In body-fixed frame the Dirac operator of the rotating soliton $\tilde {\slash D}_E(\bar U)$ can be expressed in the form:

$$\tilde{\slash D}_E(\bar U) =  R^{\dagger} \
{\slash D}_E(\tilde U) \ R \ ,
\eqno(\DIRFULp)$$

where $\tilde{\slash D}_E$ is defined by:

$$\tilde{\slash D}_E(\bar U) = \gamma_4 \left[ \partial_4 - h(\bar U) +
i \Omega - i A_4 \, R^{\dagger} \hat Q R +
i \gamma_4 \gamma_k A_k \, R^{\dagger} \hat Q R \right]
\ . \eqno(\DIRFUL)$$

The operator $\Omega$ is Hermitian in both, Euclidean and Minkowski space-time:

$$\Omega^{\dagger}_E = \Omega_E, \quad\quad
\Omega^{\dagger}_M = \Omega_M
\ . \eqno(\OMHER)$$

Taking into account () and () we can write down the Hermitian conjugate to ():

$$\,\tilde{\slash D}_E^{\dagger}(\bar U) =
\left[ - \partial_4 - h(\bar U)  - i \Omega + i A_4^{\dagger} \,
R^{\dagger} \hat Q R + i \gamma_4 \gamma_k A_k^{\dagger}
\,  R^{\dagger} \hat Q R \, \right]
\gamma_4
\ , \eqno(\DIRCON)$$

where the collective coordinate $\Omega_M$ is will be quantized according to the canonical quantization rule, which in Minkowski space-time are:

$$\Omega^A \rightarrow {1\over I} \  \hat{T}^A \equiv
{1\over 2I} \  \tau^A \ .
\eqno(\OMQ)$$

Here $\hat{T}^A$ are the $SU(2)$ group generators. The moment of inertia $I$ has been computed numerically in \[\].

