Download The Power of Perturbation Theory

The Power of Perturbation Theory
Marco Serone, SISSA, Trieste
based on 1612.04376 and to appear, with G. Spada and G. Villadoro
February 2017
1
Main result:
All observables in a one-dimensional quantum
mechanical system with a bounded potential can be
entirely computed from a single perturbative series
2
Introduction and Motivation
It is known that perturbative expansions in QM and QFT
are only asymptotic with zero radius of convergence
[Dyson, 1952]
In special cases, like the anharmonic oscillator in QM and 42,3 QFT
it has been proved that perturbation theory is Borel resummable
and leads to the exact results (no non-perturbative contributions)
[Loeffel et al, 1969; Simon and Dicke, 1970; Eckmann, Magnen and
Seneor, 1975; Magnen and Seneor, 1977]
Most perturbative expansions are not Borel resummable
Technically, this is due to the appearance of singularities of the Borel
function along an integral needed to perform to get the answer
The singularity can be avoided by deforming the contour
but the result present an ambiguitiy of order exp( a/ n )
3
If one is able to find a semi-classical instanton like configuration (and its whole
series) leading to the same factors, one might hope to remove the ambiguity
A systematic way to proceed along these lines uses the theory of resurgence
[Ecalle, 1981]
Unfortunately, one encounters various difficulties using this approach
• Even if Borel summability is somehow proven, convergence to
the exact result requires the knowledge of some analytic
properties of the observable, in general hard to verify
•
Beyond the weak coupling regime, one needs to consider an
infinite number of instantons (impractical)
4
Alternative geometric picture starts from path integral
Perturbation theory is infinite dimensional generalization of
steepest-descent method to evaluate ordinary integrals
Understanding which instantons contribute to a given observable amounts to
understand which saddles of the action should be considered in the path integral
We can hope to address this point using generalization to
infinite dimensions of known mathematical methods.
[Witten, 2011]
They provide a geometric alternative to the resurgence program
Key question:
Under what conditions only one saddle point (trivial vacuum)
contributes so that perturbation theory gives the full answer?
We answered this question in QM.
5
Plan
One dimensional integrals
Lefschetz thimbles
Borel sums
Exact perturbation theory
Path integral
Examples
Conclusions
6
One dimensional integrals
1
Z( ) ⌘ p
Convergent for any
Z
1
dx g(x) e
f (x)/
1
0
We can compute the integral exactly using steepest-descent methods:
1.
2.
Determine the saddle points contributing to Z
Resum the “perturbative” expansion around each saddle
Consider first point 1.
Analytically continue in complex plane
1
Z( ) = p
Z
dz g(z) e
Cx
7
f (z)/
Saddle points z : f 0 (z ) = 0
For each saddle we call downward and upward flows J and K the
lines where Re F decrease and increase, respectively, and Im F is constant
F (z) ⌘
f (z)/
J is the path of steepest descent. If it flows to Re F =
1
it is called a Lefschetz thimble (or just thimble).
By construction, the integral over a thimble is always well defined and convergent.
If J hits another saddle point, the flow splits int two branches
and ambiguity arises (Stokes line).
Picard-Lefschetz theory: deform the contour Cx in a contour C :
C=
h. . .i
X
n = hCx , K i
J n
denotes intersection pairings
hJ , K⌧ i =
8
⌧
[Witten, 1001.2933]
If J = K⌧ intersection not well defined.
Ambiguity resolved by assigning a small imaginary part to
Z( ) =
1
Z ( )⌘ p
X
Z
n Z ( )
dz g(z) e
f (z)/
J
Not all saddles contribute to the integral, only those with n 6= 0
Example: i)
1 2 1 4
f (x) = x + x ,
2
4
g(x) = 1
Three saddles: z0 = 0, z± = ±i
Degenerate situation. Give a small imaginary part to
9
1
Im
1
>0
Im
4
4
J+
K+
K+
2
J+
2
z+
J0
z0
0
z≠
K0
≠2
≠2
J≠
0
J0
z≠
≠2
K≠
K0
z+
z0
0
≠4
≠4
<0
2
≠4
≠4
4
n0 = 1, n± = 0
J≠
≠2
0
K≠
2
Only saddle at the origin contributes to the integral.
No need to deform the original contour of integration (thimble itself)
No “non-perturbative” contributions
10
4
1 2 1 4
Example: ii)
f (x) =
x + x ,
2
4
Three saddles: z0 = 0, z± =1 ±1
Im
4
2
>0
Im
0
J≠
z0
K0
≠2
C+ = J
J+
z+
≠2
K+
J0
0
0
2
≠4
≠4
4
J0 + J+
<0
K0
2
z≠
≠4
≠4
Again degenerate situation. 1
4
K≠
≠2
g(x) = 1
J≠
z≠
J0
z0
K+
z+
J+
K≠
≠2
0
2
C = J + J0 + J+
The integral is on a Stokes line and the intersection numbers depend on the
deformation. Ending result of the integral will eventually be unambiguous
There are “non-perturbative” contributions
11
4
Borel Sums
1
Z ( )⌘ p
Z
f (z)/
dz g(z) e
J
can be computed using a saddle-point expansion. Resulting series is asymptotic
Z( )
N
X
Zn
n
n=0
= O(
N +1
),
!0
as
Borel transform to reconstruct function: assume Zn ⇠ n!an nc
Bb Z(t) =
1
X
n=0
Zn
tn
(n + b + 1)
ZB ( ) =
(Borel Le Roy function)
Z
1
0
dt e t tb Bb Z(t )
If no singularities for t>0 and under certain assumptions
ZB ( ) = Z( )
12
Analytic structure of Borel function close to the origin can be
determined by the large-order behaviour of the series coefficients
n
for Zn ⇠ n!a , B0 Z(t) =
P
n
(at)
⇠
n
1
1 at
Singularity dangerous or not depending on the sign of a:
a>0 (same sign series) singularity for t>0
☹
a<0 (alternating series) singularity for t < 0
😀
Lateral Borel summation: move the contour off the real positive
axis to avoid singularity for t>0.
Result is ambiguous and ambiguity signals presence of
non-perturbative contributions to Z not captured by ZB
Deformation to define the lateral Borel sum corresponds to
the one we did to avoid Stokes line
13
Thimbles are Borel Resummable
Z
1
Z ( )⌘ p
dz g(z) e
f (z)/
J
is Borel resummable to the exact result.
Proof: Change variable
t=
f (z)
f (z )
[Berry, Howls 1991]
For any z 2 J , t is real and non-negative
Z ( )=e
f (z )
Z
1
dt t
1/2
t
e B ( t)
0
X g(zk ( t)) p
B ( t) ⌘
t .
0
|f (zk ( t)|
k=1,2
By definition, on a thimble f 0 (z) 6= 0 for z 6= z
=) B (t) regular for t > 0
But this is exactly the expression one would get by
considering the Borel transform with b=-1/2!
q.e.d.
14
Summarizing
1
Z( ) ⌘ p
Z
1
dx g(x) e
f (x)/
=
1
X
n Z ( )
Each Z ( ) is Borel resummable to the exact result
1 saddle contributes: Z( ) reconstructable from perturbation theory
More saddles contribute: Z( ) given by multi-series
(non-perturbative corrections)
1 2 1 4
f (x) = x + x
2
4
Example first case:
Zn =
p
2( )
(2n +
n!
n
Z
1
0
dt t
1
2)
1
2
B
1/2 Z(t)
=
s
p
1 + 1 + 4t
1 + 4t
⇣ 1 ⌘
1
1
e t B 1/2 Z( t) = p e 8 K 14
8
2
15
Exact result
Example second case:
Coefficients at minima:
f (x) =
Z±,n =
1/2 Z± (
For real
2n +
n!
