Abstract
A mathematically rigorous relativistic quantum Yang–Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is nonperturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers Dynin ([11] and [12]). The principal result is established anew: due to the quartic self-interaction term in the Yang–Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang–Mills energy spectrum has a positive mass gap. Furthermore, the quantum Yang–Mills Hamiltonian has a countable orthogonal eigenbasis in a Fock space, so that the quantum Yang–Mills spectrum is point and countable. In addition, a fine structure of the spectrum is elucidated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Quantum Yang–Mills Theory (http://www.claymath.org/prizeproblems/index.html).
C. S. Agarwal and E. Wolf, “Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics,” Phys. Rev. D 2, 2161–2225 (1970).
M. S. Agranovich, Elliptic Boundary Problems (Partial differential operators, IX Encyclopaedia of Mathematical Sciences, Springer, 1997).
V. Arnold et al. (editors), Mathematics: Frontiers and Perspectives (American Mathematical Society, Providence, 2000).
F. Berezin, The Method of Second Quantization (Nauka, Moscow, 1965; Academic Press, 1966).
F. A. Berezin, “Wick and Anti-Wick Symbols of Operators,” Math. USSR Sb. 15, 577–606 (1971); Math. USSR-Sbornik 86 (128), 577–606 (1971).
F. A. Berezin, “Covariant and Contravariant Symbols of Operators,” Izv. Akad. Nauk USSR 36, 1134–1167 (1972); Math. SSSR-Izvestiya 6, 1117–1151 (1972).
F. A. Berezin and M. A. Shubin, The Schrödinger Equation (Kluwer Academic Publishers, 1991).
P. J. Boland, “Holomorphic Functions on Nuclear Spaces,” Trans. Amer. Math. Soc. 209, 275–280 (1975).
J. Colombeau, “Differential Calculus and Holomorphy,” North-Holland Math. Stud. 84 (North-Holland Publishing Co., Amsterdam–New York, 1982).
A. Dynin, “Quantum Yang–Mills-Weyl Dynamics in the Schroedinger Paradigm,” Russ. J. Math. Phys. 21 (2), 169–188 (2014).
A. Dynin, “On the Yang–Mills Quantum Mass Gap Problem,” Russ. J. Math. Phys. 21 (2), 326–328 (2014).
G. Dell’Antonio and D. Zwanziger, “Every Gauge Orbit Passes inside the Gribov Horizon,” Comm. Math. Phys. 138, 259–299 (1991).
L. Faddeev and A. Slavnov, Gauge Fields, Introduction to Quantum Theory (Addison-Wesley, 1991).
G. Folland, Harmonic Analysis in Phase Space (Princeton University Press, 1989).
I. M. Gelfand and R. A. Minlos, “Solution of Quantum Field Equations,” I. M. Gelfand, Collected Papers 1, 462–465 (1987) (Springer).
I. Gelfand and N. Vilenkin, Generalized Functions (4, Academic Press, 1964).
R. T. Glassey and W. A. Strauss, “Decay of Classical Yang–Mills Fields,” Comm. Math. Phys. 65 (1), 1–13 (1979).
J. Glimm and A. Jaffe, “An Infinite Renormalization of the Hamiltonian is Necessary,” J. Math. Phys. 10, 2213–2214 (1969).
J. Glimm and A. Jaffe, Quantum Physics: A Functional Point of View (Springer, 1987).
M. V. Goganov and L. V. Kapitanskii, “Global Solvability of the Initial Problem for Yang–Mills-Higs Equations,” Zap. LOMI 147, 18–48 (1985); J. Sov. Math. 37, 802–822 (1987).
C. Itzykson and J.-B. Zuber, Quantum Field Theory (Dover Publications, 2005).
P. Kree, “Calcule symbolique et seconde quantification des fonctions sesquiholomorphes,” C. R. Acad. Sc. Paris Sè. A, 284 (1), A25–A28 (1977).
P. Kree, “Méthodes holomorphe et méthodes nucléaires en analyse de dimension infinie et la théorie quantique des champs,” Lecture Notes in Math. 644, 212–254 (1977).
P. Kree and R. Raczka, “Kernels and Symbols of Operators in Quantum Field Theory,” Ann. Inst. H. Poincaré, Section A XXVIII, 41–73 (1978).
H.-H. Kuo, Gaussian Measures in Banach Spaces (Lecture Notes in Math. 463, Springer, 1975).
Ch. Nash and S. Sen, Topology and Geometry for Physicists (Dover Publications, 2011).
X. S. Raymond, “Simple Nash–Moser Implicit Function Theorem,” Enseign. Math. 35, 2170–226 (1989).
M. Reed and B. Simon, I. Methods of Modern Mathematical Physics (Academic Press, 1972).
M. Reed and B. Simon, II. Methods of Modern Mathematical Physics (Academic Press, 1975).
B. Simon, “Some Quantum Operators with Discrete Spectrum But Classically Continuous Spectrum,” Ann. Physics 146, 209–220 (1983).
M. Shubin, Pseudodifferential Operators and Spectral Theory (Springer, 1987).
F. Treves, Topological Vector Spaces, Distributions and Kernels (Academic Press, 1967).
E. Witten, Magic, Mystery, and Matrix, in Arnold, V., et al, Mathematics: Frontiers and Perspectives (Amer. Math. Soc., Providence, 2000, pp. 343–352).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dynin, A. Mathematical quantum Yang–Mills theory revisited. Russ. J. Math. Phys. 24, 19–36 (2017). https://doi.org/10.1134/S1061920817010022
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920817010022