Scalar field energy density

Question

Considering a classical scalar field theory, I can find the canonical energy momentum tensor and if I calculate the $00$ component I get:

$$T^{00}= \frac{1}{2} \dot \phi^2 + \frac{1}{2} (\partial_i) \phi^2 + V(\phi) $$

and it should be an energy density. My issue is that this looks like energy but I don't see why it is an energy density. For example we have $V(\phi)$ and not the potential divided by the volume.

And as a side question, what is the intuitive meaning of the $\frac{1}{2} (\partial_i) \phi^2$ term?

Answer 1

The only meaningful definition of energy is as the conserved Noether charge associated to time translation invariance. The energy density is defined to be the object which is integrated to obtain the Noether charge, namely the Noether current.

This object (together with the other components of the stress tensor) satisfy these requirements. If this makes you uncomfortable, note that in point particle physics the Hamiltonian is the only meaningful definition of the energy. But performing a canonical transformation it might no longer "look" like whatever you might think energy should look like. Doesn't make it any less conserved or any less the Hamiltonian, and hence makes it no worse a definition for the work "energy," it's just different.