  It is more subtle to obtain the chiral vortical effect (CVE) than chiral
magnetic effect (CME) in quantum transport approach. To investigate the subtlty
of the CVE we present two different derivation in the Wigner function approach.
The first one is based on the method in our previous work \cite{Gao:2012ix} in
which the CVE was derived under static-equilibrium conditions without details.
We provide a detailed derivation using a more transparent and powerful method,
which can be easily generalized to higher order calculation. In this derivation
of the CVE current, there is an explicit Lorentz covariance. The second
derivation is based on a more general chiral kinetic theory in a semi-classical
expansion of the Wigner function without assuming static-equilibrium conditions
\cite{Gao:2018wmr}. In this derivation, there is a freedom to choose a
reference frame for the CVE current, so the explicit Lorentz covariance seems
to be lost. Howerver, under static-equilibrium conditions, we show that the CVE
current in this derivation can be decomposed into two parts, identified as the
normal and magnetization current. Each part depends on the reference frame, but
the sum of two parts does give the total CVE current which is independent of
the reference frame. In the comoving frame of the fluid, it can be shown that
the normal and magnetization current give one-third and two-thirds of the total
CVE current respectively. This gives a natural solution to the 'one-third'
puzzle in the CVE current in three-dimensional version of the chiral kinetic
theory in the literature.
