Infinite Class of $\mathcal{PT}$-Symmetric Theories from One Timelike Liouville Lagrangian

Logarithmic timelike Liouville quantum field theory has a generalized $\mathcal{PT}$ invariance, where $\mathcal{T}$ is the time-reversal operator and $\mathcal{P}$ stands for an $S$-duality reflection of the Liouville field $\varphi$. In Euclidean space, the Lagrangian of such a theory $L=\frac{1}{2}(\nabla \varphi {)}^2-ig\varphi \exp (ia\varphi )$ is analyzed using the techniques of $\mathcal{PT}$-symmetric quantum theory. It is shown that $L$ defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer $n$. In one-dimensional space (quantum mechanics), the energy spectrum is calculated in the semiclassical limit and the $m$th energy level in the $n$th sector is given by $E_{m,n}\sim (m+1/2{)}^2{a}^2/(16n^2)$.

Figure 1

Integration paths terminating in Stokes wedges (shaded regions) for Hermitian ${\varphi }^4$ (left panel) and $\mathcal{PT}$-symmetric $-{\varphi }^4$ (right panel) interactions.

Figure 2

Integration paths in complex-$\varphi$ space for which the integral (4) converges. The paths terminate in infinitely thin Stokes wedges and define unitarily inequivalent theories.

Figure 3

Complex classical paths for the potential $V=-ix$ with $E=1$. The paths are parabolic and thus do not close.

Figure 4

Classical paths for $H=\frac{1}{2}{p}^2+e^{ix}$ at $E=1$. The classical paths are $2\pi$ periodic and open.

Figure 5

Graphical solution to Eq. (11). The solid curve is the left side of Eq. (11), and the dashed curve is the right side.

Figure 6

Left panel: Four complex classical paths for the $p^2+x^2$ oscillator with energy $E=1$. The family of paths encloses the turning points (dots) at $x=\pm 1$ and fills the entire complex plane. Right panel: Two examples of each of the three families of classical paths for the $p^2+x^6$ oscillator for $E=1$. Each family encloses one pair of turning points, and together the three families fill the complex plane.

Figure 7

Classical paths (solid curves) and separatrices (dashed curves) for $\mathcal{H}$ in Eq. (9) for $a=1$, $g=1$, and $E=1$. The central paths enclose the $n=\pm 1$ pair of turning points (dots); the next family encloses the $n=\pm 2$ pair, and so on.