# Exploring Weak measurements within the Einstein-Dirac Cosmological framework

Williams Dhelonga-Biarufu<EMAIL_ADDRESS>Dominique Lambert
<EMAIL_ADDRESS>Namur Insitute for Complex Systems (naXys),
Department of Mathematics, University of Namur, Rue de Bruxelles 61, B-5000
Namur, Belgium

###### Abstract

Our study applies the _Two-State Formalism_ alongside weak measurements within
a spatially homogeneous and isotropic cosmological framework, wherein Dirac
spinors are intricately coupled to classical gravity. To elucidate this, we
provide detailed formulations for computing the weak values of the energy-
momentum tensors, the Z component of spin, and the characterization of pure
states. Weak measurements appear to be a generalization and extension of the
computation already made by Finster an Hainzl, [1], in _A spatially
homogeneous and isotropic Einstein-Dirac cosmology_. Our analysis reveals that
the acceleration of the Universe expansion can be understood as an outcome of
postselection, underscoring the effectiveness of weak measurement as a
discerning approach for gauging cosmic acceleration.

General Relativity, Quantum Mechanic, weak measurements, Geometric phase

††preprint: APS/123-QED

##  I INTRODUCTION

Measuring involves assigning numerical values to the attributes or
characteristics of a phenomenon. The impact of the observer on measurement
results is a complex consideration in scientific research. In quantum
mechanics, the alteration of a quantum state after its measurement is well
known, but its interpretation is not obvious. For example, the famous double-
slit experiment demonstrates that particles like electrons can behave as
particles and waves, and measurement prescribes their behavior [2, 3].

Weak measurements offer a procedure to measure the system state that minimizes
its disturbance with minimal perturbation, achieved through a weak coupling
between the measurement apparatus and the system’s state. Notably, these
measurements enable the characterization of a particle’s trajectory in the
double-slit experiment without disrupting the interference pattern, as
demonstrated in previous studies [3, 4, 5, 6]. Unlike classical or ideal
measurements, which result in the stochastic collapse of the system state into
one of the eigenstates of the measured observable [7], weak measurements do
not induce a collapse of the state vector. Instead, they introduce a small
angular bias to the state vector, and the measurement device exhibits a
superposition of multiple values rather than a clear eigenvalue [4].
Consequently, weak measurements unveil unconventional weak values, including
complex numbers.

In literature, some authors have explored weak measurement and _Two-state
vectors formalism_ within the theoretical framework of cosmology and ontology.
George Ellis and Rotman [8] proposed a paradigm that challenges conventional
notions of time and reality, termed the _Crystallizing Block Universe (CBU)_ ,
an extension of the _Emergent Block Universe (EBU)_. According to their
perspective, past, present, and future coalesce, amalgamating space-time into
a singular entity. The future is conceptualized as a superposition of myriad
possibilities, while the past remains immutable. The transition between these
temporal states predominantly unfolds in the present, albeit in a non-uniform
manner. Within the CBU framework, certain discrete patches of quantum
indeterminacy endure and are resolved later on. Furthermore, CBU models
integrate the _Two-time interpretation_ [9] of quantum mechanics.
Nevertheless, this theoretical and ontological paradigm lies beyond the
purview of our current study.

Davies [10] is the first to propose applying weak measurements theory combined
with pre-and-post-selection to quantum cosmology and to explore the potential
large-scale cosmological effects arising from this new sector of quantum
mechanics. He illustrated the theory with a two-spacetime dimensional toy
model of a scalar field with mass $m$ propagating in an expanding Universe
with the scale factor $a(t)$ and metric $ds^{2}=dt^{2}-a(t)dx^{2}$. He
resolved a debate regarding the coupling of pre- and postselected quantum
fields to the gravity and proposed an experimental test for weak values in
cosmology. He observed that in the literature, in equation
$G_{\mu\nu}+\text{higher order terms in curvature }=\frac{\langle
0_{fin}|T_{\mu\nu}|0_{in}\rangle}{\langle 0_{fin}|0_{in}\rangle}$ originally
derived by DeWitt, who adapted the Schwinger effective action theory of
quantum electrodynamics to the gravitational case, the source term is nothing
else than the weak values of the stress-energy-momentum tensor $T_{\mu\nu}$,
[11]. This equation was widespread in 1970, [12], [13], [14].

After Davies’s paper [10], no research papers have been pursued in the
direction of the theory of weak measurements in Cosmology until now. The
subject seems to arouse interest. Charis Anastopoulos has just published on
ar$\chi$iv a paper on _Final States in Quantum Cosmology: Cosmic Acceleration
as a Quantum Postselection Effect_ , [15] wherein he argues that there is no
compelling physical reason to preclude a probability assignment with a final
quantum state at the cosmological level and analyses its implications in
quantum cosmology. One significant result is that cosmic acceleration emerges
as a quantum postselection effect.

We aim to examine weak measurements of Dirac particles within the framework of
time symmetry as applied to the Einstein-Dirac system in a homogeneous and
isotropic space such as the Friedman-Lemaître-Robertson-Walker( FLRW) space.
To the best of our knowledge, no prior research has explored this direction.
Drawing upon insights from the paper of Finster and Hainzl, titled _A
Spatially Homogeneous and Isotropic Einstein-Dirac Cosmology_ [1], we endeavor
to extend certain findings to the domain of weak measurements. Among the
outcomes of our investigation are the computation of weak values, including
those pertaining to energy-momentum, pure states, and the Z component of spin.
Additionally, we demonstrate that the epochs of accelerated expansion in the
Universe may be regarded as consequences of postselection. This corroborates
earlier studies that have explored the potential of spinor fields to elucidate
phenomena such as the inflationary period in the early universe and,
subsequently, the concept of dark energy. Notably, Anastopoulos arrived at a
similar conclusion, albeit without invoking a spinor field or dark energy,
thus emphasizing alternative avenues for understanding the observed
cosmological acceleration. Ribas reached the same conclusion without resorting
to weak measurements or postselection techniques. References to relevant
literature supporting these assertions include [16, 17, 18, 19, 20, 21].

In the time-symmetric formulation of quantum mechanics [22], a quantum system
is not characterized by a single state vector but by Two-State vectors at a
specific time. The preselected state is determined by measurements conducted
on the system at times prior to $t_{0}$, while the postselected state by
measurements at times $t>t_{0}$. The postselected state is conceptualized as a
quantum state evolving backward in time. Weak measurements occur in the
interval between these conventional measurements through a weak coupling
between the measurement apparatus and the system [4].

Formally, if $\hat{A}$ is a Hermitian operator on the system $S$,
$|\psi_{in}\rangle$ and $|\psi_{out}\rangle$ (or $|\psi_{fin}\rangle$) are the
preselected and postselected state vectors in the Hilbert space of $S$, and
$|\phi(x)\rangle$ is the state vector of the needle of the measurement device,
$g(t)$ a coupling impulse such that $\int_{0}^{T}g(t)dt=1$, $T$ the coupling
time and $\hat{P}_{d}$ the momentum conjugated operator to $\hat{Q}_{d}$ the
position operator, then weakly measuring an ensemble of preselected vectors
states $|\psi_{in}\rangle$ with $\hat{H}=g(t)\hat{A}\otimes\hat{P}_{d}$ the
interaction Hamiltonian yields the following amplitude to get the postselected
state vector: $\langle\psi_{fin}|e^{(-i\hat{H}T)}|\psi_{in}\rangle$. In
detail, this is obtained as follows: let
$\hat{P}=|\psi_{fin}\rangle\langle\psi_{fin}|\otimes\hat{I}$ be an operator
which projects onto the postselected state $|\psi_{fin}\rangle$, and
$|\psi_{w}\rangle=e^{(-i\hat{H}T)}|\psi_{in}\rangle\otimes|\phi(x)\rangle$ the
entangled state of the system with the collapsed state of the measurement
device [4],[23]. Then we get

$\displaystyle\hat{P}|\psi_{w}\rangle$
$\displaystyle=|\psi_{fin}\rangle\langle\psi_{fin}|e^{(-i\hat{H}T)}|\psi_{in}\rangle\otimes|\phi(x)\rangle$

Assuming that $\hat{P}_{d}$ is distributed around 0 with low variance, its
conjugate variable $\hat{Q}_{d}$ will have high variance, and the measurement
will be weak. Hence, we shall get:

$\displaystyle|\psi_{fin}\rangle\langle\psi_{fin}|(1-i\hat{A}\otimes\hat{P}_{d}\cdot
T)|\psi_{in}\rangle\otimes|\phi(x)\rangle$
$\displaystyle=|\psi_{fin}\rangle\otimes\langle\psi_{fin}|\psi_{in}\rangle
e^{-i\left(\langle\hat{A}\rangle_{w}\hat{P}_{d}T\right)}$

where
$\langle\hat{A}\rangle_{w}=\frac{\langle\psi_{fin}|\hat{A}|\psi_{in}\rangle}{\langle\psi_{fin}|\psi_{in}\rangle}$
is the so-called weak measurement of the observable $\hat{A}$. The
probabilities to get this result is
$|\langle\psi_{fin}|\psi_{in}\rangle|^{2}$.

Weak values are weak measurement results and must be understood as statistical
averages. They are also members of a decomposition of eigenvalues weighted by
the relative probabilities of finding the out states among the ensemble of
states vectors identically prepared in $|\psi_{in}\rangle$. Weak measurement
should then be understood as a conditional probability where one of the events
undergoes a perturbation. Dressel [24] clarifies this probabilistic
understanding of weak values. He shows that weak values characterize the
relative correction to a detection probability
$|\langle\psi_{fin}|\psi_{in}\rangle|^{2}$ due to a small intermediate
perturbation $\hat{U}(\epsilon)$ that results in a modified detection
probability $|\langle\psi_{fin}|\hat{U}(\epsilon)\psi_{in}\rangle|^{2}$.
Alternatively, in a simple way, he defines weak values as complex numbers that
one can assign to the powers of a quantum observable operator $\hat{A}$ using
two states: an initial state $|\psi_{in}\rangle$ and a final state
$|\psi_{fin}\rangle$. Weak values might lie outside the spectrum of
eigenvalues and can be complex numbers. One interesting phenomenon is the
amplification obtained when the preselected state and the postselected one are
almost orthogonal [4, 10].

In cosmology [10], all observations and measurements are considered weak in
the quantum sense due to the nature of the processes involved. For instance,
when observing the redshift of a galaxy, the measurement is conducted by
observing the light emitted from a large number of photons originating from
numerous sources within the galaxy. While the emission of a single photon from
an atom may not be considered weak from the atom’s perspective, the use of a
large ensemble of photons to measure a property of the entire galaxy
constitutes a weak measurement in the quantum sense. This is because the
quantum back-reaction of the photons on the relevant physical variable of the
entire galaxy, such as its momentum, is negligible. Therefore, the large-scale
nature of cosmological observations and measurements, involving statistical
averages over a large ensemble of photons, results in weak values in the
quantum sense.

In the laboratory, the Two-State vectors require human interventions. In
quantum cosmology, there are many proposals, as noticed by Davies: Hartle and
Hawking came up with the no-boundary wave function, which is the state of the
Universe before the Planck epoch [25]. In a semi-classical approach, in the
framework of the theory of quantum fields, the initial state is taken to be a
vacuum state. As for the final state, this can be anything. In weak
measurement, one will ensure that the final state is not orthogonal to the in-
state.

For the Einstein-Dirac in FLRW context, the preselected state will be the
spinor solution of the Einstein-Dirac equation in the limit of the massless
Universe, that is, the radiation Universe, whereas the final state will be the
spinor solution of the Einstein-Dirac equation for the dust Universe. The
implementation of weak measurement in this specific context is intended to
acquire geometrical information about the Universe. The subsequent
computational analysis is made feasible by the association between weak
measurement and the Berry phase [7]. This has never been explored before, as
far as we know.

Regarding the Einstein-Dirac system, it investigates the interaction between
particles with a spin of $1/2$ and the gravitational field. The Einstein field
equations essentially relate the geometry of spacetime to the distribution of
matter and energy within that spacetime. In simpler terms, they describe how
matter and energy, represented here by the stress tensor applied on spinors,
influence the curvature of spacetime and how the curvature of spacetime
influences the motion of matter and energy. Finster [1] investigated the
nonlinear coupling of gravity to matter in a time-dependent, spherically
symmetric Einstein-Dirac system. He showed that quantum oscillations of the
Dirac wave functions can prevent the formation of a big bang or big crunch
singularity.

