Why the trace of the Canonical Stress Tensor (Generalization of the Hamiltonian) implies that the photons have mass?

Question

In Chapter 12.10 B) of “Classical Eletrodynamics” of Jackson, it is said that

The two final criticisms of $T^{\alpha \beta}$ are that it involves the potentials explicitly, and so is not gauge invariant, and its trace ( $T^{\alpha}_{\alpha}$) is not zero, as required for zero mass photons.

Could someone explain me why the fact that the trace of the tensor have to be zero in order that the photons have zero mass?

Answer 1

Set $\mu_0=c=1$. Without a current, a massive photon's Lagrangian density can be taken as$$\mathcal{L}=-\frac14F_{ab}F^{ab}-\frac12m^2A_aA^a,$$from which one can show (proof is an exercise)$$T^{ab}=-F^{ac}F^b_{\;\;c}+\frac14\eta^{ab}F_{cd}F^{cd}+m^2(\tfrac12\eta^{ab}A_cA^c-A^aA^b),$$so in $4$-dimensional spacetime$$T^a_{\;\;a}=m^2A_aA^a.$$