共形对称真的是反常的吗？
Is conformal symmetry really anomalous?
摘要：
相互作用量子理论的共形反常（也称为应力-能量迹线反常）与量子效应破坏外尔（共形）对称性有关，如果我们赋予该理论一个与矢量场耦合的伸缩流，这个矢量场是局域外尔对称变换的规范连接，那么这个理论就可以得到修正。这种外尔联系的自然候选者是几何挠率张量的迹，特别是如果我们回想起具有挠率的纯（Cartan-Einstein）引力是共形的。我们首先指出，规范和路径积分量化尊重外尔对称性。量子效应违反共形对称性的唯一方法是正则化过程。然而，如果我们从共形不变的经典理论中，通过使用符合Weyl对称性的正则化过程来计算有效作用量，则共形沃德恒等式将被满足。在这个意义上，外尔对称性不会被量子效应破坏。这项工作表明，外尔对称性可以与规范对称性和引力同等对待，因为无限的沃德恒等式可以保证它们不会被量子效应破坏。
英文摘要：
The conformal anomaly (also known as the stress-energy trace anomaly) of an interacting quantum theory, associated with violation of Weyl (conformal) symmetry by quantum effects, can be amended if one endows the theory with a dilatation current coupled to a vector field that is the gauge connection of local Weyl symmetry transformations. The natural candidate for this Weyl connection is the trace of the geometric torsion tensor, especially if one recalls that pure (Cartan-Einstein) gravity with torsion is conformal. We first point out that both canonical and path integral quantisation respect Weyl symmetry. The only way quantum effects can violate conformal symmetry is by the process of regularization. However, if one calculates an effective action from a conformally invariant classical theory by using a regularisation procedure that is conform with Weyl symmetry, then the conformal Ward identities will be satisfied. In this sense Weyl symmetry is not broken by quantum effects. This work suggests that Weyl symmetry can be treated on equal footing with gauge symmetries and gravity, for which an infinite set of Ward identities guarantees that they remain unbroken by quantum effects.
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