Effective Field calculations of the Energy Spectrum of the $\mathcal{P}\mathcal{T}$-Symmetric ($-x^4$) Potential

Abstract

In this work, we show that the traditional effective field approach can be applied to the $\mathcal{P}\mathcal{T}$-symmetric wrong sign ($-x^4$) quartic potential. The importance of this work lies in the possibility of its extension to the more important $\mathcal{P}\mathcal{T}$-symmetric quantum field theory while the other approaches which use complex contours are not willing to be applicable. We calculated the effective potential of the massless $-x^4$ theory as well as the full spectrum of the theory. Although the calculations are carried out up to first order in the coupling, the predicted spectrum is very close to the exact one taken from other works. The most important result of this work is that the effective potential obtained, which is equivalent to the Gaussian effective potential, is bounded from below while the classical potential is bounded from above. This explains the stability of the vacuum of the theory. The obtained quasi-particle Hamiltonian is non-Hermitian but $\mathcal{P}\mathcal{T}$-symmetric and we showed that the calculation of the metric operator can go perturbatively. In fact, the calculation of the metric operator can be done even for higher dimensions (quantum field theory) which, up till now, can not be calculated in the other approaches either perturbatively or in a closed form due to the possible appearance of field radicals. Moreover, we argued that the effective theory is perturbative for the whole range of the coupling constant and the perturbation series is expected to converge rapidly (the effective coupling $g_{eff}=1/6$).

arXiv comment14 pages, 5 figures

References in corpus (5)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• The disappearing $Q$ operator
• A Novel phase in the phase structure of the $(g{\varphi }^4+h{\varphi }^6{)}_{1+1}$ field theoretic model

Cited by in corpus (5)

• Non-perturbative tests for the Asymptotic Freedom in the $\mathcal{P}\mathcal{T}$-symmetric $(-{\varphi }^4{)}_{3+1}$ theory
• New Ansatz for Metric Operator Calculation in Pseudo-Hermitian Field Theory
• Isomorphic Hilbert spaces associated with different Complex Contours of the $\mathcal{P}\mathcal{T}$-Symmetric $(-x^4)$ Theory
• Vacuum Stability of the $\mathcal{P}\mathcal{T}$-Symmetric $\left(-{\varphi }^4\right)$ Scalar Field Theory
• Vacuum Stability of the wrong sign $(-{\varphi }^6)$ Scalar Field Theory