リサージェンス理論に基づく非摂動効果の理解 - 中央大学...Non-Borel...

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リサージェンス理論に基づく非摂動効果の理解

2017/06/15 21:07Research and Education Center for Natural Sciences, Keio University

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Chuo U. physics colloquium@Kasuga campus, Chuo U. 08/28/19

はじめに：摂動論とリサージェンス理論

摂動論：H0の固有状態に基づいて，量子揺らぎを計算

非摂動解析：摂動パラメタの大きな領域ではHを対角化し厳密な固有値を得る必要あり

量子論における摂動論と非摂動解析

H = H0 + g2 H � g2 � 1 H = H0 + g2 H � g2 � 1

自明な固定点(摂動的真空)

摂動級数

経路積分と固定点

非自明な固定点

非摂動的寄与

�S

��= 0

e�Ssol ⇠ e�Ag2

Z =

ZD� exp(�S[�]) =

X

�2saddlesZ�

Z0 =1X

q=0

aqg2q

Z� /

摂動級数と非摂動的寄与の関係

X

q=0

aqg2q

exp

� Ag2

�

❌

「摂動的寄与と非摂動的寄与は関連付かない異なる寄与」というのが一般的な見方

摂動級数非摂動的寄与

本当にそうだろうか？

摂動計算とボレル和[29]. The divergence encodes physical information about the saddles of ordinary integrals, orpath integrals of quantum mechanics and quantum field theory, as a consequence of Darboux’s

theorem [1, 3]. We recall a few relevant definitions and motivate (known) generalizations of

those definitions by using simple quantum mechanics.

Let P (g2) denote a perturbative asymptotic series that satisfies the “Gevrey-1” condition:

P (g2) =⇥�

q=0

aqg2q, Gevrey � 1 : |aq| ⇥ CRqq! (6.1)

for some positive constants C and R [5, 7]. Known examples of perturbative series that arise

in quantum mechanics and QFT satisfy the “Gevrey-1” condition [29]. We denote the Borel

transform of P (�) by BP (t) and define it as

BP (t) :=⇥�

q=0

aqq!tq. (6.2)

The formal Borel transform determines “a germ of a holomorphic function” at t = 0, with

a finite radius of convergence. Next, one analytically continues the obtained germ BP (t)

to the whole complex t-plane, called the Borel plane. We also assume that the analytic

continuation of the Borel transform BP (t) is “endlessly continuable”. That roughly means

that the function is represented by an analytic function with a discrete set of singularities

(poles or cuts) on its Riemann surface. The Borel resummation of P (g2), when it exists, is

defined as the Laplace transform of the analytic continuation of the germ:

B(g2) = 1g2

⇥ ⇥

0BP (t)e�t/g

2dt . (6.3)

In quantum theories with multiple-degenerate vacua, (but no instability of any kind), per-

turbation theory is typically a non-alternating Gevrey-1 series, hence non Borel resummable

[20, 21, 24, 26, 27, 29]. Non-Borel summability means that there is no unique answer in

perturbation theory; i.e., resummed perturbation theory does not produce a unique answer

for a physical observable which ought to be unique, for example, the ground state energy. Of

course, this is senseless. This means that perturbation theory (re-summed or otherwise) is

insu⇤cient to define the theory.

In certain cases, a perturbative sum which is not Borel summable becomes Borel summable

upon continuation g2 ⇤ �g2, see Fig. 2. In simple quantum mechanics, let us mention anexample that is directly relevant for our purpose [21]. Perturbation theory for the peri-

odic potential V (x) = 1g2 sin2(gx) is non-Borel summable, whereas perturbation theory for

V (x) = 1g2 sinh2(gx) is Borel summable. [Recall and compare with the 0-dimensional parti-

tion functions discussed in Section 1.6]. Both series are, of course, asymptotic and divergent.

The di�erence between the two is that the asymptotic series which arises in the first case is

non-alternating, whereas the series in the latter is just the alternating version of the former.

Let us refer to the Borel resummed series for the latter, Borel resummable series, as B0(g2).

– 50 –

2

I. INTRODUCTION

d2ψ

dx2=

2m(V (x)− E)!2 ψ (1)

[H0 + g

2Hpert]ψ(x) = Eψ(x) (2)

S =

∫dt

[m

2

(dx

dt

)2− V (x)

](3)

SE =

∫dτ

[m

2

(dx

dτ

)2+ V (x)

](4)

⟨x = a|e−iHt/!|x = b⟩ =∫

d[x(t)] eiS[x(t)]/! (5)

⟨x = a|e−Hτ/!|x = b⟩ =∫

d[x(τ)] e−SE [x(τ)]/! (6)

P (a → b) ≈ e−1!∫ ba dx

√2mV (x) (7)

ボレル変換：有限の収束半径を持つ級数に変換ボレル和：元の摂動級数を漸近級数として持つ解析関数

摂動級数（漸近級数と仮定）は一般に階乗発散し収束半径0 aq / q!

高次まで摂動計算を行っても意味のある情報は得られなさそうだが…

摂動計算とボレル和

2

I. INTRODUCTION

d2ψ

dx2=

2m(V (x)− E)!2 ψ (1)

[H0 + g


S =

∫dt

[m

2

(dx

dt

)2− V (x)

](3)

SE =

∫dτ

[m

2

(dx

dτ

)2+ V (x)

](4)

⟨x = a|e−iHt/!|x = b⟩ =∫

d[x(t)] eiS[x(t)]/! (5)

⟨x = a|e−Hτ/!|x = b⟩ =∫

d[x(τ)] e−SE [x(τ)]/! (6)

P (a → b) ≈ e−1!∫ ba dx

√2mV (x) (7)

[29]. The divergence encodes physical information about the saddles of ordinary integrals, or

path integrals of quantum mechanics and quantum field theory, as a consequence of Darboux’s




P (g2) =⇥�

q=0





BP (t) :=⇥�

q=0

aqq!tq. (6.2)








B(g2) = 1g2

⇥ ⇥

0BP (t)e�t/g

2dt . (6.3)
















– 50 –

of the Borel transform. The Borel transform method is applicable to the following class ofdivergent series (called Gevrey-1)

P (g2) =1X

q=0

aq(g2)q, |aq| ∑ Cq!µ

1A

∂q, (12)

where C,A are constants. The Borel transform BP (t) is defined as

BP (t) =1X

q=0

aqq!

tq, (13)

and the Borel resummation B(g2) is defined as

B(g2) =Z 1

0

dt

g2e°t/g

2BP (t). (14)

One can easily see that the Borel resummation B(g2) reproduces the original sum P (g2) correctlywhenever one can exchange the integral and the sum. Otherwise, we need to define the sum interms of the Borel resummation.

As a simplified toy model, let us consider a factorially divergent series of the following onewith alternating signs

P (g2) = C1X

q=0

q!µ°g2

A

∂q. (15)

Then the Borel transform becomes an analytic function without singularities on the positive realaxis

BP (t) = C1X

q=0

µ°tA

∂q=

CA

A + t. (16)

Therefore the Borel resummation is well-defined as an integral along the positive real axis

B(g2) =Z 1

0

dt

g2e°t/g

2 CA

A + t. (17)

This altenating factorially divergent series is a typical example of Borel summable divergentseries.

On the other hand, if perturbation series is not alternating, the factorially divergent seriesgives the Borel transform with singularities on positive real axis and the Borel resummmationhas imaginary ambiguities. For instance, suppose that the perturbation series Ppert(g2) givesnon-alternating factorially divergent series like

Ppert(g2) = C1X

q=0

q!µ

g2

A

∂q. (18)

The Borel transform has a singularity on positive real axis

BPpert(t) = C1X

q=0

µt

A

∂q=

CA

A ° t , (19)

Bpert(g2) =Z 1

0

dt

g2e°t/g

2 CA

A ° t . (20)

ボレル和






P (g2) =⇥�

q=0





BP (t) :=⇥�

q=0

aqq!tq. (6.2)








B(g2) = 1g2

⇥ ⇥

0BP (t)e�t/g

2dt . (6.3)
















– 50 –

ボレル変換



2

I. INTRODUCTION

d2ψ

dx2=

2m(V (x)− E)!2 ψ (1)

[H0 + g


S =

∫dt

[m

2

(dx

dt

)2− V (x)

](3)

SE =

∫dτ

[m

2

(dx

dτ

)2+ V (x)

](4)

⟨x = a|e−iHt/!|x = b⟩ =∫

d[x(t)] eiS[x(t)]/! (5)

⟨x = a|e−Hτ/!|x = b⟩ =∫

d[x(τ)] e−SE [x(τ)]/! (6)

P (a → b) ≈ e−1!∫ ba dx

√2mV (x) (7)

of the Borel transform. The Borel transform method is applicable to the following class ofdivergent series (called Gevrey-1)

P (g2) =1X

q=0

aq(g2)q, |aq| ∑ Cq!µ

1A

∂q, (12)

where C,A are constants. The Borel transform BP (t) is defined as

BP (t) =1X

q=0

aqq!

tq, (13)

and the Borel resummation B(g2) is defined as

B(g2) =Z 1

0

dt

g2e°t/g

2BP (t). (14)

One can easily see that the Borel resummation B(g2) reproduces the original sum P (g2) correctlywhenever one can exchange the integral and the sum. Otherwise, we need to define the sum interms of the Borel resummation.

As a simplified toy model, let us consider a factorially divergent series of the following onewith alternating signs

P (g2) = C1X

q=0

q!µ°g2

A

∂q. (15)

Then the Borel transform becomes an analytic function without singularities on the positive realaxis

BP (t) = C1X

q=0

µ°tA

∂q=

CA

A + t. (16)

Therefore the Borel resummation is well-defined as an integral along the positive real axis

B(g2) =Z 1

0

dt

g2e°t/g

2 CA

A + t. (17)

This altenating factorially divergent series is a typical example of Borel summable divergentseries.

On the other hand, if perturbation series is not alternating, the factorially divergent seriesgives the Borel transform with singularities on positive real axis and the Borel resummmationhas imaginary ambiguities. For instance, suppose that the perturbation series Ppert(g2) givesnon-alternating factorially divergent series like

Ppert(g2) = C1X

q=0

q!µ

g2

A

∂q. (18)

The Borel transform has a singularity on positive real axis

BPpert(t) = C1X

q=0

µt

A

∂q=

CA

A ° t , (19)

Bpert(g2) =Z 1

0

dt

g2e°t/g

2 CA

A ° t . (20)

厳密結果






P (g2) =⇥�

q=0





BP (t) :=⇥�

q=0

aqq!tq. (6.2)








B(g2) = 1g2

⇥ ⇥

0BP (t)e�t/g

2dt . (6.3)
















– 50 –

幾つかの例ではボレル和が厳密結果を与える！



2

I. INTRODUCTION

d2ψ

dx2=

2m(V (x)− E)!2 ψ (1)

[H0 + g


S =

∫dt

[m

2

(dx

dt

)2− V (x)

](3)

SE =

∫dτ

[m

2

(dx

dτ

)2+ V (x)

](4)

⟨x = a|e−iHt/!|x = b⟩ =∫

d[x(t)] eiS[x(t)]/! (5)

⟨x = a|e−Hτ/!|x = b⟩ =∫

d[x(τ)] e−SE [x(τ)]/! (6)

