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INSTANTONS AND LARGE ORDER PERTURBATION THEORY IN THE 1-DQUANTUM MECHANICAL QUARTIC OSCILLATOR
A thesis submitted to the faculty ofSan Francisco State University
In partial fulfilment ofThe Requirements for
The Degree
Master of ScienceIn
Physics
by
Scott Shermer
San Francisco, CA
May 2013
Copyright by
Scott W. Shermer
2013
CERTIFICATION OF APPROVAL
I certify that I have read Instantons and Large Order Perturbation Theory in the 1-D
Quantum Mechanical Quartic Oscillator by Scott Shermer, and that in my opinion
this work meets the criteria for approving a thesis submitted in partial fulfillment of
the requirements for the degree: Master of Science in Physics at San Francisco State
University.
Mithat Unsal
Assistant Professor
Jeff Greensite
Professor
Joseph Barranco
Assistant Professor
INSTANTONS AND LARGE ORDER PERTURBATION THEORY IN THE 1-DQUANTUM MECHANICAL QUARTIC OSCILLATOR
Scott ShermerSan Francisco, California
2013
The purpose of this paper is to provide an exposition of techniques and subtleties
associated with obtaining the ground state energy for a quantum quartic oscillator
with the following Hamiltonian:
H = − ~2
2m
d2
dx2+
mω2x2
2+ g
x4
4
It will be shown that the application of perturbation theory to the calculation of
the ground state energy of this system yields a divergent asymptotic series. The
divergence of this series must be interpreted if a sensible, physical description
of the measurable ground state energy is to be extracted. It will be shown that
for a positive coupling parameter g the series is Borel summable, and a sensible
result is thereby obtained. For negative coupling however, we show that an
imaginary ambiguity is induced when the Borel prescription is applied. This
ambiguity will be seen to correspond directly with an ambiguity arising from
a non-perturbative treatment of the system. Instantons methods are employed
to explore non-perturbative, exponentially small corrections to the ground state
energy. To perform these calculations, analytic continuation of the coupling
constant becomes necessary and an imaginary ambiguity arises that is shown to
precicely cancel with the ambiguity induced by resummation of the large order
asymptotics of the pertubative series.
I certify that the Abstract is a correct representation of the content of this thesis.
Chair, Thesis Committee Date
TABLE OF CONTENTS
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1. Introduction 1
2. The Harmonic Oscillator 1
3. Pertubation theory and the quartic oscillator 8
3.1. General Properties of the Hamiltonian and the Ground State Energy 9
3.2. The Ground State at Large Orders from Functional Methods 11
3.3. Asymptotics and Borel Summability 16
3.4. Dispersion Relations and Large Order Asymptotics 20
4. Non-perturbative Phenomena 21
4.1. The Stokes Equation, Airy Functions and Leading Order Asymptotics 21
4.2. Borel Summability and the Stokes Phenomenon 29
5. The Zero-Dimensional Quartic Potential 32
5.1. The Trivial Saddle for p = 0 37
5.2. The Non-trivial Saddles 39
6. Nonperturbative Methods and the Quartic Oscillator 40
6.1. The Single Instanton Contribution 42
6.2. Instanton Anti-Instanton Configurations 44
7. The Perturbative Expansion for Negative Coupling and Ambiguities 51
8. Discussion 55
v
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vi
LIST OF FIGURES
Figure Page
1 The Simple Harmonic Oscillator Potential for g = 0 2
2 The Quartic Oscillator Potential for g = +0.1 8
3 The Quartic Oscillator Potential for g = −0.1 9
4 First ten terms in perturbative corrections En to the quartic oscillator groundstate energy for g = +0.01 15
5 Divergence of En for late terms in perturbative expansion for g = +0.01 16
6 Regions of convergence (shaded), steepest descent contours and saddles(circled) for the Airy function 23
7 Steepest descent contours and the second saddle contribution at θ = 4π3
26
8 Steepest descent contours and the second saddle contribution at θ = 2π3
28
9 Shift of branch cut for continuation from postive to negative coupling 30
10 Choice of integration path in Borel plane for negative coupling 31
11 Regions of convergence (shaded), steepest descent contours and saddles(circled) for the supersymmetric potential; λ postive,p = 0, v = 1 37
12 Added contribution of saddles at ±√λ for negative coupling 39
13 The Quartic Oscillator Potential for m = −1, g = 0.1 41
14 The Shifted Quartic Oscillator Potential for m = −1, g = 0.1 42
15 Instanton solution; g = 0.1 43
16 Instanton/Anti-Instanton solution for large separation; g = 0.1 45
vii
17 ”Molecular” Instanton/Anti-Instanton solution for small separation; g = 0.1 46
18 Coordinate Tranformation for Evaluation of the IAI Action 48
19 Dispersion ”Keyhole” Contour 52
viii
1
1. Introduction
The simple harmonic oscillator is likely the most thoroughly investigated mathe-
matical model in the whole of physics. Many natural systems exhibit near-perfect
periodicity. The forces of interaction between different components of a system or
between different systems often give rise to periodic behavior as well. It comes as
little surprise then that the harmonic oscillator is the paradigm upon which both
the mechanical and field theoretical manifestations of quantum physics have become
founded. The aim of this paper is to examine the ground state energy of the quantum
mechanical quartic anharmonic oscillator. The system under question possesses the
following Hamiltonian:
H = − ~2
2m
d2
dx2+mω2x2
2+ g
~x4
4m2ω3. (1.1)
Qualitatively, this system behaves like two coupled harmonic oscillators with a finite
potential barrier between them. Tunneling between these minima is expected, the
frequency of which is directly correlated with the strength of the dimensionless cou-
pling parameter g. In field theory, particles are viewed as excitations in a spacetime
framework of harmonic oscillators. In this context, the quartic term represents a min-
imal type of interaction between these excitations. While this paper will focus on the
quantum mechanical aspects of the system, many of the results have been generalized
and extended to field theory with great success.
2. The Harmonic Oscillator
The purpose of this paper is to derive a set of results for the quartic anharmonic
oscillator. In the limit of zero coupling in the parameter g, the potential of the
system maps to that of the simple harmonic oscillator. It makes sense then to first
review in some detail the behavior of the harmonic oscillator. We will then proceed
to view the weak coupling limit as a perturbation of the SHO Hamiltonian. Our
primary objective in this section will be to derive the energy levels for the quantum
2
VHxL
x
-4 -2 2 4
2
4
6
8
10
12
Figure 1. The Simple Harmonic Oscillator Potential for g = 0
SHO through the application path integral techniques. Depending on the focus of
the calcuation at hand, we will often set ~ = m = ω = 1.
In the Schrodinger picture, the energy levels of a quantum system are obtained via
the kernel given (in the position basis) by
K (xi, ti;xf , tf ) = 〈xf | e−iH(tf−ti)
~ |xi〉
=∑n
〈xf |n〉 〈n| e−iH(tf−ti)
~ |xi〉
=∑n
e−iEn(tf−ti)
~ 〈xf |n〉 〈n| xi〉.
Time translation invariance allows us to write this as
K = 〈xf | e−iHt~ |xi〉
=
∫ ∞−∞〈xf | p〉 〈p| e−
iHt~ |xi〉 dp. (2.1)
If we break the trajectory into small time steps such that t = Nε and maintain this
relation in the limit as ε→ 0, N →∞, we obtain the following result for one interval:
3
〈x2ε| e−iHε~ |xε〉 '
1
2π~
∫ ∞−∞
e−iHε~ e
i~p(x2ε−xε)dp.
Making the substitutions x2ε−xεε
= x and p′ = p− x for the Hamiltonian
H = p2
2+ V (x) yields the Gaussian integral
〈x2ε| e−iHε~ |xε〉 '
1
2π~
∫ ∞−∞
e−iε~ [ 12p′
2+V (x)− 12x2]dp′
= (2πi~ε)−1/2eiε~ [ 12 x2−V (x)]. (2.2)
Inserting N-1 completness relations into (2.1)
∫dxN−1
∫dxN−2 〈xf | x(N−1)ε
⟩ ⟨x(N−1)ε
∣∣ e− iHε~ ∣∣x(N−2)ε
⟩ ⟨x(N−2)ε
∣∣ e− iHε~ · ··〈x2ε| e−
iHε~
∫dxε |xε〉 〈xε| e−
iHε~ |xi〉,
we obtain in the appropriate limits
K = 〈xf | e−iHt~ |xi〉 = lim
ε→0(2πi~ε)−1/2
∫ N−1∏n=0
dxn (2πi~ε)−1/2 ei~S(xf ,xi)
≡∫D [x (t)] e
i~S(x,x), (2.3)
S =
tf∫ti
dt
[1
2
(dx
dt
)2
− V (x)
]. (2.4)
In this hasty derivation of the path integral represetation, we have ignored some
subtleties regarding the commutativity of the potential V (x) with p. For our treat-
ment it can be shown that corrections of O(ε2) should be included, but it can be
further demonstrated that these corrections do indeed vanish in the ε→ 0 limit [12].
