--- title: Discrete Symmetries — C, P, T, and CPT module: Symmetries and Conservation Laws moduleNumber: 3 lessonNumber: 2 order: 302 summary: > Parity reflects space, charge conjugation swaps particle for antiparticle, and time reversal runs the clock backward. Each assigns multiplicative quantum numbers that act as selection rules — intrinsic parities, the photon's C = −1, the C-parity argument fixing the pion's two-photon decay. Their product CPT is a theorem of any local relativistic field theory, forcing particle and antiparticle to share mass and lifetime. topics: [Symmetries and Conservation Laws] draft: false sources: - book: Griffiths ref: "Ch. 4 — Symmetries; §4.4 Parity, §4.5 Charge Conjugation, §4.6 CP, §4.7 Time Reversal and the TCP Theorem" - book: Thomson ref: "§4.5–4.7; Ch. 14 opening" - book: Perkins ref: "Ch. 3 — Invariance Principles and Conservation Laws" - book: Tong ref: "The Standard Model (Cambridge Part III), §1.4, §2.1" --- 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: $$ P\,\lvert \vec{p}, s\rangle = \eta_P\,\lvert -\vec{p}, s\rangle . $$ 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.[^tong-parity] For a system of two particles with relative orbital angular momentum $\ell$, the spatial wavefunction is a spherical harmonic $Y_\ell^m(\theta,\phi)$, which under $\vec{r}\to-\vec{r}$ picks up $(-1)^\ell$. The total parity multiplies the two intrinsic pieces with this orbital factor: $$ P = \eta_1\,\eta_2\,(-1)^\ell . $$ $$ % caption: The parity of a composite state factorizes into the intrinsic parity % of each constituent times an orbital factor (minus one) to the power of the % relative angular momentum. Each factor is plus or minus one and they multiply. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \node[draw, acc, minimum width=20mm, minimum height=9mm, align=center] (a) at (0,0) {intrinsic\\parity 1}; \node[font=\large] at (2.0,0) {$\times$}; \node[draw, acc, minimum width=20mm, minimum height=9mm, align=center] (b) at (4.0,0) {intrinsic\\parity 2}; \node[font=\large] at (6.0,0) {$\times$}; \node[draw, black, minimum width=24mm, minimum height=9mm, align=center] (c) at (8.4,0) {orbital factor\\(sign of L)}; \node[font=\large] at (10.7,0) {$=$}; \node[draw, acc, very thick, minimum width=20mm, minimum height=9mm, align=center] (d) at (12.7,0) {total\\parity}; \end{tikzpicture} $$ 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 $$ P(\pi) = (+1)(-1)(-1)^0 = -1 . $$ 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 $P(\pi^-) = -1$.[^griffiths-parity] > **Definition (Intrinsic parity).** The intrinsic parity $\eta_P$ of a particle > is the eigenvalue by which its state is multiplied under $\vec{r}\to-\vec{r}$ > once its motion has been accounted for. It is $+1$ for the standard fermions by > convention, $-1$ for their antifermions, $-1$ for the photon, and $-1$ for the > pseudoscalar mesons. 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: $$ C\,\lvert \text{particle};\vec{p}, s\rangle = \eta_C\,\lvert \text{antiparticle};\vec{p}, s\rangle . $$ 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 $$ C = (-1)^{\ell + s} . $$ $$ % caption: The C-parity argument for the neutral pion. The pion has C-parity plus % one; each photon carries C-parity minus one, so a final state of n photons has % C equal to minus one to the n. Two photons give plus one (allowed); three give % minus one (forbidden). The observed decay is to two photons. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % pion vertex \draw[acc, very thick, dashed] (-2.6,0) -- (-0.6,0); \node[acc, left, font=\scriptsize] at (-2.6,0) {neutral pion}; \fill[acc] (-0.6,0) circle (2.2pt); % two photons out (allowed) \draw[->, acc, thick] (-0.6,0) -- (1.4,0.9); \draw[->, acc, thick] (-0.6,0) -- (1.4,-0.9); \node[right, font=\scriptsize] at (1.4,0.9) {photon}; \node[right, font=\scriptsize] at (1.4,-0.9) {photon}; \node[align=center, font=\scriptsize] at (0.4,-1.9) {two photons: C = plus one\\allowed}; % divider \draw[black, dashed] (3.6,-2.3) -- (3.6,1.5); % three photons out (forbidden) \draw[acc, very thick, dashed] (4.4,0) -- (5.6,0); \fill[acc] (5.6,0) circle (2.2pt); \draw[->, black, thick] (5.6,0) -- (7.4,1.0); \draw[->, black, thick] (5.6,0) -- (7.6,0.0); \draw[->, black, thick] (5.6,0) -- (7.4,-1.0); \node[align=center, font=\scriptsize] at (6.3,-1.9) {three photons: C = minus one\\forbidden}; \end{tikzpicture} $$ 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 $C$.