Transcript stochastic quantization on the computer

STOCHASTIC QUANTIZATION
ON THE COMPUTER
Enrico Onofri
Southampton, January 2002
Plan of the talk:
1. Probabilistic methods and
Quantum Theory (M. Kac, EQFT,
the classical era ’50-’70)
2. Stochastic Quantization (Parisi and
Wu, Parisi, the modern era ’80-’90)
3. The Numerical Stochastic Perturbation
Theory approach: results and
problems until 1999.99
4. Recent results and programs Next talk
•Feynman-Kac formula:
a bridge between
diffusion processes and quantum (field) theory
t
 x |e
tH
| y  e

( x y )
2t

 V ( w ( s )) ds
2
E (e
0
|w( 0 ) x ,w( t ) y )
H  p 2 / 2  V ( q), w(.)  standard Brownian process
M.Kac, 1950 beautiful results relating potential
theory, quantum mechanics and stochastic
processes. Main emphasis: probability theory
gives powerful estimates applicable in
mathematical physics
Refs.: M. Kac, “Lezioni Fermiane”, SNS 1980;
“Probability and related topics in the physical
sciences”, Interscience; B.Simon, “Functional
integration and Quantum Theory”; E. Nelson
“Dynamical theories of Brownian motion”; ….
Modern era: probability theory can provide
powerful algorithms, not necessarily the most
efficient, but worth considering for some special
applications.
Parisi & Wu, Sci. Sinica 24 (1981)
Parisi, Nucl.Phys. B180 (source method)
Barnes & Daniell, (brownian motion with
approximate ground state)
Duane & Kogut, (Hybrid method)
Kuti & Polonyi, (stochastic method for lattice
determinants)
Parisi-Wu (1980)
• Diffusion process in the Euclidean field
configuration space with asymptotic
distribution exp(-S)/Z
d
S[ ]
 ( x , )  

d

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Parisi 1981: let S  S- l(0)
  ( x )  l   ( x )  0  l   (0) ( x )  conn O ( l2 )
 ( x, )   0  l1  l  2  ...
d
S
 0 ( x , )  

d
0 ( x, )
2
d
 2S
1 ( x ,  )   
1 ( y , )dy   ( x )
d
0 ( x, )0 ( y , )
 1 ( x, )   ( x ) (0) 
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Around 1990 G. Marchesini suggested to
merge the two ideas into one and try doing
perturbation theory entirerly on the computer
At that time Monte Carlo was synonim of
NON-perturbative algorithm, so the
idea seemed somewhat bizarre. A first
trial was nonetheless performed
(G.M. and E.O.) on the scalar ^4 theory.
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1
1 2 2 1 4
2
S  (  )  m0  l
2
2
4
d
 ( x, )  (   m02 ) ( x, )  l ( x, )3   ( x, )
d
 ( x, )   0 ( x, )  l1 ( x, )  l2 2 ( x, )  l3 3 ( x, )  ...
d
 0 ( x, )  (   m02 ) 0 ( x, )   ( x, )
d
d
1 ( x, )  (   m02 )1 ( x, )   0 ( x, )3
d
d
 2 ( x, )  (   m02 ) 2 ( x, )  3 0 ( x, )2 1 ( x, )
d
....................................................
d
 n ( x, )  (   m02 ) n ( x, )  klm
 k l m
k  l  m  n 1
d
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Every Green’s function can be expanded in such a
way that its n-th order is assigned a stochastic
estimator
1 T
  ( x ) ( y )    ( x, ) ( y, )d
T 0
1 T
  ( 0 ( x, ) 0 ( y, )  l ( 0 ( x, )1 ( y, )  1 ( x, ) 0 ( y, ))  ...
T 0
The infinite-dimensional system for { j }
can be truncated at any order with no approximation
involved.
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Doing P.T. to order n requires introducing n+1 copies of the
lattice fields, which may be rather demanding on your
computer’s memory. However, on a 1990 VAX750 or SUN3
the limit was speed: statistics was too poor to get meaningful
results.
Soon after suitable machines were available (CM2, APE100)
and, more important, new brainpower!
(Di Renzo, Marenzoni, Burgio, Scorzato, Alfieri in Parma,
and later Butera, Comi, Pepe in Milano)
It was time to try to apply the idea to LGT!
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INGREDIENTS:
•Langevin algorithm (Cornell group)
•Stochastic gauge fixing (Zwanziger)
U  e
'
 F [U , H ]
U
F [U , H ] 
U  ( x)  e
''
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
4N
 (U
P  e
w[U ]( x )
'
P

