Abstract
We present an overview of some key results obtained in a recent series devoted to non-Hermitian quantum field theories for which we systematically modify the underlying symmetries. Particular attention is placed on the interplay between the continuous symmetry group that we alter from global to local, from Abelian to non-Abelian, from rank one to generic rank N, and the discrete anti-linear modified CPT-symmetries. The presence of the latter guarantees the reality of the mass spectrum in a certain parameter regime. We investigate the extension of Goldstone’s theorem and the Higgs mechanism, which we demonstrate to work in the conventional fashion in the CPT-symmetric regime, but which needs to be modified technically at the standard exceptional points of the mass spectrum and entirely fails at what we refer to as zero exceptional points as well as in the broken CPT-symmetric regime. In the full non-Hermitian non-Abelian gauge theory we identify the t’Hooft-Polyakov monopoles by means of a fourfold Bogomol’nyi-Prasad-Sommerfield (BPS) limit. We investigate this limit further for other types of non-Hermitian field theories in 1+1 dimensions that possess complex super-exponential and inverse hyperbolic kink/anti-kink solutions and for 3+1 dimensional Skyrme models for which we find new types of complex solutions, that all have real energies due to the presence of different types of CPT-symmetries.
Highlights
Abelian and non-Abelian gauge field theories as manifested in quantum electrodynamics and the standard model, respectively, play a central role in particle physics and quantum field theory
As shown in detail in [4], in the CPT -symmetric regime the Goldstone theorem holds in the usual sense and the fields corresponding to the Goldstone bosons can be identified explicitly
In [7] we showed that the mechanism is extendable to non-Hermitian theories, di↵erently in di↵erent regimes characterized by the CPT -symmetry as we shall discuss
Summary
Abelian and non-Abelian gauge field theories as manifested in quantum electrodynamics and the standard model, respectively, play a central role in particle physics and quantum field theory. We report on a series of studies [4, 5, 6, 7, 8, 9] that explore certain sectors of the standard model to allow for the possibility of non-Hermitian gauge field theories. This approach in field theory is inspired by the success of non-Hermitian extensions to quantum mechanics. We start by investigating how the central concepts in standard Hermitian quantum field theory, such as how the Higgs mechanism builds on the Goldstone theorem, are extended to non-Hermitian theories.
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