Weak measurements present numerous advantages, foremost among them being the
capability to detect exceedingly subtle effects while minimizing disturbance
to the system state. Onur Hosten and Paul Kwiat [26] exemplified the utility
of weak measurement techniques in amplifying the spin Hall effect, Meng-Jun Hu
[27] discussed in his paper the proposal for Weak Measurements Amplification
based Laser Interferometer Gravitational-wave Observatory (WMA-LIGO) to detect
gravitational waves using weak measurements to amplify ultra-small phase
signals, while Dixon et al. [28] applied these methods to enhance the
detection of minute transverse deflections in an optical beam. This facet of
weak measurements proves highly advantageous in cosmology, enabling the
amplification and measurement of wave functions originating in the remote
past, close to singularities.

This paper is organized as follows: After the introduction, we give a summary
of the Einstein-Dirac system and then find the spinor solutions for the
radiation Universe and the dust one. In section III, we give the results of
our main computations on weak measurements on the Einstein-Dirac solutions.In
section IV, we give a conclusion. The appendices provide some details of our
computations.

##  II The Einstein-Dirac Solution in Friedmann-Lemaître-Robertson-Walker
Space

In this section, we provide an overview of the Einstein-Dirac solution within
the Friedmann-Lemaître-Robertson-Walker (FLRW) space framework. For a more
comprehensive treatment, readers are encouraged to refer to [1] and [29].

The Einstein-Dirac (ED) equations are given by

$R^{i}_{j}-\frac{1}{2}R\delta^{i}_{j}=8\pi\kappa
T^{i}_{j},\quad(\mathcal{D}-m)\Psi=0,$ (1)

where $\kappa$ represents the gravitational constant, $T^{i}_{j}$ the energy-
momentum tensor of the Dirac particles, $\mathcal{D}$ the Dirac operator, $m$
the mass of the Dirac particles, and $\Psi$ the Dirac wave function.

Our Einstein-Dirac equations are formulated within the FLRW space, a spacetime
manifold with a (1,3) signature. This simple cosmological model is
characterized by a metric described by the homogeneous and isotropic line
element

$ds^{2}=dt^{2}-R^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(2)

where $t\in\mathbb{R}$ represents time,
$(\theta,\phi)\in(0,\pi)\times[0,2\pi)$ are angular coordinates, $r$ is the
radial coordinate, $R(t)$ the scale function, $d\Omega^{2}$ the line element
on $S^{2}$, and $k$ can take values of -1, 0, or 1, corresponding to open,
flat, and closed universes, respectively. We adopt ”Planck units” where
$c=\hbar=G=1$.

In this article, we specifically consider the case of a closed universe with
$k=1$, although similar computations can be extended to other cases. In this
scenario, the line element simplifies to

$ds^{2}=dt^{2}-R^{2}(t)\left(\frac{dr^{2}}{1-r^{2}}+r^{2}d\Omega^{2}\right).$
(3)

In this case, the coordinate $r$ will vary in the interval $(0,1)$. After
conducting detailed calculations, outlined in the appendices of [1] and [29],
the Dirac operator takes the following form

$\left[i\gamma\left(\partial_{t}+\frac{3}{2}\frac{\dot{R}}{R}\right)-R(t)m+\begin{pmatrix}0&\mathcal{D}_{\mathcal{H}}\\\
-\mathcal{D}_{\mathcal{H}}&0\end{pmatrix}\right]\Psi=0,$ (4)

where $\mathcal{D}_{\mathcal{H}}$ represents the purely spatial operator on
$S^{3}$, given by

$\mathcal{D}_{\mathcal{H}}=i\sigma^{r}\left(\partial_{r}+\frac{f-1}{rf}\right)+i\sigma^{\theta}\partial_{\theta}+\sigma^{\phi}\partial_{\phi},$
(5)

where $\sigma^{r},\sigma^{\theta}\text{and}\sigma^{\phi}$ are linear
combinations of the Pauli matrixes defined as follows

$\displaystyle\left\\{\begin{array}[]{lll}\sigma^{r}:=f(r)\left(\cos{\theta}\sigma^{3}+\sin{\theta}\cos{\phi}\sigma^{1}+\sin{\theta}\sin{\phi}\sigma^{2}\right)\\\
\sigma^{\theta}:=\frac{1}{r}\left(-\sin{\theta}\sigma^{3}+\cos{\theta}\cos{\phi}\sigma^{1}+\cos{\theta}\sin{\phi}\sigma^{2}\right)\\\
\sigma^{\phi}:=\frac{1}{r\sin{\theta}}\left(-\sin{\phi}\sigma^{1}+\cos{\phi}\sigma^{2}\right).\end{array}\right.$
(9)

and the function

$\displaystyle\begin{array}[]{lll}f(r)&=&\sqrt{1-r^{2}},\hskip
28.45274ptr\in(0,1)\\\ \end{array}$ (11)

This spatial operator $\mathcal{D}_{\mathcal{H}}$ has a purely discrete
spectrum

$\sigma(\mathcal{D}_{\mathcal{H}})=\left\\{\pm\frac{3}{2},\pm\frac{5}{2},\pm\frac{7}{2},\ldots\right\\},$
(12)

and the dimension of the corresponding eigenspace is given as:

$\text{dim}\;\text{ker}\left(\mathcal{D}_{\mathcal{H}}-\lambda\right)=\lambda^{2}-\frac{1}{4}.$
(13)

An orthonormal eigenvector basis in terms of spherical harmonics and Jacobi
polynomials is provided in the appendix of [29] and is denoted by
$\psi^{\pm}_{njk}$, where $n\in\mathbb{N}_{0}$,
$j\in\mathbb{N}_{0}+\frac{1}{2}$, and $k\in\\{-j,-j+1,\ldots,j\\}$.

The spatial differential operator on the eigenvector can be represented as
follows

$\mathcal{D}_{\mathcal{H}}\psi^{\pm}_{njk}=\lambda\psi^{\pm}_{njk},\quad\text{where}\quad\lambda=\pm(n+j+1).$
(14)

We represent the normalized eigenfunction of $\mathcal{D}_{\mathcal{H}}$
related to the eigenvalue $\lambda$ by $\psi_{\lambda}\in L^{2}(S^{3})^{2}$.

To further analyze the system, we employ a separation ansatz for

$\Psi=\frac{1}{R(t)^{\frac{3}{2}}}\begin{pmatrix}\alpha(t)\psi_{\lambda}(r,\theta,\phi)\\\
\beta(t)\psi_{\lambda}(r,\theta,\phi)\end{pmatrix}.$ (15)

This ansatz enables the derivation of a coupled system of ordinary
differential equations (ODEs) for the complex-valued functions $\alpha(t)$ and
$\beta(t)$:

$i\frac{d}{dt}\begin{pmatrix}\alpha(t)\\\
\beta(t)\end{pmatrix}=\begin{pmatrix}m&-\frac{\lambda}{R(t)}\\\
-\frac{\lambda}{R(t)}&-m\end{pmatrix}\begin{pmatrix}\alpha(t)\\\
\beta(t)\end{pmatrix}.$ (16)

The spinors are normalized according to:

$|\alpha|^{2}+|\beta|^{2}=\lambda^{2}-\frac{1}{4}.$ (17)

These spinors enter the Einstein equation via the energy-momentum tensor of
the wave function, ensuring a coupling with the Dirac equation. The non-
vanishing components of the energy-momentum tensor are computed in Appendix B
of [1] and are given by

$\displaystyle T^{0}_{0}$
$\displaystyle=R^{-3}\left[m\left(|\alpha|^{2}-|\beta|^{2}\right)-\frac{2\lambda}{R}\text{Re}\left(\alpha\bar{\beta}\right)\right],$
(18) $\displaystyle T^{r}_{r}$
$\displaystyle=T^{\theta}_{\theta}=T^{\phi}_{\phi}=R^{-3}\frac{2\lambda}{3R}\text{Re}\left(\alpha\bar{\beta}\right).$
(19)

A short calculation for the Einstein tensor $G^{j}_{k}$, given in [1], yields

$\displaystyle G_{0}^{0}$
$\displaystyle=3\left(\frac{\dot{R}^{2}+1}{R^{2}}\right),$ (20) $\displaystyle
G_{r}^{r}$
$\displaystyle=G_{\theta}^{\theta}=G_{\phi}^{\phi}=2\frac{\ddot{R}}{R}+\frac{\dot{R}^{2}+1}{R^{2}}.$
(21)

Knowing that :

$G^{i}_{j}=8\pi\kappa T^{i}_{j},\quad\text{where}\quad
G^{i}_{j}=R^{i}_{j}-\frac{1}{2}R\delta^{i}_{j},$

we set $\kappa=\frac{3}{8\pi}$ for a convenient computation as in [1]. This
yields the following expression, the Einstein equation

$\dot{R}^{2}+1=R^{2}T^{0}_{0},$ (22)

which in terms of two level system is given as

$\dot{R}^{2}+1=\frac{m}{R}\left(|\alpha|^{2}-|\beta|^{2}\right)-\frac{2\lambda}{R^{2}}\text{Re}\left(\alpha\bar{\beta}\right).$
(23)

We derive also the acceleration of the Universe represented here by the second
derivative with respect to time of the scale function given firstly in terms
of the energy-momentum tensors as

$\ddot{R}=\frac{R}{2}\left(3T^{j}_{j}-T^{0}_{0}\right).$ (24)

Then in terms of spinors we shall have

$\ddot{R}=-\frac{m}{2R^{2}}\left(|\alpha|^{2}-|\beta|^{2}\right)+\frac{2\lambda}{R^{3}}\text{Re}\left(\alpha\bar{\beta}\right).$
(25)

One sees immediately that the acceleration of the Universe is caused by the
spinor field, and this is consistent along the lines of [16, 17, 18, 19, 20,
21]. We will later on show some behaviours of the acceleration of the scale
function.

At this stage, we are left with two differential equations, equations (16) and
(23), in which the spinor $\begin{pmatrix}\alpha\\\ \beta\end{pmatrix}$ is
considered as a two-level quantum state.

As suggested by Finster and Hainzl in [1], the Einstein-Dirac equation can be
rewritten in terms of a _Bloch vector_ $\vec{v}$, where

$\displaystyle\vec{v}$
$\displaystyle=\begin{pmatrix}\langle\xi|\sigma^{1}|\xi\rangle\\\
\langle\xi|\sigma^{2}|\xi\rangle\\\
\langle\xi|\sigma^{3}|\xi\rangle\end{pmatrix},$ (26) $\displaystyle\vec{b}$
$\displaystyle=2\frac{\lambda}{R}\vec{e}_{1}-2m\vec{e}_{3}.$ (27)

Here, $\xi$ represents the spinor $\begin{pmatrix}\alpha\\\
\beta\end{pmatrix}$, $\sigma^{i}$ are the Pauli matrices, and $\vec{e}_{i}$
are the standard basis vectors in $\mathbb{R}^{3}$ ($i\in\\{1,2,3\\}$).

The Einstein-Dirac equations can then be expressed as

$\dot{\vec{v}}=\vec{b}\wedge\vec{v},\quad\dot{R}^{2}+1=-\frac{1}{2R}\vec{b}\cdot\vec{v}.$
(28)

Here, ‘$\wedge$’ and ‘$\cdot$’ denote the cross and scalar product in
Euclidean space $\mathbb{R}^{3}$, respectively. We rewrite also the scale
acceleration in terms of the Bloch vector components as

$\ddot{R}=-\frac{m}{2R^{2}}v^{3}+\frac{\lambda}{R^{3}}v^{1}.$ (29)

To simplify the analysis, a rotation“U” around the $\vec{e}_{2}$-axis is
applied to $\vec{b}$, making $\vec{b}$ parallel to $\vec{e}_{1}$. This results
in a transformed vector $\vec{w}$

$\vec{w}=U\vec{v}=\begin{pmatrix}w^{1}\\\ w^{2}\\\ w^{3}\end{pmatrix}$ (30)

where the components of $\vec{w}$ are given by

$\displaystyle w^{1}$
$\displaystyle=\frac{1}{\sqrt{\lambda^{2}+m^{2}R^{2}}}\left(2\lambda\text{Re}\left(\alpha\bar{\beta}\right)-mR\left(|\alpha|^{2}-|\beta|^{2}\right)\right),$
(31) $\displaystyle w^{2}$
$\displaystyle=2\text{Im}\left(\alpha\bar{\beta}\right),$ (32) $\displaystyle
w^{3}$
$\displaystyle=\frac{1}{\sqrt{\lambda^{2}+m^{2}R^{2}}}\left(2m\text{Re}\left(\alpha\bar{\beta}\right)+\lambda\left(|\alpha|^{2}-|\beta|^{2}\right)\right).$
(33)

At this stage, the Einstein-Dirac equations reduce to a system of ODEs
involving the scale function $R(t)$ and the complex functions $\alpha(t)$ and
$\beta(t)$

$\dot{\vec{w}}=\vec{d}\wedge\vec{w},\quad\dot{R}^{2}+1=-\frac{1}{R^{2}}\sqrt{\lambda^{2}+m^{2}R^{2}}w^{1},$
(34)

where $\vec{d}$ is defined as

$\vec{d}:=\frac{2}{R}\sqrt{\lambda^{2}+m^{2}R^{2}}\vec{e}_{1}-\frac{\lambda
mR}{\lambda^{2}+m^{2}R^{2}}\frac{\dot{R}}{R}\vec{e}_{2}.$ (35)

The length of the Bloch vector is constant, and the normalization convention
is

$|\vec{w}|=\lambda^{2}-\frac{1}{4}=N.$ (36)

Within weak measurement and the _Two Vectors State Formalism_ this research
primarily explores solutions to the Einstein-Dirac equation under specific
conditions. These conditions involve both the mass parameter ($m$) and the
coupling constant ($\lambda$) approaching zero. In this regime, we distinguish
between preselected states and postselected states, the solutions of (34) for
respectively $m=0$ and $\lambda=0$.