P traceless
U )
U  ( x )e
i H
w[U ]( x  e )
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Next, substitute the Lie algebra field A(x):
and expand
1
A ( x ) 
A ( x )

1/ 2
2

A ( x )

U  ( x)  e
A ( x )
3

A ( x )

3/ 2
 ...
The algorithm splits into a cascade of updating rules for
all auxiliary fields:
A(1)  A(1)  F(1)
1
A( 2 )  A( 2 )  F( 2 )  [ F(1) , A(1) ]
2
1
1
A( 3)  A( 3)  F( 3)  [ F(1) , A( 2 ) ]  [ F( 2 ) , A(1) ] 
2
2
1
1
 [ F(1) , [ F(1) , A(1) ]]  [ A(1) , [ F(1) , A(1) ]]
12
12
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Results (’94-’95):
Plaquette SU(3) 4-dim:
1x1 1
2
3
4
5
6
7
8
9
10
NSPT
1.9994
1.2206
2.9523
9.345
33.97
134.6
565.3
2480
11240
52270
(6)
(16)
(58)
(27)
(14)
(7)
(34)
(18)
(10)
(520)
2.
1.218(7)
2.9602
2x2 1
2
3
4
5
6
7
8
NSPT
5.465(12)
-4.338(50)
0.0(1)
3.04(36)
16.8(1.4)
85(6)
413(25)
1952(127)
Exact
5.47563
-4.3342
Exac
t
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High order coefficients have been analysed from the point of
view of renormalons. Unconventional L^2 behaviour detected.
See Di Renzo and Scorzato,
JHEP 0110:038,2001 (hep-lat/0011067).
Another seminar! Controversial issue. Another speaker!
Hereafter: Statistical analysis using toy models for which long
expansions are available and fast simulations possible.
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This study was triggered by an observation of M.Pepe (Thesis,
Milano ’96). Studying O(3) s-model he discovered unexpected
large deviations from the known perturbative coefficients.
We studied three different toy models (random variables, the last
is Weingarten’s “pathological” model):
1 2 1 4
S    l , (  )
2
4
2
S  (1  cos( l )) / l , (  [ ,  ))
1
1
2
S     l , (  ,      )
2 
4 

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Algorithm’s details: we tried to reduce the
algorithmic error by:
1. Exact representation of free field
(Ornstein-Uhlenbeck)
2. Trapezoidal rule and a variant of
Simpson’s rule for higher orders.

2 1 / 2
 0 (t   )  e  0 (t )  (1  e ) 
A typical history (averaged over 1K
histories in parallel)
At high orders it is always the case that
large fluctuations dominate the final
average – effectively discontinuous (stiff)
behaviour
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Such stiff behaviour being rather misterious,
an independent calculation was performed,
based on Langevin equation, but avoiding
power expansion of the diffusion process
(suggested by G.Jona-Lasinio). The method
relies on Girsanov’s formula
dx (t )   Ax(t )dt  b( x (t ), t )dt  dw(t )
dy (t )   Ay (t )dt  dw(t )

E ( ( x (T )))  E ( ( y (T )) e T )
w
w
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If A is the free inverse propagator and b(x(t)) is
the drift due to the interaction, Girsanov’s
formula gives a closed form for the
perturbative expansion (Gellmann-Low
theorem). The results are consistent with
previous method. Some intrinsic property of
statistical estimators are at the basis of the
phenomenon.
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Our conclusion is that these cases are
characterised by distributions very far from
normality (Gaussian). Some non-parametric
analysis may help
An example of Bootstrap analysis,
a second example (3-d
Weingarten’s model)
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Conclusions
1. NSPT has been applied to LGT for several years
and it appears to give consistent results (also
finite size scaling turns out to be consistent, see
FDR
2. NSPT should be the option in cases where
analytic calculations require an unacceptable
cost in brainpower.
3. High order coeff’s should be analyzed with care
from the viewpoint of Pepe’s effect. This turns
out NOT to be a problem for SU(3) LGT, at least
up to ^10.
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