybelow and above 0, yellow: 0.637π, magenta: 0.1961π, black: the intervall [−π, π]line we need to integrate along (sometimes vertically shifted).

4-68

5 Expansions for QM eigenvalues: from WKB to resurgence

Schrodinger equation in 1d (for 2m = 1):

−~2ψ′′(x) + V (x)ψ(x) = Eψ(x) (5.1)

~2ψ′′(x) = Q(x)ψ(x) Q(x) = V (x)− E (5.2)

where Q ≷ 0 in the classically forbidden/allowed region E ≶ V (x)

writing −Q(x) = E−V (x) = p(x)2 (to compare to [GD]), p(x) can be viewed as some ‘locallydefined momentum’

the aim is to get the eigenvalues in the famous WKB approximation [Wentzel, Kramers,Brillouin 1926, Liouville, Green 1837, Jeffreys 1923]; here shortest and hopefully still com-prehensible derivation

5.1 WKB basics

ansatz

ψ(x) = eS(x) ⇒ ~2(S′ 2 + S′′) = Q(x) (5.3)

the latter is a Riccati (nonlinear ordinary differential) equation

want to cancel the ~ on the left hand side: S′′ ∝ 1/~2 ruled out [OC, 72]

S′ ≈ ±√Q

~⇒ S = ±

∫ x√Q

~+ ~0S1 (5.4)

next differential equation at O(~):

±~(√

Q′+ 2√QS′1

)+ ~2

(S′ 21 + S′′1

)= 0 (5.5)

S′1 ≈ −1

2(log

√Q)′ ⇒ S = ±

∫ x√Q

~− 1

4logQ+ ~1S2 (5.6)

ψ(x) ≈ 1

Q(x)1/4exp

(± 1

~

∫ x

xdy√Q)

(5.7)

where x is an arbitrary but fixed integration point [CMSO, 487] (changing it changesthe constant in front)

higher terms [CMSO, 486]:

ψ(x) ∼ exp[1

~

∞∑n=0

~nSn(x)], (5.8)

S2 = ±∫ x [ Q′′

8Q3/2− 5(Q′)2

32Q5/2

], S3 = − Q′′

16Q2+

5(Q′)2

64Q3etc. (5.9)

only after the first two terms the series could converge in ~, since the correction ise~S2 = 1 +O(~S2)

but WKB expansions are typically only asymptotic expansions (~→ 0), related is:

5-69

WKB is a ‘singular perturbation theory’, since ~ multiplies the highest derivative in thedifferential equation (5.2) and setting it to zero changes the order of it

when exact? S2,3,... = 0 for Q′ = 0: piecewise constant potentials

5.2 Turning points and uniform WKB

WKB breaks down at a turning point Q(x) = 0, i.e. where V (x) = E and the classicalparticle is reflected

not only ~S1 6S0 but Q = 0 spoils all higher Sn

for more on validity of WKB, see [CMSO, 493]

let Q(x) be negative (class. allowed) for x < 0 and positive (class. forbidden) for x > 0,for technical reasons Q(x) 1/x2 (x → ∞), and linear with positive slope Q = ax +O(x2), a > 0 near the origin = linear turning point

and let’s analyze ψ(x) as ~→ 0 with boundary condition ψ(+∞) = 0

long version [CMSO, 505]:

(1) use ψ from (5.7) away from the turning point: negative sign in S0 for large posi-tive x (boundary condition!) and both signs for largely negative x; around the originapproximate Q and use the exactly known Airy functions Ai and Bi

(2) expand the outer approximations in ‘small’ x and the inner approximations in ‘large’x, now the functional forms of those wave function pieces agree and one can match them,i.e. fix the normalization constants; actually no Bi-part

to be precise, the matching is done in the window ~2/3 |x| ~2/5, which enlarges as~→ 0

short result [CMSO, 510] = uniform asymptotic approximation [Langer, 1935]

