Charge Conservation and Legendre Transformations

As a follow-up (sort of, but not exactly) to my previous post on the matter, I would like to post a few updates and new questions, using Einstein summation throughout for convenience. The first has to do with why $p^{\mu} = \int T^{(0, \mu)} d^3 x$ is a Lorentz-contravariant vector. Apparently Noether's theorem says that if some Noether current $J^{\mu}$ generates a symmetry and satisfies \[ \partial_{\mu} J^{\mu} = 0 \] then the quantity \[ q = \int J^{(0)} d^3 x \] called the Noether charge is Lorentz-invariant and conserved. The first is not easy to show, but apparently some E&M textbooks do it for the example of electric charge. The second is fairly easy to show: using the condition that $\int \partial_{j} J^{j} d^3 x = \int_{\partial \mathbb{R}^3} J^{j} d\mathcal{S}_{j} = 0$ (from the divergence theorem applied to all of Euclidean space) in conjunction with $\partial_{\mu} J^{\mu} = 0$, the result $\dot{q} = 0$ follows.

As an example, let us consider the generator of rotations and Lorentz boosts for a general energy distribution: that is the 3-index angular momentum tensor \[ M^{\mu \nu \sigma} = x^{\mu} T^{\nu \sigma} - x^{\nu} T^{\mu \sigma} .\] Given that $\partial_{\nu} T^{\mu \nu} = 0$ then $\partial_{\sigma} M^{\mu \nu \sigma} = (\partial_{\sigma} x^{\mu})T^{\nu \sigma} + x^{\mu} \partial_{\sigma} T^{\nu \sigma} - (\partial_{\sigma} x^{\nu})T^{\mu \sigma} - x^{\nu} \partial_{\sigma} T^{\mu \sigma}$ $= \delta_{\sigma}^{\; \mu} T^{\nu \sigma} - \delta_{\sigma}^{\; \nu} T^{\mu \sigma} = T^{\nu \mu} - T^{\mu \nu} = 0$. Therefore the 3-index angular momentum is a proper Noether current. Its corresponding conserved charge is the 2-index angular momentum integrated over spatial directions: \[ L^{\mu \nu} = \int M^{(\mu \nu, 0)} d^3 x \] (except for perhaps a sign because $M^{\mu \nu \sigma}$ is antisymmetric in its indices) and it should be easy now to show that $\dot{L}^{\mu \nu} = 0$, which is cool. The only remaining question I have is whether it is more correct to say $L^{\mu \nu} = x^{\mu} p^{\nu} - x^{\nu} p^{\mu}$ where $p^{\mu} = \int T^{(\mu, 0)} d^3 x$ as before or if the better definition is the one integrating $M^{(\mu \nu, 0)}$ over space.

Now I have an even bigger question looming ahead of me though. The Noether current generating spacetime translational symmetry is exactly the stress-energy tensor derivable as the Legendre transformation of the Lagrangian. The term involving the conjugate momenta is easy, but the term involving the Lagrangian is confusing. For a scalar field $\phi$ (and for a vector field this is easily replaced with $A^{\sigma}$), what I have seen is \[ T^{\mu \nu} = \frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)} \partial^{\nu} \phi - \mathcal{L} B^{\mu \nu} .\] The problem is that the tensor $B^{\mu \nu}$ seems to depend on either the field $\phi$ used or on the notation consistently used. Sometime $B^{\mu \nu} = \delta^{\mu \nu}$, while other times $B^{\mu \nu} = \eta^{\mu \nu}$. I'm not really sure which it is supposed to be, as sometimes for scalar fields $B = \delta$ is used, while for the electromagnetic field $B = \eta$ is used, and sometimes the notation isn't even that consistent. The issue is that either one would properly specify a Lorentz-contravariant 2-index tensor, but only one of them actually defines the translational symmetry Noether current properly. Which one is it? The issue appears to be akin to the problem of two grammatically correct sentences where one carries meaning and makes sense while the other makes no sense at all (e.g. "colorless green ideas sleep furiously").

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