Nonlinear Dirac Equations

Wei Khim Ng ; Rajesh R. Parwani

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.

Détail

Keywords: nonlinear Dirac equation; Lorentz violation.

Bibliographie

[1] Schrödinger E., Collected papers on wave mechanics, Blackie Son, London, 1928

[2] Bolinger J. J., Heinzen D. J., Itano W. M., Gilbert S. L., Wineland D. J., “Test of the linearity of quantum mechanics by rf spectroscopy of the $^9\mathrm Be^+$ ground state”, Phys. Rev. Lett., 63 (1989), 1031–1034 | DOI

[3] Chupp T., Hoare R., “Coherence in freely precessing $^{21}\mathrm Ne$ and a test of linearity of quantum mechanics”, Phys. Rev. Lett., 64 (1990), 2261–2264 | DOI

[4] Walsworth R. L., Silvera I. F., Mattison E. M., Vessot R. F. C., “Test of the linearity of quantum mechanics in an atomic system with a hydrogen maser”, Phys. Rev. Lett., 64 (1990), 2599–2602 | DOI

[5] Majumder P. K., Venema B. J., Lamoreaux S. K., Heckel B. R., “Test of the linearity of quantum mechanics in optically pumped $^{201}\mathrm Hg$”, Phys. Rev. Lett., 65 (1990), 2931–2934 | DOI

[6] Akhmediev N. N., Ankiewicz A., Solitons: nonlinear pulses and beams, Chapman and Hall, 1997

[7] Pethick C., Smith H., Bose–Einstein condensation in dilute gases, Cambridge University Press, 2001

[8] Heisenberg W., “Quantum theory of fields and elementary particles”, Rev. Mod. Phys., 29 (1957), 269–278 | DOI | MR | Zbl

[9] Ionescu D. C., Reinhardt R., Muller B., Greiner W., “Nonlinear extensions of the Dirac equation and their implications in QED”, Phys. Rev. A, 38 (1988), 616–620 | DOI

[10] Zecca A., “Dirac equation in space-time with torsion”, Internat. J. Theoret. Phys., 41 (2002), 421–428 | DOI | MR | Zbl

[11] Esteban M. J., Sere E., “An overview on linear and nonlinear Dirac equations”, Discrete Contin. Dyn. Syst., 8 (2002), 381–397 | DOI | MR | Zbl

[12] Esteban M. J., Sere E., “Stationary states of the nonlinear Dirac equation: a variational approach”, Comm. Math. Phys., 171 (1995), 323–350 | DOI | MR | Zbl

[13] Bialynicki-Birula I., Mycielski J., “Nonlinear wave mechanics”, Ann. Physics, 100 (1976), 62–93 | DOI | MR

[14] Kibble T. W. B., “Relativistic models of nonlinear quantum mechanics”, Comm. Math. Phys., 64 (1978), 73–82 | DOI | MR

[15] Weinberg S., “Testing quantum mechanics”, Ann. Physics, 194 (1989), 336–386 | DOI | MR

[16] Doebner H. D., Goldin G. A., Nattermann P., “Gauge Transformations in quantum mechanics and the unification of nonlinear Schrödinger equations”, J. Math. Phys., 40 (1999), 49–63 ; quant-ph/9709036 | DOI | MR | Zbl

[17] Parwani R., “Why is Schrödinger's equation linear?”, Braz. J. Phys., 35 (2005), 494–496 ; quant-ph/0412192 | DOI

[18] Parwani R., “An information theoretic link between space-time symmetries and quantum linearity”, Ann. Physics, 315 (2005), 419–452 ; hep-th/0401190 | DOI | MR | Zbl

[19] Parwani R., “Information measures for inferring quantum mechanics and its deformations”, J. Phys. A: Math. Gen., 38 (2005), 6231–6237 ; quant-ph/0408185 | DOI | MR | Zbl

[20] Parwani R., “A physical axiomatic approach to Schrödinger's equation”, Internat. J. Theoret. Phys., 45 (2006), 1901–1933 ; quant-ph/0508125 | DOI | MR

[21] Doebner H.D., Zhdanov R., Nonlinear Dirac equations and nonlinear gauge transformations, quant-ph/0304167

[22] Fushchich W. I., Zhdanov R. Z., “Symmetry and exact solutions of nonlinear spinor equations”, Phys. Rep., 172 (1989), 123–174 | DOI | MR

