Conserving Color Charge Marian J. R. Gilton University of California, Irvine Gauge Symmetry Charge U(1) Electric charge SU(3) Color charge Noether’s Theorem Symmetry Conserved Quantity $ 1/51 Noether’s Theorem Symmetry Conserved Quantity $ Gauge Symmetry Charge U(1) Electric charge SU(3) Color charge 1/51 Electric Charge 2/51 Electric Charge 3/51 Electric Charge 4/51 μ Noether Theorem: ∂μj = 0. d dtQ = 0. Noether’s Theorem Electric Charge 0 Charge-current density jμ = j + ~j. ∫︀ 0 3 Total charge Q = j d x. 5/51 Noether’s Theorem Electric Charge 0 Charge-current density jμ = j + ~j. ∫︀ 0 3 Total charge Q = j d x. μ Noether Theorem: ∂μj = 0. d dtQ = 0. 5/51 Color Charge 6/51 conservation of red conservation of blue conservation of green Noether’s Theorem Color Charge Current for color charge? 7/51 Noether’s Theorem Color Charge Current for color charge? conservation of red conservation of blue conservation of green 7/51 Question: How does Jaμ relate to red, blue, and green? Currents μ Electric Charge ∂μj = 0 • aμ Color Charge ∂μj = 0 • 8/51 Currents μ Electric Charge ∂μj = 0 • aμ Color Charge ∂μj = 0 • Question: How does Jaμ relate to red, blue, and green? 8/51 Noether’s theorem does show conservation of • a relational quantity of combinations of color and anti-color. Claims Noether’s theorem does not show conservation • of red, blue, and green. 9/51 Claims Noether’s theorem does not show conservation • of red, blue, and green. Noether’s theorem does show conservation of • a relational quantity of combinations of color and anti-color. 9/51 Outline 1. Scalar Electrodynamics 2. Scalar Chromodynamics 3. Group Representations 4. What is Charge? 10/51 Outline 1. Scalar Electrodynamics 2. Scalar Chromodynamics 3. Group Representations 4. What is Charge? 11/51 ϕ is a matter field • Dμ = ∂μ + iqAμ • Scalar Electrodynamics charged scalar field μ ∗ ∗ LE = Dμϕ(D ϕ) mϕϕ − 12/51 Scalar Electrodynamics charged scalar field μ ∗ ∗ LE = Dμϕ(D ϕ) mϕϕ − ϕ is a matter field • Dμ = ∂μ + iqAμ • 12/51 Scalar Electrodynamics Noether’s Theorem μ ∗ ∗ LE = Dμϕ(D ϕ) mϕϕ − μ ∗ μ μ ∗ j = iq(ϕ D ϕ ϕ(D ϕ) ) − 13/51 Q gives total amount of electric charge. Scalar Electrodynamics Noether’s Theorem μ ∗ μ μ ∗ j = iq(ϕ D ϕ ϕ(D ϕ) ) − ∫︁ 3 0 Q = d xj . 14/51 Scalar Electrodynamics Noether’s Theorem μ ∗ μ μ ∗ j = iq(ϕ D ϕ ϕ(D ϕ) ) − ∫︁ 3 0 Q = d xj . Q gives total amount of electric charge. 14/51 μ ∗ μ ∗ μ ∗ Dμϕ(D ϕ) = iqA (ϕ ∂μϕ + ϕ∂ ϕ ) − 2 μ ∗ μ ∗ q AμϕA ϕ + ∂μϕ∂ ϕ . − Scalar Electrodynamics Interactions μ ∗ ∗ LE = Dμϕ(D ϕ) mϕϕ − 15/51 Scalar Electrodynamics Interactions μ ∗ ∗ LE = Dμϕ(D ϕ) mϕϕ − μ ∗ μ ∗ μ ∗ Dμϕ(D ϕ) = iqA (ϕ ∂μϕ + ϕ∂ ϕ ) − 2 μ ∗ μ ∗ q AμϕA ϕ + ∂μϕ∂ ϕ . − 15/51 Scalar Electrodynamics Interactions μ ∗ μ ∗ 2 μ ∗ iqA (ϕ ∂μϕ + ϕ∂ ϕ ) q AμϕA ϕ − 16/51 Outline 1. Scalar Electrodynamics 2. Scalar Chromodynamics 3. Group Representations 4. What is Charge? 