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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.

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After some amount of on and off thinking here's what I have come up with. Please pardon the coarse picture.

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.

Comments and corrections are very welcome.

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  • $\begingroup$ since there has been no further suggestions/comments on this reasoning, I'm accepting this as a correct answer to close this thread. $\endgroup$ – vik Nov 7 '13 at 22:14

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