Isomorphic Hilbert spaces associated with different Complex Contours of the PT\mathcal{PT}-Symmetric (−x4)(-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.Comment: In this version, we have added many details and omitted confusing statements. Also, the way we present the main idea has been changed and added two figure

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