Post on 08-Jul-2020
Conserving Color Charge
Marian J. R. GiltonUniversity of California, Irvine
Noether’s Theorem
Symmetry ↔ Conserved Quantity
Gauge Symmetry Charge
U(1) Electric chargeSU(3) Color charge
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Noether’s Theorem
Symmetry ↔ Conserved Quantity
Gauge Symmetry Charge
U(1) Electric chargeSU(3) Color charge
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Electric Charge
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Electric Charge
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Electric Charge
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Noether’s TheoremElectric Charge
Charge-current density jμ = j0 + ~j.
Total charge Q =∫j0d3x.
Noether Theorem: ∂μjμ = 0.ddtQ = 0.
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Noether’s TheoremElectric Charge
Charge-current density jμ = j0 + ~j.
Total charge Q =∫j0d3x.
Noether Theorem: ∂μjμ = 0.ddtQ = 0.
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Color Charge
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Noether’s TheoremColor Charge
Current for color charge?
conservation of redconservation of blueconservation of green
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Noether’s TheoremColor Charge
Current for color charge?conservation of redconservation of blueconservation of green
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Currents
• Electric Charge ∂μjμ = 0
• Color Charge ∂μjaμ = 0
Question: How does Jaμ relate to red, blue,and green?
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Currents
• Electric Charge ∂μjμ = 0
• Color Charge ∂μjaμ = 0
Question: How does Jaμ relate to red, blue,and green?
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Claims
• Noether’s theorem does not show conservationof red, blue, and green.
• Noether’s theorem does show conservation ofa relational quantity of combinations of colorand anti-color.
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Claims
• Noether’s theorem does not show conservationof red, blue, and green.
• Noether’s theorem does show conservation ofa relational quantity of combinations of colorand anti-color.
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Outline
1. Scalar Electrodynamics
2. Scalar Chromodynamics
3. Group Representations
4. What is Charge?
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Outline
1. Scalar Electrodynamics
2. Scalar Chromodynamics
3. Group Representations
4. What is Charge?
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Scalar Electrodynamicscharged scalar field
LE = Dμϕ(Dμϕ)∗ − mϕϕ∗
• ϕ is a matter field
• Dμ = ∂μ + iqAμ
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Scalar Electrodynamicscharged scalar field
LE = Dμϕ(Dμϕ)∗ − mϕϕ∗
• ϕ is a matter field
• Dμ = ∂μ + iqAμ
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Scalar ElectrodynamicsNoether’s Theorem
LE = Dμϕ(Dμϕ)∗ − mϕϕ∗
jμ = iq(ϕ∗Dμϕ− ϕ(Dμϕ)∗)
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Scalar ElectrodynamicsNoether’s Theorem
jμ = iq(ϕ∗Dμϕ− ϕ(Dμϕ)∗)
Q =∫
d3xj0.
Q gives total amount of electric charge.
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Scalar ElectrodynamicsNoether’s Theorem
jμ = iq(ϕ∗Dμϕ− ϕ(Dμϕ)∗)
Q =∫
d3xj0.
Q gives total amount of electric charge.14/51
Scalar ElectrodynamicsInteractions
LE = Dμϕ(Dμϕ)∗ − mϕϕ∗
Dμϕ(Dμϕ)∗ = −iqAμ(ϕ∗∂μϕ+ ϕ∂μϕ∗)
− q2AμϕAμϕ∗ + ∂μϕ∂μϕ∗.
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Scalar ElectrodynamicsInteractions
LE = Dμϕ(Dμϕ)∗ − mϕϕ∗
Dμϕ(Dμϕ)∗ = −iqAμ(ϕ∗∂μϕ+ ϕ∂μϕ∗)
− q2AμϕAμϕ∗ + ∂μϕ∂μϕ∗.