P (a → b) ≈ e−1!∫ ba dx

√2mV (x) (7)






P (g2) =⇥�

q=0





BP (t) :=⇥�

q=0

aqq!tq. (6.2)








B(g2) = 1g2

⇥ ⇥

0BP (t)e�t/g

2dt . (6.3)
















– 50 –

正実軸上の特異点のため積分路に不定性






P (g2) =⇥�

q=0





BP (t) :=⇥�

q=0

aqq!tq. (6.2)








B(g2) = 1g2

⇥ ⇥

0BP (t)e�t/g

2dt . (6.3)
















– 50 –

一般的にはボレル変換が正の実軸上に特異点を持つ


B(g2e±i✏)Z 1e±i✏

0

dt

g2e� t

g2 BP (t)


2

I. INTRODUCTION

d2ψ

dx2=

2m(V (x)− E)!2 ψ (1)

[H0 + g


S =

∫dt

[m

2

(dx

dt

)2− V (x)

](3)

SE =

∫dτ

[m

2

(dx

dτ

)2+ V (x)

](4)

⟨x = a|e−iHt/!|x = b⟩ =∫

d[x(t)] eiS[x(t)]/! (5)

⟨x = a|e−Hτ/!|x = b⟩ =∫

d[x(τ)] e−SE [x(τ)]/! (6)

P (a → b) ≈ e−1!∫ ba dx

√2mV (x) (7)

積分路の不定性に付随して符合の不定性を持つ虚部が出現

この摂動ボレル和の不定虚部こそ非摂動寄与の情報を含む！






P (g2) =⇥�

q=0





BP (t) :=⇥�

q=0

aqq!tq. (6.2)








B(g2) = 1g2

⇥ ⇥

0BP (t)e�t/g

2dt . (6.3)
















– 50 –

Im[B(g2)] ⇡ e�Ag2


B(g2e±i✏) = Re[B]± iIm[B]

第1部：常微分方程式と積分におけるリサージェンス構造

形式的解 at z=∞

E(1)np O(g)∂ϵE|ϵ=1

E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z

ϵ S±Φ0 ≡∫∞e±iϵ0 e

−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)

(Alien calculus) 3, 16, 17)

2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018

�0 =1X

q=0

n!z�n�1 &

摂動的非摂動的

リサージェンス構造が存在．しかしなぜ？


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018

Ecalle (81)


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018

t=1

オイラー方程式のリサージェンス構造

ボレル和を通して関係づいている！

z ⇠ 1g2

• 常微分方程式の解は各漸近級数のボレル和の総和 = トランス級数

• arg[z]=0でトランス級数パラメタ σ が不連続 = ストークス現象

• 解の連続性から各形式的解が結びつく

形式的解

：ストークス定数


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018

�0 =1X

q=0

n!z�n�1

s = 2⇡i

ボレル和

&

◆オイラー方程式


◆エイリアン解析


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


E(2)np O(g)

ω

12∂

2ϵE|ϵ=1

2 N = (2, 0) CPN−112, 13, 14, 15)

(N = 2 )

8)

N = 2

N > 2

2

7.

(ODE)

(ODE ) 3)

z = ∞

ϕ′(z) + ϕ(z) =1

z(37)

Φ0 =∑∞

n=0 n! z−n−1

BΦ0[t] = 11−t8 σ ∈ Rϕ(z;σ) = Φ0+σe−z


−ztBΦ0(t)

8 z 1/g

9

S±ϕ(z;σ) = S±Φ0(z) + σe−z. (38)

σ

z = |z|eiθ θ = 0

σ s = 2πi

S± ±ϵ θθ = 0 S+ϕ(z;σ) → S−ϕ(z;σ+ s)

S+ϕ(z;σ) = S−ϕ(z;σ + s)

S+Φ0(z)− S−Φ0(z) = 2πie−z (39)

“ ” Φ0 e−z “

” (Ecalle)

ODE

3)


2

S

S+ = S− ◦S, S = exp[e−z∆

](40)

∆ t = 1

[e−z∆, ∂σ] = 0

ϕ(z;σ)

e−z∆ϕ(z;σ) = s ∂σϕ(z;σ) (41)

s

∆Φ0 = s

∆e−z = 0 (40)

SΦ0 = Φ0 + se−z, Sϕ(z;σ) = ϕ(z;σ + s) (42)

S− (40)S+ϕ(z;σ) = S−ϕ(z;σ+ s)

9 ODE

7 Vol. 1, No. 1, 2018


・特異点方向上下のボレル和を繋ぐ群作用：Stokes automorphism

・各特異点について異微分作用素：Alien derivative

・Alien calculusと通常の微分を関係付ける方程式：Bridge equation

・Bridge eq.の左右辺比較により各漸近級数間の関係が判明！

[e�z�, @z] = 0 [@�, @z] = 0

'00 � z' = 0ex.) エアリー方程式

' = Ai(z) ⇡ e� 23 z32 S±

Xanz

� 32n + � e23 z

32 S±

Xbnz

� 32n

(z = ∞に不確定特異点)

0 arg[z] 2⇡

Re[Ai(z)]

摂動計算から非摂動寄与の抽出

ある種のリサージェンス構造は量子論にも存在すると考えられ，摂動ボレル和が非摂動的寄与の情報を含む．

摂動ボレル和の虚部不定性非摂動的寄与

摂動級数から非摂動寄与を原理的には求めることが可能！

・コーシーの積分定理による摂動-非摂動関係の抽出

Tatsuhiro Misumi

I. RESURGENCE

F [z,ϕ(z), ...,ϕ(k)(z)] = 0 (1)

z ∼ 1g2

(2)

Φ0(z) =∑

q

aqz−q (3)

e−nAzΦn(z) (4)