[23] Fushchych W. I., Zhdanov R. Z., Symmetries and exact solutions of nonlinear Dirac equations, Mathematical Ukraina Publisher, Kyiv, 1997; math-ph/0609052

[24] Bogoslovsky G. Yu., “Lorentz symmetry violation without violation of relativistic symmetry”, Phys. Lett. A, 350 (2006), 5–10 ; hep-th/0511151 | DOI

[25] Bogoslovsky G. Yu., Goenner H. F., “Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation”, Phys. Lett. A, 323 (2004), 40–47 ; hep-th/0402172 | DOI | MR | Zbl

[26] Bogoslovsky G. Yu., “A special-relativistic theory of the locally anisotropic space-time”, Nuovo Cimento B, 40 (1977), 99–134 | DOI

[27] Itzykson C., Zuber J. B., Quantum field theory, McGraw-Hill International Book Co., New York, 1980 | MR

[28] Kostelecky V. A., “Gravity, Lorentz violation, and the standard model”, Phys. Rev. D, 69 (2004), 105009, 20 pp., ages ; hep-th/0312310 | DOI

[29] Bohm D., Hiley B. J., The undivided universe. An ontological interpretation of quantum theory, Routledge, London, 2003 | MR

[30] Moshinsky M., Nikitin A., “The Many body problem in relativistic quantum mechanics”, Rev. Mexicana Fís., 50 (2005), 66–73 ; hep-ph/0502028 | MR

[31] Goldstein S., Tumulka R., “Opposite arrows of time can reconcile relativity and nonlocality”, Classical Quantum Gravity, 20 (2003), 557–564 ; quant-ph/0105040 | DOI | MR | Zbl

[32] Svetlichny G., “On linearity of separating multiparticle differential Schrödinger operators for identical particles”, J. Math. Phys., 45 (2004), 959–964 ; quant-ph/0308001 | DOI | MR | Zbl

[33] Goldin G. A., Svetlichny G., “Nonlinear Schrödinger equations and the separation property”, J. Math. Phys., 35 (1994), 3322–3332 | DOI | MR | Zbl

[34] Parwani R., Tan H. S., “Exact solutions of a non-polynomially nonlinear Schrödinger equation”, Phys. Lett. A, 363 (2007), 197–201 ; quant-ph/0605123 | DOI | MR

[35] Parwani R., Tabia G., “Universality in an information-theoretic motivated nonlinear Schrödinger equation”, J. Phys. A: Math. Gen., 40 (2007), 5621–5635 ; quant-ph/0607222 | DOI | MR | Zbl

[36] Greenberg O. W., “CPT violation implies violation of Lorentz invariance”, Phys. Rev. Lett., 89 (2002), 231602, 4 pp., ages ; hep-ph/0201258 | DOI

[37] Colladay D., Kostelecky V. A., “CPT violation and the standard model”, Phys. Rev. D, 55 (1997), 6760–6774 ; hep-ph/9703464 | DOI

[38] Lehnert R., “Dirac theory within the standard-model extension”, J. Math. Phys., 45 (2004), 3399–3412 ; hep-ph/0401084 | DOI | MR | Zbl

[39] Jacobson T., Liberate S., Mattingly D., “Lorentz violation at high energy: concepts, phenomena, and astrophysical constraints”, Ann. Physics, 321 (2006), 150–196 ; astro-ph/0505267 | DOI | Zbl

[40] Vucetich H., Testing Lorentz invariance violation in quantum gravity theories, gr-qc/0502093

[41] Ng W. K., Parwani R., Nonlinear Dirac equations,, arXiv:0707.1553v1 | MR

[42] Ng W. K., Parwani R., Nonlinear Schrödinger–Pauli equations, arXiv:0807.1877

[43] Ng W. K., Parwani R., Probing quantum nonlinearities through neutrino oscillations, arXiv:0805.3015

[44] Kent A., “Nonlinearity without superluminality”, Phys. Rev. A, 72 (2005), 012108, 4 pp., ages ; quant-ph/0204106 | DOI | MR

[45] Holman M., On arguments for linear quantum dynamics, quant-ph/0612209

[46] Geim A. K., Novoselov K. S., “The rise of graphene”, Nature Materials, 6 (2007), 183–191 ; cond-mat/0702595 | DOI