American Institute of Mathematical Sciences

Advanced
March  2015, 35(3): 1103-1138. doi: 10.3934/dcds.2015.35.1103

Unified field equations coupling four forces and principle of interaction dynamics

Tian Ma 1, and  Shouhong Wang 2,

1. 

Department of Mathematics, Sichuan University, Chengdu
2. 

Department of Mathematics, Indiana University, Bloomington, IN 47405

Received  January 2013 Revised  July 2014 Published  October 2014

The main objective of this article is to postulate a principle of interaction dynamics (PID) and to derive field equations coupling the four fundamental interactions based on first principles. PID is a least action principle subject to div$_A$-free constraints for the variational element with $A$ being gauge potentials. The Lagrangian action is uniquely determined by 1) the principle of general relativity, 2) the $U(1)$, $SU(2)$ and $SU(3)$ gauge invariances, 3) the Lorentz invariance, and 4) the principle of representation invariance (PRI), introduced in [11]. The unified field equations are then derived using PID. The field model spontaneously breaks the gauge symmetries, and gives rise to a new mass generation mechanism. The unified field model introduces a natural duality between the mediators and their dual mediators, and can be easily decoupled to study each individual interaction when other interactions are negligible. The unified field model, together with PRI and PID applied to individual interactions, provides clear explanations and solutions to a number of outstanding challenges in physics and cosmology, including e.g. the dark energy and dark matter phenomena, the quark confinement, asymptotic freedom, short-range nature of both strong and weak interactions, decay mechanism of sub-atomic particles, baryon asymmetry, and the solar neutrino problem.
Keywords: unified field equations, dual particle fields, solar neutrino problem, Higgs mechanism, dark energy, short-range nature of strong and weak interactions, baryon asymmetry., Principle of interaction dynamics (PID), fundamental interactions, asymptotic freedom, quark confinement, principle of representation invariance, gauge symmetry breaking, dark matter.
Mathematics Subject Classification: 81T13, 83E9.
Citation: Tian Ma, Shouhong Wang. Unified field equations coupling four forces and principle of interaction dynamics. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1103-1138. doi: 10.3934/dcds.2015.35.1103
References:
[1]

F. Englert and R. Brout, Broken symmetry and the mass of gauge vector mesons, Physical Review Letters, 13 (1964), 321-323. doi: 10.1103/PhysRevLett.13.321.  Google Scholar

[2]

D. Griffiths, Introduction to Elementary Particles, Wiley-Vch, 2008. doi: 10.1002/9783527618460.  Google Scholar

[3]

G. Guralnik, C. R. Hagen and T. W. B. Kibble, Global conservation laws and massless particles, Physical Review Letters, 13 (1964), 585-587. doi: 10.1103/PhysRevLett.13.585.  Google Scholar

[4]

F. Halzen and A. D. Martin, Quarks and Leptons: An Introductory Course in Modern Particle Physics, John Wiley and Sons, New York, NY, 1984. Google Scholar

[5]

P. W. Higgs, Broken symmetries and the masses of gauge bosons, Physical Review Letters, 13 (1964), 508-509. doi: 10.1103/PhysRevLett.13.508.  Google Scholar

[6]

M. Kaku, Quantum Field Theory, A Modern Introduction, Oxford University Press, 1993.  Google Scholar

[7]

G. Kane, Modern Elementary Particle Physics, vol. 2, Addison-Wesley Reading, 1987. Google Scholar

[8]

T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812701152.  Google Scholar

[9]

_______, Duality theory of strong interaction, Electronic Journal of Theoretical Physics, 11:31(2014), 101-124. Google Scholar

[10]

_______, Duality theory of weak interaction, Indiana University Institute for Scientific Computing and Applied Mathematics Preprint Series, #1302: http://www.indiana.edu/~iscam/preprint/1302.pdf, (2012). Google Scholar

[11]

_______, Unified field theory and principle of representation invariance, arXiv:1212.4893; version 1 appeared in Applied Mathematics and Optimization, 69 (2014), 359-392. doi: 10.1007/s00245-013-9226-0.  Google Scholar

[12]

_______, Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34 (2014), 335-366; see also arXiv:1206.5078v2. Google Scholar

[13]

_______, Weakton model of elementary particles and decay mechanisms, Indiana University Institute for Scientific Computing and Applied Mathematics Preprint Series, #1304: http://www.indiana.edu/~iscam/preprint/1304.pdf, (May 30, 2013). Google Scholar

[14]

Y. Nambu, Quasi-particles and gauge invariance in the theory of superconductivity, Phys. Rev., 117 (1960), 648-663. doi: 10.1103/PhysRev.117.648.  Google Scholar

[15]

Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev., 122 (1961), 345-358. doi: 10.1103/PhysRev.122.345.  Google Scholar

[16]

_______, Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev., 124 (1961), 246-254. Google Scholar

[17]

C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, 2nd Edition, Princeton Unversity Press, 2013. Google Scholar

show all references

References:
[1]

F. Englert and R. Brout, Broken symmetry and the mass of gauge vector mesons, Physical Review Letters, 13 (1964), 321-323. doi: 10.1103/PhysRevLett.13.321.  Google Scholar

[2]

D. Griffiths, Introduction to Elementary Particles, Wiley-Vch, 2008. doi: 10.1002/9783527618460.  Google Scholar

[3]

G. Guralnik, C. R. Hagen and T. W. B. Kibble, Global conservation laws and massless particles, Physical Review Letters, 13 (1964), 585-587. doi: 10.1103/PhysRevLett.13.585.  Google Scholar

[4]

F. Halzen and A. D. Martin, Quarks and Leptons: An Introductory Course in Modern Particle Physics, John Wiley and Sons, New York, NY, 1984. Google Scholar

[5]

P. W. Higgs, Broken symmetries and the masses of gauge bosons, Physical Review Letters, 13 (1964), 508-509. doi: 10.1103/PhysRevLett.13.508.  Google Scholar

[6]

M. Kaku, Quantum Field Theory, A Modern Introduction, Oxford University Press, 1993.  Google Scholar

[7]

G. Kane, Modern Elementary Particle Physics, vol. 2, Addison-Wesley Reading, 1987. Google Scholar

[8]

T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812701152.  Google Scholar

[9]

_______, Duality theory of strong interaction, Electronic Journal of Theoretical Physics, 11:31(2014), 101-124. Google Scholar

[10]

_______, Duality theory of weak interaction, Indiana University Institute for Scientific Computing and Applied Mathematics Preprint Series, #1302: http://www.indiana.edu/~iscam/preprint/1302.pdf, (2012). Google Scholar

[11]

_______, Unified field theory and principle of representation invariance, arXiv:1212.4893; version 1 appeared in Applied Mathematics and Optimization, 69 (2014), 359-392. doi: 10.1007/s00245-013-9226-0.  Google Scholar

[12]

_______, Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34 (2014), 335-366; see also arXiv:1206.5078v2. Google Scholar

[13]

_______, Weakton model of elementary particles and decay mechanisms, Indiana University Institute for Scientific Computing and Applied Mathematics Preprint Series, #1304: http://www.indiana.edu/~iscam/preprint/1304.pdf, (May 30, 2013). Google Scholar

[14]

Y. Nambu, Quasi-particles and gauge invariance in the theory of superconductivity, Phys. Rev., 117 (1960), 648-663. doi: 10.1103/PhysRev.117.648.  Google Scholar

[15]

Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev., 122 (1961), 345-358. doi: 10.1103/PhysRev.122.345.  Google Scholar

[16]

_______, Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev., 124 (1961), 246-254. Google Scholar

[17]

C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, 2nd Edition, Princeton Unversity Press, 2013. Google Scholar

[1]

Tian Ma, Shouhong Wang. Gravitational Field Equations and Theory of Dark Matter and Dark Energy. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 335-366. doi: 10.3934/dcds.2014.34.335
[2]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016
[3]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Corrigendum to "On small data scattering of Hartree equations with short-range interaction" [Comm. Pure. Appl. Anal., 15 (2016), 1809-1823]. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1939-1940. doi: 10.3934/cpaa.2017094
[4]

Mathias Schäffner, Anja Schlömerkemper. On Lennard-Jones systems with finite range interactions and their asymptotic analysis. Networks & Heterogeneous Media, 2018, 13 (1) : 95-118. doi: 10.3934/nhm.2018005
[5]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 785-826. doi: 10.3934/dcds.2009.23.785
[6]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309
[7]

Yong Zhou, Jishan Fan. Regularity criteria of strong solutions to a problem of magneto-elastic interactions. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1697-1704. doi: 10.3934/cpaa.2010.9.1697
[8]

Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317
[9]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[10]

Shijin Ding, Bingyuan Huang, Xiaoyan Hou. Strong solutions to a fluid-particle interaction model with magnetic field in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021042
[11]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[12]

Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic & Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005
[13]

Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155
[14]

Matteo Negri. Crack propagation by a regularization of the principle of local symmetry. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 147-165. doi: 10.3934/dcdss.2013.6.147
[15]

Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193
[16]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002
[17]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[18]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505
[19]

H.M. Byrne, S.M. Cox, C.E. Kelly. Macrophage-tumour interactions: In vivo dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 81-98. doi: 10.3934/dcdsb.2004.4.81
[20]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic & Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (112)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy