arXiv subjects

Abouzeid Shalaby

Publications and source records attributed to Abouzeid Shalaby.

4 recordsLinked to original sources

arXiv2015

Linked $\mathcal{PT }$-symmetry to Supersymmetry in a class of non-Hermitian Hamiltonians

We introduce and study a class of non-Hermitian Hamiltonians which have velocity dependent potentials. Since stability can not be advocated directly from the classical potential, we show that the energy spectra are real and bounded from below which proves the stability of the spectra of all members in the class. We find that the introduced class of non-Hermitian Hamiltonians do have a corresponding superpartner class of non-Hermitian Hamiltonians. We were able to introduce supercharges which in conjunction with the corresponding super Hamiltonians constitute a closed super algebra. Among the introduced Hamiltonians, we show that non-$\mathcal{PT }$-symmetric Hamiltonians can be transformed into their corresponding superpartner Hamiltonians via a specific canonical transformation while the $\mathcal{PT }$-symmetric ones failed to be mapped to their corresponding superpartner Hamiltonians via the same canonical transformation. Since canonical transformations preserve the spectrum, we conclude that non-$\mathcal{PT }$-symmetric Hamiltonians out of the introduced class of Hamiltonians have the same spectrum as the corresponding superpartner Hamiltonians and thus Susy is broken for such Hamiltonians. This kind of intertwining of $\mathcal{PT }$-symmetry and Supersymmetry is new as all the so far discussed cases concentrate on Hamiltonians of broken $\mathcal{PT }% $-symmetry that have broken Supersymmetry too while we showed that Susy can be also broken for non-$\mathcal{PT }$-symmetric and non-Hermitian Hamiltonians .

quant-ph↗

arXiv2014

Isomorphic Hilbert spaces associated with different Complex Contours of the $\mathcal{PT}$-Symmetric $(-x^{4}) $ Theory

In this work, we stress the existence of isomorphisms which map complex contours from the upper half to contours in the lower half of the complex plane. The metric operator is found to depend on the chosen contour but the maps connecting different contours are norm-preserving. To elucidate these features, we parametrized the contour $z=-2i\sqrt{1+ix}$ considered in Phys.Rev.D73:085002 (2006) for the study of wrong sign $x^{4}$ theory. For the parametrized contour of the form $z=a\sqrt{b+i c x}$, we found that there exists an equivalent Hermitian Hamiltonian provided that $a^{2} c$ is taken to be real. The equivalent Hamiltonian is $b$-independent but the metric operator is found to depend on all the parameters $a$, $b$ and $c$. Different values of these parameters generate different metric operators which define different Hilbert spaces . All these Hilbert spaces are isomorphic to each other even for parameters values that define contours with ends in two adjacent wedges. As an example, we showed that the transition amplitudes associated with the contour $z=-2i\sqrt{1+ix}$ are exactly the same as those calculated using the contour $z=\sqrt{1+ix}$, which is not $\mathcal{PT}$-Symmetric and has ends in two adjacent wedges in the complex plane.

math-ph↗

arXiv2010

Non-perturbative tests for the Asymptotic Freedom in the $\mathcal{PT}% $-symmetric $(-ϕ^{4})_{3+1}$ theory

In the literature, the asymptotic freedom property of the $(-ϕ^{4})$ theory is always concluded from real-line calculations while the theory is known to be a non-real-line one. In this article, we test the existence of the asymptotic freedom in the $(-ϕ^{4})_{3+1}$ theory using mean field approach. In this approach and contrary to the original Hamiltonian, the obtained effective Hamiltonian is rather a real-line one. Accordingly, this work resembles the first reasonable analysis for the existence of the asymptotic freedom property in the $\mathcal{PT}$-symmetric $(-ϕ^{4})$ theory. In this respect, we calculated three different amplitudes of different positive dimensions (in mass units) and find that all of them goes to very small values at high energy scales (small coupling) in agreement with the spirit of the asymptotic freedom property of the theory. To test the validity of our calculations, we obtained the asymptotic behavior of the vacuum condensate in terms of the coupling, analytically, and found that the controlling factor $Λ$ has the value $\frac{(4 π)^{2}}{6}= 26. 319$ compared to the result $Λ=26.3209$ from the literature which was obtained via numerical predictions. We assert that the non-blow up of the massive quantities at high energy scales predicted in this work strongly suggests the possibility of the solution of the famous hierarchy puzzle in a standard model with $\mathcal{PT}$-symmetric Higgs mechanism.

hep-th↗

arXiv2009

Hermiticity Breaking and Restoration in the $(gϕ^4 + hϕ^6)_{1+1}$ Field Theoretic Model

We introduce hermiticity as a new symmetry and show that when starting with a model which is Hermitian in the classical level, quantum corrections can break hermiticity while the theory stay physically acceptable. To show this, we calculated the effective potential of the ($gϕ^{4}+hϕ^{6}$)$_{1+1}$ model up to first order in $g$ and $h$ couplings which is sufficient as the region of interest has finite correlation length for which mean field calculation may suffice. We show that, in the literature, there is a skipped phase of the theory due to the wrong believe that the theory in the broken hermiticity phase is unphysical. However, in view of recent discoveries of the reality of the spectrum of the non-Hermitian but $PT$ symmetric theories, in the broken hermiticity phase the theory possesses $PT$ symmetry and thus physically acceptable. In fact, ignoring this phase will lead to violation of universality when comparing this model predictions with other models in the same class of universality.

hep-th↗