# Time-energy uncertainty relation for neutrino oscillations: historical
development, applications and future prospects

G. G. Luciano<EMAIL_ADDRESS>Department of Chemistry,
Physics and Environmental and Soil Sciences, Escola Politècnica Superior,
Universitat de Lleida, Av. Jaume II, 69, 25001 Lleida, Spain  L. Smaldone
<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di Salerno,
Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy.  INFN Gruppo
Collegato di Salerno - Sezione di Napoli c/o Dipartimento di Fisica,
Università di Salerno, Italy.

###### Abstract

Time-energy uncertainty relation (TEUR) plays a fundamental role in quantum
mechanics, as it allows to grasp peculiar aspects of a variety of phenomena
based on very general principles and symmetries of the theory. Using the
Mandelstam-Tamm method, TEUR has been recently derived for neutrino
oscillations by connecting the uncertainty on neutrino energy with the
characteristic time-scale of oscillations. Interestingly enough, the
suggestive interpretation of neutrinos as unstable-like particles has proved
to naturally emerge in this context. Further aspects have been later discussed
in semiclassical gravity by computing corrections to the neutrino energy
uncertainty in a generic stationary curved spacetime, and in quantum field
theory, where the clock observable turns out to be identified with the non-
conserved flavor charge operator. In the present work, we give an overview on
the above achievements. In particular, we analyze the implications of TEUR and
explore the impact of gravitational and non-relativistic effects on the
standard condition for neutrino oscillations. Correlations with the quantum-
information theoretic analysis of oscillations and possible experimental
consequences are qualitatively discussed.

Time-energy uncertainty relation, quantum theory, general relativity,
neutrino, flavor oscillations, unstable particles

##  I Introduction

Uncertainty relations constitute a cornerstone of quantum theory. On one hand,
the Heisenberg principle sets a fundamental lower limit on how precisely we
can perform simultaneous measurements of position and momentum of any quantum
system. A direct consequence is that an exactly specified (classical-like)
phase-space trajectory cannot be defined for these systems. Another such pair
of incompatible variables are energy and time. An informal meaning of the
time-energy uncertainty relation (TEUR) is that quantum states that only exist
for a relatively short time cannot have a sharply determined energy.

Despite formal similarities, the analogy between TEUR and the Heisenberg
principle cannot be pursued all the way because of the lack of a time operator
in quantum mechanics (QM). As a matter of fact, the proper formalization and
general validity of TEUR have been controversial issues since the advent of
quantum theory. For a comprehensive review on the subject, one can refer to
[1], while a recent discussion on the non-uniqueness of TEUR formulation
appears in [2]. In what follows, we shall basically resort to the most general
Mandelstam–Tamm version of TEUR, which is based on the only assumption that
the evolution of Heisenberg operators in quantum theory is ruled by the
Hamiltonian $H$ [3]. Using natural units $\hbar=c=G=1$, TEUR in this framework
takes the form

$\Delta E\,\Delta t\,\geq\frac{1}{2}\,,$ (1)

where $\Delta E\equiv\sigma_{H}$ is interpreted as the standard deviation of
the energy on a generic quantum state $|\psi\rangle$ and

$\Delta t\equiv\sigma_{O}/\left|\frac{\mathrm{d}\langle
O(t)\rangle}{\mathrm{d}t}\right|\,.$ (2)

Here $O(t)$ denotes the “clock observable”, whose dynamics quantifies temporal
changes of the system and $\Delta t$ is the characteristic time-scale over
which the mean value $\langle
O(t)\rangle\equiv\langle\psi|O(t)|\psi\rangle=\langle\psi(t)|O|\psi(t)\rangle$
varies by a standard deviation.

One of the most common applications of TEUR concerns the understanding of the
decay of excited states of atoms. In this scenario, the minimum time it takes
for an atom to decay to its ground state and radiate is related to the energy
uncertainty of the excited state [4]. Based on the Mandelstam-Tamm method, in
[5] TEUR was derived for neutrino flavor oscillations in the QM picture.
Neutrino oscillations is a phenomenon characterized by a finite time scale -
the oscillation time [6] \- and, thus, particularly suited for a description
in terms of Eq. (1). By identifying $\Delta t$ with the time during which a
significant change of the flavor content happens for a given neutrino state,
the model of [5] came to the conclusion that a non-vanishing neutrino energy
uncertainty $\Delta E$ is actually needed for oscillations to occur.

Interesting advances in the study of TEUR for neutrino oscillations were later
achieved from both the phenomenological and theoretical points of view. In the
former context, the applicability of TEUR to the Mössbauer neutrino experiment
was analyzed with conflicting results [7, 8]. On the theoretical side, a
generalized TEUR was obtained in [9] for a generic stationary curved
spacetime, while in [10] Eq. (1) was recast in terms of a flavor-energy
uncertainty relation by using the quantum field theoretical (QFT) formalism of
[11, 12]. Remarkably, the QFT approach of [10] brings out the interpretation
of TEUR for flavor neutrinos as fundamental bounds on energy-variances, in
close analogy with the case of unstable particles. All of the above
considerations make it clear that a careful study of TEUR might reveal
valuable information about the true nature of neutrinos and the related
mechanism of flavor oscillations, which are still subject of active
investigation.

Based on the outlined premises, here we review recent progress in the
description of TEUR for neutrino oscillations. After a brief historical
overview, we summarize basic concepts and results in literature, with focus on
the future prospects and research lines that can be inspired by the present
analysis. In detail, the structure of the work is as follows: in Section II,
we go through the derivation of the original QM TEUR for both neutrinos and
unstable particles. Section III is devoted to explore gravitational effects on
TEUR, while the extension to QFT is investigated in Section IV. Conclusions
and discussion are finally summed up in Section V.

##  II TEUR for oscillating and unstable particles

Let us start by presenting TEUR for neutrino oscillations as first derived by
Bilenky [5]. The result is compared to the corresponding formula for unstable
particles [13, 14], emphasizing conceptual similarities and differences.

###  II.1 TEUR for neutrino oscillations (à-la Bilenky)

Following [5], we set $O=P_{l}=|\nu_{l}\rangle\langle\nu_{l}|$ in the
Mandelstam-Tamm inequality (1), where $|\nu_{l}\rangle$ ($l=e,\mu,\tau$) is
the flavor neutrino state
($\langle\nu_{l}|\nu_{l^{\prime}}\rangle=\delta_{ll^{\prime}}$) and $P_{l}$
the projection operator on this state. It is straightforward to check that the
average value of $P_{l}$ on a generic state $|\psi(t)\rangle$ is nothing but
the probability to find the flavor neutrino $\nu_{l}$ in $|\psi(t)\rangle$,
i.e. $\langle P_{l}(t)\rangle=|\langle\nu_{l}|\psi(t)\rangle|^{2}$. Hence,
assuming $|\psi(0)\rangle=|\nu_{l}\rangle$, one has that $\langle
P_{l}(t)\rangle$ coincides with the survival probability
$P_{\nu_{l}\rightarrow\nu_{l}}(t)$ of the neutrino $\nu_{l}$ at time $t$.
Clearly, the following relations hold true:

$P_{\nu_{l}\rightarrow\nu_{l}}(0)=1\,,\quad\,\,P_{\nu_{l}\rightarrow\nu_{l}}(t)\leq
1\,\,\,\mathrm{for\,\,any}\,\,t>0\,.$ (3)

Projection operators are idempotent, which means they satisfy $P^{2}=P$. From
this condition, it follows that the standard deviation $\Delta P_{l}$ obeys

$\Delta
P_{l}(t)=\sqrt{P_{\nu_{l}\rightarrow\nu_{l}}(t)-P^{2}_{\nu_{l}\rightarrow\nu_{l}}(t)}\,.$
(4)

By substitution into Eq. (1), we derive

$\Delta E\geq\frac{1}{2}\hskip
0.56905pt\frac{|\frac{d}{dt}P_{\nu_{l}\rightarrow\nu_{l}}(t)|}{\sqrt{P_{\nu_{l}\rightarrow\nu_{l}}(t)-P^{2}_{\nu_{l}\rightarrow\nu_{l}}(t)}}\,.$
(5)

Let us now consider the survival probability
$P_{\nu_{l}\rightarrow\nu_{l}}(t)$ in the time interval $0\leq t\leq
t_{1min}$, where $t_{1min}$ is the time when
$P_{\nu_{l}\rightarrow\nu_{l}}(t)$ reaches the first minimum. In this range,
$P_{\nu_{l}\rightarrow\nu_{l}}(t)$ is a monotonically decreasing function [5,
7, 8, 15]. By integration of Eq. (5) from $0$ to $t$, we finally get

$\Delta
E\,t\geq\frac{1}{2}\left[\frac{\pi}{2}-\sin^{-1}\left(2P_{\nu_{l}\rightarrow\nu_{l}}(t)-1\right)\right].$
(6)

It is useful to apply the above inequality to some specific cases of
experimental interest. For instance, atmospheric neutrinos are produced by
cosmic rays interacting in the upper atmosphere of the Earth. In turn, the
resulting pions and kaons decay into muon-neutrinos and muons, many of which
give rise to electrons and a muon-neutrino electron-neutrino pair. A crude
estimation indicates that muon-neutrinos are roughly two times more abundant
than electron-neutrinos [16, 17]. In the atmospheric Long Baseline region, the
survival probability $P_{\nu_{\mu}\rightarrow\nu_{\mu}}(t)$ is driven by the
neutrino mass-squared difference $\Delta m_{23}^{2}$ (for more details on the
theoretical treatment of neutrino mixing, see Sec. III and Sec. IV). By taking
into account that $P_{\nu_{\mu}\rightarrow\nu_{\mu}}(t^{(23)}_{1min})\simeq
0$, where $t^{(23)}_{1min}=2\pi E/\Delta m_{23}^{2}$, we obtain the following
TEUR for ultrarelativistic neutrinos ($L\simeq t$) [5]

$\Delta E\,t_{osc}^{(23)}\geq\pi\,,$ (7)

where $t_{osc}^{(23)}=2t^{(23)}_{1min}$ denotes the period of oscillations in
the atmospheric-Long Baseline region. As explained in [5], the above relation
provides a necessary condition for atmospheric neutrino oscillations to be
observed.