1
2
iZ0,n
Coefficients at maximum:
B
1 2 1 4
x + x
2
4
t) =
B
s
p
1+ 1 4 t
2(1 4 t)
1/2 Z
Recall we need Im
not Borel resummable
6= 0 to avoid Stokes line
The thimble decomposition instructs us how to consistently perform lateral
Borel sums and consistently sum up the different contributions:
Im
Im
>0
<0
Z ( )
Z0 ( ) + Z+ ( )
1
⇡
= p e8
4
Z ( ) + Z0 ( ) + Z+ ( )

I
1
4
✓
1
8
◆
+ I 14
✓
1
8
◆
Exact result
16
Exact Perturbation Theory
When the decomposition in thimbles is non-trivial, the
intersection numbers n have to be determined
This is easy for one-dimensional integrals, but very complicated in
the path integral case, where in general they will be infinite
Key idea: the thimble decomposition can be modified by a simple trick
The decomposition of the integral Z in terms of thimbles
is governed by the function f(z) and not by g(z). Define
1
p
Ẑ( , 0 ) ⌘
fˆ(x) ⌘ f (x) + f (x) ,
Z
ĝ(x,
1
dx e
fˆ(x)/
0)
0
0)
1
⌘ g(x)e
f (x)/
By construction
Ẑ( ,
ĝ(x,
= ) = Z( )
17
0
lim
|x|!1
f (x)
=0
f (x)
Saddle-point expansion in
at fixed
0:
f in fˆ: “classical” deformation
f in ĝ: “quantum” deformation”
Thimble decomposition is determined by fˆ
By a proper choice of f , we can generally trivialize the thimble decomposition
=) one real saddle and no need to deform the contour of integration
Series expansion of Ẑ( ,
0)
in
at fixed
0:
Exact Perturbation Theory (EPT)
Non-perturbative contributions of Z are all contained
in the perturbative expansion of Ẑ
18
Consider again
and choose
1 2 1 4
x + x
2
4
f (x) =
f (x) = x2
1 2 1 4
ˆ
f (x) = x + x ,
2
4
so that
ĝ(x,
0)
= exp
⇣
x2 ⌘
0
Only saddle at origin contributes. We have
s
p
p
1 + 1 + 4 t 1+4 t
0
B 1/2 Ẑ( t, 0 ) =
e
1+4 t
Z
1
0
dt t
1
2
e tB
1/2 Ẑ( t, )
1
⇡
= p e8
4

I
1
4
✓
1
8
Exact result is now completely perturbative
19
1
◆
+ I 14
✓
1
8
◆
Path Integral
These results can be generalized to higher-dimensional integrals
n/2
Z ( )=
Z ( )=e
f (z )/
B ( t) ⌘ ( t)1
= ( t)
n/2
1 n/2
Z
1
Z0
Z
J
⌦
Z
dt tn/2
dz g(z) e
f (z)/
J
1
e
t
B ( t) ,
dz g(z) [f (z)
g(z)
d⌃
|rf (z)|
t
20
f (z )
t]
We can take the continuum for the path integral case.
Consider quantum mechanics where divergencies are all reabsorbed by the
path integral measure (UV limit not expected to be problematic)
Z( ) =
Z
S[x] =
Dx(⌧ ) G[x(⌧ )]e
Z
d⌧

S[x(⌧ )]/
1 2
ẋ + V (x)
2
V (x) analytic function of x
Discrete spectrum
V (x) ! 1 for |x| ! 1
Key result:
If the action S[x(⌧ )] has only one real saddle point x0 (⌧ ), the formal series
expansion of Z( ) around = 0 is Borel resummable to the exact result.
21
Remarks:
1. Number of saddles depends on boundary conditions
and the infinite time limit
2. Depending on the system, some observables are, and
some other are not, Borel reconstructable
If V admits only one minimum, the above subtleties do not arise.
Like in the one-dimensional case, we can define an EPT as follows
Suppose V = V0 +
V such that
1. V0 has a single non-degenerate critical point (minimum);
2. lim|x|!1
Ẑ( ,
0)
=
Z
V /V0 = 0 .
Dx G[x] e
R
d⌧
V
0
e
S0
,
Ẑ( , ) = Z( )
22
S0 ⌘
Z

1 2
d⌧
ẋ + V0
2
Ẑ is guaranteed to be Borel resummable to the exact answer
We have then proved the statement made at the beginning:
All observables in a quantum mechanical system
with a bounded potential can be entirely computed
from a single perturbative series (EPT)
(provided points 1. and 2. above apply)
No need of resurgence, thimble decomposition, trans-series.
Even observables in systems known to have instanton corrections
will be reconstructed by a single perturbative series
Large arbitrariness in the choice of EPT. In principle all choices equally
good, although in numerical studies some choices better than others
We will denote by Standard Perturbation Theory (SPT) the usual
perturbative computation (instantons included)
Time to show explicit results
23
Examples
Numerical analysis to test our results. We compute N orders using package
[Sulejmanpasic, Unsal, 1608.08256]
At fixed order N of perturbative terms, we use Padé approximants
for the Borel function, that is then numerically integrated.
Results are compared with other methods (such as
Rayleigh-Ritz) to get exact answers in QM
p 3
1 2
V (x; ) = x + ↵ x + x4
2
2
p
For |↵| < 2 2/3 V has a unique minimum at x = 0
In principle no need of EPT, being SPT
Borel resummable
Yet, EPT greatly improves the accuracy of result at a
given order in perturbation theory
24
1 2
V0 = x + x 4 ,
2
2
3/2 3
V =↵
x
=5
= 1/20
1
10-21 10-10
SPT
10-31
10
300
400
N
500
10-20
-41
200
SPT
EPT
E0/E0
E0/E0
10-11
EPT
10-30
600
200
300
400
N
500
At weal coupling SPT better than EPT, viceversa at strong
coupling, where SPT is inaccurate
25
600
Symmetric double well
V =
2
✓
x2
1
4
◆2
Prototypical system where instanton occurs. Write
⇣ 1
⌘
⇣
⌘ x2
1
0 2
V0 =
+ x + x4 ,
V =
0+
32
2
2
2 2
Already at small coupling, EPT is able to resolve the
instanton contribution to the ground state energy.
