# Neutrinos in curved space-time: particle mixing and flavor oscillations

A. Capolupo<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello”
Università di Salerno, and INFN – Gruppo Collegato di Salerno, Via Giovanni
Paolo II, 132, 84084 Fisciano (SA), Italy  G. Lambiase<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello” Università di Salerno, and INFN –
Gruppo Collegato di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA),
Italy  A. Quaranta<EMAIL_ADDRESS>Dipartimento di Fisica “E.R.
Caianiello” Università di Salerno, and INFN – Gruppo Collegato di Salerno, Via
Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy

###### Abstract

We present a quantum field theoretical approach to the vacuum neutrino
oscillations in curved space, we analyze the non–trivial interplay between
quantum field mixing and field quantization in curved space and derive new
oscillation formulae. We compute the formulae explicitly in the spatially flat
FLRW metrics for universes dominated by a cosmological constant and by
radiation. We evaluate the transition probabilities in the Schwarzschild black
hole metric, and we show that the Hawking radiation affects the oscillations
of neutrinos. We show that our results are consistent with those of previous
analyses when the quantum mechanical limit is considered.

##  I Introduction

Since they were theoretically proposed by Pauli Pauli , neutrinos have proven
to be among the most enigmatic particles in the universe. Until the discovery
of flavor oscillations NeutrinoOscillations1 ; NeutrinoOscillations2 , whose
theory was pioneered by Pontecorvo Pontecorvo ; Bilenky , neutrinos were
believed to be massless. Today it is accepted that neutrinos are massive
particles, and that they oscillate among three flavors
$\nu_{e},\nu_{\mu},\nu_{\tau}$ corresponding to the companion charged leptons
$e,\mu,\tau$. This peculiarity renders neutrinos unique among the known
elementary particles and puts them beyond the scope of the standard model of
particles Mohapatra . In many respects, neutrinos are forerunners of a new
physics, as several issues, including the origin of their mass NeutrinoMass
and their fundamental nature NeutrinoNature , are still open to the present
day.

On the other hand, the relevance of neutrinos in astrophyisical and
cosmological contexts has grown dramatically during the last years. They
figure as a valuable source of information, along with gravitational waves and
electromagnetic radiation, in the ever–growing field of multi–messenger
astronomy Franckowiak . The study of neutrinos of astrophysical origin can
indeed provide fundamental insights on the source that produced them. In
addition, neutrinos are expected to play an important role in the first phases
of the universe Buchmuller ; CosmologicalNeutrinos , and the detection of the
cosmic neutrino background, pursued in experiments as PTOLEMY PTOLEMY , could
represent an essential test for the standard cosmological model
CosmologicalNeutrinos . Mass varying neutrinos have also been proposed as a
possible explanation for Dark Energy Mavans .

This state of affairs requires a careful investigation of neutrino
oscillations on a curved spacetime. The topic has been discussed in several
works , where it was found that gravitational fields may alter both the
oscillations in vacuum and in matter Grossman ; Piriz ; Cardall .

Here we wish to go beyond the heuristic treatment of ref. Cardall , and
present a quantum field theoretical approach, based on the field quantization
in curved space-time, to evaluate the effects of gravitational fields on
neutrino oscillations. We derive general oscillation formulae for flavor
fields in curved space-time, which represent our main result. We discuss the
particle interpretation of the fields in presence of gravity and study how the
mixing changes when moving from a mass field representation to another. We
demonstrate the invariance of local observables, which are represented by
expectation values on flavor states of local operators constructed from the
flavor fields. We show that the oscillation probabilities, on the other hand,
do in general depend on the representation of the mass fields, since they are
not a local observable and involve the comparison between particles in
different spacetime regions. We establish the conditions which have to be
satisfied in order that the resulting transition probabilities are invariant
under changes of mass field representation.

We also compute explicitly the oscillation formulae for two examples of
spatially flat Friedmann–Lemaitre–Robertson–Walker spacetimes, corresponding
to a cosmological constant–dominated and a radiation–dominated universe
respectively. In these cases, exact analytical solutions to the Dirac equation
are available, and the formalism here introduced can be applied directly.
Moreover, we give an estimation of the oscillation formulae for neutrinos
propagating from the past infinity to the future infinity in a stationary
Schwarzschild spacetime. We introduce a method to extract the oscillation
formulae on spacetimes with asymptotically flat regions without resorting to
the exact solutions of the Dirac equation. We then employ this strategy to
compute the formulae on the Schwarzschild black hole spacetime, for neutrinos
propagating from the past infinity to the future infinity. We show how the
Hawking radiation is naturally embedded in the resulting transition
probabilities.

Our results generalize those of the previous treatments Cardall , and are
consistent with the latter when the suitable limits are considered. In our
computation, for simplicity, we limit our analysis to the vacuum oscillations,
therefore considering the sole effect of gravity.

The paper is organized as follows: in section II we provide the setting for
the description of the mass fields in curved space; in section III we develop
field mixing and find the oscillation probabilities in curved spacetime, with
a thorough analysis of their features; in section IV we apply the formalism to
some spacetimes of interest, including the spatially flat FLRW metric for a
radiation-dominated universe and for a cosmological constant-dominated
universe, and the Schwarzschild black hole metric, where we show the impact of
the Hawking effect on neutrino oscillations; finally in section V we draw our
conclusions.

##  II Mass neutrino fields in curved Space

To evaluate the oscillation formulae for neutrinos on a curved spacetime, it
is necessary to consider both the effects of curvature and mixing on the
(free) mass fields. Let $M$ be a globally hyperbolic spacetime, and let
$\tau\in\mathbb{R}$ label a foliation of $M$ by Cauchy surfaces. Consider the
tetrad fields $e^{\mu}_{a}(x)$ satisfying
$\eta^{ab}e^{\mu}_{a}(x)e^{\nu}_{b}(x)=g^{\mu\nu}(x)$. Here $\eta^{ab}\equiv
diag(1,-1,-1,-1)$ is the Minkowski metric tensor, while $g^{\mu\nu}(x)$ is the
contravariant metric $g^{\mu\nu}(x)g_{\nu\rho}(x)=\delta^{\mu}_{\rho}$ on $M$
in a given coordinate system. The massive neutrino fields satisfy the Dirac
equations:

$(i\gamma^{\mu}(x)D_{\mu}-m_{i})\psi_{i}=0$ (1)

where $\gamma^{\mu}(x)=e^{\mu}_{a}(x)\gamma^{a}$, $\gamma^{a}$ being the usual
flat space Dirac matrices, and
$D_{\mu}=\partial_{\mu}-\frac{i}{4}\omega_{\mu}^{ab}\sigma_{ab}$. The spin
connection is defined as
$\omega_{\mu}^{ab}=e^{a}_{\nu}\Gamma^{\nu}_{\rho\mu}e^{\rho
b}+e^{a}_{\nu}\partial_{\mu}e^{\nu b}$, whereas $\sigma_{ab}$ are the
commutators of flat Dirac matrices
$\sigma_{ab}=\frac{i}{2}[\gamma^{a},\gamma^{b}]$. In equation (1), the index
$i=1,2,...,N$ ranges over the number of neutrino species $N$. For the sake of
simplicity we focus on the case $N=2$, though the generalization to $N=3$ is
straightforward. In general equation (1) cannot be solved exactly. Even if one
is able to find exact solutions, these do not play the same prominent role as
their flat spacetime counterpart. It is well–known, indeed, that the positive
frequency solutions of equation cannot be defined univocally, and that,
consequently, there is no natural (nor unique) particle interpretation for the
corresponding Quantum Field Theory Birrell ; Wald . Nevertheless, the
canonical quantization of the Dirac field proceeds along the same lines as in
Minkowski spacetime.

To perform a field expansion, one must find a set of positive $\zeta_{k,i}$
and negative $\xi_{k,i}$ frequency solutions for each of the equations (1). In
general the bipartition of the solutions to eq. (1) makes sense only locally,
while there is no natural global definition of positive and negative frequency
modes. Anyway, one is free to choose a set of modes Note1
$\\{\zeta_{k,i},\xi_{k,i}\\}$ , deemed to be positive/negative frequency modes
according to some specified observer, and expand the field with respect to
them, provided that they form a complete (and orthonormal) set of solutions
under the inner product

$(a_{i},b_{i})=\int_{\Sigma(\tau)}\sqrt{-g}d\Sigma_{\mu}(\tau)\bar{a}_{i}\gamma^{\mu}(x)b_{i}\
$ (2)

with $a_{i},b_{i}$ any solution to equation (1) with mass $m_{i}$ and
$\bar{b}_{i}=b^{\dagger}_{i}\gamma^{0}(x)$. Here
$d\Sigma^{\mu}(\tau)=n^{\mu}(\tau)dV_{\tau}$ denotes the volume element on the
surface $\tau$ with unit timelike normal $n^{\mu}(\tau)$. This has to hold
separately for each $i=1,2$. As it is easy to prove, for $a_{i},b_{i}$
solutions of the (same) Dirac equation, the inner product (2) does not depend
on the hypersurface chosen for the integration. In particular, it is
independent on the foliation by Cauchy hypersurfaces employed. The fields can
then be expanded as

$\psi_{i}(x)=\sum_{k,s}\left(\gamma_{k,s;i}\zeta_{k,s;i}(x)+\epsilon_{k,s;i}^{\dagger}\xi_{k,s;i}(x)\right)$
(3)

with the operator coefficients $\gamma_{k,s,i},\epsilon_{k,s,i}$ satisfying
the usual canonical anticommutation relations, $k$ momentum index and $s$
helicity index. The annihilators are also required to anticommute for $i\neq
j$, and, in particular
$\\{\gamma_{k,s,i},\gamma^{\dagger}_{k,s,j}\\}=\delta_{ij}$,
$\\{\epsilon_{k,s,i},\epsilon^{\dagger}_{k,s,j}\\}=\delta_{ij}$. In equation
(3) we prefer to keep any space-time dependence within the modes, for ease of
treatment with a general metric. The expansions (3) define the mass Hilbert
space $\mathcal{H}_{m}=H_{1}\otimes H_{2}$, which is constructed out of the
vacuum $|0_{m}\rangle=|0_{1}\rangle\otimes|0_{2}\rangle$. Here $\ket{0_{i}}$
is defined, as usual, by
$\gamma_{k,s,i}\ket{0_{i}}=0=\epsilon_{k,s,i}\ket{0_{i}}$ for each $k,s,i$.

