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Stochastic Lévy differential operators and Yang–Mills equations

Boris O. Volkov

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow 119991, Russia

E-mail Address: borisvolkov1986@gmail.com

https://doi.org/10.1142/S0219025717500084Cited by:9 (Source: Crossref)

Abstract

The relationship between the Yang–Mills equations and the stochastic analogue of Lévy differential operators is studied. The value of the stochastic Lévy–Laplacian is found by means of Cèsaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang–Mills equations and the Lévy–Laplace equation for such Laplacian are not equivalent in contrast to the deterministic case. An equation equivalent to the Yang–Mills equations is obtained. The equation contains the Lévy divergence. It is proved that the Yang–Mills action functional can be represented as an infinite-dimensional analogue of the Direchlet functional of a chiral field. This analogue is also derived using Cèsaro averaging.

Communicated by O. Smolyanov

Keywords:

AMSC: 70S15, 81S25

References

1. L. Accardi, Yang–Mills Equations and Lévy–Laplacians, Dirichlet Forms and Stochastic Processes, Beijing, 1993 (de Gruyter, 1995), pp. 1–24. Google Scholar
2. L. Accardi and V. I. Bogachev , The Ornstein–Uhlenbeck process associated with the Lévy–Laplacian and its Dirihlet form, Probab. Math. Statist. 17 (1997) 95–114. Google Scholar
3. L. Accardi, P. Gibilisco and I. V. Volovich , Yang–Mills gauge fields as harmonic functions for the Lévy–Laplacian, Russ. J. Math. Phys. 2 (1994) 235–250. Google Scholar
4. L. Accardi, P. Gibilisco and I. V. Volovich , The Lévy–Laplacian and the Yang–Mills equations, Rend. Lincei 4 (1993) 201–206, doi: 10.1007/BF03001574. Crossref, Google Scholar
5. L. Accardi, U. C. Ji and K. Saitô , The exotic (higher order Lévy) Laplacians generate the Markov processes given by distribution derivatives of white noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013) 1350020, doi: 10.1142/S0219025713500203. Link, Web of Science, Google Scholar
6. L. Accardi, U. C. Ji and K. Saitô , Higher order multi-dimensional extensions of Cesàro theorem, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (2015) 1550030, doi: 10.1142/S0219025715500307. Link, Web of Science, Google Scholar
7. L. Accardi and O. G. Smolianov [Smolyanov], On Laplacians and traces, Conf. Semin. Univ. Bari 250 (1993) 1–25. Google Scholar
8. L. Accardi and O. G. Smolyanov , Classical and nonclassical Levy Laplacians, Dokl. Math. 76 (2007) 801–805, doi: 10.1134/S1064562407060014. Crossref, Web of Science, Google Scholar
9. I. Ya. Arefieva and I. V. Volovich , Higher order functional conservation laws in gauge theories, Proc. Int. Conf. Generalized Functions and Their Applications in Mathematical Physics (Academy of Sciences of the USSR, 1981), pp. 43–49, in Russian. Google Scholar
10. M. Arnaudon, R. O. Bauer and A. Thalmaier , A probabilistic approach to the Yang–Mills heat equation, J. Math. Pures Appl. 81 (2002) 143–166, doi: 10.1016/S0021-7824(02)01254-0. Crossref, Web of Science, Google Scholar
11. M. Arnaudon and A. Thalmaier , Yang–Mills fields and random holonomy along Brownian bridges, Ann. Probab. 31 (2003) 769–790, doi: 10.1214/aop/1048516535. Crossref, Web of Science, Google Scholar
12. V. I. Averbukh, O. G. Smolyanov and S. V. Fomin , Generalized functions and differential equations in linear spaces. II, Differential operators and their Fourier transform, Trudy Moskov. Mat. Obshch. 27 (1972) 247–262, in Russian, doi: 10.1070/RM1967v022n06ABEH003761. Google Scholar
13. R. O. Bauer , Characterizing Yang–Mills fields by stochastic parallel transport, J. Funct. Anal. 155 (1998) 536–549, doi: 10.1006/jfan.1997.3238. Crossref, Web of Science, Google Scholar