At
E0
⇡ 10
E0
=
0
= 1/25, N = 200
E1
⇡ 4 · 10
E1
8
E0inst
2
p
⇡
e
⇡
26
1
6
⇡ 0.087
14
At larger values of the coupling, instanton computation breaks down
EPT works better and better
No need of many perturbative terms.
With just 2 (using a conformal mapping method) at
= 1 we get
E0
' 3%
E0
We have also computed higher energy states and eigenfunctions.
In all cases EPT give excellent agreement with other methods. No
signal of missing non-perturbative contributions.
27
SUSY double well
Notable tilted double well:
V (x; ) =
2
⇣
x2
1 ⌘2 p
+
x
4
Potential one obtains integrating out fermions in SUSY QM
E0 = 0
to all orders in SPT due to SUSY
Yet, E0 6= 0 by instanton effects that dynamically break supersymmetry
28
We can approach this system in three ways:
1. ordinary instanton methods, SPT
[Jentschura, Zinn-Justin, hep- ph/0405279]
2. Turn the quantum tilt into classical. Alternative
perturbation theory (APT), sort of inverse of EPT
VAPT =
2
⇣
x2
1 ⌘2
0
+p x
4
3. EPT
⇣ 1
⌘
0 2
4
V0 =
+ x + x ,
32
2
2
0
V =p x
29
⇣
1 ⌘ x2
0+
2 2
1
E0
10 -5
+
10 -10
+
×
+
×××
×
+
+×
+
+×
+×
+×
+×
+×
+×
+
+
+
×
×
+
×
+
×
+
×
+
×
+
×
+
×
×
+
×
+
×
+
×
+
×
+×
+×
+×
×
+
×
+
×
+
×
+
10 -15 +
ΔE 0 /E 0
1
10
-10
10
-20
10
-30
+ + + + +××
++++++
10 -40 -2
10
SPT
××
××
++++++++++++++
××
××
0.1
××
××
1
λ
N=200
30
++++++++
× ×EPT
××××××
10
APT
×
×
××
×
×
×
100
False vacuum
1.0
fv
E0
0.5
fv
V (x;1,∞)
V (x)
0.0
V fv (x;1)
- 0.5
fv
E0
fv
- 1.0
E0 (EPT)
- 1.5
- 2.0
-1
0
1
x
31
2
Pure anharmonic potentials:
H = p2 + x2l
Usual perturbation theory breaks down
k
(4)
Ek
(4)
(4)
Ek /Ek
0
1.0603620904
3 · 10
1
3.7996730298
2 · 10
2
7.4556979379
9 · 10
3
11.6447455113
4 · 10
4
16.2618260188
4 · 10
(6)
Ek
(6)
45
0.5724012268
44
2 · 10
2.1692993557
37
3 · 10
4.5365422804
36
9 · 10
7.4675848174
36
7 · 10
10.8570827110
2 · 10
Table 2: Energy eigenvalues Ek(2`) and the corresponding accuracies
(6)
Ek /Ek
(2`)
19
19
17
16
16
(2`)
Ek /Ek of the first five levels
of the pure anharmonic x4 and x6 potentials, computed using EPT with N = 200. Only the first ten
digits after the comma are shown (no rounding on the last digit).
of the pure x4 and x6 oscillators computed comparing EPT to the results from RR methods.
We used N = 200 orders of perturbation theory and in eq.(4.19) we chose c1 = 2 for ` = 2 and
(4)
c1 = 4, c2 = 2 for ` = 3. Notice the accuracy of E0 up to 45 digits! At fixed N , similarly to the
32
Conclusions and future perspectives
We have introduced a new perturbation theory in quantum mechanics
(EPT) that allows us to compute observables perturbatively
No need of trans-series and to worry for possibly ambiguous results
(possible addressed by resurgence or by thimble decomposition)
Is it possible to get the same results directly in Minkowski space?
Generalize to higher dimensional QM systems
33
What about QFT?
Is renormalization an issue?
UV problem
Infinite volume limit, spontaneous symmetry breaking? IR problem
4
Borel summability of 2,3 theories suggest a promising
development in QFT and is the next thing to do
A new approach to strongly coupled physics has begun …
34
Thank You
35