<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:math>-symmetric<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo>−</mml:mo><mml:mi>g</mml:mi><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>theory
Wen-Yuan Ai, Carl M. Bender, Sarben Sarkar
Abstract
The scalar field theory with potential $V(\ensuremath{\varphi})=\frac{1}{2}{m}^{2}{\ensuremath{\varphi}}^{2}\ensuremath{-}\frac{1}{4}g{\ensuremath{\varphi}}^{4}$ ($g>0$) is ill defined as a Hermitian theory but in a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric framework it is well defined, and it has a positive real energy spectrum for the case of spacetime dimension $D=1$. While the methods used in the literature do not easily generalize to quantum field theory, in this paper the path-integral representation of a $\mathcal{P}\mathcal{T}$-symmetric $\ensuremath{-}g{\ensuremath{\varphi}}^{4}$ theory is shown to provide a unified formulation for general $D$. A new conjectural relation between the Euclidean partition functions ${Z}^{\mathcal{P}\mathcal{T}}(g)$ of the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric theory and ${Z}_{\text{Herm}}(\ensuremath{\lambda})$ of the $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ ($\ensuremath{\lambda}>0$) Hermitian theory is proposed: $\mathrm{log}{Z}^{\mathcal{P}\mathcal{T}}(g)=\frac{1}{2}\mathrm{log}{Z}_{\text{Herm}}(\ensuremath{-}g+\mathrm{i}{0}^{+})+\frac{1}{2}\mathrm{log}{Z}_{\text{Herm}}(\ensuremath{-}g\ensuremath{-}\mathrm{i}{0}^{+})$. This relation ensures a real energy spectrum for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric $\ensuremath{-}g{\ensuremath{\varphi}}^{4}$ field theory. A closely related relation is rigorously valid in $D=0$. For $D=1$, using a semiclassical evaluation of ${Z}^{\mathcal{P}\mathcal{T}}(g)$, this relation is verified by comparing the imaginary parts of the ground-state energy ${E}_{0}^{\mathcal{P}\mathcal{T}}(g)$ (before cancellation) and ${E}_{0,\mathrm{Herm}}(\ensuremath{-}g\ifmmode\pm\else\textpm\fi{}\mathrm{i}{0}^{+})$.