University of California, Irvine · Semi-simple Lie algebras and their representations. Courier...

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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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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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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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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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




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 ElectrodynamicsNoether’s Theorem


jμ = iq(ϕ∗Dμϕ− ϕ(Dμϕ)∗)

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Q =∫

d3xj0.

Q gives total amount of electric charge.

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Q =∫

d3xj0.

Q gives total amount of electric charge.14/51

Scalar ElectrodynamicsInteractions


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




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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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




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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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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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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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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We distinguish between inequivalentrepresentations using this weight space.

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• i = r, b, g fundamental representation

• a = 1, 2, . . .8 adjoint representation

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• 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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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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Jμa = ig(ϕ∗i Dμϕi − ϕi(Dμϕi)∗)

= ig(ϕ∗i (∂μ + igAaμ)ϕi − ϕi(∂μ − igAa

μ)ϕ∗i )

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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




4. What is Charge?

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Color Charge and Electric Charge



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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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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.

Studies in History and Philosophy of Science Part B: Studies in History andPhilosophy of Modern Physics, 33(1):3–22.

[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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