In the case where $m=0$, the Einstein-Dirac equation simplifies to the well-
known FLRW equation, describing the Universe’s behaviour in the radiation-
dominated phase. Conversely, in the second scenario, where $\lambda=0$,
corresponds to a dust-dominated Universe. Notably, for sufficiently large
values of the scale factor $R$, the Universe exhibits classical behaviour
similar to the dust-dominated case.

The above assumption could be seen easily from the complete and classical
Friedmann general equation with $k=1,0,-1$ corresponding to closed, flat and
opened universe respectively

$\dot{R}^{2}+k=-\left(\frac{\rho_{m}}{R}+\frac{\rho_{r}}{R^{2}}-\Lambda
R^{2}\right),$ (37)

where $\rho_{m}$ is the density of matter, $\rho_{r}$ is the density of
radiation and $\Lambda$ is the vacuum energy. Just after the Big Bang, the
Universe was too small. For $R\ll 1$, whatever the terms in the right side of
the expression (37), the radiative one would dominate in the evolution of the
Universe. The expression becomes

$\dot{R}^{2}+k=-\left(\frac{\rho_{r}}{R^{2}}\right).$ (38)

While, ignoring the vacuum energy, if $R\gg 1$, the mass term would dominate
and the expression (37) becomes

$\dot{R}^{2}+k=-\left(\frac{\rho_{m}}{R}\right),$ (39)

This equation describes a dust-dominated Universe.

Figure 1: Preselection and Postselection à $t_{in}$ and $t_{out}$

It is essential to note that the radiation-dominated Universe pertains to a
brief period near the initial singularity. In contrast, the dust-dominated
universe extends over a much longer timeframe, including the present day.

###  II.1 $\mid\Psi_{in}\rangle$: Solution of the Einstein-Dirac equation for
$m=0$.

When $m=0$, the ODEs in (34) becomes

$\displaystyle\left\\{\begin{array}[]{lll}\dot{w}^{1}&=&0\\\
\dot{w}^{2}&=&-\frac{2\lambda}{R}w^{3}\\\
\dot{w}^{3}&=&\frac{2\lambda}{R}w^{2},\end{array}\right.$ (43)

$\displaystyle\left(\frac{dR}{dt}\right)^{2}+1=-\frac{1}{R^{2}}\lambda w^{1}.$
(44)

We set the following conditions. The expression 44 must very that at a certain
time $t=t_{B},\frac{dR}{dt}|_{t_{B}}=0\,\,\text{and}R(t_{B})=R_{0}$. This
implies that $w^{1}=-\frac{R^{2}_{0}}{\lambda}$. The meaning is that, at time
$t_{B}$, the radiative energy ceases to be the driving force and the matter
takes over. Of course, the solution is a simplification of the big picture.
The solutions to the differential scale function equation is

$\displaystyle R(t)=\sqrt{-\left(t-R_{0}\right)^{2}+R^{2}_{0}}.$ (45)

The form of this solution is confirmed by scale function for the radiation
dominated Universe proposed by Plebanski and Krasinski in [30] in page 289. It
reads as

$\displaystyle R(t)$ $\displaystyle=$
$\displaystyle\sqrt{-k\left(t-t_{B}-\frac{1}{k}\sqrt{\frac{\epsilon_{0}}{3}}\right)^{2}+\frac{\epsilon_{0}}{3k}},\quad\text{when
}k\neq 0,$ $\displaystyle R(t)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\epsilon_{0}}{3}(t-t_{B})},\quad\text{when }k=0,$

where $\epsilon_{0}$ is the radiation density, $k$ takes values
$1\;\text{or}-1$.

Figure 2:  $\frac{dR(t)}{dt}$ from time $t=0$ to $t=R_{0}$. If one considers
the today scale function is set $R=1$, the scale function being without unity,
we may calibrate time so that, when the radiative forces cease to be the
driving one, the corresponding time would be $t=R_{0}$. In the literature we
know that the radiation dominated Universe epoch elapsed $\approx 72000years$
[31], page 331.

Having the initial condition to the Bloch vector differential equation

$\vec{w}(0)=\left(-\frac{R^{2}_{0}}{\lambda},\sqrt{N^{2}-\frac{R^{4}_{0}}{\lambda^{2}}}\sin(\phi_{0}),\sqrt{N^{2}-\frac{R^{4}_{0}}{\lambda^{2}}}\cos(\phi_{0})\right),$
(46)

and substituting $R(t)$ into (43), and based on the relationship between
spherical and hyperbolic trigonometry, we can find the subsequent solutions:

$\displaystyle\left\\{\begin{array}[]{lll}{w}^{1}&=&-\frac{R^{2}_{0}}{\lambda},\\\
{w}^{2}&=&-\sqrt{N^{2}-\frac{R^{4}_{0}}{\lambda^{2}}}\sin(4\lambda G(t)),\\\
{w}^{3}&=&\sqrt{N^{2}-\frac{R^{4}_{0}}{\lambda^{2}}}\cos\left(4\lambda
G(t)\right),\end{array}\right.$ (50)

where $G(t)=\tan^{-1}\left(\frac{t}{R(t)}\right)$ and $\phi_{0}=0$ . We can
notice that the Bloch vector is a function of the scale function $R(t)$.

Figure 3:  $R(t)$ increasing from time $t=0$ to $t=R_{0}$.

Subtsituting $m=0$ into the components of $\vec{w}$ in (31), (32) and (33), we
obtain

$\displaystyle\left\\{\begin{array}[]{lll}{w}^{1}&=&2\text{Re}\left(\alpha\bar{\beta}\right)\\\
{w}^{2}&=&2\text{Im}\left(\alpha\bar{\beta}\right)\\\
{w}^{3}&=&|\alpha|^{2}-|\beta|^{2}.\end{array}\right.$ (54)

Replacing $\alpha$ by $\rho_{in_{1}}e^{i\eta_{in_{1}}}$ and $\beta$ by
$\rho_{in_{2}}e^{i\eta_{in_{2}}}$ into (54), where $\rho$ is the modulus of
$\alpha$ and $\eta$ its argument. We differentiate modulus and arguments from
preselected and postselected states with indices. We obtain

$\displaystyle\left\\{\begin{array}[]{lll}{w}^{1}&=&2\rho_{in_{1}}\rho_{in_{2}}\cos(\eta_{in_{1}}-\eta_{in_{2}}),\\\
{w}^{2}&=&2\rho_{in_{1}}\rho_{in_{2}}\sin(\eta_{in_{1}}-\eta_{in_{2}}),\\\
{w}^{3}&=&\rho_{in_{1}}^{2}-\rho_{in_{2}}^{2}.\end{array}\right.$ (58)

These results constitute additional constraints for the solutions (50).
Consequently ${w}^{1},{w}^{2},{w}^{3}$ must be real numbers. For $G(t)$ to be
real, It has to be in the interval $]-\pi/2,\pi/2[$.

From the expression (58) we obtain

$\displaystyle\eta_{in_{1}}-\eta_{in_{2}}$ $\displaystyle=$
$\displaystyle-\arctan\left(\frac{\sqrt{\lambda^{2}N^{2}-R^{4}_{0}}\sin(4\lambda
G(t))}{R^{2}_{0}}\right).$ (59)

Also from (58) and using the norm from equation (36), we obtain

$\displaystyle\rho_{in_{1}}$
$\displaystyle=\frac{\sqrt{2}}{2}\sqrt{N+\sqrt{N^{2}-\frac{R^{4}_{0}}{\lambda^{2}}}\cos\left(4\lambda
G(t)\right)},$ (60) $\displaystyle\rho_{in_{2}}$
$\displaystyle=\frac{\sqrt{2}}{2}\sqrt{N-\sqrt{N^{2}-\frac{R^{4}_{0}}{\lambda^{2}}}\cos\left(4\lambda
G(t)\right)}.$ (61)

To sum up, $\mid\Psi_{in}\rangle$ is represented by the knowledge of the
components of the three-vector $\vec{w}$ or
$\alpha_{\text{in}}=\rho_{1}e^{i\eta_{1}}$ and
$\beta_{\text{in}}=\rho_{2}e^{i\eta_{2}}$ at a certain $t_{in}$ chosen such
that the scale function is real.

For the scale function to be real, $R(t_{in})$ must be real. This implies that
$0\leq t_{\text{in}}\leq R_{0}$. Within this period, the scale function
increases with time. In our approximation the scale function stops to increase
at $R_{0}$. But in the general picture, the driving force due to matter takes
over and the Universe continues to expand. Let us point out also that the
period within the radiation-dominated Universe is very short compared to the
lifespan of the Universe. This period is approximated as $72000$ years vs
$13,9$Gyrs today [31]. We also remind that within this period, the scale
function is very small. The radiative period ceases where the scale function R
approximately attains the value $R\approx 3.10^{-4}$. Finster in [1] shows a
bouncing behaviour of the scale function around singularity.

We can illustrate the behaviour of the scale acceleration in the presence of
the fermions field. This graphic is constructed only for a Universe without
matter.

Figure 4: The scale acceleration in the Universe dominated by fermions.

###  II.2 $\mid\Psi_{out}\rangle$: Solution of the Einstein-Dirac equation for
$\lambda=0$.

When $\lambda=0$, the ODEs in (34) becomes:

$\displaystyle\left\\{\begin{array}[]{lll}\dot{w}^{1}&=&0\\\
\dot{w}^{2}&=&-2m{w}^{3}\\\ \dot{w}^{3}&=&2m{w}^{2},\end{array}\right.$ (65)

$\displaystyle\left(\frac{dR}{dt}\right)^{2}+1=-\frac{1}{R}mw^{1}.$ (66)

We differ from Finster in [1] and we set that at
$t_{max},\frac{dR}{dt}|_{t_{max}}=0$, and $R(t_{max})=R_{max}$. This implies
that $w^{1}=-\frac{R_{max}}{m}$. One easily sees that the two differential
equations decouple. The block vector solution will not depend on the scale
function. These equations yield the following solutions

$\displaystyle R(g)$ $\displaystyle=$
$\displaystyle\frac{R_{max}}{2}\left(1-\cos(g)\right),$ (67) $\displaystyle
t(g)$ $\displaystyle=$
$\displaystyle\frac{R_{max}}{2}\left(g-sin\left(g\right)\right),$ (68)

where g is a parameter varying between $0$ and $2\pi$.

Figure 5:  This parametric solution is set for $R_{max}=10$. This value is
taken from Finster in [1]. The time is derived from the formula
$t(g)=\frac{R_{max}}{2}\left(g-sin\left(g\right)\right)$.

The solution to the Bloch vector differential equation for the following
conditions

$\vec{w}(t_{\text{max}})=\left(w^{1}_{\text{max}},\rho\cos(\phi_{\text{max}}),\rho\sin(\phi_{\text{max}})\right),$
(69)

gives the following solution:

$\displaystyle\left\\{\begin{array}[]{lll}{w}^{1}&=&-\frac{R_{max}}{m}\\\
{w}^{2}&=&\sqrt{N^{2}-\frac{R^{2}_{max}}{m^{2}}}\cos(2mt+\phi_{\text{max}}-2mt_{\text{max}})\\\
{w}^{3}&=&\sqrt{N^{2}-\frac{R^{2}_{max}}{m^{2}}}\sin(2mt+\phi_{\text{max}}-2mt_{\text{max}}),\end{array}\right.$
(73)

where $m$ is the mass of Dirac particle.

