CPT Theorem
Any quantum field theory that is local, Lorentz invariant, and fully consistent with the principles of quantum mechanics is invariant under the combined CPT transformation.
In practical terms, this means that if CP symmetry is violated in a physical process, time-reversal symmetry T must also be violated.
The conservation of CPT therefore establishes a direct conceptual link between time-reversal symmetry and CP symmetry.
$$ T = (CP)^{-1} $$
This relation expresses the fact that any violation of CP must be accompanied by a corresponding violation of T, ensuring that the overall CPT symmetry remains preserved.
For example, experiments have conclusively shown that CP symmetry is violated in weak interactions. Since the CPT theorem requires CPT invariance, it follows unavoidably that time-reversal symmetry T must also be violated. The violation of T is thus not merely an empirical finding, but a logically necessary consequence of the CPT theorem.
Why is it important?
The CPT theorem is one of the most general and far-reaching results in modern physics and quantum field theory. It concerns the fundamental symmetries of the laws governing elementary particles and imposes exceptionally strong constraints on the structure of physically admissible theories.
The theorem involves three discrete transformations:
- T (time reversal)
Reverses the direction of time, \( t \rightarrow -t \), and corresponds to describing a physical process as if it were observed running backward in time. - C (charge conjugation)
Replaces a particle with its antiparticle, reversing electric charge and all additive internal quantum numbers. - P (parity)
Inverts spatial coordinates, \( \vec x \rightarrow -\vec x \), corresponding to a mirror reflection.
The symmetries C, P, and T may each be violated individually, whereas their combined CPT symmetry must always be conserved.
As a result, a violation of CP necessarily implies a violation of T.
Note. The CPT theorem is not an empirical assumption, but a logical consequence of very general principles, including the validity of quantum mechanics, Lorentz invariance, the locality of interactions, and the quantum field theoretic framework itself. If any of these assumptions were to fail, the theorem might no longer apply. For this reason, a genuine violation of CPT would have deep and revolutionary implications for fundamental physics.
The CPT theorem plays a central role because it provides a consistency criterion for fundamental theories and reveals deep connections between symmetries that might otherwise appear unrelated.
It therefore allows physicists to infer properties that are not directly accessible to observation and guides the search for new physics beyond established models, by excluding theories that fail to satisfy CPT invariance.
In practice, the CPT theorem does not describe how nature behaves, but rather delineates how it cannot behave. For this reason, it stands as one of the most powerful and restrictive results in theoretical physics.
Among its fundamental consequences is the requirement that a particle and its antiparticle have exactly the same mass and the same lifetime.
A concrete example
In the system of neutral kaons, experiments clearly demonstrate that CP symmetry is violated.
In particular, the long-lived neutral kaon ( $ K_L $ ) can decay into two pions ( $ \pi $ ):
$$ K_L \rightarrow 2\pi $$
This decay would be forbidden if CP were an exact symmetry, because the initial and final states carry different CP eigenvalues (see further discussion).
If CP symmetry were exact, the long-lived neutral kaon would decay exclusively into three pions:
$$ K_L \rightarrow 3\pi $$
The fact that this decay does occur, albeit with a small probability, into two pions demonstrates that CP symmetry is violated in the weak interaction.
Since the CPT theorem requires the combined CPT transformation to be an exact symmetry, the violation of CP cannot occur in isolation. It must necessarily be accompanied by a violation of time-reversal symmetry T, so that the overall CPT symmetry remains intact.
Note. Starting from an experimental fact, namely the violation of CP symmetry, one can theoretically infer that time-reversal invariance T cannot be an exact symmetry of the weak interaction responsible for the decay of long-lived neutral kaons.
And so on.
