手征反常,，英文名叫chiral anomaly, 又叫 quantum anomaly, axial anomaly, 還有叫 Adler-Bell-Jackiw (ABJ) anomaly和triangle anomaly[17]. 這些豐富的叫法都挺合適的，而且基本都沿用至今,，但是對于初學者來說卻是丈二和尚摸不著頭腦,，很是頭疼的。物理上,，手征反常是有準確定義的,，用英文來說就是” The chiral anomaly is a quantum term that violates the classical conservation of the chiral current”[25]。解釋起來就是說,，手征反常是一種量子效應,，但有趣的是并沒有經(jīng)典對應。一般為了得到一個量子理論,，我們會先得到一個經(jīng)典理論（Lagrangian）然后通過量子化的手續(xù)得到相應的量子理論,。在這個過程中，經(jīng)典體系中的對稱性也得到一個量子版本,，根據(jù)Noether定理每種對稱變換給出一個相應的守恒流（或守恒荷）,，比如熟知的Lorentz對稱性或CPT對稱等等。這些都是量子場論教科書入門的知識,，就不多贅述,。但是一個經(jīng)典的對稱性就一定存在一個量子的對稱性嗎？或者換個說法，經(jīng)典系統(tǒng)中存在兩個對稱性,，那么量子化之后,，這個兩個對稱依然可以自洽共存，不打架,？物理的世界就是充滿著矛盾,，這時手征對稱性就扮演這樣的'壞分子’。對于一個無質量的Dirac Lagrangian,，系統(tǒng)存在一個經(jīng)典場論下的手征對稱性,，也就是其左右手費米子數(shù)是分別守恒的；一旦量子化,，這個手征對稱性就會本征自發(fā)地破壞，其結果就是左右手費米子數(shù)不再保持不變,，同時手征守恒流多出個源項[6],，
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