Subtsituting $\lambda=0$ into the components of $\vec{w}$ in (31-33), we
obtain

$\displaystyle\left\\{\begin{array}[]{lll}{w}^{1}&=&-\left(|\alpha|^{2}-|\beta|^{2}\right)\\\
{w}^{2}&=&2\text{Im}\left(\alpha\bar{\beta}\right)\\\
{w}^{3}&=&2\text{Re}\left(\alpha\bar{\beta}\right).\end{array}\right.$ (77)

Replacing $\alpha$ by $\rho_{out_{1}}e^{i\eta_{out_{1}}}$ and $\beta$ by
$\rho_{out_{2}}e^{i\eta_{out_{2}}}$ into (54), we obtain

$\displaystyle\left\\{\begin{array}[]{lll}{w}^{1}&=&-\left(\rho_{out_{1}}^{2}-\rho_{out_{2}}^{2}\right)\\\
{w}^{2}&=&2\rho_{out_{1}}\rho_{out_{2}}\sin(\eta_{out_{1}}-\eta_{out_{2}})\\\
{w}^{3}&=&2\rho_{out_{1}}\rho_{out_{2}}\cos(\eta_{out_{1}}-\eta_{out_{2}}).\end{array}\right.$
(81)

These results constitute additional constraints for the solutions (50).
Consequently ${w}^{1},{w}^{2},{w}^{3}$ must be real numbers.

From the expression (81) we obtain

$\displaystyle\eta_{out_{1}}-\eta_{out_{2}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}-2mt-\phi_{\text{max}}+2mt_{\text{max}}.$ (82)

We also obtain from (81) and the norm (36):

$\displaystyle\rho_{out_{1}}$
$\displaystyle=\frac{\sqrt{2}}{2}\sqrt{N+\frac{R_{max}}{m}},$ (83)
$\displaystyle\rho_{out_{2}}$
$\displaystyle=\frac{\sqrt{2}}{2}\sqrt{N-\frac{R_{max}}{m}}.$ (84)

Figure 6:  Rate of change the scale function in an Universe dominated by
matter.  Figure 7:  The acceleration of the scale function in an Universe
dominated by matter.

To sum up, $\mid\Psi_{out}\rangle$ is represented by the knowledge of the
components of the three-vector $\vec{w}$ or
$\alpha_{\text{out}}=\rho_{out_{1}}e^{i\eta_{out_{1}}}$ and
$\beta_{\text{out}}=\rho_{out_{2}}e^{\eta_{out_{2}}}$ at a certain $t_{out}$
chosen such that the scale function is real.

##  III Weak measurement on the Einstein-Dirac solutions

The general formula of a weak measurement of any observable is given by

$\langle\hat{A}\rangle_{w}=\frac{\langle\psi_{out}(t)|\hat{A}|\psi_{in}(t)\rangle}{\langle\psi_{out}(t)|\psi_{in}(t)\rangle}.$
(85)

wherein,
$\langle\psi_{out}(t)|=\langle\psi_{out}(t_{out})|U^{\dagger}(t,t_{out})$ and
$|\psi_{in}(t)\rangle=U(t,t_{in})|\psi_{in}(t_{in})\rangle$.

A pictorial description of a weak measurement in our context is like in figure
8.

Figure 8: Weak measurement at time t.

The solutions of Einstein-Dirac equation $|\psi_{in}(t_{in})\rangle$ and
$|\psi_{out}(t_{out})\rangle$ are defined at the respective times $t_{in}$ and
$t_{out}$. We must construct the evolutionary operator to allow the weak
measurement to occur at time t between $t_{in}$ and $t_{out}$.

Because of the time dependency of the scale function, an exact evolutionary
operator is impossible to compute. Still, it can be approximated by using the
_Wentzel–Kramers–Brillouin (WKB)_ method. Following Finster [29], the
smallness of the Compton wavelength compared to the lifetime of the Universe
justifies the use of WKB-type approximation. Therefore, we define the
approximated unitary evolutionary operator as

$U(t,t_{0})=U^{-1}(t)\left(\begin{array}[]{cc}e^{-iF(t,t_{0})}&0\\\
0&e^{iF(t,t_{0})}\end{array}\right)U(t_{0}).$ (86)

wherein
$F(t,t_{0})=-F(t_{0},t)=\int^{t}_{t_{0}}\frac{\sqrt{R^{2}(t)m^{2}+\lambda^{2}}}{R(t)}dt$.
$U(t)$ is defined such that

$U(t)\begin{pmatrix}m&-\frac{\lambda}{R(t)}\\\
-\frac{\lambda}{R(t)}&-m\end{pmatrix}U(t)^{-1}=\frac{\sqrt{R^{2}(t)m^{2}+\lambda^{2}}}{R(t)}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (87)

The equation (87) yields $g(t)=\frac{1}{2}\arctan(\frac{\lambda}{mR(t)})$ for
the unitary matrix given by

$U(t)=\begin{pmatrix}\cos(g(t))&-\sin(g(t))\\\
\sin(g(t))&\cos(g(t))\end{pmatrix}.$ (88)

The computation of $\left(\begin{array}[]{cc}e^{-iF(t,t_{0})}&0\\\
0&e^{iF(t,t_{0})}\end{array}\right)$ follows Finster in [29].

Let us point out here that $U(t,t_{\text{in}})$ is given by

$U^{-1}(t)\left(\begin{array}[]{cc}e^{-iF(t,t_{\text{in}})}&0\\\
0&e^{iF(t,t_{\text{in}})}\end{array}\right)U(t_{in})$

wherein the preselected state is taken from the radiation-dominated Universe
with $m=0$, which yields
$g(t_{\text{in}})=\frac{1}{2}\arctan(\frac{\lambda}{m*R(t_{\text{in}})})=\frac{\pi}{4}+k\pi$
for $m=0$. For $k=0$, the matrix will be given by

$U(t_{\text{in}})=\frac{\sqrt{2}}{2}\begin{pmatrix}1&-1\\\ 1&1\end{pmatrix}$

.

Whereas the postselected state is taken from the dust-dominated Universe
wherein $\lambda=0$. This will yield
$g(t)=\frac{1}{2}\arctan(\frac{\lambda}{m*R(t_{\text{in}})})=0+k\pi$ for
$\lambda=0$, and for $k=0$ this results in $U(tout)$ is the identity matrix

$U(t_{out})=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right).$

Knowing that $\left.U^{\dagger}(t,t_{\text{out}})=U(t_{\text{out}},t)\right.$,
this later is given by

$U^{-1}(t_{\text{out}})\left(\begin{array}[]{cc}e^{-iF(t_{\text{out}},t)}&0\\\
0&e^{iF(t_{\text{out}},t)}\end{array}\right)U(t).$

The unitary operator approximated, we can now compute the weak measurement of
some observable starting from the Energy-momentum Tensor.

###  III.1 Weak measurement of the Energy-momentum tensor .

In this section, we compute the weak measurement of the energy-momentum tensor
given by

$\langle
T_{\mu\nu}\rangle_{w}=\frac{\langle\psi_{out}(t)|T_{\mu\nu}|\psi_{in}(t)\rangle}{\langle\psi_{out}(t)|\psi_{in}(t)\rangle}.$
(89)

More explicitly, we have

$\langle
T_{\mu\nu}\rangle_{w}=\frac{1}{2}\frac{\langle\psi_{\text{out}}(t)|\left(iG_{\mu}D_{\nu}+iG_{\nu}D_{\mu}\right)|\psi_{\text{in}}(t)\rangle}{\langle\psi_{\text{out}}(t)|\psi_{\text{in}}(t)\rangle}.$
(90)

where $G^{\mu}$ are the linear combinations of the Dirac matrices of Minkowski
space given by:

$\displaystyle G^{0}=\gamma^{0},$ (91) $\displaystyle
G^{r}=\frac{f(r)}{R(t)}\left(\cos\vartheta\gamma^{3}+\sin\vartheta\cos\varphi\gamma^{1}+\sin\vartheta\sin\varphi\gamma^{2}\right),$
(92) $\displaystyle
G^{\vartheta}=\frac{1}{rR(t)}\left(-\sin\vartheta\gamma^{3}+\cos\vartheta\cos\varphi\gamma^{1}+\cos\vartheta\sin\varphi\gamma^{2}\right),$
(93) $\displaystyle
G^{\varphi}=\frac{1}{rR(t)\sin\vartheta}\left(-\sin\varphi\gamma^{1}+\cos\varphi\gamma^{2}\right),$
(94)

with $f(r)=\sqrt{1-r^{2}}$. Note that $G_{\mu}=g_{\mu\nu}G^{\nu}$.

The Dirac matrices are given as

$\gamma^{0}=\begin{pmatrix}I&0\\\
0&-I\end{pmatrix},\quad\gamma^{\alpha}=\begin{pmatrix}0&\sigma^{\alpha}\\\
-\sigma^{\alpha}&0\end{pmatrix},$

where $I$ is the $2\times 2$ identity matrix and $\sigma^{\alpha}$ are the
Pauli matrices:

$\sigma^{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma^{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma^{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$

The differential operator is given by $D_{\mu}=\partial_{\mu}-iE_{\mu}\text{,
}$ and $G^{\mu}$ obeys to the anti-commutation rule.

$\left\\{G^{\mu},G^{\nu}\right\\}\equiv
G^{\mu}G^{\nu}+G^{\nu}G^{\mu}=2g^{\mu\nu}\mathbf{1}_{4}.$

The so-called spin coefficients are given by $E_{\mu}$.

For an orthogonal metric, the combination $G^{\mu}E_{\mu}$ takes the simple
form (for more details see [1])

$G^{\mu}E_{\mu}=\frac{i}{2\sqrt{|g|}}\partial_{\mu}\left(\sqrt{|g|}G^{\mu}\right)\quad\text{
with }\quad g=\operatorname{det}g_{ij},$

making it unnecessary to compute the spin connection coefficients or even the
Christoffel symbols.

To compute the numerator part of the weak measurement of the energy-momentum
tensor, following Finster in[29], we adapt the object
$P(t,\vec{x};t^{\prime},\vec{x^{\prime}})$ to our need

$=\left(R(t)R(t^{\prime})\right)^{-\frac{3}{2}}E_{\lambda}(\vec{x},\vec{x^{\prime}})\otimes\begin{pmatrix}\alpha_{\text{in}}(t)\overline{\alpha_{\text{out}}(t^{\prime})}-\alpha_{\text{in}}(t)\overline{\beta_{\text{out}}(t^{\prime})}\\\
\beta_{\text{in}}(t)\overline{\alpha_{\text{out}}(t^{\prime})}-\beta_{\text{in}}(t)\overline{\beta_{\text{out}}(t^{\prime})}\end{pmatrix}.$
(95)

wherein $E_{\lambda}$ denote the spectral projectors of $\mathcal{D}_{S^{3}}$,
given by
$E_{\pm\mid\lambda\mid}=\displaystyle\sum^{\mid\lambda\mid-\frac{3}{2}}_{n=0}\sum^{j}_{k=-j}\psi^{\pm}_{njk}(x)\overline{\psi^{\pm}_{njk}(\acute{x})}$.
Then, the numerator part of the energy-momentum tensor,
$\langle\psi_{\text{out}}|T_{\mu\nu}|\psi_{\text{in}}\rangle(t,\vec{x})$ can
be expressed in terms of $P$ by

$\begin{split}=\frac{1}{2}\operatorname{Tr}_{\mathbb{C}^{4}}\Big{\\{}&\left(iG_{\mu}D_{\nu}+iG_{\nu}D_{\mu}\right)P\left(t,\vec{x};t^{\prime},\vec{x}^{\prime}\right)\Big{\\}}\Bigg{|}_{t^{\prime}=t,\vec{x}^{\prime}=\vec{x}}.\end{split}$
(96)

Because of the homogeneity and the isotropy of the FLRW space, only the
diagonal energy-momentum tensor components are different from 0. We get the
following results after computations and normalization

$\displaystyle\langle
T_{0}^{0}\rangle_{w}=\frac{\langle\psi_{out}(t)|T_{0}^{0}|\psi_{in}(t)\rangle}{\langle\psi_{out}(t)|\psi_{in}(t)\rangle}=\frac{\Bigl{[}mR(t)\left(\alpha_{\text{in}}(t)\overline{\alpha_{\text{out}}}(t)-\beta_{\text{in}}(t)\overline{\beta_{\text{out}}}(t)\right)-\lambda\left(\alpha_{\text{in}(t)}\overline{\beta_{\text{out}}}(t)+\beta_{\text{in}}(t)\overline{\alpha_{\text{out}}}(t)\right)\Bigr{]}}{R(t)\left(\overline{\alpha_{\text{out}}}(t)\alpha_{\text{in}}(t)+\overline{\beta_{\text{out}}}(t)\beta_{\text{in}}(t)\right)},$
(97)

and

$\displaystyle\langle
T_{j}^{j}\rangle_{w}=\frac{\langle\psi_{out}(t)|T_{j}^{j}|\psi_{in}(t)\rangle}{\langle\psi_{out}(t)|\psi_{in}(t)\rangle}=\frac{\lambda\left(\alpha_{\text{in}}(t)\overline{\beta_{\text{out}}}(t)+\beta_{\text{in}}(t)\overline{\alpha_{\text{out}}}(t)\right)}{3R(t)\left(\overline{\alpha_{\text{out}}}(t)\alpha_{\text{in}}(t)+\overline{\beta_{\text{out}}}(t)\beta_{\text{in}}(t)\right)},$
(98)

where j represents $r,\vartheta,\varphi$. Due to spherical symmetry and
homogeneity of the space,
$T_{r}^{r}=T_{\vartheta}^{\vartheta}=T_{\varphi}^{\varphi}$ are all equal.

