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Abstract
We generalise previous studies on the extension of Goldstone's theorem from Hermitian to non-Hermitian quantum field theories with Abelian symmetries to theories possessing a glocal non-Abelian symmetry. We present a detailed analysis for a non-Hermitian field theory with two complex two component scalar fields possessing a SU(2)-symmetry and indicate how our finding extend to the general case. In the PT-symmetric regime and at the standard exceptional point the Goldstone theorem is shown to apply, although different identification procedures need to be employed. At the zero exceptional points the Goldstone boson can not be identified. Comparing our approach, based on the pseudo-Hermiticity of the model, to an alternative approach that utilises surface terms to achieve compatibility for the non-Hermitian system, we find that the explicit forms of the Goldstone boson fields are different.
Highlights
Noting that the change in the complex scalar fields is δφkj 1⁄4 iαaTkalφlj, with the generators Ta of the symmetry transformation taken to be standard Pauli matrices σa, a 1⁄4 1, 2, 3, we directly identify the infinitesimal changes for the real component fields as δφ1j
At the standard exceptional point, i.e., when K2 1⁄4 −2L and λbþ 1⁄4 λb−, the two eigenvectors v− and vþ coalesce so that the matrix U is no longer invertible and the Goldstone boson fields may take on a different form as found in [19]
We note that it is by far not obvious that the Goldstone boson fields acquire the same form in the PT -symmetric regime as at the exceptional point
Summary
The extension from quantum field theories with Hermitian actions to those with non-Hermitian actions has been addressed recently for various concrete systems, such as scalar field theory with imaginary cubic selfinteraction terms [1,2], field theoretical analogs to the deformed harmonic oscillator [3], non-Hermitian versions with a field theoretic Yukawa interaction [4,5,6,7], free fermion theory with a γ5-mass term and the massive Thirring model [8], PT -symmetric versions of quantum electrodynamics [9,10], and PT -symmetric quantum field theories in higher dimensions [11]. The physical subspace is bounded by the values for which the eigenvalues of the mass squared matrix acquire an exceptional point, a singularity or becomes zero It is the interplay between these two types of symmetries, continuous and discrete, that produce very interesting and novel behavior when compared to the standard Hermitian setting. While some features are the same in both approaches, e.g., both versions predict the same number of massless Goldstone bosons that is expected from Goldstone’s theorem, they differ in several aspects
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