ψ(x) = #(3∫ x

0

√Q

2~

)1/6 1

Q(x)1/4Ai((3

∫ x0

√Q

2~

)2/3)(~→ 0) (5.10)

watch the prominent role of the Airy function, to which it reduces exactly, when Qis linear (

∫ x√Q ∝ x3/2 etc., we know that Ai is the solution of that problem decaying

at large x)

valid everywhere: at the turning point x = 0 (note that we fixed the integral range to

start there), both( ∫ x

0

√Q)1/6

and Q(x)1/4 go like x1/4; for negative x√Q has phase

e3πi/2 (is purely imaginary) such that its power ..2/3 is negative ⇒ Ai oscillates

no quantization of energy, since we are in the continuous spectrum for that potential(out of the two constants in the second order differential equation, only one is fixed atpos. x, no constraint at negative x)

5-70

two turning points x1 < x2 with Q < 0 in between [CMSO, 519] quantize energies

1

~

∫ x2

x1

√−Q(x) dx =

(n+

1

2

)π +O(~) (5.11)

by matching one-turning-point WKBs which are oscillatory between the two turningpoints, this means matching sines which are identical up to multiples of π

this generalizes the Bohr-Sommerfeld quantization ← only the nπ term53

in view of conventional WKB (5.7) and (5.10) the following ansatze (derived in moredetail in [GD]) are uniform WKBs:

ψ(x) =1√R′(x)

exp

(h−1R(x)

)Ai(h−2/3R(x)

)D− 1

2

(h−1/2R(x)

) (5.12)

where D is the parabolic cylinder function obeying D′′ν(z) +(ν + 1

2 −z2

4

)Dν(z) = 0

again, the Schrodinger equation becomes a nonlinear differential equation for the ansatzfunction R(x):

~2(3(R′′)2

4(R′)4− R′′′

2(R′)3

)+ (R′)2

1

R

R2/4

= Q (5.13)

to leading order in ~ this is solved by (check)

R =

±∫ x√

Q(± 3

2

∫ x√Q)2/3(

± 4∫ x√

Q)1/2 (5.14)

this is actually the exact solution, iff R′′ = R′′′ = 0, i.e. for const./linear/quadratic Q

thus the latter ansatz is best suited for quadratic turning points

note that the harmonic oscillator (recall 2m = 1 and set ω = 2) has Q quadratic plusconstant (E)

−~2ψ′′ + (x2 − E)ψ = 0 ⇒ ψ ∝ D− 12± E

2~

(±√

2

~x)

(5.15)

and thus D has variable index ν; for integer ν, i.e. for the known energy eigenvaluesE = (ν + 1/2)2~, reduces to gaussian times Hermite polynomial

53BS is known to miss the ground state energy of the harmonic oscillator and thus the 1/2 on the right handside, that comes form S1

5-71

5.3 The double well (mexican hat) potential

definition

V (x) =1

2x2(1− gx)2 (5.16)

is positive and symmetric around x = 1/2g, has minima at x = 0 and x = 1/g and becomesa harmonic oscillator around the origin, V (x) ≈ x2/2, in the limit of weak coupling g → 0

rescaling y = gx the Schrodinger equation becomes:

~2

mg4ψ′′(y) =

(y2(1− y)2 − 2Eg2

)︸ ︷︷ ︸Q(y)

ψ(y) (5.17)

where we have set m = 1 and ~ = 1, since we will treat the coupling g2 as the constantthat goes to zero in a WKB-approximation (again a singular perturbation); in this weakcoupling limit, the potential will become harmonic at the same time

in view of the discussion above, Dunne/Unsal [16] make an ansatz

ψ(y) =1√u′(y)

Dν

(1

gu(y)

)(5.18)

again a differential equation for u(y) follows, in leading order this is conventional WKB

the philosophy is to simultaneously expand u(y) and ν and thus E in small coupling g2

these series are divergent, non-alternating and thus not Borel-resummable

the late term behavior was conjectured long ago [19] and numerically checked [20]

