Combined C and P Warning, this drawing contains a common error.

Helicity--- h = 2S·p/(ℏp).  No ordinary fundamental particle with mass should ever be a helicity eigenstate, for obvious reasons!   But....

Intrinsic parity is something I usually have to look up in a table. However, some usages are obvious. For example, a "pseudoscalar" particle is one which has spin zero, but negative parity. [We would normally associate scalars with positive parity!] Hence, a 0⁻ particle is called a pseudoscalar particle.

It was realized in 1957 that weak interactions are not P-symmetric, but PC was still thought to be a good symmetry. However, in 1964 it was found that neutral K (497.6 MeV) decays violate CP conservation. [See H & G, 9.6, 9.7, 9.8] However, this is so-called “indirect CP violation.” The observed K mesons are oscillating mixtures of the particle and antiparticle (CP eigenvalues -1 and +1) so that both -1 and +1 decays can be seen from the “same” particle. What physicists needed to see desperately was “direct CP violation,” in which a pure -1 state decays directly into a +1 state. Such decays are less than 1 in 10⁶ and were seen for neutral kaons only in 1999. They were then seen for neutral B mesons (5.279 GeV) in 2001 and for neutral D mesons (1.864 GeV) in 2011.  Perhaps the most disturbing thing is that there is NOTHING in the Standard Model description of the strong interaction that results in CP conservation, yet so far only ONE strong process that does not conserve CP has been seen!

Particle-antiparticle mixing diagram.

In the earliest moments of the universe, antiparticles MUST have behaved differently than particles to some extent, in order to produce an excess of particles after the mutual annihilation of particle-antiparticle pairs produced by decay of unknown heavy bosons or other now obscure processes. Direct observations and Big Bang nucleosynthesis measurements show that the baryon asymmetry parameter η (the ratio of baryons to photons) is about 6 × 10⁻¹⁰, meaning there was roughly one extra baryon (proton or neutron) per billion that survived the annihilations to photons. That single extra baryon per billion annihilations, plus a similar asymmetry for leptons, is all that remained to form the atoms, stars, and galaxies we see today.  But in the lab we have still not seen processes that violate CP to even this slight amount.   The key CP-nonconserving decays would have occurred at the quark-lepton level.

The incredible sensitivity of these experiments to the mass difference, found to astonishing precision!  It is possible because of the very large number of oscillations observed in the decaying states.

Explicit representation of a discrete symmetry operator can be very difficult. Even simple discrete symmetries give much food for thought. For example, the time-reversal operator has to be anti-unitary! The problem is that for fundamental processes, the time reversal operator should commute with the Hamiltonian. But the Hamiltonian operator in time representation is iℏ(∂/∂t). So the time reversal operator will not commute with the Hamiltonian unless it complex conjugates as well as changes t to −t.

The CKM matrix describes the unfortunate fact that the weak interaction does not see "pure" flavors of quarks, but rather sees quarks that are a linear flavor combination, with slight admixtures of other flavors to the predominant flavor. We will discuss this later.  It was first thought that this idea might serve as a guide to finding strong CPV processes, but that didn't work out and instead interest turned to very heavy neutral mesons such as the B meson.