Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories

Abstract

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 P T and possess a continuous global symmetry. 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, Ellis, Millington and Seynaeve our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. We extend our analysis to a continuous local symmetry and find that 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. E * . Thus as originally noted by Wigner in his study of time reversal invariance, energies can thus be real or appear in complex conjugate pairs with complex conjugate eigenfunctions. It is often the case that one can move between these two realizations by a change in the parameters in H. There will thus be a transition point (known as an exceptional point) at which the switch over occurs. However, at this transition point the two complex conjugate wave functions (|ψ and A|ψ ) have to collapse into a single common wave function as there are no complex conjugate pairs on the real energy side. Since this collapse to a single common wave function reduces the number of energy eigenfunctions, at the transition point the eigenspectrum of the Hamiltonian becomes incomplete, with the Hamiltonian then being of non-diagonalizable Jordan-block form, the thus third possible realization of antilinear symmetry.While the above analysis would in principle apply to any antilinear symmetry, because of its H = p 2 +ix 3 progenitor, the antilinear symmetry program is conventionally referred to as the P T -symmetry program. However, P T symmetry can actually be selected out for a different reason, namely it has a connection to spacetime. Specifically, it was noted in [11] and emphasized in [3] that for the spacetime coordinates the linear part of a P T transformation is the same as a particular complex Lorentz transformation, while in [7,12] it was noted that for spinors the linear part of a CP T transformation is the same as that very same particular complex Lorentz transformation, where C denotes charge conjugation. 2 Then in [7,12] it was shown that if one imposes only two requirement...

Pseudo-Hermitian approach to Goldstone’s theorem in non-Abelian non-Hermitian quantum field theories

“…While in the former proposition Noether's theorem is evaded the latter is based on the standard variational principle leading to standard Noether currents. Moreover with regard to the Higgs mechanism the "surface term approach" predicts that the gauge particle becomes massive in the local case [20], whereas the "pseudo-Hermitian approach" leads to a theory in which the gauge particle remains massless at the exceptional point [18]. Here we also find that the explicit form of the Goldstone bosons differs.…”

“…So far two distinct alternative propositions have been made to overcome this issue. Alexandre, Ellis, Millington and Seynaeve proposed to apply a non-standard variational principle by keeping some non-vanishing surface terms [17,20] or, in line with the pseudo-Hermitian/PT -symmetric quantum mechanical approach [24,25,26,27], one may seek a consistent equivalent similarity transformed Hermitian action, as pursued by Mannheim and the present authors [18,19]. While some features are the same in both approaches, e.g.…”

“…At first sight such type of theories appear to be inconsistent as the two sets of equations of motion obtained by functionally varying the action I separately with respect to the fields φ i and φ * i , δI n /δφ i = 0 and δI n /δφ * i = 0, are in general incompatible when U = I. One may, however, overcome this problem by using a non-standard variational principle combined with keeping some non-vanishing surface terms [17,20] or alternatively by exploiting the fact that the content of the theory is unaltered as long as the equal time commutation relations are preserved and carry out a similarity transformation that guarantees that feature [8,18,19]. Hence, in the latter approach one seeks a Dyson map η, named this way in analogy to its quantum mechanical counterpart [28], to construct a new equivalent actionÎ…”

“…The generalisations also include Goldstone's theorem [12,13] and the Higgs mechanism [14,15,16] [17,18,19,20]. Both of these mechanisms are governed by the continuous symmetries of the theories, global or local, respectively, that might by spontaneously broken by some vacuum states.…”

PT -symmetric non-Hermitian quantum field theories with supersymmetry

“…It has also been suggested that non-Hermitian quantum field theory might also have applications in fundamental physics, e.g., to neutrino physics [8][9][10], dark matter [11] Higgs decays [12] and particle mixing [13]. It has been shown that it is possible to carry over to PT -symmetric non-Hermitian theories familiar concepts from Hermitian quantum field theory such as the spontaneous breaking of global symmetries [14][15][16] and the Englert-Brout-Higgs mechanism in gauge theories [17][18][19], despite the appearance of subtleties [20,21] in the relationship between current conservation, Lagrangian symmetries and Noether's theorem [22] in PT -symmetric theories.…”