17/51 a D ∂ igA μ = μ + μ • – a is for Lie algebra – adjoint representation of SU(3) Color Charge charged scalar field L D Dμ ∗ m ∗ C = μϕi( ϕi) ϕiϕi − ϕi matter field • – i is for red, blue, or green – first fundamental representation of SU(3) 18/51 Color Charge charged scalar field L D Dμ ∗ m ∗ C = μϕi( ϕi) ϕiϕi − ϕi matter field • – i is for red, blue, or green – first fundamental representation of SU(3) a D ∂ igA μ = μ + μ • – a is for Lie algebra – adjoint representation of SU(3) 18/51 Color Charge charged scalar field L D Dμ ∗ m ∗ C = μϕi( ϕi) ϕiϕi − Jμa ig ∗Dμ Dμ ∗ = (ϕi ϕi ϕi( ϕi) ) − ig ∗ igAa igAa ∗ = (ϕi (∂μ + )ϕi ϕi(∂μ )ϕi ) μ − − μ 19/51 Group Representations Group representations hardwired into • the expressions for the current Current transforms according to the • adjoint representation 20/51 Outline 1. Scalar Electrodynamics 2. Scalar Chromodynamics 3. Group Representations 4. What is Charge? 21/51 Concretely: representations. • A representation ρ : G GL(V). • ! The dimension of ρ is the dimension of V. • Groups Representations Abstractly: (G, ). • Lie groups have· lie algebras. • 22/51 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. • 22/51 For SU(3) two fundamental representations: color and anti-color Representations of SU(N) For SU(N), there are N 1 many fundamental representations. − 23/51 Representations of SU(N) For SU(N), there are N 1 many fundamental representations. − For SU(3) two fundamental representations: color and anti-color 23/51 Representations of SU(3) The Fundamental Reps ⎛1⎞ ⎛0⎞ ⎛0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ r = ⎝0⎠, b = ⎝1⎠, g = ⎝0⎠. 0 0 1 3 These define a basis for C . 24/51 Representations of SU(3) A basis for the Lie algebra: ⎛ ⎞ ⎛0 1 0⎞ ⎛0 i 0⎞ ⎛1 0 0⎞ 1 ⎜ ⎟ 2 ⎜ − ⎟ 3 ⎜0 -1 0⎟ λ = ⎝1 0 0⎠ , λ = ⎝ i 0 0⎠ , λ = ⎝0 -1 0⎠, 0 0 0 0 0 0 0 0 0 ⎛0 0 1⎞ ⎛0 0 i⎞ 4 ⎜ ⎟ 5 ⎜ − ⎟ λ = ⎝0 0 0⎠ , λ = ⎝0 0 0 ⎠ , 1 0 0 i 0 0 ⎛ ⎞ ⎛0 0 0⎞ ⎛0 0 0 ⎞ ⎛1 0 0 ⎞ 6 7 8 1 λ ⎜0 0 1⎟ , λ ⎜0 0 i⎟ , λ ⎜0 1 0 ⎟ = ⎝ ⎠ = ⎝ ⎠ = p3 ⎝ ⎠ 0 1 0 0 i −0 0 0 2 − 25/51 Cartan Subalgebra ⎛ ⎞ ⎛ ⎞ ⎛1 0 0⎞ ⎛1 0 0 ⎞ 3 8 1 λ ⎜0 -1 0⎟, λ ⎜0 1 0 ⎟ = ⎝ ⎠ = p3 ⎝ ⎠ 0 0 0 0 0 2 − Distinguish representations of SU(3) by the subalgebra’s eigenvalues, called weights. 26/51 We say that the weight of b is 1 1 . ( , 3 ) − p Representations of SU(3) The Fundamental Reps ⎛ ⎞ ⎛1 0 0⎞⎛0⎞ ⎛0⎞ 3 ⎜0 -1 0⎟⎜ ⎟ ⎜ ⎟ λ (b) = ⎝0 -1 0⎠⎝1⎠ = -1⎝1⎠ 0 0 0 0 0 ⎛ ⎞ ⎛1 0 0 ⎞⎛0⎞ ⎛0⎞ 8 1 1 λ b ⎜0 1 0 ⎟⎜1⎟ ⎜1⎟ ( ) = p3 ⎝ ⎠⎝ ⎠ = p3 ⎝ ⎠ 0 0 2 0 0 − 27/51 Representations of SU(3) The Fundamental Reps ⎛ ⎞ ⎛1 0 0⎞⎛0⎞ ⎛0⎞ 3 ⎜0 -1 0⎟⎜ ⎟ ⎜ ⎟ λ (b) = ⎝0 -1 0⎠⎝1⎠ = -1⎝1⎠ 0 0 0 0 0 ⎛ ⎞ ⎛1 0 0 ⎞⎛0⎞ ⎛0⎞ 8 1 1 λ b ⎜0 1 0 ⎟⎜1⎟ ⎜1⎟ ( ) = p3 ⎝ ⎠⎝ ⎠ = p3 ⎝ ⎠ 0 0 2 0 0 − We say that the weight of b is 1 1 . ( , 3 ) − p 27/51 Representations of SU(3) The Fundamental Reps 28/51 Representations of SU(3) The Fundamental Reps For anti-colors we set: ⎛0⎞ ⎛0⎞ ⎛1⎞ ⎜ ⎟ ¯ ⎜ ⎟ ⎜ ⎟ ¯r = ⎝0⎠, b = ⎝1⎠, g¯ = ⎝0⎠. 