No Access

Lévy differential operators and Gauge invariant equations for Dirac and Higgs fields

Boris O. Volkov

Bauman Moscow State Technical University, 2-ya Baumanskaya st. 5/1, Moscow 105005, Russia

E-mail Address: borisvolkov1986@gmail.com

https://doi.org/10.1142/S0219025719500012Cited by:3 (Source: Crossref)

Abstract

We study the Lévy infinite-dimensional differential operators (differential operators defined by the analogy with the Lévy Laplacian) and their relationship to the Yang–Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated connection is a solution to the Yang–Mills equations. This system is an analogue of the equations of motion of chiral fields and contains the Lévy divergence. The systems of infinite-dimensional equations containing Lévy differential operators, that are equivalent to the Yang–Mills–Higgs equations and the Yang–Mills–Dirac equations (the equations of quantum chromodynamics), are obtained. The equivalence of two ways to define Lévy differential operators is shown.

Communicated by Luigi Accardi

Dedicated to the memory of Sergei Starodubov

Keywords:

AMSC: 70S15

References

1. L. Accardi, Yang–Mills equations and Lévy-Laplacians, in Dirichlet Forms and Stochastic Processes (Beijing, 1993; de Gruyter, 1995), pp. 1–24. Google Scholar
2. L. Accardi, P. Gibilisco and I.V. Volovich, The Lévy Laplacian and the Yang–Mills equations, Rend. Lincei 4 (1993) 201–206. https://doi.org/10.1007/BF03001574 Google Scholar
3. L. Accardi, P. Gibilisco and I. V. Volovich, Yang–Mills gauge fields as harmonic functions for the Levy-Laplacians, Russ. J. Math. Phys. 2 (1994) 235–250. Google Scholar
4. 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), Article ID: 1350020, 26 pp. https://doi.org/10.1142/S0219025713500203 Link, Web of Science, Google Scholar
5. L. Accardi, U.C. Ji and K. Saitô, Higher order multi-dimensional extensions of Cesàro theorem, Infini. Dimens. Anal. Quantum Probab. Relat. Top. 18 (2015), ArticleID: 1550030 14 pp.. https://doi.org/10.1142/S0219025715500307 Web of Science, Google Scholar
6. L. Accardi and O. G. Smolyanov, On Laplacians and traces, Conf. Semin. Univ. Bari 250 (1993) 1–25. Google Scholar
7. L. Accardi and O. G. Smolyanov, Feynman formulas for evolution equations with Levy Laplacians on infinite-dimensional manifolds, Dokl. Math. 73 (2006) 252–257. https://doi.org/10.1134/S106456240602027X Web of Science, Google Scholar
8. L. Accardi and O. G. Smolyanov, Classical and nonclassical Levy Laplacians, Dokl. Math. 76 (2007) 801–805. https://doi.org/10.1134/S1064562407060014 Web of Science, Google Scholar
9. L. Accardi and O. G. Smolyanov, Generalized Lévy Laplacians and Cesaro means, Dokl. Math. 79 (2009) 90–93. https://doi.org/10.1134/S106456240901027X Web of Science, Google Scholar
10. I. Ya. Arefieva and I. V. Volovich, Higher order functional conservation laws in gauge theories, in 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
11. 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. https://doi.org/10.1016/S0021-7824(02)01254-0 Web of Science, Google Scholar
12. M. Arnaudon and A. Thalmaier, Yang–Mills fields and random holonomy along Brownian bridges, Ann. Probab. 31 (2003) 769–790. https://doi.org/10.1214/aop/1048516535 Web of Science, Google Scholar
13. V. I. Averbukh, O. G. Smolyanov and S. V. Fominm, Generalized functions and differential equations in linear spaces. II, Differential operators and their Fourier transform, Tr. Mosk. Mat. Obs. 27 (1972) 247–262 (in Russian). Google Scholar
14. R. O. Bauer, Characterizing Yang–Mills fields by stochastic parallel transport, J. Funct. Anal. 155 (1998) 536–549. Web of Science, Google Scholar