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Scalar ElectrodynamicsInteractions
iqAμ(ϕ∗∂μϕ+ ϕ∂μϕ∗) − q2AμϕAμϕ∗
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Outline
1. Scalar Electrodynamics
2. Scalar Chromodynamics
3. Group Representations
4. What is Charge?
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Color Chargecharged scalar field
LC = Dμϕi(Dμϕi)∗ − mϕiϕ∗i
• ϕi matter field– i is for red, blue, or green– first fundamental representation of SU(3)
• Dμ = ∂μ + igAaμ
– a is for Lie algebra– adjoint representation of SU(3)
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Color Chargecharged scalar field
LC = Dμϕi(Dμϕi)∗ − mϕiϕ∗i
• ϕi matter field– i is for red, blue, or green– first fundamental representation of SU(3)
• Dμ = ∂μ + igAaμ
– a is for Lie algebra– adjoint representation of SU(3)
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Color Chargecharged scalar field
LC = Dμϕi(Dμϕi)∗ − mϕiϕ∗i
Jμa = ig(ϕ∗i Dμϕi − ϕi(Dμϕi)∗)
= ig(ϕ∗i (∂μ + igAaμ)ϕi − ϕi(∂μ − igAa
μ)ϕ∗i )
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Group Representations
• Group representations hardwired intothe expressions for the current
• Current transforms according to theadjoint representation
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Outline
1. Scalar Electrodynamics
2. Scalar Chromodynamics
3. Group Representations
4. What is Charge?
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GroupsRepresentations
• Abstractly: (G, ·).• Lie groups have lie algebras.
• Concretely: representations.
• A representation ρ : G→ GL(V).
• The dimension of ρ is the dimension of V.
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GroupsRepresentations
• Abstractly: (G, ·).• Lie groups have lie algebras.
• Concretely: representations.
• A representation ρ : G→ GL(V).
• The dimension of ρ is the dimension of V.
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Representations of SU(N)
For SU(N), there are N− 1 many fundamentalrepresentations.
For SU(3) two fundamental representations: colorand anti-color
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Representations of SU(N)
For SU(N), there are N− 1 many fundamentalrepresentations.
For SU(3) two fundamental representations: colorand anti-color
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Representations of SU(3)The Fundamental Reps
r =
⎛⎜⎝100
⎞⎟⎠, b =
⎛⎜⎝010
⎞⎟⎠, g =
⎛⎜⎝001
⎞⎟⎠.
These define a basis for C3.
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Representations of SU(3)
A basis for the Lie algebra:
λ1 =
⎛⎜⎝0 1 01 0 00 0 0
⎞⎟⎠ , λ2 =
⎛⎜⎝0 −i 0i 0 00 0 0
⎞⎟⎠ , λ3 =
⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠,
λ4 =
⎛⎜⎝0 0 10 0 01 0 0
⎞⎟⎠ , λ5 =
⎛⎜⎝0 0 −i0 0 0i 0 0
⎞⎟⎠ ,
λ6 =
⎛⎜⎝0 0 00 0 10 1 0
⎞⎟⎠ , λ7 =
⎛⎜⎝0 0 00 0 −i0 i 0
⎞⎟⎠ , λ8 = 1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠25/51
Cartan Subalgebra
λ3 =
⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠, λ8 = 1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠
Distinguish representations of SU(3) by the subalgebra’seigenvalues, called weights.
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Representations of SU(3)The Fundamental Reps
λ3(b) =
⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝010
⎞⎟⎠ = -1
⎛⎜⎝010
⎞⎟⎠
λ8(b) = 1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠⎛⎜⎝010
⎞⎟⎠ = 1p3
⎛⎜⎝010
⎞⎟⎠
We say that the weight of b is (−1, 1p3).
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Representations of SU(3)The Fundamental Reps
λ3(b) =
⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝1 0 00 -1 00 0 0
⎞⎟⎠⎛⎜⎝010
⎞⎟⎠ = -1
⎛⎜⎝010
⎞⎟⎠
λ8(b) = 1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠1p3
⎛⎜⎝ 1 0 00 1 00 0 −2
⎞⎟⎠⎛⎜⎝010
⎞⎟⎠ = 1p3
⎛⎜⎝010
⎞⎟⎠
We say that the weight of b is (−1, 1p3).
