What is energy in $z \neq 1$ theories?

In a critical theory with dynamical critical exponent $z \neq 1$, which amongst frequency, $\omega$, and dispersion, $E(\vec{k})$, may be referred to as ''energy''? I'm confused about this since in general $\omega$ and $E(\vec{k})$ can have different scaling dimensions. Some clarification would be very appreciated.

The interpretation of the dispersion as energy is applicable to non-interacting particle. In general, for interacting particles, $E(\vec{k})$ cannot be interpreted as energy (of?). However, frequency $\omega$ is always proportional to energy of the system. One could see it in the following way: The Schrödinger’s equation, $$i \frac{\partial}{\partial t} \psi = \hat{H} \psi,$$ on Fourier transforming is given by, $$\omega \psi = \hat{H} \psi.$$ Therefore, the set of (discrete) frequency $\omega$ is the set of eigenvalues of the Hamiltonian operator $\hat{H}$. So the conjugate variable to time $t$ should always correspond to energy of the system.
In non-interacting case $\omega \propto E(\vec{k})$. But, generically in the presence of interactions it should not be the case.