Discrete Symmetries — C, P, T, and CPT

Beyond the continuous symmetries that generate additive conservation laws, three discrete operations act on every process: parity $P$ reflects space, charge conjugation $C$ exchanges each particle for its antiparticle, and time reversal $T$ runs the motion backward. Each is its own inverse, so its eigenvalues are $\pm 1$ — multiplicative quantum numbers that multiply across a state rather than adding. Where a discrete symmetry holds, its eigenvalue is conserved and forbids any transition that would change it; the strong and electromagnetic interactions respect all three, and the pattern of which ones the weak interaction breaks is the story of the next two lessons. The product $CPT$, however, is not an empirical symmetry but a theorem: no local, Lorentz-invariant quantum field theory can violate it.

Parity

The parity operator reflects the spatial coordinates through the origin, $\vec{r}\to -\vec{r}$. Acting on a single-particle momentum eigenstate it reverses the momentum but leaves the spin — an axial vector built from $\vec{r}\times \vec{p}$, in which both factors flip — unchanged:

The phase ${\eta }_P=\pm 1$ is the particle's intrinsic parity. Because two reflections restore the original coordinates, $P^2=1$ and ${\eta }_P^2=1$. The intrinsic parity of a particle is a fixed, measurable label, conventionally set to $+1$ for the proton, neutron, and electron. The Dirac equation then forces a fermion and its antifermion to carry opposite intrinsic parity, while a boson and its antiboson carry the same intrinsic parity.¹

For a system of two particles with relative orbital angular momentum $\ell$, the spatial wavefunction is a spherical harmonic $Y_{\ell }^m(\theta ,\varphi )$, which under $\vec{r}\to -\vec{r}$ picks up $(-1{)}^{\ell }$. The total parity multiplies the two intrinsic pieces with this orbital factor:

The pseudoscalar mesons are the key example. The pion is a bound state of a quark and an antiquark with total spin $0$ and $\ell =0$; the quark and antiquark carry opposite intrinsic parity, so

The pion is a pseudoscalar: spin $0$, parity $-1$, written $J^P=0^{-}$. This value is measured, not assumed — the capture reaction ${\pi }^{-}+d\to n+n$ from a ${\pi }^{-}$ atom in an $\ell =0$ orbit, together with the known deuteron and dineutron quantum numbers, fixes $P({\pi }^{-})=-1$.²

Parity is conserved in the strong and electromagnetic interactions, so a strong or electromagnetic reaction cannot change the total parity of the state. The pion's odd parity thus constrains which final states it can reach strongly. That parity is not a symmetry of the weak interaction is the subject of the next lesson.

Charge conjugation

Charge conjugation $C$ replaces every particle by its antiparticle, reversing all internal quantum numbers — electric charge, baryon number, lepton number, strangeness — while leaving momentum and spin untouched:

Because $C$ moves a state to a different particle, only a system that is its own antiparticle can be an eigenstate of $C$ — the photon, the ${\pi }^0$, the $\eta$, and neutral $q\bar{q}$ or ${\ell }^{+}{\ell }^{-}$ systems. For everything else $C$ is a symmetry of the dynamics without a conserved eigenvalue.

The photon's $C$-parity follows from electromagnetism. The field $A^{\mu }$ is sourced by electric charge, and under $C$ every charge reverses sign, so $A^{\mu }\to -A^{\mu }$ and the photon carries ${\eta }_C(\gamma )=-1$. A state of $n$ photons therefore has $C=(-1{)}^n$. For a fermion-antifermion pair with orbital angular momentum $\ell$ and total spin $s$, exchanging the two particles (which is what $C$ effects, up to the spatial and spin exchange) gives

The neutral pion is observed to decay ${\pi }^0\to \gamma \gamma$, which fixes its $C$-parity: two photons give $C=(-1{)}^2=+1$, so ${\eta }_C({\pi }^0)=+1$. The same bookkeeping then forbids ${\pi }^0\to \gamma \gamma \gamma$, which would require $C=-1$; the measured branching ratio for three-photon decay is below $10^{-8}$, a clean confirmation that the electromagnetic interaction conserves $C$.³ Positronium tells the same story: the spin-singlet para-positronium ($\ell =0$, $s=0$, $C=+1$) annihilates to two photons, while the spin-triplet ortho-positronium ($s=1$, $C=-1$) must go to three, and its lifetime is correspondingly longer.

