超越微扰理论：参量微扰理论
Going beyond perturbation theory: Parametric Perturbation Theory
摘要：
我们设计了一种{ \sl非微扰}方法，称为{ \sl参数微扰理论}（PPT），它是普通微扰理论的替代。该方法依赖于一个简单的原则，可观察到的解决方案，这是限制在一定的线性（非物理）参数。微扰展开是在这个参数中进行的，而不是在物理耦合中（如在普通微扰理论中）。我们提供了一些非平凡的例子，我们的方法是能够恢复的发散微扰系列，{ \sl提取}领先的渐近（强耦合）的行为和{ \sl预测}具有高精度的系数的微扰系列。在零维场论的情况下，我们证明了PPT可以用来提供解决方案的虚部，当问题解析地继续负耦合。在$ϕ^4$晶格模型1+1和弹性理论的情况下，我们已经证明，用PPT表示的观测值在耦合的有限值处显示分支点，标志着从稳定到亚稳态的转变。我们还将该方法应用于二维和三维硬球气体维里系数的预测;在这个例子中，我们还发现用PPT表示的解在有限密度下具有奇异性。对未知维里系数进行了预测。
英文摘要：
We devise a {\sl non--perturbative} method, called {\sl Parametric Perturbation Theory} (PPT), which is alternative to the ordinary perturbation theory. The method relies on a principle of simplicity for the observable solutions, which are constrained to be linear in a certain (unphysical) parameter. The perturbative expansion is carried out in this parameter and not in the physical coupling (as in ordinary perturbation theory). We provide a number of nontrivial examples, where our method is capable to resum the divergent perturbative series, {\sl extract} the leading asymptotic (strong coupling) behavior and {\sl predict} with high accuracy the coefficients of the perturbative series. In the case of a zero dimensional field theory we prove that PPT can be used to provide the imaginary part of the solution, when the problem is analytically continued to negative couplings. In the case of a $ϕ^4$ lattice model 1+1 and of elastic theory we have shown that the observables resummed with PPT display a branch point at a finite value of the coupling, signaling the transition from a stable to a metastable state. We have also applied the method to the prediction of the virial coefficients for a hard sphere gas in two and three dimensions; in this example we have also found that the solution resummed with PPT has a singularity at finite density. Predictions for the unknown virial coefficients are made.
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