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Representations of SU(3)The Fundamental Reps
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Representations of SU(3)The Fundamental Reps
For anti-colors we set:
r =
⎛⎜⎝001
⎞⎟⎠, b = −
⎛⎜⎝010
⎞⎟⎠, g =
⎛⎜⎝100
⎞⎟⎠.
The representation on this space is ρ(g) = −ρ(g)tr
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Representations of SU(3)The Fundamental Reps
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Representations of SU(3)The Fundamental Reps
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We distinguish between inequivalentrepresentations using this weight space.
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Representations of SU(3)
LC = Dμϕi(Dμϕi)∗ − mϕiϕ∗i
• i = r, b, g fundamental representation
• a = 1, 2, . . .8 adjoint representation
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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 ρadj is conjugation: ρadj(g)(v) = gvg−1.
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Adjoint RepresentationSU(3)
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
10000000
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
01000000
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, . . .
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
00000001
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.
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The Adjoint Representation
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Representations of SU(3)
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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 inthis basis
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Interpretation
Lie algebra valued quantities are combinations ofcolor and anti-color.
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Color Chargecharged scalar field
LC = Dμϕi(Dμϕi)∗ − mϕiϕ∗i
Jμa = ig(ϕ∗i Dμϕi − ϕi(Dμϕi)∗)
= ig(ϕ∗i (∂μ + igAaμ)ϕi − ϕi(∂μ − igAa
μ)ϕ∗i )
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Noether’s TheoremColor Charge
Given SU(3) gauge symmetry, Noether’s thoeremgive us
∂μJμa = 0.
The conserved quantity is a combination of colorand anti-color.
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Outline
1. Scalar Electrodynamics
2. Scalar Chromodynamics
3. Group Representations
4. What is Charge?
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Color Charge and Electric Charge
LE = Dμϕ(Dμϕ)∗ − mϕϕ∗
LC = Dμϕi(Dμϕi)∗ − mϕiϕ∗i
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U(1)
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Representations of U(1)
ρn(eiθ) = einθ.
• Each ρn has C as its carrier space.
• The adjoint representation is ρ0.
• ρ0 = ρ−1+1.
• The Lie algebra is R.
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Representations of U(1)
ρn(eiθ) = einθ.
• Each ρn has C as its carrier space.
• The adjoint representation is ρ0.
• ρ0 = ρ−1+1.
• The Lie algebra is R.
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Representations of U(1)
ρn(eiθ) = einθ.
• Each ρn has C as its carrier space.
• The adjoint representation is ρ0.
• ρ0 = ρ−1+1.
• The Lie algebra is R.
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Charge exchange
r g
gr46/51
Electric charge
-1 -1
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Conclusions
• Red, blue, and green are not individuallyconserved.
• The conserved quantity is a combination ofcharge and anti-charge.
• Conservation of charge is relational.
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Thank you!
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References I
[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.
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[Cahn, 2014] Cahn, R. N. (2014).
Semi-simple Lie algebras and their representations.
Courier Corporation.
[Forger and Römer, 2004] Forger, M. and Römer, H. (2004).
Currents and the energy-momentum tensor in classical field theory: a fresh lookat an old problem.
Annals of Physics, 309(2):306–389. 50/51
References II[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 ofParticle Physics.
Springer.
[Muta, 2009] Muta, T. (2009).
Foundations of Quantum Chromodynamics: An Introduction to PerturbativeMethods 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,±p3/2 = (1/2)(λ4 ± iλ5) red states� green states.E∓1/2,±p3/2 = (1/2)(λ6 ± iλ7) blue states� green states.
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SU(3) Adjoint
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Yang-Mills Theory
LYM = −1
4FaμνFaμν
Jaμ =1
g2DμFaμν
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Charged Gauge Field
LC = Dμϕi(Dμϕi)∗ − mϕiϕ∗i −
1
4FaμνFaμν
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Covariant Derivative
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