As a further example, we consider the $\Delta m_{23}^{2}$-driven survival
probability of $\bar{\nu}_{e}$. According to CHOOZ experiment [18], this
probability is close to the maximal value of unity. Using the condition
$P_{\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}}(t_{1min}^{(23)})=1-\sin^{2}2\theta_{13}$
along with the constraint $\sin^{2}2\theta_{13}\lesssim 2\cdot 10^{-1}$, where
$\theta_{13}$ is the mixing angle, Eq. (6) becomes (up to terms of order
$\sin^{2}2\theta_{13}$)

$\Delta E\,t_{osc}^{(23)}\geq 2\sin 2\theta_{13}\,,$ (8)

which is less stringent than the bound (7).

###  II.2 TEUR and unstable particles

We here digress briefly to discuss TEUR for unstable particles. Let us stress
that such systems provide one of the most interesting examples where TEUR is
typically applied, see e.g. [19]. Toward this end, we stick to the original
treatment of [13] and the recent review in [14].

The analysis of [13], which inspired the derivation of TEUR for neutrino
oscillations, is built upon the same considerations leading to Eq. (5), with
$P_{\nu_{l}\rightarrow\nu_{l}}$ being now replaced by the so-called quantum
non-decay probability $P_{t}$. For convenience, let us recast the inequality
(5) as

$\left|\frac{d}{dt}P_{t}\right|\leq 2\Delta E\sqrt{P_{t}-P_{t}^{2}}\,,$ (9)

which allows to identify the particle half-life $T_{h}$, for which $P_{t}=1/2$
and the right-hand side attains its maximum value. Then, Eq. (9) gives the
weaker inequality

$\left|\frac{d}{dt}P_{t}\right|\leq\Delta E\,,$ (10)

which sets a time-limit to the instability of decaying quantum systems (see
below for more quantitative discussion).

Since for decaying particles $P_{t=0}=1$ and
$P_{t\rightarrow\infty}\rightarrow 0$, we can infer the following features
that are likely to be of interest in the asymptotic regimes of decay

$\displaystyle\left|\frac{d}{dt}P_{t}\right|=0\,,\quad\,\,\mathrm{for}\,\,\,t=0\,,$
(11) $\displaystyle\frac{d}{dt}\cos^{-1}P_{t}\leq 2\Delta
E\,,\quad\,\,\mathrm{for}\,\,\,t\rightarrow 0\,,$ (12)
$\displaystyle\frac{d}{dt}P_{t}^{1/2}\leq\Delta
E\,,\quad\,\,\mathrm{for}\,\,\,t\rightarrow\infty\,,$ (13)

which can be unified as

$\frac{d}{dt}\cos^{-1}P_{t}^{1/2}\leq\Delta
E\,,\quad\,\,\mathrm{for}\,\,\,0\leq t\leq\infty\,.$ (14)

Integration of the the latter inequality gives

$\Delta E\,t\geq\cos^{-1}P_{t}^{1/2}\,,$ (15)

which is, in fact, the analogue of Eq. (6) for unstable particles. At this
stage, it becomes clear that the parallelism between neutrinos and unstable
particles has a merely formal meaning: indeed, although a TEUR can be
associated to both systems, the phenomenon of oscillations is not strictly
equivalent to a decay, since the oscillation probability does not vanish at
asymptotically long time. Notice also that Eq. (15) fixes the minimum-time
limit to $\pi/(2\Delta E)$.

Now, for $t$ equal to the half-life $T_{h}$ defined earlier, Eq. (15) leads
straightforwardly to

$\Delta
E\,T_{h}\geq\frac{\pi}{4}\,\,\quad\,\,\mathrm{for}\,\,\,\,0<T_{h}\leq\frac{\pi}{2\Delta
E}\,.$ (16)

On the other side, if $T_{h}>\pi/(2\Delta E)$, the non-negativity of $P_{t}$
allows us to write

$P_{t}\geq 0>1-\frac{2\Delta
E\,t}{\pi}\,,\quad\,\,\mathrm{for}\,\,\,t>\frac{\pi}{2\Delta E}$ (17)

which again implies the inequality (16). Combining these two results together,
we finally obtain the Mandelstam-Tamm version of TEUR for a decaying quantum
system

$\Delta E\,T_{h}\geq\frac{\pi}{4}\,.$ (18)

The above derivation has been recently revised in [14], leading to a more
stringent bound. Based on the approach of [20], the basic idea is to regard a
system of unstable particles as an open quantum system. This entails that a
comprehensive analysis cannot be limited to the system alone, but must
necessarily involve its surrounding “environment” too. In a field theoretical
language, such a prescription amounts to saying that it is not sufficient to
describe the system as excitations of some fields on a single vacuum state,
but one should take into account extra (artificial) degrees of freedom. A
typical strategy consists in doubling each physical degree of freedom,
resulting into an enlargement of the Hilbert space. Although this construction
was originally proposed to develop a QFT at finite temperature - Thermo Field
Dynamics [21, 22, 23], various applications have been later considered in QFT
on curved spacetime [24] and quantum brain model [25], among others.

For practical purposes, let us consider a set of canonical fermionic operators
$a_{\textbf{k}}$, $a^{{\dagger}}_{\textbf{k}}$, which annihilate and create a
decaying particle of momentum k at $t=0$, respectively. The associated vacuum
is denoted by $|0\rangle$, i.e. $a_{\textbf{k}}|0\rangle=0$. Following the
standard notation [21, 22], we indicate the corresponding set of _doubled_
operators by $\tilde{a}_{\textbf{k}}$, $\tilde{a}^{{\dagger}}_{\textbf{k}}$
and the related vacuum by $|\tilde{0}\rangle$, so that
$\tilde{a}_{\textbf{k}}|\tilde{0}\rangle=0$. Notice that such new operators
obey the same (anti-) commutation relations as $a_{\textbf{k}}$,
$a^{{\dagger}}_{\textbf{k}}$, i.e.
$\left\\{a_{\textbf{k}},a^{\dagger}_{\textbf{p}}\right\\}=\left\\{\tilde{a}_{\textbf{k}},\tilde{a}^{\dagger}_{\textbf{p}}\right\\}=\delta\left(\textbf{k}-\textbf{p}\right)$
.

Using the above tools, it is natural to define an enlarged Fock space having
the tensor-product structure between the Fock spaces associated to
$a_{\textbf{k}}$ and $\tilde{a}_{\textbf{k}}$, respectively. In this setting,
the underlying vacuum state
$|0\rangle\\!\rangle=|0\rangle\otimes|\tilde{0}\rangle$ is defined in such a
way that

$a_{\textbf{k}}|0\rangle\\!\rangle\ =\
\tilde{a}_{\textbf{k}}|0\rangle\\!\rangle\ =\ 0\,.$ (19)

The core idea of [14] is to construct a state $|0(\varphi)\rangle$, such that
the expectation value of the number operator $N_{\textbf{k}}\ =\
a^{{\dagger}}_{\textbf{k}}a_{\textbf{k}}$ satisfies

$\mathcal{N}_{\textbf{k}}(t)\ \equiv\ \langle
0(\varphi)|N_{\textbf{k}}|0(\varphi)\rangle\ =\
\exp\left(-\Gamma_{\textbf{k}}\,t\right)\,,$ (20)

where $\Gamma_{\textbf{k}}$ is the inverse lifetime of the unstable particle
(the physical meaning of the parameter $\varphi$ will become clear later).
Inspired by Thermo Field Dynamics [21], one can introduce the new set of
ladder operators via the Bogoliubov transformations

$\displaystyle a_{\textbf{k}}(\varphi)$ $\displaystyle=$
$\displaystyle\,\cos\varphi_{\textbf{k}}\,a_{\textbf{k}}\,-\,\sin\varphi_{\textbf{k}}\,\tilde{a}^{\dagger}_{\textbf{k}}\,,$
(21) $\displaystyle\tilde{a}_{\textbf{k}}(\varphi)$ $\displaystyle=$
$\displaystyle\,\cos\varphi_{\textbf{k}}\,\tilde{a}_{\textbf{k}}\,+\,\sin\varphi_{\textbf{k}}\,a^{\dagger}_{\textbf{k}}\,.$
(22)

The related vacuum is defined by the usual condition
$a_{\textbf{k}}(\varphi)\,|0(\varphi)\rangle=\tilde{a}_{\textbf{k}}(\varphi)\,|0(\varphi)\rangle=0$.
By explicit computation, it is possible to show that [21]

$|0(\varphi)\rangle\ =\
\prod_{\textbf{k}}\left(\cos\varphi_{\textbf{k}}\,+\,\sin\varphi_{\textbf{k}}a^{\dagger}_{\textbf{k}}\,\tilde{a}^{\dagger}_{\textbf{k}}\right)\,|0\rangle\\!\rangle\,.$
(23)

Combining Eqs. (20) and (23), we infer

$\mathcal{N}_{\textbf{k}}(t)\ =\ \sin^{2}\varphi_{\textbf{k}}(t)\ =\
\exp\left(-\Gamma_{\textbf{k}}t\right)\,,$ (24)

which displays the meaning of
$\varphi_{\textbf{k}}\equiv\varphi_{\textbf{k}}(t)$ in terms of the inverse
lifetime $\Gamma_{\textbf{k}}$.

To have more physical intuition on $|0(\varphi)\rangle$, let us consider the
asymptotic regions $t\rightarrow 0$ and $t\rightarrow\infty$. In the first
case, we have

$\left.|0(\varphi)\rangle\right|_{t=0}\ =\
\prod_{\textbf{k}}\,|0(\varphi_{\textbf{k}})\rangle\,,$ (25)

where

$|0(\varphi_{\textbf{k}})\rangle\ =\
a^{\dagger}_{\textbf{k}}\,\tilde{a}^{\dagger}_{\textbf{k}}\,|0\rangle\\!\rangle\,,$
(26)

is the state describing an unstable particle of momentum k. On the other hand,
the limit

$\left.|0(\varphi)\rangle\right|_{t=\infty}\ =\ |0\rangle\\!\rangle\,,$ (27)

gives the zero-particle state.

