Massless Scalar Field Theory
Free, massless quantized scalar field theory has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the Gaussian fixed point.
However, even though the classical massless φ4 theory is scale-invariant in, the quantized version is not scale-invariant. We can see this from the beta-function for the coupling parameter, g.
Even though the quantized massless φ4 is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the Wilson-Fisher fixed point.
Read more about this topic: Scale Invariance, Scale Invariance in Quantum Field Theory
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