1 − 0 0 tr The representation on this space is ρ¯(g) = ρ(g) − 29/51 Representations of SU(3) The Fundamental Reps 30/51 Representations of SU(3) The Fundamental Reps 31/51 We distinguish between inequivalent representations using this weight space. 32/51 Representations of SU(3) L D Dμ ∗ m ∗ C = μϕi( ϕi) ϕiϕi − i = r, b, g fundamental representation • a = 1, 2, ...8 adjoint representation • 33/51 Representations of SU(3) A representation ρ : G GL(V) • ! We call V the “carrier space.” • The adjoint representation: V is the Lie algebra. The action is conjugation: g v gvg 1. ρadj ρadj( )( ) = − 34/51 Adjoint Representation SU(3) ⎛1⎞ ⎛0⎞ ⎛0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜1⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜0⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜0⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟, ⎜ ⎟,... ⎜ ⎟ . ⎜0⎟ ⎜0⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜0⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝0⎠ ⎝0⎠ ⎝0⎠ 0 0 1 35/51 The Adjoint Representation 36/51 Representations of SU(3) 37/51 The Adjoint Representation The weights correspond to basis vectors of V • The carrier space V is the Lie algebra • So: any element of Lie algebra can be written in • this basis 38/51 Interpretation Lie algebra valued quantities are combinations of color and anti-color. 39/51 Color Charge charged scalar field L D Dμ ∗ m ∗ C = μϕi( ϕi) ϕiϕi − Jμa ig ∗Dμ Dμ ∗ = (ϕi ϕi ϕi( ϕi) ) − ig ∗ igAa igAa ∗ = (ϕi (∂μ + )ϕi ϕi(∂μ )ϕi ) μ − − μ 40/51 Noether’s Theorem Color Charge Given SU(3) gauge symmetry, Noether’s thoerem give us μa ∂μJ = 0. The conserved quantity is a combination of color and anti-color. 41/51 Outline 1. Scalar Electrodynamics 2. Scalar Chromodynamics 3. Group Representations 4. What is Charge? 42/51 Color Charge and Electric Charge μ ∗ ∗ LE = Dμϕ(D ϕ) mϕϕ − L D Dμ ∗ m ∗ C = μϕi( ϕi) ϕiϕi − 43/51 U(1) 44/51 Each ρn has C as its carrier space. • The adjoint representation is ρ0. • ρ0 = ρ 1+1. • − The Lie algebra is R. • Representations of U(1) iθ inθ ρn(e ) = e . 45/51 ρ0 = ρ 1+1. • − The Lie algebra is R. • Representations of U(1) iθ inθ ρn(e ) = e . Each ρn has C as its carrier space. • The adjoint representation is ρ0. • 45/51 Representations of U(1) iθ inθ ρn(e ) = e . Each ρn has C as its carrier space. • The adjoint representation is ρ0. • ρ0 = ρ 1+1. • − The Lie algebra is R. • 45/51 Charge exchange rg g¯r 46/51 Electric charge -1 -1 0 47/51 Conclusions Red, blue, and green are not individually • conserved. The conserved quantity is a combination of • charge and anti-charge. Conservation of charge is relational. • 48/51 Thank you! 49/51 ReferencesI [Bleecker, 2013] Bleecker, D. (2013). Gauge Theory and Variational Principles. Dover Books on Mathematics. Dover Publications. [Brading, 2002] Brading, K.