Problems

1. Show that $S_F(x^{\prime},x)$ reduces to the free-particle retarded propagator of the Schördinger equation in the nonrelativistic limit.

2. Calculate $S_F(x)$ explicitly. How does it behave as $x\rightarrow\infty$, as $x\rightarrow 0$, and on the light cone?

3. Show that the forms

   
   

   $$S_F(x^{\prime}-x) = -i \int \frac{d^3p}{(2\pi)^3} e^{i\vec{p... ...ot\vec{\gamma} + m}{2E} \quad\textrm{for}\quad t^{\prime} > t$$
   
   

   and

   
   

   $$S_F(x^{\prime}-x) = -i \int \frac{d^3p}{(2\pi)^3} e^{i\vec{p... ...ot\vec{\gamma} + m}{2E} \quad\textrm{for}\quad t^{\prime} < t$$
   
   

   can be combined into a single expression by using projection operators.

4. Verify

   
   

   $$\theta(t^{\prime}-t)\psi^{(+)}(x^{\prime}) = i\int d^3x S_F(x^{\prime}-x) \gamma_0 \psi^{(+)}(x)$$
   
   

   and

   
   

   $$\theta(t-t^{\prime})\psi^{(-)}(x^{\prime}) = -i\int d^3x S_F(x^{\prime}-x) \gamma_0 \psi^{(-)}(x)$$
   
   

   and derive analogous relations for the adjoint solutions $\overline{\psi}^{(+)}$ and $\overline{\psi}^{(-)}$.