we choose bosonic commutation relations for the annihilation and creation

We choose bosonic commutation relations for the

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Copyright © 2010 University of Cambridge. Not to be quoted or reproduced without permission.with all other commutators vanishing. Note the strange minus sign for thebopera-tors. Repeat the calculation of Question 5 to show that these are equivalent to thecommutation relations 8.Using the methods presented in the lectures, find an expression for the Feynmanpropagator of a Dirac fieldSF(xy)≡ (0|Tψ(x)ψ(y)|0)(15)in terms of theθ-function and integrals over 3-momentum.Deduce, by evaluating a suitable contour integral, thatSF(xy) =iintegraldisplayd4p(2π)4eip.(xy)γ.p+mp2m2+.(16)Verify thatSFis a Green’s function for the Dirac operator.9.The HamiltonianHfor a quantum system can be split into two parts thus:H=H0+Hint(17)whereH0is the Hamiltonian for a known (simple) system andHintis the interactionHamiltonian. Define the Interaction picture for the system. Show that the evolutionoperator for states in the Interaction picture between timest0andt1isU(t1,t0) where 3
Copyright © 2010 University of Cambridge. Not to be quoted or reproduced without permission.

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