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. A. (2002). Which symmetry? noether, weyl, and conservation of electric charge. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 33(1):3–22.
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[Forger and Römer, 2004] Forger, M. and Römer, H. (2004). Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem.
Annals of Physics, 309(2):306–389. 50/51 ReferencesII [Hall, 2015] Hall, B. C. (2015). Lie groups, Lie algebras, and representations: an elementary introduction, volume 222. Springer.
[Hamilton, 2017] Hamilton, M. J. (2017). Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics. Springer.
[Muta, 2009] Muta, T. (2009). Foundations of Quantum Chromodynamics: An Introduction to Perturbative Methods in Gauge Theories, volume 78. World Scientific Publishing Co Inc.
[Sardanashvily, 2016] Sardanashvily, G. (2016). Noether’s Theorems. Springer. 51/51 SU(3) Adjoint
Three pairs of raising and lowering operators:
E 1,0 = (1/2)(λ1 iλ2) red states blue states. ± E 1 2 ± i red states green states. 1/2, p3/2 = ( / )(λ4 λ5) ± ± E 1 2 ± i blue states green states. 1/2, p3/2 = ( / )(λ6 λ7) ∓ ± ±
52/51 SU(3) Adjoint
53/51 Yang-Mills Theory
1 aμν LYM = F Faμν − 4 1 Jaμ D Faμν = g2 μ
54/51 Charged Gauge Field
1 L D Dμ ∗ m ∗ FaμνF C = μϕi( ϕi) ϕiϕi aμν − − 4
55/51 Covariant Derivative
56/51