Emmy-Noether-theorem

For each symmetry in nature, there is one conserved observable quantity!

symmetry transformation(U is unitary matrix)

G: generator of symmetry group

G is hermitian operator, thus related to a physics observable

physics invariant under U transformation, thus [U,H] = 0 → [G,H] = 0

G is operator of a conserved physics observable

infinitesimal small transformation finite transformation

One example: SU(2) Isospin symmetry

Heisenberg(1932)

Proton and neutron are same particles with respect to strong IA (ignoring charge)!This concept was then later extended to u, d quarks.

physics invariant under rotation in isospin space

U: 2 x2 unitary matrix → 4 complex parameters → 8 real parameters + 4 unitary conditions → 4 independent parameters, thus 4 generators of U(2) group

physics is symmetric under an overall phase transformation it is however not a rotation in isospin space

→ three generators are left which form SU(2) [subgroup of U(2) with det(U)=1]→ generators G of SU(2) are traceless matrices

2D representation of generators of SU(2) are Pauli matrices

Isospin SU(2) – 2 D representation

states

symmetry transformation

generators

commutator relations

isopsin operators

3. component

Casimir operator

ladder operator

Isospin SU(2) – 3 D representation

states

generators

commutator relations

isopsin operators

3. component

Casimir operator

ladder operator

To descripe set of total isospin = J, need (2J+1)-dimensional representation of SU(2)

For any representation of SU(2), following relations are valid:

Gell-Mann Matrices

SU(3) fundamental representation

anti quarks have opposite quantum numbers

“-” is a definition in the choice of GellMann matrices for anti-quarks

Y

I3

Combining two quarks to mesons

What is composition of Y=0, I3=0 state?

Exploit ladder operators

2 independent states, third state not part of multiplet

Combining two quarks to mesons

Charged and neutral pions are member of same isospin doublet, thus should be in same flavour multiplet as well.

singulet state must be symmetric in flavour

exploiting orthogonality:

You can apply any ladder operator on singuletand it yields 0.

Cross-section of e+e- → hadrons

e+e- - 2 jet events

Colour SU(3)

Confinment and Colour Singlet

QCD Interaction

Connection between Lagrangien, Feynman rules and Symmetries

To each Lagrangian, there corresponds a set of Feynman rules. Once theserules are identified, the connection is made.

If physics is symmetric under some transformation, the Lagrangian must reflect thissymmetry. Thus it need to be extended by (interaction) terms.

Symmetries define Lagrangien, Lagrangien defines interaction

Excursion to Field theory: reminder Lagrange formalism

difference of kinetic and potential energy of system

generalized coordiates

Euler-Lagrange equation to determine equation of motion.

Particle are described by wave functions which depend on 4-momenta

Lagrangian of Spin = ½ particle

Dirac equation for adjoint operators

Invariance under U(1) Symmetry transformation

local (depending on x) phase transformation

breaks invariance

Introduce modified (covariant) derivatives with following translation behaviour

This can be accomplished by

were transforms like:

Invariant under phase transformation

Local phase invariance not possible for free particle!

vertex factorcoupling of fermion currentto photon field

If A is associated to photon field, kinetic energy of photon field is missing in Lagrangiengeneral form of kinetic energy

Additional term needs to be invariant under

Phase transformation invariant Lagrangian → QED (conserved property: el. charge)

L = “ψψ” + e”ψψA” + “A2”

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Study Lagrangien under SU(3) transformation

again local phase spoils invariance

study on small translation

apply same trick:

coupling of quark current to vector field

Up to hear analogous to QED, however need to take care of non-abelian gauge theory!

Apply second translation on Lagrangian

matrices do not commute, need additional gauge term

Now still need kinematical term for each of the (gauge/gluon) fields which is invariant under above transformation

L = “qq” + “G2” + g”qqG” + g”G3” + g2”G4”

Non-abelian symmetry group resulted in 3 and 4 gluon vertices.

The symmetry group determines the interaction!

Now we can go ahead and use Feynman rules to compute processes.