The inclusion of fermions – J=1/2 particles Weyl spinors Dirac spinor 2-component spinors of SU(2) velocity Left-handed m=0 Right-handed The spinor structure follows from the representation structure of the Lorentz Group …

The Lorentz transformations form a group, G Rotations Angular momentum operator

} } The Lorentz group Generate the group SO(3,1) To construct representations a more convenient (non-Hermitian) basis is }

} The Lorentz group Generate the group SO(3,1) To construct representations a more convenient (non-Hermitian) basis is Representations

Weyl spinors Dirac spinor 2-component spinors of SU(2) Rotations and Boosts Dirac spinor Can combine to form a 4-component “Dirac” spinor Lorentz transformations where Weyl basis

Weyl spinors Dirac spinor 2-component spinors of SU(2) Rotations and Boosts Dirac spinor Can combine to form a 4-component “Dirac” spinor Lorentz transformations where (Dirac gamma matrices, …new 4-vector ) Note :

The Dirac equation Fermions described by 4-cpt Dirac spinors Lorentz invariant New 4-vector The Lagrangian Dimension? From Euler Lagrange equation obtain the Dirac equation Feynman rules U(1) symmetry

Fermion masses Lorentz invariant Dirac mass Lorentz invariant Only term allowed by charge conservation is Majorana mass

Fermions and the weak interactions Fermi theory of decay

} V-A Fermions and the weak interactions Fermi theory of decay W+ p n

Weak Interactions Symmetry : Local conservation of SU(2) local gauge theory Symmetry : Local conservation of 2 weak isospin charges Weak coupling, α2 u d Gauge boson (J=1) Wa=1..3 A non-Abelian (SU(2)) local gauge field theory e Neutral currents

Massive vector propagator (W, Z bosons) Free particle solution Helicity polarisation vectors

} Propagation of unstable scalar particle No decay iJ iJ .. Particle decays into final state n …. iJ Optical theorem – conservation of probability, time evolution is unitary }

Fermi theory (‘40s) Dimensional analysis The hard part!

In μ decay

Have Fun! Fundamental principles of particle physics Introduction - Fundamental particles and interactions Symmetries I - Relativity Quantum field theory - Quantum Mechanics + relativity Theory confronts experiment - Cross sections and decay rates Symmetries II – Gauge symmetries, the Standard Model Fermions and the weak interactions The Standard Model and Beyond Have Fun!

velocity Left-handed m=0 Right-handed