ϕ±(z;σ) = S±Φ0(z) +∑

n

σne−nAzS±Φn(z) (5)

S+Φ0(z)− S−Φ0(z) ≈ se−AzSΦ1(z) (6)

Sθ = Id−Discθ = exp[∑

e−ωθz∆ωθ

](7)

e−ωθz∆ωθϕ(z;σ) ∝ ∂σϕ(z;σ) (8)

SθΦn = exp[e−Az∆A]Φn =

∞∑

l=0

(n+ ln

)s1e

−lAzΦn+l (9)

· エアリー積分

Ai(g�2) =

Z 1

�1d� exp

�i

✓�3

3+

�

g2

◆�

⇡r

g

4⇡exp

✓� 23g2

◆

1/g2 = 1

1/g2 = i/2

積分におけるリサージェンス構造

0次元積分における最急降下法では積分径路を変形し複素固定点に繋がる径路(thimble)に分解

経路積分においても複素固定点を考えるのは自然

Re[e�i(�3/3+�/g2)]

�

i

g

� ig

arg[g2] = 0+

最急降下法(Thimble分解)における複素固定点の寄与


複素平面上の2つの複素固定点

Ai(g�2) =

Z 1

�1d� exp

�i

✓�3

3+

�

g2

◆�

� = ± ig



複素平面上の2つの複素固定点

�

i

g

� ig

arg[g2] = 0+

最急降下法：元の積分径路を，固定点を通り，虚部一定の最急降下径路に分解

C =X

�

n�J�最急降下径路分解

= Thimble分解

Ai(g�2) =

Z 1

�1d� exp

�i

✓�3

3+

�

g2

◆�

� = ± ig



arg[g2] = 0+

Re[S] Re[S0]


J� 最急降下径路

n� = hK�, Ci 最急上昇径路Kと元の径路との交叉数

Ai(g�2) =

Z 1

�1d� exp

�i

✓�3

3+

�

g2

◆�

C =X

�

n�J�


・

・

Im[S] = Im[S0]

�

i

g

� ig

J�



n+ = hK+, Ci = 0n� = hK�, Ci = 1

C = J�C =X

�

n�J�

arg[g2] = 0+


arg[g2] =2⇡

3�


arg[g2] = 0+K�

J+

K+

J�まで有効な分解


C =X

�

n�J�

ストークス現象：特定のarg[g^2]でthimble分解が不連続変化


arg[g2] =2⇡

3+

n+ = hK+, Ci = 1n� = hK�, Ci = 1

C = J+ + J�


arg[g2] =2⇡

3+

K�

J+

K+

J�

arg[g2] = �2⇡3

! ⇡ストークス現象· エアリー積分

C = J�

cf.) Real-time formalism, Finite-density lattice system


C = J+ + J�

arg[g2] =2⇡

3+arg[g2] =

2⇡

3�


Thimble分解がストークス線で不連続に変化エアリー関数自体はストークス線でも連続

J�2⇡

3

��= J�

2⇡

3

+�+ J+

arg[g2] = �2⇡3

! ⇡ストークス現象

摂動ボレル和の不定性はストークス線上でのthimble分解の不定性に対応！


C = J�

cf.) Real-time formalism, Finite-density lattice system


C = J+ + J�

arg[g2] =2⇡

3+arg[g2] =

2⇡

3�


Thimble分解がストークス線で不連続に変化エアリー関数自体はストークス線でも連続

· 変形ベッセル積分

arg[λ]=0でのストークス現象

・各Thimbleがトランス級数の各セクター（摂動+非摂動）に対応・摂動寄与　が虚部を持ち， arg[λ]=0± で不定になる・適切にThimbleの寄与を加えることで不定性のない結果が得られる

C = J0(0�) + J1(0�)

C = J0(0+)� J1(0+)arg[�] = 0+

arg[�] = 0�

J0

Z(�) =1p�

Z ⇡2

�⇡2dz e�

12� sin

2 z =⇡e�

14�

p�

I0

✓1

4�

◆z0 = 0z1 = ±⇡/2

固定点：


第2部：量子論におけるリサージェンス構造

・2重井戸型量子力学の摂動論

正の実軸上に特異点

aq = �3q+1

⇡q!

Zinn-Justin(81)Bender-Wu(73)

Lipatov(77)Bogomolny, Fateyev(77)

BPpert(t) =1

⇡

1

t� 1/3

H =p2

2+

x2(1� gx)2

2LE =

ẋ2

2+

x2(1� gx)2

2( )

E0,pert =X

q=0

aqg2q

q � 1( )

2重井戸型量子力学系

Im[E(g2e±i✏)] = Im"Z 1e±i✏

0

dt

g2e� t

g2

⇡

1

t� 1/3

#= ⌥e

� 13g2

g2


SI =1

6g2

Zinn-Justin(81)Bender-Wu(73)


BPpert(t) =1

⇡

1

t� 1/3

H =p2

2+

x2(1� gx)2

2LE =

ẋ2

2+

x2(1� gx)2

2( )

x(⌧) =1

2g

✓1 + tanh

⌧ � ⌧I2

◆インスタントン解

2倍！

aq = �3q+1

⇡q!E0,pert =

X

q=0

aqg2q

q � 1( )




0

dt

g2e� t

g2

⇡

1

t� 1/3

#= ⌥e

� 13g2

g2

Zinn-Justin(81)

摂動的ボレル和に含まれる不定虚部は非摂動的インスタントン寄与を示しているのか？

Bender-Wu(73)


BPpert(t) =1

⇡

1

t� 1/3

H =p2

2+

x2(1� gx)2

2LE =

ẋ2

2+

x2(1� gx)2

2( )

aq = �3q+1

⇡q!E0,pert =

X

q=0

aqg2q

q � 1( )





0

dt

g2e� t

g2

⇡

1

t� 1/3

#= ⌥e

� 13g2

g2

インスタントン-反インスタントン配位 = Bion

10

C. 1 instanton + 1 anti-instanton

The amplitude of one instanton and one anti-instanton amplitude is composed of two configura-

tions [IĪ] and [ĪI], as shown in Fig. 2. In these cases, the interaction between the two constituents

is attractive, and the quasi moduli integral is ill-defined. Therefore we introduce the Bogomolnyi–

Zinn-Justin (BZJ) prescription [29, 30]: we first evaluate the integral by taking −g2 > 0, and then

we analytically continue the result from −g2 > 0 back to g2 > 0 in the complex g2 plane. This

procedure provides the imaginary ambiguity depending on the path of the analytic continuation

as −g2 = e∓iπg2.