In order to extract TEUR, we now switch to the Heisenberg representation. In
this picture, the expectation value (24) is kept unchanged, while the vacuum
$|0(\varphi)\rangle$ must be set as in Eq. (25). In turn, the number operator
$N_{\textbf{k}}$ defined above Eq. (20) gains a non-trivial time-dependence.
With this setting, we can choose $O(t)=N_{\textbf{k}}(t)$ in Eqs. (1) and (2)
to obtain

$\sigma^{2}_{N}\ =\ \langle
0(\varphi)|N^{2}_{\textbf{k}}(t)|0(\varphi)\rangle\ -\ \langle
0(\varphi)|N_{\textbf{k}}(t)|0(\varphi)\rangle^{2}\ =\
\mathcal{N}_{\textbf{k}}(t)\left(1-\mathcal{N}_{\textbf{k}}(t)\right)\,,$ (28)

which yields

$\left|\frac{\mathrm{d}\mathcal{N}_{\textbf{k}}(t)}{\mathrm{d}t}\right|\,\leq\,2\Delta
E\,\sqrt{\mathcal{N}_{\textbf{k}}(t)\left(1-\mathcal{N}_{\textbf{k}}(t)\right)}\,.$
(29)

Notice that this inequality is the same as Eq. (9), with the non-decay
probability $P_{t}$ being now formally replaced by the vacuum density
$\mathcal{N}_{\textbf{k}}(t)$. For the characteristic time $T_{h}$ that
maximizes the right side, we have $\mathcal{N}_{\textbf{k}}(T_{h})=1/2$ and

$\Delta
E\,\geq\,\left|\frac{\mathrm{d}\mathcal{N}_{\textbf{k}}(t)}{\mathrm{d}t}\right|\,,$
(30)

to be compared with Eq. (10).

The relation (30) can be further manipulated by resorting to the triangular
inequality and integrating both sides from $0$ to $T$ to obtain

$\Delta E\,T\ \geq\
\int^{T}_{0}\\!\\!\mathrm{d}t\,\left|\frac{\mathrm{d}\mathcal{N}_{\textbf{k}}(t)}{\mathrm{d}t}\right|\,\geq\,\left|\int^{T}_{0}\\!\\!\mathrm{d}t\,\frac{\mathrm{d}\mathcal{N}_{\textbf{k}}(t)}{\mathrm{d}t}\right|\,,$
(31)

which yields

$\Delta E\,T\geq 1-\mathcal{N}_{\textbf{k}}(T)\,.$ (32)

At $T=T_{h}$, we then get the Heisenberg-like lower bound

$\Delta E\,T_{h}\geq\frac{1}{2}\,,$ (33)

that is stronger than the one in Eq. (18).

As a final remark, it is worth noting from Eq. (24) that $T_{h}=\log
2/{\Gamma_{\textbf{k}}}=\tau_{\textbf{k}}\,\log 2$, where
$\tau_{\textbf{k}}=1/{\Gamma_{\textbf{k}}}$ is the particle life-time.
Therefore, TEUR (33) can be rearranged as

$\Delta E\geq\frac{1}{\tau_{\textbf{k}}\,\log 4}\,.$ (34)

This inequality sets an intrinsic lower bound on the energy distribution width
of unstable quantum particles. In the same way, by carrying on the analogy
between neutrinos and unstable particles, we can interpret Eq. (5) as a
fundamental limit on the width of the neutrino energy distribution.

##  III Gravitational effects on TEUR

In [9] TEUR for neutrino oscillations has been derived for a generic
stationary curved spacetime to study how gravity affects the characteristic
oscillation length. To review such a formalism, we first take a step back and
consider the Bilenky-like (Minkowski) form (6) of TEUR, here rewritten
equivalently as

$2\Delta
E{\sqrt{P_{\nu_{l}\rightarrow\nu_{l}}(x)-P^{2}_{\nu_{l}\rightarrow\nu_{l}}(x)}}\geq\left|\frac{d}{dt}P_{\nu_{l}\rightarrow\nu_{l}}(x)\right|\,,$
(35)

where, for more generality, we have assumed that the neutrino survival
probability $P_{\nu_{l}\rightarrow\nu_{l}}$ depends on $t$ through the
spacetime coordinate $x\equiv x(t)$. For simplicity, from now on we restrict
to a toy model involving two flavors only, but the same considerations can be
extended to the more realistic case of three neutrino generations.

As observed in Sec. II.2, the left side of Eq. (35) reaches its maximum when
$P_{\nu_{l}\rightarrow\nu_{l}}=1/2$. Integrating both sides between some
initial time $t_{0}$ (such that $P_{\nu_{l}\rightarrow\nu_{l}}(x(t_{0}))=1$)
and $t$, and employing once more the triangular inequality, we are led to [9]

$\Delta E\,T\geq P_{\nu_{l}\rightarrow\nu_{l^{\prime}}}\hskip
1.42262pt(x(t))\,,$ (36)

where $T\equiv\int_{t_{0}}^{t}dt^{\prime}=t-t_{0}$ is the neutrino flight-time
and $P_{\nu_{l}\rightarrow\nu_{l^{\prime}}}\hskip
1.42262pt(x(t))=1-P_{\nu_{l}\rightarrow\nu_{l}}\hskip 1.42262pt(x(t))$ the
oscillation probability.

Gravitational corrections to flavor oscillations typically enter the neutrino
phase [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44]. To find out the form of these extra terms, we write explicitly the
generic flavor eigenstate $|\nu_{l}\rangle$ as a mixture of the mass
eigenstates $|\nu_{k}\rangle$ ($k=1,2$) according to [6]

$|\nu_{l}(x)\rangle=\sum_{k=1,2}U_{{lk}}\hskip
0.28453pt(\theta)\,|\nu_{k}(x)\rangle\,,\quad\,\,l=e,\mu\,,$ (37)

where $U_{{lk}}\hskip 0.28453pt(\theta)$ is the generic element of the mixing
matrix [6].

Since the states $|\nu_{k}(x)\rangle$ are Hamiltonian eigenstates, in the
standard quantum theory on flat background their evolution is governed by the
usual relation

$|\nu_{k}({x})\rangle=\exp[{-i\varphi_{k}(x)}]\hskip
0.56905pt|\nu_{k}\rangle,\quad(k=1,2)\,,$ (38)

where

$\varphi_{k}(x)\equiv E_{k}\hskip 1.13809ptt-\textbf{p}_{k}\cdot\textbf{x}$
(39)

denotes the QM phase of the $k^{th}$ eigenstate having mass $m_{k}$, energy
$E_{k}$ and three-momentum $\textbf{p}_{k}$, respectively (not to be confused
with the parameter $\varphi_{\textbf{k}}$ appearing in the Bogoliubov
transformations (21) and (22)). These three quantities are connected via the
flat mass-shell relation $E^{2}_{k}=m^{2}_{k}+{|\textbf{p}_{k}|}^{2}$.

For relativistic neutrinos traveling along the $x$-axis from the source point
$x_{0}$ to the detector point $x$ (with $x>x_{0}$), the phase (39) can be
simplified to

$\varphi_{k}\simeq\frac{m^{2}_{k}}{2E}\hskip 0.56905ptL_{p}\,,$ (40)

where we have assumed the mass eigenstates to have roughly the same energy
$E$. Furthermore, we have denoted the proper distance traveled by neutrinos by
$L_{p}=x-x_{0}$, which can be approximated to the flight-time $T$ in the
relativistic limit.

Now, substitution of Eq. (40) into (37) and (38) gives

$|\nu_{l}({x})\rangle=\sum_{k=1,2}U_{{lk}}\hskip 0.28453pt(\theta)\hskip
0.56905pt\exp\left({-i\hskip 0.85358pt\frac{m^{2}_{k}}{2E}\hskip
0.85358ptL_{p}}\right)|\nu_{k}\rangle\,,\quad\,\,l=e,\mu\,.$ (41)

In turn, the probability that a neutrino $|\nu_{l}\rangle$ undergoes a flavor
transition to $|\nu_{l^{\prime}}\rangle$ after propagating over the distance
$L_{p}$ is [9]

$\displaystyle P_{\nu_{l}\rightarrow\nu_{l^{\prime}}}(x)$ $\displaystyle=$
$\displaystyle|\langle\nu_{l^{\prime}}(x)|\nu_{l}(x_{0})\rangle|^{2}$ (42)
$\displaystyle=$
$\displaystyle\sin^{2}(2\theta)\,\sin^{2}\left(\frac{\varphi_{12}}{2}\right)$
$\displaystyle=$ $\displaystyle\sin^{2}(2\theta)\sin^{2}\left(\frac{\pi\hskip
0.56905ptL_{p}}{L^{osc}}\right)\,,\quad\,\,\,l\neq l^{\prime}\,,$

where in the second step $\varphi_{12}\equiv\varphi_{1}-\varphi_{2}$ denotes
the phase-shift acquired by the mass eigenstates during the propagation, while
in the last step we have introduced the characteristic _oscillation length_
$L^{osc}$

$L^{osc}\equiv\frac{4\pi E}{\Delta m^{2}}\,,\quad\,\,\,\Delta
m^{2}=|m_{1}^{2}-m_{2}^{2}|\,.$ (43)

Accordingly, TEUR in Eq. (36) takes the form

$\Delta E\,T\geq\sin^{2}(2\theta)\sin^{2}\left(\frac{\pi\hskip
0.56905ptL_{p}}{L^{osc}}\right).$ (44)

Since this inequality holds for any spacetime point, we can set
$L_{p}=L^{osc}/2$ so as to maximize the right side and obtain the neutrino
oscillation condition

$\Delta E\geq\frac{2\,\sin^{2}(2\theta)}{L^{osc}}\,.$ (45)

Two comments are in order here: first, we notice that for $\theta=0$ and/or
$L^{osc}\rightarrow\infty$, the lower bound on the neutrino energy uncertainty
vanishes, as expected in the absence of mixing and/or oscillations. Moreover,
in [45] $\Delta E$ is interpreted as the minimum energy transferred to
neutrinos in scattering processes, which is necessary to reveal them once the
oscillation occurred.

###  III.1 TEUR in curved spacetime

Let us extend the above formalism to stationary curved spacetime. For this
kind of metrics, we remark that TEUR can be consistently defined, since there
always exists a global timelike Killing vector field $K^{\mu}$, such that
$K\equiv\int K_{\mu}\,T^{\mu\nu}\,\mathrm{d}\Sigma_{\nu}$ does not depend on
the spacelike hypersurface $\Sigma$, where $T^{\mu\nu}$ is the (conserved)
stress-energy tensor [46]. We are then allowed to introduce a coordinate $t$,
such that the metric is independent of it and with respect to which
$K^{\mu}=(1,0,0,0)$. One has
$K\equiv\int\sqrt{-g}:T^{0}_{\,\,\,0}:\mathrm{d}^{3}x=H$, where
$:T^{0}_{\,\,\,0}:$ denotes the normal-ordered operator, $g$ is the
determinant of the metric tensor $g^{\mu\nu}$ and $H$ the Hamiltonian of the
considered system. More generally, the role of the above Killing vector can be
taken by Kodama vector for dynamical metrics [47].

