CPT symmetry

), the transformation is given by $$\Theta \psi(t, \vec{x}) \Theta^{-1} = \psi^*(-t, -\vec{x}),$$Θψ(t,x

)Θ−1=ψ∗(−t,−x

), where $\Theta$Θ is antiunitary, satisfying $\Theta i = -i \Theta$Θi=−iΘ and $\Theta^\dagger = \Theta^{-1}$Θ†=Θ−1. This equation ensures that the field at a point in spacetime is mapped to its complex conjugate at the inverted point, maintaining the reality conditions and canonical commutation relations $[\psi(t, \vec{x}), \pi(t, \vec{y})] = i \delta^3(\vec{x} - \vec{y})$[ψ(t,x

),π(t,y

​)]=iδ3(x

−y

​), where $\pi$π is the conjugate momentum. The preservation arises because $\Theta$Θ conjugates the fields, reversing the roles of creation and annihilation operators: specifically, $\Theta a_{\vec{p}} \Theta^{-1} = a^\dagger_{-\vec{p}}$Θap

​​Θ−1=a−p

​†​ for bosons, up to phases determined by spin, thus interchanging particles and antiparticles while preserving $[a_{\vec{p}}, a^\dagger_{\vec{q}}] = \delta^3(\vec{p} - \vec{q})$[ap

​​,aq

​†​]=δ3(p

​−q

​).^[3] This transformation extends to general fields, including spinors and vectors, by incorporating appropriate phase factors and index contractions to respect the spin-statistics theorem and Lorentz representations. For Dirac fields, the CPT operator maps $\psi(t, \vec{x})$ψ(t,x

) to $\gamma^5 \psi^*(-t, -\vec{x})$γ5ψ∗(−t,−x

) (up to conventions), ensuring anticommutation relations $\{\psi_\alpha(t, \vec{x}), \psi^\dagger_\beta(t, \vec{y})\} = \delta_{\alpha\beta} \delta^3(\vec{x} - \vec{y})${ψα​(t,x

),ψβ†​(t,y

​)}=δαβ​δ3(x

−y

​) are invariant. The hermiticity of the Lagrangian density $\mathcal{L}^\dagger = \mathcal{L}$L†=L guarantees that $\Theta$Θ commutes with the time evolution operator $U(t)$U(t), yielding $\Theta H \Theta^{-1} = H$ΘHΘ−1=H, where $H$H is the Hamiltonian, thus establishing dynamical invariance under CPT. A complementary proof demonstrates CPT invariance of the S-matrix using perturbation theory. The S-matrix elements are expanded via the Dyson series in the interaction picture: $S = T \exp\left(-i \int_{-\infty}^\infty H_I(t) dt\right)$S=Texp(−i∫−∞∞​HI​(t)dt), where $T$T denotes time-ordering and $H_I$HI​ is the interaction Hamiltonian. Due to the antiunitary nature of $\Theta$Θ, CPT invariance implies that S-matrix elements satisfy $\langle f | S | i \rangle = \langle \Theta i | S | \Theta f \rangle^*$⟨f∣S∣i⟩=⟨Θi∣S∣Θf⟩∗. This holds term-by-term in the expansion, as each normal-ordered interaction :$\mathcal{L}_I(-x)$LI​(−x): maps to itself under CPT due to the field transformations and the assumptions of locality and Lorentz invariance, preserving the unitarity of the S-matrix.^[15][3] In the Feynman–Stückelberg interpretation, this framework interprets antiparticles as particles propagating backward in time: a positive-energy solution for an antiparticle corresponds to a negative-energy solution for the particle under time reversal, combined with charge conjugation via CPT to yield forward-propagating antiparticles with reversed momentum and helicity. This aligns with the field transformations, where incoming particle states map to outgoing antiparticle states under $\Theta$Θ. Individual charge conjugation (C), parity (P), and time reversal (T) transformations often fail to be symmetries due to explicit breaking in weak interactions or the need for antiunitary extensions that disrupt hermiticity or locality separately; however, their composition CPT succeeds universally because it leverages the full Lorentz invariance and locality to restore the original structure without such inconsistencies.^[3]

Experimental Status

Precision Tests

Searches for Violations

Implications

Predictions for Particles and Antiparticles

Extensions Beyond Standard Model