Abstract
Solitary waves of nonlinear Dirac, Maxwell–Dirac and Klein–Gordon–Dirac equations are considered. We deduce some virial identities and check that the energy-momentum relation for solitary waves coincides with the Einstein energy-momentum relation for point particles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
S. Abenda, ‘‘Solitary waves for Maxwell–Dirac and Coulomb–Dirac models,’’ Ann. Inst. H. Poincaré Phys. Théor. 68, 229–244 (1998).
M. Balabane, T. Cazenave, A. Douady, and F. Merle, ‘‘Existence of excited states for a nonlinear Dirac field,’’ Commun. Math. Phys. 119, 153–176 (1988).
M. Balabane, T. Cazenave, and L. Vazquez, ‘‘Existence of standing waves for Dirac fields with singular nonlinearities,’’ Commun. Math. Phys. 133, 53–74 (1990).
H. A. Bethe, Intermediate Quantum Mechanics (W. A. Benjamin, New York, Amsterdam, 1964).
J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).
T. Cazenave and L. Vazquez, ‘‘Existence of localized solutions for a classical nonlinear Dirac field,’’ Comm. Math. Phys. 105, 35–47 (1986).
J. Chadam, ‘‘Global solutions of the Cauchy problem for the (classical) coupled Maxwell–Dirac system in one space dimension,’’ J. Funct. Anal. 13, 173–184 (1973).
J. Chadam and R. Glassey, ‘‘On certain global solutions of the Cauchy problem for the (classical) coupled Klein–Gordon–Dirac equations in one and three space dimensions,’’ Arch. Rational Mech. Anal. 54, 223–237 (1974).
J. Chadam and R. Glassey, ‘‘On the Maxwell–Dirac equations with zero magnetic field and their solutions in two space dimension,’’ J. Math. Anal. Appl. 53, 495–507 (1976).
G. H. Derrick, ‘‘Comments on nonlinear wave equations as models for elementary particles,’’ J. Math. Phys. 5, 1252–1254 (1964).
T. V. Dudnikova, A. I. Komech, and H. Spohn, ‘‘Energy-momentum relation for solitary waves of relativistic wave equation,’’ Russ. J. Math. Phys. 9, 153–160 (2002).
F. Dyson, Advanced Quantum Mechanics (World Scientific, Singapore, 2007).
A. Einstein and G. Grommer, ‘‘Allgemeine Relativitätstheorie und Bewegungsgesetz,’’ Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., 2–13 (1927).
A. Einstein, L. Infeld, and B. Hoffmann, ‘‘The gravitational equations and the problem of motion,’’ Ann. Math. 39, 65–100 (1938).
M. Esteban, V. Georgiev, and E. Séré, ‘‘Stationary solutions of the Maxwell–Dirac and the Klein–Gordon–Dirac equations,’’ Calc. Var. Partial Differ. Equat. 4, 265–281 (1996).
M. J. Esteban, M. Lewin, and E. Séré, ‘‘Variational methods in relativistic quantum mechanics,’’ Bull. (New Ser.) AMS 45, 535–593 (2008).
M. Esteban and E. Séré, ‘‘Stationary states of the nonlinear Dirac equation: A variational approach,’’ Commun. Math. Phys. 171, 323–350 (1995).
M. Esteban and E. Séré, ‘‘An overview on linear and nonlinear Dirac equation,’’ Discrete Contin. Dynam. Syst. 8, 381–397 (2002).
L. Gross, ‘‘The Cauchy problem for the coupled Maxwell and Dirac equations,’’ Comm. Pure Appl. Math. 19, 1–5 (1966).
D. J. Gross and A. Neveu, ‘‘Dynamical symmetry breaking in asymptotically free field theories,’’ Phys. Rev. D 10, 3235–3253 (1974).
S. Y. Lee, T. K. Kuo, and A. Gavrielides, ‘‘Exact localized solutions of two-dimensional fields theories of massive fermions with Fermi interactions,’’ Phys. Rev. D 12, 2249–2253 (1975).
A. G. Lisi, ‘‘A solitary wave solution of the Maxwell–Dirac equations,’’ J. Phys. A 28, 5385–5392 (1995).
F. Merle, ‘‘Existence of stationary states for nonlinear Dirac equations,’’ J. Differ. Equat. 74, 50–68 (1988).
H. Ounaies, ‘‘Perturbation method for a class of nonlinear Dirac equations,’’ Differ. Integr. Equat. 13, 707–720 (2000).
S. I. Pohozaev, ‘‘On the eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\),’’ Sov. Math. Dokl. 6, 1408–1411 (1965).
A. F. Ranada, ‘‘Classical nonlinear Dirac field models of extended particles,’’ in Quantum Theory, Groups, Fields and Particles, Ed. by A. O. Barut (Reidel, Dordrecht, 1983), pp. 271–291.
A. F. Ranada and L. Vazquez, ‘‘Classical system of nonlinear Dirac and Klein–Gordon fields,’’ Prog. Theor. Phys. 56, 311–323 (1976).
N. Rosen, ‘‘A field theory of elementary particles,’’ Phys. Rev. 55, 94–101 (1939).
G. Rosen, ‘‘Particlelike solutions to nonlinear complex scalar field theories with positive-definite energy densities,’’ J. Math. Phys. 9, 996–998 (1968).
G. Rosen, ‘‘Charged particle-like solutions to nonlinear complex scalar field theories with positive-definite energy densities,’’ J. Math. Phys. 9, 999–1002 (1968).
S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and Co., Evanston, IL, 1961).
L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968).
M. Soler, ‘‘Classical, stable, nonlinear spinor fields with positive rest energy,’’ Phys. Rev. D1, 2766–2769 (1970).
B. Thaller, The Dirac Equation, Texts and Monographs in Physics, Ed. by W. Beiglböck, E. H. Lieb, and W. Thirring (Springer, Berlin, Heidelberg, New York, 1992).
M. Wakano, ‘‘Intensely localized solutions of the classical Dirac–Maxwell field equations,’’ Progr. Theor. Phys. 35, 1117–1141 (1966).
H. Yukawa, ‘‘On the interaction of elementary particles. I,’’ Proc. Phys.-Math. Soc. Jpn. 17, 48–57 (1935).
Funding
This work was supported by the Russian Science Foundation (grant no. 19-71-30004).
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by A. I. Aptekarev)
Rights and permissions
About this article
Cite this article
Dudnikova, T.V. Virial Identities and Energy–Momentum Relation for Solitary Waves of Nonlinear Dirac Equations. Lobachevskii J Math 41, 956–981 (2020). https://doi.org/10.1134/S1995080220060074
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080220060074