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Generalizations of the Klein–Gordon and the Dirac Equations from Non-standard Lagrangians

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Abstract

We present various generalizations of the Klein–Gordon and Dirac formalisms based on non-standard Lagrangians within the framework of the calculus of variations characterized by a power-law Lagrangian \( {\mathsf{L}}^{1 + \gamma},\,\gamma \) being a free parameter. In the case of the bosonic scalar field, the modified dispersion relation has been derived and based on this, it was observed that for a particular choice of non-standard Lagrangians, the new field theory forbid the presence of massless particles. Besides, the Klein–Gordon equation is modified and becomes similar to the Barut equation which is a second order equation in the \( ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2},0) \oplus (0,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}) \) representation of the Lorentz group, which explains the splitting of leptons. For the case of the spinor scalar field, the Barut-like equation was derived from a non-standard Lagrangian as well. For some specific class of non-standard Lagrangians, the modified dispersion is modified and prohibits the presence of massless particles.

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References

  1. El-Nabulsi RA (2012) Non-linear dynamics with non-standard Lagrangians Qual. Theory Dyn. Syst. doi: 10.1007/s12346-012-0074-0

  2. El-Nabulsi RA (2012) Quantum field theory from and exponential action functional. Indian J Phys 87:379–383

    Google Scholar 

  3. Alekseev AI, Arbuzov BA (1984) Classical theory of Yang-Mills field for nonstandard. Lagrangians Theor Math Phys 59:372–378

    Article  MathSciNet  Google Scholar 

  4. Musielak ZE (2008) Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J Phys A: Math Theor 41:055205–055223

    Article  MathSciNet  ADS  Google Scholar 

  5. Musielak ZE (2009) General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems. Chaos, Solitons Fractals 42:2645–2652

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Chandrasekar VK, Pandey SN, Senthilvelan M, Lakshmanan M (2006) Simple and unified approach to identify integrable nonlinear oscillators and systems. J Math Phys 47:023508–023554

    Article  MathSciNet  ADS  Google Scholar 

  7. Chandrasekar VK, Senthilvelan M, Lakshmanan M (2005) On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator. Phys Rev E72:066203–066222

    ADS  Google Scholar 

  8. Carinena JF, Ranada MF, Santander M (2005) Lagrangian formalism for nonlinear second order Riccati Systems: one-dimensional integrability and two-dimensional superintegrability. J Math Phys 46:062703–062728

    Article  MathSciNet  ADS  Google Scholar 

  9. Renaux-Petel S, Tasinato G (2009) Nonlinear perturbations of cosmological scalar fields with non-standard kinetic terms. J Cosmo Astropart Phys 0901:012–033

    ADS  Google Scholar 

  10. Bender CM, Holm DC, Hook DW (2007) Complex trajectories of a simple pendulum. J Phys A: Math Gen 40:F81–F89

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Barut AO, Cordero P, Ghirardi GC (1970) A unified treatment of leptons. Nuovo Cim A66:36–46

    Article  ADS  Google Scholar 

  12. Barut AO (1978) The mass of the muon. Phys Lett B73:310–312

    Article  MathSciNet  ADS  Google Scholar 

  13. Barut AO (1979) Lepton mass formula Phys. Rev. Lett. 42:1251–1255; (1979) Erratum-ibid. 43:1057–1059

    Google Scholar 

  14. Dvoeglazov VV (2008) The Barut second-order equation: lagrangian, dynamical invariants and interactions. Adv Appl Clifford Algebra 18:579–585

    Article  MathSciNet  MATH  Google Scholar 

  15. Kruglov SI (2004) On the generalized Dirac equation for fermions with two mass states, Annales Fond Broglie 29:1005–1016

    Google Scholar 

  16. Kruglov SI (2006) On the Hamiltonian form of generalized Dirac equation for fermions with two mass states. Elec J Theor Phys 10:11–16

    Google Scholar 

  17. Sprenger M, Nicolini P, Bleicher M (2012) Physics on smallest scales-an Introduction to minimal length phenomenology. Eur J Phys 33:853–862

    Article  MATH  Google Scholar 

  18. Weinberg S (1996) The quantum theory of fields Vol. I and II. Cambridge University Press, Cambridge

    Book  Google Scholar 

  19. Bogoslovsky G, Goenner H (2004) Generalized Lorentz symmetry and nonlinear spinor fields in a flat Finslerian space-time. Proc Inst Math NAS Ukraine 50:637–644

    MathSciNet  Google Scholar 

  20. Nozari K, Karami M (2005) Minimal length and generalized Dirac equation. Mod Phys Lett A20:3095–3104

    Article  MathSciNet  ADS  Google Scholar 

  21. Nozari K, Mehdipour SH (2007) Implications of minimal length scale on the statistical mechanics of ideal gas. Chaos Solitons Fractals 32:1637–1644

