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The φ4 theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be anomalous. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called cosmic inflation, as long as the theory can be studied through perturbation theory.

The φ4 theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be anomalous. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called cosmic inflation, as long as the theory can be studied through perturbation theory.

上面的 φ4理论例子表明，即使相应的经典场论是尺度不变的(或共形不变的) ，量子场论的耦合参数也可以是尺度相关的。如果是这种情况，经典尺度(或共形)不变性被称为反常的。一个经典的尺度不变场理论，其中尺度不变性被量子效应打破，提供了早期宇宙近乎指数膨胀的解释，称为宇宙暴涨，只要该理论可以通过摄动理论研究。

上面的φ⁴理论例子表明，量子场论的耦合参数可以是标度依赖的，即使相应的经典场论是标度不变(或共形不变)。如果是这种情况，则称经典标度(或共形)不变性为异常。经典的标度不变场论，当量子效应打破其中的标度不变性，可以为接近指数级膨胀的早期宇宙提供了一种解释，即为宇宙膨胀，只要该理论可以通过微扰理论研究。

==Phase transitions 相变==

==Phase transitions 相变==