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Symmetries in Particle Physics  Emmy Noether Theorem: Any Symmetry Result in an Conserved Observable

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Symmetries in Particle Physics  Emmy Noether Theorem: Any Symmetry Result in an Conserved Observable Particle Physics WS 2012/13 (30.11.2012) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 226, 3.101 Content of Today Symmetries in Particle Physics Emmy Noether Theorem: any symmetry result in an conserved observable Symmetry of isospin SU(2): „u and d quark are not distinguishable in strong IA“ [this is only an approximative symmetry e.g. m(u)~m(d)] conservation of isospin in strong IA Flavour symmetry SU(3): „ u, d and s quark are not distinguishable in strong IA“ [this is an even more approximative symmetry: m(s) ~ 150 MeV, m(u)~ few MeV] conservation of hypercharge in strong IA Colour symmetry SU(3): „quarks of different colour are not distinguishable among each other“ this is an exact symmetry conservation of colour in all IA Experimental evidence for Colour 2 Reminder: Emmy-Noether Theorem Suppose physics is invariant under (linear) transformation: ψ → ψ‘ = U ψ † T* U = U † 4 † † 4 Ensure normalization: ψ ψ 푑 푥 = 1 ψ 푈 푈ψ 푑 푥 = 1 † 푈 푈 = 1 U has to be unitary To ensure physics is invariant under transformation: † † † ψ † 퐻 ψ 푑4푥 = ψ′ 퐻 ψ′푑4푥 = ψ 푈 퐻 푈 ψ 푑4푥 H = U† H U UH = HU [H, U] = 0 Now consider infinitesimal transformation: U = 1 + i ε G G: generator of transformation U†U = (1 – iεG†)(1+iεG) = 1 + iε(G-G†) + O(ε2) † † 푈 푈 = 1 G = G G is hermitian, thus coresponds to an observable quantity g [H,U] = 0 [H,G] = 0 g is a conserved observable quantity! For each infinite symmetry transformation, there is an conserved observable quantity! 3 Reminder: Emmy-Noether Theorem Example: infinite 1D spatial translation: x → x + δx 휕ψ(푥) 휕 Ψ(x) → ψ‘(x) = ψ(x + δx) = ψ(x) + δx = (1+ δx ) ψ(x) 휕푥 휕푥 휕 U P = -i ψ‘(x) = (1-i δx P )ψ(x) x 휕푥 x U x component of momentum is conserved! In general the symmetry operation may depend on more than one parameters: U = 1 + i ε 퐺 in our example: 푟 → 푟 + δ푥 U = 1 – i δ푥 푃 A finite transformation (α) can be expressed as a series of infinitesimal transformations: 푛 α U(α) = lim 1 + 푖 퐺 = 푒푖α퐺 푛→∞ 푛 Emmy Noether Theorem: For each symmetry transformation, there is an conserved quantity4! Symmetries in Particle Physics: Isospin Proposed by Heisenberg (1932) : Proton and neutron are same particles with respect to strong IA! Indications: m(p) ~ m(n) and nuclear binding energies (if one ignores Coulomb IA) are similar 1 0 Proton and neutron are two states of single particle: the nucleon p= ; n= ; 0 1 In analogy to spin, call this isospin. Expect physics to be invariant under rotation in this space Later