E0 ∼∑n

3nn!g2n (5.19)

in contrast, the perturbative series for other systems is divergent, but alternating andthus Borel resummable, e.g. the Zeeman effect E =

∑n(−1)nΓ(2n+ 3/2)( 4

π )5/2B2n [22]

reason: real g are Stokes lines, imaginary g give the leading term y4 a negative prefactor(in the real part of V (y)): instability

lateral Borel sums around positive real g2 give an imaginary, nonpertubative am-biguity i exp(− 1

3g2

)to the lowest (and all other) eigenvalues

(along the lines of (3.36) with x = 1/g2 and positive β = 3: the residue is 1 at t = 1/3g2)

in the framework of (5.18) one needs the asymptotics of Dν(z) for large complex z; notjust zν exp(−z2/4) (times a series in 1/z2) as valid for large real z, but another termz−ν−1 exp(z2/4) is present near the real axis

these are expansions around the local minima and one has to do a matching again

the ground and first excited state are even resp. odd around 1/2g and thus obey vonNeumann resp. Dirichlet boundary conditions there

5-72

these boundary conditions have to be anlyzed consistently for the entire Borel resum-mation and a quantization condition follows [16, 20] that contains perturbative andnonperturbative functions

5.3.1 Instantons and their effects

in the Euclidean action of the model, here m = 1

L =1

g2

( y2

2+ V (y)

), V (y) =

1

2y2(y − 1)2 (5.20)

there are classical solutions representing tunneling from one minimum to the other orback

Euclidean action amounts to motion in the inverted potential V = −V ≤ 0, which nowhas two hills

∃ solutions that ‘roll’ from say y = 0 in the infinite past to y = 1 in the infinite future

in a finite time range they visit the valley y = 1/2, the former maximum

the conserved energy y2/2 + V (y) for them is zero (indeed y ≥ 0 for V ≤ 0) ⇒ firstorder differential equation ⇒

explicit solution

y(t) =1

2(1± tanh

t− t02

) =1

1 + exp(∓ (t− t0)

) (5.21)

where the sign distinguishes forward from backward motion between the maxima andt0 is the time of this event or ‘(anti)instanton’ or ‘(anti)kink’, where the minimum isvisited

they approach their initial and final value exponentially with a decay constant that alsoappears as the slope at t0 (inverse mass)

action of kinks∫ ∞−∞

dt y(t)2 zero energy=

∫ ∞−∞

dt (−V (y)) =

∫ ∞−∞

dt V (y) =1

12, S =

1

6g2(5.22)

one can see the similarity to WKB-quantization (5.11) with two turning points y = 0, 1:∫ ∞−∞

dtdy

dt

√V (y) =

∫ y(t=∞)=1

y(t=−∞)=0dy√V (y) (5.23)

the energy difference of the two lowest states, which in the harmonic approximationwould be degenerate, can be shown to be proportional to this action [17, 20]

∆E = 2a, a =1√π|g|

exp(− 1

6g2

)(5.24)

5-73

systematically: Euclidean path integral with periodic boundary conditions

the energy eigenvalues can be obtained by ‘spectroscopy’ Z(T )T→∞−→ e−E0T

perturbation theory around the vacuum (= trivial static solution at one minimum) donetherein

for that one would need a solution starting and ending at the same minimum

(correct argument? check against (46) in [16]!)

