在量子力學和量子場論中的Resurgence

Abstract

本論文總結了關於應用resurgence trans-series 在量子力學和量子場論的知識。我們重現了在微擾級數中的虛數項和瞬子-反瞬子對中的虛數項的相消。結合所有來自非微擾半古典組態的貢獻之後，就可以構築出所謂的trans-series。因為紅外重整子的存在，在量子場論中構建trans-series被認為是無法實現的。近年來，因為在某些被緊湊化後的量子場論中發現了新的半古典組態，這些半古典組態被認為可能對應到紅外重整子。利用trans-series來給部分量子場論在弱藕荷常數極限下一個非微擾的定義可能是可以實現的。

This thesis reviews the idea of resurgence trans-series and its application on Quantum mechanics and Quantum field theory. The cancellation between the imaginary part of the perturbative series and the imaginary part of the instanton-anti-instanton configuration in quantum mechanics is shown. Combining all the contribution from the non-perturbative semiclassical configuration, the so-called trans-series can be constructed. Because of the existence of IR renormalon, the construction of trans-series in QFT was not thought to be valid. In recent years, new semiclassical configurations have been found in some compactified theories. Those configurations are thought to be corresponded to the elusive IR renormalons. A non-perturbative definition of a class of field theories by the trans-series in the weak coupled limit may be possible.