One sees here that if $|\psi_{in}(t)\rangle=|\psi_{out}(t)\rangle$, the weak
measurement of the energy-momentum tensor will coincide with the one computed
in [1]. In this regard, weak measurements are generalizations of classical
ones.

Another quick observation is that if

$\overline{\alpha_{\text{out}}}(t)\alpha_{\text{in}}(t)+\overline{\beta_{\text{out}}}(t)\beta_{\text{in}}(t)\approx
0,$

an amplification phenomenon occurs. This is realized when
$\overline{\alpha_{\text{out}}}(t)\approx-k\beta_{\text{in}}(t)$ and
$\overline{\beta_{\text{out}}}(t)\approx k\alpha_{\text{in}}(t)$, where $k$ is
a factor or a phase.

For better analysis, we construct a _complex Bloch vector_ since it is
constructed in the same manner as a Bloch vector, but the components are
complex functions. Aharonov and Gruss constructed a density operator from the
Two-State vectors in their paper _Two-time interpretations_ of quantum
mechanic [9]. This complex Bloch vector is linked to that density operator.
One should note that if the preselected state is equal to the postselected
one, then we shall get the same construction as in the paper of Finster and
Hainzl on page 14 in [1].

$\vec{v}_{f,i}=\begin{pmatrix}v_{f,i}^{1}\\\ v_{f,i}^{2}\\\
v_{f,i}^{3}\end{pmatrix}=\begin{pmatrix}\langle\xi_{\text{out}}(t)|\sigma^{1}|\xi_{\text{in}}(t)\rangle\\\
\langle\xi_{\text{out}}(t)|\sigma^{2}|\xi_{\text{in}}(t)\rangle\\\
\langle\xi_{\text{out}}(t)|\sigma^{3}|\xi_{\text{in}}(t)\rangle\end{pmatrix}$

wherein $\xi_{\text{out}}(t)=\left(\begin{array}[]{c}\alpha_{\text{out}}(t)\\\
\beta_{\text{out}}(t)\end{array}\right)$ and
$\xi_{\text{in}}(t)=\left(\begin{array}[]{c}\alpha_{\text{in}}(t)\\\
\beta_{\text{in}}(t)\end{array}\right)$.

We also define $v_{f,i}^{0}$ as $\langle\xi_{\text{out}}(t)|I_{2\times
2}|\xi_{\text{in}}(t)\rangle$ where $I_{2\times 2}$ is the 2 dimensional
identity matrix. This will yield
$\overline{\alpha_{\text{out}}}(t)\alpha_{\text{in}}(t)+\overline{\beta_{\text{out}}}(t)\beta_{\text{in}}(t)$.
We can now redefine the weak energy-momentum tensor as

$\begin{split}&\langle
T_{0}^{0}\rangle_{w}=\frac{\Bigl{[}mR(t)\left(v_{f,i}^{3}\right)-\lambda\left(v_{f,i}^{1}\right)\Bigr{]}}{R(t)\left(v_{f,i}^{0}\right)}\\\
&\langle
T_{j}^{j}\rangle_{w}=\frac{\lambda\left(v_{f,i}^{1}\right)}{3R(t)\left(v_{f,i}^{0}\right)}\end{split}$
(99)

Let us express the energy-momentum tensor weak values in  terms of
$\alpha_{\text{in}}(t_{\text{in}})$ and $\alpha_{\text{out}}(t_{\text{out}})$
wherein
$A=e^{i\left(F(t_{\text{out}},t_{\text{in}})\right)}$,
$\bar{A}=e^{-i\left(F(t_{\text{out}},t_{\text{in}})\right)}$,
$B=e^{i\left(F(t,t_{\text{in}})+F(t,t_{\text{out}})\right)}$ and
$\bar{B}=e^{-i\left(F(t,t_{\text{in}})+F(t,t_{\text{out}})\right)}$. Because
of the lengthiness of the expression, we break it as follows

$\begin{split}&v_{f,i}^{0}=\frac{\sqrt{2}}{2}\left(\overline{\alpha_{\text{out}}}(t_{\text{out}})\alpha_{\text{in}}(t_{\text{in}})-\overline{\alpha_{\text{out}}}(t_{\text{out}})\beta_{\text{in}}(t_{\text{in}})\right)\bar{A}+\frac{\sqrt{2}}{2}\left(\overline{\beta_{\text{out}}}(t_{\text{out}})\beta_{\text{in}}(t_{\text{in}})+\overline{\beta_{\text{out}}}(t_{\text{out}})\alpha_{\text{in}}(t_{\text{in}})\right)A,\end{split}$
(100)

$\begin{split}&v_{f,i}^{3}=\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\overline{\alpha_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}B+\cos{(2g(t))}\bar{A}\right]+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\overline{\beta_{\text{out}}}(t_{\text{out}})\left[-\sin{(2g(t))}\bar{B}-\cos{(2g(t))}A\right]\\\
&+\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\overline{\beta_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}\bar{B}-\cos{(2g(t))}A\right]+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\overline{\alpha_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}B-\cos{(2g(t))}\bar{A}\right],\end{split}$
(101)

$\begin{split}&v_{f,i}^{1}=\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\overline{\alpha_{\text{out}}}(t_{\text{out}})\left[-\sin{(2g(t))}\bar{A}+\cos{(2g(t))}B\right]+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\overline{\beta_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}A-\cos{(2g(t))}\bar{B}\right]\\\
&+\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\overline{\beta_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}A+\cos{(2g(t))}\bar{B}\right]+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\overline{\alpha_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}\bar{A}+\cos{(2g(t))}B\right].\end{split}$
(102)

We can deduce the orthogonality condition for $v_{f,i}^{0}\approx 0$. This is
obtained if $\overline{\alpha_{\text{out}}}(t_{\text{out}})\approx$:

$\begin{split}&-\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\left[-\sin{(g(t_{\text{out}}))}e^{-iF(t_{\text{out}},t_{\text{in}})}+\cos{(g(t_{\text{out}}))}e^{iF(t_{\text{out}},t_{\text{in}})}\right]\\\
&-\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\left[\sin{(g(t_{\text{out}}))}e^{-iF(t_{\text{out}},t_{\text{in}})}+\cos{(g(t_{\text{out}}))}e^{iF(t_{\text{out}},t_{\text{in}})}\right].\end{split}$
(103)

and $\overline{\beta_{\text{out}}}(t_{\text{out}})\approx$

$\begin{split}&\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\left[\cos{(g(t_{\text{out}}))}e^{-iF(t_{\text{out}},t_{\text{in}})}+\sin{(g(t_{\text{out}}))}e^{iF(t_{\text{out}},t_{\text{in}})}\right]\\\
&+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\left[-\cos{(g(t_{\text{out}}))}e^{-iF(t_{\text{out}},t_{\text{in}})}+\sin{(g(t_{\text{out}}))}e^{iF(t_{\text{out}},t_{\text{in}})}\right].\end{split}$
(104)

As it is difficult to have a simple expression of the weak measurement of the
energy-momentum tensor to make an interpretation, let us compute a concrete
example with values that make the computation easy. Let us consider
$\lambda=3/2$, the eigenvalue of the Dirac operator
$\mathcal{D}_{\mathcal{H}}$, $t_{\text{in}}\approxeq 0$ and $R_{0}=10^{-3}$.
Theses conditions imply $G(t_{\text{in}})=0$ in (50). They will yield the
following expression: $\rho_{in_{1}}\approx 2$ and $\rho_{in_{2}}\approx 0$
and $\eta_{in_{1}}-\eta_{in_{2}}=0$. For simplicity, we shall consider
$\eta_{in_{1}}=\eta_{in_{2}}=0$.

We also remind that the preselected state is taken from the radiation-
dominated Universe with $m=0$ which yields
$g(t_{\text{in}})=\frac{1}{2}\arctan(\frac{\lambda}{0*R(t_{\text{in}})})=\frac{\pi}{4}+k\pi$.
For $k=0$, the matrix will be given by
$U(t_{\text{in}})=\frac{\sqrt{2}}{2}\begin{pmatrix}1&-1\\\ 1&1\end{pmatrix}$.
Therefore the preselected state at $t_{\text{in}}$ will be
$\alpha_{\text{in}}(t_{\text{in}})=2$ and
$\beta_{\text{in}}(t_{\text{in}})=0$.

As for the postselected state vector, the scale function and the time
expressed $R(g)=\frac{R_{\text{max}}}{2}\left(1-\cos(g)\right)$,
$t(g)=\frac{R_{\text{max}}}{2}\left(g-\sin(g)\right)$, we have the following
observations

  * •

A maximal $R(g)$ will be $R_{\text{max}}$, and the maximal $t$, that is
$t_{\text{out}}$ will be $R_{\text{max}}k\frac{\pi}{2}$.

  * •

We shall have: $\rho_{\text{out}1}=\sqrt{2+\frac{R_{\text{max}}}{m}}$,
$\rho_{\text{out}2}=\sqrt{2-\frac{R_{\text{max}}}{m}}$ and
$\eta_{\text{out}1}-\eta_{\text{out}2}=\frac{\pi}{2}-\phi_{\text{max}}$.

We remind that the postselected state is taken from the dust-dominated
Universe wherein $\lambda=0$. This will yield
$g(t)=\frac{1}{2}\arctan(\frac{0}{m*R(t_{\text{in}})})=0+k\pi$ . This will
lead, for $k=0$, to $U(t_{out})=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)$, an identity matrix.

The postselected state $t_{\text{out}}$ will be

$\begin{split}&\alpha_{\text{out}}(t_{\text{out}})=\sqrt{2+\frac{R_{\text{max}}}{m}}e^{i\left(\frac{\pi}{2}-\phi_{\text{max}}\right)},\\\
&\beta_{\text{out}}(t_{\text{out}})=\sqrt{2-\frac{R_{\text{max}}}{m}}.\end{split}$
(105)

For simplicity, let us consider a weak measurement taking place at $t=t_{out}$
Universe and $R_{\text{max}}=10$ and $m=21.5$. This implies that

$\left.\left(\begin{array}[]{c}\alpha_{\text{in}}(t_{\text{out}})\\\
\beta_{\text{in}}(t_{\text{out}})\end{array}\right)=U(t_{\text{out}},t_{\text{in}})\left(\begin{array}[]{c}\alpha_{\text{in}}(t_{\text{in}})\\\
\beta_{\text{in}}(t_{\text{in}})\end{array}\right)\right.,$

where

$U(t_{\text{out}},t_{\text{in}})=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}e^{-i\pi
m\text{Rmax}}&-e^{-i\pi m\text{Rmax}}\\\ e^{i\pi m\text{Rmax}}&e^{i\pi
m\text{Rmax}}\\\ \end{array}\right),$

and $U(t_{\text{out}},t_{\text{out}})=I$.
$F(t_{\text{out}},t_{\text{in}})=mt_{\text{out}}$. These elements yield the
following state vectors

$\begin{split}&\alpha_{\text{in}}(t_{\text{out}})=\frac{1}{\sqrt{2}}(\alpha_{\text{in}}-\beta_{\text{in}})e^{-i\pi
m\text{Rmax}},\\\
&\beta_{\text{in}}(t_{\text{out}})=\frac{1}{\sqrt{2}}(\alpha_{\text{in}}+\beta_{\text{in}})e^{i\pi
m\text{Rmax}}.\end{split}$ (106)

Since the weak measurement is taking place at $t=t_{\text{out}}$, the
postselected stated is fixed.

