LAUSR

In recent work Alexandre, Ellis, Millington and Seynaeve have extended the Goldstone theorem to non-Hermitian Hamiltonians that possess a discrete antilinear symmetry such as $PT$. They restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here we extend the discussion to the two other realizations possible with antilinear symmetry, namely energies in complex conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at all. In particular, we show that under certain circumstances it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-block Hamiltonians. While we discuss the same model as Alexandre et al. our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. In their paper Alexandre et al. presented a variational procedure for the action in which the surface term played an explicit role, to thus suggest that one has to use such a procedure in order to establish the Goldstone theorem in the non-Hermitian case. However, by taking certain fields that they took to be Hermitian to actually either be anti-Hermitian or be made so by a similarity transformation, we show that we are then able to obtain a Goldstone boson using a completely standard variational procedure. Since we use a standard variational procedure we can readily extend our analysis to a continuous local symmetry by introducing a gauge boson. We show that when we do this the gauge boson acquires a non-zero mass by the 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.