# CP-symmetry and Ward identities and finite temperature

I have a few questions about Ward-identities which I summarize here. For each I am very greateful for answers and references to literature.

Wikipedia states about Ward-identities:

The Ward-Takahashi identity is a quantum version of the classical Noether's theorem, and any symmetries in a quantum field theory can lead to an equation of motion for correlation functions.

1. Taken seriously this means that Ward-Takahashi identities also exist for discrete symmetries like CP (if present in the Lagrangian) in contrast to a conserved Noether current. If so, I wonder if CP relations between amplitudes, such as $|\mathcal{M}|_{S_i\rightarrow S_f} =|\mathcal{M}|_{\bar{S}_i\rightarrow \bar{S}_f}$ which I can get by computing the amplitudes in perturbation theory and comparing, can be considered Ward-identities. $\bar{S}_{i,f}$ are the CP-conjugated states of ${S}_{i,f}$ respectively.

2. Since Ward-identities are considered to hold for the quantized theory, I wonder if they are invariant under the renormalization group running of parameters. If so, how does RGE running take these restrictions into account.

3. Finally I wonder if there is an established finite-temperature generalization.

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