Superfields and Fermi-Bose symmetry

The realization of supergauge transformations on fields defined over an 8-dimensional space whose points are labeled by x_μ and the anticommuting Majorana spinor θ_α is described. The covariant derivative is defined and applied to the problem of decomposing superfields into irreducible (chiral) parts and to the problem of constructing "supersymmetric" Lagrangians. Further, it is shown how to build internal symmetries (both global and local) into these Lagrangians. An example is discussed in which the internal (global) symmetry is spontaneously violated, giving rise to a supermultiplet of Goldstone particles (including fermions). When a local symmetry is broken the Higgs mechanism (for bosons and fermions) is shown to be operative. A possible solution to the problem of defining a conserved fermion number is indicated.