University:
Boston University
Course:
CAS PY 211 | General Physics I
Academic year:

2023
Views:

446

Pages:

19
Author:

Tara Guerrero

Scalar Electrodynamics Interactions LE = Dμ ϕ(Dμ ϕ)∗ − mϕϕ∗ Dμ ϕ(Dμ ϕ)∗ = −iqAμ (ϕ∗ ∂μ ϕ + ϕ∂μ ϕ∗ ) − q2Aμ ϕAμ ϕ∗ + ∂μ ϕ∂μ ϕ∗ . Scalar Electrodynamics Interactions iqAμ (ϕ∗ ∂μ ϕ + ϕ∂μ ϕ∗ ) − q2 A μ ϕ A μ ϕ ∗ Color Charge charged scalar ﬁeld LC = Dμ ϕi (Dμ ϕi )∗ − mϕi ϕi∗ • ϕi matter ﬁeld – i is for red, blue, or green – ﬁrst fundamental representation of SU(3) Color Charge charged scalar ﬁeld LC = Dμ ϕi (Dμ ϕi )∗ − mϕi ϕi∗ • ϕi matter ﬁeld – i is for red, blue, or green – ﬁrst fundamental representation of SU(3) • Dμ = ∂μ + igAa μ – a is for Lie algebra – adjoint representation of SU(3) Color Charge charged scalar ﬁeld LC = Dμ ϕi (Dμ ϕi )∗ − mϕi ϕi∗ Jμa = ig(ϕi∗ Dμ ϕi − ϕi (Dμ ϕi )∗ ) = ig(ϕi∗ (∂μ + igAaμ )ϕi − ϕi (∂μ − igAaμ )ϕi∗ ) Group Representations • Group representations hardwired into the expressions for the current • Current transforms according to the adjoint representation Groups Representations • Abstractly: (G, ·). • Lie groups have lie algebras. Groups Representations • Abstractly: (G, ·). • Lie groups have lie algebras. • Concretely: representations. • A representation ρ : G → GL(V). • The dimension of ρ is the dimension of V. Representations of SU(N) For SU(N), there are N − 1 many fundamental representations. Representations of SU(N) For SU(N), there are N − 1 many fundamental representations. For SU(3) two fundamental representations: color and anti-color Representations of SU(3) The Fundamental Reps ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ r = ⎝0⎠, b = ⎝ 1 ⎠, g = ⎝0⎠. 0 0 1 These deﬁne a basis for C3 . Representations of SU(3) A basis for the Lie algebra: ⎞ ⎛ 0 1 0 ⎟ ⎜ λ 1 = ⎝ 1 0 0⎠ , 0 0 0 ⎛ 0 ⎜ 4 λ = ⎝0 1 ⎛ ⎞ 0 0 0 ⎟ ⎜ 6 λ = ⎝0 0 1 ⎠ , 0 1 0 ⎛ ⎞ ⎛ 1 0 0 −i 0 ⎜ ⎟ ⎜ 3 2 0 0⎠ , λ = ⎝0 -1 λ = ⎝i 0 0 0 0 0 ⎛ ⎞ ⎞ 0 1 0 0 −i ⎟ ⎜ ⎟ 5 0 0 ⎠ , λ = ⎝0 0 0 ⎠ , 0 0 i 0 0 ⎛ ⎞ ⎛ 0 0 0 1 ⎜ ⎟ 1 ⎜ 7 8 p 0 λ = ⎝0 0 −i⎠ , λ = ⎝ 3 0 i 0 0 ⎞ 0 ⎟ 0⎠ , 0 ⎞ 0 0 ⎟ 1 0⎠ 0 −2 Cartan Subalgebra ⎞ 1 0 0 ⎟ ⎜ λ3 = ⎝0 -1 0⎠ , 0 0 0 ⎛ ⎞ 1 0 0 ⎟ 1 ⎜ 0⎠ λ8 = p ⎝0 1 3 0 0 −2 ⎛ Distinguish representations of SU(3) by the subalgebra’s eigenvalues, called weights. Representations of SU(3) The Fundamental Reps ⎛ 1 ⎜ 3 λ (b) = ⎝0 0 ⎛ 1 1 ⎜ 8 p 0 λ (b) = ⎝ 3 0 0 -1 0 0 1 0 ⎛ ⎞ ⎞⎛ ⎞ 0 0 0 ⎜ ⎟ ⎟⎜ ⎟ 0⎠ ⎝ 1 ⎠ = -1⎝ 1 ⎠ 0 0 0 ⎞⎛ ⎞ ⎛ ⎞ 0 0 0 ⎟⎜ ⎟ ⎜ ⎟ 0 ⎠ ⎝ 1 ⎠ = p13 ⎝ 1 ⎠ −2 0 0 Representations of SU(3) The Fundamental Reps ⎛ 1 ⎜ 3 λ (b) = ⎝0 0 ⎛ 1 1 ⎜ 8 p 0 λ (b) = ⎝ 3 0 0 -1 0 0 1 0 ⎛ ⎞ ⎞⎛ ⎞ 0 0 0 ⎜ ⎟ ⎟⎜ ⎟ 0⎠ ⎝ 1 ⎠ = -1⎝ 1 ⎠ 0 0 0 ⎞⎛ ⎞ ⎛ ⎞ 0 0 0 ⎟⎜ ⎟ ⎜ ⎟ 0 ⎠ ⎝ 1 ⎠ = p13 ⎝ 1 ⎠ −2 0 0 We say that the weight of b is (−1, p13 ). Representations of SU(3) The Fundamental Reps Representations of SU(3) The Fundamental Reps For anti-colors we set: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ r̄ = ⎝0⎠, b̄ = − ⎝ 1 ⎠, ḡ = ⎝0⎠. 1 0 0 The representation on this space is ρ̄(g) = − ρ(g)tr Representations of SU(3) The Fundamental Reps Representations of SU(3) The Fundamental Reps