Conservation Laws and Symmetries

2.1 Exactly Conserved Quantities

The following are conserved in all known interactions:

• Energy $E$
• Momentum $\mathbf{p}$
• Angular momentum $\mathbf{L}$
• Electric charge $Q$
• Colour charge
• Baryon number $B$ (each quark has $B = 1/3$)
• Lepton family numbers $L_e$, $L_\mu$, $L_\tau$ (each lepton has $L = +1$Each antilepton $L = -1$)

2.2 Approximate or Partially Conserved Quantities

• Isospin $I$: Conserved in strong interactions, violated by electromagnetic and weak. $I_3$ determines the electric charge via the Gell-Mann—Nishijima formula.
• Strangeness $S$: Conserved in strong and electromagnetic, violated by weak (hence “strange” particles are produced in pairs but decay via weak interaction).
• Parity $P$: Conserved in strong and electromagnetic, maximally violated in weak interactions.
• Charge conjugation $C$: Conserved in strong and electromagnetic, violated in weak.
• CP: Conserved in most interactions; violated in weak interactions (observed in $K$ and $B$ meson systems). CP violation is necessary for the matter-antimatter asymmetry.

2.3 The Gell-Mann—Nishijima Formula

The Gell-Mann—Nishijima formula relates the electric charge of a hadron to its isospin Projection $I_3$Baryon number $B$And strangeness $S$:

$Q = I_3 + \frac{B + S}{2}$

This can be generalised to include charm $C$Bottomness $B"$And topness $T$:

$Q = I_3 + \frac{1}{2}(B + S + C + B' + T)$

Derivation. The formula follows from the definition of the hypercharge $Y = B + S$ and the Relation $Q = I_3 + Y/2$Which is a direct consequence of the embedding of $\mathrm{U}(1)_{\mathrm{em}}$ Within $\mathrm{SU}(3)$ flavour. For the electroweak theory, the hypercharge is generalised to $Y = B + S + C + B' + T$ when additional flavours are included.

2.4 Parity Violation

Parity transformation $P$ reverses the sign of all spatial coordinates: $\mathbf{x} \to -\mathbf{x}$. Under parity, polar vectors change sign while axial vectors do not.

In 1956, Lee and Yang proposed that parity might not be conserved in weak interactions. This was Confirmed by the Wu experiment (1957), which measured the angular distribution of electrons Emitted in the beta decay of polarised $^{60}$Co nuclei. The electrons were emitted preferentially Opposite to the nuclear spin direction, a clear parity-violating asymmetry.

The weak interaction maximally violates parity: only left-handed fermions (and right-handed Antifermions) participate in charged-current weak interactions. This is encoded in the $V - A$ structure of the weak current:

$J^\mu_{\mathrm{weak} = \bar{\psi}\gamma^\mu(1 - \gamma^5)\psi}$

Where the $(1 - \gamma^5)$ projector selects the left-handed chirality component.

2.5 Worked Examples: Conservation Laws in Decays

2.6 Symmetry and Noether’s Theorem

Noether’s Theorem: Every continuous symmetry of the action corresponds to a conserved quantity.

Proof (sketch). Consider an infinitesimal transformation $x^\mu \to x^\mu + \delta x^\mu$ $\phi \to \phi + \delta\phi$. If the action $S = \int \mathcal{L}\,d^4x$ is invariant, then the Current $j^\mu = \frac{\partial \mathcal{L}}{\partial(\partial_\mu\phi)}\delta\phi - \mathcal{L}\delta x^\mu$ satisfies $\partial_\mu j^\mu = 0$, yielding a conserved charge $Q = \int j^0\,d^3x$. $\blacksquare$

Discrete symmetries ($P$, $C$, $T$) do not arise from Noether’s theorem but are still powerful Constraints. The CPT theorem states that any Lorentz-invariant local quantum field theory is Invariant under the combined transformation $CPT$.