4
Let us take a few steps towards evaluting the path integral given in (2.3). In order
to avoid convergence issues, we first Wick rotate into Euclidean space through the
substitution t = itE. This serves to convert the oscillatory piece of the integrand into
a decaying exponential and at the same time takes V (x) to −V (x).
SE =
tf∫ti
dt
[1
2
(dx
dt
)2
+ V (x)
](2.5)
Our ultimate goal here is to obtain the energy levels for the SHO with Euclidean
Lagrangian density
LE =1
2
d2x
dt2+
1
2ω2x2.
The classical path, which dominates in the S >> ~ regime, is found by extremizing
the variational derivative to obtain the classical equation of motion:
0 =δS
δx= − d
dt
∂S
∂ (dx/dt)+∂S
∂x
0 =d2x
dt2+ V ′ (x)
⇒ x = ω2x2. (2.6)
The solution satisfying the boundary conditions x(ti) = xi and x(tf ) = xf is:
xcl (t) =xi sinhω (tf − t) + xf sinhω (t− ti)
sinh (ωT )(2.7)
where tf − ti = T . The integral for the classical action then gives
5
SE (xcl) =
tf∫ti
dt
[1
2
d2xcldt2
+ ω2xcl2
]
=ω
2
(xi2 + xf
2) cosh (ωT )− 2xixfsinh (ωT )
. (2.8)
This result is sufficient in the classical limit. The classical path evidently depicts a
particle oscillating at a resonant frequency in one dimension. However, for the regime
S ∼ ~, quantum effects are no longer negligible and it become necessary to take into
account all the paths contributing to the path integral. We will clean up our notation
a bit and rewrite x(t) in terms of the classical path as follows:
x (t) = x (t) +∑n
cnxn (t) (2.9)
where x ≡ xcl and each xn represents an element of a set of orthonormal functions.
Any function satisfying the boundary conditions specified above can be decomposed
into such a set. Accordingly,
∫ T/2
−T/2dt xn (t)xm (t) = δmn (2.10)
where we have symmetrized the limits for convenience. If we choose these functions
to be eigenfunctions of the second variational derivative of the action,
δ2S
δx2
∣∣∣∣x
=
(− d
dt+ V ′′ (x)
),
we obtain the following eigenvalue equation:
(d2
dt2+ V ′′ (x)
)xn = λnxn. (2.11)
6
Also, since the cn coefficients uniquely determine such an eigenfunction expansion,
we can invoke the equivalency in measure:
[dx] = N ′∏n
dcn. (2.12)
Using the definition ℵn ≡∑n
cnx(t)n, we insert the complete orthonormal set for x(t)
and Taylor expand the action to second order, obtaining
S =
∫dt
[1
2
(d
dt(x+ ℵ)
)2
+ V (x+ ℵ)
]
≈∫ T/2
−T/2dt
[1
2
(dx
dt
)2
+ V (x) + V ′ (x)ℵn +dx
dt
dℵndt
+1
2
dℵndt
dℵmdt
+1
2V ′′ (x)ℵnℵm
].
Using integration by parts, we are left with
S = Scl + ℵndx
dt
∣∣∣∣T/2−T/2
+1
2ℵndℵmdt
∣∣∣∣T/2−T/2
+
∫ T/2
−T/2dt
[(−d
2x
dt2+ V ′ (x)
)ℵn + ℵm
(−1
2
d2
dt2+ V ′′ (x)
)ℵn].
Applying the boundary conditions for x and the orthonormality constraints on xn,
we finally obtain
S = Scl +
∫ T/2
−T/2dt (λnℵnℵm)
= Scl + λnδmn∑
cnxn∑
cmxm
= Scl + λncn2.
Our path integral thus becomes a simple product of Gaussian integrals, and we are
left with
7
K(x, t) = N
∫ ∏n
dcne−S
= Ne−Scl∏n
λn−1/2 (1 +O (~)) . (2.13)
For the harmonic oscillator V (x) is a second order polynomial, so the Taylor ex-
pansion of the actions terminates at second order and these results are exact. The
eigenfunctions and eigenvalues for a Fourier decomposition of the non-classical paths
are
xn =
√2
Tsin(nπTT)
(2.14)
λn, =(nπT
)2
− ω2. (2.15)
The calculation of the normalization factor is a bit involved, but it can be done using
determinant methods [12]. After rotating back to real time, the final result is
K
(xf ,
T
2;xi,−
T
2
)=( ω
2πi~ sinωt
)1/2
eiScl(x,x). (2.16)
The partition function is defined as
Z =
∫K (xf = xi, T ) dx (2.17)
which yields, upon subsitution of (2.16),
Z =1
2i sin (ωT/2)
=e−iωT/2
1− e−2iωT/2
=∞∑n=0
e−iTω(n+ 12). (2.18)
8
We have thus obtained the correct and expected energy levels for the quantum har-
monic oscillator.
3. Pertubation theory and the quartic oscillator
We will now apply a perturbation to the harmonic oscillator of the form g x4
4(see
(1.1)). Assuming g is small, we should be able to express the ground state energy for
the perturbed system as a power a series in the ”coupling parameter” g.
E0(g) =∑n
angn (3.1)
For positive coupling it will be seen that such a perturbative expansion is alternating.
This is not the case for negative coupling. In either case, however, the series obtained
are asymptotic and divergent. Although it would appear that for small coupling,
successive terms would grow smaller with n, this simply does not happen. The di-
vergence for the case of negative coupling can be related to the fact that although
the minimum of the potential at the origin is stable locally, tunneling renders such a
state metastable and a particle will inevitably tunnel to infinity. The behavior of the
system for positive and negative coupling is depicted in Figures 2 and 3.
VHxL
x
-4 -2 2 4
5
10
15
20
25
Figure 2. The Quartic Oscillator Potential for g = +0.1
9
VHxL
x
-4 -2 2 4
-3
-2
-1
1
2
Figure 3. The Quartic Oscillator Potential for g = −0.1
Since these series are asymptotic and divergent in nature, it will be necessary to
consider methods for extracting meaningful results from them. This will be accom-
plished through examinination of the large order asymptotics and a technique known
as Borel resummation.
3.1. General Properties of the Hamiltonian and the Ground State Energy.
We first note an important scaling property of the Hamiltonian under scrutiny. Mak-
ing the tranformation x (t)→ g−1/2x (t) yields the following transformed action:
S =
∫dt
[1
2
(dx
dt
)2
+x2
2+ g
x4
4
]→∫dt
[1
g
(dx
dt
)2
+1
g
x2
2+
g
g2
x4
4
]
=1
gS[g−1/2x (t)
].
This observation will later justify our employment of semiclassical methods for ob-
taining energy levels of the systems to be discussed since small g assumes the role of
~.
In fact, scaling can tell us even more about the ground state energy. Let us first
present an observation, due to Symanzik [13]. For the moment, let us work with the
more general Hamiltonian
10
H = p2 + αx2
2+ g
x4
4.
Following [13], we will designate the energy levels for this system by
En (α, g) .
We implement the following unitary transformation:
[U (c) f ] (x) = c1/4f(c1/2x
).
Since the transformation is unitary, the enery spectrum is preserved:
UHU−1 = c
(p2 + αc−2x
2
2+ gc−3x
4
4
)
En (α, g) = λEn(αc−2, gc−3
)For g = c−3 and α = 1, we are left with
En (α, g) = g1/3En(g−2/3, 1
).
Making the replacement λ = 1g2
, we can also write this as
En (α, λ) = λ−1/6En(λ1/3, 1
).
These results appear to indicate that any En will have at best a third order branch
point at g = 0 since En behaves like g1/3. We also see that a study the small λ limit
corresponds to a study of the large λ limit.
11
Simon proves that the singularity at g =∞ cannot be isolated. A corollary to this
result is that En does have a global cubic branch point at g = 0. We will thus define
the energy levels to be analytic in a cut plane; for the ground state energy the cut
will be taken along the negative real axis.
3.2. The Ground State at Large Orders from Functional Methods. A stan-
dard strategy for obtaining an expansion for the ground state energy of a quantum
system is the application of functional methods. Using this method, we will derive
the first order perturbative correction to the ground state energy of the anharmonic
oscillator. Treating the quartic term as a small perturbation of the harmonic system,
we write
LE = L0 + ∆L.
For the purposes of this section, ∆L = λx4
4!. We introduce a source ”forcing term”
that alters the Lagrangian in the following manner:
L (x, x)→ L (x, x) + J (τ)x (τ) .
where we have omitted the Euclidean subscript. The partition function is then given
by
Z [β, J ] ≡ K (xf , β/2;xi,−β/2)
=
∫D [x] e−
∫ β/2−β/2 L−J(τ)x(τ)dτ (3.2)
with xf = xi. We now proceed to decompose the Lagrangian density into its pertur-
bative and non-perturbative parts and series expand the perturbative component.
12
Z [β, J ] =
∫D [x] exp
(−∫ β/2
−β/2L0 +
λx4
4!− J (τ)x (τ) dτ
)
=
∫D [x]
∞∑n=0
1
n!
∫ (λ
4!
