_{{\bf k}}|\; \beta^{r}_{-{\bf
k},1}(t)\;+\;\epsilon^{r}\; |V_{{\bf k}}|\; \alpha^{r\dag}_{{\bf
k},1}(t)\right),
\eea
where $\epsilon^{r}=(-1)^{r}$ and
\begin{eqnarray}}\newcommand{\eea}{\end{eqnarray}
 |U_{{\bf k}}|\equiv u^{r\dag}_{{\bf k},i}
u^{r}_{{\bf k},j} = v^{r\dag}_{-{\bf k},i} v^{r}_{-{\bf k},j} \,
,\qquad  i \neq j\,; \qquad \qquad |V_{{\bf k}}|\equiv
\varepsilon_{ij} \epsilon^{r}\; u^{r\dag}_{{\bf k},i} v^{r}_{-{\bf
k},j}, \quad {\rm no \,\, summation} \eea
with $\varepsilon_{ij}= 0,1,-1$ for $i=j,~ i<j,  ~i>j$,
respectively.
We have:
  \begin{eqnarray}}\newcommand{\eea}{\end{eqnarray}
\nonumber}\def\ot{\otimes |U_{{\bf
k}}|=\left(\frac{\omega_{k,1}+m_{1}}{2\omega_{k,1}}\right)^{\frac{1}{2}}
\left(\frac{\omega_{k,2}+m_{2}}{2\omega_{k,2}}\right)^{\frac{1}{2}}
\left(1+\frac{{\bf
k}^{2}}{(\omega_{k,1}+m_{1})(\omega_{k,2}+m_{2})}\right)
\\
\label{Vk}|V_{{\bf
k}}|=\left(\frac{\omega_{k,1}+m_{1}}{2\omega_{k,1}}\right)^{\frac{1}{2}}
\left(\frac{\omega_{k,2}+m_{2}}{2\omega_{k,2}}\right)^{\frac{1}{2}}
\left(\frac{k}{(\omega_{k,2}+m_{2})}-\frac{k}{(\omega_{k,1}+m_{1})}\right)
\eea
\begin{eqnarray}}\newcommand{\eea}{\end{eqnarray}
|U_{{\bf k}}|^{2}+|V_{{\bf k}}|^{2}=1\,.
\eea

The  number of  condensate neutrinos for each $\bf k$ is given
by
\begin{eqnarray}}\newcommand{\eea}{\end{eqnarray} \label{density} _{e,\mu}\langle 0| \alpha}\def\bt{\beta}\def\ga{\gamma_{{\bf k},i}^{r \dag}
\alpha}\def\bt{\beta}\def\ga{\gamma^r_{{\bf k},i} |0\rangle_{e,\mu}\,= \;_{e,\mu}\langle 0|
\bt_{{\bf k},i}^{r \dag} \bt^r_{{\bf k},i} |0\rangle_{e,\mu}\,=\,
\sin^{2}\te\; |V_{{\bf k}}|^{2} \;,
 \eea
with $i=1,2\,$. Equivalently, $ _{1,2}\langle 0| \alpha}\def\bt{\beta}\def\ga{\gamma_{{\bf
k},\sigma}^{r \dag} \alpha}\def\bt{\beta}\def\ga{\gamma^r_{{\bf k},\sigma} |0\rangle_{1,2}\,=
\;_{1,2}\langle 0| \bt_{{\bf k},\sigma}^{r \dag} \bt^r_{{\bf
k},\sigma} |0\rangle_{1,2}\,=\, \sin^{2}\te\; |V_{{\bf k}}|^{2}
\;,$ with $\sigma = e,\mu\,$.