Combined CP and time reversal

Neither $C$ nor $P$ is a symmetry of the weak interaction, but their product $CP$ comes close. Under $CP$ a left-handed particle maps to a right-handed antiparticle, and to good approximation the weak interaction treats the two the same — so $CP$ is almost conserved. Its small violation, discovered in the neutral kaon system in 1964, is deferred to the weak-interaction module; here we note only that $CP$ is the finer near-symmetry that survives the separate breaking of $C$ and $P$.

Time reversal $T$ sends $t\to -t$, reversing every momentum and spin while leaving positions fixed. Its implementation in quantum mechanics is subtle. A symmetry that reversed $t$ but preserved the Schrödinger equation would have to leave $i{\partial }_t\psi =H\psi$ invariant, yet flipping the sign of $t$ alone flips the left-hand side. The resolution, due to Wigner, is that $T$ acts antiunitarily: it complex-conjugates as it reverses time, so that

and $\psi (\vec{r},t)\to {\psi }^{\ast }(\vec{r},-t)$ solves the same equation. An antiunitary operator has no eigenvalue spectrum of the usual kind, so $T$ does not assign a conserved multiplicative quantum number the way $P$ and $C$ do. Instead its consequences appear as relations between rates — the principle of detailed balance, equating the amplitude for a reaction and its time-reverse — and as the requirement that a nonzero particle electric dipole moment would signal $T$ violation.⁴

The CPT theorem

Where $C$, $P$, and $T$ are each empirical symmetries that the weak interaction may break, their combined product is guaranteed. The CPT theorem states that any quantum field theory that is local, Lorentz-invariant, and has a Hermitian Hamiltonian is invariant under the product $CPT$ applied in any order.⁵

The corollary is the most tested statement in particle physics. $CPT$ maps a particle at rest onto its antiparticle at rest, so the two must have identical rest energy; it maps a decaying state onto the antiparticle's decay, so the total widths — and hence the lifetimes — match. The neutral kaon system provides the sharpest mass test: the fractional mass difference between $K^0$ and ${\bar{K}}^0$ is bounded below $10^{-18}$, and the electron and positron $g$-factors agree to parts in $10^{12}$. No violation of $CPT$ has ever been seen.

Because the weak interaction violates $P$ maximally and violates $CP$ slightly, the $CPT$ theorem forces compensating violations elsewhere: with $CP$ broken and $CPT$ exact, $T$ must also be broken by the same amount, and the direct observation of $T$ violation in kaon and $B$-meson mixing confirms this. $CPT$ is thus the rigid scaffold within which the weak interaction is free to break the individual discrete symmetries.

G-parity

For the non-strange mesons a further multiplicative quantum number sharpens the selection rules of strong decays. G-parity combines charge conjugation with a rotation by $\pi$ about the second axis in isospin space,

chosen so that $G$ is an eigenvalue not just of the neutral member of an isospin multiplet but of the whole multiplet — the isospin rotation carries $I_3\to -I_3$, undoing the charge flip so that a charged pion, too, is a $G$ eigenstate.⁶ For a $q\bar{q}$ meson of isospin $I$, orbital angular momentum $\ell$, and spin $s$,

The pion has $G=-1$, so a state of $n$ pions carries $G=(-1{)}^n$. Since the strong interaction conserves $G$, the number of pions in a strong decay is fixed in parity: the $\rho$ meson ($G=+1$) decays to two pions, the $\omega$ meson ($G=-1$) to three, and no strong process can convert an even number of pions into an odd number. G-parity extends the reach of the discrete symmetries into the purely hadronic sector, where $C$ itself applies only to neutral states.

The discrete symmetries thus divide cleanly. $P$, $C$, and $G$ furnish conserved multiplicative quantum numbers for the strong and electromagnetic interactions; $CP$ is the near-symmetry the weak interaction almost respects; and $CPT$ is the theorem binding every particle to its antiparticle. What the weak interaction does to $P$ and $C$ individually — break them, and maximally — is the subject of the next lesson.