= Solution
The <CPT theorem> is a structural consequence of relativistic <quantum field theory>. In flat spacetime, assume a positive physical state space, the <positive-energy spectrum condition>, <Lorentz invariance>, <microcausality>, the usual <Spin-statistics theorem>, and a Hermitian dynamics. For ordinary field theories these are realized by a local Hermitian Lorentz-scalar <Lagrangian density>. There is then an <antiunitary operator> $\Theta$ implementing the combined <charge conjugation>, <parity symmetry in quantum field theory> and <time-reversal symmetry>, with a <CPT>-invariant vacuum. \b[The combined symmetry is required even when the three separate symmetries fail.]
Under the combined spacetime transformation, $x\mapsto-x$. A complex <scalar field> can be taken to transform into its adjoint at $-x$; a Hermitian vector field transforms into minus itself at $-x$. A derivative contributes a minus sign through differentiation of its reflected argument. For a <Dirac field> the transformation supplied in the paper may be written
$$
\Theta\psi(x)\Theta^{-1}=\gamma_5^T\psi^\dagger(-x)^T.
$$
Field phases are conventional and do not affect the physical conclusions. Crucially, <antiunitarity> means $\Theta i\Theta^{-1}=-i$ and conjugates every numerical coefficient. The reversal of time is compatible with positive energies because this conjugation changes $e^{-iHt}$ to $e^{iHt}$ while leaving the Hamiltonian invariant. Treating <time reversal> as a unitary change of coordinates would instead give incorrect signs.
One can see explicitly how the <CPT transformation of Dirac bilinears> works. Let $B_\Gamma(x)=:\psi^\dagger\gamma^0\Gamma\psi:$ be a Hermitian <fermion bilinear>, with $\Gamma^\dagger=\gamma^0\Gamma\gamma^0$. The colons indicate normal ordering so that the graded reordering below has no vacuum contact term. Using the reflected fields and conjugating $\gamma^0\Gamma$, then interchanging the two <fermion> factors, gives
$$
\begin{aligned}
\Theta B_\Gamma(x)\Theta^{-1}
&=:\psi^T\gamma_5^*(\gamma^0\Gamma)^*\gamma_5^T\psi^{\dagger T}:_{-x}\\
&=-:\psi^\dagger\gamma_5(\gamma^0\Gamma)^\dagger\gamma_5\psi:_{-x}\\
&=:\overline\psi\gamma_5\Gamma\gamma_5\psi:_{-x}.
\end{aligned}
$$
The last step uses the Dirac-Hermiticity condition and $\{\gamma^0,\gamma_5\}=0$. The identities for the <gamma matrices> then give
$$
\begin{array}{c|c|c}
\text{bilinear}&\Gamma&\text{CPT sign}\\\hline
\text{scalar}&1&+\\
\text{pseudoscalar}&i\gamma_5&+\\
\text{vector}&\gamma^\mu&-\\
\text{axial vector}&\gamma^\mu\gamma_5&-\\
\text{tensor}&\frac{i}{2}[\gamma^\mu,\gamma^\nu]&+
\end{array}
$$
The pseudoscalar's sign is particularly instructive: its explicit $i$ must also be conjugated. The minus sign from exchanging <fermions> is equally essential. These signs depend on the number of free vector indices, rather than on a field's separate parity assignment.
This supplies a <local Lagrangian argument for CPT invariance>. Hermitian local field combinations with $r$ vector indices transform with sign $(-1)^r$, including the appropriate reordered derivative terms. A <Lorentz scalar> contracts those indices in pairs or with four-index invariant tensors, giving a positive total sign. For charged terms, the reflected conjugate fields and conjugated coefficients exchange a term with its Hermitian-conjugate partner. Thus
$$
\boxed{\Theta\mathcal L(x)\Theta^{-1}=\mathcal L(-x),\qquad
\Theta S_{\mathrm{action}}\Theta^{-1}=S_{\mathrm{action}}.}
$$
The integration variable $x\mapsto-x$ has unit absolute Jacobian. The general field-index argument is discussed in https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf[Srednicki's treatment of discrete symmetries, section 40].
For example, the scalar mass term $\overline\psi\psi$ is even. The <Dirac electromagnetic current> and vector potential are both odd, making their contracted interaction even. The Hermitian Dirac kinetic term $(i/2)\overline\psi\gamma^\mu\overleftrightarrow{\partial}_\mu\psi$ is also even: reflection and reordering interchange the two derivative placements, while <antiunitarity> conjugates its $i$. A complex <Yukawa coupling> and its Hermitian conjugate are exchanged rather than requiring the coupling itself to be real. Hence <CP violation> in the <CKM matrix> is compatible with exact <CPT>. Renormalizability is not necessary: local Hermitian higher-dimensional Lorentz-scalar operators satisfy the same argument.
The axiomatic form of the <CPT theorem> does not start by assuming a polynomial <Lagrangian density>. The <positive-energy spectrum condition> provides analytic continuations of <Wightman functions>, and complexified <Lorentz transformations> relate configurations to their total spacetime inversions. At suitable spacelike configurations, <microcausality> permits reversal of the order of field operators with the signs dictated by the <Spin-statistics theorem>. Uniqueness of <analytic continuation> extends this relation to the correlators generally, from which the Hilbert-space <antiunitary operator> is obtained. This explains why locality and positive energy are genuine hypotheses, not incidental features of a particular interaction. The elementary argument above displays the mechanism for the scalar, spinor and vector fields relevant to the <Standard Model>; it is not a substitute for all the analytic steps of the axiomatic proof.
For one-particle states, <CPT> reverses additive charges and spin projections. The spatial momentum is unchanged by the combination: <parity symmetry in quantum field theory> and <time reversal> each reverse it. Thus, up to phases,
$$
\Theta|\boldsymbol p,s,q\rangle
=|\boldsymbol p,-s,-q\rangle.
$$
Since the energy-momentum spectrum is preserved, the <CPT constraints on particle properties> give \b[equal particle-antiparticle masses, equal spin magnitudes, and opposite conserved additive charges]. Relating the corresponding two-point functions also fixes equal pole positions, so unstable conjugate particles have equal total <decay widths>, hence equal lifetimes. Magnetic moments at the same chosen spin orientation have opposite signs. These are precise experimental tests of <CPT>, not tests that every individual discrete symmetry is respected.
For scattering, <CPT> interchanges incoming and outgoing boundary conditions. The <antiunitary operator> identity $\langle a|b\rangle=\langle\Theta b|\Theta a\rangle$ relates $\langle f,\mathrm{out}|i,\mathrm{in}\rangle$ to the amplitude with transformed $f$ incoming and transformed $i$ outgoing. Thus the reversed process with antiparticles and reversed spins has the same transition probability. This is not, in general, equality with the forward CP-conjugate process. In particular, <CP violation> can make corresponding partial <decay widths> different while their complete sums remain equal. With <CPT> preserved, genuine <CP violation> requires corresponding <time-reversal symmetry> violation.
Finally, a proposed <CPT> violation requires revisiting at least one assumption: for example, a nonlocal interaction, a preferred spacetime direction violating <Lorentz invariance>, or loss of unitary Hermitian dynamics. A generic curved background need not possess the global spacetime inversion used here. Those observations delimit the theorem; they do not weaken its prediction for the ordinary local <Standard Model> in Minkowski spacetime.
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