15. R. O. Bauer, Random Holonomy for Yang–Mills Fields: Long-Time Asymptotics, Potential Anal. 18 (2003) 43–57. https://doi.org/10.1023/A:1020529721290 Web of Science, Google Scholar
16. B. Driver, Classifications of bundle connection pairs by parallel translation and lassos, J. Funct. Anal. 83 (1989) 185–231. https://doi.org/10.1016/0022-1236(89)90035-9 Web of Science, Google Scholar
17. 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 Publisher, 1986) (In Russian). Google Scholar
18. M. N. Feller, The Lévy Laplacian, Cambridge Tracts in Mathematics, Vol. 166 (Cambridge Univ. Press, 2005). Google Scholar
19. L. Gross, A Poincarè lemma for connection forms, J. Funct. Anal. 63 (1985) 1–46. https://doi.org/10.1016/0022-1236(85)90096-5 Web of Science, Google Scholar
20. H.-H. Kuo, N. Obata and K. Saitô, Lévy-Laplacian of Generalized Functions on a Nuclear Space, J. Funct. Anal. 94 (1990) 74–92. https://doi.org/10.1016/0022-1236(90)90028-J Web of Science, Google Scholar
21. R. Leandre and I. V. Volovich, The stochastic levy laplacian and Yang–Mills equation on manifolds, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001) 151–172. https://doi.org/10.1142/S0219025701000449 Link, Web of Science, Google Scholar
22. P. Lévy, Problèmes Concrets d’analyse Fonctionnelle, (Gautier-Villars, 1951). Google Scholar
23. A. M. Polyakov, String representations and hidden symmetries for gauge fields, Phys. Lett. B 82 (1979) 247–250. https://doi.org/10.1016/0370-2693(79)90747-0 Web of Science, Google Scholar
24. A. M. Polyakov, Gauge fields as rings of glue, Nuclear Phys. B. 164 (1980) 171–188. https://doi.org/10.1016/0550-3213(80)90507-6 Web of Science, Google Scholar
25. H. H. Schaefer with M. P. Wolff, Topological Vector Spaces, 2nd edn. Graduate Texts in Mathematics (Book 3) (Springer, 1999). Google Scholar
26. Y. M. Shnir, Magnetic Monopoles, (Springer-Verlag, 2005). Google Scholar
27. O. G. Smolyanov, Linear differential operators in spaces of measures and functions on Hilbert space, Uspekhi Mat. Nauk 28 (1973) 251–252 (in Russian). Google Scholar
28. B. O. Volkov, Lévy-Laplacian and the gauge fields, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (2012) 1250027. https://doi.org/10.1142/S0219025712500270 Link, Web of Science, Google Scholar
29. B. O. Volkov, Hierarchy of Lévy-Laplacians and quantum stochastic processes, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013) 1350027. https://doi.org/10.1142/S0219025713500276 Link, Web of Science, Google Scholar
30. B. O. Volkov, Lévy Laplacians and related constructions, PhD thesis (Lomonosov Moscow State University, 2014) (in Russian). Google Scholar
31. 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). https://doi.org/10.14498/vsgtu1372 Google Scholar
32. B. O. Volkov, Lévy Laplacians and instantons, Proc. Steklov Inst. Math. 290 (2015) 210–222. https://doi.org/10.1134/S037196851503019X Web of Science, Google Scholar
33. B. O. Volkov, Stochastic Lévy differential operators and Yang–Mills equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 20 (2017), ArticleID: 1750008, 23 pp.. https://doi.org/10.1142/S0219025717500084 Link, Web of Science, Google Scholar
34. B. O. Volkov, Lévy Laplacians in Hida calculus and Malliavin calculus, Proc. Steklov Inst. Math. 301 (2018) 11–24. https://doi.org/10.1134/S0081543818040028 Web of Science, Google Scholar
35. B. O. Volkov, Lévy Laplacians and annihilation process, Uchenye Zapiski Kazanskogo Univ. Ser. Fiz.-Mat. Nauki 160 (2018) 399–409. Web of Science, Google Scholar
36. R. S. Ward and R. O. Wells, Jr., Twistor Geometry and Field Theory (Cambridge Univ. Press, 1990). Google Scholar