The values in the weak measurement of the energy-momentum tensor in the
following expression

$\begin{split}&\langle
T_{0}^{0}\rangle_{w}=\frac{\Bigl{[}mR(t_{\text{out}})\left(v_{f,i}^{3}\right)-\lambda\left(v_{f,i}^{1}\right)\Bigr{]}}{R(t_{\text{out}})\left(v_{f,i}^{0}\right)}\\\
&\langle
T_{j}^{j}\rangle_{w}=\frac{\lambda\left(v_{f,i}^{1}\right)}{3R(t_{\text{out}})\left(v_{f,i}^{0}\right)}\end{split}$
(107)

will be:

$\begin{split}&v_{f,i}^{0}=1.6e^{i\pi
mR_{\text{max}}}-(2.3i)e^{i\phi_{\text{max}}-i\pi mR_{\text{max}}}\\\
&v_{f,i}^{1}=(2.3i)e^{i(\pi mR_{\text{max}}+\phi_{\text{max}})}+1.6e^{-i\pi
mR_{\text{max}}}\\\ &v_{f,i}^{3}=(2.3i)\left(-e^{i\phi_{\text{max}}-i\pi
mR_{\text{max}}}\right)-1.6e^{i\pi mR_{\text{max}}}\end{split}$ (108)

What we can conclude is that weak measurements reveal unusual values, such as
complex numbers. In our case, for fixed $m$ and $R_{\text{max}}$, the
numerical result will depend on $\phi_{max}$.

In (24) the acceleration of the Universe is given in terms of the energy
momentum tensor. In (29), the acceleration of the Universe is given in terms
of the Bloch vector. But in the _Two-State Formalism_ , the Hopf
transformation yields a complex Bloch vector. The _Two-State Formalism_
generalizes the _One-State Formalism_ since it suffices to consider the same
state vector in the _Two-State Formalism_ to have the classical Bloch vector.
We propose to give here the acceleration of the universe in terms of weak
measurements of the energy-momentum tensors and derive it in terms of Complex
Bloch vector. The acceleration of the Universe would then be given as

$\begin{split}\ddot{R}&=\frac{R}{2}\left(3\langle T_{j}^{j}\rangle_{w}-\langle
T_{0}^{0}\rangle_{w}\right),\\\
\ddot{R}&=\frac{\Bigl{[}-mR\left(v_{f,i}^{3}\right)+2\lambda\left(v_{f,i}^{1}\right)\Bigr{]}}{2\left(v_{f,i}^{0}\right)}.\end{split}$
(109)

In this formula, one clearly sees that the acceleration of the Universe may be
comprehended from the _Two-State Formalism_ and weak measurements theories.
Assuming a different postselected state vector may change the shape of the
acceleration of the Universe. We remind here that the real and the imaginary
parts of a weak value can be measured. The real part of a weak value may be
interpreted as the conditioned average associated with an observable in Two
Vector Formalism, [24], [32], and they represent the shift of the average
detected position due to postselection. The imaginary part of the weak value
represents the shift of the average impulsion due to postselection.

The computation of the acceleration of the Universe with the above condition
would yield

$\begin{split}&\ddot{R}=Re\left(\frac{\Bigl{[}-mR\left(v_{f,i}^{3}\right)+2\lambda\left(v_{f,i}^{1}\right)\Bigr{]}}{2\left(v_{f,i}^{0}\right)}\right)\end{split}$
(110)

This example shows that the acceleration is not Zero and depends here on the
postselection.

Figure 9: The real part of the weak values of the acceleration of the Universe
in function of time for $m=21.5,R_{\text{max}}=10,\phi_{\text{max}}=0$ and
$\lambda=\frac{3}{2}$

Figure (9) is about the weak measurement of the acceleration of the Universe
as a function of time in a matter Dominated Universe.

Figure 10: The real part of the weak values of the acceleration of the
Universe in function of time for
$m=21,R_{\text{max}}=10,\phi_{\text{max}}=\frac{\pi}{4}$ and
$\lambda=\frac{3}{2}$.

The difference between Figure 9 and 10 shows how the acceleration of the
Universe may change with a different postselection. We have postselected a
state vector with $\phi_{\text{max}}=0$ in Figure 9 and
$\phi_{\text{max}}=\frac{\pi}{4}$ in Figure 10. It is clear that, this
difference causes more acceleration in Figure 10 than in Figure 9. In
conclusion, we have shown how to compute the weak measurement of the Energy-
momentum tensor. We derived how to amplify weak values of the energy-momentum
tensor using an appropriate state vector almost orthogonal to the preselected
one. We have finally derived the weak values of the acceleration of the
Universe and have showed how sensitive it is compared to classical
acceleration.

###  III.2 Weak measurement of $\sigma^{z}$ operator.

In this section, we delve into the weak measurement of the $\sigma^{z}$
operator which is given by the following expression

$\sigma^{z}_{w}=\frac{\langle\psi_{out}(t)|\sigma^{z}|\psi_{in}(t)\rangle}{\langle\psi_{out}(t)|\psi_{in}(t)\rangle}.$
(111)

Following Kofman, Cormann and Ferraz in [33], [34] and [7] respectively, the
weak measurement of the generator of $SU(2)$ is given by

$\sigma_{r,w}=\frac{\vec{f}.\vec{r}+\vec{r}.\vec{i}+i\vec{f}.(\vec{r}\wedge\vec{i})}{\frac{1}{2}(1+\vec{f}.\vec{i})}.$
(112)

In our case, $\vec{r}=(\begin{array}[]{ccc}0&0&1\end{array})$ to get
$\sigma^{z}$. Following Kofman in [33], the vectors $\vec{f}$ and $\vec{i}$
are computed as follows

$\vec{w_{f}}=\begin{pmatrix}\langle\xi_{\text{out}}(t)|\sigma^{1}|\xi_{\text{out}}(t)\rangle\\\
\langle\xi_{\text{out}}(t)|\sigma^{2}|\xi_{\text{out}}(t)\rangle\\\
\langle\xi_{\text{out}}(t)|\sigma^{3}|\xi_{\text{out}}(t)\rangle\end{pmatrix},$

where $\xi_{\text{out}}(t)=\left(\begin{array}[]{c}\alpha_{\text{out}}(t)\\\
\beta_{\text{out}}(t)\end{array}\right)$ is expressed as

$\begin{split}\overline{\alpha_{\text{out}}}(t)=&\overline{\alpha_{\text{out}}}(t_{\text{out}})\cos{(g(t))}e^{-iF(t_{\text{out}},t)}+\\\
&\overline{\beta_{\text{out}}}(t_{\text{out}})\sin{(g(t))}e^{iF(t_{\text{out}},t)},\end{split}$
(113)

and

$\begin{split}\overline{\beta_{\text{out}}}(t)=&-\overline{\alpha_{\text{out}}}(t_{\text{out}})\sin{(g(t))}e^{-iF(t_{\text{out}},t)}+\\\
&\overline{\beta_{\text{out}}}(t_{\text{out}})\cos{(g(t))}e^{iF(t_{\text{out}},t)}.\end{split}$
(114)

In the same way, the preselected state is given by:

$\vec{w_{i}}=\begin{pmatrix}\langle\xi_{\text{in}}(t)|\sigma^{1}|\xi_{\text{in}}(t)\rangle\\\
\langle\xi_{\text{in}}(t)|\sigma^{2}|\xi_{\text{in}}(t)\rangle\\\
\langle\xi_{\text{in}}(t)|\sigma^{3}|\xi_{\text{in}}(t)\rangle\end{pmatrix},$

where $\xi_{\text{in}}(t)=\left(\begin{array}[]{c}\alpha_{\text{in}}(t)\\\
\beta_{\text{in}}(t)\end{array}\right)$ is expressed as

$\begin{split}\alpha_{\text{in}}(t)=&\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\left[\cos{(g(t))}e^{-iF(t,t_{\text{in}})}+\sin{(g(t))}e^{iF(t,t_{\text{in}})}\right]+\\\
&\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\left[-\cos{(g(t))}e^{-iF(t,t_{\text{in}})}+\sin{(g(t))}e^{iF(t,t_{\text{in}})}\right],\end{split}$
(115)

and

$\begin{split}\beta_{\text{in}}(t)=&\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\left[-\sin{(g(t))}e^{-iF(t,t_{\text{in}})}+\cos{(g(t))}e^{iF(t,t_{\text{in}})}\right]+\\\
&\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\left[\sin{(g(t))}e^{-iF(t,t_{\text{in}})}+\cos{(g(t))}e^{iF(t,t_{\text{in}})}\right].\end{split}$
(116)

For simplicity we shall consider $t_{\text{in}}=0$ which implies that
$\eta_{\text{in1}}-\eta_{\text{in2}}=0$. We will set
$\eta_{in_{1}}-\eta_{in_{2}}=\Delta_{in}$ when necessary. This implies
$\Delta_{\text{in}}=0$ and $t_{\text{out}}=t_{\text{max}}$. We obtain the
following Postselected and Preselected vectors $\vec{w_{f}}$ and $\vec{w_{i}}$
whose components are given as $\vec{w_{f}}=$

$\displaystyle\left\\{\begin{array}[]{lll}&{w_{f}}^{1}=-\frac{R_{\text{max}}}{m}\sin{(2g(t))}+\\\
&\sqrt{N^{2}-\frac{R^{2}_{max}}{m^{2}}}\cos{\left(\frac{\pi}{2}-\phi_{\text{max}}-2F(t_{\text{out}},t)\right)}\cos{(2g(t))}\\\
&{w_{f}}^{2}=\sqrt{N^{2}-\frac{R^{2}_{max}}{m^{2}}}\sin{(\frac{\pi}{2}-\phi_{\text{max}}-(2F(t_{\text{out}},t)))}\\\
&{w_{f}}^{3}=\frac{R_{\text{max}}}{m}\cos{(2g(t))}+\\\
&\sqrt{N^{2}-\frac{R^{2}_{max}}{m^{2}}}\cos{(\frac{\pi}{2}-\phi_{\text{max}}-2F(t_{\text{out}},t))}\sin{(2g(t))}\\\
\end{array}\right.,$ (122)

and  $\vec{w_{i}}=$

$\displaystyle\left\\{\begin{array}[]{lll}w_{i}^{1}&=&\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\cos{(2g(t))}\cos{(2F(t,t_{\text{in}}))}+\frac{R_{0}^{2}}{\lambda}\sin{(2g(t))}\\\
w_{i}^{2}&=&-\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\sin{(2F(t,t_{\text{in}}))}\\\
w_{i}^{3}&=&\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\sin{(2g(t))}\cos{(2F(t,t_{\text{in}}))}-\frac{R_{0}^{2}}{\lambda}\cos{(2g(t))}\end{array}\right..$
(126)

The vector $\vec{f}$ is equal to $\frac{\vec{w_{f}}}{|\vec{w_{f}}|},\text{and
}\vec{i}=\frac{\vec{w_{i}}}{|\vec{w_{i}}|}$. The length of both Bloch vectors
are

$|\vec{w_{f}}|=|\vec{w_{i}}|=\lambda^{2}-\frac{1}{4}.$

They represent the unit vectors on the Bloch sphere of post-selected and pre-
selected state. In terms of the components, the $\sigma^{z}$ weak measurement
is given by

$\sigma_{r,w}=\frac{{w_{f}}^{3}+{w_{i}}^{3}+i(-{w_{f}}^{1}{w_{i}}^{2}+{w_{f}}^{2}{w_{i}}^{1})}{\frac{1}{2}(1+{w_{f}}^{1}{w_{i}}^{1}+{w_{f}}^{2}{w_{i}}^{2}+{w_{f}}^{3}{w_{i}}^{3})}.$
(127)

The real part of the numerator of the weak value of $\sigma^{z}$ operator in
general terms with the above conditions ($t_{\text{in}}=0$,
$\Delta_{\text{in}}=0$ and $t_{\text{out}}=t_{\text{max}}$), $\sigma_{r,w}$,
is given by

${w_{f}}^{3}+{w_{i}}^{3}=$

$\begin{split}&\sin(2g(t))\left(\Delta\rho_{\text{in}}^{2}\cos\left(2F\left(t,t_{\text{in}}\right)\right)+2\rho_{out_{1}}\rho_{out_{2}}\cos\left(\Delta_{\text{out}}\right)\right)+\\\
&\left(\Delta\rho_{\text{out}}^{2}-2\rho_{in_{1}}\rho_{in_{2}}\right)\cos(2g(t)).\end{split}$
(128)

In terms of the preselected and postselected we have computed, it would be

$\begin{split}&\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\cos\left(2F\left(t,t_{\text{in}}\right)\right)\sin(2g(t))+\frac{\left(-mR_{0}^{2}+\lambda
R_{\max}\right)\cos(2g(t))}{m\lambda}+\\\
&\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}\sin(2g(t))\sin\left(2F\left(t_{\text{out}},t\right)+\phi_{\max}\right).\end{split}$
(129)

In the same way, the imaginary part of the numerator of the weak value of
$\sigma^{z}$ operator is expressed, in general term with the above conditions
as,