kink-antikink-profile

y(t) = y+,−t0(t) + y−,t0(t)− 1 =1

1 + exp(−(t+ t0))+

1

1 + exp(t− t0)− 1 (5.25)

where we have fixed the center of mass to 0

alternative: y(t) = y+,−t0(t) for t < 0 and y−,t0(t) for t > 0 [17]

becomes a solution in the limit of large separations θ ≡ 2t0 →∞

at large but finite θ the equations of motion are spoiled by the exponential tails and theaction is twice 1/6g2 plus some ‘interaction’

eventually, at small θ the overlap is strong, the other minimum is not reached anymoreand the configuration becomes a fluctuation around one minimum

stability: while a single (anti)kink is protected by its boundary conditions y(t→∞)−y(t → ∞) = ±1 (changing the boundary condition one leaves the minimum of thepotential and the action integral gives rise to an infinite ‘barrier’), the kink-antikink isnot: it has ‘topological charge’ zero, i.e. is topologically neutral, like the vacuum thathas lowest possible action 0

consequently, the interaction potential is exp. small and attractive:

Uint(θ) = − 2

g2e−2θ (5.26)

the contribution to the path integral:

Zint ∝ a2

∫dθ e−Uint(θ) = dilute gas +

∫ ∞0dθ(

exp( 2

g2e−2θ

)− 1)

(5.27)

the integrand at vanishing θ, exp(−2/g2), diverges in the small coupling limit and thusthe integral in this limit is dominated by small θ, in which the approximation breaksdown

complexify coupling to g2 = −λ2 < 0∫ ∞0dθ(

exp(− 2

λ2e−2θ

)− 1)≈ −γE

2+

1

2log

λ2

2(5.28)

since the integrand is −1 in a θ-interval from 0 to log λ

ambiguous imaginary part (from log) when λ2 → g2 = −λ2 [GD]

5-74

this multiplies the kink-antikink amplitude a2 ∝ exp(

13g2

), and thus gives the same

imaginary nonperturbative ambiguity as the lateral Borel sums of the per-turbation series (5.19)

⇒ perturbation theory encodes nonperturbative information = resurgence 23.7.15

for another (original) approach to resurgence in this model see Bogomolny [17]

the expansion of energies not only contains exp(−1/6g2), exp(−2/6g2), etc., but alsologarithms in 1/g2 from quasi-zero modes: changing the center of mass of the configu-ration (parameter) does not change the action at all = trans-series

6 Outlook

similar resurgence mechanisms occur for

double well in SUSY QM [23]

sine-Gordon potential = splitting into bands [18]

perturbation theory around kink-antikink⇒ canceled by ambiguity in kink-kink-antikink-antikink ⇒ resurgence tower [16]

systems where instantons are not real, but occur at complexified couplings [24]

quantum field theory?

CP(N-1) models in 2d← asympotically free and contain instantons, that fractionalize[25] and whose interaction becomes exponential in the distance, when phase boundaryconditions are required in the compactified time-like direction: resurgence shown in thesemi-classical limit of small radius

systems without topologically protected instantons, but with saddle points like O(N)[26] and principal chiral models [27]

SU(N) gauge theories?? instantons exist and can also fractionalize into magnetic monopoleswhen one direction is compact [28]; additional problems due to (UV-)renormalization:‘renormalons’ [29]

6-75

7 Appendix

7.1 How to get the series (1.11) in the Stirling formula

plausibility only

Taylor expand the logarithm of the Gamma function:

log Γ(1 + x) = −γx+∞∑k=2

ζ(k)(−1)k

kxk (7.1)

with the Riemann zeta function

ζ(s) =∞∑n=1

n−s (7.2)

(converging for Re s > 1)

the second derivative, also called polygamma ψ(1), is then

d2

dx2log Γ(1 + x) =

∞∑k=2

ζ(k)(−1)k(k − 1)xk−2 = . . .

=

∞∑n=1

1

(x+ n)2= − 1

x2+

∞∑n=0

1

(x+ n)2(7.3)

use the Euler-Maclaurin formula, an improved trapezoidal rule for integration, as aformula for a sum

b∑n=a

f(n) =

∫ b

ady f(y) +

f(b) + f(a)

2+

∞∑k=1

B2k

(2k)!

