Causality and the Green's Function in Classical EM

[merrypark3]

Homework Statement

I am not sure about the problem set up.

For (a), Using Equation of motion, need to express Lagrangian in terms of only J?

I got, $$L=-\frac{1}{2 \Box^2 }(\partial_\mu J_\nu)^2 - \frac{{J_\mu}^2}{\Box}$$, using lorentz gauge

(b) $$\partial_\mu J_\mu =0$$ means $$k_\mu J_\mu =0 ?$$

For (d), I can't understand what it's asking. In momentum space, the term without time derivative is the first term in (a)??

Homework Equations

The Attempt at a Solution

 

[vanhees71]

How should we help you, if you don't even post the question?

 

[WannabeNewton]

For (a), Using Equation of motion, need to express Lagrangian in terms of only J?

Yes.

(b) $$\partial_\mu J_\mu =0$$ means $$k_\mu J_\mu =0 ?$$

Yes.

For (d), I can't understand what it's asking. In momentum space, the term without time derivative is the first term in (a)??

$\omega \rightarrow \partial_t$ when going from Fourier space to configuration space. Causally propagating terms have $\partial_t$ to some order e.g. $\partial^2_t$. Non-causal ones only have spatial derivatives e.g. $\nabla^2$ with no $\partial^2_t$ present. The question is asking why the latter are considered non-causal. What are your thoughts on this? As a simple example, consider interactions in Newtonian gravity $\nabla^2 \varphi = 4\pi \rho$. In what sense is this non-causal?

 

[merrypark3]

Yes.



Yes.



$\omega \rightarrow \partial_t$ when going from Fourier space to configuration space. Causally propagating terms have $\partial_t$ to some order e.g. $\partial^2_t$. Non-causal ones only have spatial derivatives e.g. $\nabla^2$ with no $\partial^2_t$ present. The question is asking why the latter are considered non-causal. What are your thoughts on this? As a simple example, consider interactions in Newtonian gravity $\nabla^2 \varphi = 4\pi \rho$. In what sense is this non-causal?

For Newton's gravity, the green function wouldn't contain time, which means can't distinguish source and field point for time

 

[WannabeNewton]

For Newton's gravity, the green function wouldn't contain time, which means can't distinguish source and field point for time

Yes exactly so there is no retardation of the Green's function. Causality of field propagation is codified by the retarded Green's function of a given field equation. But there is an important distinction to be made between the case of Newtonian gravity and classical EM, the latter of which is considered in the problem. In Newtonian gravity the non-causal mode is physical, albeit incorrect because Newtonian gravity is not an inherently relativistic theory. On the other hand, in EM the non-causal mode, which in this case corresponds to $\frac{J_0}{k^2} \rightarrow \frac{J_0}{\nabla^2}$, is unphysical because it is just an artifact of the choice of gauge.