)nx4 (τ1)x4 (τ2) · · · x4 (τn) e−
∫ β/2−β/2 L0−J(τ)x(τ)dτdτ1 · · · dτn
=∑n
1
n!
∫ (λ
4!
)nδ4
δJ4 (τ1)· · · δ4
δJ4 (τn)Z0 [β, J ] dτ1 · · · dτn
≡ e−∫λ4!
δ4
δJ4(τ)Z0 [β, J ]
(3.3)
where δδJ
represents a functional derivative with respect to the ”source” functional.
We will now explore methods for evaluating Z0 [β, J ]. We begin by recognizing that
the harmonic oscillator partition function can be expressed in terms of differential
operators. Through integration by parts and application of the boundary conditions
x (−β/2) = x (β/2) we obtain
S =
∫ β/2
−β/2dτ
[1
2
(dx
dτ
)2
+ω2x2
2+ Jx
]
=
∫ β/2
−β/2dτ
[1
2x
(− d2
dτ 2+ ω2
)x+ Jx
]=
∫ β/2
−β/2dτ
[1
2xOx+ Jx
](3.4)
where O =(− d2
dτ2+ ω2
). We can define the inverse of the operator O as
OG (τ1, τ2) ≡(−∂2
τ1+ ω
)G (τ1, τ2) = δ (τ1 − τ2) (3.5)
where G (τ1, τ2) = G (τ2, τ1) and G obeys the boundary conditions G (τ, β/2) =
G (τ,−β/2). Following [11], we make the variable shift
13
x′ (τ) = x (τ) +
∫G (τ, τ ′) J (τ ′) dτ ′. (3.6)
This yields
1
2x′Ox′ + Jx′ =Jx+ J
∫G (τ, τ ′) J (τ ′) dτ ′
+1
2
[xOx+ xO
∫G (τ, τ ′) J (τ ′) dτ ′ +
(∫G (τ, τ ′) J (τ ′) dτ ′
)Ox
]+
1
2
(∫G (τ, τ ′) J (τ ′) dτ
)O
(∫G (τ, τ ′) J (τ ′) dτ ′
)]
= Jx+ J ·G · J +1
2xOx+ Jx+
1
2J ·G · J
where J ·G ·J ≡∫dτ[J∫G (τ, τ ′) J (τ ′) dτ ′
]. Using (3.4), this indicates that we can
rewrite the integrand in the action integral as follows:
1
2xOx+ Jx =
1
2x′Ox′ − 1
2J ·G · J. (3.7)
It becomes evident that
Z0 [β, J ] = Z0 [β, 0] e12
∫ ∫J(τ)G(τ,τ ′)J(τ ′)dτdτ ′ (3.8)
By solving (3.5) subject to periodic boundary conditions, it can be shown that the
Green’s function of the differential equation is [12]
G (τ, τ ′) =1
2ω
cosh[ω(β2− |τ − τ ′|
)]sinh
[ωβ2
] . (3.9)
The ground state energy can be obtained from the partition function in the limit
limβ→∞
− 1
βlnZ [β, 0] = E0. (3.10)
14
To actually perform the necessary calculation for the first order correction to the
ground state energy, we expand the exponential containing the functional derivatives
to first order in λ = g. This correction takes the form
E(1)0 = − g
4!
β/2∫−β/2
dτ
(δ
δJ (τ)
)4
e12J ·G·J
∣∣∣∣∣J=0
. (3.11)
The functional derivatives can be computed by first noting that
δ
δJ (τ1)
δ
δJ (τ2)e
12J ·G·J =
δ
δJ (τ1)exp
[1
2
∫dτ
∫dτ ′J (τ)G (τ, τ ′) J (τ ′)
]× 1
2
[∫G (τ, τ2) J (τ) dτ +
∫G (τ ′, τ2) J (τ ′) dτ ′
].
It follows that
δ
δJ (τ1)
δ
δJ (τ2)e
12J ·G·J
∣∣∣J=0
=1
2G (τ1, τ2) +
1
2G (τ1, τ2) .
Using Wick’s theorem [11, 12], this result can be extended to the four derivative case.
δ
δJ (τ1)
δ
δJ (τ2)
δ
δJ (τ3)
δ
δJ (τ4)e
12J ·G·J
∣∣∣J=0
= G (τ1, τ2, τ3, τ4)
In terms of Feynman diagrams, it is the interaction vertex that determines the shift
in the ground state energy. This corresponds to all the τ ’s in the Green’s function
being equal. Accordingly,
E(1)0 = −3g
4!
β/2∫−β/2
G2 (τ, τ) dτ (3.12)
where the factor of 3 comes from the symmetry of the contributing Feynman diagram.
15
En
n
2 4 6 8 10
0.002
0.004
0.006
0.008
0.010
Figure 4. First ten terms in perturbative corrections En to the quarticoscillator ground state energy for g = +0.01
Using (3.9), the final result for the first perturbative correction to the ground state
energy is
E(1)0 =
g
64ω2. (3.13)
It would appear at first glance that successive corrections to the ground state en-
ergy should grow smaller at each step with increasing powers of g, eventually yielding
a convergent series. However, adopting a Feynman diagram perspective of the per-
turbative expansion allows us to see that this is not the case. The order g correction
to the ground state corresponds to a single connected Feynman diagram. Were we
to pursue the computation of the order g2 correction, we would see that two new
diagrams would contribute. While the early terms in the perturbative series grow
smaller through increasing powers of g, eventually so many Feynman diagrams con-
tribute to the perturbative corrections at large orders that the terms would begin to
grow like n! for an expansion in gn.
In their seminal paper, Bender and Wu [2, 3] examined the large order behavior
of the quartic oscillator using both Feynman diagram techniques and WKB methods
to obtain the large order asymptotic behavior of the perturbative expansion. They
discovered at large order that the perturbative expansion coefficients for positive
coupling grow approximately like
16
En
n
96 97 98 99 100
1000
2000
3000
4000
5000
6000
7000
Figure 5. Divergence of En for late terms in perturbative expansionfor g = +0.01
an ∼ (−1)n+1 3n+1/2
2π3/2Γ
(n+
1
2
)(3.14)
It should be noted that our normalization here differs slightly from their result. The
Gamma function makes the n!-like divergence immediately evident. This behavior is
depicted in figures 4 and 5.
3.3. Asymptotics and Borel Summability. By applying basic peturbation theory
to the quartic oscillator, we have uncovered a divergent, asymptotic series. In order for
perturbation theory to maintain itself as a valid endeavor in this context, we must find
some way to interpret this series. One method for extracting a physically meaningful
sum from such a divergent one is to extend the boundaries of convergence through
an analytical continuation of the series to a region outside the radius of convergence.
Borel summation is a specific techinque for accomplishing this [6]. Before we describe
this procedure, we will give a brief outline of the nature and structure of asymptotic
series. The discussion is taken from [8].
The following integral cannot be expressed in closed form.
I (x) =
∫ ∞x
e−u2
du
17
We can approximate the integral as a series by noting that the integrand is domi-
nated by the lower limit of the integration region. We make the subsitution y = u2−x2
and thereby obtain
I (x) =e−x
2
2
∫ ∞0
e−y(
1 +y
x2
)−1/2
dy.
By expanding the root in a taylor series about y = 0 and integrating the result
term by term, we obtain the following relationship:
I (x) =e−x
2
x√π
∞∑n=0
(−1)n(n− 1/2)!
x2n
where we have used the expansion
√1 + x =
1√π
∞∑n=0
(n− 1/2)!
n!(−xn) . (3.15)
The series thus obtained for I(x) is alternating for Re [x] > 0, and for large values
of x, the terms in the series will decrease up to n ∼ |x2|. After this, the n! growth
will beging to dominate, and the series asymptotic series will diverge. The origin of
this divergence, as Dingle points out, is in fact due to extension of (3.15) beyond its
radius of convergence. The function (1 + x)−1/2 possesses a square root branch point
singularity at x = −1. The presence of this singularity directly indictates that the
expansion is invalid for values of x greater than unity. Application of the expansion
to the region outside the radius of convergence is responsible for the asymptotically
divergent nature of the resultant series. The divergence of the series expansion for
the ground state energy of the quartic oscillator is due to the cubic branch point
singularity structure of the theory. The standard presciption for obtaining a mean-
ingful result from such a divergent series is to artificially terminate the series at the
smallest term. This is at the very least unsatisfying. We will now adopt a simpler
example in order to explore another method for obtaining a meaningful result from
an asymptotic series.
We start with the sum
18
I(g) =∞∑n=0
angn.
In the event of an n!-like growth of the an coefficients, we can perform a Borel trans-
form which is defined as
BTI (g) ≡∞∑n=0
ann!tn. (3.16)
If we note the integral definition of n!,
Γ (n+ 1) = n! =
∫ ∞0
xne−xdx,
we see that the Borel transform is essentially equivalent to an inverse Laplace trans-
form. The Borel sum, through which we obtain a physically meaningful series, is then
given by
BI (g) =
∫ ∞0
e−tBT (tg) dt (3.17)
which is clearly the Laplace transform of BTI(g).
