12.4 Discrete symmetries: parity, time reversal, and charge conjugation

Discrete symmetries are fundamental to understanding particle behavior in quantum mechanics. Parity, time reversal, and charge conjugation transformations reveal deep insights into the nature of physical laws and particle interactions.

These symmetries shape the allowed interactions in particle physics and have far-reaching consequences. From parity violation in weak interactions to CP violation's role in the universe's matter-antimatter asymmetry, discrete symmetries are crucial to our understanding of nature.

Fundamental Discrete Symmetries

Parity, time reversal, and charge conjugation

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• Parity (P)
  • Spatial inversion of coordinates $\vec{r} \rightarrow -\vec{r}$ flips handedness of coordinate system like mirror reflection
  • Eigenvalues $\pm 1$ correspond to even or odd parity states (spherical harmonics)
  • Conserved in electromagnetic and strong interactions, violated in weak interactions
• Time reversal (T)
  • Reverses time direction $t \rightarrow -t$ and motion $\vec{v} \rightarrow -\vec{v}$ like rewinding a video
  • Complex conjugates wavefunctions $\psi \rightarrow \psi^*$ affects quantum phase
  • Subtle symmetry related to microscopic reversibility and detailed balance
• Charge conjugation (C)
  • Interchanges particles with antiparticles swapping internal quantum numbers (electron → positron)
  • Preserves mass, spin, and momentum maintaining physical properties
  • Symmetry between matter and antimatter in most interactions

Behavior under discrete symmetry transformations

• Parity transformations
  • Scalar quantities unchanged (energy, mass) retain their values
  • Vectors change sign (position, momentum) reverse direction
  • Pseudovectors unchanged (angular momentum, magnetic field) maintain orientation
• Time reversal transformations
  • Position unchanged $\vec{r} \rightarrow \vec{r}$ spatial coordinates stay the same
  • Momentum changes sign $\vec{p} \rightarrow -\vec{p}$ velocity reverses
  • Angular momentum changes sign $\vec{L} \rightarrow -\vec{L}$ rotation direction flips
• Charge conjugation transformations
  • Electric charge changes sign $q \rightarrow -q$ (proton → antiproton)
  • Magnetic moment changes sign $\vec{\mu} \rightarrow -\vec{\mu}$ field orientation reverses
  • Baryon number changes sign $B \rightarrow -B$ matter becomes antimatter

Consequences of discrete symmetries

• Noether's theorem
  • Continuous symmetries lead to conserved quantities (translation → momentum)
  • Discrete symmetries impose constraints on physical laws shaping allowed interactions
• Parity conservation
  • Electromagnetic and strong interactions conserve parity maintain mirror symmetry
  • Weak interactions violate parity demonstrated in beta decay experiments
• Time reversal invariance
  • Most fundamental laws T-invariant processes reversible microscopically
  • Exceptions in weak interactions (CP violation) linked to matter-antimatter asymmetry
• Charge conjugation symmetry
  • Electromagnetic interactions C-invariant photons interact equally with particles and antiparticles
  • Weak interactions violate C symmetry neutrinos always left-handed

Discrete symmetries in particle physics

• CPT theorem
  • Combined symmetry of C, P, and T conserved in all interactions fundamental principle
  • Guarantees equal masses and lifetimes for particles and antiparticles
• CP violation
  • Observed in certain weak interactions (K and B meson decays)
  • Explains matter-antimatter asymmetry in the universe baryon asymmetry problem
• Neutrino oscillations
  • Violation of individual lepton number conservation flavor mixing
  • Implications for neutrino mass and mixing non-zero masses required
• Symmetry breaking
  • Spontaneous symmetry breaking in electroweak theory unified electromagnetic and weak forces
  • Higgs mechanism generates masses for fundamental particles W and Z bosons