As hinted above, the field expansions (3) are somewhat arbitrary, as opposed
to the flat spacetime case, where there is no ambiguity in the definition of
positive and negative frequency modes. Any other basis
$\\{\tilde{\zeta}_{k,s,i},\tilde{\xi}_{k,s,i}\\}$ can be used to expand the
fields
$\psi_{i}=\sum_{k,s}(\tilde{\gamma}_{k,s,i}\tilde{\zeta}_{k,s,i}(x)+\tilde{\epsilon}_{k,s,i}^{\dagger}\tilde{\xi}_{k,s,i}(x))$.
Since both the sets $\\{\zeta_{k,s,i},\xi_{k,s,i}\\}$ and
$\\{\tilde{\zeta}_{k,s,i},\tilde{\xi}_{k,s,i}\\}$ form a basis for the space
of solutions of eqs. (1), one can write the modes of a set in terms of the
other, for each $i$:

$\displaystyle\tilde{\zeta}_{k^{\prime},s^{\prime},i}$ $\displaystyle=$
$\displaystyle\sum_{k,s}\left(\Gamma_{k^{\prime},s^{\prime};k,s;i}^{*}\zeta_{k,s,i}+\Sigma_{k^{\prime},s^{\prime};k,s;i}^{*}\xi_{k,s,i}\right)$
$\displaystyle\tilde{\xi}_{k^{\prime},s^{\prime},i}$ $\displaystyle=$
$\displaystyle\sum_{k,s}\left(\Gamma_{k^{\prime},s^{\prime};k,s;i}\xi_{k,s,i}-\Sigma_{k^{\prime},s^{\prime};k,s;i}\zeta_{k,s,i}\right)$
(4)

where
$\Gamma_{k^{\prime},s^{\prime};ks;i}=(\tilde{\zeta}_{k^{\prime},s^{\prime},i},\zeta_{k,s,i})=(\xi_{k,s,i},\tilde{\xi}_{k^{\prime},s^{\prime},i})$
and
$\Sigma_{k^{\prime},s^{\prime};ks;i}=(\tilde{\zeta}_{k^{\prime},s^{\prime},i},\xi_{k,s,i})=-(\zeta_{k,s,i},\tilde{\xi}_{k^{\prime}s^{\prime},i})$.
This is a fermionic Bogoliubov transformation, for which
$\sum_{q,r}\left(\Gamma_{k,s;q,r;i}^{*}\Gamma_{k^{\prime},s^{\prime};q,r;i}+\Sigma_{k,s;q,r;i}^{*}\Sigma_{k^{\prime},s^{\prime};q,r;i}\right)=\delta_{k,k^{\prime}}\delta_{s,s^{\prime}}$
for each $i$. The corresponding relation between the two sets of annihilators
is given by

$\displaystyle\tilde{\gamma}_{k,s,i}$ $\displaystyle=$
$\displaystyle\sum_{k^{\prime},s^{\prime}}\left(\Gamma_{k,s;k^{\prime}s^{\prime};i}\gamma_{k^{\prime},s^{\prime},i}+\Sigma_{k,s;k^{\prime},s^{\prime};i}\epsilon^{\dagger}_{k^{\prime},s^{\prime},i}\right)$
$\displaystyle\tilde{\epsilon}_{k,s,i}$ $\displaystyle=$
$\displaystyle\sum_{k^{\prime},s^{\prime}}\left(\Gamma_{k,s;k^{\prime},s^{\prime};i}\epsilon_{k^{\prime},s^{\prime},i}-\Sigma_{k,s;k^{\prime},s^{\prime};i}\gamma^{\dagger}_{k^{\prime},s^{\prime},i}\right)\
.$ (5)

It is often the case that the Bogoliubov coefficients
$\Gamma_{k,s;k^{\prime},s^{\prime};i},\Sigma_{k,s;k^{\prime},s^{\prime};i}$
can be written as
$\Gamma_{k,s;k^{\prime},s^{\prime};i}=\delta_{k,k^{\prime}}\delta_{s,s^{\prime}}\Gamma_{k,i}$,
$\Sigma_{k,s;k^{\prime},s^{\prime};i}=\delta_{k,k^{\prime}}\delta_{s,s^{\prime}}\Sigma_{k,i}$,
with $\Gamma_{k,i}$ and $\Sigma_{k,i}$ depending on $k$ alone. In this
occurrence, they admit the parametrization
$\Gamma_{k,i}=e^{i\eta_{k,i}}\cos(\theta_{k,i})$
,$\Sigma_{k,i}=e^{i\phi_{k,i}}\sin(\theta_{k,i})$, with
$\eta_{k,i},\phi_{k,i},\theta_{k,i}$ real functions of $k$. We remark that the
Bogoliubov transformations (II) can be recast in terms of the generators
$J_{i}=e^{\sum_{k,k^{\prime},s,s^{\prime}}\left[(\lambda_{k,k^{\prime},s,s^{\prime},i}^{*}\gamma_{k,s,i}^{\dagger}\epsilon_{k^{\prime}s^{\prime},i}^{\dagger}-\lambda_{k,k^{\prime},s,s^{\prime},i}\epsilon_{k,s,i}\gamma_{k^{\prime},s^{\prime},i})\right]}$,
with
$\lambda_{k,k^{\prime},s,s^{\prime},i}=Arctan(\frac{\Sigma_{k,s;k^{\prime},s^{\prime};i}}{\Gamma_{k,s;k^{\prime},s^{\prime};i}})$,
as

$\tilde{\gamma}_{k,s,i}=J_{i}^{-1}\gamma_{k,s,i}J_{i}\;,\qquad\qquad\ \
\tilde{\epsilon}_{k,s,i}=J_{i}^{-1}\epsilon_{k,s,i}J_{i}\ .$ (6)

The maps $J_{i}:\tilde{\mathcal{H}}_{i}\rightarrow\mathcal{H}_{i}$ interpolate
between the Fock spaces $\mathcal{H}_{i}$ built from the
$\gamma_{k,s,i},\epsilon_{k,s,i}$ and the Fock space $\tilde{\mathcal{H}}_{i}$
built from the $\tilde{\gamma}_{k,s,i},\tilde{\epsilon}_{k,s,i}$. In
particular, one has for the vacuum states
$|\tilde{0}_{i}\rangle=J_{i}^{-1}|0_{i}\rangle$. As for the untilded
representation, the mass Hilbert space in the tilded representation is the
tensor product
$\tilde{\mathcal{H}}_{m}=\tilde{\mathcal{H}}_{1}\otimes\tilde{\mathcal{H}}_{2}$.
It is convenient to define a unique generator of Bogoliubov transformations
$J:\tilde{\mathcal{H}}_{m}\longrightarrow\mathcal{H}_{m}$ on
$\tilde{\mathcal{H}}_{m}$ as the tensor product $J=J_{1}\otimes J_{2}$. Then

$\tilde{\gamma}_{k,s,i}=J^{-1}\gamma_{k,s,i}J\;,\qquad\qquad\ \
\tilde{\epsilon}_{k,s,i}=J^{-1}\epsilon_{k,s,i}J$ (7)

for $i=1,2$. The expansions of the two fields $\psi_{1}$ and $\psi_{2}$ must
be compatible with each other, i.e., each of the modes
$\zeta_{k,s,2},\xi_{k,s,2}$ must be obtainable from the corresponding modes
$\zeta_{k,s,1},\xi_{k,s,1}$ by the substitution $m_{1}\leftrightarrow m_{2}$,
and vice–versa. In the context of mixing, this ensures that the same kind of
particle, described by the same set of quantum numbers, is being mixed. This,
of course, does not undermine the arbitrariness in the choice of the modes;
these can be any complete set of solutions to the Dirac equation, provided
that the same choice is made for the two fields.

##  III Neutrino mixing and oscillation formulae in curved space-time

In this Section, we show new oscillation formulae for flavor fields in curved
space-time and we present general considerations on the infinitely many
unitarily inequivalent representations of the canonical anticommutation
relations which characterize the quantization of mixed fields and of fields in
curved space.

###  III.1 Oscillation Formulae

As discussed above, the QFT of free Dirac fields in curved space is
characterized by infinitely many unitarily inequivalent representations of the
canonical anticommutation relations. The phenomenon of mixing, even in
Minkowski space, suffers from an analogous ambiguity, in that the flavor and
the mass representations are unitarily inequivalent Capolupo1 . The effects of
such inequivalence have been analyzed in flat space time Capolupo:2006et  and
the possibility to reveal them in experimental setup has been recently
proposed Capolupo:2019gmn . Let us start by fixing the mass field expansions
(3) and describe the mixing in a given representation of the mass fields. The
flavor fields are defined as
$\psi_{e}=\cos(\theta)\psi_{1}+\sin(\theta)\psi_{2}$ and
$\psi_{\mu}=\cos(\theta)\psi_{2}-\sin(\theta)\psi_{1}$ with $\theta$ the
(2-flavor) mixing angle. Just like the Bogoliubov transformations (7), the
rotation to flavor fields can be cast in terms of a generator
$\mathcal{I}_{\theta}(\tau)$. This is given by

$\mathcal{I}_{\theta}(\tau)=e^{\theta[(\psi_{1},\psi_{2})_{\tau}-(\psi_{2},\psi_{1})_{\tau}]}\
.$ (8)

where the scalar products $(\psi_{i},\psi_{j})$ _do_ depend on the
hypersurface chosen for the integration, since they are solutions to different
Dirac equations. Then, by definition, the flavor fields are expressed as

$\psi_{e}=\mathcal{I}_{\theta}^{-1}(\tau)\psi_{1}\mathcal{I}_{\theta}(\tau)\;,\qquad\qquad\
\
\psi_{\mu}=\mathcal{I}_{\theta}^{-1}(\tau)\psi_{2}\mathcal{I}_{\theta}(\tau).$
(9)