We will now illustrate the method of Borel summation through a simple example.
Take the geometric series
I (g) =∞∑n=0
gn.
For g < 1 we know that this series converges to
I(g) =1
1− g.
19
As has already been noted, the divergence of this series for g > 1 is due to the simple
pole at g = −1. The Borel transform is given by
BTI (g) =∞∑n=0
tn
n!
and the Borel sum yields
BI (g) =
∫ ∞0
e−t∞∑n=0
(tg)n
n!dt
=
∫ ∞0
e−tetgdt
=1
1− g.
We thus obtain the analytical continuation of the original series into the region outside
the unit radius of convergence. Now imagine if we alter the phase of g = |g|eiθ so
that θ = π. This gives an alternating series for any magnitude of g:
I (g) =∞∑n=0
(−1)ngn.
The Borel sum and Borel transform are
BTI (g) =∞∑n=0
(−1)n
n!tn
BI (g) =
∫ ∞0
e−t∞∑n=0
(−1)n
n!(tg)ndt
=
∫ ∞0
e−te−tgdt =1
1 + g.
20
As expected, this analytic continuation is convergent for any value of g. Throughout
the rest of this paper we will find it necessary to apply these techniques a number
of times. We will postpone their application to the quartic oscillator until we have
obtained some non-perturbative results.
3.4. Dispersion Relations and Large Order Asymptotics. Bender and Wu were
able to infer the large order asymptotic behavior of the quartic oscillator from ob-
servations of the behavior of Feynman diagram contributions at successive orders.
Through numerical analysis at each order, they were able to correctly surmise the
large order growth. They were also able to obtain the same results through another
method which we will proceed to outline.
We begin by shifting the sum in (3.1) and define a new series as follows [15]:
E (g) =1
2+∑n=1
bngn
F (g) =1
g
[E (g)− 1
2
]=
1
g
∞∑n=1
bngn+1 =
∞∑n=0
bn+1gn.
From the complex dispersion relation
Re f (g) =1
π
∫C
Im f (g′)
g′ − gdg′ (3.18)
we obtain the following relationship:
ReE (g) =1
2+
1
π
∫C
ImE (g′)
g′ (g′ − g)dg′ (3.19)
where the contour C is to be determined from the analytic properties of the funtion
in question. We have already defined the ground state energy to be analytic in the
21
complex plane cut along the negative real axis, so the relevant contour is a keyhole
contour passing just above and just below the branch cut. Using the fact that
1
g′ − g=∞∑k=0
gk
g′k+1,
we are left with
∞∑n=0
bngn =
1
2+
1
π
∞∑k=0
gk∫C
ImE (g′)
g′k+1dg′
bn =1
π
∫C
ImE (g′)
g′n+1dg′ for n > 0. (3.20)
Bender and Wu used precisely this method to obtain their large order asymptotics.
In order for this approach to be of any use, it is obviously necessary for us to know
something about the imaginary part of the pertubative expansion. Recognizing that
tunneling amplitudes in the quartic oscillator correspond to exponentially small imag-
inary contributions to the energy levels, Bender and Wu were able to employ complex
WKB techniques to determine the behavior of an at large orders.
4. Non-perturbative Phenomena
Having explored perturbative methods for obtaining the ground state energy of
the anharmonic oscillator, we will now embark upon a program to understand the
nonperturbative aspects of the same system. It will ultimately be seen that there is an
intimate connection between these two ostensibly unrelated theoretical treatments.
We begin by looking at a useful heuristic example.
4.1. The Stokes Equation, Airy Functions and Leading Order Asymptotics.
The following differential equation is the Stokes equation; the solutions to this equa-
tion embody the simplest case of what is known as the Stokes phenomenon:
22
d2y
dλ2= λy. (4.1)
For complex λ, we can find a contour integral solution of the form
y (λ) =
∫ b
a
eλxf (x) dx. (4.2)
Plugging this solution into the differential equation and integrating by parts yields
∫ b
a
(x2 − λ
)eλxf (x) dx = 0,
−eλxf (x)∣∣ba
+
∫ b
a
(x2f (x) +
df (x)
dx
)eλxdx = 0.
When the first term vanishes at the upper and lower limits, the following relationship
is satisfied:
x2f (x) +df (x)
dx= 0
⇒ f (x) = Ae−x3/3. (4.3)
It is immediately seen that in order for the first term to vanish, the condition Rex =
Re (|x|eiϕ) > 0 has to hold such that :
lim|x|→∞
e−x3/3+λx = 0
⇒ 2πn− π
2< 3ϕ < 2πn+
π
2(4.4)
There are three regions in the complex plane that obey this condition (see figure 6).
23
Re[x]
Im[x]
-4 -2 0 2 4
-4
-2
0
2
4
Figure 6. Regions of convergence (shaded), steepest descent contoursand saddles (circled) for the Airy function
The Airy functions are defined by equations (4.2) and (4.3). We can approximate
the value of the Airy function using the saddle point method. At large values of x
in the specified region the integrand decays rapidly and it is actually the case that
the integral is dominated by ”saddles” (2D minima) of the integrand. The steepest
descent paths through the these minima are given by the countours of constant phase
that pass through these saddles. For the Airy case, the saddles are given by
d
dx
(−x
3
3+ λx
)= 0 = −x2 + λ
⇒ x± = ±√λ. (4.5)
There are three contours of constant phase that pass through these saddles. How-
ever, two of them can be continuously deformed into the third indicating that there
are only two unique solutions to the Stokes equation (as expected for a second order
differential equation). These contours are depicted in figure 6. The Airy function
24
Ai(x) is defined by the contour originating and terminating in quadrants III and II,
respectively. The second contour defines the Airy function Bi(x), the second solution
to the Stokes equation.
After finding the saddle points, we expand the argument of the exponent in a
Taylor series about any saddles the contour is capable of passing through and add
the contributions. For the Airy function with positive real values of λ the contour
only passes through one saddle.
I1 =
∫C
e−x3/3+λxdx (4.6)
I1 ≈∫C
exp
[∞∑n=0
xn
n!
dn
dxn
(−x
3
3+ λx
)∣∣∣∣−√λ
]dx (4.7)
I1 = ef0∫ b
a
e−σ2 dx
dσdσ (4.8)
where f0 is the first (constant) term in the Taylor series and
−σ2 =
(x+√λ)2
2!
d2f
dx2
∣∣∣∣−√λ
+
(x+√λ)3
3!
d3f
dx3
∣∣∣∣−√λ
(4.9)
f (x) = −x3
3+ λx (4.10)
Since σ is real and positive along the lines of steepest descent, the integration contour
must begin and end at either infinity or a singularity of the integrand (of which there
are none in this case). The integral is dominated by the region around σ = 0, the
neighborhood of the saddle point. In order to evaluate it, we must first find dxdσ
. If
we neglect the cubic term, we can immediately invert the expression for σ2 and find
(x+√λ) as a function of σ. We obtain
25
σ = ±(x2
2f ′′ (x)|−√λ
)1/2
= ±(x+√λ)λ1/4, (4.11)
dσ
dx= λ1/4 ⇒ dx
dσ= e−iθ/4|λ|−1/4, (4.12)
where the sign is generally dependent upon the contour’s direction through the saddle.
The extra phase θ has been introduced in order to allow variation in the phase of λ.
Combining all these results for the Ai(x) contour, with f0 = −23λ3/2 and θ = 0, the
value of the integral is
I1 ≈π1/2
λ1/4e−(2/3)λ3/2 (4.13)
The Airy function is real for all real values of λ = |λ|eiθ, so it should be real for
negative as well as positive values. However, if the phase of λ is rotated from 0 to −π,
it is easily seen that solution (4.13) becomes complex. In order to compensate for this,
the behavior of the approximation (4.13) must change discontinuously at some point
between θ = 0 and θ = π. This discontinuous change corellates directly with the
addition of a second saddle point to the approximation as the phase of λ changes. It
can be shown by evaluating the steepest descent curves that end in the correct domain
for the Airy function that as θ is varied across the ray at θ = 2π/3 the integration
contour picks up a second contribution from the saddle at x = +√λ. As the phase is
varied through θ = 4π/3 the contribution from the first saddle disappears and only
the second contributes. Recall that their are two solutions to the Stokes equation,
so the manifestation of different behavior in different regions of the complex plane
corresponds directly with the fact that the Stokes equation has different solutions for
different phases of λ.
Using the technology developed above, we can immediately obtain the contribution
from the second saddle. In this case, f0 = e+(2/3)λ3/2 . In taking the root and inverting
to find dxdσ
, however, we obtain a factor of i since the second derivative evaluated at
the saddle is positive. Combining all of this, we have
26
Im[x]
Re[x]
-4 -2 0 2 4
-4
-2
0
2
4
Figure 7. Steepest descent contours and the second saddle contribu-tion at θ = 4π
3
I2 ≈ iπ1/2
λ1/4e−(2/3)λ3/2 (4.14)
Along the ray at θ = 2π/3, the exponential in (4.14) is minimized at infinity and (4.13)
is maximized at infinity. Since the asymptotic representation of the Airy function
must change discontinuously across this ray, and since the Stokes equation must still
be satisfied, it can only pick up extra terms proportional to e±(2/3)λ3/2 . The lines
across which disconinuites are induced when the phase of λ is varied are known as
Stokes lines. Since e−(2/3)λ3/2 is maximized along the Stokes line, its asymptotic
approximation will pick up an extra term proportional to the associated function
minimized along the Stokes line, e+(2/3)λ3/2 . It can be shown that the discontinuity
induced by crossing a Stokes line is less than the error introduced by the saddle point
approximation [4].
