CPT Symmetry: Proving the Theorem & Lorentz Transformation

[fxdung]

Some books prove CPT theorem basing on scalars,vectors, tensors building from 4-spinor of fermion and gamma matrices.Why can they do that?Because a general Lagrangian can contain bose scalar,bose vector,bose tensor fields and spinor fields.
The CPT theorem says CPT symmetry is a strictly correct.What about the PT symmetry,is it also strictly correct because it is Lorentz transformation?

 

[vanhees71]

In Weinberg, Quantum Theory of Fields, vol. 1 you find a proof for fields of arbitrary spin.

Then you should note that Lorentz invariance (or better Poincare) invariance refers to the continuous part of the Poincare group connected with the identity, i.e., the semidirect product of space-time translations with the proper orthochronous Lorentz group $\mathrm{SO}(1,3)^{\uparrow}$. Poincare invariance just dictates invariance under this group due to the spacetime structure of special relativity. There's no need a priori that the theory should be invariant under any of the discrete transformations $P$, $T$, and $C$. The $CPT$ theorem, however, tells you that any local relativistic QFT with a stable ground state (Hamiltonian bounded from below) is also automatically symmetric under $CPT$. In the Standard Model all other combinations are violated by the weak interaction, and this is experimentelly checked for each of them, i.e., nature is not symmetric under each of the transformations $P$ (e.g., Wu experiment), $CP$ (neutral-kaon system, Cronin&Fitch), $T$ (I forgot who did the independent experimental proof on some B decays first; it was some recent LHC experiment).