2019/04/01 by Alireza Beygi, S. P. Klevansky, Carl M. Bender +1 · 1 citation
Mathematics · Physics and Astronomy · #Dirac equation #Dispersion relation #Eigenvalues and eigenvectors #Hamiltonian (control theory) #Hermitian matrix #Invariant (physics) #Mathematical physics #Mathematics #Neutrino Physics Research #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1103/physreva.99.062117
published as Phys. Rev. A 99, 062117 (2019) · 16 pages, 16 figures
arxiv created 2019/04/01 · openalex created_date 2019/04/11 · openalex publication_date 2019/06/24 · arxiv updated 2019/07/03 · openalex updated_date 2026/08/05
Relativistic PT-symmetric fermionic interacting systems are studied in 1+1 and 3+1 dimensions. The noninteracting Dirac equation is separately P and T invariant. The objective here is to include non-Hermitian PT-symmetric interaction terms that give rise to real spectra. Such interacting systems could be physically realistic and could describe new physics. The simplest such non-Hermitian Lagrangian density is L=L0+Lint=\ensuremathψ(i\ensuremath∂\phantom\rule-4pt0ex/\ensuremath-m)\ensuremathψ\ensuremath-g\ensuremathψ\ensuremathγ5\ensuremathψ. The associated relativistic Dirac equation is PT invariant in 1+1 dimensions and the associated Hamiltonian commutes with PT. However, the dispersion relation p2=m2\ensuremath-g2 shows that the PT symmetry is broken (the eigenvalues become complex) in the chiral limit m\ensuremath→0. For field-theoretic interactions of the form Lint=\ensuremath-g(\ensuremathψ\ensuremathγ5\ensuremathψ)N with N=2,\phantom\rule0.16em0ex3, which we can only solve approximately, we also find that if the associated (approximate) Dirac equation is PT invariant, the dispersion relation always gives rise to complex energies in the chiral limit m\ensuremath→0. Other models are studied in which x-dependent PT-symmetric potentials such as ix3, \ensuremath-x4, i\ensuremathκ/x, Hulth'en, or periodic potentials are coupled to the fermionic field \ensuremathψ using vector or scalar coupling schemes or combinations of both. For each of these models the classical trajectories in the complex-x plane are examined. Some combinations of these potentials can be solved numerically, and it is shown explicitly that a real spectrum can be obtained. In 3+1 dimensions, while the simplest system L=L0+Lint=\ensuremathψ(i\ensuremath∂\phantom\rule-4pt0ex/\ensuremath-m)\ensuremathψ\ensuremath-g\ensuremathψ\ensuremathγ5\ensuremathψ resembles the 1+1-dimensional case, the Dirac equation is not PT invariant because T2=\ensuremath-\mathbb1. This explains the appearance of complex eigenvalues as m\ensuremath→0. Other Lorentz-invariant two-point and four-point interactions are considered that give non-Hermitian PT-symmetric terms in the Dirac equation. Only the axial vector and tensor Lagrangian interactions Lint=\ensuremath-i\ensuremathψ\stackrel\ifmmode \else \~\fiB_\ensuremathμ\ensuremathγ5\ensuremathγ^\ensuremathμ\ensuremathψ and Lint=\ensuremath-i\ensuremathψT_\ensuremathμ\ensuremathν\ensuremathσ^\ensuremathμ\ensuremathν\ensuremathψ fulfill both requirements of PT invariance of the associated Dirac equation and non-Hermiticity. The dispersion relations show that both interactions lead to complex spectra in the chiral limit m\ensuremath→0. The effect on the spectrum of the additional constraint of self-adjointness of the Hamiltonian with respect to the PT inner product is investigated.