New Ansatz for Metric Operator Calculation in Pseudo-Hermitian Field Theory

In this work, a new ansatz is introduced to make the calculations of the metric operator in Pseudo-Hermitian field theory simpler. The idea is to assume that the metric operator is not only a functional of the field operators $\phi$ and its conjugate field $\pi$ but also on the field gradient $\nabla\phi$. Rather than the locality of the metric operator obtained, the ansatz enables one to calculate the metric operator just once for all dimensions of the space-time. We calculated the metric operator of the $i\phi^{3}$ scalar field theory up to first order in the coupling. The higher orders can be conjectured from their corresponding operators in the quantum mechanical case available in the literature. We assert that, the calculations existing in literature for the metric operator in field theory are cumbersome and are done case by case concerning the dimension of space-time in which the theory is investigated. Moreover, while the resulted metric operator in this work is local, the existing calculations for the metric operator leads to a non-local one. Indeed, we expect that the new results introduced in this work will greatly lead to the progress of the studies in Pseudo-Hermitian field theories where there exist a lack of such kind of studies in the literature. In fact, with the aid of this work a rigorous study of a $\mathcal{PT}$-symmetric Higgs mechanism can be reached.Comment: In this version, for a more illustrative presentation, we used the i\phi^3 theory to show that the new ansatz introduced is applicabl

Vacuum Stability of the PT\mathcal{PT}-Symmetric (−ϕ4)\left( -\phi^{4}\right) Scalar Field Theory

In this work, we study the vacuum stability of the classical unstable $\left( -\phi^{4}\right)$ scalar field potential. Regarding this, we obtained the effective potential, up to second order in the coupling, for the theory in $1+1$ and $2+1$ space-time dimensions. We found that the obtained effective potential is bounded from below, which proves the vacuum stability of the theory in space-time dimensions higher than the previously studied $0+1$ case. In our calculations, we used the canonical quantization regime in which one deals with operators rather than classical functions used in the path integral formulation. Therefore, the non-Hermiticity of the effective field theory is obvious. Moreover, the method we employ implements the canonical equal-time commutation relations and the Heisenberg picture for the operators. Thus, the metric operator is implemented in the calculations of the transition amplitudes. Accordingly, the method avoids the very complicated calculations needed in other methods for the metric operator. To test the accuracy of our results, we obtained the exponential behavior of the vacuum condensate for small coupling values, which has been obtained in the literature using other methods. We assert that this work is interesting, as all the studies in the literature advocate the stability of the $\left( -\phi^{4}\right)$ theory at the quantum mechanical level while our work extends the argument to the level of field quantization.Comment: 20 pages, 4 figures, appendix added and more details have been added to

Effective Field calculations of the Energy Spectrum of the PT\mathcal{PT}% -Symmetric (−x4-x^{4}) Potential

In this work, we show that the traditional effective field approach can be applied to the $\mathcal{PT}$-symmetric wrong sign ($-x^{4}$) quartic potential. The importance of this work lies in the possibility of its extension to the more important $\mathcal{PT}$-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 $\mathcal{PT}$-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 $g_{eff}={1/6}$).Comment: 14 pages, 5 figure

Possible treatment of the Ghost states in the Lee-Wick Standard Model

Very recently, the Lee-Wick standard model has been introduced as a non-SUSY extension of the Standard model which solves the Hierarchy problem. In this model, each field kinetic term attains a higher derivative term. Like any Lee-Wick theory, this model suffers from existence of Ghost states. In this work, we consider a prototype scalar field theory with its kinetic term has a higher derivative term, which mimics the scalar sector in the Lee-Wick Standard model. We introduced an imaginary auxiliary field to have an equivalent non-Hermitian two-field scalar field theory. We were able to calculate the positive definite metric operator $\eta$ in quantum mechanical and quantum field versions of the theory in a closed form. While the Hamiltonian is non-Hermitian in a Hilbert space with the Dirac sense inner product, it is Hermitian in a Hilbert space endowed by the inner product $$ as well as having a correct-sign propagator (no Lee-Wick fields). Besides, the obtained metric operator also diagonalizes the Hamiltonian in the two fields (no mixing). Moreover, the Hermiticity of $\eta$ constrained the two Higgs masses to be related as $M>2m$, which has been obtained in another work using a very different regime and thus supports our calculation. Also, an equivalent Hermitian (in the Dirac sense) Hamiltonian is obtained which has no Ghost states at all, which is a forward step to make the Lee-Wick theories more popular among the Physicists.Comment: 11 pages, 3 figure

Exact Critical Exponents of the Yang-Lee Model from Large-Order Parameters

Based on the Large-Order behavior of the perturbation series of the ground state energy of Yang-Lee model we suggested a Hypergeometric approximants that can mimic the same Large-order behavior of the given series. Near the branch point, the Hypergeometric function $_{p}F_{p-1}$ has a power law behavior from which the critical exponent and critical coupling can be extracted. While the resummation algorithm shows almost exact predictions for the ground state energy from law orders of perturbation series as input, we found that the exact critical exponents are solely determined by one of the parameters in the large order behavior of the series. Based on this result we conjecture that the Large-order parameters might know the exact critical exponents. Since the ground state energy is the generating functional of the 1-P irreducible amplitudes, one gets all the critical exponents via functional differentiation with respect to the external magnetic field.Comment: 13 pages, two table