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

Download Quantum Field Theory - Math Tripos - Past Exam and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Thursday 30 May 2002 9 to 12 PAPER 62 QUANTUM FIELD THEORY Attempt THREE questions There are four questions in total The questions carry 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 A hermitian scalar, φ(x), has a Lagrangian density L(x) = 1 2 (∂φ(x))2 − 1 2 m2φ2(x) . Explain briefly the procedure for quantising the field theory and show that in the Heisenberg Picture, [φ(x, t), φ̇(x′, t)] = iδ(3)(x− x′) . Construct the Hamiltonian for the field theory and show that the Heisenberg fields obey (∂2 +m2)φ(x) = 0 , and hence that φ(x) can be expressed in terms of mode operators, a(p) and a†(p), in the form φ(x) = ∫ d3p (2π)32Ep ( a(p)e−ip.x + a†(p)eip.x ) , where p2 = m2 and Ep = √ p2 +m2. Write down and justify the commutation relations satisfied by the mode operators. Express the Hamilitonian in terms of the mode operators and show that [H, a(p)] = −Ep a(p) , and [H, a†(p)] = Ep a†(p) . Explain the particle interpretation of the theory. The Feynman propagator, ∆F (x− y), for the field φ(x) is defined by the equation i∆F (x− y) = 〈0|T (φ(x)φ(y))|0〉 , where the symbol T indicates the time ordered product. Show that ∆F (x− y) = ∫ d4p (2π)4 1 p2 −m2 + i e−ip.(x−y) , and hence that (∂2 +m2)∆F (x− y) = −δ(4)(x− y) . Paper 62