$\mathcal{P}\mathcal{T}$-symmetric $-g\varphi^4$ theory

• URL: http://arxiv.org/abs/2209.07897v4
• Date: Thu, 5 Jan 2023 08:26:09 GMT
• Title: $\mathcal{P}\mathcal{T}$-symmetric $-g\varphi^4$ theory
• Authors: Wen-Yuan Ai, Carl M. Bender and Sarben Sarkar
• Abstract summary: Hermitian theory is proposed: $log ZmathcalPmathcalT(g)=textstylefrac12 log Z_rm Herm(-g+rm i 0+rm i 0+$. A new conjectural relation between the Euclidean partition functions $ZmathcalPmathcalT$-symmetric theory and $Z_rm Herm(lambda)$ of the $lambda varphi4$ is presented.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The scalar field theory with potential $V(\varphi)=\textstyle{\frac{1}{2}} m^2\varphi^2-\textstyle{\frac{1}{4}} g\varphi^4$ ($g>0$) 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 $-g\varphi^4$ 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 $Z_{\rm Herm}(\lambda)$ of the $\lambda \varphi^4$ ($\lambda>0$) Hermitian theory is proposed: $\log Z^{\mathcal{P}\mathcal{T}}(g)=\textstyle{\frac{1}{2}} \log Z_{\rm Herm}(-g+{\rm i} 0^+)+\textstyle{\frac{1}{2}}\log Z_{\rm Herm}(-g-{\rm i} 0^+)$. This relation ensures a real energy spectrum for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric $-g\varphi^4$ 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 $E_{0,\rm Herm}(-g\pm {\rm i} 0^+)$.

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