Energy current for field satisfying KG equation

[etotheipi]

Homework Statement

To find the energy current density $\vec{j}$ corresponding to the energy density $\mathcal{E} = \frac{1}{2} \left[ \phi_t^2 + c^2 (\nabla \phi)^2 + m^2 c^4 \phi^2 \right] = 0$, where $\phi$ satisfies $\phi_{tt} - c^2 \nabla^2 \phi + m^2 c^4 \phi = 0$

Relevant Equations

N/A

First to compute the time derivative of $\mathcal{E}$,$$\mathcal{E}_t= \phi_t \phi_{tt} + c^2 (\nabla \phi_t) \cdot (\nabla \phi) + m^2 c^4 \phi \phi_t = \phi_t \left[ \phi_{tt} m^2 c^4 + \phi \right] + c^2 (\nabla \phi_t) \cdot (\nabla \phi)$$Then we switch out $\phi_{tt} + m^2 c^4 \phi$ for $c^2 \nabla^2 \phi$ as per the KG equation,$$\mathcal{E}_t = c^2 \left[\phi_t \nabla^2 \phi + (\nabla \phi_t) \cdot (\nabla \phi) \right]$$This must satisfy the continuity equation with zero sources or sinks, and so$$\nabla \cdot \vec{j} = - \mathcal{E}_t = - c^2 \left[\phi_t \nabla^2 \phi + (\nabla \phi_t) \cdot (\nabla \phi) \right]$$The RHS looks like it might be able to be cleaned up with a vector calculus identity or something, but I can't see it. Wondered if anyone had a hint? Thanks!

 

[TeethWhitener]

Homework Statement:: To find the energy current density $\vec{j}$ corresponding to the energy density $\mathcal{E} = \frac{1}{2} \left[ \phi_t^2 + c^2 (\nabla \phi)^2 + m^2 c^4 \phi^2 \right] = 0$, where $\phi$ satisfies $\phi_{tt} - c^2 \nabla^2 \phi + m^2 c^4 \phi = 0$
Relevant Equations:: N/A

The RHS looks like it might be able to be cleaned up with a vector calculus identity or something, but I can't see it. Wondered if anyone had a hint?

Product rule?

 

[etotheipi]

Ahh okay, I see it now looking at it a second time. It's going to be$$\phi_t \nabla^2 \phi + \nabla \phi_t \cdot \nabla \phi = \nabla \cdot (\phi_t \nabla \phi)$$We can prove this relation by looking at the components, starting from the RHS (and using the summation convention)$$\begin{align*}

\nabla \cdot (\phi_t \nabla \phi) = \partial_i (\phi_t \nabla \phi)_i &= \partial_i(\phi_t \vec{e}_j \partial_j \phi)_i \\

&= \partial_i (\phi_t \partial_i \phi) \\

&= (\partial_i \phi_t) (\partial_i \phi) + \phi_t \partial_i \partial_i \phi \\

&= (\partial_i \phi_t) \vec{e}_i \cdot (\partial_j \phi) \vec{e}_j + \phi_t \nabla^2 \phi \\

&= \nabla \phi_t \cdot \nabla \phi + \phi_t \nabla^2 \phi

\end{align*}$$So back to the actual question, we're left with$$\nabla \cdot \vec{j} = -c^2 \nabla \cdot (\phi_t \nabla \phi)$$which would suggest$$\vec{j} = -c^2 \phi_t \nabla \phi$$Does that look okay?

 

[TeethWhitener]

The math looks right. I admit, I’ve never seen that expression before, but field theory isn’t really my strong suit. It kind of looks like half of the probability current, which I guess makes sense?