14. V. I. Bogachev , Gaussian Measures (Amer. Math. Soc., 1998). Crossref, Google Scholar
15. B. A. Dubrovin, S. P. Novikov and A. T. Fomenko , Sovremennaja geometrija: Metody i prilozhenija. 2-e izd. Modern Geometry: Methods and Applications, 2nd edn. (Nauka, 1986), in Russian. Google Scholar
16. M. N. Feller , The Lévy–Laplacian, Cambridge Tracts in Math., Vol. 166 (Cambridge Univ. Press, 2005). Crossref, Google Scholar
17. T. Hida , Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes, No. 13 (1975). Google Scholar
18. T. Hida and S. Si , Lectures on White Noise Functionals (World Scientific, 2008). Link, Google Scholar
19. H.-H. Kuo, N. Obata and K. Saitô , Lévy–Laplacian of generalized functions on a nuclear space, J. Funct. Anal. 94 (1990) 74–92, doi: 10.1016/0022-1236(90)90028-J. Crossref, Web of Science, Google Scholar
20. H.-H. Kuo , White Noise Distribution Theory (CRC Press, 1996). Google Scholar
21. R. Leandre and I. V. Volovich , The stochastic Lévy–Laplacian and Yang–Mills equation on manifolds, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001) 151–172, doi: 10.1142/S0219025701000449. Link, Web of Science, Google Scholar
22. P. Lévy , Problèmes Concrets d’analyse Fonctionnelle (Gautier-Villars, 1951). Google Scholar
23. N. G. Marchuk , Demonstration representation and tensor products of clifford algebras, in Proc. of the Steklov Institute of Mathematics, Vol. 290 (2015) 143–154, doi: 10.1134/S0081543815060139. Google Scholar
24. N. G. Marchuk and D. S. Shirokov , General solutions of one class of field equations, Rep. Math. Phys. 78 (2016) 305–326, doi: 10.1016/S0034-4877(17)30011-3. Crossref, Web of Science, Google Scholar
25. D. Nualart , The Malliavin Calculus and Related Topics, 2nd edn. (Springer-Verlag, 2006). Google Scholar
26. E. M. Polishchuk , Continual Means and Boundary Value Problems in Function Spaces (Birkhäuser, 1988). Crossref, Google Scholar
27. A. M. Polyakov , String representations and hidden symmetries for gauge fields, Phys. Lett. B 82 (1979) 247–250, doi: 10.1016/0370-2693(79)90747-0. Crossref, Web of Science, Google Scholar
28. A. M. Polyakov , Gauge fields as rings of glue, Nucl. Phys. B 164 (1980) 171–188, doi: 10.1016/0550-3213(80)90507-6. Crossref, Web of Science, Google Scholar
29. B. Simon , Functional Integration and Quantum Physics (Academic Press, 1979). Google Scholar
30. B. O. Volkov , Lévy–Laplacian and the gauge fields, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (2012) 1250027, doi: 10.1142/S0219025712500270. Link, Web of Science, Google Scholar
31. B. O. Volkov , Hierarchy of Lévy–Laplacians and quantum stochastic processes, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013) 1350027, doi: 10.1142/ S0219025713500276. Link, Web of Science, Google Scholar
32. B. O. Volkov , Lévy–Laplacians and instantons, in Proc. of the Steklov Institute of Mathematics, Vol. 290 (2015) 210–222, doi: 10.1134/S008154381506019X. Google Scholar
33. B. O. Volkov , Stochastic Lévy divergence and Maxwell’s equations, Math. Math. Model. Bauman MSTU (5) (2015) 1–16, in Russian, doi: 10.7463/mathm.0515.0820322. Google Scholar
34. B. O. Volkov , Stochastic Lévy–Laplacian and d’Alambertian and Maxwell’s equations, Math. Math. Model. Bauman MSTU (6) (2015) 1–16 in Russian, doi: 10.7463/mathm.0615.0822138. Google Scholar
35. B. O. Volkov , Lévy d’Alambertians and their application in the quantum theory, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki. 19 (2015) 241–258, in Russian, doi: 10.14498/vsgtu1372. Crossref, Google Scholar
36. V. V. Zharinov , Backlund transformations, Theor. Math. Phys. 189 (2016) 1681–1692, doi: 10.1134/S0040577916120011. Crossref, Web of Science, Google Scholar