I want to show explicitly that the Hamiltonian $$ H = -\Omega V + \int d\tilde{\textbf{k}}\ \omega (a^\dagger(\textbf{k}) a(\textbf{k}) + b(\textbf{k}) b^\dagger(\textbf{k}) ) $$ is Lorentz covariant. ($\Omega$ and $V$ are constants, $a$ and $b$ are the operators coming from the canonical quantization of the Klein Gordon equation, like what is done in (3.30) of Srednicki's book Quantum Field Theory). This made me realize my lack of understanding of Lorentz transformations:
I think that the definition of Lorentz invariance goes as follows: a certain object $A$ which depends on the spacetime coordinates $x$ is Lorentz invariant if $U(\Lambda)^{-1} A(x) U(\Lambda)= A(\Lambda^{-1}x)$ where $U$ is the appropriate representation of $\Lambda$ (it gives a map $U(\Lambda)$ that acts on the space to which $A$ belongs).
Now, can I state a similar definition for Lorentz covariance and use it to solve my problem? I know that the derivative is Lorentz covariant: $$ \partial_{\mu'} = (\Lambda^{-1})^\nu_{\ \mu}\partial_\nu $$
But a definition of the type "$A$ is Lorentz covariant if $A_{\mu'} = (\Lambda^{-1})^\nu_{\ \mu}A^\nu$" only makes sense if $A$ is a four component object, and that's not the case for the Hamiltonian. I wonder how one can state a definition of covariance for a more general case, maybe more on the lines of the definition of Lorentz invariance that I gave, and how to apply it to this case.
I am finding it hard to find a structured approach to Lorentz transformations. If you could help me untangle this a bit, that would be great.