Quantum Field Theory - Math Tripos - Exam, Exams of Mathematics

Download Quantum Field Theory - Math Tripos - Exam and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Friday 1 June 2001 9 to 12 PAPER 59 QUANTUM FIELD THEORY Attempt THREE questions. The questions are of equal weight. You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 The Dirac equation for a particle of mass m is (iγµ∂µ −m)ψ(x) = 0 , where ψ(x) is the spinor wave function for the particle and the matrices, {γµ} are given by γ0 = ( 1 0 0 −1 ) , γj = ( 0 σj −σj 0 ) . Here 1 is the 2× 2 unit matrix and {σj} are the Pauli matrices which satisfy σj2 = 1 and σjσk = −σkσj = iσl where (j, k, l) is a cyclic permutation of (1, 2, 3). Verify that {γµ, γν} = 2gµν , where gµν is the standard Lorentz metric. Use this result to show that ψ(x) satisfies the Klein-Gordan equation for a relativistic particle of mass m. Given that, for a Lorentz transformation Lµ ν , there exists a 4×4 matrix S(L) such that S−1(L)γµS(L) = Lµ νγ ν , show that the Dirac equation is invariant under Lorentz transformations. An infinitesimal Lorentz transformation can be expressed in the form Lµν = δµν + ωµν , where ωµν = −ωνµ. The corresponding form for S(L) is S(L) = 1− i 4 σµνωµν , where σµν = i 2 [γµ, γν ] . Show that i 4 [σµνωµν , γρ] = ωρνγν , Obtain the solutions of the Dirac equation that represent a particle with 4- momentum p = (E,p) and explain why the particle has spin 12 . Paper 59