The Bogoliubov coefficient $|V_{{\bf k}}|^{2}$ appearing in
Eq.(\ref{density}) can be written as a function of the dimensionless
momentum $p=\frac{|{\bf k}|}{\sqrt{m_1 m_2}}$ and dimensionless
parameter $a= \frac{(m_{2}-m_{1})^2}{m_1 m_2}$, as follows,
\begin{eqnarray}}\newcommand{\eea}{\end{eqnarray} |V(p,a)|^2 & =& \frac{1}{2}\left}\def\lrar{\leftrightarrow(1-\frac{p^2+1} {\sqrt{(p^2 +
1)^2 + a p^2}}\right}\def\ti{\tilde) ~.\label{Vpa}
 \eea

From Fig.1 we see that the effect is maximal when $p=1$, i.e. for
$|{\bf k}|^2 = m_{1} m_{2}$, the natural scale of the neutrino
mixing.  $|V_{{\bf k}}|^2$ goes to zero for large momenta, i.e. for
$|{\bf k}|^2\gg m_{1} m_{2}$,  as $|V_{{\bf k}}|^2 \approx
\frac{(\Delta m)^2}{4 k^2}$.   It acts as a ``form factor'' in the
$\bf k$ space controlling the neutrino vacuum condensate. We thus
find that the Pontecorvo's formalism is nothing but the relativistic
limit of the QFT formalism: $|V_{{\bf k}}|^2 \rightarrow}\def\Rar{\Rightarrow 0$ and $|U_{{\bf
k}}|^2 \rightarrow}\def\Rar{\Rightarrow 1$ for  $|{\bf k}|^2 \gg m_{1} m_{2}$. In the
Pontecorvo's formalism the non-perturbative contributions from the
vacuum condensate are thus missing. This is the meaning of the
approximation made in the QM treatment of the mixing. Missing these
condensate contributions is of course of no relevance for the
experimental observation of neutrino oscillations at today
instrumentation resolution. These contributions might play, however,
a relevant role in the study of the cosmological background. The
fact that $|V_{{\bf k}}|^2$ contributes maximally for low energies
suggests indeed to us that the contribution of the mixing phenomenon
may be taken as a good candidate in the study of dark energy.

\begin{figure}
\centering \resizebox{10cm}{!}{\includegraphics{ploty6.eps}}
\caption{The fermion condensation density $|V(p,a)|^2$ as a
function of $p$ for $a=0.98$ (solid line) and $a=0.5$ (dashed
line).} \label{Fig: 1}
\end{figure}


We point out that, since $|0\rangle_{1,2}$ and $|0\rangle_{e,\mu}$
are unitary inequivalent states in the infinite volume limit, two
different normal orderings must be defined, respectively with
respect to the vacuum $|0\rangle_{1,2}$ for fields with definite
masses, denoted as usual  by $:...:$,  and with respect to the
vacuum for fields with definite flavor $|0\rangle_{e,\mu}$, denoted
by $::...::$~. The Hamiltonian normal ordered with respect to the
vacuum $|0\rangle_{1,2}\,$ is given by
\begin{eqnarray}}\newcommand{\eea}{\end{eqnarray} \label{Hnorm}:H: \,=\, H - _{1,2} \langle 0|H|0\rangle_{1,2} = H +
\, 2\int d^{3}{\bf k} \, (\omega_{k,2} + \omega_{k,1}) =
\sum_{i}\sum_{r}\int d^{3}{\bf k}\,\omega_{k,i}[\alpha_{{\bf
k},i}^{r \dag}\alpha_{{\bf k},i}^{r}+ \beta_{{\bf k},i}^{r
\dag}\beta_{{\bf k},i}^{r}] ~,
\eea
and the Hamiltonian normal ordered with respect
to the vacuum $|0\rangle_{e,\mu}$ is
\begin{eqnarray}}\newcommand{\eea}{\end{eqnarray}\label{Hflav} :\;\!\!\;\!\!: H :\;\!\!\;\!\!:\, \equiv \,  H \, -\, {}_{e,\mu}\langle}\def\lar{\leftarrow
0(t)| H | 0(t) \ran_{e,\mu}\, = H \, +\, 2\int d^{3}{\bf k} \,
(\omega_{k,2} + \omega_{k,1}) \,(1 - 2\,|V_{\bf k}|^{2}
\sin^{2}\theta)\,. \eea
 Note that the difference of
energy between $|0\rangle_{e,\mu}$ and $|0\rangle_{1,2}$ represents
the energy of the condensed neutrinos given in Eq.(\ref{density})
\begin{eqnarray}}\newcommand{\eea}{\end{eqnarray}\label{energyCond} {}_{e,\mu}\langle}\def\lar{\leftarrow 0(t)| :H: | 0(t)
\ran_{e,\mu