U(1) Abelian Symmetry The Lagrangian is invariant under the phase transformation of the field operator: invariant

If A,B,C become complex, they carry charges! The interaction is invariant only if U(1) symmetry is related to charge conservation!

The Dirac Fermion Lagrangian is also invariant under U(1)

SU(N) Non-Abelian Symmetry Assume there are N kinds of fields If they are similar, we have a SU(N) symmetry! are invariant under SU(N)!

量子力學下互換群卻變得更大！ 量子力學容許量子態的疊加 u a u + b d u d d c u + d d 古典 量子 u-d 互換對稱

They are invariant under SU(N)!

Gauge symmetry

Global Symmetry Gauge (Local) symmetry Kinetic energy is not invariant under gauge transformation!

Could we find a new “derivative” that works as if the transformation is global? To get rid of the extra term, we introduce a new vector field:

Global Symmetry Gauge (Local) symmetry Replacing derivative with covariant derivative, is invariant under gauge transformation!

The scalar photon interaction vertices

To force it to be gauge invariant, you only need to replace derivative with coariant derivative. is gauge invariant!

This gauge invariant Lagrangian gives a definite interaction between fermions and photons

This form is forced upon us by gauge symmetry! It is really a Fearful Symmetry! Tony Zee Tyger! Tyger! burning bright In the forests of the night What immortal hand or eye Could frame thy fearful symmetry! William Blake

Let there be light! In the name of gauge symmetry!

Hermann Weyl, 1885-1955

Yang and Mills

SU(N) Non-Abelian Symmetry Assume there are N kinds of fields If they are similar, we have a SU(N) symmetry! are invariant under SU(N)!

Non-Abelian Gauge Symmetry We need one gauge field for each generator. Gauge fields transform as: is invariant under gauge transformation!

2 × 2 matrices

Vacua happen at: Choose: For infinitesimal transformation: SU(2)χU(1)Y is broken into U(1)EM

Z become massive W become massive Photon is massless.