!!!!

!!!!

[IĪ]

!!!!

!!!!

[ĪI]

FIG. 2: A schematic figure of an example of one-instanton and one anti-instanton amplitude ([IĪ], [ĪI]).

Each horizontal line stands for the vacuum in the sine-Gordon potential.

The amplitude of one-instanton and one anti-instanton configuration [IĪ] corresponding to the

left of Fig. 2 is obtained as

[IĪ]ξ−2 =∫ ∞

0dR exp

(

−2

−g2e−R − ϵR

)

|g2|≪1−→(

−g2

2

)ϵ

Γ(ϵ)

−g2=e∓iπg2−→ −(

γ + log2

e∓iπg2

)

+ O

(

1

ϵ

)

+ O(ϵ)

= −(

γ + log2

g2

)

∓ iπ + O(

1

ϵ

)

+ O(ϵ) , (32)

where we perform the integral in the first line by considering −g2 > 0, and in the second line

analytically continue −g2 > 0 back to g2 > 0 in the complex g2 plane [29, 30]. The third line

shows a two-fold ambiguous expression of −g2 depending on the path of analytic continuation as

−g2 = e∓iπg2. As with the two-instanton case, we have subtracted the divergent part O(1/ϵ) while

the O(ϵ) term disappears in the ϵ → 0 limit.

インスタントン-反インスタントン間の引力ポテンシャル

31

s(x1, x2)

x

y

0

2

4

420

0

1

−1

−2−4

x1

x2

x1

)

0

2

4

−4 40 2−2

s(x1, x2)

FIG. 11: The euclidean action density s(x1, x2) of neutral bion configurations for λ1 = 1/1000,λ2 = 1/1000

and φ = π/4 in the CP1model on R

1× S

1. The same action density is depicted in two ways, as a function

of x1, x2 (left) and x1 (right). There is no x2 dependence in the action density, with x2 being a coordinate

of the compactied dimension.

In that case, we obtain the angular coordinate fields of S2 as

Φ(x1, x2) = φ1 −πx2L

, cotΘ(x1, x2)

2= λ1e

−πx1L ∓ λ2e

πx1L . (108)

This configuration starts from N at x1 = −∞. For the upper sign, it goes through S at

x1 = −Lπ log(λ1λ2) and reaches to N with Θ = 2π at x1 = ∞, namely it winds once around the

great circle. The configuration represents the double instanton configuration of the sine-Gordon

quantum mechanics as shown in Fig.1. For the lower sign, the configuration returns back to N

with Θ = 0 at x1 = ∞ approaching but never reaching S at any point in −∞ < x1 < ∞. This

clearly represents the instanton and anti-instanton configuration [IĪ] of the sine-Gordon quantum

mechanics, as shown in the left panel of Fig. 2. The sine-Gordon quantum mechanics captures

only field configurations that can cover the (part of) S2 in the following specific fashion : When

x1 is varied with fixed x2, Θ goes along the great circle (namely fixed Φ), whereas x2 variation

with fixed x1 makes a rotation of Φ with the constant velocity by an amount π at fixed Θ. The

first homotopy group π1 for the sine-Gordon model is one for the upper sign and zero for the lower

sign, but the second homotopy group π2 for the CP 1 model is zero for the both cases. In Fig. 12,

we show the instanton–anti-instanton and instanton-instanton configurations in the sine-Gordon

quantum mechanics corresponding to ei(φ2−φ1) = ∓1 in Eq. (107), and how the corresponding

configuration of the CP 1 model in Eq.(108) cover the sphere S2. Here, each of fractional instanton

31

s(x1, x2)

x

y

0

2

4

420

0

1

−1

−2−4

x1

x2

x1

)

0

2

4

−4 40 2−2

s(x1, x2)

FIG. 11: The euclidean action density s(x1, x2) of neutral bion configurations for λ1 = 1/1000,λ2 = 1/1000

and φ = π/4 in the CP 1 model on R1 × S1. The same action density is depicted in two ways, as a function

of x1, x2 (left) and x1 (right). There is no x2 dependence in the action density, with x2 being a coordinate

of the compactied dimension.

