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Lecture 3 – Su(2)

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Lecture 3 – Su(2) LECTURE 3 – SU(2) Contents • 2D Representations • 3D Representations • 2D Transformations • Rotation Matrices • Gauge Transformations and the Adjoint • Hadronic Isospin • Weak Isospin • Conjugate States • Combining States • Hadron Wavefunctions Messages • SU(2) describes spin angular momentum . • SU(2) is isomorphic to the description of angular momentum – SO(3) . • SU(2) also describes isospin – for nucleons, light quarks and the weak interaction. • We see how to describe hadrons in terms of several quark wavefunctions . Symmetries & Conservation Laws Lecture 3, page1 2D Representation of the Generators [3.1, 3.2, 3.3] SU(2) corresponds to special unitary transformations on complex 2D vectors. The natural representation is that of 2 ×2 matrices acting on 2D vectors – nevertheless there are other representations, in particular in higher dimensions. 2 There are 2 −1 parameters, hence 3 generators: {J 1, J 2, J 3}. The generators are traceless and Hermitian. It is easy to show that the matrices have the form: a b * Special where a is real. There are three parameters: a, Re(b), Im(b). b − a A suitable (but not unique) representation is provided by the Pauli spin matrices: J i = σi/2 where 0 1 0 − i 1 0 σ1 = ,σ2 = ,σ3 = 1 0 i 0 0 −1 These have the properties that: σi σj = δij + i εijk σk The Lie algebra is: [J i,J j] = i εijk Jk This is a sum over k, but in this case is trivial: the commutator just gives the “other” generator. E.g. [J1,J 2] = i J 3 Symmetries & Conservation Laws Lecture 3, page2 Quantum Numbers [3.2, 3.3] A Casimir Operator is one which commutes with all other generators. 2 2 2 2 In SU(2) there is just one Casimir: J = J 1 + J 2 + J 3 2 Since [J ,J 3] = 0, they can have simultaneous observables and can provide suitable QM eigenvalues by which to label states. We can define Raising & Lowering Operators : J ± = J 1 ± iJ2 Can show [J 3,J ±] = ±J± Define eigenstates |j m > with their corresponding eigenvalues : J2 |j m > = j(j+1) |j m > J3 |j m > = m |j m > Recall from angular momentum in QM, J ± generates different states within a multiplet: J3 J±|j m > = (m ±1) J±|j m > and J± |j m > = j( m m j)( ± m + )1 |j m ±1> Exercise 2 Prove [J ,J i] = 0 and [J 3,J ±] = ±J± And J 3 J±|j m > = (m ±1) J ±|j m > and J ± |j m > = j( m m j)( ± m + )1 |j m ±1> And J ± |j ±j> = 0 Symmetries & Conservation Laws Lecture 3, page3 2D Representation [3.2, 3.3] While we started in 2D to motivate the group structure (which is defined originally in 2D), we have derived the Lie algebra and scheme for labelling eigenstates of the generators … which can be expressed in various vector spaces. Further, we recalled results from angular momentum theory, yet we have not explicitly restricted ourselves to angular momentum. SU(2) is motivated in 2D (although groups of transformations can operate in higher dimension vector spaces). We can form a 2D representation by choosing two orthogonal states as base vectors: 1 0 1 1 1 1 ≡| 2 + 2 > and ≡| 2 − 2 > 0 1 By inspection 1 0 J = 2 3 1 0 − 2 1 1 1 1 1 1 To find J +, consider: J+ | 2 + 2 >= 0 and J+ | 2 −2 >= | 2 + 2 > 0 1 ⇒ J+ = 0 0 Recalling the definition of J ± and the fact that the generators are Hermitian: H H 0 0 J− = J1 − iJ 2 = J( 1 + iJ 2 ) = J+ = 1 0 Symmetries & Conservation Laws Lecture 3, page4 Hence: 0 1 0 −i J = 1 J( + J ) = 2 and J = 1 J( − J ) = 2 1 2 + − 1 2 i2 + − i 2 0 2 0 and we recover J i = σi/2 Symmetries & Conservation Laws Lecture 3, page5 Alternative 2D Representation The previous result arose from a particular choice (the most natural) of base states. What if we made another choice such that 1 0 1 1 1 1 1 1 1 1 1 1 ≡ (| 2 + 2 > + | 2 − 2 >) and ≡ (| 2 + 2 > − | 2 − 2 >) ? 