Should the energy-momentum tensor be invariant under gauge transformations? (SOLVED)
For example, consider the electromagnetic theory given by \begin{align} I=-\frac{1}{4}\int d^4x\, F_{\mu\nu}F^{\mu\nu}, \end{align} where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. The action has a symmetry given by the space-time translations \begin{align} \delta A_\mu=-\epsilon^\alpha\partial_\alpha A_\mu, \end{align} with $\epsilon^\alpha$ the transformation parameter. In this case the energy-momentum tensor will be given by \begin{align} T^{\alpha}\nu=F^{\mu\alpha}\partial\nu A_\mu-\frac{1}{4}\delta^{\alpha}\nu F{\lambda\rho}F^{\lambda\rho}. \end{align} Everything seems to be ok, but note $T^{\alpha}\nu$ will not be gauge-invariant and this is because $\delta A\mu=-\epsilon^\alpha\partial_\alpha A_\mu$ is not gauge-invariant. This seems to be a problem, so one can do some sleight-of-hand and do an 'improved translation', which is a space-time translation + a particular gauge transformation \begin{align} \delta A_\mu=-\epsilon^{\alpha}\partial_\alpha A_\mu+\partial_\mu(\epsilon^{\alpha}A_\alpha)=F_{\mu\alpha}\epsilon^\alpha. \end{align} Note the transformation will be gauge-invariant. For completeness, the (gauge-invariant) energy-momentum tensor will be \begin{align} T^{\alpha}\nu=-F^{\alpha\beta}F{\nu\beta}+\frac{1}{4}\delta^{\alpha}\nu F^{\gamma\delta}F{\gamma\delta}. \end{align} So this was a happy example.
My questions are:
1. What would happen if I can't do the 'improved translation' always?
2. 818
Energy-momentum tensor should always be gauge-invariant? What does imply this? (Furthermore, what would be the consequences at the quantum theory?)
3. There's some examples of not gauge-invariant energy-momentum tensor? (or perhaps some loss of - some more (or less) "dramatic" - symmetry?)
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I will answer with a concrete example.
"Energy-momentum tensor should always be gauge-invariant? What does imply this? (Furthermore, what would be the consequences at the quantum theory?)". We want it to be gauge-invariant because energy and momentum will be a measure and it must be the ideal case, nevertheless, it'll not always be gauge-invariant and it's fine. Consider the gravitational wave theory, whose action is the linearised general relativity action. The energy-momentum tensor $t_{\mu\nu}$ will not be gauge invariant. As a consequence we will consider the mean value of it $\langle t_{\mu\nu}\rangle$ in order to obtain a dependence of the physical modes $h_{ij}^{TT}$ [1]. $\langle t_{\mu\nu}\rangle$ will be the observable indeed, and not $t_{\mu\nu}$.
"There's some examples of not gauge-invariant energy-momentum tensor? (or perhaps some loss of - some more (or less) "dramatic" - symmetry?)". Yup, the same example exposed in [1] for gravitational waves, which is a nice example because was measured in 2016.
[1] <PERSON>, Michele - Gravitational Waves, Vol. 1: Theory and Experiments (equation (1.136)) 298