$-{w_{f}}^{1}{w_{i}}^{2}+{w_{f}}^{2}{w_{i}}^{1}=$

$\begin{split}&2\rho_{out_{1}}\rho_{out_{2}}\left(\Delta\rho_{\text{in}}^{2}\cos(2g(t))\sin\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)+\right.\\\
&\left.\rho_{in_{1}}\rho_{in_{2}}\sin(2g(t))\sin\left(\Delta_{\text{out}}\right)\right)-\Delta\rho_{\text{in}}^{2}\Delta\rho_{\text{out}}^{2}\sin(2g(t))\sin\left(2F\left(t,t_{\text{in}}\right)\right).\end{split}$
(130)

It would explicitly be given in terms of the preselected and postselected
state as

$\begin{split}&-\frac{\sin\left(2F\left(t,t_{\text{in}}\right)\right)\sin(2g(t))\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}R_{\max}}{m}+\\\
&\frac{1}{\lambda}\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}\left(R_{0}^{2}\cos\left(2F\left(t_{\text{out}},t\right)+\phi_{\max}\right)\sin(2g(t))+\right.\\\
&\left.\lambda\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\cos(2g(t))\cos\left(2F\left(t,t_{\text{in}}\right)-2F\left(t_{\text{out}},t\right)-\phi_{\max}\right)\right).\end{split}$
(131)

Finally, the denominator part will be given in general terms by:
$(1+{w_{f}}^{1}{w_{i}}^{1}+{w_{f}}^{2}{w_{i}}^{2}+{w_{f}}^{3}{w_{i}}^{3})=$

$\begin{split}&2\Delta\rho_{\text{in}}^{2}\rho_{out_{1}}\rho_{out_{2}}\cos\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)-2\Delta\rho_{\text{out}}^{2}\rho_{in_{1}}\rho_{in_{2}}+1.\end{split}$
(132)

It would explicitly be given by:

$\begin{split}&\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}\cos\left(a\right)-\frac{R_{0}^{2}R_{\max}}{\lambda
m}+1\end{split}.$ (133)

wherein $a=2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}$.

One can see that the amplification phenomenon is obtained in many ways. One
way is

$\begin{split}\cos(a)\approx-\frac{1-\frac{R_{0}^{2}R_{\max}}{\lambda
m}}{\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}}\end{split}.$
(134)

We remind that
$\Delta_{\text{out}}=(\eta_{out_{1}}-\eta_{out_{2}}+2F(t_{\text{out}},t))$. By
adjusting adequately the parameters appearing in $a$, one can amplify the weak
values.

It is essential to note that any arbitrary observable in a 2-level system can
be expressed as
$\hat{A}=a_{I}\hat{I}_{2}+a_{L}\vec{\alpha}.\hat{\vec{\sigma}}$, where $a_{I}$
and $a_{L}$ are constants, $\vec{\alpha}$ is a unit vector in three
dimensions, and $\sigma$ denotes the Pauli matrices.

In this subsection we have derived the computation of the weak value of the Z
component of the Spin of fermions in the context of cosmology. We have shown
how it is possible to amplify the signal by choosing adequate parameters.

###  III.3 Berry phase from the weak measurement of a pure state.

Let us now compute the weak measurement of a pure state which is given as:

$\hat{\Pi}_{r,w}=\frac{\langle\psi_{out}(t)|\hat{\Pi}_{r}|\psi_{in}(t)\rangle}{\langle\psi_{out}(t)|\psi_{in}(t)\rangle}.$
(135)

where $\hat{\Pi}_{r}$ is a pure state given as
$\frac{1}{2}\left(\hat{I}+\vec{r}\vec{\hat{\sigma}}\right)$. The complete
formula to compute the weak values of a pure state is found in [33] or the
appendix of [34] and [7]. It reads as

$\hat{\Pi}_{r,w}=\frac{1+\vec{f}.\vec{r}+\vec{r}.\vec{i}+\vec{f}.\vec{i}+i\vec{f}.(\vec{r}\wedge\vec{i})}{\frac{1}{2}(1+\vec{f}.\vec{i})}.$
(136)

The vector $\vec{r}$ is defined as

$\left(\begin{array}[]{c}\sin(\theta)\cos(\phi)\\\ \sin(\theta)\sin(\phi)\\\
\cos(\theta)\end{array}\right).$ (137)

For readability and simplicity we shall consider
$\theta=\varphi=\frac{\pi}{2}$,
$a={\left(mt_{\text{out}}-\frac{\pi}{2}-F(t_{\text{out}},t_{\text{in}})\right)}$
and $b=2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}$.
Then the real part of the numerator of $\hat{\Pi}_{r,w}$ will be given in
general terms by

$\begin{split}&-\Delta\rho_{\text{in}}^{2}\sin\left(2F\left(t,t_{\text{in}}\right)\right)+2\rho_{out_{1}}\rho_{out_{2}}\sin\left(\Delta_{\text{out}}\right)+1.\end{split}$
(138)

In terms of the preselected and postselected state computed, it would be

$\begin{split}-\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\sin\left(2F\left(t,t_{\text{in}}\right)\right)+\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}\sin\left(\Delta_{\text{out}}\right)+1\end{split}.$
(139)

The imaginary part of numerator of the $\hat{\Pi}_{r,w}$,
$1+\vec{f}.\vec{r}+\vec{r}.\vec{i}+\vec{f}.\vec{i}$, obtained is in general
terms

$\begin{split}&-\Delta\rho_{\text{in}}^{2}\Delta\rho_{\text{out}}^{2}\cos\left(2F\left(t,t_{\text{in}}\right)\right)-4\rho_{in_{1}}\rho_{in_{2}}\rho_{out_{1}}\rho_{out_{2}}\cos\left(\Delta_{\text{out}}\right).\end{split}$
(140)

In terms of our particular solutions for the preselected and postselected
state, it would be given by

$\begin{split}&-\frac{R_{\max}\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}}{m}\cos\left(2F\left(t,t_{\text{in}}\right)\right)\\\
&-\frac{R_{0}^{2}\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}}{\lambda}\cos\left(\Delta_{\text{out}}\right).\end{split}$
(141)

In the denominator, $1+\vec{f}.\vec{i}$ remains the same

$\begin{split}&2\Delta\rho_{\text{in}}^{2}\rho_{out_{1}}\rho_{out_{2}}\cos\left(b\right)-2\Delta\rho_{\text{out}}^{2}\rho_{in_{1}}\rho_{in_{2}}+1\end{split}.$
(142)

Explicitly we have

$\begin{split}\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}\cos\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)-\frac{R_{0}^{2}R_{\max}}{\lambda
m}+1\end{split}.$ (143)

The amplification phenomenon is obtained as explained in the case of the weak
value of $\sigma^{z}$.

Let us compute now the argument of the weak value of the pure state. It is
given from ([7]) as

$\arg\hat{\Pi}_{r,w}=\arctan\frac{\vec{f}.(\vec{r}\wedge\vec{i})}{1+\vec{f}.\vec{r}+\vec{r}.\vec{i}+\vec{f}.\vec{i}}$
(144)

The berry phase would be given by $\arg\hat{\Pi}_{r,w}=$

$\arctan{\left(\frac{-\frac{R_{\max}\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}}{m}\cos\left(2F\left(t,t_{\text{in}}\right)\right)-\frac{R_{0}^{2}\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}}{\lambda}\cos\left(\Delta_{\text{out}}\right)}{-\sqrt{N^{2}-\frac{R_{0}^{4}}{\lambda^{2}}}\sin\left(2F\left(t,t_{\text{in}}\right)\right)+\sqrt{N^{2}-\frac{R_{\max}^{2}}{m^{2}}}\sin\left(\Delta_{\text{out}}\right)+1}\right)}.$
(145)

As a conclusion, this berry phase confirms the fact that the Universe
considered here is not flat since its argument is not 0 or a multiple of
$2\pi$ in all cases .

##  IV CONCLUSION

Our objectives in this paper were to apply the theory of weak measurement in
Einstein-Dirac system in the _Friedman-Lemaître-Robertson-Walker_ space. We
have elucidated numerous conclusions, particularly regarding the enlarged
scope of weak measurements in contrast to classical measurement paradigms.
Notably, our analysis revealed that the weak value of the energy-momentum
tensor extends beyond what is described classical energy-momentum concepts,
exhibiting unusual values as complex numbers in certain instances.

We have also shown that it is possible to amplify measurements with less
expensive equipments through the weak measurements process by strategically
assuming appropriate initial and final state vector. Furthermore, weak
measurements serve as a powerful tool for shedding some lights to the
geometric characteristics of the underlying space. By virtue of its
association with the Berry phase, weak measurements enabled us to ascertain
that the underlying space exhibits non-flat geometry, as evidenced by the weak
measurement of fermion wave functions.

Our analytical framework also demonstrates the effectiveness of weak
measurements in detecting the acceleration of the Universe, leveraging the
Two-State vector formalism and weak measurement techniques. Notably, we
established that the scale function may be understood as an outcome of the
postselection process.

We end this paper by noticing that it is possible to use this tool in
cosmology. This should be the next step for some future researches.

###### Acknowledgements.

We gratefully acknowledge the Fund for Scientific Research (FNRS) grant and
the Society of Jesus to have funded our research stay at the University of
Regensburg to finalize this article. We extend our deep gratitude to Professor
Felix Finster for his invaluable insights and engaging discussions during the
finalization of this work. We thank the Department of Mathematics at the
University of Regensburg for providing a conducive environment during our
research. Additionally, we thank Ramachandran Chittur Anantharaman and Charles
Modera for checking the orthographies and grammar of this paper.

##  Appendix A  Weak measurements of the Energy momentum

Let us make the computations of the energy momentum more explicit where
$\langle
T_{0}^{0}\rangle_{w}=\frac{\langle\psi_{out}(t)|T_{0}^{0}|\psi_{in}(t)\rangle}{\langle\psi_{out}(t)|\psi_{in}(t)\rangle}$
and $\langle
T_{j}^{j}\rangle_{w}=\frac{\langle\psi_{out}(t)|T_{j}^{j}|\psi_{in}(t)\rangle}{\langle\psi_{out}(t)|\psi_{in}(t)\rangle}.$
They are given as follows

$\begin{aligned} &\langle
T_{0}^{0}\rangle_{w}=\frac{\Bigl{[}mR(t)\left(\alpha_{\text{in}}(t)\overline{\alpha_{\text{out}}}(t)-\beta_{\text{in}}(t)\overline{\beta_{\text{out}}}(t)\right)-\lambda\left(\alpha_{\text{in}(t)}\overline{\beta_{\text{out}}}(t)+\beta_{\text{in}}(t)\overline{\alpha_{\text{out}}}(t)\right)\Bigr{]}}{R(t)\left(\overline{\alpha_{\text{out}}}(t)\alpha_{\text{in}}(t)+\overline{\beta_{\text{out}}}(t)\beta_{\text{in}}(t)\right)}\\\
\end{aligned},$ (146)

and

$\begin{aligned} &\langle
T_{j}^{j}\rangle_{w}=\frac{\lambda\left(\alpha_{\text{in}}(t)\overline{\beta_{\text{out}}}(t)+\beta_{\text{in}}(t)\overline{\alpha_{\text{out}}}(t)\right)}{3R(t)\left(\overline{\alpha_{\text{out}}}(t)\alpha_{\text{in}}(t)+\overline{\beta_{\text{out}}}(t)\beta_{\text{in}}(t)\right)}\\\
\end{aligned}.$ (147)

###  A.1 Computation of the preselected state at time t

The spinor at time $t$ is obtained by applying the evolutionary operator on a
spinor defined at a specific time. The case of preselected state at time $t$
is the preselected state at time $t_{\text{in}}$ on which the unitary operator
is applied. It is given by

$\left.\left(\begin{array}[]{c}\alpha_{\text{in}}(t)\\\
\beta_{\text{in}}(t)\end{array}\right)=U(t,t_{\text{in}})\left(\begin{array}[]{c}\alpha_{\text{in}}(t_{\text{in}})\\\
\beta_{\text{in}}(t_{\text{in}})\end{array}\right)\right.$

.
$U(t,t_{\text{in}})$ is given by
$U^{-1}(t)\left(\begin{array}[]{cc}e^{-iF(t,t_{\text{in}})}&0\\\
0&e^{iF(t,t_{\text{in}})}\end{array}\right)U(t_{in})$

The specific case wherein the preselected state is taken in the radiation
dominated Universe with $m=0$ will yield
$g(t_{\text{in}})=\frac{1}{2}\arctan(\frac{\lambda}{0*R(t_{\text{in}})})=\frac{\pi}{4}+k\pi$.
For $k=0$, the matrix will be given by
$U(t_{\text{in}})=\frac{\sqrt{2}}{2}\begin{pmatrix}1&-1\\\ 1&1\end{pmatrix}$.

