stochastic quantization on the computer enrico onofri southampton, january 2002
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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
M.Kac, 1950 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”; ….
process Brownianstandard(.)),(2/
)|(||2
)(,)0(
))((2
)(
0
2
wqVpH
eEeyex ytwxw
dsswVt
yxtH
t
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)
January 23, 2002 Southampton - LGT workshop 5
Parisi-Wu (1980)
• Diffusion process in the Euclidean field configuration space with asymptotic distribution exp(-S)/Z
][),(
Sx
d
d
January 23, 2002 Southampton - LGT workshop 6
Parisi 1981: let S S-
)0()(),(
)(),(),(),(
),(
),(),(
...),(
)()()0()()(
1
100
2
1
00
22
10
20
xx
xdyyyx
Sx
d
d
x
Sx
d
d
x
Oxxx conn
January 23, 2002 Southampton - LGT workshop 7
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.
January 23, 2002 Southampton - LGT workshop 8
mlnmlk
klm knn xmxd
d
xxxmxd
d
xxmxd
d
xxmxd
d
xxxxx
xxxmxd
d
mS
1
20
12
02202
301
201
0200
33
22
10
320
4220
2
),()(),(
....................................................
),(),(3),()(),(
),(),()(),(
),(),()(),(
...),(),(),(),(),(
),(),(),()(),(
4
1
2
1)(
2
1
January 23, 2002 Southampton - LGT workshop 9
Every Green’s function can be expanded in such a way that its n-th order is assigned a stochastic estimator
...)),(),(),(),((),(),((1
),(),(1
)()(
01100 00
0
yxyxyxT
dyxT
yx
T
T
The infinite-dimensional system for
can be truncated at any order with no approximation involved.
}{ j
January 23, 2002 Southampton - LGT workshop 10
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!
January 23, 2002 Southampton - LGT workshop 11
)]([')]([''
traceless
],['
)()(
)(4
],[
exUwxUw
PeP
P
HUF
exUexU
HiUUN
HUF
UeU
INGREDIENTS:
•Langevin algorithm (Cornell group)
•Stochastic gauge fixing (Zwanziger)
January 23, 2002 Southampton - LGT workshop 12
Next, substitute the Lie algebra field A(x):
and expand
The algorithm splits into a cascade of updating rules for all auxiliary fields:
)()( xAexU
...)()()(
)(2/3
32
2/1
1
xAxAxAxA
]],[,[12
1]],[,[
12
1
],[2
1],[
2
1
],[2
1
)1()1()1()1()1()1(
)1()2()2()1()3()3()3(
)1()1()2()2()2(
)1()1()1(
AFAAFF
AFAFFAA
AFFAA
FAA
January 23, 2002 Southampton - LGT workshop 13
Results (’94-’95):
Plaquette SU(3) 4-dim:
1x1 1 2 3 4 5 6 7 8 9 10NSPT 1.9994
(6)
1.2206(16)
2.9523(58)
9.345(27)
33.97(14)
134.6(7)
565.3(34)
2480(18)
11240(10)
52270(520)
Exact
2. 1.218(7) 2.9602
2x2 1 2 3 4 5 6 7 8NSPT 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
January 23, 2002 Southampton - LGT workshop 14
High order coefficients have been analysed from the point of view of renormalons. Unconventional ^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.
January 23, 2002 Southampton - LGT workshop 15
This study was triggered by an observation of M.Pepe (Thesis, Milano ’96). Studying O(3) -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):
),(,4
1
2
1
)),[(,/))cos(1(
)(,4
1
2
1
2
2
42
S
S
S
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.
2/1200 )1()()( etet
January 23, 2002 Southampton - LGT workshop 17
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
January 23, 2002 Southampton - LGT workshop 18
January 23, 2002 Southampton - LGT workshop 19
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
)))((()))(((
)()()(
)()),(()()(
TeTyw
ETxw
E
tdwdttAytdy
tdwdtttxbdttAxtdx
January 23, 2002 Southampton - LGT workshop 20
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.
January 23, 2002 Southampton - LGT workshop 21
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)
January 23, 2002 Southampton - LGT workshop 22
Conclusions1. 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.