Problem Set 6 Solutions - Quantum Field Theory | PHY 396K, Assignments of Physics

Download Problem Set 6 Solutions - Quantum Field Theory | PHY 396K and more Assignments Physics in PDF only on Docsity! PHY–396 K. Problem set #6. Due October 23, 2000. 1. The first problem is about Dirac’s γ matrices. (a) Verify [ Sκλ, Sµν ] = i ( gλµSκν − gλνSκµ − gκµSλν + gκνSλµ ) . (b) Verify M−1(L)γµM(L) = Lµνγν for L = exp(θ) (i.e., L µ ν = δ µ ν + θ µ ν + 1 2θ µ λθ λ ν + · · ·) and M(L) = exp ( − i2θαβS αβ ) (c) Calculate {γρ, γλγµγν}, [γρ, γκγλγµγν ] and [Sρσ, γλγµγν ]. (d) Show that γαγα = 4, γ αγνγα = −2γν , γαγµγνγα = 4gµν and γαγλγµγνγα = −2γνγµγλ. Hint: use γαγν = 2gνα − γνγα repeatedly. (e) Consider the electron’s spinor field Ψ(x) in an electomagnetic background. Show that the gauge-covariant Dirac equation ( iγµDµ +m ) Ψ(x) = 0 implies( m2 +D2 + qFµνS µν ) Ψ(x) = 0. 2. The second problem is about the Lorentz group and its generators Ĵµν . In 3–index nota- tions, Ĵ ij = ij`Ĵ` generate ordinary rotations while Ĵ0i = −Ĵ i0 = K̂i generate the Lorentz boosts. Let Ĵ± = 1 2 ( Ĵ ± iK̂ ) . (1) (a) Show that the Ĵ+ and the Ĵ− commute with each other and that each satisfies the commutations relations of an angular momentum, [Ĵk±, Ĵ ` ±] = i k`mĴm± . The “angular momentum” Ĵ+ is non-hermitian and hence its finite irreducible representa- tions are non-unitary analytic continuations of the spin–j representations of a hermitian Ĵ. The same is true for the Ĵ− = Ĵ † +, so altogether, the finite irreducible representations of the Lorentz algebra are specified by two integer or half-integer ‘spins’ j+ and j−. The simplest non-trivial representations of the Lorentz algebra are the Weyl spinor (j+ = 1 2 , j− = 0) — a doublet where Ĵ acts as 1 2~σ and K̂ as − i 2~σ and the congugate Weyl ‘anti- spinor’ (j+ = 0, j− = 1 2) where Ĵ also acts as 1 2~σ but K̂ acts as + i 2~σ. Together the Weyl spinor and the Weyl antispinor comprise the Dirac spinor. 1 (b) Show that for any infinitesimal combination of a Lorentz boost ~b and rotation ~θ ≡ θn, Ψ′(x′) = Ψ(x) + ( − i2(~θ − i~b) · ~σ 0 0 − i2(~θ + i~b) · ~σ ) Ψ(x), (2) which means that a Dirac spinor indeed decomposes into a Weyl spinor and a Weyl antispinor. Finite Lorentz transformations act on Weyl spinors as complex, unimodular (det = 1) but non-unitary two-by-two matrices. The group SL(2,C) of such matrices is actually isomorphic to the Spin(3, 1) — the double cover of the continuous Lorentz group. (This is similar to Spin(3) ∼= SU(2).) Any (j+, j−) representation of the Spin(3, 1) becomes in the SL(2,C) terms a tensor Φa1...a(2j+),ȧ1...ȧ(2j−) , totally symmetric in its 2j+ un-dotted in- dices a1, . . . , a(2j+) and separately totally symmetric in its 2j− dotted indices ȧ1, . . . , ȧ(2j−), transforming according to Φ′a1...a(2j+),ȧ1...ȧ(2j−) = U b1a1 · · ·U b(2j+) a(2j+) U∗ ḃ1ȧ1 · · ·U ∗ ḃ(2j−) ȧ(2j−) Φb1...b(2j+),ḃ1...ḃ(2j−) . (3) The vector representation of the Lorentz group has j+ = j− = 1 2 . To cast the action of the Lorentz group in SL(2,C) terms (3), consider Xµσµ = T −X · ~σ. (Here σ0 = 1 while σ1, σ2 and σ3 are the Pauli matrices.) Let X ′µσµ ≡ Lµν(U)Xνσµ = U ( Xµσµ ) U †. (4) (c) Show that for any SL(2,C) matrix U , eq. (4) indeed defines a Lorentz transform. (Hint: prove and use det(Xµσµ) = X 2 ≡ XµXµ). Also verify the group law, L(U2U1) = L(U2)L(U1). (d) Verify explicitly that for U = exp ( − i2θn ·~σ ) , L(U) is a rotation by angle θ around axis n while for U = exp ( −12rn · ~σ ) , L(U) is a boost of rapidity r (β = tanh r, γ = cosh r) in the direction n. 2