[f (2k−1)(b)− f (2k−1)(a)

](7.4)

for the function f(y) = 1(x+y)2 and a = 0, b =∞,

∞∑n=1

1

(x+ n)2=

∫ ∞0dy

1

(x+ y)2+

1(x+∞)2 + 1

(x+0)2

2

+∞∑k=1

B2k

(2k)!(−1)2k−1(2k)!

[ 1

(x+∞)2k+1− 1

(x+ 0)2k+1

](7.5)

=1

x+

1

2x2+∞∑k=1

B2k

x2k+1(7.6)

integrating twice we obtain, with (7.3)

log Γ(1 + x) = const. + linear + x log x− x+log x

2+∞∑k=1

B2k

2k(2k − 1)x2k−1(7.7)

which upon x! = Γ(1 + x) gives (1.11)

7-76

7.2 How to derive integral representations of the Airy functions

start from the differential equation y′′ − zy = 0 and represent

y(z) =

∫Cdt F (t)eizt (7.8)

the complex contour will be fixed after F (t) has been; representing z by a partial differentiationwrt. t and integration by parts we demand

iF (t)eizt∣∣∣C−∫Cdt (t2 + i

d

dt)F (t) · eizt = 0 (7.9)

the integral vanishes for

F = #eit3/3 (normalization) choice: # =

1

2π(7.10)

and the boundary contribution (the contour C cannot be closed, since the integrand has nopoles)

1

2πei(t

3/3+zt)|C (7.11)

vanishes, iff for large |t| the imaginary part of (..) is negative, i.e. – neglecting the secondsubdominant summand –

sin(3 arg t) > 0 arg t ∈ (0,π

3) ∪ (

2π

3, π) ∪ (

4π

3,5π

3) (7.12)

the same ‘good’ wedges as in Eq. (1.49) and Fig. 8

one can choose two rays slightly above the real axis, arg t = ε and arg t = π − ε, into whichthe contour for Ai with real z could have been deformed, thus these are smoothly connectedto the Airy function originally defined for real z in (1.44)

Ai(z) =1

2π

∫ ∞−∞

dt ei(t3

3+zt) =

1

π

∫ ∞0dt cos(

t3

3+ zt) (7.13)

using the third wedge around the negative imag. axis one defines Bi:

Bi(z) =1

2π

( ∫ −i∞−∞

+

∫ −i∞∞

)ei(

t3

3+zt) =

1

π

∫ ∞0dt[

exp(− t3

3+ zt) + sin(

t3

3+ zt)

](7.14)

note that the definitions in [CMSO, 313-314] use different integrands and contours

7-77

-30 -25 -20 -15 -10 -5 5

1

2

3

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

Fig. 17: Summary of the Airy function Bi. Left: real arguments, middle and right: absolutevalue (log) and phase for complex argument (cf. Fig. 10 for Ai).

7-78

Paper references

[1] M. Berry, “Stokes’ phenomenon; smoothing a Victorian discontinuity”, Publ. Math. In-stitut des Hautes tudes scientifique, 68 (1989) 211,https://michaelberryphysics.files.wordpress.com/2013/07/berry190.pdf

[2] M. Berry, “Infinitely many Stokes smoothings in the Gamma function”, Proc. R. Soc. A434 (1991) 465,https://michaelberryphysics.files.wordpress.com/2013/07/berry222.pdf

[3] M. Berry, “Uniform asymptotic smoothing of Stokess discontinuities”, Proc. R. Soc. A422 (1989) 7,https://michaelberryphysics.files.wordpress.com/2013/07/berry181.pdf

[4] Joel Feldman, Supplementary Notes to Math 320, University of British Columbia Van-couver, 2008, http://www.math.ubc.ca/∼feldman/m320/

[5] J.D. Jackson, “From Alexander of Aphrodisias to Young and Airy”, Phys.Rep. 320 (1999)27

[6] J.C. LeGuillou, J. Zinn-Justin, “Large order behavior of Perturbation Theory”, Elsevier,1990 [collection of papers with introduction]

[7] P. Pisani, ”Spectral functions in QFT”, 1505.04237.