27
Adding up the contributions for each saddle, the total value of the integral (4.2) in
the region 2π/3 < θ < 4π/3 is
I (λ) = I1 + I2 ≈π1/2
λ1/4
[e−(2/3)λ3/2 + ie+(2/3)λ3/2
](4.15)
and i is the constant multiplier characterizing the discontinuity across the θ = 2π/3
Stokes line. This constant is a Stokes constant. When λ is rotated through θ = 4π/3,
the first saddle no longer contributes obtain the following approximation from the
second saddle.
I (λ) = I2 ≈ iπ1/2
λ1/4e−(2/3)λ3/2 (4.16)
The situation has changed across this second Stokes line. While the I2 was dominated
by I1 at θ = 2π/3, the signs of the exponential arguments have now flipped, and their
roles are now reversed. Since the discontinuity must occur with the subdominant
term, it is seen that the I1 disappears at θ = 4π/3.
28
Im[x]
Re[x]
-4 -2 0 2 4
-4
-2
0
2
4
Figure 8. Steepest descent contours and the second saddle contribu-tion at θ = 2π
3
There can be no discontinuous change in the coefficient of the solution’s dominant
term across the θ = 0 line (since the Airy function is defined purely by the subdomi-
nant solution here), so how do we reconcile (4.13), which should be valid in the region
−2π/3 < θ < 2π/3, with (4.16). It turns out that (4.16) is only valid in the region
−2π/3 < θ < 0, but it continuously connects with (4.13) at θ = 0. It is easily verified
that
I2
(|λ|e2iπ
)= I1
(|λ|e0iπ
)If we approach the region with both exponential contributions from the θ = −2π/3
direction then the Stokes discontinuity again occurs across that ray and the Stokes
constant is again i, but in that vicinity the solution (4.16) is equivalent to the second
term in (4.15) so the discontinuity again occurs with respect to the subdominant
term. This phenomenon of discontinuties in solutions to differential equations across
29
Stokes lines dividing different regions in the complex plane is known generally as the
Stokes phenomenon.
4.2. Borel Summability and the Stokes Phenomenon. In the last section we
uncovered the leading order asymptotic behavior of the Airy solution to the Stokes
equation using steepest descent methods. The full asymptotic behavior of the Airy
solution to the Stokes equation can be written as [8]
I1(λ) =e−(2/3)λ3/2
4π (−1/6)! (−5/6)!
∞∑n=0
(n− 1/6)! (n− 5/6)!
n![− (4/3)λ3/2]n . (4.17)
This series diverges like n!. The Borel transform is obtained through simple division
by n!. The Borel sum is
B1 (λ) =e−(2/3)λ3/2
4π (−1/6)! (−5/6)!
∞∑n=0
(n− 1/6)! (n− 5/6)!
n!n!(−4/3)n
∞∫0
e−t(t
λ
)3n/2
dt
This equation can be reformulated in terms of a hypergeometric function as follows:
B1 (λ) =e−(2/3)λ3/2
4π
∞∫0
e−tF
(5
6,1
6, 1;− 3t
4λ3/2
)dt (4.18)
where the hypergeometric function is defined as [10]
F (a, b, c;x) ≡ Γ (c)
Γ (a) Γ (b)
∞∑n=0
Γ (a+ n) Γ (b+ n)
Γ (c+ n)
xn
n!(4.19)
For λ3/2 > 0 this integral is well-defined. The hypergeometric function has a branch
cut along the real axis from x = 1 to −∞. The simplest way to see this is to examine
the singularity structure of the differential equation that the hypergeometric function
satisfies. Another more heuristic approach is to set b = c in the definition of the
series. In this case,
30
g-plane
Figure 9. Shift of branch cut for continuation from postive to negative coupling
F (±a, b, b; z) =1
(1∓ z)a.
We see that the singularity structure illustrated in (3.3) is recovered. For integer
values of a, the function has a simple pole structure. Generally the hypergeometric
arguments will take on non-integer values in which case the function exhibits branch
point singularity structure.
For the argument above the cut traces the negative real axis. We could have
also anticipated this by noting that the series generated is alternating for λ3/2 > 0.
However, if we allow the phase of the coupling to vary, the branch cut can rotate its
alignment from the negative side of the real axis to the positive as depicted in figure
9. In this event, an ambiguity is invoked since we can evaluate the integral directly
above (C1) or directly below (C2) the cut (see figure 10). In general, the difference
between the principle branches on either side of the cut is [10]
F (a, b, c;x+ iε)−F (a, b, c;x− iε) =2πi
Γ (a) Γ (b)(x− 1)c−a−bF
(c− a, c− bc− a− b+ 1
; 1− x).
(4.20)
31
Borel Plane
C1
C2
Figure 10. Choice of integration path in Borel plane for negative coupling
For the Airy Borel sum , we have
Babove −Bbelow =iπ1/2e+(2/3)λ3/2
2π1/2Γ (1/6) Γ (5/6)
∞∫4λ3/2/3
e−tF
(5
6,1
6, 1; 1− 3t
4λ3/2
)dt.
Making the variable subsitution x = t− 4λ3/2
3and noting the symmetry of the hyper-
geometric function under interchange of a and b leaves us with
Babove −Bbelow = (i)e−(2/3)λ3/2
4π (−1/6)! (−5/6)!
∞∫0
e−xF
(5
6,1
6, 1;− 3x
4λ3/2
)dx. (4.21)
The factor in parentheses is the only difference from the original Borel sum for the the
alternating, convergent series. The Airy function represents a decaying solution to
the Stokes equation for 0 < Arg(λ) < 2π/3. The discontinuity produced in the Borel
sum at the upper limit of this phase range corresponds precisely to the discontinuous
change in the form of the Airy function at the Stokes line along Arg(λ) = 2π/3 that
we have already shown to exist. We have also verified the previous result that the
Stokes constant activated across the Stokes line for the more precise asymptotic form
of the Airy function is i.
The Stokes equation offers a simple context for the visualization of the Stokes phe-
nomenon whereby discontinuities are observed when crossing between various regions
32
in the complex plane divided by Stokes lines. The cases to be examined in the re-
mainder of this paper will become increasingly complicated, but the same principles
will apply. It will be seen in the full one-dimensional quantum theory that the Stokes
line discontinuity is related to the cubic branch structure of the ground state energy
5. The Zero-Dimensional Quartic Potential
It is our ultimate goal to study the quartic oscillator with Hamiltonian (1.1). We
take a step closer to this goal by analyzing the structure of the zero-dimensional inte-
gral counting the number of Feynman diagrams contributing to the energy levels for
the quartic case as well as for a related potential arising in supersymmetric quantum
mechanics. For the latter case we will proceed to investigate the asymptotic structure
of the series thereby derived.
The integral that counts the number of Feynman diagrams for the quartic potential
is [15]
I (g) =
∫ ∞−∞
e−(x2/2+gx4/4)dx (5.1)
Taylor expanding the integrand about x = 0, we obtain
e−gx4
4 =∞∑n=0
(−1)ngnx4n
4nn!,
I (g) = 2
∫ ∞0
(−1)n
n!
(gx4
4
)ne−x
2/2dx,
Performing the Gaussian integration yields the following expression:
33
In (g) = (−1)n22ngn
4nn!(−1)2n
∫ ∞0
e−αx2/2dx
= (−1)n√π
2
22ngn
4nn!(−1)2n ∂
2n
∂α2nα−1/2
= (−1)n√π
2
gn
n!
Γ (2n+ 1/2)
Γ (1/2),
In (g) = (−1)ngn
21/2n!(2n− 1/2)!. (5.2)
Again, we have applied a series expansion to a region exterior to its radius of conver-
gence and the n! divergence is immediately self-evident.
We will now attempt to understand a quartic potential through the application of
saddle point methods. As warm up for the one-dimensional case, we will examine
a potential arising in supersymmetric quantum mechanics and field theory. For the
simplest supersymmetric theory, the corresponding action is given by [1]
S (x) =
∫ T/2
−T/2dt
(1
2x2 +
1
2λ2(x2 − v2
)2+ pλx
)(5.3)
where λ = 1g2
. The actual supersymmetric case is for p = 1. For the p = 0 case, (5.3)
can be obtained from (5.1) as follows. We first complete the square:
V (x) =x2
2+gx4
4=
(√gx2
2+
v2
2√g
)2
− v2
4g.
We proceed by scaling and shifting the potential in the following manner:
V → V
2+v2
4g.