extended idea to u, d quark: m(u) ~ m(d) 1 0 푢′ 푢 푢 푢 푢 u = , d = , = U = 11 12 0 1 푑′ 푑 푢21 푢22 푑 rotation in 2D u-d quark space Strong IA is invariant to rotation in isospin space, U: 2x2 unitary matrix 4 complex matrix elements 8 real parameters + 4 unitary conditions 4 parameters In group theory these matrices form U(2) group unitary matrix 2D 5 Reminder: Definition of a Group * is operation which connects elements of a group There is a neutral element e, that for all group elements a: e*a = a = a*e Completeness: a, b element of the group → a*b as well element of the group For each element a there is an inverse element a-1, so that a*a-1 = a-1*a = e Example 1: entire numbers, “+” is group operation, “0” is neutral element Example 2: rational numbers without 0, “*” is group operation, “1” is neutral element Example 3: element a: rotation around 60o, “*” is group operation, 2 3 4 5 6 complete list of group elements: {a,a ,a ,a ,a ,a =e} → finite group, dimension 6 Subgroup: element a: rotation around 120° 2 4 6 {a ,a ,a =e} → finite group, dimension 3 6 Isospin 1 0 1 0 U = 푒푖φ; U †= 푒−푖φ; U U †=1 Ψ‘ = U1ψ for all wave functions 1 0 1 1 0 1 1 1 not distinbuishable from original system U1 ϵ U(2), however U1 has no physical relevance: absolute phases cannot be measured Left with three independent matrices which form SU(2) subgroup of U(2) special All unitary matrices have |det(U)| = 1, all elements of SU(2) have det(U)=1. 1 + 푖ε퐺11 푖ε퐺12 2 U = 1 + iεG det(U) = det = 1 + iε(G11+G22) + O(ε ) 푖ε퐺21 1 + 푖ε퐺22 G11+G22 = Trace(G) = 0 Generators of rotation in isospin space are traceless 2x2 matrices A set of linear independent traceless 2x2 matrices are e.g. the Pauli matrices 0 1 0 −푖 1 0 σ = ; σ = ; σ = Note: Isospin has NOTHING 1 1 0 2 푖 0 3 0 −1 to do with spin – JUST SAME MATH 1 2D represenation of generator of Isospin symmetry group: 퐼 = 퐺 = σ 2 7 Isospin SU(2) – 2D representation 1 0 states: |u> = |d> = 0 1 symmetry transformation: U = 푒푖ασ commutator relations: [σi, σj] = 2iεijkσk generators: σ1, σ2, σ3 σ σ isospin operators : 푇 = ; 푇 = 푖 i=1,2,3 2 푖 2 1 1 0 1 1 1 1 0 0 1 3. component: T |u> = = |푢 > T |d> = = − |푑 > 3 2 0 −1 0 2 3 2 0 −1 1 2 1 1 3 casimir operator: 푇2 |푢 > = (σ 2 + σ 2 + σ 2) = |푢 > 4 1 2 3 0 4 1 0 3 푇2 |푑 > = (σ 2 + σ 2 + σ 2) = |푑 > 4 1 2 3 1 4 2 Commutation properties: [푇 , Ti] for i=1,2,3 = 0 [Tj, Ti] ≠ 0 for i≠j only two observables simultanously measurable 2 Choose common eigenstages of 푇 and T3 : |I,I3> 2 푇 퐼, 퐼3 > = 퐼 퐼 + 1 퐼, 퐼3 > T3 퐼, 퐼3 > = 퐼3 퐼, 퐼3 > 8 Isospin SU(2) – 2D representation I I I I 3 1 3 0 states: |u> = |1/2, 1/2> = |d> = |1/2, -1/2> = 0 1 0 1 0 0 ladder operator: T = T + i T = T = T - i T = + 1 2 0 0 - 1 2 1 0 T+|u> = 0 T+|d> = |u> T_|u> = |d> T_|d> = 0 T+ T_ d u -1/2 +1/2 I3 Isospin is generator