If we let the generator (8) act on the mass annihilators, we obtain the flavor
annihilators for curved space

$\gamma_{k,s,e}(\tau)=\mathcal{I}_{\theta}^{-1}(\tau)\gamma_{k,s,1}\mathcal{I}_{\theta}(\tau)=\cos(\theta)\gamma_{k,s,1}+\sin(\theta)\sum_{q,r}\bigg{[}\Lambda^{*}_{q,r;k,s}(\tau)\gamma_{q,r,2}+\Xi_{q,r;k,s}(\tau)\epsilon_{q,r,2}^{\dagger}\bigg{]}\,.$
(10)

And similar for
$\gamma_{k,s,\mu}(\tau),\epsilon_{k,s,e}(\tau),\epsilon_{k,s,\mu}(\tau)$. The
Bogoliubov coefficients are provided by the inner products of the solutions to
the _curved space_ Dirac equation with mass $m_{1}$ and $m_{2}$, that is,
$\Lambda_{q,r;k,s}(\tau)=(\zeta_{q,r,2},\zeta_{k,s,1})_{\tau}=(\xi_{k,s,1},\xi_{q,r,2})_{\tau}$
and
$\Xi_{q,r;k,s}(\tau)=(\zeta_{k,s,1},\xi_{q,r,2})_{\tau}=-(\zeta_{q,r,2},\xi_{k,s,1})_{\tau}$.
The mixing coefficients always satisfy

$\sum_{q,r}\left(\Lambda_{k,s;q,r}^{*}(\tau)\Lambda_{k^{\prime},s^{\prime};q,r}(\tau)+\Xi_{k,s;q,r}^{*}(\tau)\Xi_{k^{\prime},s^{\prime};q,r}(\tau)\right)\\!=\\!\delta_{k,k^{\prime}}\delta_{s,s^{\prime}}\
.$ (11)

Since the mass expansions are compatible, the mixing coefficients are often
diagonal, namely of the form

$\displaystyle\Lambda_{q,r;k,s}(\tau)$ $\displaystyle=$
$\displaystyle\delta_{q,k}\delta_{r,s}\Lambda_{k,s}(\tau)$
$\displaystyle\Xi_{q,r;k,s}(\tau)$ $\displaystyle=$
$\displaystyle\delta_{q,k}\delta_{r,s}\Xi_{k,s}(\tau)$ (12)

with $\Lambda_{k,s}(\tau)$, $\Xi_{k,s}(\tau)$ depending on $k$ and $s$ alone
Note3 . Exceptions to this arise when we consider expansions of the mass
fields in terms of modes labelled by the energy. In such a case, the mixing
coefficients are non–diagonal and different from zero
$\Lambda_{\omega,\omega^{\prime}}\neq 0$ ,$\Xi_{\omega,\omega^{\prime}}\neq
0$, once $\omega$ is fixed, only for a specific value of $\omega^{\prime}$.

$|\Lambda_{k,s}(\tau)|^{2}+|\Xi_{k,s}(\tau)|^{2}=1$ (13)

for each $k,s,\tau$, and

$|\Lambda_{\omega;\omega^{\prime}}(\tau)|^{2}+|\Xi_{\omega;\omega^{\prime}}(\tau)|^{2}=1$
(14)

respectively. The mass and the flavor representations are unitarily
inequivalent. For each $\tau$ one has a distinct flavor Fock space
$\mathcal{H}_{f}(\tau)$ defined by
$\gamma_{e,\mu}(\tau),\epsilon_{e,\mu}(\tau)$. The flavor vacuum
$\ket{0_{f}(\tau)}=\mathcal{I}_{\theta}^{-1}(\tau)\ket{0_{m}}$ is a condensate
of $\psi_{1},\psi_{2}$ particle-antiparticle pairs.

In order to define the transition probabilities, we observe that the total
Lagrangian is invariant under $U(1)$ gauge transformations. Therefore the
total charge $Q=Q_{1}+Q_{2}=Q_{e}+Q_{\mu}$ is conserved Note4 , where
$Q_{i}=\sum_{k,s}\left(\gamma_{k,s,i}^{\dagger}\gamma_{k,s,i}-\epsilon_{k,s,i}^{\dagger}\epsilon_{k,s,i}\right)$
for $i=1,2,e,\mu$. It is then meaningful to define the transition
probabilities as

$P^{\rho\rightarrow\sigma}_{k,s}(\tau)=\sum_{q,r}\bigg{(}\langle\nu_{\rho,k,s}(\tau_{0})|Q^{q,r}_{\sigma}(\tau)|\nu_{\rho,k,s}(\tau_{0})\rangle-\langle
0_{f}(\tau_{0})|Q^{q,r}_{\sigma}(\tau)|0_{f}(\tau_{0})\rangle\bigg{)}.$ (15)

Here $\rho,\sigma=e,\mu$, the state
$|\nu_{\rho,k,s}(\tau_{0})\rangle=\gamma^{\dagger}_{k,s,\rho}(\tau_{0})\ket{0_{f}(\tau_{0})}$
is the state with a single neutrino of flavor $\rho$, momentum $k$ and
helicity $s$ on the reference hypersurface $\tau=\tau_{0}$ . The second term
on the rhs of (15) is just the implementation of the normal ordering with
respect to $\ket{0_{f}(\tau_{0})}$. By construction $P^{e\rightarrow
e}_{k,s}(\tau)+P^{e\rightarrow\mu}_{k,s}(\tau)=1$ and $P^{\mu\rightarrow
e}_{k,s}(\tau)+P^{\mu\rightarrow\mu}_{k,s}(\tau)=1$ for each $\tau$. A
straightforward calculation yields, in the general case (accounting for both
diagonal and non–diagonal mixing coefficients) the result

$P^{e\rightarrow\mu}_{k,s}(\tau)=2\cos^{2}(\theta)\sin^{2}(\theta)\times\bigg{[}1-\sum_{q,r}\Re\bigg{(}\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)+\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)\bigg{)}\bigg{]}\
.$ (16)

Equation (16) is the central result of the paper. When equations (III.1) hold,
this reduces to

$P^{e\rightarrow\mu}_{k,s}(\tau)=2\cos^{2}(\theta)\sin^{2}(\theta)\times\bigg{[}1-\Re\bigg{(}\Lambda_{k,s}^{*}(\tau_{0})\Lambda_{k,s}(\tau)+\Xi_{k,s}^{*}(\tau_{0})\Xi_{k,s}(\tau)\bigg{)}\bigg{]}\
.$ (17)

In both cases one has

$\displaystyle P^{e\rightarrow
e}_{k,s}(\tau)=1-P^{e\rightarrow\mu}_{k,s}(\tau).$ (18)

###  III.2 Mixing on a curved background and gravity-induced ambiguity in the
particle interpretation

Up to now, we have worked within a fixed, but arbitrary, representation of the
mass fields. The question arises about the other possible representations, and
how the mixing changes when moving from a representation to another. For the
definition (15) to make sense, we must determine if and how the probabilities
vary when the mass representation is changed. We take as a guideline the
principle of covariance, so that the _local_ physical observables should be
independent of the underlying representation. In moving from a given
representation $\\{\gamma_{1},\epsilon_{1}\\},\\{\gamma_{2},\epsilon_{2}\\}$
to another
$\\{\tilde{\gamma}_{1},\tilde{\epsilon}_{1}\\},\\{\tilde{\gamma}_{2},\tilde{\epsilon}_{2}\\}$,
we know how to connect the mass Fock spaces, namely via the generator (7)
$J^{-1}:\mathcal{H}_{m}\rightarrow\tilde{\mathcal{H}}_{m}$. For each mass
representation, we can proceed as we did above and build the corresponding
flavor annihilators and flavor spaces
$\mathcal{H}_{f}(\tau),\tilde{\mathcal{H}}_{f}(\tau)$, together with the
mixing generators
$\mathcal{I}_{\theta}(\tau):\mathcal{H}_{f}(\tau)\rightarrow\mathcal{H}_{m}$,
$\tilde{\mathcal{I}}_{\theta}(\tau):\tilde{\mathcal{H}}_{f}(\tau)\rightarrow\tilde{\mathcal{H}}_{m}$.
It is useful, at this point, to determine the relations among the mixing
coefficients $\Lambda(\tau),\Xi(\tau)$ and
$\tilde{\Lambda}(\tau),\tilde{\Xi}(\tau)$ that appear in the explicit form of
the two generators $\mathcal{I}_{\theta}(\tau)$ and
$\tilde{\mathcal{I}}_{\theta}(\tau)$. By definition we have

$\tilde{\Lambda}_{q,r;k,s}(\tau)=(\tilde{\zeta}_{q,r;2},\tilde{\zeta}_{k,s;1})_{\tau}=\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{(}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}^{*}\zeta_{q^{\prime},r^{\prime};2}+\Sigma_{q,r;q^{\prime},r^{\prime};2}^{*}\xi_{q^{\prime},r^{\prime};2}\bigg{]},\bigg{[}\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\zeta_{k^{\prime},s^{\prime};1}+\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\xi_{k^{\prime},s^{\prime};1}\bigg{]}\bigg{)}_{\tau}\,.$
(19)

Here the first equality is just the definition of $\tilde{\Lambda}(\tau)$, the
second follows from the Bogoliubov transformations (II). By using the
properties of the inner product (2), in the general case (again, accounting
for both the diagonal and non–diagonal mixing coefficients), we obtain

$\displaystyle\tilde{\Lambda}_{q,r;k,s}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime}r^{\prime};2}\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}(\zeta_{q^{\prime},r^{\prime};2},\zeta_{k^{\prime},s^{\prime};1})_{\tau}+\
\
\Gamma_{q,r;q^{\prime},r^{\prime};2}\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}(\zeta_{q^{\prime},r^{\prime};2},\xi_{k^{\prime},s^{\prime};1})_{\tau}$
(20) $\displaystyle+$ $\displaystyle\ \
\Sigma_{q,r;q^{\prime}r^{\prime};2}\Gamma_{k,s;k^{\prime}s^{\prime};1}^{*}(\xi_{q^{\prime},r^{\prime};2},\zeta_{k^{\prime},s^{\prime};1})_{\tau}+\
\
\Sigma_{q,r;q^{\prime},r^{\prime};2}\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}(\xi_{q^{\prime},r^{\prime};2},\xi_{k^{\prime},s^{\prime};1})_{\tau}\bigg{]}\,,$

and, finally, from the definition of $\Lambda(\tau)$ and $\Xi(\tau)$, we have

$\displaystyle\tilde{\Lambda}_{q,r;k,s}(\tau)=\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r,q^{\prime}r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)-\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)\right)$
$\displaystyle+\
\Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime};k^{\prime}s^{\prime}}^{*}(\tau)+\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{*}(\tau)\right)\bigg{]}\
.$ (21)

Similarly we have

$\displaystyle\tilde{\Xi}_{q,r;k,s}(\tau)=\sum_{q^{\prime},k^{\prime};r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)+\Sigma_{k,s;k^{\prime},s^{\prime};1}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)\right)$
$\displaystyle-\
\Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{*}(\tau)-\
\
\Sigma_{k,s;k^{\prime},s^{\prime};1}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{*}(\tau)\right)\bigg{]}\
.$ (22)

When equations (III.1) hold for both the representations
$\\{q,r\\},\\{q^{\prime},r^{\prime}\\}$, the equations reduce to

$\displaystyle\tilde{\Lambda}_{q,r}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{q^{\prime},r^{\prime}}\bigg{[}\Gamma_{q,r,q^{\prime}r^{\prime};2}\left(\Gamma_{q,r;q^{\prime},r^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime}}(\tau)-\Sigma_{q,r;q^{\prime},r^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime}}(\tau)\right)$
(23) $\displaystyle+$ $\displaystyle\
\Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{q,r;q^{\prime},r^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime}}^{*}(\tau)+\Sigma_{q,r;q^{\prime},r^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime}}^{*}(\tau)\right)\bigg{]}.$

and

$\displaystyle\tilde{\Xi}_{q,r}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{q^{\prime},r^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{q,r;q^{\prime},r^{\prime};1}\Xi_{q^{\prime},r^{\prime}}(\tau)+\Sigma_{q,r;q^{\prime},r^{\prime};1}\Lambda_{q^{\prime},r^{\prime}}(\tau)\right)$
(24) $\displaystyle-$ $\displaystyle\
\Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{q,r;q^{\prime},r^{\prime};1}\Lambda_{q^{\prime},r^{\prime}}^{*}(\tau)-\Sigma_{q,r;q^{\prime},r^{\prime};1}\Xi_{q^{\prime},r^{\prime}}^{*}(\tau)\right)\bigg{]}\
.$

The equations (III.2, III.2, 23, 24) provide an explicit relation between
$\mathcal{I}_{\theta}(\tau)$ and $\tilde{\mathcal{I}}_{\theta}(\tau)$, and
show how the mixing coefficients change, in moving from a mass representation
to another, in order to ensure covariance. In particular, the tilded
coefficients turn out to be a linear combination of the untilded coefficients
weighted by the coefficients of the Bogoliubov transformations between the two
mass representations. A slightly modified version of the eqs. (III.2) and
(III.2) will be expedient in the calculation of the transition probabilities
in a number of interesting cases.

It remains to establish how the flavor operators
$\gamma_{\rho}(\tau),\epsilon_{\rho}(\tau)$ and the flavor vacuum
$|0_{f}(\tau)\rangle$ transform under a change of mass representation. We
focus on the vacuum state $|0_{f}(\tau)\rangle\in\mathcal{H}_{f}(\tau)$. First
we employ the generator
$\mathcal{I}_{\theta}(\tau):\mathcal{H}_{f}(\tau)\rightarrow\mathcal{H}_{m}$
to get the mass vacuum $|0_{m}\rangle$. Then we apply the generator of
Bogoliubov transformations (7)
$J^{-1}:\mathcal{H}_{m}\rightarrow\tilde{\mathcal{H}}_{m}$ to obtain
$|\tilde{0}_{m}\rangle$. Finally, the generator of mixing in the tilde
representation
$\tilde{\mathcal{I}}_{\theta}^{-1}(\tau):\tilde{\mathcal{H}}_{m}\rightarrow\tilde{\mathcal{H}}_{f}(\tau)$
is employed to get $|\tilde{0}_{f}(\tau)\rangle$. We conclude that the two
flavor vacua are related by the transformation

$|\tilde{0}_{f}(\tau)\rangle=J_{f}^{-1}(\tau)|0_{f}(\tau)\rangle\doteq\tilde{\mathcal{I}}_{\theta}^{-1}(\tau)J^{-1}\mathcal{I}_{\theta}(\tau)|0_{f}(\tau)\rangle\
$ (25)

where we have defined the inverse $J_{f}^{-1}(\tau)$ for convenience. The
flavor operators must then transform as

$\displaystyle\gamma_{k,s,\rho}(\tau)$ $\displaystyle\rightarrow$
$\displaystyle J_{f}^{-1}(\tau)\gamma_{k,s,\rho}(\tau)J_{f}(\tau)$
$\displaystyle\epsilon_{k,s,\rho}(\tau)$ $\displaystyle\rightarrow$
$\displaystyle J_{f}^{-1}(\tau)\epsilon_{k,s,\rho}(\tau)J_{f}(\tau)$ (26)

and similarly for the creation operators. Equations (25) and (III.2) ensure
the invariance of local observables in the form of expectation values
$\bra{\psi_{f}(\tau)}F(\psi_{e}(\tau),\psi_{\mu}(\tau))\ket{\psi_{f}(\tau)}$
with $\ket{\psi_{f}(\tau)}\in\mathcal{H}_{f}(\tau)$ and
$F(\psi_{e}(\tau),\psi_{\mu}(\tau))$ any operator constructed from the fields
$\psi_{e}(\tau),\psi_{\mu}(\tau)$.

###  III.3 Transition probabilities and the mass representation

The oscillation probabilities are not a local observable, since they involve
the comparison between particles at different values of $\tau$, as it is
evident from the definition (15). In general these quantities _do_ depend on
the representation of the mass fields, and this is because distinct
representations might assign a different meaning to the quantum numbers $k,s$.
For example, one might consider two expansions of the mass fields, one in
terms of plane waves, labelled by the three–momenta $\\{\boldsymbol{k}\\}$,
and one in terms of localized wave packets, labelled by a suitable set of
quantum numbers $\\{q\\}$. It is clear that the two expansions describe
particles with different physical properties; the first describes particles
with definite momentum, the second describes particles for which momentum and
position are definite to some extent. Therefore the probabilities
$P^{\rho\rightarrow\sigma}_{k,s}$ and $P^{\rho\rightarrow\sigma}_{q,s}$ refer
to the oscillations of different particles, and have a different
interpretation. It would make no sense, in such a case, to require the
equivalence of the two. This, of course, would be true even in flat space.

What is meaningful to require, is that the transition probabilities
$P^{\rho\rightarrow\sigma}_{k,s}$ be the same for each compatible
representation, i.e., for each representation that refers to the same kind of
particle, and therefore agrees on the meaning of the quantum numbers $k,s$. In
mathematical terms, any two such representations shall be connected by
diagonal Bogoliubov transformations

$\displaystyle\tilde{\zeta}_{k,s,i}$ $\displaystyle=$
$\displaystyle\Gamma_{k,s;i}^{*}\zeta_{k,s,i}+\Sigma_{k,s;i}^{*}\xi_{k,s,i}$
$\displaystyle\tilde{\xi}_{k,s,i}$ $\displaystyle=$
$\displaystyle\Gamma_{k,s;i}\xi_{k,s,i}-\Sigma_{k,s;i}\zeta_{k,s,i}\ $ (27)

where it is understood that
$\Gamma_{k,s;q,r;i}=\delta_{k,q}\delta_{s,r}\Gamma_{k,s;i}$ and
$\Sigma_{k,s;q,r;i}=\delta_{k,q}\delta_{s,r}\Sigma_{k,s;i}$, In this case the
transition probabilities $P^{\rho\rightarrow\sigma}_{k,s}$ are indeed the
same, and this can be proven explicitly, by writing out equation (17) for the
two representations

$\displaystyle
P^{e\rightarrow\mu}_{k,s}(\tau)=2\cos^{2}(\theta)\sin^{2}(\theta)\bigg{[}1-\sum_{q,r}\Re\bigg{(}\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)+$
$\displaystyle\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)\bigg{)}\bigg{]}$
(28)

and

$\displaystyle\tilde{P}^{e\rightarrow\mu}_{k,s}(\tau)=2\cos^{2}(\theta)\sin^{2}(\theta)\times\bigg{[}1-\sum_{q,r}\Re\bigg{(}\tilde{\Lambda}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Lambda}_{k,s;q,r}(\tau)+\tilde{\Xi}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Xi}_{k,s;q,r}(\tau)\bigg{)}\bigg{]}$
(29)

With the aid of equations (III.2) , (III.2) and (45) we find

$\displaystyle\tilde{\Lambda}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Lambda}_{k,s;q,r}(\tau)+\tilde{\Xi}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Xi}_{k,s;q,r}(\tau)=$
$\displaystyle+\left(\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)+\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)\right)\left[|\Gamma_{k,s,2}|^{2}|\Gamma_{q,r,1}|^{2}+|\Gamma_{k,s,2}|^{2}|\Sigma_{q,r,1}|^{2}\right]$
$\displaystyle+\left(\Lambda_{k,s;q,r}(\tau_{0})\Lambda_{k,s;q,r}^{*}(\tau)+\Xi_{k,s;q,r}(\tau_{0})\Xi_{k,s;q,r}^{*}(\tau)\right)\left[|\Sigma_{k,s,2}|^{2}|\Gamma_{q,r,1}|^{2}+|\Sigma_{k,s,2}|^{2}|\Sigma_{q,r,1}|^{2}\right]\
.$ (30)

Each of the terms in the square brackets is real. Considered that

$\displaystyle\Lambda_{k,s;q,r}(\tau_{0})\Lambda_{k,s;q,r}^{*}(\tau)=\left(\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)\right)^{*}\,,\qquad\Xi_{k,s;q,r}(\tau_{0})\Xi_{k,s;q,r}^{*}(\tau)=\left(\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)\right)^{*}\,,$
(31)

and that the Bogoliubov coefficients satisfy
$|\Gamma_{k,s,i}|^{2}+|\Sigma_{k,s,i}|^{2}=1$ for each $k,s,i$, we finally get

$\displaystyle\Re[\tilde{\Lambda}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Lambda}_{k,s;q,r}(\tau)+\tilde{\Xi}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Xi}_{k,s;q,r}(\tau)]=\Re[\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)+\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)]\,,$
(32)

which proves the invariance of (28).

In the most general case, as the quantum numbers $k,s$ and
$k^{\prime},s^{\prime}$ have a different phyiscal meaning, the probabilities
$P^{\rho\rightarrow\sigma}_{k,s}$ and
$\tilde{P}^{\rho\rightarrow\sigma}_{k^{\prime},s^{\prime}}$ have different
interpretations. Different representations of the mass fields do indeed assign
a different meaning to such indices, so that the probabilities (15) have no
invariant meaning. In order to make sense of the probabilities in eqs. (15) in
the most general case, a representation of the mass fields must be fixed on
the grounds of physical relevance. When the underlying spacetime $M$ possesses
non trivial symmetries, as time translational invariance or spherical
symmetry, there is no doubt that the representation should be fixed so to take
them into account. In these cases ”good quantum numbers” are suggested by the
symmetries themselves (for instance, the energy $\omega$ for stationary
metrics, the angular momentum $l,m$ for spherically symmetric spacetimes). In
any case, the probabilities in a given mass representation can always be
related to the probabilites in any other mass representation with the aid of
equations (23) and (24). As a final remark, we stress that the issue discussed
here has nothing to do with the diffeomorphism covariance of the theory. All
the probabilities (17) are (generally covariant) scalars, as it is evident
from the definitions.

##  IV Neutrino oscillation formulae in FLRW metrics and in presence of a
Schwarzschild black hole

In this section we apply the formalism developed above to some cases of
interest. After an analysis of the flat space limit, we consider two
cosmologically relevant FLRW metrics, corresponding to a cosmological
constant–dominated and a radiation–dominated universe respectively. In these
cases, exact analytical solutions to the Dirac equation are available, and it
is possible to employ equation (16) directly. We then introduce a method to
extract the oscillation formulae on spacetimes with asymptotically flat
regions without resorting to the exact solutions of the Dirac equation. We
employ this strategy to compute the formulae on the Schwarzschild black hole
spacetime, for neutrinos propagating from the past infinity to the future
infinity. We show how the Hawking radiation is naturally embedded in the
resulting transition probabilities.

###  IV.1 Flat spacetime limit

As a first, trivial, application of the formulae (17), let us check the flat
spacetime limit. We can see at once that the equations (17) reduce to the
ordinary oscillation formulae. Indeed, in this case, we can choose the cauchy
hypersurfaces to be the $t=constant$ surfaces in a given Minkowskian
coordinate system, while the modes
$\\{\zeta_{\boldsymbol{k},s,i}(x),\xi_{\boldsymbol{k},s,i}(x)\\}$ are just the
plane wave solutions to the flat Dirac equation with definite momentum, so
that $\Lambda_{q,r;k,s}(t)\rightarrow U_{q,r;k,s}(t)$ and
$\Xi_{q,r;k,s}(t)\rightarrow V_{q,r;k,s}(t)$, where $U_{q,r;k,s}$ and
$V_{q,r;k,s}$ are the usual mixing coefficients in flat space Capolupo1 .
Assuming, without loss of generality, $k$ along the $z$ direction, the
helicity indices decouple
$U_{q,r;k,s}=\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\delta_{r,s}U_{\boldsymbol{k}}$,
$V_{q,r;k,s}=\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\delta_{r,s}(-1)^{s}V_{\boldsymbol{k}}$.
Since
$U_{\boldsymbol{k}}(t)=U_{\boldsymbol{k}}(0)e^{i(\omega_{k,2}-\omega_{k,1})t}$
and
$V_{\boldsymbol{k}}(t)=V_{\boldsymbol{k}}(0)e^{i(\omega_{k,2}+\omega_{k,1})t}$,
we get

$\displaystyle\sum_{k^{\prime},r^{\prime},q,r}\Re[U^{*}_{k,s;k^{\prime},r^{\prime}}(0)U_{k^{\prime},r^{\prime};q,r}(t)$
$\displaystyle+$ $\displaystyle
V^{*}_{k,s;k^{\prime},r^{\prime}}(0)V_{k^{\prime},r^{\prime};q,r}(t)]\;=\;\Re[|U_{\boldsymbol{k}}(0)|^{2}e^{i(\omega_{k,2}-\omega_{k,1})t}+|V_{\boldsymbol{k}}(0)|^{2}e^{i(\omega_{k,2}+\omega_{k,1})t}]$
(33) $\displaystyle=$
$\displaystyle|U_{\boldsymbol{k}}(0)|^{2}\cos[(\omega_{k,2}-\omega_{k,1})t]+|V_{\boldsymbol{k}}(0)|^{2}\cos[(\omega_{k,2}+\omega_{k,1})t].$

Substituting this result in eqs. (17) yields the flat space oscillation
formulae, which further reduce to the Pontecorvo oscillation formulae in the
quantum mechanical limit ($V_{\boldsymbol{k}}=0$). Flat space also offers the
possibility to illustrate some points discussed above in the simplest possible
context. One might well expand the mass fields in terms of modes with definite
energy and angular momentum $\zeta_{\omega,\kappa_{j},m_{j};i}$,
$\xi_{\omega,\kappa_{j},m_{j},s;i}$ instead of considering modes with definite
cartesian three–momentum $\boldsymbol{k}$. The former shall be suitable
combinations of spherical spinors AngularMom . An interesting aspect, is that
in such a representation, the mixing coefficients are no longer diagonal.
Indeed one has

$\Lambda_{\omega^{\prime}\kappa^{\prime}_{j}m^{\prime}_{j};\omega\kappa_{j}m_{j}}(t)=\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}\delta_{\kappa^{\prime}_{j},\kappa_{j}}\delta_{m^{\prime}_{j},m_{j}}|U_{\omega,\omega^{\prime}}|e^{i(\omega^{\prime}-\omega)t}$
(34)

with $\Delta m^{2}=m_{2}^{2}-m_{1}^{2}$ and

$|U_{\omega,\omega^{\prime}}|=\sqrt{\frac{\omega^{\prime}+m_{2}}{2\omega^{\prime}}}\sqrt{\frac{\omega+m_{1}}{2\omega}}\left(1+\sqrt{\frac{(\omega^{\prime}-m_{2})(\omega-
m_{1})}{(\omega+m_{1})(\omega^{\prime}+m_{2})}}\right)\ ,$ (35)

and similar for $\Xi$, where the exponential is
$e^{i(\omega^{\prime}+\omega)t}$ and $|U_{\omega,\omega^{\prime}}|$ is
replaced by

$|V_{\omega,\omega^{\prime}}|=\sqrt{\frac{\omega^{\prime}+m_{2}}{2\omega^{\prime}}}\sqrt{\frac{\omega+m_{1}}{2\omega}}\left(\sqrt{\frac{\omega^{\prime}-m_{2}}{\omega^{\prime}+m_{2}}}-\sqrt{\frac{\omega-
m_{1}}{\omega+m_{1}}}\right)\ .$ (36)

Here the quantum number $\kappa_{j}$ refers to a relativistic generalization
of the spin–orbit operator, which enters the Dirac equation in spherical
coordinates AngularMom ; Thaller , and takes into account both the orbital and
spin angular momentum. The index $m_{j}$ refers to one component of the total
angular momentum $\boldsymbol{J}$, and has not to be confused with the masses
$m_{i}$. Without delving into the details of the calculation, the result of
equation (34) can be understood as follows. The modes, apart from a
normalization constant, are given by

$\zeta_{\omega,\kappa_{j},m_{j};i}=e^{-i\omega
t}\sqrt{\frac{\omega+m_{i}}{2\omega
r^{2}}}\begin{pmatrix}P_{\kappa_{j}}(\lambda_{i}r)\Omega_{\kappa_{j},m_{j}}(\theta,\phi)\\\
\sqrt{\frac{\omega-
m_{i}}{\omega+m_{i}}}P_{\kappa_{j}}(\lambda_{i}r)\Omega_{-\kappa_{j},m_{j}}(\theta,\phi)\end{pmatrix}$
(37)

$\xi_{\omega,\kappa_{j},m_{j};i}=e^{i\omega
t}\sqrt{\frac{\omega+m_{i}}{2\omega r^{2}}}\begin{pmatrix}-\sqrt{\frac{\omega-
m_{i}}{\omega+m_{i}}}P_{\kappa_{j}}(-\lambda_{i}r)\Omega_{\kappa_{j},m_{j}}(\theta,\phi)\\\
P_{\kappa_{j}}(-\lambda_{i}r)\Omega_{-\kappa_{j},m_{j}}(\theta,\phi)\end{pmatrix}$
(38)

where $\Omega_{\kappa_{j},m_{j}}(\theta,\phi)$ are spherical spinors, and
$P_{\kappa_{j}}(\lambda_{i}r)$ are radial functions of the product
$\lambda_{i}r$, with the radial momentum
$\lambda_{i}=\sqrt{\omega^{2}-m_{i}^{2}}$. These are solutions to the radial
part of the Dirac equation, which turns out to be a Riccati–Bessel
equationAbramowitz . The functions $P_{\kappa_{j}}(\lambda_{i}r)$ are
combinations of spherical bessel functions $j_{n}$ of the form
$P_{\kappa_{j}}(\lambda_{i}r)=rj_{\kappa_{j}}(\lambda_{i}r)$. In computing the
inner products, the radial integration $\int
drP^{*}_{\kappa_{j};2}(\lambda_{2}r)P_{\kappa_{j};1}(\lambda_{1}r)$ will
produce a factor $\delta_{\lambda_{2},\pm\lambda_{1}}$, because of the closure
relation satisfied by the spherical Bessel functions. Since
$\lambda_{2}=\sqrt{\omega^{{}^{\prime}2}-m_{2}^{2}}$ and
$\lambda_{1}=\sqrt{\omega^{2}-m_{1}^{2}}$, this will give rise to the delta
factor appearing in (34). Notice that
$|U_{\omega,\omega^{\prime}}|,|V_{\omega,\omega^{\prime}}|$ are numerically
the same as the usual flat space coefficients
$|U_{\boldsymbol{k}}|,|V_{\boldsymbol{k}}|$, when
$k^{2}=\omega^{2}-m_{1}^{2}=\omega^{{}^{\prime}2}-m_{2}^{2}$ .This shows why
the mixing coefficients $\Lambda$, $\Xi$ are not generally diagonal, and the
flexibility of the formalism we have employed. Indeed, the non–diagonal
coefficients automatically ensure that the flavor operators
$\gamma_{\omega,\rho}$, $\epsilon_{\omega,\rho}$ take into account the mass
difference, involving operators with distinct energies
$\omega,\omega^{\prime}$ for the fields $\psi_{1}$ and $\psi_{2}$ Note5 . In
flat space, the shift between the two representations is actually of no use.
However, in a non–trivial framework, the versatility of the formalism is
essential, as there are instances in which the cartesian components of the
momentum $k_{x},k_{y},k_{z}$ are useless, while the ”spherical” quantum
numbers $\omega,l,m$ are well defined.

###  IV.2 Expanding universe with exponential growth of the scale factor

The simplest non–trivial application is to spatially flat
Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetimes. Consider the metric
$ds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2})$ with an exponential expansion
$a(t)=e^{Ht}$, $H=constant$. This is well–suited to describe a homogeneous,
spatially flat and isotropic universe dominated by a cosmological constant.
The normalized solutions to the Dirac equation for this metric were derived in
Barut . Assuming, without loss of generality, the momentum $k$ to be along the
$z$ direction, the helicities decouple as in flat space. Choosing the Cauchy
surfaces as the surfaces with $t=constant$, or equivalently with
$a(t)=constant$, the mixing coefficients read

$\displaystyle\Lambda_{k,s;q,r}$
$\displaystyle(t)=\delta_{s,r}\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\frac{\pi
ke^{-Ht}}{2H\sqrt{\cos(\frac{i\pi m_{2}}{H})\cos(\frac{i\pi
m_{1}}{H})}}\times\left[\\!J_{v_{1}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!+\\!J_{v_{1}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{v_{2}-1}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\right]$
(39) $\displaystyle\Xi_{k,s;q,r}$
$\displaystyle(t)=\delta_{s,r}(-1)^{s}\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\frac{\pi
ke^{-Ht}}{2H\sqrt{\cos(\frac{i\pi m_{2}}{H})\cos(\frac{i\pi
m_{1}}{H})}}\left[\\!J_{v_{1}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{-v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!+\\!J_{v_{1}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{1-v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\right]$
(40)

where $J_{\alpha}$ denotes the $\alpha$ Bessel function and
$v_{j}=\frac{1}{2}\left(1+\frac{2im_{j}}{H}\right)$ for $j=1,2$. Plugging
these expressions in equations (17), we obtain

$\displaystyle
P^{e\rightarrow\mu}_{k,s}(t)=2\cos^{2}(\theta)\sin^{2}(\theta)\bigg{\\{}1-\frac{\pi^{2}k^{2}e^{-H(t+t_{0})}}{4H^{2}\cos(\frac{i\pi
m_{2}}{H})\cos(\frac{i\pi m_{1}}{H})}$ (41)
$\displaystyle\times\Re\bigg{[}\left[\\!J_{v_{1}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\\!J_{v_{2}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!+\\!J_{v_{1}-1}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\\!J_{v_{2}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\right]\left[\\!J_{v_{1}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!+\\!J_{v_{1}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{v_{2}-1}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\right]$
$\displaystyle+\left[\\!J_{v_{1}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\\!J_{-v_{2}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!+\\!J_{v_{1}-1}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\\!J_{1-v_{2}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\right]\left[\\!J_{v_{1}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{-v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!+\\!J_{v_{1}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{1-v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\right]\bigg{]}\bigg{\\}}$

###  IV.3 Expanding universe dominated by radiation

Here we consider the FLRW metric for a radiation dominated universe
$a(t)=a_{0}t^{\frac{1}{2}}$. Notice that since $a(t)$ has to be adimensional,
$a_{0}$ has dimension $[t]^{-\frac{1}{2}}=[m]^{\frac{1}{2}}$. As before,
without loss of generality, we assume the neutrino momentum $k$ along the $z$
direction to decouple the helicities, and consider a foliation by the
$t=constant$ hypersurfaces. The solutions to the Dirac equation for this
metric are again found in Barut , and yield the mixing coefficients

$\displaystyle\Lambda_{k,s;q,r}(t)=\delta_{s,r}\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\frac{1}{\sqrt[4]{4m_{1}m_{2}t^{2}}}e^{-\frac{\pi
k^{2}(m_{1}+m_{2})}{4m_{1}m_{2}a_{0}^{2}}}\bigg{\\{}W^{*}_{\kappa_{2},\frac{1}{4}}(-2im_{2}t)W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)$
(42)
$\displaystyle+\frac{4}{m_{1}m_{2}a_{0}^{2}t}\bigg{[}W^{*}_{\kappa_{2},\frac{1}{4}}(-2im_{2}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{2}a_{0}^{2}}\right)W^{*}_{\kappa_{2}-1,\frac{1}{4}}(-2im_{2}t)\bigg{]}\bigg{[}W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t)\bigg{]}\bigg{\\}}$
$\displaystyle\Xi_{k,s;q,r}(t)=\delta_{s,r}\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\frac{k}{\sqrt[4]{2m_{1}(2m_{2})^{3}a_{0}^{2}t}}e^{-\frac{\pi
k^{2}(m_{1}+m_{2})}{4m_{1}m_{2}a_{0}^{2}}}\bigg{\\{}W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)W_{-\kappa_{2},\frac{1}{4}}(-2im_{2}t)$
(43)
$\displaystyle+\frac{1}{m_{1}m_{2}a_{0}^{2}t}\bigg{[}W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W^{*}_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t)\bigg{]}\bigg{[}W_{-\kappa_{2},\frac{1}{4}}(2im_{2}t)+\frac{2im_{2}a_{0}^{2}}{k^{2}}W_{-\kappa_{2}+1,\frac{1}{4}}(2im_{2}t)\bigg{]}\bigg{\\}}$

where $W_{\kappa,\mu}(z)$ are the Whittaker functions Abramowitz  and
$\kappa_{j}=\frac{1}{4}\left(1+\frac{2ik^{2}}{a_{0}^{2}m_{j}}\right)$ for
$j=1,2$. Insertion in eqs. (17) gives the transition probabilities
($t_{0},t>0$)

$\displaystyle
P^{e\rightarrow\mu}_{k,s}(t)=2\cos^{2}(\theta)\sin^{2}(\theta)\bigg{\\{}1+\Re\bigg{[}\frac{1}{\sqrt[2]{4m_{1}m_{2}t_{0}t}}e^{-\frac{\pi
k^{2}(m_{1}+m_{2})}{2m_{1}m_{2}a_{0}^{2}}}\bigg{\\{}W_{\kappa_{2},\frac{1}{4}}(-2im_{2}t_{0})W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t_{0})$
$\displaystyle+\frac{4}{m_{1}m_{2}a_{0}^{2}t_{0}}\left(W_{\kappa_{2},\frac{1}{4}}(-2im_{2}t_{0})-\frac{1}{8}\left(1+\frac{ik^{2}}{m_{2}a_{0}^{2}}\right)W_{\kappa_{2}-1,\frac{1}{4}}(-2im_{2}t_{0})\right)\left(W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t_{0})-\frac{1}{8}\left(1+\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W^{*}_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t_{0})\right)\bigg{\\}}$
$\displaystyle\times\bigg{\\{}W^{*}_{\kappa_{2},\frac{1}{4}}(-2im_{2}t)W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)$
$\displaystyle+\frac{4}{m_{1}m_{2}a_{0}^{2}t}\left(W^{*}_{\kappa_{2},\frac{1}{4}}(-2im_{2}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{2}a_{0}^{2}}\right)W^{*}_{\kappa_{2}-1,\frac{1}{4}}(-2im_{2}t)\right)\left(W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t)\right)\bigg{\\}}$
$\displaystyle+\frac{k^{2}}{\sqrt[2]{2m_{1}(2m_{2})^{3}a_{0}^{4}t_{0}t}}e^{-\frac{\pi
k^{2}(m_{1}+m_{2})}{2m_{1}m_{2}a_{0}^{2}}}\bigg{\\{}W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t_{0})W^{*}_{-\kappa_{2},\frac{1}{4}}(-2im_{2}t_{0})$
$\displaystyle+\frac{1}{m_{1}m_{2}a_{0}^{2}t_{0}}\left(W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t_{0})-\frac{1}{8}\left(1+\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t_{0})\right)\left(W^{*}_{-\kappa_{2},\frac{1}{4}}(2im_{2}t_{0})-\frac{2im_{2}a_{0}^{2}}{k^{2}}W^{*}_{-\kappa_{2}+1,\frac{1}{4}}(2im_{2}t_{0})\right)\bigg{\\}}$
$\displaystyle\times\bigg{\\{}W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)W_{-\kappa_{2},\frac{1}{4}}(-2im_{2}t)$
$\displaystyle+\frac{1}{m_{1}m_{2}a_{0}^{2}t}\left(W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W^{*}_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t)\right)\left(W_{-\kappa_{2},\frac{1}{4}}(2im_{2}t)+\frac{2im_{2}a_{0}^{2}}{k^{2}}W_{-\kappa_{2}+1,\frac{1}{4}}(2im_{2}t)\right)\bigg{\\}}\bigg{]}\bigg{\\}}$
(44)

###  IV.4 Spacetimes with asymptotically flat regions

The FLRW spacetimes considered above are among the few non–trivial metrics for
which the Dirac equation (1) can be solved analytically. More often one does
not have an exact solution at his disposal, and the implementation of eqs.
(17) is a complicated task. In many cases of interest, however, the spacetime
$M$ admits asymptotically flat regions $\Omega_{A},\Omega_{B}\subset M$,
usually in the far past and in the far future. $\Omega_{A}$ and $\Omega_{B}$
are separated by a region with non-trivial curvature, where the Dirac equation
is usually unsolvable analytically. In $\Omega_{A}$ and $\Omega_{B}$ the
solutions to the Dirac equation are the flat space modes
$\\{u^{I}_{k,s,i}(x),v^{I}_{k,s,i}\\}$ with $I=A,B$, and one has a natural
choice for the positive frequency modes. Because of the non-trivial curvature,
the two sets of solutions $A,B$ are distinct. When limited to one of the two
regions, eqs. (17) reduce to the ordinary flat space formulae. When the
intermediate curvature region is involved, a direct application of eqs. (17)
is prohibitive. Nevertheless, if one is able to provide the relation between
the two sets $A,B$, in the form of a Bogoliubov transformation, it is possible
to derive oscillation formulae for the propagation from $\Omega_{A}$ to
$\Omega_{B}$. Assume that $\Omega_{A}=\bigcup_{\tau\leq\tau_{A}}\Sigma_{\tau}$
and $\Omega_{B}=\bigcup_{\tau\geq\tau_{B}}\Sigma_{\tau}$ for some values
$\tau_{B}>\tau_{A}$, and consider $\tau_{0}\leq\tau_{A}$ , $\tau\geq\tau_{B}$.
Suppose also that the $B$ modes are given in terms of the $A$ modes as

$\displaystyle u^{B}_{k^{\prime},s^{\prime},i}$ $\displaystyle=$
$\displaystyle\sum_{k,s}\left(\Gamma_{k^{\prime},s^{\prime};k,s;i}^{*}u^{A}_{k,s,i}+\Sigma_{k^{\prime},s^{\prime};k,s;i}^{*}v^{A}_{k,s,i}\right)$
(45) $\displaystyle v^{B}_{k^{\prime}s^{\prime},i}$ $\displaystyle=$
$\displaystyle\sum_{k,s}\left(\Gamma_{k^{\prime},s^{\prime};k,s;i}v^{A}_{k,s,i}-\Sigma_{k^{\prime},s^{\prime};k,s;i}u^{A}_{k,s,i}\right)\
.$ (46)

Here the Bogoliubov coefficients
$\Gamma_{k^{\prime},s^{\prime};k,s;i},\Sigma_{k^{\prime},s^{\prime};k,s;i}$
are again provided by the inner products
$(u^{A}_{i},u^{B}_{i}),(u^{A}_{i},v^{B}_{i})$, yet their significance is
slightly different from those in (II). While equation (II) describes a general
and arbitrary change of basis, the transformation of equation (45) is dictated
by the circumstance of having a natural choice for the modes in $\Omega_{A}$
and $\Omega_{B}$, with a well–defined physical meaning. Indeed, the Bogoliubov
coefficients take into account the effect of the curvature in the intermediate
region. Then we can specialize equations (III.2), (III.2) to get

$\displaystyle\Lambda^{B}_{q,r;k,s}(\tau)=\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A}(\tau)-\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A}(\tau)\right)$
$\displaystyle+\Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A*}(\tau)+\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A*}(\tau)\right)\bigg{]}\
.$ (47)
$\displaystyle\Xi^{B}_{q,r;k,s}(\tau)=\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}\Xi^{A}_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)+\Sigma_{k,s;k^{\prime},s^{\prime};1}\Lambda^{A}_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)\right)$
(48)
$\displaystyle-\Sigma_{q,r;q^{\prime},r^{\prime};2}\times\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A*}(\tau)-\Sigma_{k,s;k^{\prime},s^{\prime};1}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A*}(\tau)\right)\bigg{]}\
.$

For $\tau<\tau_{A}$ $\Lambda^{A}(\tau),\Xi^{A}(\tau)$ are trivial, and vice-
versa, for $\tau>\tau_{B}$, $\Lambda^{B}(\tau),\Xi^{B}(\tau)$ are trivial.
Choosing, for instance, the $B$ representation, one would have a trivial
expression for $\Lambda^{B}(\tau),\Xi^{B}(\tau)$ and one can make use of eqs.
(IV.4), (48) to obtain $\Lambda^{B}(\tau_{0}),\Xi^{B}(\tau_{0})$ in terms of
$\Lambda^{A}(\tau_{0}),\Xi^{A}(\tau_{0})$, which are also trivial. Then one
can plug $\Lambda^{B}(\tau,\tau_{0})$ and $\Xi^{B}(\tau,\tau_{0})$ in equation
(17) to obtain $P^{e\rightarrow\mu}_{k,s\rightarrow q,r}(\tau)$ for
$\tau\geq\tau_{B}$ and the reference hypersurface $\tau_{0}\leq\tau_{A}$.

####  IV.4.1 Scwharzschild black hole

As a realization of this scheme, consider the (static) Schwarzschild metric

$\displaystyle
ds^{2}=\left(1-\frac{2GM}{r}\right)dt^{2}-\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}-r^{2}d\Omega,$
(49)

and two sequences of spacelike hypersurfaces Note6
$\\{\Sigma_{n}^{+}\\}_{n\in\mathbb{N}}$,
$\\{\Sigma_{n}^{-}\\}_{n\in\mathbb{N}}$ approaching, respectively, the future
$\mathcal{I}^{+}$ and the past null infinity $\mathcal{I}^{-}$ as
$n\rightarrow\infty$. We require that $\Sigma_{n}^{+}$ and $\Sigma_{n}^{-}$ be
Cauchy surfaces respectively for the causal past $J^{-}(\mathcal{I}^{+})$ of
$\mathcal{I}^{+}$ and the causal future $J^{+}(\mathcal{I}^{-})$of
$\mathcal{I}^{-}$ for each $n$. For $n$ large enough, these surfaces span an
approximately flat portion of the Schwarzschild spacetime. On the surfaces
$\Sigma_{n}^{-}$, as $n$ approaches infinity, we expand the massive fields in
terms of the incoming solutions, with frequency defined with respect to the
schwarzschild time $t$ and with definite angular momentum,

$\displaystyle\zeta^{IN}_{\omega,\kappa_{j},m_{j};i}(t,r,\theta,\phi)\propto
e^{-i\omega
t}\,,\qquad\qquad\xi^{IN}_{\omega,\kappa_{j},m_{j};i}(t,r,\theta,\phi)\propto
e^{i\omega t}.$ (50)

These modes reduce to the flat space solutions (37) and (38) as
$\mathcal{I}^{-}$ is approached. Omitting the irrelevant angular and spin
quantum numbers, as $n\rightarrow\infty$, we get

$\displaystyle\Lambda^{IN}_{\omega;\omega^{\prime}}(\Sigma^{-}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}|U_{\omega,\omega^{\prime}}|e^{i\varphi^{-}(\omega,n)}$ (51)
$\displaystyle\Xi^{IN}_{\omega;\omega^{\prime}}(\Sigma^{-}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}|U_{\omega,\omega^{\prime}}|e^{i\rho^{-}(\omega,n)}$ (52)

with $|U_{\omega,\omega^{\prime}}|$ and $|V_{\omega,\omega^{\prime}}|$ flat
space spherical mixing coefficients as defined in (35) and (36), and
$\varphi^{-}(\omega,n),\rho^{-}(\omega,n)$ phase factors depending on $\omega$
and $n$. A similar reasoning can be carried on for the _outgoing_ modes
emerging at $\mathcal{I}^{+}$,
$\zeta^{OUT}_{\omega,\kappa_{j},m_{j};i}(t,r,\theta,\phi)$ ,
$\xi^{OUT}_{\omega,\kappa_{j},m_{j};i}(t,r,\theta,\phi)$, so to yield as
$n\rightarrow\infty$

$\displaystyle\Lambda^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}|U_{\omega,\omega^{\prime}}|e^{i\varphi^{+}(\omega,n)}$ (53)
$\displaystyle\Xi^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}|V_{\omega,\omega^{\prime}}|e^{i\rho^{+}(\omega,n)}\,.$ (54)

Because of the Black Hole, the $IN$ and $OUT$ modes do not coincide. Fermion
creation by the Schwarzschild black hole has been studied, via the tunneling
method, in Kerner . There it has been shown that the Hawking temperature
$T_{H}=\frac{1}{8\pi GM}$ is recovered for the emission of spin–$\frac{1}{2}$
particles. We then infer that the $IN$ and $OUT$ modes are related by a
thermal Bogoliubov transformation at the Hawking temperature $T_{H}$,
corrected for fermions:

$\displaystyle\zeta^{OUT}_{\omega,\kappa_{j},m_{j};i}=\sqrt{\frac{e^{\frac{\omega}{k_{B}T_{H}}}}{e^{\frac{\omega}{k_{B}T_{H}}}+1}}\zeta^{IN}_{\omega,\kappa_{j},m_{j};i}+\sqrt{\frac{1}{e^{\frac{\omega}{k_{B}T_{H}}}+1}}\xi^{IN}_{\omega,\kappa_{j},m_{j};i}$
(55)
$\displaystyle\xi^{OUT}_{\omega,\kappa_{j},m_{j};i}=\sqrt{\frac{e^{\frac{\omega}{k_{B}T_{H}}}}{e^{\frac{\omega}{k_{B}T_{H}}}+1}}\xi^{IN}_{\omega,\kappa_{j},m_{j};i}-\sqrt{\frac{1}{e^{\frac{\omega}{k_{B}T_{H}}}+1}}\zeta^{IN}_{\omega,\kappa_{j},m_{j};i}\
,$ (56)

It is understood that these equations hold as long as the spacetime is
stationary (eternal black hole). For a body that collapses to a Schwarzschild
black hole, as considered in the original paper by Hawking Hawking , we expect
a slightly modified version of the Bogoliubov transformations, comprising
non–diagonal thermal coefficients $\Gamma_{\omega,\omega^{\prime};i}$ and
$\Sigma_{\omega,\omega^{\prime};i}$ with $\omega\neq\omega^{\prime}$. We pick
the ingoing representation to calculate the probabilities (of course, the
outgoing representation yields the same result, as the Bogoliubov
transformations are diagonal), and employ eqs. (IV.4, 48) to obtain, with
$F_{H}(\omega)=\frac{1}{e^{\frac{\omega}{k_{B}T_{H}}}+1}$,

$\displaystyle\Lambda^{IN}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})$
$\displaystyle=$
$\displaystyle\bigg{\\{}\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}\Lambda^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})-\
\
\sqrt{F_{H}(\omega)\left[1-F_{H}(\omega^{\prime})\right]}\Xi^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})$
(57) $\displaystyle+$
$\displaystyle\sqrt{F_{H}(\omega^{\prime})\left[1-F_{H}(\omega)\right]}\left(\Xi^{OUT}_{\omega;\omega^{\prime}}\right)^{*}(\Sigma^{+}_{n})+\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}\left(\Lambda^{OUT}_{\omega;\omega^{\prime}}\right)^{*}(\Sigma^{+}_{n})\bigg{\\}}\,,$

Considered that
$\Lambda^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}|U_{\omega,\omega^{\prime}}|e^{i\varphi^{+}(\omega,n)}$ and
$\Xi^{+}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}|V_{\omega,\omega^{\prime}}|e^{i\rho^{+}(\omega,n)}$ as
$n\rightarrow\infty$, we obtain

$\displaystyle\Lambda^{IN}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})$
$\displaystyle=$
$\displaystyle\bigg{\\{}\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}|U_{\omega;\omega^{\prime}}|e^{i\varphi^{+}(\omega,n)}-\sqrt{F_{H}(\omega)\left[1-F_{H}(\omega^{\prime})\right]}|V_{\omega;\omega^{\prime}}|e^{i\rho^{+}(\omega,n)}$
(58) $\displaystyle+$
$\displaystyle\sqrt{F_{H}(\omega^{\prime})\left[1-F_{H}(\omega)\right]}|V_{\omega;\omega^{\prime}}|e^{-i\rho^{+}(\omega,n)}+\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}|U_{\omega,\omega^{\prime}}|e^{-i\varphi^{+}(\omega,n)}\bigg{\\}}\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}$

and

$\displaystyle\Xi^{IN}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})$
$\displaystyle=$
$\displaystyle\bigg{\\{}\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}|V_{\omega,\omega^{\prime}}|e^{i\rho^{+}(\omega,n)}+\sqrt{F_{H}(\omega)\left[1-F_{H}(\omega^{\prime})\right]}|U_{\omega,\omega^{\prime}}|e^{i\varphi^{+}(\omega,n)}$
(59)
$\displaystyle-\sqrt{F_{H}(\omega^{\prime})\left[1-F_{H}(\omega)\right]}|U_{\omega,\omega^{\prime}}|e^{-i\varphi^{+}(\omega,n)}+\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}|V_{\omega,\omega^{\prime}}|e^{-i\rho^{+}(\omega,n)}\bigg{\\}}\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta
m^{2}}}\ .$

Choosing as reference hypersurface $\Sigma^{-}_{m}$ for large $m$, we can now
compute the probabilities (17) for a neutrino propagating from
$\Sigma^{-}_{m}$ to $\Sigma^{+}_{n}$, i.e. from $\mathcal{I}^{-}$ to
$\mathcal{I}^{+}$ in the limit $m,n\rightarrow\infty$. We find, for
$m,n\rightarrow\infty$

$\displaystyle P^{e\rightarrow\mu}_{\omega}(m,n)$ $\displaystyle\approx$
$\displaystyle 2\cos^{2}\theta\sin^{2}\theta\
\bigg{(}1-\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}\left[|U_{\omega;\omega^{\prime}}|^{2}\cos(\Delta^{-}_{\omega;m,n})+|V_{\omega;\omega^{\prime}}|^{2}\cos(\Phi^{-}_{\omega;m,n})\right]$
(60) $\displaystyle+$
$\displaystyle\sqrt{F_{H}(\omega)\left[1-F_{H}(\omega^{\prime})\right]}|U_{\omega;\omega^{\prime}}||V_{\omega;\omega^{\prime}}|\left[\cos(\Theta^{-}_{\omega;m,n})-\cos(\Psi^{-}_{\omega;m,n})\right]$
$\displaystyle+$
$\displaystyle\sqrt{F_{H}(\omega^{\prime})\left[1-F_{H}(\omega)\right]}|U_{\omega;\omega^{\prime}}||V_{\omega;\omega^{\prime}}|\left[\cos(\Psi^{+}_{\omega;m,n})-\cos(\Theta^{+}_{\omega;m,n})\right]$
$\displaystyle-$
$\displaystyle\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}\left[|U_{\omega;\omega^{\prime}}|^{2}\cos(\Delta^{+}_{\omega;m,n})+|V_{\omega;\omega^{\prime}}|^{2}\cos(\Phi^{+}_{\omega;m,n})\right]\bigg{)}\
.$

with $\Delta^{\pm}_{\omega;m,n}=\varphi^{+}(\omega,n)\pm\varphi^{-}(\omega,m)$
, $\Phi^{\pm}_{\omega;m,n}=\rho^{+}(\omega,n)\pm\rho^{-}(\omega,m)$,
$\Psi^{\pm}_{\omega;m,n}=\rho^{+}(\omega,n)\pm\varphi^{-}(\omega,m)$ and
$\Theta^{\pm}_{\omega;m,n}=\varphi^{+}(\omega,n)\pm\rho^{-}(\omega,m)$ . In
particular, for large energies of the mass fields $\omega,\omega^{\prime}$,
$|V_{\omega,\omega^{\prime}}|\rightarrow 0$ and
$|U_{\omega,\omega^{\prime}}|\rightarrow 1$, thus

$\displaystyle P^{e\rightarrow\mu}_{\omega}(m,n)\approx
2\cos^{2}\theta\sin^{2}\theta\
\bigg{(}1-\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}\cos(\Delta^{-}_{\omega,\kappa_{j};m,n})-\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}\cos(\Delta^{+}_{\omega,\kappa_{j};m,n})\bigg{)}\
,$ (61)

where it is understood that $\omega^{{}^{\prime}2}=\omega^{2}+\Delta m^{2}$.
If the limit $T_{H}\rightarrow 0$ is taken in equation (61), one obtains,
apart from a phase factor, the usual Pontecorvo oscillation formulae. To
compute the oscillation formulae on a Schwarzschild background for an
arbitrary propagation, since exact analytical solutions are unavailable, one
has to resort to approximate solutions to the Dirac equation. In all the
applications considered, the oscillation formulae do not depend on the
helicity $s$ of neutrinos. However, when additional complications due to
frame–dragging and non–conservation of angular momentum arise (see e.g.
Lambiase1 ; Lambiase2 ), the formulae for left-handed and right-handed
neutrinos can differ.

###  IV.5 Quantum mechanical limit

It is a general feature of equations (17) that when all the quantum field
theoretical effects are negligible, the oscillation formulae are modified only
for a phase factor. Indeed, when one can neglect $\Xi_{k,s}$ in equation (17),
one immediately has $|\Lambda^{*}_{k,s}(\tau)|=1$,
($|\Lambda_{\omega,\omega^{\prime}}(\tau)|=1$ respectively, in the
non–diagonal case) for each $\tau$ from equation (III.1). Then the product
$\Lambda_{k,s}(\tau_{0})^{*}\Lambda_{k,s}(\tau)$,
($\Lambda_{\omega,\omega^{\prime}}(\tau_{0})^{*}\Lambda_{\omega,\omega^{\prime}}(\tau)$
respectively) is just a phase $e^{i\varphi(\tau_{0},\tau)}$ and the net effect
is a phase shift with respect to the Pontecorvo oscillation formulae,
consistently with previous results obtained in a heuristic treatmentCardall .
Of course, the explicit value of the phase $\varphi(\tau_{0},\tau)$ depends on
the metric and the surfaces $\tau$ considered, as well as on the mode
expansion chosen for the mass fields. When the gravitational fields are weak
enough, the phase can be computed by means of geometrical optics
considerations Cardall ; Lambiase1 .

##  V Conclusions

We have developed a quantum field theoretical approach to the vacuum neutrino
oscillations in curved space, discussing the transition probabilities, and
their behaviour under changes of mass representation. We have analyzed the
non–trivial interplay between quantum field mixing and field quantization in
curved space, and have found that the former has a remarkably richer structure
when compared to its flat space counterpart. In particular, the formalism has
to be versatile enough to deal with the existence of infinite unitarily
inequivalent representations of the mass fields, which is to say that no
preferred notion of particle does generally exist. In the spirit of general
covariance, we have determined the effect on flavor fields of a shift in the
expansion of the mass fields, and established under which conditions the
resulting transition probabilities are the same. We have then computed the
oscillation formulae in three example metrics, including two FRLW spacetimes
and the static Schwarzschild black hole. In the latter we have found that the
Hawking radiation affects, although very slightly, the oscillations for
neutrinos propagating from the asymptotic past infinity to the asymptotic
future infinity. As a general result, it is found that when all the quantum
field theoretical effects on neutrino mixing can be neglected, the
gravitational background only affects the phase of the oscillations,
constistently with previous analyses.

## Acknowledgements

Partial financial support from MIUR and INFN is acknowledged. A.C. and G.L.
also acknowledge the COST Action CA1511 Cosmology and Astrophysics Network for
Theoretical Advances and Training Actions (CANTATA).

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