$\mathcal{PT}% $ -对称（$-x^{4}$）势能谱的有效场计算
Effective Field calculations of the Energy Spectrum of the $\mathcal{PT}% $-Symmetric ($-x^{4}$) Potential
摘要：
在这项工作中，我们表明，传统的有效场方法可以应用到$\mathcal{PT}$ -对称错误符号（$-x^{4}$）四次势。这项工作的重要性在于它的可能性扩展到更重要的$\mathcal{PT}$ -对称量子场论，而其他方法，使用复杂的轮廓是不愿意适用。我们计算了无质量$-x^{4}$理论的有效势以及理论的全谱。虽然计算进行了一阶耦合，预测的光谱是非常接近的确切的一个从其他作品。本文最重要的结果是得到的等效于高斯有效势的有效势是从下有界的，而经典势是从上有界的。这就解释了真空理论的稳定性。所得到的准粒子哈密顿量是非厄米但$\mathcal{PT}$ -对称的，并且我们证明了度规算符的计算可以微扰进行。事实上，度规算符的计算甚至可以在更高的维度（量子场论）中进行，到目前为止，由于可能出现场根，这些维度在其他方法中无法微扰计算或以封闭形式计算。此外，我们认为，有效的理论是微扰的耦合常数的整个范围内和微扰级数预计迅速收敛（有效耦合$g_{eff}={1/6}$）。
英文摘要：
In this work, we show that the traditional effective field approach can be applied to the $\mathcal{PT}$-symmetric wrong sign ($-x^{4}$) quartic potential. The importance of this work lies in the possibility of its extension to the more important $\mathcal{PT}$-symmetric quantum field theory while the other approaches which use complex contours are not willing to be applicable. We calculated the effective potential of the massless $-x^{4}$ theory as well as the full spectrum of the theory. Although the calculations are carried out up to first order in the coupling, the predicted spectrum is very close to the exact one taken from other works. The most important result of this work is that the effective potential obtained, which is equivalent to the Gaussian effective potential, is bounded from below while the classical potential is bounded from above. This explains the stability of the vacuum of the theory. The obtained quasi-particle Hamiltonian is non-Hermitian but $\mathcal{PT}$-symmetric and we showed that the calculation of the metric operator can go perturbatively. In fact, the calculation of the metric operator can be done even for higher dimensions (quantum field theory) which, up till now, can not be calculated in the other approaches either perturbatively or in a closed form due to the possible appearance of field radicals. Moreover, we argued that the effective theory is perturbative for the whole range of the coupling constant and the perturbation series is expected to converge rapidly (the effective coupling $g_{eff}={1/6}$).
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