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accelerator amplitudes angle angular momentum approximation assumes the form axis Bessel function betatron circular polarization classical commutation relations components condition confine our attention constant of motion constant uniform magnetic coordinates corresponding cos2 cosh delta function derivative describe Dirac equation direction electromagnetic field electron moves emission emitted energy loss energy tensor evaluated following expression formula four-dimensional frequency given Green's function Hamiltonian harmonic helical Hence integration with respect introduce Klein-Gordon equation Lagrangian Laguerre functions Laguerre polynomials linear polarization longitudinal Lorentz magnetic field matrix elements nonrelativistic obtain the following operator orbit particles photon plane pseudovector quantity quantization quantum number quantum theory radiated intensity radiative radius readily relativistic right-hand side satisfy scalar sin2 solution spin storage rings Substituting synchrotron radiation theory of synchrotron tion transformation ultra-relativistic uniform magnetic field unit vector vacuum vector potential velocity wave wavefunction wavevector zero