In that case, we obtain the angular coordinate fields of S2 as

Φ(x1, x2) = φ1 −πx2L

, cotΘ(x1, x2)

2= λ1e

−πx1L ∓ λ2e

πx1L . (108)

This configuration starts from N at x1 = −∞. For the upper sign, it goes through S at

x1 = −Lπ log(λ1λ2) and reaches to N with Θ = 2π at x1 = ∞, namely it winds once around the

great circle. The configuration represents the double instanton configuration of the sine-Gordon

quantum mechanics as shown in Fig.1. For the lower sign, the configuration returns back to N

with Θ = 0 at x1 = ∞ approaching but never reaching S at any point in −∞ < x1 < ∞. This

clearly represents the instanton and anti-instanton configuration [IĪ] of the sine-Gordon quantum

mechanics, as shown in the left panel of Fig. 2. The sine-Gordon quantum mechanics captures

only field configurations that can cover the (part of) S2 in the following specific fashion : When

x1 is varied with fixed x2, Θ goes along the great circle (namely fixed Φ), whereas x2 variation

with fixed x1 makes a rotation of Φ with the constant velocity by an amount π at fixed Θ. The

first homotopy group π1 for the sine-Gordon model is one for the upper sign and zero for the lower

sign, but the second homotopy group π2 for the CP 1 model is zero for the both cases. In Fig. 12,

we show the instanton–anti-instanton and instanton-instanton configurations in the sine-Gordon

quantum mechanics corresponding to ei(φ2−φ1) = ∓1 in Eq. (107), and how the corresponding

configuration of the CP 1 model in Eq.(108) cover the sphere S2. Here, each of fractional instanton

寄与は次の積分で主に与えられる

2

I. INTRODUCTION

τ (1)

0 ≤ φ < 2π (2)

θ2 ≡ φ = 0 (3)

V [R] = −4κLg2

cos θ2 e−κR (4)

d2ψ

dx2=

2m(V (x)− E)!2 ψ (5)

[H0 + g


S =

∫dt

[m

2

(dx

dt

)2− V (x)

](7)

SE =

∫dτ

[m

2

(dx

dτ

)2+ V (x)

](8)

⟨x = a|e−iHt/!|x = b⟩ =∫

d[x(t)] eiS[x(t)]/! (9)

⟨x = a|e−Hτ/!|x = b⟩ =∫

d[x(τ)] e−SE [x(τ)]/! (10)

P (a → b) ≈ e−1!∫ ba dx

√2mV (x) (11)

Bion

VIĪ(R) = �2

g2exp[�R] + ✏R

2

I. INTRODUCTION

V (x) (1)

x (2)

τ (3)

0 ≤ φ < 2π (4)

θ2 ≡ φ = 0 (5)

V [R] = −4κLg2

cos θ2 e−κR (6)

d2ψ

dx2=

2m(V (x)− E)!2 ψ (7)

[H0 + g


S =

∫dt

[m

2

(dx

dt

)2− V (x)

](9)

SE =

∫dτ

[m

2

(dx

dτ

)2+ V (x)

](10)

2

I. INTRODUCTION

V (x) (1)

x (2)

τ (3)

0 ≤ φ < 2π (4)

θ2 ≡ φ = 0 (5)

V [R] = −4κLg2

cos θ2 e−κR (6)

d2ψ

dx2=

2m(V (x)− E)!2 ψ (7)

[H0 + g


S =

∫dt

[m

2

(dx

dt

)2− V (x)

](9)

SE =

∫dτ

[m

2

(dx

dτ

)2+ V (x)

](10)

R

xIĪ =1

2g

tanh

⌧ � ⌧I2

� tanh ⌧ � ⌧Ī2

�

Z 1

0e�VIĪ(R)dR

・Bogomolny-Zinn-Justin(BZJ)処方

yield an imaginary part for [BB], but does yield an imaginary part for [BB]. The quasi-zeromode integrals are of the form

I(g2) =

� ⇤

0d⌃ exp (�V (⌃)) for [BB], and (6.36)

⇥I(g2) =� ⇤

0d⌃ exp (+V (⌃)) for [BB], (6.37)

where

V (⌃) = (µB, µB)8⌅

g2e�⇥⇤ (6.38)

and µB = �i � �j ⌃ �⇧r is the charge of the bion Bij .This type of integral, as noted earlier, is addressed in bosonic quantum mechanics by

Bogomolny [19]. Both integrals are divergent at large separation, and the latter is dominated

by ⌃ ⇧ 0 where molecular configurations are meaningless. The first of these problems is dueto double-counting of the uncorrelated [B]-[B] or [B]-[B] events, and is subtracted o⇥.

C̃�

C̃+

g2

Figure 11. Defining left (right) bion-anti-bion amplitude [BijBij ]�=0± , we proceed as in the construc-tion of left (right) Borel resummation B0,�=0± .

The short-distance domination of ⇥I(g2) can be taken care of by modifying the integrationcontour, or by rotating g2 ⇧ �g2, where the bion-anti-bion interaction becomes repulsive,and continuing the integral back to positive |g2|+i0±. The result, as was the case with (6.34),is two-fold ambiguous:

[BijBij ]�=0± = Re [BijBij ] + i Im [BijBij ]�=0± ⌅ e�4S0 ± i⇧ e�4S0 (6.39)

Consider a typical observable in CPN�1 theory with Nf ⇤ 1 fermions. We expect that thisobservable will receive contributions to all orders in perturbation theory, as well as non-

perturbative contributions. Denote the lateral Borel summation for perturbation theory by

B0,�=0± . Then write g2 = |g2|ei�, where ⇥ is the phase of the complexified coupling. For QFTto make sense, these two ambiguities must cancel:

ImB0,�=0± + Im [BB]�=0± = 0 , up to e�6S0 (6.40)

– 62 –

Bogomolny(79) Zinn-Justin(81)

Bion配位からの寄与の計算

1. を正の実数とみなす 2. 準モジュライ積分を実行 3. という解析接続実行

10









!!!!

!!!!

[IĪ]

!!!!

!!!!

[ĪI]





[IĪ]ξ−2 =∫ ∞

0dR exp

(

−2

−g2e−R − ϵR

)

|g2|≪1−→(

−g2

2

)ϵ

Γ(ϵ)

−g2=e∓iπg2−→ −(

γ + log2

e∓iπg2

)

+ O

(

1

ϵ

)

+ O(ϵ)

= −(

γ + log2

g2

)

∓ iπ + O(

1

ϵ

)

+ O(ϵ) , (32)






10









!!!!

