Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions

Abstract

Relativistic PT-symmetric fermionic interacting systems are studied in 1+1 and 3+1 dimensions. The objective is to include non-Hermitian PT-symmetric interaction terms that give {\it real} spectra. Such interacting systems could describe new physics. The simplest non-Hermitian Lagrangian density is $L=L_0+L_{int}=\bar{\psi }(i/\partial -m)\psi -g\bar{\psi }{\gamma }^5\psi$. The associated relativistic Dirac equation is PT invariant in 1+1 dimensions and the associated Hamiltonian commutes with PT. However, the dispersion relation $p^2=m^2-g^2$ shows that the PT symmetry is broken in the chiral limit $m\to 0$. For interactions $L_{int}=-g(\bar{\psi }{\gamma }^5\psi {)}^N$ with N=2,3, if the associated Dirac equation is PT invariant, the dispersion relation gives complex energies as $m\to 0$. Other models are studied in which x-dependent PT-symmetric potentials such as $i{x}^3$, $-x^4$, $i\kappa /x$, Hulthén, or periodic potentials are coupled to $\psi$ and the classical trajectories plane are examined. Some combinations of these potentials give a real spectrum. In 3+1 dimensions, the simplest system $L=L_0+L_{int}=\bar{\psi }(i/\partial -m)\psi -g\bar{\psi }{\gamma }^5\psi$ resembles the 1+1-dimensional case but the Dirac equation is not PT invariant because $T^2=-1$. This explains the appearance of complex eigenvalues as $m\to 0$. Other Lorentz-invariant 2-point and 4-point interactions give non-Hermitian PT-symmetric terms in the Dirac equation. Only the axial vector and tensor Lagrangian interactions $L_{int}=-i\bar{\psi }{\tilde{B}}_{\mu }{\gamma }^5{\gamma }^{\mu }\psi$ and $L_{int}=-i\bar{\psi }{T}_{\mu \nu }{\sigma }^{\mu \nu }\psi$ fulfil both requirements of PT invariance of the associated Dirac equation and non-Hermiticity. Both models give complex spectra as $m\to 0$. The effect on the spectrum of the additional constraint of selfadjointness of the Hamiltonian with respect to the PT inner product is investigated.

16 pages, 16 figures

References in corpus (4)

• PT-symmetry breaking and laser-absorber modes in optical scattering systems
• Bifurcation diagram and pattern formation in superconducting wires with electric currents
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Symmetries and conservation laws in non-Hermitian field theories

Cited by in corpus (17)

• On the consistency of a non-Hermitian Yukawa interaction
• PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation
• Spontaneous Non-Hermiticity in Nambu-Jona-Lasinio model
• Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions
• Discrete spacetime symmetries, second quantization, and inner products in a non-Hermitian Dirac fermionic field theory
• Non-Hermitian BCS-BEC crossover of Dirac fermions
• Poincaré symmetries and representations in pseudo-Hermitian quantum field theory
• The phase diagram and vortex properties of PT-symmetric non-Hermitian two-component superfluid
• Non-Hermitian Yukawa interactions of fermions with axions: potential microscopic origin and dynamical mass generation
• Non-Hermitian Quantum Quenches in Holography
• Spontaneous non-Hermiticity in the (2+1)-dimensional Gross-Neveu model
• Spontaneous non-Hermiticity in the (2+1)-dimensional Thirring model
• Fermion and meson mass generation in non-Hermitian Nambu--Jona-Lasinio models
• Wilsonian approach to the interaction ${\varphi }^2(i\varphi {)}^{\epsilon }$
• Thermodynamic properties of non-Hermitian Nambu--Jona-Lasinio models
• Continuous quantum phase transition in the fermionic mass solutions of the Nambu-Jona-Lasinio model
• Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory