Classical propagator for massive spin 1 particle

[kelly0303]

Homework Statement

Calculate the classical propagator for a massive spin 1 particle by inverting the equations of motion to the form $$A_\mu=\Pi_{\mu\nu}J_\nu$$

Homework Equations

The Attempt at a Solution

By solving the lagrangian for a massive spin 1 particle one gets $$(\Box + m^2)A_\mu=J_\mu$$ I thought of going to momentum space, so this becomes $$(-p^2+m^2)A_\mu=J_\mu$$ So I need to invert this equation, which would give for the propagator $$\frac{1}{(-p^2+m^2)}$$ But this seems wrong as it doesn't look like a second rank tensor and I assume (based on W boson propagator) my answer should look something like $$\frac{g_{\mu\nu}+\frac{p_\mu p_\nu}{m^2}}{-p^2+m^2}$$. What am I doing wrong? Thank you!

 

[mark726]

Thank you for your question regarding the calculation of the classical propagator for a massive spin 1 particle. Your approach is correct, but there is one important step missing in your calculation.

After going to momentum space, you correctly obtain the equation $$(-p^2+m^2)A_\mu=J_\mu$$ However, in order to solve for $A_\mu$, you need to invert this equation by multiplying both sides by the inverse of the operator $(-p^2+m^2)$. This operator is a second rank tensor, which can be written as $g_{\mu\nu}+\frac{p_\mu p_\nu}{m^2}$. Therefore, the correct solution for $A_\mu$ is given by $$A_\mu=\frac{g_{\mu\nu}+\frac{p_\mu p_\nu}{m^2}}{-p^2+m^2}J_\nu$$

I hope this helps clarify your confusion. Keep up the good work!