[8] T. Blum, A. Denig, I. Logashenko, E. de Rafael, B. Lee Roberts, T. Teubner, G. Venan-zoni, “The Muon (g-2) Theory Value: Present and Future”, 1311.2198.

[9] T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, “Complete Tenth-Order QED Contri-bution to the Muon g-2”, Phys. Rev. Lett. 109 (2012) 111808, 1205.5370.

[10] J. S. Schwinger, “On gauge invariance and vacuum polarization”, Phys. Rev. 82 (1951)664.

[11] F. J. Dyson, “Divergence of perturbation theory in quantum electrodynamics”, Phys.Rev. 85 (1952) 631.

[12] W. Heisenberg and H. Euler, “Consequences of Dirac’s theory of positrons” [“Fol-gerungen aus der Diracschen Theorie des Positrons”] Z. Phys. 98 (1936) 714, arxiv:physics/0605038.

[13] G. V. Dunne, “The Heisenberg-Euler Effective Action: 75 years on”, Int. J. Mod. Phys.A 27 (2012) 1260004, 1202.1557.

[14] G. V. Dunne, “Heisenberg-Euler effective Lagrangians: Basics and extensions”, in Shif-man, M. (ed.) et al.: From fields to strings, vol. 1, 445, hep-th/0406216

[15] A. D. Sokal, “An improvement of Watson’s theorem on Borel summability” J. Math.Phys. 21 (1980) 261

[16] G. V. Dunne and M. Unsal, “Uniform WKB, Multi-instantons, and Resurgent Trans-Series’,’ Phys. Rev. D 89 (2014) 10, 105009, 1401.5202.

[17] E. B. Bogomolny, “Calculation Of Instanton - Anti-instanton Contributions In Quantum

7-79

Mechanics”, Phys. Lett. B 91 (1980) 431.

[18] J. Zinn-Justin, “Multi - Instanton Contributions in Quantum Mechanics”, Nucl. Phys. B192 (1981) 125; J. Zinn-Justin, “Multi - Instanton Contributions in Quantum MechanicsII”, Nucl. Phys. B 218 (1983) 333.

[19] E. Brezin, G. Parisi and J. Zinn-Justin, “Perturbation Theory at Large Orders for Po-tential with Degenerate Minima”, Phys. Rev. D 16 (1977) 408.

[20] J. Zinn-Justin and U. D. Jentschura, “Multi-instantons and exact results I: Conjectures,WKB expansions, and instanton interactions”, Annals Phys. 313 (2004) 197, quant-ph/0501136.

[21] A. Cherman, D. Dorigoni and M. Unsal, “Decoding perturbation theory using resurgence:Stokes phenomena, new saddle points and Lefschetz thimbles”, 1403.1277

[22] J. E. Avron et al., “Bender-Wu Formula, the SO(4,2) Dynamical Group, and the ZeemanEffect in Hydrogen”, Phys. Rev. Lett. 43 (1979) 691.

[23] I. I. Balitsky and A. V. Yung, “Instanton Molecular Vacuum in N = 1 SupersymmetricQuantum Mechanics”, Nucl. Phys. B 274 (1986) 475.

[24] G. Basar, G. V. Dunne and M. Unsal, “Resurgence theory, ghost-instantons, and analyticcontinuation of path integrals”, JHEP 1310 (2013) 041, 1308.1108.

[25] F. Bruckmann, “Instanton constituents in the O(3) model at finite temperature”, Phys.Rev. Lett. 100 (2008) 051602, 0707.0775

[26] G. V. Dunne and M. Unsal, “Resurgence and Dynamics of O(N) and GrassmannianSigma Models”, 1505.07803.

[27] A. Cherman, D. Dorigoni, G. V. Dunne and M. nsal, “Resurgence in Quantum FieldTheory: Nonperturbative Effects in the Principal Chiral Model”, Phys. Rev. Lett. 112(2014) 021601, 1308.0127.