To generalize TEUR for flavor oscillations to curved spacetime, it proves
convenient to rephrase the evolution (41) of the neutrino state in a
manifestly covariant way as [27]

$|\nu_{l}({\lambda})\rangle=\sum_{k=1,2}U_{{lk}}\hskip 0.28453pt(\theta)\hskip
0.56905pt\exp(-i\Phi)|\nu_{k}\rangle\,,\quad\,\,\,l=e,\mu\,,$ (46)

where $\lambda$ is the neutrino world-line parameter and $\Phi$ generalizes
the definition (39) of neutrino phase to

$\Phi=\int_{\lambda_{0}}^{\lambda}P_{\mu}\frac{\mathrm{d}x^{\mu}_{null}}{\mathrm{d}\lambda^{\prime}}\hskip
0.56905pt\mathrm{d}\lambda^{\prime}\,.$ (47)

Notice that $P_{\mu}$ and $\mathrm{d}x^{\mu}_{null}/d\lambda$ in the above
integral are the generator of spacetime translations of neutrino mass states
and the null tangent vector to the neutrino worldline, respectively. For
$g_{\mu\nu}=\eta_{\mu\nu}$, one can check that Eq. (47) reproduces the
evolution (41) in flat spacetime, as expected [9].

For a generic stationary curved spacetime, the operator $P_{\mu}$ is defined
by the generalized mass-shell relation [27]

$\left(P^{\mu}+\frac{A^{\mu}\gamma^{5}}{2}\right)\left(P_{\mu}+\frac{A_{\mu}\gamma^{5}}{2}\right)=M^{2}\,,$
(48)

where $M=\mathrm{diag}[m_{1},m_{2}]$ is the diagonal neutrino mass-matrix and
$\gamma_{\mu}$ are the Dirac matrices obeying the Clifford algebra
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}$
($\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$). Furthermore, the
potential-like term

$A^{\mu}=\frac{\sqrt{g}}{4}\hskip 0.85358pt\hskip
0.85358pte^{\mu}_{\hat{a}}\hskip
0.85358pt\epsilon^{{\hat{a}}{\hat{b}}{\hat{c}}{\hat{d}}}\left(\partial_{\sigma}e_{{\hat{b}}\nu}-\partial_{\nu}e_{{\hat{b}}\sigma}\right)e^{\nu}_{\hat{c}}e^{\sigma}_{\hat{d}}$
(49)

enters the spin-connection in the Dirac equation on curved spacetime [27, 9].
Here $e^{\mu}_{\hat{a}}$ are the tetrads connecting general curvilinear and
locally inertial coordinates, denoted by greek and (hatted) latin indices,
respectively. The symbol $\epsilon^{{\hat{a}}{\hat{b}}{\hat{c}}{\hat{d}}}$
stands for the totally antisymmetric Levi-Civita tensor of component
$\epsilon^{{\hat{0}}{\hat{1}}{\hat{2}}{\hat{3}}}=+1$, while $\partial$
indicates ordinary derivative.

As shown in [27], the integral in (47) can be simplified for relativistic
neutrinos moving along a null trajectory by neglecting $\mathcal{O}(A^{2})$
and $\mathcal{O}{(AM^{2})}$-terms. Under these assumptions, we obtain [9]

$\Phi=\int_{\lambda_{0}}^{\lambda}\left(\frac{M^{2}}{2}-\frac{\mathrm{d}x^{\mu}_{null}}{\mathrm{d}\lambda^{\prime}}A_{\mu}\gamma^{5}\,\right)\hskip
0.56905pt\mathrm{d}\lambda^{\prime}\,,$ (50)

which can be further manipulated by rewriting the line element $d\lambda$ in
terms of the differential proper distance $\mathrm{d}\ell$ at constant time
for null trajectories as

$\mathrm{d}\lambda=\mathrm{d}\ell{\left(-g_{ij}\frac{\mathrm{d}x^{i}}{\mathrm{d}\lambda}\frac{\mathrm{d}x^{j}}{\mathrm{d}\lambda}\right)}^{-\frac{1}{2}}=\mathrm{d}\ell{\left[g_{00}\left(\frac{\mathrm{d}t}{\mathrm{d}\lambda}\right)^{2}+2\hskip
0.85358ptg_{0i}\hskip 0.85358pt\frac{\mathrm{d}t}{\mathrm{d}\lambda}\hskip
0.85358pt\frac{\mathrm{d}x^{i}}{d\lambda}\right]}^{-\frac{1}{2}}.$ (51)

Equation (50) provides the crucial ingredient toward reformulating TEUR in
curved spacetime. To this end, we recast Eq. (36) in terms of $\lambda$ to
obtain

$\Delta
E\,T(\lambda)\geq\mathcal{P}_{\nu_{l}\rightarrow\nu_{l^{\prime}}}\hskip
0.85358pt(x(\lambda))\,,$ (52)

where the neutrino flight-time is given by $T(\lambda)\equiv
t(\lambda)-t(\lambda_{0})$ for a fixed $\lambda_{0}$. From Eq. (42), it
follows that

$\Delta
E\geq\frac{\mathrm{sin}^{2}\left(2\theta\right)}{T(\lambda)}\,\mathrm{sin}^{2}\bigg{[}\frac{{\varphi_{12}(x(\lambda))}}{2}\bigg{]},\quad{\varphi}_{12}\equiv{\varphi}_{1}-{\varphi}_{2}\,,$
(53)

where the neutrino phase $\varphi_{k}$ ($k=1,2$) is now generally defined as
the eigenvalue of $\Phi$ in Eq. (50) with respect to the $k^{th}$ mass
eigenstate, i.e. $\Phi|\nu_{k}\rangle=\varphi_{k}|\nu_{k}\rangle$.

Before moving onto the evaluation of the above inequality for some specific
metrics, it is worth observing that $\Delta E$ appearing in Eq. (53)
represents the uncertainty on the neutrino energy as measured by an asymptotic
observer (see [9] for more details). However, the truly measurable quantity in
neutrino oscillation experiments is the local energy uncertainty $\Delta
E_{\ell}$, which represents the uncertainty for a local observer temporarily
at rest in the curved background (and, thus, with respect to the oscillation
experiment). The connection between these two quantities is established via
the vierbein fields

$E_{\ell}(x)\equiv P_{\hat{0}}(x)=e^{\nu}_{\hat{0}}(x)\,P_{\nu}\,.$ (54)

Clearly, this transformation must also be implemented into the expression of
the phase-shift $\varphi_{12}$, along with the usage of the proper distance
traveled by neutrinos. This is an essential step to make a consistent
comparison with the TEUR (45) in the limit when
$g_{\mu\nu}\rightarrow\eta_{\mu\nu}$.

####  III.1.1 Schwarzschild spacetime

The Schwarzschild metric in isotropic coordinates $(t,x,y,z)$ and (linearized)
weak-field approximation is

$\mathrm{d}s^{2}=\left(1+2\phi\right)\mathrm{d}t^{2}-(1-2\phi)\,(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2})\,,$
(55)

where $\phi(r)=-M/r\equiv-M/\sqrt{x^{2}+y^{2}+z^{2}}$ is the gravitational
potential and $M$ the spherically symmetric source mass. The only non-trivial
components of the tetrads are $e^{0}_{\hat{0}}=1-\phi$ and
$e^{i}_{\hat{j}}=(1+\phi)\,\delta^{i}_{j}$, where $\delta^{i}_{j}$ is the
Kronecker delta.

Because of the particular symmetry of the metric, the potential $A_{\mu}$ in
Eq. (49) vanishes. From Eqs. (50) and (51), the resulting neutrino phase-shift
$\varphi_{12}$ takes the simple form [9]

$\varphi_{12}=\frac{\Delta
m^{2}}{2}\int^{\ell}_{\ell_{0}}\frac{1+\phi}{E}\,\mathrm{d}\ell^{\prime}\,,$
(56)

with $E\equiv P_{0}=g_{00}\,\mathrm{d}t/\mathrm{d}\lambda$.

If we set the frame axes so that the neutrino propagates radially (e.g. along
the $x$-direction), we have $\mathrm{d}\ell=(1-\phi)\,\mathrm{d}x$. Then, by
taking into account that $E$ is constant along a null trajectory, the integral
(56) can be solved straightforwardly to give the following form of TEUR

$\Delta
E\geq\frac{\mathrm{sin}^{2}\left(2\theta\right)}{T}\,\mathrm{sin}^{2}\left[\frac{\Delta
m^{2}\left(x-x_{0}\right)}{4E}\right].$ (57)

In terms of the proper distance

$L_{p}\equiv\int^{x}_{x_{0}}\sqrt{-g_{11}}\,\mathrm{d}x^{\prime}=x-x_{0}+M\ln\left(\frac{x}{x_{0}}\right)$
(58)

and the local energy (54), we obtain

$\frac{\Delta
E_{\ell}}{1-\phi}\geq\frac{\mathrm{sin}^{2}\left(2\theta\right)}{T}\,\mathrm{sin}^{2}\left(\frac{\pi
L_{p}}{L^{osc}_{\phi}}\right),$ (59)

where we have introduced the Schwarzschild-corrected proper oscillation length
(at the leading order in $\phi$) [29]

$L^{osc}_{\phi}\equiv\frac{4\pi E_{\ell}}{\Delta
m^{2}}\left[1+\phi+\frac{M}{x-x_{0}}\ln\left(\frac{x}{x_{0}}\right)\right].$
(60)

Here, the gravitational potential is to be regarded as $\phi(x)=-M/x$.

As emphasized below Eq. (44), the right side of the inequality (60) is
maximized for $L_{p}=L^{osc}_{\phi}/2$, which in turn gives the following
relation between the time interval $T$ and the oscillation length
$L^{osc}_{\phi}$ [9]

$T=\frac{L^{osc}_{\phi}}{2}\left[1+\frac{M}{x-x_{0}}\mathrm{ln}\left(\frac{x}{x_{0}}\right)\right].$
(61)

By plugging into Eq. (59), we can finally cast the TEUR in Schwarzschild
spacetime as

$\Delta
E_{\ell}\geq\frac{2\,\mathrm{sin}^{2}(2\theta)}{L^{{osc}}_{eff}(M)}\,,$ (62)

where

$L^{{osc}}_{eff}(M)\equiv
L^{osc}_{\phi}\left[1+\phi+\frac{M}{x-x_{0}}\mathrm{ln}\left(\frac{x}{x_{0}}\right)\right],$
(63)

is the _effective oscillation length_.