Making the substitutions u = g1/2x and g = 12λ2
leaves us with
34
V (u) =1
2λ2(u2 − v2
). (5.4)
For the supersymmetric case p 6= 0, the integral that counts the number of Feynman
diagrams is given by
I (λ) =
∫ ∞−∞
e−[12λ2(u2−v2)
2−pλu
]dx. (5.5)
The condition necessary to enforce convergence of the integrand is Re (λ2x4) ≥ 0.
The maximum is movable since the convergence depends on both λ and x. Making
the variable change u = λ1/2y eliminates this inconvenience and leaves us with
I (λ) =1√λ
∫ ∞−∞
e−[12(y2−λv2)
2−pλ1/2y
]dy. (5.6)
This integral converges in regions for which Re (y4) ≥ 0 holds. These regions are
−π8
+nπ
2< arg (y) <
π
8+nπ
2(5.7)
where n is an integer.
We will now continue to examine our case for p = 0 and λ positive and real. The
potential is simply
V (y) =1
2
(y2 + λ
)2, (5.8)
and the integral counting the numer of Feynman diagrams is
I (λ) =1√λ
∫ ∞−∞
e−12(y2+λ)
2
dy. (5.9)
Evalutating this integral directly in Mathematica yields the following result in terms
of modified Bessel functions:
35
I (λ) =e−λ
2/4K−1/4 (λ2/4)√2
. (5.10)
As in the Airy case, this integral represents the solution to a second order differential
equation. We can derive this equation as follows. The modified Bessel’s equation for
the function given above is
u2 d2z
du2+ u
dz
du−(u2 +
1
16
)z = 0 (5.11)
where u = λ2/4. Substituting z (u) =√
2e+uI (u) into the differential equation gives
λ2I ′′ + λ(1 + λ2
)I ′ +
(λ2 − 1
4
)I = 0. (5.12)
The most general solution to this differential equation in terms of Hermite polynomials
Hν (z) and modified Bessel functions Iν (z) is given by
I (λ) =C1e
−λ2/4I−1/4 (λ2/4)√2
+C2e
−λ2/2H−1/2
(λ/√
2)
√x
. (5.13)
If we were to attempt to derive the Stokes equation from it’s integral solution, we
could simply employ parameter differentiation.
Iairy =
∫ ∞−∞
e−x3
3+λxdx
d2Iairydλ2
− λIairy =
∫ (x2 − λ
)e−
x3
3+λxdx = 0
We can express this as
∫δS
δxe−S(x)dx = 0.
36
This is a basic action extremization principle. When equated to zero, the functional
form of δSδx
is actually the equation for the saddle points of the integral. For the
quartic potential, the saddles are given by
y = 0,±i√λ. (5.14)
While the differential equation satisfied by the quartic integral cannot be so easily
derived from a minimization principle, the presence of three saddles would suggest
that the integral should satisfy a third order differential equation in λ. However, the
following integral representation of the modified Bessel function makes it clear that
only two saddles uniquely contribute to the action of the theory [10]:
Kν (z) =zν
2ν+1
∫ (t−ν−1e−t−
z2
4t
)dt. (5.15)
We can look at an integral passing through any of the saddle points. In general
the expression to be evaluated is
I (λ) ≈ 1√λ
∫C
exp
[−∞∑n=0
yn
n!
(dn
dynV (y)
)∣∣∣∣ysaddle
]dy (5.16)
where the contour is a line of steepest decent passing through a given saddle and
terminating at infinity in a region where the integrand goes to zero. The ”good
regions” and contours of constant phase through the saddles are depicted in figure
11.
37
ImHyL
ReHyL-4 -2 0 2 4
-4
-2
0
2
4
Figure 11. Regions of convergence (shaded), steepest descent con-tours and saddles (circled) for the supersymmetric potential; λpostive,p = 0, v = 1
5.1. The Trivial Saddle for p = 0. For positive coupling, only one saddle at the
origin contributes dominates the integrand. The saddle point expansion yields pre-
cisely the original form of the integral. Again, we can expand the integrand in a
Taylor series about y = 0 and evaluate the resultant Gaussian integral. The result is
I0 (λ) =e−λ
2/2
λ
∞∑n=0
(−1)n
2nn!
2 (2n+ 1/2)!
4n+ 1λ−2n. (5.17)
The Borel transform is
BTI0 (t) =∞∑m=0
1
m!I1 (t)
and the Borel sum is given by
38
BI0 (λ) =
∫ ∞0
e−tBI1
(t
λ
)dt,
BI0 (λ) =e−λ
2/2
λ
∞∫0
e−t∞∑n=0
2 (2n+ 1/2)!
n!n!
(−1)n
(4n+ 1)
(t
2λ2
)ndt.
Again, we can rewrite this in terms of the hypergeometric function:
BI0 (λ) =e−λ
2/2
λ
∞∫0
e−t∞∑n=0
[Γ (1/2 + 2n)
Γ (1 + n)
1
n!
(− t
2λ2
)n]dt.
Reformulating this equation using Γ (2z) = 22z−1√π
Γ (z) Γ (z + 1/2), we obtain
BI0 (λ) =(2π)1/2 e−λ
2/2
λ
∞∫0
e−tF
(1
4,3
4, 1;− 2t
λ2
)dt. (5.18)
The integral is well defined in this case for λ2 > 0. Rotating the phase of λ by π2
places
the branch singularity on the positive real axis. For the Borel sum, the ambiguity
across the cut is given by
BI0 (λ+ iε)−BI0 (λ− iε) =i(2π)1/2e+λ2/2
λ
∞∫λ2/2
e−tF
(3
4,1
4, 1; 1− 2t
λ2
)dt.
Making the variable subsitution x = t− λ2
2and noting the symmetry of the hyperge-
ometric function under interchange of a and b leaves us with
BI0 (λ+ iε)−BI0 (λ− iε) = (i)(2π)1/2
λ
∞∫0
e−xF
(1
4,3
4, 1;−2x
λ2
)dx. (5.19)
The appearance of the first term in parentheses indicates that we have reached a
Stokes line by rotating the phase of λ. An alternative way of seeing this is the fact
that varying the phase of λ by π2
serves to shift the imaginary saddles onto the real axis
39
ReHyL
ImHyL
-4 -2 0 2 4
-4
-2
0
2
4
Figure 12. Added contribution of saddles at ±√λ for negative coupling
at y = ±√λ. Accordingly, the expansion picks up contributions from both of these
saddles (see figure 12). Additionally, the series becomes non-alternating, and therefore
it is not Borel summable in the strict sense. The series diverges approximately like
n! in this case. The ambiguity across the cut is directly related to this divergence.
We have already seen in the Airy case that crossing a Stokes line evokes or annihi-
lates an exponentially small term. This term necessarily satisfies the same differential
equation as its complementary solution in another region of the complex plane. It is
also known that the Borel sum of a solution to a differential equation satisfites the
same differential equation [7]. Accordingly, it can be verified that the discontinuity
obtained in (5.19) satisfies the same differential equation (5.12) as the original series
obtained in (5.18). The simplest way to do this is by direct matching of the hyperge-
ometric series to Hermite polynomials and modified Bessel functions using identities
and integral representations.
5.2. The Non-trivial Saddles. The other saddles produce the following integral
40
I± (λ) ≈ 1√λ
∫C
e−(λ2+λ3+y4/2)e−4y2|λ|e∓2iy3√|λ|dy. (5.20)
Since the Gaussian is even and sine is odd, the imaginary piece disappears. For I+(λ),
we are left with
I± (λ) ≈ e−(λ2+λ3)√λ
×∞∫
−∞
cos(
2y3√|λ|)e−(4y2|λ|+y4/2)dy. (5.21)
We can rearrange the integral as follows:
∞∫−∞
cos(
2y3√|λ|)e−(4y2+y4/2)dy =
∫e−4y2
∞∑n=0
(−1)n
(2y3√|λ|)2n
(2n)!
∞∑m=0
(−1)m(y4)
m
2mm!dy
=∞∑n=0
∞∑m=0
(−1)n+m(
2√|λ|)2n
2m (2n)! (m)!
i∞∫−∞
y6n+4me−4y2dy.
The final result is
I± (λ) =e−(λ2+λ3)
λ
∞∑n=0
∞∑m=0
(−1)n+m
2n+5m+1 (2n)! (m)!
Γ(3n+ 2m+ 1
2
)λ2m+2n
. (5.22)
It is clear that this sum diverges factorially as expected. One could obtain the Borel
sum of the series by performing a double Borel transform and applying an inverse
Laplace tranform of the result. We do not pursue this computation here, but we will
come back to it in the discussion.
6. Nonperturbative Methods and the Quartic Oscillator
It is now our goal to apply semiclassical techniques to obtain nonperturbative
corrections to the ground state energy of the quartic double well potential. These
41
VHxL
x
-4 -2 2 4
-2
-1
1
2
3
Figure 13. The Quartic Oscillator Potential for m = −1, g = 0.1
corrections are nonperturbative because, as we will demonstrate, they are exponen-
tially small and vanish at any order in perturbation theory. Our strategy will be
to apply the saddle point methods developed above to the full 1D quartic oscillator.