[28] T. C. Kraan and P. van Baal, “Periodic instantons with nontrivial holonomy”, Nucl.Phys. B 533 (1998) 627, hep-th/9805168; K. M. Lee and C. h. Lu, “SU(2) calorons andmagnetic monopoles”, Phys. Rev. D 58 (1998) 025011, hep-th/9802108.

[29] M. Beneke, “Renormalons”, Phys. Rept. 317 (1999) 1, hep-ph/9807443.

Contents

1 Introduction by examples 1-11.1 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.1.1 Eight examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11.1.2 Going complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

1.2 Tunnelling in Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . 1-31.3 Stirling’s formula (Gamma function) . . . . . . . . . . . . . . . . . . . . . . . 1-3

1.3.1 Raw asymptotics from Laplace method . . . . . . . . . . . . . . . . . 1-41.3.2 Improved asymptotics and its limitations . . . . . . . . . . . . . . . . 1-4

7-80

1.3.3 Going complex: a puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . 1-71.4 The error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8

1.4.1 Definition and asymptotic series . . . . . . . . . . . . . . . . . . . . . 1-81.4.2 Going complex: the Stokes phenomenon . . . . . . . . . . . . . . . . . 1-101.4.3 A magic computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12

1.5 The Airy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-131.5.1 Rainbows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-131.5.2 Definition and some basic properties . . . . . . . . . . . . . . . . . . . 1-151.5.3 Saddle points and steepest descent integration . . . . . . . . . . . . . 1-171.5.4 Going complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-191.5.5 Asymptotic series expansion . . . . . . . . . . . . . . . . . . . . . . . . 1-221.5.6 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25

2 Asymptotic series 2-292.1 Remainder and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-292.2 Some properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-312.3 An integral counting Feynman diagrams . . . . . . . . . . . . . . . . . . . . . 2-332.4 Particle in magnetic field: Zeta function regularization and asymptotic series 2-35

2.4.1 Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-362.4.2 Zeta function, heat kernel, Mellin transform, Schwinger proper time . 2-382.4.3 Result and interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 2-40

2.5 Asymptotic series in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-442.5.1 Good news: the anomalous magnetic dipole moment of the muon (‘g-2’) 2-442.5.2 Bad news: the Dyson instability . . . . . . . . . . . . . . . . . . . . . 2-452.5.3 Good news again: Euler-Heisenberg effective action and pair creation 2-46

3 Borel resummation and transform 3-493.1 Idea and nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-493.2 Watson’s lemma and Laplace’s method . . . . . . . . . . . . . . . . . . . . . . 3-503.3 Some properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-523.4 The Watson-Nevanlinna-Sokal Theorem . . . . . . . . . . . . . . . . . . . . . 3-56

3.4.1 Illustration by examples . . . . . . . . . . . . . . . . . . . . . . . . . . 3-58

4 An(other) example towards resurgence 4-604.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-604.2 Perturbative expansion(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-614.3 Borel resummation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-644.4 Going complex: saddles and Lefschetz thimbles . . . . . . . . . . . . . . . . . 4-66

5 Expansions for QM eigenvalues: from WKB to resurgence 5-695.1 WKB basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-695.2 Turning points and uniform WKB . . . . . . . . . . . . . . . . . . . . . . . . 5-705.3 The double well (mexican hat) potential . . . . . . . . . . . . . . . . . . . . . 5-72

5.3.1 Instantons and their effects . . . . . . . . . . . . . . . . . . . . . . . . 5-73

6 Outlook 6-75

7-81

7 Appendix 7-767.1 How to get the series (1.11) in the Stirling formula . . . . . . . . . . . . . . . 7-767.2 How to derive integral representations of the Airy functions . . . . . . . . . . 7-77

Paper references 7-79

7-82