Write down the greens function g0omega for free propagation

Question

Assignment-

Problem 1 - The Klein-Gordon equation for a bosonic particle is given by

(∇² - (1/c²)(∂²/∂t²))ψ = (m²c²/¯h²)ψ.

(a) Write down the Green's function G⁰(ω) for free propagation of the boson.

(b) Let us now consider the boson to be a particle mediating a force. The boson is created by the scattering of a particle experiencing this force. If we take the coupling strength 4πg, what is the potential in momentum space in the static limit?

(c) What is the potential in real space? Give a physical interpretation of the potential.

Problem 2 - Let us consider a relativistic particle with spin up moving with momentum p_z in the z direction.

(a) Write the four-momentum in matrix form using the Dirac unit matrices. Use hyperbolic functions with angle α₀ to express the components.

(b) Give the spinor |p⁺ ↑) for this particle (use again hyperbolic functions with angle α₀).

(c) Calculate the norm of the spinor (throughout the question, use addition formulas for hyperbolic functions whenever possible).

(d) Calculate the expectation value of

mc = mc_µ=0Σ³e_µe^µ.

Express the result in terms of p_z and the energy E_p of the particle.

(e) Write down a Lorentz boost R_zt(α/2) in the z direction.

(f) Perform a Lorentz boost in the z direction on the spinor.

(g) In the Dirac basis basis, the scalar is given by γ₀ instead of 1₄. A Lorentz boost should leave the scalar unchanged. If the scalar is given by 1₄, this can be achieved by defining the Lorentz transformation as 1′₄ = R_zt(α/2)1₄R^-1_zt (α/2) = R_zt(α/2)R^-1_zt (α/2) = 1₄, which indeed leaves the scalar matrix unchanged. Derive the transformation when the scalar matrix is γ₀.

(h) Perform a Lorentz boost in the z direction on the matrix of four momentum p from (a).