Symmetries and particle physics Exercises

Transcription

1 Symmetries and particle physics Exercises Stefan Flörchinger SS 017 1

2 Lecture From the lecture we know that the dihedral group of order has the presentation D = a, b a = e, b = e, bab 1 = a 1. Moreover we introduced the quaternion group Q which has a representation {1, e 1, e, e, 1, e 1, e, e } on the vector space of quaternions H where (1, e 1, e, e ) is a basis of H. They obey the quaternion algebra e 1 e = e e 1 = e e i = 1 i {1,, }, e e 1 = e 1 e = e e e = e e = e 1. Exercise.1: Write down the multiplication table of the D. Exercise.: Give a presentation of the dihedral group of order n. Exercise.: defines a norm on the vector space H. N : H R 0 q q q Lecture Let G be a group and R a representation of G on some vector space V. Let M(g) R denote the action of g on V. The representation is reducible if M M(g) = 1 (g) 0 N(g) M (g) with M 1 (g) GL(V 1 ) acting on a subspace V 1 V, M (g) GL(V1 ) acting on the orthorgonal complemet V1 V and N some ( dim(v ) dim(v 1 ) ) dim(v 1 ) matrix. Exercise.1: Let G be a group and H its maximal normal subgroup. G /H is simple. Exercise.: the alternating group A 5 is the first simple non-abelian alternating group.

3 Exercise.: W = 1 ord(g) M (g 1 ) N(g) G g diagonalizes M(g) g G such that 1 0 M 1 (g) 0 W 1 N(g) M (g) 1 0 M = 1 (g) 0 W 1 0 M (g). Exercise.4: Let R a R b be two irreducible representations of a group G on vector spaces V a and V b respectivley. For g G let M a (g) denote the action on V a and M b (g) the action on V b. Define S := g G M a (g) N M b (g 1 ) where N is an arbitrary dim(v a ) dim(v b ) matrix and show that holds. M a (g) S = S M b (g) Lecture 5 In our study of the su() Lie algebra we considered representations on Hilbert spaces with states c, m which are labelled by eigenvalues of the Casimir operator C := (T 1 ) + (T ) + (T ) and the third generator of the Lie algebra, T. The generators are defined in terms of raising and lowering operators T 1 := 1 ( T + T +), T := i (T T +) where d (±) m T ± c, m = d (±) m c, m ± 1 R is such that the states form an orthonormal basis of the Hilbert space. Since the Casimir is positive definite there need to be a the lowest weight state c, j and highest weight state c, k which are subject to T + c, j = 0, T c, k = 0. Since the value of the Casimir c is determined by the Dynkin label j we label the states j, m. Further we introduced the groups generated by the exponentiation U(θ) = e iθ AT A, θ := θ 1 θ R θ where in the fundamental representation the group SU() is generated which group elements are of the form θ θ U(θ) = cos 1 + iˆθ A σ A sin where θ = θ A θ A, ˆθA = θ A θ and σ A, A {1,, } are the Pauli matrices. In the adjoint representation the group elements are of the form R(θ) BC = exp(θ Aε ABC ) generating the group of rotations in -dimensional Euclidean space, SO().

4 Exercise 5.1: Show for the lowest weight state C c, k = k(k 1) c, k. Exercise 5.: d (+) m = (j m)(j + m + 1) and d ( ) m = (j + m)(j m + 1). Exercise 5.: the direct product representation of su() decomposes to j + 1 k + 1 = (j + k) + 1 (j + k 1) (j k) + 1 where w.l.o.g. j k. Exercise 5.4: = 4. Exercise 5.5: Work out U(θ) U(η) in the fundamental representation. Exercise 5.6: ( R(θ) ) BC = δ BC cos(θ) + ε BCA ˆθA sin(θ) + ˆθ B ˆθC (1 cos(θ)). Lecture 6 The quantum mechanical Bohr atom is described by the Hamiltonian H = ˆp ˆp m eˆr where ˆr = ˆx ˆx with position and momentum operators ˆx and ˆp respectively, charge e and mass m. The angular momentum operator ˆL i = ε ijk ˆx j ˆp k commutes with the Hamiltonian H, ˆL i = 0 and in order to solve the Bohr atom we introduced the quantized version of the Laplace-Runge-Lenz vector  i = ε ijk ˆp j ˆLk me ˆx i ˆr iˆp i. Exercise 6.1: ˆL  = 0 =  ˆL ˆL i, Âj = iε ijk  k H, Âj = 0 holds. Exercise 6.: Show Âi, Âj = iε ijk ˆLk ( mh). 4

5 Lecture 7 In the lecture we mentioned that there is an isomorphism between the group of conformal Bogoliubov transformations SU(1, 1) and the symplectic group Sp(, R). We identified the two groups by looking at the conformal transformations of a bosonic harmonic oscillator with creation and annihilation operators a and a respectively, a, a = 1 and the symmetry group which leaves the canonical commutation relation of the position operator ˆx and momentum operator ˆp ˆx, ˆp = i invariant. The conformal Bogoliubov transformations form the group SU(1, 1) which elements are of the form u v U = v u where u, v C whereas the symmetry group of the canonical commutation relation is the symplectic group Sp(, R) which is evident by rewriting them ˆq i, ˆq j = iω ij, i, j {1, } with (ˆxˆq) ˆq =, Ω = ( 0 ) Exercise 7.1: SU(1, 1) is isomorphic to Sp(, R). Lecture 8 Exercise 8.1: Calculate the self-dual and antiself-dual of the electromagnetic field tensor and show it has three independent components. Exercise 8.: In the fundamental representation the generators of SO(N) are given by the hermitian antisymmetric matrices they obey the algebra J ij (mn) = i( δ mi δnj δ mj δ ni). J (mn), J (pq) = i ( δ mp J (nq) δ mq J (np) + δ nq J (mp) δ np J (mq) ). Exercise 8.: ϕ ij j transforms like ϕ ij j U i kϕ kj j under SU(N). 5

6 Exercise 8.4: Show the Pauli matrix identity σ σ Aσ = σ A holds A {1,, }. Lecture 9 In the Cartan-Weyl basis the commutation relations of the generators of the su() algebra are given by T, I ± = ±I ±, T, U ± = 1 U ±, T, V ± = ± 1 V ± T 8, I ± = 0, T 8, U ± = ± U ±, T 8, V ± = ± V ± I +, I = T, U +, U = T 8 T, V +, V = T 8 + T where the states are labelled by the eigenstates of T and T 8, t, t 8. In the fundamental representation we identified the three states with the up quark, down quark and strange quark, u := 1, 1 d := 1, 1 s := 0, 1. Exercise 9.1: Work out I ± t, t 8, U ± t, t 8, V ± t, t 8. Exercise 9.: Work out the proportionality factors I u d, V u s I + d u, U d s U + s d, V + s u. Lecture 10 Exercise 10.1: Decompose = Lecture 11 Exercise 11.1: Decompose (8 8) s = and (8 8) a = Exercise 11.: Show tr(φφ diag(0, 0, 1)) = K K + + K 0 K 0 + η 6

7 Exercise 11.: Test the Gell-Mann Okubo formula with masses from PDG Exercise 11.4: Show δω µν = δω νµ Lecture 1 Exercise 1.1: Construct inverse map to SO(, 1) SL(, C) Exercise 1.: Check σ j σ + σ (σ j ) T = 0 j {1,, } 7