internal symmetry :SU(3) c ×SU(2) L ×U(1) Y standard model ( 標準模型 ) quarks SU(3) c leptons L Higgs scalar Lorentzian invariance, locality, renormalizability, SU(3) c :color, SU(2) L :weak iso spin U(1) Y : hypercharge gauge symmetry hypercharge requirements: fields : SU(3) c : SU(2) L : U(1) Y gauge bosons 3 fermions R 1 SU(2) L 2 11 1/   2/3  4/ L R matter fields

U(1) symmetry of phase transformations Symmetry Groups U(1)  group of multiplications by U  e  i  with real . f → f '  e  i  f The transformations are commutative. Abelian group Fields are transformed as Infinitesimal transformation the only irreducible representation is one dimensional. f → f '  f  i  f  f  i  f f † f is invariant under global and gauge transformation. f † → f † '  e i  f † Then f † f → f † ' f '  e i  f † e  i  f  f † f ∴ called commutative group or Invariants under all the infinitesimal transformation is invariant under the whole connected part of the group Because the transformations are commutative.

Example 2: O(3) symmetry of space rotations X : infinitesimal rotation  angular momentum generator commutators representations on Fock space commutators irreducible representations are specified by a harf integer j 2 j +1dimensional representation O(3)  group of 3×3 real matrices A with AA t  1(orthogonal) o(3)  Lie Algebra of X such that A  e  iX ∊ O(3) space rotation of o(3) The transformations are not commutative. non-Abelian group non-commutative group

Example 3: SU(2) isospin symmetry  isospin generator commutators representations on Fock space commutators irreducible representations are specified by a harf integer i 2 i +1dimensional representation SU(2)  group of 2×2 complex matrices U with UU † =1 (unitary) & det U = 1 (special) SU(2) is homomorphic to O(3) su(2)  Lie Algebra of X such that e  iX ∊ SU(2)  i Pauli matrices non-commutative group

Example 4: SU(3) unitary symmetry  unitary spin generator commutators representations on Fock space commutators irreducible representations are specified by two integers SU(3)  group of complex 3×3 matrices U with UU † =1 (unitary) & det U = 1 (special) su(3)  Lie Algebra of X such that e  iX ∊ SU(3) i Gell-mann matrices *815 26*15*27 310* non-commutative group

Example: U(1) symmetry Global Symmetry and Gauge Symmetry transformations are f → f '  e  i  f f † f is invariant under global and gauge transformation. Fields are transformed as the spacetime coordinates. independent of dependent on Global Gauge but not invariant under global transformation, because ∂  f →∂  f '  e  i  ∂  f  i ∂   e  i  f ∂  f ' † ∂  f '  (∂  f  if ∂   )(∂  f  if ∂   ) ≠ ∂  f † ∂  f  e  i  (∂  f  if ∂   ) ∂  f † ∂  f is invariant under global transformation, f † → f † '  e i  f †

gauge invariant Lagrangian covariant derivative gauge field sistem with

standard model ( 標準模型 )