  We discuss the extension of the Goldstone and Englert-Brout-Higgs mechanisms
to non-Hermitian Hamiltonians that possess an antilinear PT symmetry. We study
a model due to Alexandre, Ellis, Millington and Seynaeve and show that for the
spontaneous breakdown of a continuous global symmetry we obtain a massless
Goldstone boson in all three of the antilinear symmetry realizations:
eigenvalues real, eigenvalues in complex conjugate pairs, and eigenvalues real
but eigenvectors incomplete. In this last case we show that it is possible for
the Goldstone boson mode to be a zero-norm state. For the breakdown of a
continuous local symmetry the gauge boson acquires a non-zero mass by the
Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry,
except the one where the Goldstone boson itself has zero norm, in which case,
and despite the fact that the continuous local symmetry has been spontaneously
broken, the gauge boson remains massless.