The last form of TEUR (62) allows for a direct comparison with the
corresponding flat expression (45). Remarkably, we can see that gravitational
corrections increase $L^{osc}_{eff}$ with respect to the standard oscillation
length (43), resulting into a more stringent bound on $\Delta E_{\ell}$. For
consistency, one can check that $L^{{osc}}_{eff}(M)\rightarrow L^{osc}$ for
$M\rightarrow 0$ and the inequality (45) is recovered. On the other hand, in
the limit $L^{{osc}}_{eff}(M)\to\infty$, the right side of Eq. (62) vanishes,
in line with the discussion below Eq. (45).

####  III.1.2 Lense-Thirring spacetime

We can now apply the same line of reasoning to extract TEUR in Lense-Thirring
background, which can be interpreted as the exterior gravitational field of a
rotating spherical source. Notice also that this metric is usually adopted to
describe gravitomagnetic frame-dragging effects, see for instance [48].

In the weak-field approximation, the line element on the equatorial plane
reads [49]

$\mathrm{d}s^{2}=\left(1+2\phi\right)\mathrm{d}t^{2}-\frac{\phi\hskip
0.56905pt\Omega}{r^{2}}\left(x\,\mathrm{d}y-y\,\mathrm{d}x\right)\,\mathrm{d}t-(1-2\phi)\,(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}),$
(64)

where $\Omega\equiv 4R^{2}\omega/5$, $\omega$ is the angular velocity of the
source of radius $R$ and $\phi$ the gravitational potential defined below Eq.
(55). Without loss of generality, we assume the rotation takes place around
the $z$ axis.

Following the same calculations as in Sec. III.1.1, we now arrive to [9]

$\Delta E_{\ell}\geq\frac{2\sin^{2}(2\theta)}{L^{osc}_{eff}(M,\Omega)}\,,$
(65)

where the effective oscillation length exhibits a non-trivial dependence on
both the mass and angular velocity of the source as

$L^{osc}_{eff}(M,\Omega)\equiv
L^{osc}\left[1+\frac{M}{x-x_{0}}\mathrm{ln}\left(\frac{r+x}{r_{0}+x_{0}}\right)+\frac{M\,\Omega}{2b\left(x-x_{0}\right)}\left(\frac{x}{r}-\frac{x_{0}}{r_{0}}\right)+\phi\left(1+\frac{\Omega
b}{r^{2}}\right)\right]\,.$ (66)

with $b>R$ being the impact parameter. As discussed for Schwarzschild
geometry, the effective oscillation length is increased with respect to the
flat case. The latter is however restored for $M,\Omega\rightarrow 0$, as it
should be.

####  III.1.3 Rindler spacetime

As last example, we discuss TEUR for an observer undergoing constant proper
acceleration $a>0$ (Rindler observer). It is interesting to note that,
although the corresponding Rindler metric is flat, it well-describes the
static gravitational field that would be appropriate for a spacetime with
fictitious “infinite flat earth” [50]. Assuming the acceleration along the
$x$-axis, the (linearized) line element in Rindler-Fermi coordinates is [51]

$\mathrm{d}s^{2}\ =\
\left(1+ax\right)\mathrm{d}t^{2}-(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2})\,.$
(67)

TEUR for Rindler observer can be cast in the same form as Eq. (62) (or Eq.
(65)), with the acceleration-dependent oscillation length being now defined by
[9]

$L^{osc}_{eff}(a)\equiv
L^{osc}\left[1+\frac{a}{4}\left(x-x_{0}\right)\right].$ (68)

Once again, the effective oscillation length is increased with respect to the
flat case, giving rise to a more stringent bound on the energy uncertainty
$\Delta E_{\ell}$. As expected, the standard formula in Minkowski spacetime is
reproduced for $a\rightarrow 0$. On other other hand, the limit of
$L^{osc}_{eff}(a)\rightarrow\infty$ for fixed $a$ cannot be considered in the
present case, as it invalidates the linearized approximation under which Eq.
(68) is derived [51].

##  IV TEUR in Quantum Field Theory

We start this section by reviewing the TEUR in the Pontecorvo formalism for
neutrino mixing and oscillations [52, 53, 54, 55, 56] as developed in [10].
This analysis gives similar results to the ones shown in Sec. II and III.
Although the Pontecorvo approach is experimentally successful, it has however
several conceptual drawbacks and is widely recognized to be only applicable to
relativistic neutrinos [57, 58, 11, 59, 60, 61]. These issues motivate us to
introduce the main aspects of the so-called _flavor Fock space_ approach to
neutrino mixing in QFT [62, 11, 63, 64, 65, 66, 67, 68, 69, 70], which extends
the Pontecorvo formalism to the non-relativistic domain. We then show how TEUR
can be studied in this generalized framework [10]. For consistency with the
original notation [11], in what follows we label neutrino fields with definite
masses by latin indices, while greek indices are reserved for definite flavor
fields.

###  IV.1 Time-energy uncertainty relations for neutrino oscillations:
Pontecorvo flavor states

In Section II.2 we have re-formulated the TEUR for unstable particles in a
second quantization language. The same procedure can be repeated for the
Minkowski form of TEUR for neutrino oscillations.

The fields $\nu_{j}(x)$, $j=1,2$, describe Dirac neutrinos with definite
masses $m_{j}$:

$\nu_{j}(x)=\sum_{r}\,\int\\!\\!\frac{\mathrm{d}^{3}k}{(2\pi)^{\frac{3}{2}}}\,\left[u_{{\bf
k},j}^{r}(t)\,\alpha_{{\bf k},j}^{r}+\ v_{-{\bf k},j}^{r}(t)\,\beta_{-{\bf
k},j}^{r\dagger}\right]e^{i{\bf k}\cdot{\bf x}}\,,$ (69)

with $x\equiv({\bf x},t)$, $\;u^{r}_{{\bf
k},j}(t)=e^{-i\omega_{\textbf{k},j}t}\,u^{r}_{{\bf k},j}$, $\;v^{r}_{{\bf
k},j}(t)=e^{i\omega_{\textbf{k},j}t}\,v^{r}_{{\bf k},j}$ and
$\omega_{\textbf{k},j}=\sqrt{|\textbf{k}|^{2}+m_{j}^{2}}$. The ladder
operators $\alpha^{r}_{\textbf{k},j}$ and $\beta_{{\bf k},j}^{r}$ annihilate
the massive neutrino vacuum $|0\rangle_{1,2}$, i.e.

$\alpha^{r}_{\textbf{k},j}|0\rangle_{1,2}=0=\beta_{{\bf
k},j}^{r}|0\rangle_{1,2}\ .$ (70)

The anticommutation relations are

$\\{\nu^{\alpha}_{i}(x),\nu^{\beta{\dagger}}_{j}(y)\\}_{t_{x}=t_{y}}=\delta^{3}({\bf
x}-{\bf y})\delta_{\alpha\beta}\delta_{ij}\,,$ (71) $\\{\alpha^{r}_{{\bf
k},i},\alpha^{s{\dagger}}_{{\bf q},j}\\}=\delta_{\bf
kq}\delta_{rs}\delta_{ij}\,,\,\,\quad\,\,\\{\beta^{r}_{{\bf
k},i},\beta^{s{\dagger}}_{{\bf q},j}\\}=\delta_{\bf
kq}\delta_{rs}\delta_{ij}\,,$ (72)

while the orthonormality and completeness relations are fixed as

$\displaystyle u^{r{\dagger}}_{{\bf k},i}u^{s}_{{\bf
k},i}=v^{r{\dagger}}_{{\bf k},i}v^{s}_{{\bf
k},i}=\delta_{rs}\,,\quad\,\,u^{r{\dagger}}_{{\bf k},i}v^{s}_{-{\bf
k},i}=0\,,\quad\,\,\sum_{r}(u^{r\alpha*}_{{\bf k},i}u^{r\beta}_{{\bf
k},i}+v^{r\alpha*}_{-{\bf k},i}v^{r\beta}_{-{\bf
k},i})=\delta_{\alpha\beta}\;.$ (73)

Similarly to the case of unstable particles, we define the number operator for
flavor (Pontecorvo) neutrinos as

$N_{P,\sigma}(t)\ =\
\sum_{\textbf{k},r}\alpha^{r{\dagger}}_{P,\textbf{k},\sigma}(t)\alpha^{r}_{P,\textbf{k},\sigma}(t)\,,$
(74)

where the flavor ladder operators $\alpha^{r}_{P,\textbf{k},\sigma}$ are just
linear combinations of the mass operators

$\alpha^{r}_{P,\textbf{k},\sigma}\ =\
\sum^{2}_{j=1}\,U_{\sigma\,j}\,\alpha^{r}_{\textbf{k},j}\,.$ (75)

Then, Pontecorvo flavor states can be built as

$|\nu^{r}_{\textbf{k},\sigma}\rangle_{P}\ =\
\alpha^{r{\dagger}}_{P,\textbf{k},\sigma}|0\rangle_{1,2}\,,$ (76)

which coincide with Eq. (41) of the QM treatment when one assumes equal
momenta.

The flavor survival probability can be thus computed as the expectation value
of the number operator on a reference flavor state

$\displaystyle\mathcal{P}_{\sigma\rightarrow\sigma}(t)\ =\ \langle
N_{P,\sigma}(t)\rangle_{\sigma}\ =\
1-\sin^{2}(2\theta)\sin^{2}\left(\frac{\Omega_{\textbf{k}}^{{}_{-}}}{2}t\right)\,,$
(77)

where, in this subsection,
$\langle\cdots\rangle_{\sigma}={}_{P}\langle\nu^{r}_{\textbf{k},\sigma}|\cdots|\nu^{r}_{\textbf{k},\sigma}\rangle_{P}$.
Moreover, we have introduced the shorthand notation
$\Omega_{\textbf{k}}^{{}_{\pm}}\equiv\omega_{\textbf{k},1}\pm\omega_{\textbf{k},2}$.

It is clear that the number of flavor neutrinos is a good candidate as clock-
observable, i.e. $O(t)={N_{P,\sigma}(t)}$. Noting that

$\displaystyle\sigma^{2}_{N}\ =\ \langle N^{2}_{P,\sigma}(t)\rangle_{\sigma}\
-\ \langle N_{P,\sigma}(t)\rangle_{\sigma}^{2}\ =\ \
\mathcal{P}_{\sigma\rightarrow\sigma}(t)\left(1-\mathcal{P}_{\sigma\rightarrow\sigma}(t)\right)\,,$

we get

$\left|\frac{\mathrm{d}\mathcal{P}_{\sigma\rightarrow\sigma}(t)}{\mathrm{d}t}\right|\,\leq\,2\Delta
E\,\sqrt{\mathcal{P}_{\sigma\rightarrow\sigma}(t)\left(1-\mathcal{P}_{\sigma\rightarrow\sigma}(t)\right)}\,,$
(78)

which is in fact equivalent to Eq. (35). An important remark is that
$\sigma_{N}$ quantifies the dynamic flavor entanglement of the neutrino state
(76) (see [71, 72]) and it naturally appears in the study of Leggett-Garg
inequalities in the context of neutrino oscillations [73, 74, 75].

