November 2012
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Day November 5, 2012

PT-Symmetric and non-Hermitian theories outside their Stokes Wedges

Abouzeid Shalaby

Based on the realization that, in \(\mathcal{PT}\)-symmetric quantum mechanics, the analytic continuation of the eigen value problem into the complex plane is equivalent to the known canonical point transformation, we raise the question why then a theory selects some specific canonical transformations represented by contours within the Stokes wedges of the theory and rejects others represented by contours outside the Stokes wedges? To answer this question, we show that the transition amplitudes are the same either calculated within or out of the Stokes wedges but with related metric operators. To illustrate our idea, we reinvestigated the \(\mathcal{PT}\)-symmetric \(-x^{4}\) theory by selecting a complex contour outside the Stokes wedges. Following orthogonal polynomials studies, we were able to reproduce exactly the same equivalent Hermitian Hamiltonian obtained before in the literature. Since the metric is implicit in algorithms employing the Heisenberg picture, we assert the importance of this trend for the research in \(\mathcal{PT}\)-symmetric field theories. Regarding this, we select a simple \(Z_{2}\) symmetry breaking contour, regardless of being inside or outside the Stokes wedges, to investigate the \(\mathcal{PT}\)-symmetric \(-\phi^{4}\) field theory. We follow the famous effective action approach, up to two loops, to obtain very accurate results for the vacuum energy and vacuum condensate compared to previous calculations carried out for the equivalent Hermitian theory.
Mathematical Physics (math-ph)