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

The Solution of the Relativistic Schrodinger Equation for the δ′\delta'-Function Potential in 1-dimension Using Cutoff Regularization

We study the relativistic version of Schr\"odinger equation for a point particle in 1-d with potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultra-violet divergent, and the resultant expression cannot be renormalized in the usual sense. Therefore a general procedure has been developed to derive different physical properties of the system. The procedure is used first on the non-relativistic case for the purpose of clarification and comparisons. The results from the relativistic case show that this system behaves exactly like the delta function potential, which means it also shares the same features with quantum field theories, like being asymptotically free, and in the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point.Comment: 32 pages, 5 figure

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

Process for phosphonylating the surface of an organic polymeric preform

A process for phosphonylating the surface of an organic polymeric preform and the surface-phosphonylated preforms produced thereby are provided. Organic polymeric preforms made from various polymers including polyethylene, polyetheretherketone, polypropylene, polymethylmethacrylate, polyamides, and polyester, and formed into blocks, films, and fibers may have their surfaces phosphonylated in a liquid-phase or gas phase reaction. Liquid phase phosphonylation involves the use of a solvent that does not dissolve the organic polymeric preform but does dissolve a phosphorus halide such as phosphorus trichloride. The solvent chosen must also be nonreactive with the phosphorus halide. Such solvents available for use in the present process include the fully-halogenated liquid solvents such as carbon tetrachloride, carbon tetrabromide, and the like. Gas phase phosphonylation involves treating the organic polymeric preform with a gaseous phosphorus halide such as phosphorus trichloride and oxygen. The inventive processes allow for surface phosphonylation of the organic polymeric preform such that up to about 30 percent of the reactive carbon sites in the polymer are phosphonylated. The inventive phosphonylated organic polymers are particularly useful as orthopedic implants because hydroxyapatite-like surfaces which can be subsequently created on the organic implants allow for co-crystallization of hydroxyapatite to form chemically-bound layers between prosthesis and bone tissue