The only added complication is the kinetic term. In order to obtain nonperturbative
corrections to the ground state energy, we need to apply steepest descent analysis to
the partition function
Tr[e−βH
]=
∫D [x] e−S[x] (6.1)
where the action is given by
S =
∫ T/2
−T/2dt
[−1
2x2 − 1
2x2 +
g
4x4
](6.2)
satisfying the periodic boundary conditions
x (−T/2) = xi,
x (T/2) = xf ,
x (−T/2) = x (T/2) = 0. (6.3)
42
x
V HxL
Instanton HIL
Anti - Instanton HIL
Molecular Instanton HIIL
-4 -3 -2 -1 1
0.5
1.0
1.5
Figure 14. The Shifted Quartic Oscillator Potential for m = −1, g = 0.1
In order to make (6.1) more suitable for saddle point methods and instanton cal-
culus, we first improve the symmetry of the potential by shifting its zero (following
[14]).
V (x)→ V (x)− 1
4g=g
4
(i√g
+ x
)2(i√g− x)2
We then make the coordinate translation x → ix + i√g
and implement the scaling
g → 4g, leaving us with the following Euclidean potential:
V (x) = x2(1 +√gx)2. (6.4)
The scale transformation x (t) → g−1/2x (t) is still valid for demonstrating the
applicability of semiclassical approximations.
6.1. The Single Instanton Contribution. The addition of the kinetic term to the
Hamiltonian adds a complication to our semiclassical analysis. In this case the action
is actually vanishing at any of the saddle points. However, by recognizing that the
action contains the equation of motion for a particle moving in a potential −V (x)
(see Figure 3), we can obtain the following relationship:
43
xH2 tL
2 t
-4 -2 2 4
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
Figure 15. Instanton solution; g = 0.1
1
2
(dx
dt
)2
− V (x) = 0
⇒∫ tf
ti
dt =
∫ xf
xi
dx√2V (x)
= β (6.5)
Since the energy levels are proportional to e−Hβ, we seek solutions to this equation
that are finite in the β → ∞ limit. For the potential given in (6.4), (6.5) can be
integrated directly to obtain the following:
xI (t) =1
√g (1 + e±t)
. (6.6)
This is an instanton (anti-instanton) solution. The solution can be viewed as a tun-
neling event in Euclidean space between degenerate minima as depicted in Figure 15.
The tunneling occurs over very short ”time” scales (hence the designation instanton).
The contribution to the action is given by
44
SI =
∫ β/2
−β/2
1
2
(dxIdt
)2
+ V (xI)
=
∫ β/2
−β/2
(dxIdt
)2
=
∫ a
−adx(2V (xI))
2. (6.7)
Evaluating this using either the kinetic or potential term and inserting the instanton
solution yields, in the β →∞ limit,
SI =1
6g. (6.8)
This is the single instanton contribution to the action.
6.2. Instanton Anti-Instanton Configurations. We can also calculate the con-
tribution to the action of a distantly separated instanton-anti-instanton pair. First
of all, we can express the instanton solution in terms of the anti-instanton solution
as follows [15]:
x (t) = g−1/2(1 + et
)−1= g−1/2
[1−
(1 + e−t
)−1]
= g−1/2 (1− x (−t)) . (6.9)
Combining an instanton and an anti-instanton separated by a distance 2a gives us
m (t) ≡ x (t− a) + x (−t− a)− g−1/2 = x (t− a)− x (t+ a) (6.10)
Using the following definitions:
u (t) ≡ x (t− a) , (6.11)
v (t) ≡ u (t+ 2a) , (6.12)
45
2 t
xH2 tL
-15 -10 -5 5 10 15
-6
-5
-4
-3
-2
-1
Figure 16. Instanton/Anti-Instanton solution for large separation;g = 0.1
we can write this as
m (t) = u− v. (6.13)
This solution represents a tunneling event in Euclidean time from one minima of
the quartic potential to another and back (see Figures 14 and 15). In the proceeding
analysis, we will presume that the separation between instanton/anti-instanton events
is larger than the characterisic ”radius” of a single instanton event. This radius is
essentially determined by the abruptness of the instanton tunneling event as measured
by the time-width of the jump in Figure 13.
46
2 t
xH2 tL
-4 -2 2 4
-2.5
-2.0
-1.5
-1.0
-0.5
Figure 17. ”Molecular” Instanton/Anti-Instanton solution for smallseparation; g = 0.1
The action of the instanton-anti-instanton configuration is given by
SII =
∫ T/2
−T/2dt
[1
2
(dm
dt
)2
+ V (m)
]. (6.14)
Plugging in (6.4) and (6.13), we obtain
SII = 2SI +
∫ T/2
−T/2[−uv + V (u− v)− V (u)− V (v)]. (6.15)
The symmetry of u allows us to halve the range of integration. Integration by parts
yields
SII = 2SI + 2
[−v (0) u (0) +
∫ T/2
0
dt (vu+ V (u− v) + V (u) + V (v))
].
As we have already argued, u(t) satisfies the equation of motion (6.5). In the range
t > 0, v(t) is small and we can expand V in powers of v. The first correction to S is
of the order e−2a, so we only need to retain terms of order v2. The integrand becomes
47
vV ′(u) + V (u)− vV ′ (u) +v2
2V ′′ (u)− V (u)− V (0)− vV ′ (0)− v2
2V ′′ (0).
The terms linear in v immediately cancel. Also, V (0) = V ′ (0) = 0. We are left with
SII = 2SI −2e−a
g (1 + e−a)2 (1 + e2a)+
∫ ∞0
v2 [V ′′ (u)− V ′′ (0)] dt.
Since v(t) decays exponentially for large t, the integral is dominated by the t = 0
region where u ≈ 1 and accordingly v ≈ 0. In this region, V ′′ (u) ≈ V ′′ (1) = V ′′ (0)
and the integrand cancels. In the large a limit, keeping only the lowest exponential
order corrections, we finally arrive at
SII =1
3g− 2
ge−2a. (6.16)
Expression (6.16) is only valid for large separation between instantons (the dilute
instanton-gas approximation). In this regime the second term is negligible, and it
become obvious that the action of a largely separated instanton/anti-instanton pair
is just twice the action of a single instanton. We would like to know what happens
when the separation between such events become small.
The In order to calculate the action for small a, we can proceed as follows [5].
Expressing the separation distance a as |t1 − t2|, the instanton interaction energy is
given by
U(t1, t2) = (2/g) exp(−|t1 − t2|). (6.17)
This is the action of a ”molecular” instanton. Making the transformations τ1 = t1−t2and τ2 = t1 + t2 leaves us with the following partition function:
Z2 = w2
0∫−T
dτ1
T+τ1∫−T−τ1
dτ2 exp[(2/g)eτ1 ] + w2
T∫0
dτ1
T−τ1∫−T+τ1
dτ2 exp[(2/g)e−τ1 ]. (6.18)
48
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
→
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Figure 18. Coordinate Tranformation for Evaluation of the IAI Action
Perfoming the integration over τ2 results in
Z2 = w2
0∫−T
dτ1 2(T + τ1) exp[(2/g)eτ1 ] + w2
T∫0
dτ1 2(T − τ1) exp[(2/g)e−τ1 ]
= 2w2T
0∫−T
dτ1 exp[(2/g)eτ1 ] + 2w2
0∫−T
dτ1 τ1 exp[(2/g)eτ1 ]
+ 2w2T
T∫0
dτ1 exp[(2/g)e−τ1 ]− 2w2
T∫0
dτ1 τ1 exp[(2/g)e−τ1 ].
In the first two terms we make the variable change τ1 = −τ , giving
49
Z2 =2w2T
T∫0
dτ1 exp[(2/g)e−τ1 ]− 2w2
T∫0
dτ1 τ1 exp[(2/g)e−τ1 ]
+ 2w2T
T∫0
dτ1 exp[(2/g)e−τ1 ]− 2w2
T∫0
dτ1 τ1 exp[(2/g)e−τ1 ].
Combining like terms and taking into account the fact that the Jacobian of the trans-
formation is dt→ 12dτ , a detail ignored above, we are left with
Z2 = w2TI1 − w2I2 + w2T 2 − 1
2w2T 2,
Z2 =1
2w2T 2 + w2TI1 − w2I2, (6.19)
where
I1 =
∞∫0
dt [exp[(2/g)e−t]− 1], (6.20)
I2 =
∞∫0
dt t[exp[(2/g)e−t]− 1]. (6.21)
The first term corresponds to the action of the single-instanton configuration calcu-
lated above. The second term corresponds to the action of instanton anti-instanton
molecule configurations. The third term is not proportional to T and so becomes
negligible in the T →∞ limit.
