qft_final_08 - MATHEMATICAL TRIPOS Thursday 29 May 2008 Part III 9.00 to 12.00 PAPER 48 QUANTUM FIELD THEORY Attempt no more than THREE questions There

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Unformatted text preview: MATHEMATICAL TRIPOS Part III Thursday 29 May 2008 9.00 to 12.00 PAPER 48 QUANTUM FIELD THEORY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper 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 is ( iγ μ ∂ μ- m ) ψ ( x ) = 0 where the gamma matrices are given in the chiral representation by γ = 1 2 1 2 , γ i = σ i- σ i . Here σ i are the Pauli matrices and 1 2 is the unit 2 × 2 matrix. a. Show that these matrices satisfy the Clifford algebra { γ μ ,γ ν } = 2 η μν 1 4 where η μν is the Minkowski metric with signature (+1, -1, -1, -1) and 1 4 is the unit 4 × 4 matrix. b. Let M μν =-M ν μ be the generators of the Lorentz group. They satisfy the Lie algebra [ M ρσ , M τ ν ] = η στ M ρν- η ρτ M σν + η ρν M στ- η σν M ρτ . Explain how to use the Clifford algebra to construct a representation of this Lie algebra, and show that the generators do indeed satisfy the commutation relations. c. A Lorentz transformation is given by Λ = exp ( 1 2 Ω ρσ M ρσ ) ....
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