Problem 1:
(a) Write down the action for a hee scalar field Φ of mass m in Minlmwaki space including external sources linear in the field.
(b) Write a formal expression for the partition function (vacuum generating functional) of this theory as a function of the sources.
(c) What mathematical object generates the connected Green functions? Show that the n point connected Green functions vanish for n > 2 in this theory.
(d) Now add an interaction term of the form g/3!∫d4Φ3. Write down an expression for the interacting partition function in terms of the free partition function including source.
(e) Write down the Feynman diagram corresponding to the O(g2) contribution to the connected 2-point function G2(p1,p2) where pi are the external moments. What down a mathematical expression for this diagram when pi = p
(f) Argue that G2 should be expressible as a power series in the external moments
G2(p) = A + Bp2 + ....
Let's regulate G2 by imposing a UV cut-off A. Calculate the dependence of A and B on A (no need to do the integrals exactly).
Problem 2:
(a) Write down the expansion of the Dirac field ψ and its conjugate ψ in terms of creation and annihilation operators. The Feynman propagator is given by
SF(x, y) = <0|T[ψ(x)ψ¯(y)]|0>
Give a physical interpretation of this expression in terms of the creation and annihilation of electrons or positions for the cases x0 -y0 >0 and x0 - y0 < 0.
(b) What is the Lagrangian for the free Maxwell (photon) field? Show that thia Lagrangian ia invariant under a local transformation of the photon field and write down the form of this transformation.
(c) What terms must be added to the free Maxwell and Dirac Lagrangians to yield QED? How does the electron field transform under the local invariance associated with the photon field? Show that QED is invariant under this transformation.
(d) Write down the Feynman diagram(a) that are relevant to the tree level amplitude for the scattering process e+e- → e+e-. No need to give mathematical expressions for these amplitudes.
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