We can evaluate the second term by continuing 2g→ −2
gand using integration by
parts. With x = g2, we have
50
I (x) =
∫ ∞0
dτ
[exp
(−e−τ
x
)− 1
]= τ
[exp
(e−τ
x
)− 1
]∞0
+1
x
∫ ∞0
τe−τe
(− e−τx
)dτ
=1
x
∫ ∞0
dτ exp
[−e−τ
x− τ + ln τ
]= −γ + ln
1
x. (6.22)
This result can also be obtained by u-subsitution. Letting u = e−τ/x, the integral
becomes
I(u) =
∫ 1/x
0
du
(e−u
u− 1
u
).
Integrating by parts again, we are left with
I(u) = − lnu|1/x0 +e−u lnu∣∣1/x0−∫ 1/x
0
du(e−u lnu
)= ln
(e−τ/x
)∣∣∞x− exp
(−e−τ
x
)ln(e−τ/x
)∣∣∣∣∞x
+
∫ ∞x
du(e−u lnu
),
limx→0
I(x) = limx→0
(− ln (1/x) + ln x+ e−1/x ln (1/x)− lnx− γ)
= −γ + ln1
x
= −γ + ln
(−2
g
). (6.23)
We again obtain an ambiguity here. Following the observation of Bogolmony, if
−g = e±iπ|g|, we have the result that
I = −γ + ln
(2
g
)± iπ. (6.24)
51
Our final expression for the action of the instanon gas with free instantons and
anti-instantons as well as instanton-anti-instanton molecules is
SII =1
3g+
2
ge−2a. (6.25)
Combining all the results obtained thus far, at lowest order the expansion for the
ground state energy is
E0 (λ) = E0,Perturbative −1√πge−
16g +
[−γ + ln
(2
g
)± iπ
]1
πge−( 1
3g+ 2ge−2a). (6.26)
It would seem that this expansion is a haphazard combination of perturbative and
nonperturbative effects that has no formal basis. Why are we justified in combining
terms from unrelated asymptotic expansions that do not show up at any order in per-
turbation theory with pertubative terms that are obtained by a completely different
mechanism? We will postpone the answer to this question for the discussion.
7. The Perturbative Expansion for Negative Coupling and
Ambiguities
The goal of this section is to examine the large order perturbative expansion for
the ground state energy and demonstrate the presence of an imaginary ambiguity for
certain phases of the coupling constant. We proceed to relate this to the ambiguity
invoked in the non-perturbative analysis presented above. We will first obtain the
large order behavior via application of the dispersion relationship derived in section
3.4. We point out that this procedure is circular in a certain sense. The ambiguity
invoked by Borel summation of the result obtained directly through saddle point
analysis should cancel with the ambiguity invoked by the nonperturbative analysis.
However, the applicability of this analysis to the result obtained by Bender and Wu
through different methods than those employed here demonstrates the significance of
the cancellation.
At leading order, the imaginary part of the energy obtained above is
52
Figure 19. Dispersion ”Keyhole” Contour
ImE (g) =1
ge−
13g .
The relevant dispersion integral becomes
an =1
π
0∫−∞
e−1/3g
g (gn+1)dg
where the relevant contour is a ”keyhole” contour along the negative real axis, since
E(g) has a branch cut along the negative real as discussed in section 3.1 (See Figure
19).
This integral is simple to evaluate. For negative coupling we obtain
an =3n+1
πΓ (n+ 1) . (7.1)
53
Evaluation of the previous integral for positive coupling would have yielded an alter-
nating, and therefore Borel summable, series. But this system would have a unique
ground state and no non-perturbative tunneling effects. We see that for negative cou-
pling the series grows exactly like n!. We can anticipate from the discussions above
that Borel summation of this series will introduce an ambiguity.
The Borel transform of the series is given by
BTE (g) =∞∑n=0
3n+1
πgn,
and the Borel sum is
BE (g) =
∞∫0
∞∑n=0
e−t3n+1
π(tg)ndt.
Recognizing that the integrand is simpy a geometric series, the integral becomes
BE (g) =3
π
∞∫0
e−t
1− 3tgdt.
The integrand has a simple pole at t = 1/3g. The principal value can be obtained
numerically, but an ambiguity is invoked by the choice of passage above or below the
pole along the Borel contour. The result for the perturbative series is
E0,P (g) =3
π×
PV ∞∫0
e−t
1− 3tgdt± 1
2
(2πiRes
[e−t
1− 3tg, t =
1
3g
])where the the infinitesimally small semi-circle above/below the pole contributes an
amount equal to half the residue
54
Res
[t =
1
3g
]= −e
−1/3g
3g.
. The imaginary ambiguity invoked by Borel resummation of the divergent series is
thus
ImE0,P = ∓1
ge−
13g .
We see that the ambiguity invoked by Borel resummation in the case of negative
coupling cancels exactly with the ambiguity induced through inclusion of the second
order instanton/anti-instanton contribution to the action.
We will now derive the ambiguity in the perturbative sum at large orders directly
from the coefficients obtained by Bender and Wu. The Borel sum in this case is given
by
BE(g) =
√3
2π3/2
∫ ∞0
e−t∞∑0
3nΓ (n+ 1/2)
Γ (n+ 1)(gt)ndt (7.2)
=
√3
2π
∫ ∞0
e−tF
(1
2, 1, 1; 3gt
)dt (7.3)
The ambiguity invoked by the discontinuity across the branch for negative g is then
Im [BE (g)] =
√3
2π
∫ ∞1/3g
e−t(
2π
Γ (1/2)
)(3gt− 1)−1/2F
(1
2, 0,
1
2; 1− 3gt
)dt
Making the subsitution x = t− 13g
yields
Im [BE (g)] =
(√3
2π
)e−1/3g
∫ ∞0
e−x(
2π
Γ (1/2)
)(3gx)−1/2F
(1
2, 0,
1
2;−3gx
)dx
=√g × 1
ge−1/3g.
(7.4)
55
After insertion of a missing factor of g−1/2, we see that the cancellation of the imagi-
nary ambiguity induced from the nonperturbative analysis occurs again. We surmise
this factor comes from the nontrivial transformation from the potential examined by
Bender and Wu to the potential examined by Bogomolny. A full WKB analysis should
give the asymptotic behavior of the transformed potential should look like (7.1), but
such an analysis is beyond the scope of this paper. The sign difference is determined
by the direction of analytic continuation in the Borel plane. In this case, the term
needed for the cancellation of the nonperturbatively induced imaginary ambiguity is
obtained from an ostensibly unrelated examination of the large order perturbative
behavior of the system.
8. Discussion
It is not simply a miraculous coincidence that we were able to combine the per-
turbative expansion for the ground state energy of the quartic oscillator with non-
perturbative contributions computed using instanton methods and obtain from two
imaginary results a purely real expression. This cancellation is in fact embedded in
an advanced mathematical formalism known as resurgence. The approximation we
gave in (6.26) is actually the sum of lowest order terms in a transseries expansion of
the ground state energy. The full formal expansion for the energy levels to second
order in the instanton anti-instanton background is given by [9, 14, 15]
E0 (g) =∞∑l=0
algl −
∞∑n=1
(e−1/6g
√πg
)n n−1∑k=0
[ln
(−2
g
)]k ∞∑l=0
blgl (8.1)
Within this formalism, we see that the single-instanton contribution to the energy
comes from the n = 1, k = 0 term in the double sum. The double-instanton contri-
bution for large separation comes from the n = 2, k = 0 term. The contribution for
small separation comes from the n = 2, k = 1 term. The first term is the standard
(and divergent) perturbative series for the ground state energy. Borel summation of
this term yields a convergent series with an imaginary ambiguity. This ambiguity pre-
cisely cancels with the ambiguity arising from Borel resummation. The resurgence
56
formalism actually links low order semiclassical expansions in e−1/g with the large
order asymptotics of perturbation theory in a very systematic way. Resurgence ap-
plies to asyptotic series in a very general way. For one-dimensional quantum theories
the nonperturbative series expansions, known as transseries, can often be viewed as
arising from instanton effects. In our analysis of the zero-dimensional anharmonic
potential, however, these effects arose directly from saddle point expansions. It is the
goal of future work to carry out the calculation terminated in section 5.2 and compare
the results to the calculations of section 5.1. In a sense, the early transseries terms
”know” about the late term behavior of the asymptotic perturbative expansion. In
addition to this connection between perturbative and nonperturbative aspects of the
computation, the early terms in the transseries also ”know” about each other. In
the case of the zero-dimensional quartic oscillator, the imaginary saddles that didn’t
contribute to the action integral for positive coupling are actually implicitly present
in the result for the trivial saddle [4]. The shift of these saddles to the real axis with
variation of the phase of the coupling is analogous to the shift of the one-dimension
system from having a unique mimimum for positive coupling (no instanton effects,
or better, complex instantons [15]) to having two degenerate minima about which
instanton ”saddle” expansions can be made.
57
References
[1] I.I. Balitsky; A.V. Yung, Instanton Molecular Vacuum in N=1 Supersymmetric Quantum Me-chanics, Nuclear Physics, B274 475 (1986)
[2] C.M. Bender; T.T. Wu, Anharmonic Oscillator, The Physical Review, 184 1231 (1969)[3] C.M. Bender; T.T. Wu, Anharmonic Oscillator II. Study of Perturbation Theory in Large Order,
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