0 2 1 2 Then 1 1 | 1 + 1 >= 1 and | 1 − 1 >= 1 2 2 2 1 2 2 2 − 1 We find the new representation of the generators by recalling that the matrix of eigenvectors E for a matrix M relates to the diagonal matrix of eigenvalues Λ by ME = E Λ. For a Hermitian matrix M, Λ is real and E is Unitary. So M = E ΛEH Exercise By considering (ME) HE and E H(ME), show that Λ is real and E is Unitary. You will need to remember that Λ is diagonal by construction and look at components. Symmetries & Conservation Laws Lecture 3, page6 Hence 1 1 1 0 1 1 0 1 J = 1 2 1 = 2 3 2 1 2 1 1 −10 − 2 1 −1 2 0 1 1 1 1 1 1 H To find J + and J −, recall J+ | 2 + 2 >= 0 and J+ | 2 −2 >= | 2 + 2 > and J− = J+ . So by inspection: 1 1 − 1 1 1 1 J+ = and J− = 2 1 − 1 2 − 1 − 1 Hence: 1 0 0 i J = 2 and J = 2 1 1 2 −i 0 − 2 2 0 So with this basis: J 1 = σ3/2, J 2 = −σ 2/2, J3 = σ1/2 It looks like a “rotation” about the “1=3 axis”. With a different choice of basis, one would not necessarily obtain the Pauli matrices. Symmetries & Conservation Laws Lecture 3, page7 3D Representation [3.3] This corresponds to j = 1 with m = −1,0, +1. Choose 1 0 0 0 ≡ 1| +1 > 1 ≡ 10| > 0 ≡ 1| −1 > 0 0 1 Then by observation 1 0 0 J3 = 0 0 0 0 0 −1 By firstly constructing J + and J −: 0 1 0 0 −1 0 1 i J1 = 1 0 1 and J2 = 1 0 −1 2 2 0 1 0 0 1 0 So SU(n) can have representations in vector spaces of various dimensions – in each dimension, there will be an infinite number of representations. The lowest dimensional representation is in nD. The matrices which correspond to the simple base vectors (1,0,0…), (0,1,0,…) etc. are called the Fundamental Representation, of which there are (n −1). (For n>2, there are more quantum numbers and additional ways in which they can be associated to base vectors.) For SU(2), there is just one fundamental representation, namely { σ1/2, σ2/2, σ3/2}. Symmetries & Conservation Laws Lecture 3, page8 2D Transformations [3.3] Transformations under SU(2) are performed by exp(i αiJi) (implicit sum over i). In 2D, we have identified the generators {J i} with the Pauli spin matrices { σi/2} which correspond to the spin ½ angular momentum operators. Furthermore, the operators have the form we would expect from our consideration of 3D transformations of spatial wavefunctions in QM (see Lecture 1) – i.e. the form of the operators L compared to J, and the corresponding eigen-states and eigen-values. However, there are some subtle differences. Spatial Rotations Spin Rotations Vector Space ψ(x) – spatial w/f ψ – spinor Symmetry Group SO(3) SU(2) Operation exp(i θL) where L = x ×(−i∇) exp(i αJ) operator is scaler operator is matrix Operates on x – space spin There is an isomorphism between SO(3) and SU(2). Symmetries & Conservation Laws Lecture 3, page9 The transformation operator is U = exp(i αiσi/2). Writing αi = 2 ωni, where n is a unit vector, gives U = exp(i ωn⋅σ). Expanding the exponential: U = 1 + (i ωn⋅σ) + (i ωn⋅σ)2/2! + (i ωn⋅σ)3/3! … 2 2 2 (n⋅σ) = n 1 σ 1 + … + n1 n2 (σ1 σ 2 +σ2 σ 1)+ ... 2 2 2 2 2 σ1 = 1 and ( σ1 σ 2 +σ2 σ 1) = 0 ⇒ (n⋅σ) = n 1 + n 2 + n 3 = 1 So U = {1 − ω 2/2! + ω 4/4! + …} + i{ ω − ω 3/3! + ω 5/5! – …} n⋅σ = cos ω + i sin ω n⋅σ θ θ θ exp( i 2 n ⋅ σ) = cos 2 + isin 2 n σ⋅ We can now prove the Lie nature of SU(2) explicitly. We should expect that the product of two operators is a third one of a similar form, where the parameters of the result are a function of the parameters of the first two operations. exp( iαa ⋅ σ)exp( iβb ⋅ σ) = (cos α + isin αa ⋅ σ)(cos β + isin βb ⋅ σ) Using a ⋅ σ b ⋅ σ = a ⋅ b + ia × b ⋅ σ, which is derived from σi σj = δij + i εijk σk exp( iαa ⋅ σ)exp( iβb ⋅ σ) = cos α cos β − sin α sin β(a ⋅ b + ia × b ⋅ σ) + i(sin α cos β a ⋅ σ + sin βcos α b ⋅ σ) = {cos α cos β − sin α sin βa ⋅ }b + a{i × b + sin α cos β a + sin β cos α }b ⋅ σ ≡ C + iS ⋅ σ To show that this has the form cos γ + isin γc ⋅ σ , it is sufficient to show that C 2+S 2 = 1 – this straightforward, albeit tedious ! Symmetries & Conservation Laws Lecture 3, page10 Rotation Matrices Rotate the coordinated frame about the y-axis: x x′ z′ θ z A state |j m > in the first frame transforms to a different description in the second frame (same state): jm| > → Ry (θ jm|) > 2 Note: because J commutes with R y(θ), the j quantum number is unchanged.
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