The Jackiw-Rebbi model. Bose/Fermi one-loop Kink mass shifts

RESUMO

Scalar and spinor Kink fluctuations are summarized in the Jackiw-Rebbi model. Canonical quantization is conventionally achieved by promoting the coefficients of the scalar and spinor field expansions to bosonic/fermionic creation/annihilation operators. One-particle states for bosons, respectively for fermions, are supplied by the scalar eigenfunctions of the Schr$\ddot{\rm o}$dinger, respectively, the eigenspinors of the Dirac Hamiltonian, evaluated either on the ground state or the Kink background. One-loop kink mass shifts are computed induced either by bosons or fermions. Both bound and scattering states contribute when the quanta are trapped by or scattered off the Kink. Zero modes of Kink fluctuations bosonic and fermionic do not modify the Kink mass at one-loop order. The Dashed-Hasslacher-Neveu prescription, derived in the canonical quantization framework, is generalized to the fermionic realm. Alternatively, quantization à la Feynman of the Jackiw-Rebbi model in the spaces of Kink scalar and spinor fluctuations is developed. It is ensured that Berezin integration over the Grassmann coefficients of the Kink spinor fluctuations leads to identical scenario as that revealed by including Fermi statistics in the DHN paradigm, at least with respect to the computation of the Kink Casimir energy.