!!!!

[IĪ]

!!!!

!!!!

[ĪI]





[IĪ]ξ−2 =∫ ∞

0dR exp

(

−2

−g2e−R − ϵR

)

|g2|≪1−→(

−g2

2

)ϵ

Γ(ϵ)

−g2=e∓iπg2−→ −(

γ + log2

e∓iπg2

)

+ O

(

1

ϵ

)

+ O(ϵ)

= −(

γ + log2

g2

)

∓ iπ + O(

1

ϵ

)

+ O(ϵ) , (32)






摂動的ボレル和における不定虚部　　　を相殺！⌥e�2SI

g2

Eb ⇡ � lim�!1

1

�

ZbZ0

=e�2SI

⇡g2

� + log

2

g2± i⇡

�

Zb /Z 1

0dR exp

� 2�g2 e

�R � ✏R�⇡

✓�g2

2

◆✏�(✏)

⇡ �✓� + log

2

�g2

◆= �

✓� + log

2

g2

◆⌥ i⇡

g2 ⌧ 1

!

g2e±i✏に対応

どうしてBion配位が重要になるのか？

答: Bionは複素化された理論の運動方程式の解に対応している！

Behtash, Dunne, Schafer, Sulejmanpasic, Unsal(15)Fujimori, Kamata, TM, Nitta, Sakai (16)(17)

e�2SI (Re± iIm) zT , ⌧0 2 C

2

FIG. 1: Real and complex solutions in the inverted tilted double wellpotential. The inverted potential (on the real axis) is shown in black,the real bounce and associated critical and turning points are shownin red, and the pair of complex bions and turning and critical pointsare blue. The blue points correspond to zcr1 and zT ,z

⇤T in (6). Note

that the motion takes place in the real and imaginary parts of thecomplex potential, as explained in the text.

where we have used the Cauchy-Riemann equations ∂xVr =∂yVi, and ∂yVr = �∂xVi. An important aspect of (2) is that itdoes not describe an ordinary two-dimensional classical me-chanical system: the holomorphic classical mechanics is notthe same as the motion of a particle in the two-dimensionalinverted potential �Vr(x,y). Instead of the usual Newtonequations with force ~—Vr(x,y), the force in the x-direction isdue to —xVr(x,y) while the force in the y-direction is due to�—yVr(x,y). This has interesting consequences.

Supersymmetric quantum mechanics: Consider supersym-metric quantum mechanics with the superpotential W (x)

S =Z

dt� 1

2 ẋ2 + 12 (W 0)2 +[ȳẏ+ pW 00ȳy]

�, (3)

corresponding to p = 1. The parameter p will be used todeform the theory away from the supersymmetric point [9].We choose W (x) with more than one critical point, so thatthere will be real instantons. By projecting to fermion numbereigenstates one obtains a pair of Hamiltonians H± [24]:

H± = 12 p̂2 +V±(x) , V±(x) = 12 (W 0(x))2 ±

p2 W 00(x) . (4)

In the following we consider superpotentials of the formW (x) = 1gW (

pgx), and rescale x =pgx. Then the Euclideanaction takes the form SE = 1g

Rdt( 12 ẋ

2+V±(x)). We work withthe bosonized description (4). Note that compared to the orig-inal bosonic potential 12 (W

0)2 the bosonized theory containsan O(g) term that arises from integrating out the fermions.The quantum modified holomorphic equations of motion inthe inverted potential �V+(z) is

d2zdt2

=W 0(z)W 00(z)+pg2

W 000(z) . (5)

FIG. 2: Complex bion solution in supersymmetric quantum mechan-ics with a double well potential. The black and red lines show thereal and imaginary part of the solution for pg = 1 · 10�6. The char-acteristic size of the solution is Re[2t0]' 12 log

16pg . For larger values

of pg the two tunneling event merge.

Double well potential: Consider W (x) = x3/3� x, so thatV (x) is an asymmetric double well potential with an O(g)“tilt”. The ground state energy of the system is zero to all or-ders in perturbation theory, but non-perturbatively supersym-metry is spontaneously broken and the ground state energyis non-zero and positive [24]. Note that the positivity of theground state energy is a consequence of the SUSY algebra,H = 12{Q, Q̄}, where Q and Q̄ are the SUSY generators.

In the original formulation (3) this can be understood as thecontribution from approximate instanton-anti-instanton solu-tions of the bosonic potential 12 (W

0)2 [9]. In the bosonizedversion we seek classical solutions in the inverted potential�V+. However, the real equations of motion in the invertedpotential have no finite action configurations except for thetrivial perturbative saddle, and an exact (real) bounce solu-tion. But this bounce is related to the false vacuum and isnot directly relevant for ground state properties, which are de-termined by saddles starting at the global maximum of theinverted potential. But the real motion of a classical particlestarting at such a global maximum is unbounded, and has in-finite action.

On the other hand, the holomorphic Newton’s equation (5)does support finite action solutions starting from the globalmaximum. There are exact finite action complex solutionsthat start at the global maximum of the inverted potential andbounce back from one of the two complex turning points,whose real part is located near the top of the local maximum,see Fig. 1. We refer to this as the “complex bion” solution:

zcb(t) = zcr1 �zcr1 � zT

2coth

⇣wcbt02

⌘tanh

✓wcb(t + t0)

2

◆

� tanh✓

wcb(t � t0)2

◆�, (6)

where zcb(±•) = zcr1 is the global maximum of the invertedpotential, and zT =�zcr1 ± i

ppg/(�zcr1 ) are the complex turn-

zcb(⌧) = z1 �(z1 � zT )

2coth

!⌧02

tanh

!(⌧ + ⌧0)

2� tanh !(⌧ � ⌧0)

2

�

d2z

d⌧2=

@V

@z

g2 ! g2e⌥i�

複素固定点を考えるのはすでにみたように積分では自然なこと

Chapter 8

Tilted double well

superpotential W (x) = 13x3−a2x Lagrangian bosonic potential

symmetric double well fermion number projection Hamiltonian:H±∓12!W

′′ ”tilted” double well 1:

H± =1

2p2 +

1

2(x2 − a2)2 ∓ !

