# Physical significance of the canonical energy-momentum tensor

I have a question regarding the physical significance of the canonical energy momentum tensor $$T_\nu ^\mu$$ in the context of classical field theory. It is defined as

$$T_\nu ^\mu = \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \Phi^I)} \partial_\nu \Phi^I - \delta_\nu ^\mu \mathcal{L}$$, where $$\Phi^I$$ is the set of all relevant scalar, vector, or tensor fields in the Lagrangian, and $$I$$ is the corresponding indices. For example, if we consider the Lagrangian for the free gauge field in electrodynamics $$\mathcal{L}_A = - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}$$, where $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$, we have that $$\Phi^I = A_\mu$$. Accordingly, the (covariant) canonical energy momentum tensor is

$$T_{\mu \nu} = \frac{1}{4}\eta_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta} - F_{\mu \alpha} \partial _ \nu A^ \alpha$$.

It satisfies that $$\partial _\mu T_\nu ^\mu = 0$$ provided that the system, which it describes, is independent of translations in spacetime. This is a consequence of Noethers theorem.

My problem: On the one hand, it is used to define the canonical 4-momentum by

$$P_\nu = \int T_{\nu}^0 d^3 x$$,

i.e. it is used to define some physical observables of the system. Also, the canonical Hamiltonian $$\mathcal{H}$$ density is defined by $$\mathcal{H}=T_{0}^0$$

On the other hand, $$T_\nu ^\mu$$ is not in general gauge invariant - but physical observables must be gauge invariant. If the gauge field is time independent, i.e. $$\partial_0 A^\alpha = 0$$ for each $$\alpha$$, then $$T_0 ^0$$ (and thus also the energy) can be gauge independent, as is the case for $$T_0 ^0$$ for free electrodynamics defined above.

However, in general for time-dependent gauge field $$T_\nu ^\mu$$ is not gauge-invariant, which must imply that the canonical Hamiltonian density is not in general gauge independent and thus not observable.

My question: Is it possible to show that the 4-momentum is always gauge invariant and thus observable despite the fact that $$T_\nu ^\mu$$ is not?

If this is not possible, what even is the significance (and relevance) of $$T_\nu ^\mu$$ and the canonical 4-momentum?