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internal symmetry :SU(3) c ×SU(2) L ×U(1) Y

standard model ( 標準模型 ). renormalizability,. requirements:. Lorentzian invariance,. locality,. internal symmetry :SU(3) c ×SU(2) L ×U(1) Y. gauge symmetry. SU(3) c :color, SU(2) L :weak iso spin U(1) Y : hypercharge. fields. gauge bosons. : SU(3) c. : U(1) Y. : SU(2) L. SU(2) L.

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internal symmetry :SU(3) c ×SU(2) L ×U(1) Y

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  1. standard model(標準模型) renormalizability, requirements: Lorentzian invariance, locality, internal symmetry :SU(3)c×SU(2)L×U(1)Y gauge symmetry SU(3)c:color, SU(2)L:weak iso spin U(1)Y: hypercharge fields gauge bosons : SU(3)c : U(1)Y : SU(2)L SU(2)L hypercharge matter fields SU(3)c L R L R fermions quarks 4/3 1/3 3 1 2 -2/3 leptons 0 -1 1 1 2 -2 1 Higgs scalar 2 1

  2. Symmetry Groups U(1) symmetry of phase transformations U(1) = group of multiplications by U = e-ia with real a. The transformations are commutative. Abelian group called commutative group or f→f '= e-iaf Fields are transformed as Then f†→f †'= eiaf † = f †f f†f →f †'f ' = eiaf †e-iaf ∴ f †f is invariant under global and gauge transformation. df = -iaf f→f '= f - iaf Infinitesimal transformation Invariants under all the infinitesimal transformation is invariant under the whole connected part of the group Because the transformations are commutative. the only irreducible representationis one dimensional.

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

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

  5. Example 4: SU(3) unitary symmetry SU(3)=group of complex 3×3 matrices U withUU†=1 (unitary) & det U = 1 (special) non-commutative group su(3) =Lie Algebra of X such that e-iX∊SU(3) generator li Gell-mann matrices commutators = unitary spin representations on Fock space commutators irreducible representations are specified by two integers

  6. Global Symmetry and Gauge Symmetry Global Gauge independent of dependent on transformations are the spacetime coordinates. Example: U(1) symmetry f→f '=e-iaf Fields are transformed as f†→f †'= eiaf † f †f is invariant under global and gauge transformation. ∂m f †∂m f is invariant under global transformation, but not invariant under global transformation, because = e-ia (∂mf-if ∂ma) ∂mf→∂mf '= e-ia∂mf- i ∂mae-iaf ∂mf'†∂mf'=(∂mf+if ∂ma)(∂mf-if ∂ma ) ≠ ∂mf †∂mf

  7. gauge field covariant derivative sistem with gauge invariant Lagrangian

  8. standard model(標準模型)

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