Abstract
In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in $D$-dimensional Euclidean spacetime, where $\phi$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $\varepsilon$ and were calculated to first order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= \frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\mu^2\phi^2+\frac{1}{2} g\mu_0^2\phi^2\big(i\mu_0^{1-D/2}\phi\big)^\varepsilon-iv\phi$, which now includes renormalization counterterms that are linear and quadratic in the field $\phi$. The parameter $g$ is a dimensionless coupling strength and $\mu_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq D\leq2$). Renormalization is performed perturbatively to second order in $\varepsilon$ and the structure of the Green's functions is analyzed in the limit $D\to 2$. A sum of the most divergent terms is performed to {\it all} orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.
Highlights
Since the publication of the first paper on PT symmetry in 1998 [1], in which the PT -symmetric quantummechanical HamiltonianH 1⁄4 p2 þ x2ðixÞε ð1Þ was introduced, this research area has become highly active
Was introduced, this research area has become highly active. This model has been studied in detail [2], and much theoretical research has been done on the mathematical structure of non-Hermitian quantum systems [3]
Beautiful experiments have been performed in diverse areas of physics including optics, photonics, lasers, mechanical and electrical analogs, graphene, topological insulators, superconducting wires, atomic diffusion
Summary
Since the publication of the first paper on PT symmetry in 1998 [1], in which the PT -symmetric quantummechanical Hamiltonian. It may seem that the PT -symmetric quartic theory obtained by replacing g with −g is problematic because the perturbation series no longer alternates in sign: If we Borel-sum the series, we find a cut in the Borel plane, which suggests that E is complex and that the vacuum state is unstable, as one might intuitively expect with an upsidedown potential Our objective is to calculate Green’s functions for this quantum field theory as a series in powers of the parameter ε and to carry out perturbative renormalization for the two-dimensional case. This is a nontrivial extension of the earlier work in which the Green’s functions were calculated to leading order in powers of ε [18] for the dimensionless Lagrangian density (2).
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