STOCHASTIC QUANTIZATION ON THE COMPUTER

1. STOCHASTIC QUANTIZATION ON THE COMPUTER Enrico Onofri Southampton, January 2002

2. Plan of the talk: Probabilistic methods and Quantum Theory (M. Kac, EQFT, the classical era ’50-’70) Stochastic Quantization (Parisi and Wu, Parisi, the modern era ’80-’90) The Numerical Stochastic Perturbation Theory approach: results and problems until 1999.99 Recent results and programs Next talk

3. Feynman-Kac formula: a bridge between diffusion processes and quantum (field) theory 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”; ….

4. 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)

5. Parisi-Wu (1980) • Diffusion process in the Euclidean field configuration space with asymptotic distribution exp(-S)/Z Southampton - LGT workshop

6. Parisi 1981: let S  S- lf(0) 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 f^4 theory. Southampton - LGT workshop

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9. Every Green’s function can be expanded in such a way that its n-th order is assigned a stochastic estimator The infinite-dimensional system for can be truncated at any order with no approximation involved. 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! Southampton - LGT workshop

11. INGREDIENTS: • Langevin algorithm (Cornell group) • Stochastic gauge fixing (Zwanziger) 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: Southampton - LGT workshop

13. Results (’94-’95): Plaquette SU(3) 4-dim: Southampton - LGT workshop

14. 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. Southampton - LGT workshop

15. 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): Southampton - LGT workshop

16. Algorithm’s details: we tried to reduce the algorithmic error by: Exact representation of free field (Ornstein-Uhlenbeck) Trapezoidal rule and a variant of Simpson’s rule for higher orders.

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 Southampton - LGT workshop

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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 Southampton - LGT workshop

20. If A is the free inverse propagatorand 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. 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) Southampton - LGT workshop

22. Conclusions • 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 • NSPT should be the option in cases where analytic calculations require an unacceptable cost in brainpower. • 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 b^10. Southampton - LGT workshop