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I have a QFT homework to do and there I should show for a given lagrangian
$\mathscr{L} = C_1 (\partial_{\nu} A_{\mu}) (\partial^{\nu} A^{\mu}) + C_2 (\partial_{\nu} A_\mu) (\partial^{\mu} A^\nu) + C_3 A_\mu A^\mu$
That the hamiltonian
$H = \int d^3x (\pi_\nu \partial_0A^\nu - \mathscr{L})$ with $\pi_\nu = \frac{\partial \mathscr{L}}{\partial \partial_0 A^\nu}$
is unbound from above and below if $C_3$ is not 0.
So I wanted to ask, how can one prove a statement like this?
Best Regards
The Hamiltonian will be of the form $C^{\mu\nu\rho\sigma}\partial_\mu A_\nu \partial_\rho A_\sigma -C_3 A_\mu A^\mu$ for some rank-$4$ tensor $C^{\mu\nu\rho\sigma}$ you can work out. The last term is the potential energy in the Hamiltonian. Since $A_\mu A^\mu$ is unbounded, so is $-C_3 A_\mu A^\mu$.
