Why does $c_{-k,-\sigma}$ create a particle with momentum $k$? In Mudelung's book, Introduction to Solid-State Theory, I am confused by the following statement.

For many applications a further simplification is helpful. The concept of the hole presents us with the confusing situationthat a hole in the state $\mathbf{k},\sigma$ has a momentum $-ħ\mathbf{k}$, while an electron has a momentum  $ħ\mathbf{k}$. This asymmetry can be avoided by defining new quasi-particles which have momentum  $ħ\mathbf{k}$ inside and outside the Fermi sphere. When we note that the creation of a particle with  $+ħ\mathbf{k}$ is achieved by the operator $c_{\mathbf{k}\sigma}^+$ when outside the Fermi sphere and by  $c_{-\mathbf{k},-\sigma}$ when inside the Fermi sphere, it is only a small step to define the following operators:
  $$
\begin{align}
\alpha_{\mathbf{k}\sigma}^+=u_\mathbf{k}c_{\mathbf{k}\sigma}^++v_{-\mathbf{k}}c_{-\mathbf{k},-\sigma}
\\
\alpha_{\mathbf{k}\sigma}=u_\mathbf{k}c_{\mathbf{k}\sigma}+v_{-\mathbf{k}}c_{-\mathbf{k},-\sigma}^+
\end{align}
$$
  with
  $$
\begin{align}
u_\mathbf{k}=1,\; v_\mathbf{k}=0\quad\text{ for }\quad k>k_\text{F},
\\
u_\mathbf{k}=0,\; v_\mathbf{k}=1\quad\text{ for }\quad k<k_\text{F}.
\end{align}
$$

He showed the motivation of introducing Bogoliubov tansformation. But I don't understand his statement. "Creation of a particle with $+\mathbf k$ is acvieved by the operator $c^{+}_{\mathbf k,\sigma}$ when outside the Fermi shere and by $c_{-\mathbf k,-\sigma}$ when inside the Fermi sphere".
$c^{+}_{\mathbf k,\sigma}$ means create a particle with $\mathbf k$. So, why must we distinguish ouside and inside the Fermi sphere? And why does annihilation operator $c_{-\mathbf k,-\sigma}$ inside the Fermi sphere have the same effect as creation operator outside the Fermi sphere?
 A: The states inside the Fermi sphere are occupied, so $c^{+}_{k,\sigma}$ applied to any of those states gives zero.  Not a state with zero electrons, but an identically zero state function.  Not helpful in this context.
On the other hand, if one applies $c_{-k, -\sigma}$ to states inside the Fermi sphere (hence occupied), an electron having properties ($-k, -\sigma$) is destroyed leaving behind a perfectly good state function ... and for the overall state of the system $k$ and $\sigma$ have increased.  We can think of this as creating a "particle" with properties ($k, \sigma$), that is, creating a hole.
A: The Fermi Sphere
Imagine a filled Fermi Sphere, every state with $k<k_f$ is occupied and the states with $k>k_f$ are empty. Here the particles we are talking about are electrons, hence a state is occupied by an electron with a momentum $\mathbf k$ and spin $\sigma$.
$$
|F\rangle=\prod_{|\mathbf k| <k_f,\sigma}c_{\mathbf k,\sigma}^{\dagger} |0\rangle
$$
It is crucial to understand the difference between $0$ and $|0\rangle$. $|0\rangle$ is the empty state, it has no occupied state. $0$ is no state and any operator applied to $0$ gives $0$.
Electron creation or destruction
In the state $|F\rangle$ it is possible to create electrons of momentum $|\mathbf k| >k_f$ with the creation operator $c_{\mathbf k, \sigma}^{\dagger}$. It is also possible to remove one electron from the Fermi sphere with the destruction operator $c_{\mathbf k, \sigma}$.
If $|\mathbf k| >k_f$ then $c_{\mathbf k, \sigma} |F\rangle = 0$ because destroying an empty state gives $0$. If $|\mathbf k| <k_f$ then $c_{\mathbf k, \sigma}^{\dagger} |F\rangle = 0$ because of the Pauli exclusion principle. It is important to distinguish inside and outside of the Fermi sphere because of the Pauli exclusion principle.
When one electron inside the Fermi Sphere ($|\mathbf k| <k_f$) is destroyed the state $\mathbf k, \sigma$ is empty, one may call this empty state a hole. The total momentum of the system is then decreased by $\mathbf k$ and the total spin is also decreased by $\sigma$.
Hole creation or destruction
The latter hole can be seen as a particle itself. It has opposite spin and momentum. A hole is created in the Fermi sphere when an electron is destroyed in the Fermi sphere.
We can define creation and destruction operators for holes. Let's say that $a_{\mathbf k, \sigma}^{\dagger}$ creates a hole with the momentum $\mathbf k$ and spin $\sigma$ ($a_{\mathbf k, \sigma}$ destroys the latter hole). We said that the creation of a hole is the same thing as the destruction of an electron inside the Fermi shpere, then
$$
c_{\mathbf k, \sigma} = a_{-\mathbf k,-\sigma}^{\dagger}
$$
$c_{\mathbf k, \sigma}$ corresponds to a decrease of total momentum and total spin by $\mathbf k$ and $\sigma$. $a_{-\mathbf k,-\sigma}^{\dagger}$ is creating a particle, thus adding corresponding momentum and spin to the system, here the total momentum is incread by $-\mathbf k$ (and total spin by $-\sigma$).
You may have a classical picture of the problem. Imagine a set of balls going to the right with a given velocity. If you suddenly remove one ball you will see a hole going with the same velocity but in the opposite direction. Opposite momentum correspond the the opposite direction.
Quasi-particles
There is an issue though. The creation of a hole with momentum $|\mathbf k| > k_f$ is impossible:
$$
a_{\mathbf k, \sigma}^{\dagger} |F\rangle = c_{-\mathbf k, -\sigma} |F\rangle = 0
$$
As Mudelung suggests, it may be useful to get rid of asymmetry. The idea is to define new quasi-particles. Creation and destruction operators of quasi-particles are linear combinations of other creation and destruction operators. It will  be possible to create those quasi-particles wether their momentum is greater or lower than $k_f$. If the momentum is lower than the Fermi radius then we create a hole and if the momentum is greater than $k_f$ we create an electron. All this can be summarized in:

$$\begin{align}
\alpha_{\mathbf{k}\sigma}^+=u_\mathbf{k}c_{\mathbf{k}\sigma}^{\dagger}+v_{-\mathbf{k}}c_{\mathbf{k},\sigma}=u_\mathbf{k}c_{\mathbf{k}\sigma}^{\dagger}+v_{\mathbf{k}}a_{\mathbf{k},\sigma}^{\dagger}
\\
\alpha_{\mathbf{k}\sigma}=u_\mathbf{k}c_{\mathbf{k}\sigma}+v_{\mathbf{k}}c_{-\mathbf{k},-\sigma}^{\dagger}=u_\mathbf{k}c_{\mathbf{k}\sigma}+v_{\mathbf{k}}a_{\mathbf{k},\sigma}
\end{align}
$$
   with
  $$
\begin{align}
u_\mathbf{k}=1,\; v_\mathbf{k}=0\quad\text{ for }\quad k>k_\text{F},
\\
u_\mathbf{k}=0,\; v_\mathbf{k}=1\quad\text{ for }\quad k<k_\text{F}.
\end{align}
$$

Now we have operators creating quasi-particles which can be holes or electrons regarding if you are inside or outside of the Fermi surface.
