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

Abstract

The scalar field theory with potential $V(\phi )=\frac{1}{2}{m}^2{\phi }^2-\frac{1}{4}g{\phi }^4$ ($g>0$) 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 $-g{\phi }^4$ 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 $Z_{\mathrm{Herm}}(\lambda )$ of the $\lambda {\phi }^4$ ($\lambda >0$) Hermitian theory is proposed: $\log Z^{\mathcal{P}\mathcal{T}}(g)=\frac{1}{2}\log Z_{\mathrm{Herm}}(-g+\mathrm{i}{0}^{+})+\frac{1}{2}\log Z_{\mathrm{Herm}}(-g-\mathrm{i}{0}^{+})$. This relation ensures a real energy spectrum for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric $-g{\phi }^4$ 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 $E_{0,\mathrm{Herm}}(-g\pm \mathrm{i}{0}^{+})$.

7 pages, 2 figures, revtex format; v2: minor typos corrected, refs added; v3: published version; v4: two typos corrected

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• Poincaré symmetries and representations in pseudo-Hermitian quantum field theory
• Renormalisation group flows connecting a $4-\epsilon$ dimensional Hermitian field theory to a $\mathcal{P}\mathcal{T}$-symmetric theory for a fermion coupled to an axion
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• Wilsonian approach to the interaction ${\varphi }^2(i\varphi {)}^{\epsilon }$
• Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory
• Path-integral Bosonization of $d=2$ PT symmetric models
• Piecewise linear potentials for false vacuum decay and negative modes