    Article  ADS  Google Scholar 

  22. Niederle J, Nikitin AG (2001) Relativistic wave equations for interacting, massive particles with arbitrary half-integer spins. Phys Rev D64:125013–125024

    MathSciNet  ADS  Google Scholar 

  23. Niederle J, Nikitin AG (1997) Involutive symmetries, supersymmetries and reductions of the Dirac equation. J Phys A30:999–1010

    MathSciNet  ADS  Google Scholar 

  24. Niederle J, Nikitin AG (1997) Non-Lie and discrete symmetries of the Dirac equation. J Nonlin Math Phys 4:436–444

    Article  MathSciNet  MATH  Google Scholar 

  25. de Montigny M, Khanna FC, Santana AE, Santos ES, Vianna JDM (2000) Galilean covariance and the Duffin–Kemmer–Petiau equation. J Phys A: Math Gen 33:L273–L278

    Article  MATH  Google Scholar 

  26. de Montigny M, Khanna FC, Santana AE, Santos ES (2001) Galilean covariance and non-relativistic Bhabha equations. J Phys A: Math Gen 34:8901–8917

    Article  ADS  MATH  Google Scholar 

  27. Huegele R, Musielak ZE, Fry JL (2012) Fundamental dynamical equations for spinor wave functions: i. Lévy-Leblond and Schrödinger equations. J Phys A: Math Theor 45:143222–145205

    Article  MathSciNet  Google Scholar 

  28. Kaplan DB, Sun S (2012) Spacetime as a topological insulator: mechanism for the origin of the fermion generations. Phys. Rev. Letts. 108:181807–181811

    Article  ADS  Google Scholar 

  29. Kostelecky VA, Samuel S (1989) Spontaneous Breaking of Lorentz Symmetry in String Theory. Phys Rev D39:683–685

    ADS  Google Scholar 

  30. Kostelecky VA, Potting R (1991) CPT and strings. Nucl Phys B359:545–570

    Article  MathSciNet  ADS  Google Scholar 

  31. Kruglov SI (2012) Modified Dirac equation with Lorentz invariance violation and its solution for particles in an external magnetic field. Phys. Lett. B718:228–231

    Article  ADS  Google Scholar 

  32. Chang Z, Wang S (2012) Lorentz invariance violation and electromagnetic field in an intrinsically anisotropic spacetime. Eur Phys J C72:2165–2181

    Article  ADS  Google Scholar 

  33. Nielsen HB, Picek I (1982) Baryon poles in proton decay amplitudes. Phys Lett B144:141–146

    Article  ADS  Google Scholar 

  34. Nielsen HB, Picek I (1983) The Rédei-Like Model and Testing Lorentz Invariance. Nucl Phys B211:269–296

    Article  ADS  Google Scholar 

  35. Nielsen HB, Picek I (1983) Lorentz Non-Invariance. Phys Rev D27:665–667

    ADS  Google Scholar 

  36. Aringazin AK, Jannussis A, Lopez DF, Nishioka M, Veljanoski B (1991) Santilli’s Lie-Isotopic Generalization of Galilei’s and Einstein’s Relativities. Kostarakis Publishers, Athens

    Google Scholar 

  37. Nishioka M (1984) Remarks on the Lie-isotopic lifting of gauge theory. Nuovo Cimento A82:351–356

    Article  MathSciNet  ADS  Google Scholar 

  38. Nishioka M (1985) Remarks on Lie algebras appearing in the Lie-isotopic lifting of gauge theory. Nuovo Cimento A85:331–336

    Article  MathSciNet  ADS  Google Scholar 

  39. Nishioka M (1986) Applications of the Lie-isotopic lifting of gauge theory to a system of gauge fields and gravitation. Nuovo Cimento A92:132–138

    Article  MathSciNet  ADS  Google Scholar 

  40. Wio HS, Revelli JA, Deza RR, Escudero C, de la Lama MS (2010) KPZ equation: galilean-invariance violation, Consistency, and fluctuation–dissipation issues in real-space discretization. Europhys Letts 89:40008–40012

    Article  ADS  Google Scholar 

  41. Kaehler G, Wagner A (2012) Galilean invariance in fluctuating lattice Boltzmann American Physical Society, APS March Meeting, February 27-March 2, abstract #L41.014

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El-Nabulsi, R.A. Generalizations of the Klein–Gordon and the Dirac Equations from Non-standard Lagrangians. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 83, 383–387 (2013). https://doi.org/10.1007/s40010-013-0094-4

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  • DOI: https://doi.org/10.1007/s40010-013-0094-4

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