The inequality (78) can be made simpler by following similar steps to those
below Eq. (31). In so doing, we arrive to

$\Delta
E\,T\geq\mathcal{P}_{\sigma\rightarrow\rho}(T)\,,\quad\sigma\neq\rho\,,$ (79)

where $\mathcal{P}_{\sigma\rightarrow\rho}(t)\ =\
1-\mathcal{P}_{\sigma\rightarrow\sigma}(t)$ is the flavor oscillation
probability at time $t$. For $T=T_{h}$ such that
$\mathcal{P}_{\sigma\rightarrow\rho}(T_{h})=\frac{1}{2}$, we finally get the
analogue of Eq. (33) for neutrino oscillations.

###  IV.2 Flavor Fock space approach to neutrino mixing and oscillations

Consider the weak decay of a $W^{+}$ boson, $W^{+}\rightarrow e^{+}+\nu_{e}$.
This process can be described by the Lagrangian
$\mathcal{L}=\mathcal{L}_{\nu}+\mathcal{L}_{l}+\mathcal{L}_{int}$ with

$\displaystyle\mbox{}{\cal
L}_{\nu}=\overline{\nu}\left(i\gamma_{\mu}\partial^{\mu}-M_{\nu}\right)\nu\,,$
(80) $\displaystyle\mbox{}{\cal
L}_{l}=\overline{l}\left(i\gamma_{\mu}\partial^{\mu}-M_{l}\right)l\,,$ (81)
$\displaystyle\mbox{}{\cal
L}_{int}=\frac{g}{2\sqrt{2}}\left[W_{\mu}^{+}\,\overline{\nu}\,\gamma^{\mu}\,(1-\gamma^{5})\,l+h.c.\right]\,,$
(82)

where $\nu=\left(\nu_{e},\nu_{\mu}\right)^{{T}},\,l=\left(e,\mu\right)^{T}$,
and

$\displaystyle\mbox{}M_{\nu}\,=\,\left(\begin{array}[]{cc}m_{e}&m_{e\mu}\\\
m_{e\mu}&m_{\mu}\end{array}\right)\,,\qquad
M_{l}\,=\,\left(\begin{array}[]{cc}\tilde{m}_{e}&0\\\
0&\tilde{m}_{\mu}\end{array}\right).$ (87)

The off-diagonal components of $M_{\nu}$ give rise to the neutrino mixing
terms in ${\cal L}_{\nu}$. These terms can be diagonalized by means of the
field mixing transformation [52, 53, 54, 55, 56]

$\nu_{\sigma}(x)\ =\
\sum_{j}\,U_{\sigma\,j}\nu_{j}(x)\,,~{}~{}\qquad\sigma=e,\mu;~{}~{}j=1,2\,,$
(88)

wher $U$ is the mixing matrix

$U\ =\ \begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}\,,$ (89)

with $\tan 2\theta=2m_{e\mu}/(m_{\mu}-m_{e})$, so that

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{j}\,\overline{\nu}_{j}\left(i\gamma_{\mu}\partial^{\mu}-m_{j}\right)\nu_{j}\,+\,\sum_{\sigma}\,\overline{l}\left(i\gamma_{\mu}\partial^{\mu}-\tilde{m}_{\sigma}\right)l$
(90) $\displaystyle+$
$\displaystyle\frac{g}{2\sqrt{2}}\sum_{\sigma,j}\,\left[W_{\mu}^{+}(x)\,\overline{\nu}_{j}\,U^{*}_{j\sigma}\,\gamma^{\mu}\,(1-\gamma^{5})\,l_{\sigma}+h.c.\right]\,.$

This fact led to develop the idea that neutrino flavor states can be defined
as linear combinations of neutrino mass states, as discussed in the previous
sections. However, the field mixing transformation (88) does not imply the
definition (76). Then, the question arises as to how to consistently define
the flavor states $|\nu_{\sigma}\rangle$ in QFT.

In order to unveil such point, we start from the observation that the
Lagrangian $\mathcal{L}$ is invariant under the action of the global $U(1)$
transformations $\nu\rightarrow e^{i\alpha}\nu$ and $l\rightarrow
e^{i\alpha}l$, leading to the conservation of the total flavor charge
$Q_{l}^{tot}$ corresponding to the lepton-number conservation [56]. This can
be written in terms of the flavor charges for neutrinos and charged leptons

$Q_{l}^{tot}=\sum_{\sigma=e,\mu}Q_{\sigma}^{tot}(t)\,,\quad\,\,Q_{\sigma}^{tot}(t)=Q_{\nu_{\sigma}}(t)+Q_{\sigma}\,,$
(91)

with

$\displaystyle Q_{e}$ $\displaystyle=$
$\displaystyle\int\\!\\!\mathrm{d}^{3}{\textbf{x}}\,e^{{\dagger}}(x)e(x)\,,\qquad
Q_{\nu_{e}}(t)=\int\\!\\!\mathrm{d}^{3}{\textbf{x}}\,\nu_{e}^{{\dagger}}(x)\nu_{e}(x)\,,$
$\displaystyle Q_{\mu}$ $\displaystyle=$
$\displaystyle\int\\!\\!\mathrm{d}^{3}{\textbf{x}}\,\mu^{{\dagger}}(x)\mu(x)\,,\qquad
Q_{\nu_{\mu}}(t)=\int\\!\\!\mathrm{d}^{3}{\textbf{x}}\,\nu_{\mu}^{{\dagger}}(x)\nu_{\mu}(x)\,.$
(92)

Noticing that $[\mathcal{L}_{int}({\bf x},t),Q_{\sigma}^{tot}(t)]=0$, we see
that neutrinos are produced and detected with a definite flavor [58, 6].
However, $[(\mathcal{L}_{\nu}+\mathcal{L}_{l})({\bf
x},t),Q_{\sigma}^{tot}(t)]\neq 0$, leading to the flavor oscillation
phenomenon.

The second key observation is that the field mixing transformation (88) can be
exactly rewritten as [62, 11]

$\displaystyle\nu_{e}^{\alpha}(x)$ $\displaystyle=$ $\displaystyle
G^{-1}_{\theta}(t)\nu_{1}^{\alpha}(x)G_{\theta}(t)\,,$ (93)
$\displaystyle\nu_{\mu}^{\alpha}(x)$ $\displaystyle=$ $\displaystyle
G^{-1}_{\theta}(t)\nu_{2}^{\alpha}(x)\;G_{\theta}(t)$ (94)

where the  _mixing generator_ reads

$\displaystyle G_{\theta}(t)$ $\displaystyle=$
$\displaystyle\exp[\theta\left(S_{+}(t)-S_{-}(t)\right)]\,,$ (95)
$\displaystyle S_{+}(t)$ $\displaystyle\equiv$ $\displaystyle\int d^{3}{\bf
x}\,\nu_{1}^{{\dagger}}(x)\,\nu_{2}(x)\,,\qquad S_{-}(t)\ \equiv\ \int
d^{3}{\bf x}\,\nu_{2}^{{\dagger}}(x)\,\nu_{1}(x)\,.$ (96)

In fact, from the above equations we get, e.g., for $\nu_{e}$

$\frac{d^{2}}{d\theta^{2}}\,\nu^{\alpha}_{e}\,=\,-\nu^{\alpha}_{e}\,,$ (97)

with the initial conditions

$\left.\nu^{\alpha}_{e}\right|_{\theta=0}=\nu^{\alpha}_{1}\,,\quad\,\,\,\,\left.\frac{d}{d\theta}\nu^{\alpha}_{e}\right|_{\theta=0}=\nu^{\alpha}_{2}\,.$
(98)

The key point is that the vacuum $|0\rangle_{{}_{1,2}}$ is not invariant under
the action of the mixing generator $G_{\theta}(t)$. In fact, one has

$|0(t)\rangle_{e,\mu}\equiv
G^{-1}_{\theta}(t)\;|0\rangle_{{}_{1,2}}=e^{-\theta\left(S_{+}(t)-S_{-}(t)\right)}\,|0\rangle_{{}_{1,2}}\,.$
(99)

The state (99) is known as _flavor vacuum_ because it is annihilated by the
flavor ladder operators $\alpha_{\sigma}(t)$ and $\beta_{\sigma}(t)$ defined
by

$\alpha_{e}(t)|0(t)\rangle_{e,\mu}\ \equiv\
G^{-1}_{\theta}(t)\alpha_{1}{G_{\theta}(t)\;G^{-1}_{\theta}(t)}|0\rangle_{{}_{1,2}}\
=\ 0\,,$ (100)

and similarly for $\beta_{\sigma}(t)$. Their explicit form is

$\displaystyle\alpha^{r}_{{\bf k},e}(t)$ $\displaystyle=$
$\displaystyle\cos\theta\,\alpha^{r}_{{\bf k},1}+\sin\theta\left(U_{{\bf
k}}^{*}(t)\,\alpha^{r}_{{\bf k},2}+\epsilon^{r}V_{{\bf
k}}(t)\,\beta^{r{\dagger}}_{-{\bf k},2}\right)\,,$ (101)
$\displaystyle\alpha^{r}_{{\bf k},\mu}(t)$ $\displaystyle=$
$\displaystyle\cos\theta\,\alpha^{r}_{{\bf k},2}-\sin\theta\left(U_{{\bf
k}}(t)\,\alpha^{r}_{{\bf k},1}-\epsilon^{r}V_{{\bf
k}}(t)\,\beta^{r{\dagger}}_{-{\bf k},1}\right)\,,$ (102)
$\displaystyle\\!\\!\beta^{r}_{-{\bf k},e}(t)$ $\displaystyle=$
$\displaystyle\cos\theta\,\beta^{r}_{-{\bf k},1}+\sin\theta\left(U_{{\bf
k}}^{*}(t)\,\beta^{r}_{-{\bf k},2}-\epsilon^{r}V_{{\bf
k}}(t)\,\alpha^{r{\dagger}}_{{\bf k},2}\right)\,,$ (103)
$\displaystyle\\!\\!\beta^{r}_{-{\bf k},\mu}(t)$ $\displaystyle=$
$\displaystyle\cos\theta\,\beta^{r}_{-{\bf k},2}-\sin\theta\left(U_{\bf
k}(t)\,\beta^{r}_{-{\bf k},1}+\epsilon^{r}V_{{\bf
k}}(t)\,\alpha^{r{\dagger}}_{{\bf k},1}\right)\,.$ (104)

In these equations we have defined $\epsilon^{r}\equiv(-1)^{r}$, while $U_{\bf
k}$ and $V_{\bf k}$ are the _Bogoliubov coefficients_

$\displaystyle U_{{\bf k}}(t)$ $\displaystyle\equiv$ $\displaystyle
u^{r{\dagger}}_{{\bf k},2}u^{r}_{{\bf
k},1}\;e^{i(\omega_{\textbf{k},2}-\omega_{\textbf{k},1})t}\ =\
|U_{\textbf{k}}|\,e^{i(\omega_{\textbf{k},2}-\omega_{\textbf{k},1})t}\,\,,$
(105) $\displaystyle V_{{\bf k}}(t)$ $\displaystyle\equiv$
$\displaystyle\epsilon^{r}\;u^{r{\dagger}}_{{\bf k},1}v^{r}_{-{\bf
k},2}\;e^{i(\omega_{\textbf{k},2}+\omega_{\textbf{k},1})t}\ =\
|V_{\textbf{k}}|\,e^{i(\omega_{\textbf{k},2}+\omega_{\textbf{k},1})t}\,,$
(106)

where the time-independent part of the coefficients is given by

$\displaystyle|U_{\textbf{k}}|$ $\displaystyle\equiv$ $\displaystyle
u^{r{\dagger}}_{{\bf k},2}\,u^{r}_{{\bf k},1}\ =\ v^{r{\dagger}}_{-{\bf
k},1}\,v^{r}_{-{\bf k},2}$ (107) $\displaystyle=$
$\displaystyle\left(\frac{\omega_{\textbf{k},1}+m_{1}}{2\omega_{\textbf{k},1}}\right)^{\frac{1}{2}}\left(\frac{\omega_{\textbf{k},2}+m_{2}}{2\omega_{\textbf{k},2}}\right)^{\frac{1}{2}}\left(1+\frac{{\bf
k}^{2}}{(\omega_{\textbf{k},1}+m_{1})(\omega_{\textbf{k},2}+m_{2})}\right)\,.$
$\displaystyle|V_{\textbf{k}}|$ $\displaystyle=$
$\displaystyle\epsilon^{r}\;u^{r{\dagger}}_{{\bf k},1}\,v^{r}_{-{\bf k},2}\ =\
-\epsilon^{r}\,u^{r{\dagger}}_{{\bf k},2}\,v^{r}_{-{\bf k},1}$ (108)
$\displaystyle=$
$\displaystyle\frac{|\textbf{k}|}{\sqrt{4\omega_{\textbf{k},1}\omega_{\textbf{k},1}}}\left(\sqrt{\frac{\omega_{\textbf{k},2}+m_{2}}{\omega_{\textbf{k},1}+m_{1}}}-\sqrt{\frac{\omega_{\textbf{k},1}+m_{1}}{\omega_{\textbf{k},2}+m_{2}}}\right)\,.$

Notice that $|U_{\bf k}|^{2}+|V_{\bf k}|^{2}=1$. It is straightforward to
check that in the relativistic limit
$\omega_{\textbf{k},j}\approx|\textbf{k}|$, $|U_{\textbf{k}}|\rightarrow 1$
and $|V_{{\bf k}}|\rightarrow 0$. Also, $|V_{{\bf k}}|=0$ when $m_{1}=m_{2}$
and/or $\theta=0$, i.e. when no mixing occurs. $|V_{\bf k}|^{2}$ has the
maximum at $|\textbf{k}|=\sqrt{m_{1}m_{2}}$ with $|V_{\textbf{k}}|^{2}_{max}\
\rightarrow\ 1/2$ for $\frac{(m_{2}-m_{1})^{2}}{m_{1}m_{2}}\rightarrow\infty$,
and $|V_{{\bf k}}|^{2}\simeq\frac{(m_{2}-m_{1})^{2}}{4|\textbf{k}|^{2}}$ for
$|\textbf{k}|\gg\sqrt{m_{1}m_{2}}$.

The flavor fields can be thus expanded as

$\displaystyle\nu_{e}(x)$ $\displaystyle=$ $\displaystyle\sum_{{\bf
k},r}\frac{e^{i{\bf k}\cdot{\bf x}}}{\sqrt{V}}\left[u^{r}_{{\bf
k},1}(t)\,\alpha^{r}_{{\bf k},e}(t)\,+\,v^{r}_{-{\bf
k},1}(t)\,\beta^{r{\dagger}}_{-{\bf k},e}(t)\right]\,,$ (109)
$\displaystyle\nu_{\mu}(x)$ $\displaystyle=$ $\displaystyle\sum_{{\bf
k},r}\frac{e^{i{\bf k}\cdot{\bf x}}}{\sqrt{V}}\left[u^{r}_{{\bf
k},2}(t)\,\alpha^{r}_{{\bf k},\mu}(t)\,+\,v^{r}_{-{\bf
k},2}(t)\,\beta^{r{\dagger}}_{-{\bf k},\mu}(t)\right]\,,$ (110)

where $\sum_{{\bf k},r}$ is a simplified notation for the superposition over
all field modes of momentum k and polarization $r$.

A _flavor Hilbert space_ (at some reference time, say $t=0$) is defined as

${\cal
H}_{e,\mu}=\left\\{\alpha_{e,\mu}^{{\dagger}}\;,\;\beta_{e,\mu}^{{\dagger}}\;,\;|0\rangle_{e,\mu}\right\\},$
(111)

with $|0\rangle_{e,\mu}\equiv|0(t=0)\rangle_{e,\mu}$. This is a different
Hilbert space with respect to the one of mass-neutrino states. In fact, one
can verify that

$\lim_{V\rightarrow\infty}\,{}_{{}_{1,2}}\langle
0|0(t)\rangle_{e,\mu}=\lim_{V\rightarrow\infty}\,e^{\\!V\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\,\ln\,\left(1-\sin^{2}\theta\,|V_{\bf
k}|^{2}\right)^{2}}=0\,,$ (112)

i.e. flavor and massive fields belong to unitarily inequivalent
representations of the anticommutation relations.

The previous discussions suggest that flavor states
$|\nu^{r}_{\textbf{k},\sigma}\rangle\ $ can be built as one particle states of
the flavor Fock space

$|\nu^{r}_{\textbf{k},\sigma}\rangle\ =\
\alpha^{r{\dagger}}_{\textbf{k},\sigma}|0\rangle_{e,\mu}\,,$ (113)

and similarly for the antineutrino
($|\beta^{r}_{\textbf{k},\sigma}\rangle\equiv\beta^{r{\dagger}}_{\textbf{k},\sigma}|0\rangle_{e,\mu}$).
One can prove that these states are exact eigenstates of the charge operators
at the reference (production/detection) time, i.e.

$Q_{\nu_{\sigma}}(0)|\nu^{r}_{\textbf{k},\sigma}\rangle\ =\
|\nu^{r}_{\textbf{k},\sigma}\rangle\,.$ (114)

In this approach the flavor oscillation probability is computed by taking the
expectation value of the time-dependent flavor charges with respect to a
reference time flavor state [76]

$\mathcal{Q}_{\sigma\rightarrow\rho}(t)\ =\ \langle
Q_{\nu_{\rho}}(t)\rangle_{\sigma}\,,$ (115)

where
$\langle\cdots\rangle_{\sigma}\equiv\langle\nu^{r}_{\textbf{k},\sigma}|\cdots|\nu^{r}_{\textbf{k},\sigma}\rangle$,
which gives

$\displaystyle\mbox{}\mathcal{Q}_{\sigma\rightarrow\rho}(t)=\sin^{2}(2\theta)\Big{[}|U_{\textbf{k}}|^{2}\sin^{2}\left(\frac{\Omega_{\textbf{k}}^{{}_{-}}}{2}t\right)+|V_{\textbf{k}}|^{2}\sin^{2}\left(\frac{\Omega_{\textbf{k}}^{{}_{+}}}{2}t\right)\Big{]},\quad\sigma\neq\rho\,,$
(116) $\displaystyle\mbox{}\mathcal{Q}_{\sigma\rightarrow\sigma}(t)=1\ -\
\mathcal{Q}_{\sigma\rightarrow\rho}(t)\,,\quad\sigma\neq\rho\,.$ (117)

Notice the presence of the term proportional to $|V_{\textbf{k}}|^{2}$ in the
oscillation probability Eq. (116), which introduces fast oscillations that are
not present in the usual QM formula (see Eq. (77)). As already mentioned,
$|V_{\textbf{k}}|^{2}\rightarrow 0$ in the relativistic limit $|{\bf k}|\gg
m_{j}$, $j=1,2$, and the oscillation formula reduces to the standard result,
as it should be. It has been proven, in the simple case of scalar field
mixing, that the above oscillation formula is the time component of a Lorentz-
covariant formula, though the flavor vacuum breaks the Lorentz invariance
[77]. Furthermore, connections between implications of the QFT treatment of
mixing and extended (Tsallis-like) statistics have been explored in [78, 79].