of symmetry of strong IA, thus isopin (I, I3) are conserved in strong IA! 9 General Properties of Isospin Operators To describe a set of total isospin = I, need (2I+1) dimensional representation of SU(2) For any represenation of SU(2), following relation are valid: [Ti, Tj] = 2iεijkIk T3|I,I3> = I3|I,I3> 2 푇 퐼, 퐼3 > = 퐼 퐼 + 1 퐼, 퐼3 > T± |I,I3> = 퐼 ∓ 퐼3 퐼 ± 퐼3 + 1 |퐼, 퐼3 ± 1 > 2 [T3, 푇 ] = 0 10 Isospin SU(2) – 3D representation 1 0 0 states: |1, 1> = 0 |1, 0> = 1 |1,-1> = 0 e.g. π+ , π0, π- 0 0 1 0 1 0 0 −푖 0 1 0 0 1 1 generators: J = 1 0 1 J = 푖 0 −푖 J = 0 0 0 1 2 2 2 3 0 1 0 0 푖 0 0 0 −1 commutator relations: [Ji, Jj] = 2iεijkJk isospin operators : 푇 = 퐽; 푇3 = 퐽3 3. component: T3|1,1> = 1,1 > T3|1,0> = 0 T3|1,−1> = − 1, −1 > 1 0 0 2 2 2 2 casimir operator: 푇 = 퐽1 + 퐽2 + 퐽3 = 2 0 1 0 0 0 1 0 1 0 0 0 0 ladder operator: T+ = T1 + i T2 = 2 0 0 1 T- = T1 - i T2 = 2 1 0 0 0 0 0 0 1 0 11 Combining (Iso-)spins (goal: get wave function of proton) general rule (again identically to angular momentum): I3 components are added, I ≥ I3 |uu> = |1/2,1/2> |1/2, 1/2> = |1,+1> |dd> = |1/2,-1/2> |1/2, -1/2> = |1,-1> T _ |1,1> = 2 |1,0 > |푢 > How does |1,0> look like? T_ |uu> = (T_|u>) + |u>(T_ |u>) = |d>|u> + |u>|d> 1 |1,0> = (|du> + |ud>) 2 Instead of studying ladder operators consider symmetry arguments: |1,0> is linear combination of |1,1> and |1,-1> ; |1,1> and |1,-1> are symmetric |1,0> must be symmetric 1 |0,0> orthogonal to |1,0> |0,0> = (|du> - |ud> 2 triplett symmetrisch under singlett antisymmetrisch under exchange of quarks exchange of quarks |1,1> = |uu> 1 1 |1,0> = (|ud> + |du>) |0,0> = (|du> - |ud>) 2 2 |1,-1> = |dd> 12 Combining (Iso)spins From four possible combinations of isospin doublets obtain a triplet of isospin 1 states and a singlet isospin 0 state: 2 ⨂ 2 = 3 ⨁ 1 1 dd (ud+du) uu 2 1 (du-du) I3 ⨁ 2 -1 0 +1 0 I3 T± T± Can move around within multiplets using ladder operators. States with different total isospin are physically different: the isospin 1 triplet is symmetric Under interchange of quarks where as singlet is anti-symmetric. Now add additional up or down quark ddu uud 1 1 1 1 ddd (ud+du)d (ud+du)u uuu (du-du)d (du - du) u 2 2 2 2 ⨁ I3 I -3/2 -1/2 +1/2 +3/2 -1/2 +1/2 3 6 2 Use ladder operators and orthogonality to group the 6 states into isospin multiplets. 13 Combining (Iso)Spins T± |I,I3> = 푰 ∓ 푰ퟑ 푰 ± 푰ퟑ + ퟏ |푰, 푰ퟑ ± ퟏ > 14 Baryon Isospin Wave Function 2 ⨂ 2 ⨂ 2 = (3 ⨁1) ⨂ 2 = 4 ⨁ 2 ⨁ 2 Different multiplets have different symmetry properties: |3/2,3/2> = |uuu> 1 S: Sym quartett |3/2,1/2> = (|uud> + |udu> + |duu>) 3 Symmetric under interchange 1 |3/2,-1/2> = (|udd> + |dud> + |ddu>) Of any quark 3 |3/2,-3/2> = |ddd> 1 |1/2,1/2> = (2|uud> - |udu> - |duu>) 6 Ms: Mixed sym.
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