22x (8.1)

Figure 8.1: Lagrangian Bosonic potential, projection tilt

⟨x|0;±⟩ = C±e∓W (x)

! (8.2)

= C±e∓ 1!

13(x

3−a2x) (8.3)

∫ ∞

−∞| ⟨x|0;±⟩ |2 dx (8.4)

⟨x|0;±⟩ i.e. dynamical SUSY breaking0

1potential SUSY 0

55

x ! z = x+ iy

2

FIG. 1: Real and complex solutions in the inverted tilted double wellpotential. The inverted potential (on the real axis) is shown in black,the real bounce and associated critical and turning points are shownin red, and the pair of complex bions and turning and critical pointsare blue. The blue points correspond to zcr1 and zT ,z

⇤T in (6). Note

that the motion takes place in the real and imaginary parts of thecomplex potential, as explained in the text.

where we have used the Cauchy-Riemann equations ∂xVr =∂yVi, and ∂yVr = �∂xVi. An important aspect of (2) is that itdoes not describe an ordinary two-dimensional classical me-chanical system: the holomorphic classical mechanics is notthe same as the motion of a particle in the two-dimensionalinverted potential �Vr(x,y). Instead of the usual Newtonequations with force ~—Vr(x,y), the force in the x-direction isdue to —xVr(x,y) while the force in the y-direction is due to�—yVr(x,y). This has interesting consequences.

Supersymmetric quantum mechanics: Consider supersym-metric quantum mechanics with the superpotential W (x)

S =Z

dt� 1

2 ẋ2 + 12 (W 0)2 +[ȳẏ+ pW 00ȳy]

�, (3)

corresponding to p = 1. The parameter p will be used todeform the theory away from the supersymmetric point [9].We choose W (x) with more than one critical point, so thatthere will be real instantons. By projecting to fermion numbereigenstates one obtains a pair of Hamiltonians H± [24]:

H± = 12 p̂2 +V±(x) , V±(x) = 12 (W 0(x))2 ±

p2 W 00(x) . (4)

In the following we consider superpotentials of the formW (x) = 1gW (

pgx), and rescale x =pgx. Then the Euclideanaction takes the form SE = 1g

Rdt( 12 ẋ

2+V±(x)). We work withthe bosonized description (4). Note that compared to the orig-inal bosonic potential 12 (W

0)2 the bosonized theory containsan O(g) term that arises from integrating out the fermions.The quantum modified holomorphic equations of motion inthe inverted potential �V+(z) is

d2zdt2

=W 0(z)W 00(z)+pg2

W 000(z) . (5)

FIG. 2: Complex bion solution in supersymmetric quantum mechan-ics with a double well potential. The black and red lines show thereal and imaginary part of the solution for pg = 1 · 10�6. The char-acteristic size of the solution is Re[2t0]' 12 log

16pg . For larger values

of pg the two tunneling event merge.

Double well potential: Consider W (x) = x3/3� x, so thatV (x) is an asymmetric double well potential with an O(g)“tilt”. The ground state energy of the system is zero to all or-ders in perturbation theory, but non-perturbatively supersym-metry is spontaneously broken and the ground state energyis non-zero and positive [24]. Note that the positivity of theground state energy is a consequence of the SUSY algebra,H = 12{Q, Q̄}, where Q and Q̄ are the SUSY generators.

In the original formulation (3) this can be understood as thecontribution from approximate instanton-anti-instanton solu-tions of the bosonic potential 12 (W

0)2 [9]. In the bosonizedversion we seek classical solutions in the inverted potential�V+. However, the real equations of motion in the invertedpotential have no finite action configurations except for thetrivial perturbative saddle, and an exact (real) bounce solu-tion. But this bounce is related to the false vacuum and isnot directly relevant for ground state properties, which are de-termined by saddles starting at the global maximum of theinverted potential. But the real motion of a classical particlestarting at such a global maximum is unbounded, and has in-finite action.

On the other hand, the holomorphic Newton’s equation (5)does support finite action solutions starting from the globalmaximum. There are exact finite action complex solutionsthat start at the global maximum of the inverted potential andbounce back from one of the two complex turning points,whose real part is located near the top of the local maximum,see Fig. 1. We refer to this as the “complex bion” solution:

zcb(t) = zcr1 �zcr1 � zT

2coth

⇣wcbt02

⌘tanh

✓wcb(t + t0)

2

◆

� tanh✓

wcb(t � t0)2

◆�, (6)

where zcb(±•) = zcr1 is the global maximum of the invertedpotential, and zT =�zcr1 ± i

ppg/(�zcr1 ) are the complex turn-

・複素bion解

zcb(⌧) = z1 �(z1 � zT )

2coth

!⌧02

tanh

!(⌧ + ⌧0)

2� t

リサージェンス理論に基づく非摂動効果の理解 - 中央大学...Non-Borel...

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Transcript of リサージェンス理論に基づく非摂動効果の理解 - 中央大学...Non-Borel...

リサージェンス理論に基づく 非摂動効果の理解 - 中央大学...Non-Borel...

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Transcript of リサージェンス理論に基づく 非摂動効果の理解 - 中央大学...Non-Borel...

リサージェンス理論に基づく非摂動効果の理解 - 中央大学...Non-Borel...

Transcript of リサージェンス理論に基づく非摂動効果の理解 - 中央大学...Non-Borel...