Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 013, 7 pages      quant-ph/0602002      https://doi.org/10.3842/SIGMA.2006.013

Operator Gauge Symmetry in QED

Siamak Khademi a and Sadollah Nasiri a, b
a) Department of Physics, Zanjan University, P.O. Box 313, Zanjan, Iran
b) Institute for Advanced Studies in Basic Sciences, IASBS, Zanjan, Iran

Received October 09, 2005, in final form January 17, 2006; Published online January 30, 2006

Abstract
In this paper, operator gauge transformation, first introduced by Kobe, is applied to Maxwell's equations and continuity equation in QED. The gauge invariance is satisfied after quantization of electromagnetic fields. Inherent nonlinearity in Maxwell's equations is obtained as a direct result due to the nonlinearity of the operator gauge transformations. The operator gauge invariant Maxwell's equations and corresponding charge conservation are obtained by defining the generalized derivatives of the first and second kinds. Conservation laws for the real and virtual charges are obtained too. The additional terms in the field strength tensor are interpreted as electric and magnetic polarization of the vacuum.

Key words: gauge transformation; Maxwell's equations; electromagnetic fields.

pdf (167 kb)   ps (127 kb)   tex (9 kb)

References

  1. Greiner W., Müller B., Quantum mechanics: symmetries, New York, Springer-Verlag, 1989.
  2. Karatas D.L., Kowalski K.L., Noether's theorem for local gauge transformation, Am. J. Phys., 1990, V.58, 123-131.
  3. Goldestein H., Classical mechanics, 2nd ed., Massachusetts, Addison-Wesley, 1980.
  4. Cocconi G., Upper limits on the electric charge of photon, Am. J. Phys., 1992, V.60, 750-751.
  5. Cheng T.P., Li L.F., Erratum "Resource letter GI-1: gauge invariance", Am. J. Phys., 1988, V.56, 1048.
  6. Kobe D.H., Gauge transformations in classical mechanics as canonical transformation, Am. J. Phys., 1988, V.56, 252-254.
  7. Baxter C., Jaynes-Cummings Hamiltonian in a covariant gauge, Phys. Rev. A, 1991, V.44, 3178-3179.
  8. Kobe D.H., Gauge invariant in second quantization: application to Hartree-Fock and generalized random-phase approximation, Phys. Rev. A, 1979, V.19, 1876-1885.
  9. Baxter C., a-Lorentz gauge in QED, Ann. Phys., 1991, V.206, 221-236.
  10. Sakurai J.J., Advanced quantum mechanics, 11th ed., New York, Addison-Wesley, 1987.
  11. Kobe D.H., Yang K.H., Gauge transformation of the time evolution operator, Phys. Rev. A, 1985, V.32, 952-958.
  12. Kobe D.H., Gauge transformation and the electric dipole approximation, Am. J. Phys., 1982, V.50, 128-133.
  13. Zumino B., Gauge properties of propagators in quantum electrodynamics, J. Math. Phys., 1960, V.1, 1-7.
  14. Gaete P., On gauge-invariant variables in QED, Z. Phys. C, 1997, V.76, 355-361.
  15. Manoukian E.B., Action principle and quantization of gauge fields, Phys. Rev. D, 1986, V.34, 3739-3749.
  16. Manoukian E.B., Siranan S., Action principle and algebraic approach to gauge transformations in gauge theories, Internat. J. Theoret. Phys., 2005, V.44, 53-62.
  17. Mandel L., Electric dipole interaction in quantum optics, Phys. Rev. A, 1979, V.20, 1590-1592.
  18. Kobe D.H., Gray R.D., Operator gauge transformation in nonrelativistic quantum electrodynamics: application to the multipolar Hamiltonian, Nuovo Cimento Soc. Ital. Fis. B, 1985, V.86, 155-170.
  19. Healy W.P., Comment on "Maxwell's equations in the multipolar representation", Phys. Rev. A, 1982, V.26, 1798-1799.
  20. Power E.A., Thirunamachandran T., Quantum electrodynamics with nonrelativistic sources. 1. Transformation to the multipolar formalism for second-quantized electron and Maxwell interacting fields, Phys. Rev. A, 1983, V.28, 2649-2662.
  21. Haller K., Maxwell's equations in the multipolar representation, Phys. Rev. A, 1982, V.26, 26-27.
  22. Ackerhalt J.R., Milloni P.W., Interaction Hamiltonian of quantum optic, J. Opt. Soc. Am. B, 1984, V.1, 116-120.
  23. Doebner H.D., Goldin G.A., Nattermann P., Gauge transformations in quantum mechanics and the unification of nonlinear Schrödinger equation, J. Math. Phys., 1999, V.40, 49-63.
  24. Goldin G.A., Shtelen V.M., Generalization of Yang-Mills theory with nonlinear constitutive equation, J. Phys. A: Math. Gen., 2004, V.37, 10711-10718, hep-th/0401093.

Previous article   Next article   Contents of Volume 2 (2006)