量子場論中的一些課題

Abstract

量子場論中有許多值得研究的課題，在這□我們選擇其中兩個課題來進行研究。第一部份我們先利用Vilkovisky-DeWitt方法去求規範無關等效位能，若我們要利用規範相關等效位能做進一步的研究，我們就要考慮因不同的規範可能會得到不同的結果，而規範無關等效位能就沒有這問題，接下來我們研究一個簡化的標準模型，並利用此玩具模型得到的結果去說明如何求取希格斯粒子質量界限；第二部份我們從施勻格傳播子（Schwinger propagator）出發，在背景磁場為常數的情況下，求取帶電費米傳播子（charged-fermion propagator）在弱磁場極限下的展開式，而我們也發展了一個方法，可以從費曼圖一階一階的得到電子傳播子在弱磁場背景下展開式的每一項，另外我們也發現完整的電子傳播子中還包括相位因子，而相位因子會對三個頂點的圈圖產生貢獻。

There are many topics worth studying on quantum field theory. In this thesis, we select two of them to study. In the first part, we use the Vilkovisky-DeWitt method to obtain the gauge-independent effective potential. If we study further by the gauge-dependent effective potential, we must consider we might get the different results when we choose the different gauge. But there is no problem if we use the gauge-independent effective potential. We consider the simplified standard model and use the result.in this toy model to show how to get the Higgs boson mass bound. In the second part, we begin with the Schwinger propagator in a constant background field to derive the expansion of the charged-fermion propagator under the weak magnetic field limit. And we develop a method from Feynman diagram to obtain the electron propagator in the weak magnetic field order by order. Otherwise, we find there is a phase factor in the full electron propagator. The phase factor gives contribution to the loop diagram with three vertices.