###  IV.3 Time–energy uncertainty relation for neutrino oscillations in QFT

It is clear that in the QFT treatment lepton charges are natural candidates as
clock observables. In fact, starting from

$\left[Q_{\nu_{\sigma}}(t)\ ,\,H\right]\ =\
i\,\frac{\mathrm{d}Q_{\nu_{\sigma}}(t)}{\mathrm{d}t}\ \neq\ 0\,,$ (118)

we find the flavor–energy uncertainty relation

$\sigma_{H}\,\sigma_{Q}\ \geq\
\frac{1}{2}\left|\frac{\mathrm{d}\mathcal{Q}_{\sigma\rightarrow\sigma}(t)}{\mathrm{d}t}\right|.$
(119)

Proceeding as in the Pontecorvo case, one finds

$\left|\frac{\mathrm{d}\mathcal{Q}_{\sigma\rightarrow\sigma}(t)}{\mathrm{d}t}\right|\
\leq\ \Delta E\,.$ (120)

From Eq. (119) we get the simple form for TEUR

$\Delta E\,T\ \geq\
\mathcal{Q}_{\sigma\rightarrow\rho}(T)\,,\quad\sigma\neq\rho.$ (121)

When $m_{i}/|\textbf{k}|\rightarrow 0$, i.e. in the relativistic case, we get

$\displaystyle|U_{\textbf{k}}|^{2}\approx 1\ -\
\varepsilon(\textbf{k})\,,\;\;\;\;\;|V_{\textbf{k}}|^{2}\approx\varepsilon(\textbf{k})\,,$
(122)

with $\varepsilon(\textbf{k})\equiv{(m_{1}-m_{2})^{2}}/{4|\textbf{k}|^{2}}$.
In the same limit

$\Omega_{\textbf{k}}^{{}_{-}}\ \approx\ \frac{\delta m^{2}}{4|\textbf{k}|}\ =\
\frac{\pi}{L^{osc}}\,,\qquad\Omega_{\textbf{k}}^{{}_{+}}\ \approx\
|\textbf{k}|\,.$ (123)

Evaluating the inequality (121) at the leading order and for $T\approx
L=L^{osc}/2$, one finds

$\Delta E\ \geq\ \frac{2\sin^{2}(2\theta)}{L^{osc}}\,.$ (124)

Then, in such limit we recover the result (45). As we said above, such
neutrino TEUR is usually regarded as a condition for flavor oscillations to
occur [80, 5]. In other words, if one managed to measure with a great accuracy
the neutrino energy/masses, it could be inferred which massive neutrino was
produced in the weak interaction, and the oscillation would not occur. This
reasoning is based on the idea that flavor neutrinos are just a superposition
of the ‘’physical” massive neutrinos. However, the QFT approach bring us to a
different interpretation of the inequality (124). In fact, note that Eq. (112)
implies that

$\lim_{V\rightarrow\infty}\langle\nu^{r}_{\textbf{k},i}|\nu^{r}_{\textbf{k},\sigma}\rangle\
=\ 0\,,\qquad i=1,2\,,$ (125)

i.e. neutrino flavor eigenstates, which are produced in charged current weak
decays, _cannot_ be generally written as a linear superposition of single-
particle massive neutrino states. However, for Pontecorvo states (76), which
are a good approximation of the exact flavor states in the relativistic
regime, it holds

$\lim_{V\rightarrow\infty}\langle\nu^{r}_{\textbf{k},1}|\nu^{r}_{\textbf{k},e}\rangle_{P}\
=\ \cos\theta\,.$ (126)

This apparent contradiction is resolved by observing that

$\lim_{m_{i}/|\textbf{k}|\rightarrow 0}\,\,\lim_{V\rightarrow\infty}\ \neq\
\lim_{V\rightarrow\infty}\,\,\lim_{m_{i}/|\textbf{k}|\rightarrow 0}\,,$ (127)

i.e. the relativistic ${m_{i}/|\textbf{k}|\rightarrow 0}$ limit cannot be
exchanged with the “thermodynamical” limit. The relativistic approximation has
to be considered just as a single-particle approximation, which does not take
into account the intrinsic multi-particle nature of QFT. The relation (125)
should be thus understood as

$\langle\nu^{r}_{\textbf{k},i}|\nu^{r}_{\textbf{k},\sigma}\rangle\\!=\\!\
{}_{1,2}\langle
0_{\textbf{k}}|\alpha^{r}_{\textbf{k},1}\alpha^{r{\dagger}}_{\textbf{k},e}|0_{\textbf{k}}\rangle_{e,\mu}\prod_{\textbf{p}\neq\textbf{k}}\\!\\!{}_{1,2}\langle
0_{\textbf{p}}|0_{\textbf{p}}\rangle_{e,\mu}\,,$ (128)

where we have used the fact that the Hilbert spaces for both massive and
flavor fields have a tensor product structure [81]. The first factor on the
r.h.s. corresponds to (126), and it is finite. However, as said above, this
corresponds to a selection of a single particle sub-space from the complete
Hilbert space. In other words, beyond the QM single particle view, the
Pontecorvo definition of neutrino state does not work anymore. Then, the
inequality (124) should be regarded as a fundamental bound on the energy
precision which can be reached experimentally: as in the case of unstable
particles, flavor neutrinos have an intrinsic energy spread which is related
with their ‘’life-time”, i.e. $L^{osc}$.

Let us now consider the exact oscillation formula (116) at the first order in
$\varepsilon(\textbf{k})$:

$\displaystyle\mathcal{Q}_{\sigma\rightarrow\rho}(t)\approx\sin^{2}(2\theta)\Big{[}\sin^{2}\left(\frac{\pi
t}{L^{osc}}\right)\left(1-\varepsilon(\textbf{k})\right)+\varepsilon(\textbf{k})\sin^{2}\left(|\textbf{k}|t\right)\Big{]}\,,\quad\sigma\neq\rho\,.$
(129)

Evaluating the inequality (121) at $~{}T=L^{osc}/2$ we get

$\Delta E\ \geq\
\frac{2\,\sin^{2}2\theta}{L^{osc}}\,\left[1-\varepsilon(\textbf{k})\,\cos^{2}\left(\frac{|\textbf{k}|L^{osc}}{2}\right)\right]\,.$
(130)

We thus find that the energy bound is lowered with respect to (124). If we set
$m_{1}=0.0497\,{\rm eV}$, $m_{2}=0.0504\,{\rm eV}$ for neutrino masses in the
inverted hierarchy [82] and $|\textbf{k}|=1\,{\rm MeV}$, we obtain
$\varepsilon(\textbf{k})=2\times 10^{-19}$, which reveals that such correction
is negligible in the relativistic regime and Eq. (130) can be naively
approximated by the quantum mechanical version of TEUR.

The situation changes if one looks at the non-relativistic regime, e.g.,
$|\textbf{k}|=\sqrt{m_{1}m_{2}}$. In such case,

$\displaystyle|U_{\textbf{k}}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ +\ \frac{\xi}{2}\ =\ 1-|V_{\textbf{k}}|^{2}\,,$
(131) $\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{m_{1}m_{2}}}{m_{1}+m_{2}}\,,$ (132)

and we can rewrite the TEUR (121) in the form

$\Delta
E\,T\geq\frac{\sin^{2}2\theta}{2}\,\left[1-\,\cos\left(\tilde{\omega}_{1}T\right)\cos\left(\tilde{\omega}_{2}T\right)-\,\xi\sin\left(\tilde{\omega}_{1}T\right)\sin\left(\tilde{\omega}_{2}T\right)\right]\,,$
(133)

where $\tilde{\omega}_{j}=\sqrt{m_{j}(m_{1}+m_{2})}$. In order to compare it
with the previous cases, we evaluate such expression at $T=\tilde{L}^{osc}/4$,
with $\tilde{L}_{osc}=4\pi\sqrt{m_{1}m_{2}}/\delta m^{2}$, obtaining

$\Delta E\geq\frac{2\sin^{2}2\theta}{\tilde{L}^{osc}}\ \left(1-\chi\right)\,.$
(134)

Here we have introduced the shorthand notation

$\chi\ =\
\xi\,\sin\left(\tilde{\omega}_{1}\tilde{L}_{osc}/4\right)\sin\left(\tilde{\omega}_{2}\tilde{L}_{osc}/4\right)\
+\
\cos\left(\tilde{\omega}_{1}\tilde{L}_{osc}/4\right)\cos\left(\tilde{\omega}_{2}\tilde{L}_{osc}/4\right)\,.$
(135)

Using the same values for neutrino masses as above, we estimate $\chi=0.1$,
which implies that the original bound on the neutrino energy is now decreased
by $10\%$.

##  V Conclusion and Discussion

We have discussed recent advances in the study of TEUR for neutrino flavor
oscillations. In particular, we have investigated how the the original
inequality Eq. (6) derived by Bilenky gets changed in the presence of gravity
and in a field theoretical picture, respectively. In the first case and for
some specific background metrics, it has been shown that gravitational
corrections can be rearranged so as to leave the form of the generalized TEUR
unchanged, provided one defines an effective (gravity-dependent) oscillation
length (see Sec. III). On the other hand, the QFT formulation of TEUR relies
upon the identification of the (non-conserved) flavor charge operator with the
clock observable. In the latter context, we have argued that the
interpretation of neutrinos as “unstable” particles does naturally emerge, the
life-time being related to the characteristic oscillation length (see the
discussion below Eq. (128)). In passing, we mention that the analogy of
oscillating neutrinos with unstable particles has been pointed out also in the
recent [83], where the oscillation formula (116) has been independently
obtained by considering mixing as an interaction and employing the usual
perturbation expansion in the Dirac picture. In that work, the role of TEUR is
fundamental because it requires the use of finite-time QFT in order to study
flavor oscillations, which would be otherwise spoiled in the $S$-matrix
formalism ($t\to\infty$).

Further aspects are yet to be explored. In the recent analysis of [34, 84]
neutrino oscillations have been addressed in extended theories of gravity as a
testing ground for the violation of the strong equivalence principle (SEP).
Potential violation effects manifest themselves in the form of a generalized
oscillation length depending on the SEP parameter. By plugging into the
gravity-modified TEUR, one could exploit this result to delve deeper into the
interplay between neutrino oscillations and SEP, and possibly constrain SEP
corrections. Moreover, a link between cosmic scale phenomena and the QFT
properties of field mixing has been analyzed in [85] in connection with the
suggestive interpretation of the flavor vacuum condensate as a dark energy
candidate (see also [86, 87, 88, 89] for other similar dark energy models). It
would be interesting to extend our study to this context in the effort to
improve the current bounds on neutrino masses through dark energy constraints.
On the other hand, growing interest is being aroused by the study of quantum
correlations and quantum coherence in neutrino oscillations [71, 90, 91, 92,
93, 94]. Specifically, in [92] the complete complementarity relations have
been applied to neutrino flavor oscillations to fully characterize the
quantumness of such a phenomenon. In line with previous studies, the result
has been found that quantum correlations still survive after the complete
spatial separation of the wave packets composing a flavor state, revealing
that the quantum nature of mixed neutrinos goes beyond the pure flavor
oscillations. It is our intent to export TEUR paradigm to the quantum-
information theoretic analysis of oscillations to explore the time non-
classicality of this phenomenon in a relativistic domain. Preliminary results
along this direction appear in [75]. These and some other research lines are
under active investigation and will be presented elsewhere.

###### Acknowledgements.

GGL acknowledges the Spanish “Ministerio de Universidades” for the awarded
Maria Zambrano fellowship and funding received from the European Union -
NextGenerationEU. He is also grateful for participation to LISA cosmology
Working group. GGL and GL acknowledge networking support from the COST Action
CA18108 “Quantum Gravity Phenomenology in the Multimessenger Approach”.

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