I have a short question about the transformation laws of quantum fields. Although I have come across similar questions, I haven't been able to understand this explicitly.
In a QFT, we postulate that field operators transform (under e.g. Lorentz) according to \begin{align} U^\dagger(\Lambda) \phi(x) U(\Lambda) = R(\Lambda) \phi(\Lambda^{-1}(x)). \end{align} Here, $U(\Lambda)$ is a unitary rep transforming the operator $\phi(x)$. As far as I understand, $R(\Lambda)$ is the 'classical rep', i.e. the rep that would act on the classical field $\phi(x)$.
My question is that I don't quite understand how to understand the RHS. If I rewrite it with hats to denote quantum operators, we have \begin{align} U^\dagger(\Lambda) \hat{\phi}(x) U(\Lambda) = R(\Lambda) \hat{\phi}(\Lambda^{-1}(x)). \end{align} Then the $R(\Lambda)$, which is defined as a rep on some classical fields, is now acting on an operator on the Hilbert space. One way this might make sense is if what is meant by the transformation law is instead \begin{align} U^\dagger(\Lambda) \hat{\phi}(x) U(\Lambda) = \widehat{R(\Lambda) \phi}(\Lambda^{-1}(x)), \end{align} i.e. considering the operator obtained from quantising the transformed field.
So my questions are is this correct? And if not, what am I misunderstanding, and what is the resolution?