[^griffiths-C] 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 $$ T\,i\,T^{-1} = -i , $$ 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.[^tong-T] $$ % caption: The three discrete operations acting on a moving, spinning particle. % Parity reflects the position and reverses the momentum but keeps the spin. % Charge conjugation swaps the particle for its antiparticle, keeping momentum and % spin. Time reversal reverses both momentum and spin. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % original \node[font=\scriptsize] at (0,1.7) {original}; \fill[acc] (0,0) circle (3pt); \draw[->, acc, very thick] (0,0) -- (1.0,0); \draw[->, black, thick] (0,0.15) arc (-90:180:0.26); \node[font=\scriptsize] at (0.5,-0.7) {p forward}; % P \node[font=\scriptsize] at (4.6,1.7) {parity}; \fill[acc] (4.6,0) circle (3pt); \draw[->, acc, very thick] (4.6,0) -- (3.6,0); \draw[->, black, thick] (4.6,0.15) arc (-90:180:0.26); \node[font=\scriptsize] at (4.1,-0.7) {p reversed}; % C \node[font=\scriptsize] at (9.2,1.7) {charge conj.}; \draw[acc, very thick] (9.2,0) circle (3pt); \draw[->, acc, very thick] (9.2,0) -- (10.2,0); \draw[->, black, thick] (9.2,0.15) arc (-90:180:0.26); \node[font=\scriptsize] at (9.7,-0.7) {p forward}; \node[font=\scriptsize] at (9.2,-1.3) {antiparticle}; % T \node[font=\scriptsize] at (13.8,1.7) {time reversal}; \fill[acc] (13.8,0) circle (3pt); \draw[->, acc, very thick] (13.8,0) -- (12.8,0); \draw[->, black, thick] (14.08,-0.15) arc (90:-180:0.26); \node[font=\scriptsize] at (13.3,-0.7) {p reversed}; \end{tikzpicture} $$ ## 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.[^tong-cpt] > **Theorem (CPT).** Every local, Lorentz-invariant quantum field theory with a > Hermitian Hamiltonian is invariant under the combined operation $CPT$. As a > corollary, a particle and its antiparticle have exactly equal mass and lifetime, > and exactly opposite charge and magnetic moment. 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. $$ % caption: CPT as the product operation mapping matter onto antimatter. C, P, and % T are the three edges; applying all three (any order) carries a particle state % to the corresponding antiparticle state, with equal mass and lifetime and % reversed charge. Each individual edge may be broken by the weak force; the full % diagonal is a theorem. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \node[draw, acc, very thick, minimum width=26mm, minimum height=11mm, align=center] (m) at (0,0) {matter\\(particle)}; \node[draw, acc, very thick, minimum width=26mm, minimum height=11mm, align=center] (a) at (8,0) {antimatter\\(antiparticle)}; \draw[->, acc, very thick] (m) -- node[above, font=\scriptsize] {C then P then T} node[below, font=\scriptsize] {exact} (a); \node[black, font=\scriptsize, align=center] at (4,-1.5) {equal mass, equal lifetime\\opposite charge}; \node[black, font=\scriptsize, align=center] at (4,1.4) {each of C, P, T alone\\may be broken by the weak force}; \end{tikzpicture} $$ 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, $$ G = C\,e^{\,i\pi I_2} , $$ 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.[^tong-G] For a $q\bar q$ meson of isospin $I$, orbital angular momentum $\ell$, and spin $s$, $$ G = (-1)^{\ell + s + I} . $$ 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. [^tong-parity]: Tong, _The Standard Model_ (Cambridge Part III), §1.4.1 — the action of parity on Dirac spinors, giving intrinsic parity $+1$ to a fermion and $-1$ to its antifermion. http://www.damtp.cam.ac.uk/user/tong/standardmodel.html [^griffiths-parity]: Griffiths, _Introduction to Elementary Particles_, 2nd ed., §4.4 — measurement of the pion's intrinsic parity from $\pi^- d \to nn$ capture. [^griffiths-C]: Griffiths, §4.5 — the photon's charge-conjugation eigenvalue $-1$, the neutral-pion two-photon decay, and the $\pi^0 \to 3\gamma$ suppression. [^tong-T]: Tong, _The Standard Model_ (Cambridge Part III), §1.4.3 — time reversal as an antiunitary operator, $T i T^{-1} = -i$, and $\psi(\vec r,t)\to\psi^\ast(\vec r,-t)$. http://www.damtp.cam.ac.uk/user/tong/standardmodel.html [^tong-cpt]: Tong, _The Standard Model_ (Cambridge Part III), §1.4 — the CPT theorem and the equality of particle and antiparticle amplitudes it enforces. http://www.damtp.cam.ac.uk/user/tong/standardmodel.html [^tong-G]: Tong, _The Standard Model_ (Cambridge Part III), §2.1 — $G$-parity as $G = C\,e^{i\pi I_2}$ acting on the isospin multiplet. http://www.damtp.cam.ac.uk/user/tong/standardmodel.html