Title: Regularized interacting scalar quantum field theories Show Chinese title

Title: 正则化的相互作用标量量子场理论

Abstract: In this paper we consider self interacting scalar quantum field theories over a $d$ dimensional Minkowski spacetime with various interaction Lagrangians which are suitable functions of the field. The interacting field observables are represented as power series over the free theory by means of perturbation theory. The object which is employed to obtain this power series is the time ordered exponential of the interaction Lagrangian which is the $S$-matrix of the theory and thus itself a power series in the coupling constant of the theory. We analyze a regularization procedure which makes the $S$-matrix convergent to well defined unitary operators. This regularization depends on two parameters. One describes how much the high frequency contributions in the propagators are tamed and a second one which describes how much the large field contributions are suppressed in the interaction Lagrangian. We finally discuss how to remove the parameters in lower dimensional theories and for specific interaction Lagrangians. In particular, we show that in three spacetime dimensions for a $\phi^4_3$ theory one obtains sequences of unitary operators which are weakly-$*$ convergent to suitable unitary operators in the limit of vanishing parameters. The coefficients of the asymptotic expansion in powers of the coupling constant of all the possible limit points coincide and furthermore agree with the predictions of perturbation theory. Finally we discuss how to extend these results to the case of a $\phi^4_4$ theory were the final results turns out to be very similar to the three dimensional case. Show Chinese abstract

Abstract: 在本文中，我们考虑在$d$维闵可夫斯基时空上的自相互作用标量量子场论，其相互作用拉格朗日量是场的适当函数。 相互作用场可观测量通过微扰理论表示为自由理论的幂级数。 用来获得这个幂级数的对象是相互作用拉格朗日量的时间有序指数，这是理论的$S$矩阵，因此本身也是理论耦合常数的幂级数。 我们分析了一种正则化过程，使$S$矩阵收敛到定义良好的酉算子。 这种正则化依赖于两个参数。 一个参数描述了传播子中的高频贡献被抑制的程度，另一个参数描述了相互作用拉格朗日量中的大场贡献被抑制的程度。 最后，我们讨论如何在低维理论和特定的相互作用拉格朗日量中去除这些参数。 特别是，我们证明在三维时空中的$\phi^4_3$理论中，可以得到一系列弱-$*$收敛到消失参数极限下合适酉算子的酉算子。 所有可能极限点的耦合常数幂级数渐近展开的系数都一致，并且进一步与微扰理论的预测一致。 最后，我们讨论如何将这些结果扩展到$\phi^4_4$理论的情况，其中最终结果与三维情况非常相似。