This will lead to

$\begin{split}\alpha_{\text{in}}(t)=&\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\left[\cos{(g(t))}e^{-iF(t,t_{\text{in}})}+\sin{(g(t))}e^{iF(t,t_{\text{in}})}\right]+\\\
&\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\left[-\cos{(g(t))}e^{-iF(t,t_{\text{in}})}+\sin{(g(t))}e^{iF(t,t_{\text{in}})}\right].\end{split}$
(148)

and

$\begin{split}\beta_{\text{in}}(t)=&\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\left[-\sin{(g(t))}e^{-iF(t,t_{\text{in}})}+\cos{(g(t))}e^{iF(t,t_{\text{in}})}\right]+\\\
&\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\left[\sin{(g(t))}e^{-iF(t,t_{\text{in}})}+\cos{(g(t))}e^{iF(t,t_{\text{in}})}\right].\end{split}$
(149)

We remind here that
$F(t,t_{\text{in}})=\int^{t}_{t_{\text{in}}}\frac{\sqrt{R^{2}(t)m^{2}+\lambda^{2}}}{R(t)}dt$.

###  A.2 Computation of the postselected state at time t

The postselected state is taken from the dust dominated Universe wherein
$\lambda=0$. This will yield
$g(t)=\frac{1}{2}\arctan(\frac{0}{m*R(t_{\text{in}})})=0+k\pi$ . This will
lead, for $k=0$, to $U(t_{out})=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)$, the identity matrix.

Knowing that $\left.U^{\dagger}(t,t_{\text{out}})=U(t_{\text{out}},t)\right.$,

$\left.\left(\begin{array}[]{cc}\overline{\alpha_{\text{out}}}(t)&\overline{\beta_{\text{out}}}(t)\end{array}\right)\right.=\left(\begin{array}[]{cc}\overline{\alpha_{\text{out}}}(t_{\text{out}})&\overline{\beta_{\text{out}}}(t_{\text{out}})\end{array}\right)U(t_{\text{out}},t).$

$U(t_{\text{out}},t)$ is given by
$U^{-1}(t_{\text{out}})\left(\begin{array}[]{cc}e^{-iF(t_{\text{out}},t)}&0\\\
0&e^{iF(t_{\text{out}},t)}\end{array}\right)U(t).$

This will lead to:

$\begin{split}\overline{\alpha_{\text{out}}}(t)=\overline{\alpha_{\text{out}}}(t_{\text{out}})\cos{(g(t))}e^{-iF(t_{\text{out}},t)}+\overline{\beta_{\text{out}}}(t_{\text{out}})\sin{(g(t))}e^{iF(t_{\text{out}},t)}.\end{split}$
(150)

and :

$\begin{split}\overline{\beta_{\text{out}}}(t)=-\overline{\alpha_{\text{out}}}(t_{\text{out}})\sin{(g(t))}e^{-iF(t_{\text{out}},t)}+\overline{\beta_{\text{out}}}(t_{\text{out}})\cos{(g(t))}e^{iF(t_{\text{out}},t)}.\end{split}$
(151)

###  A.3  Computation of numerators and denominators of the energy-momentum
Tensor

In terms of
$\alpha_{\text{in}}(t_{\text{in}})\quad\text{and}\quad\alpha_{\text{out}}(t_{\text{out}})$
where

$\displaystyle A$
$\displaystyle=e^{i\left(F(t_{\text{out}},t_{\text{in}})\right)},\quad\bar{A}=e^{-i\left(F(t_{\text{out}},t_{\text{in}})\right)},$
$\displaystyle B$
$\displaystyle=e^{i\left(F(t,t_{\text{in}})+F(t,t_{\text{out}})\right)},\quad\bar{B}=e^{-i\left(F(t,t_{\text{in}})+F(t,t_{\text{out}})\right)}$

  * •

$\left(\overline{\alpha_{\text{out}}}(t)\alpha_{\text{in}}(t)+\overline{\beta_{\text{out}}}(t)\beta_{\text{in}}(t)\right)$
is equal to:

$\begin{split}&\frac{\sqrt{2}}{2}\left(\overline{\alpha_{\text{out}}}(t_{\text{out}})\alpha_{\text{in}}(t_{\text{in}})-\overline{\alpha_{\text{out}}}(t_{\text{out}})\beta_{\text{in}}(t_{\text{in}})\right)\bar{A}\\\
&+\frac{\sqrt{2}}{2}\left(\overline{\beta_{\text{out}}}(t_{\text{out}})\beta_{\text{in}}(t_{\text{in}})+\overline{\beta_{\text{out}}}(t_{\text{out}})\alpha_{\text{in}}(t_{\text{in}})\right)A.\end{split}$
(152)

  * •

$\left(\alpha_{\text{in}}(t)\overline{\alpha_{\text{out}}}(t)-\beta_{\text{in}}(t)\overline{\beta_{\text{out}}}(t)\right)$
is equal to:

$\begin{split}&\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\overline{\alpha_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}B+\cos{(2g(t))}\bar{A}\right]\\\
&+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\overline{\beta_{\text{out}}}(t_{\text{out}})\left[-\sin{(2g(t))}\bar{B}-\cos{(2g(t))}A\right]\\\
&+\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\overline{\beta_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}\bar{B}-\cos{(2g(t))}A\right]\\\
&+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\overline{\alpha_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}B-\cos{(2g(t))}\bar{A}\right].\end{split}$
(153)

  * •

$\left(\alpha_{\text{in}}(t)\overline{\beta_{\text{out}}}(t)+\beta_{\text{in}}(t)\overline{\alpha_{\text{out}}}(t)\right)$
is equal to:

$\begin{split}&\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\overline{\alpha_{\text{out}}}(t_{\text{out}})\left[-\sin{(2g(t))}\bar{A}+\cos{(2g(t))}B\right]\\\
&+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\overline{\beta_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}A-\cos{(2g(t))}\bar{B}\right]\\\
&+\frac{\sqrt{2}}{2}\alpha_{\text{in}}(t_{\text{in}})\overline{\beta_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}A+\cos{(2g(t))}\bar{B}\right]\\\
&+\frac{\sqrt{2}}{2}\beta_{\text{in}}(t_{\text{in}})\overline{\alpha_{\text{out}}}(t_{\text{out}})\left[\sin{(2g(t))}\bar{A}+\cos{(2g(t))}B\right].\end{split}$
(154)

##  Appendix B Weak measurements of $\sigma^{z}$ operator

The weak measurement of the generator of $SU(2)$ is given by

$\sigma_{r,w}=\frac{\vec{f}.\vec{r}+\vec{r}.\vec{i}+i\vec{f}.(\vec{r}\wedge\vec{i})}{\frac{1}{2}(1+\vec{f}.\vec{i})}.$
(155)

In our case, $\vec{r}=(\begin{array}[]{ccc}0&0&1\end{array})^{T}$ to get
$\sigma^{z}$. Following Kofman in [33], the vectors $\vec{f}$ and $\vec{i}$
are computed as follow: $\vec{w_{f}}=$

$\begin{pmatrix}\langle\xi_{\text{out}}(t)|\sigma^{1}|\xi_{\text{out}}(t)\rangle\\\
\langle\xi_{\text{out}}(t)|\sigma^{2}|\xi_{\text{out}}(t)\rangle\\\
\langle\xi_{\text{out}}(t)|\sigma^{3}|\xi_{\text{out}}(t)\rangle\end{pmatrix}=\left(\begin{array}[]{c}2\text{Re}\left(\alpha_{\text{out}}(t)\bar{\beta}_{\text{out}}(t)\right)\\\
2\text{Im}\left(\alpha_{\text{out}}(t)\bar{\beta}_{\text{out}}(t)\right)\\\
|\alpha_{\text{out}(t)}|^{2}-|\beta_{\text{out}(t)}|^{2}\end{array}\right).$

wherein $\xi_{\text{out}}(t)=\left(\begin{array}[]{c}\alpha_{\text{out}}(t)\\\
\beta_{\text{out}}(t)\end{array}\right).$

Some particular products are given as:

$\begin{split}&\alpha_{\text{out}}(t)\bar{\beta}_{\text{out}}(t)=-\left(|\alpha_{\text{out}}(t_{\text{out}})|^{2}-|\beta_{\text{out}}(t_{\text{out}})|^{2}\right)\sin(g(t))\cos(g(t))\\\
&+\overline{\alpha_{\text{out}}}(t_{\text{out}})\beta_{\text{out}}(t_{\text{out}})\cos^{2}(g(t))e^{-2iF(t_{\text{out}},t)}\\\
&-\overline{\beta_{\text{out}}}(t_{\text{out}})\alpha_{\text{out}}(t_{\text{out}})\sin^{2}(g(t))e^{2iF(t_{\text{out}},t)},\end{split}$
(156)

which in terms of
$\alpha_{\text{out}}(t_{\text{out}})=\rho_{out_{1}}e^{i\eta_{out_{1}}}$ and
$\beta_{\text{out}}(t_{\text{out}})=\rho_{out_{2}}e^{i\eta_{out_{2}}}$ will be

$\begin{split}&\text{Re}\left(\alpha_{\text{out}}(t)\bar{\beta}_{\text{out}}(t)\right)=\frac{1}{2}\left(\rho_{out_{2}}^{2}-\rho_{out_{1}}^{2}\right)\sin{(2g(t))}\\\
&+\rho_{out_{1}}\rho_{out_{2}}\cos{(2g(t))}\cos{(\eta_{out_{2}}-\eta_{out_{1}}-2F(t_{\text{out}},t))},\end{split}$
(157)

and the imaginary part in terms of
$\rho_{out_{1}},\rho_{out_{2}},\eta_{out_{2}},\eta_{out_{1}}$ will be

$\begin{split}&\text{Im}\left(\alpha_{\text{out}}(t)\bar{\beta}_{\text{out}}(t)\right)=\frac{1}{2}\left(\rho_{out_{2}}^{2}-\rho_{out_{1}}^{2}\right)\sin{(2g(t))}\\\
&+\rho_{out_{1}}\rho_{out_{2}}\sin{(\eta_{out_{2}}-\eta_{out_{1}}-2F(t_{\text{out}},t))}.\end{split}$
(158)

In the same way, the preselected state vector is computed as follows
$\vec{w_{i}}=$

$\begin{pmatrix}\langle\xi_{\text{in}}(t)|\sigma^{1}|\xi_{\text{in}}(t)\rangle\\\
\langle\xi_{\text{in}}(t)|\sigma^{2}|\xi_{\text{in}}(t)\rangle\\\
\langle\xi_{\text{in}}(t)|\sigma^{3}|\xi_{\text{in}}(t)\rangle\end{pmatrix}=\left(\begin{array}[]{c}2\text{Re}\left(\alpha_{\text{in}}(t)\bar{\beta}_{\text{in}}(t)\right)\\\
2\text{Im}\left(\alpha_{\text{in}}(t)\bar{\beta}_{\text{in}}(t)\right)\\\
|\alpha_{\text{in}(t)}|^{2}-|\beta_{\text{in}(t)}|^{2}\end{array}\right),$

wherein $\xi_{\text{in}}(t)=\left(\begin{array}[]{c}\alpha_{\text{in}}(t)\\\
\beta_{\text{in}}(t)\end{array}\right)$

$\begin{split}&\alpha_{\text{in}}(t)\bar{\beta}_{\text{in}}(t)\\\
&=\left(|\alpha_{\text{in}}(t_{\text{in}})|^{2}-|\beta_{\text{in}}(t_{\text{in}})|^{2}\right)\\\
&\quad\times\Biggl{(}\cos^{2}(g(t))e^{-2iF(t,t_{\text{in}})}-\sin^{2}(g(t))e^{2iF(t,t_{\text{in}})}\Biggr{)}\\\
&+\alpha_{\text{in}}(t_{\text{in}})\overline{\beta}_{\text{in}}(t_{\text{in}})\\\
&\quad\times\Biggl{(}\sin(2g(t))+\sin^{2}(g(t))e^{2iF(t,t_{\text{in}})}+\cos^{2}(g(t))e^{-2iF(t,t_{\text{in}})}\Biggr{)}\\\
&+\beta_{\text{in}}(t_{\text{in}})\overline{\alpha}_{\text{in}}(t_{\text{in}})\\\
&\quad\times\Biggl{(}\sin(2g(t))-\sin^{2}(g(t))e^{2iF(t,t_{\text{in}})}-\cos^{2}(g(t))e^{-2iF(t,t_{\text{in}})}\Biggr{)},\end{split}$
(159)

which in terms of
$\alpha_{\text{in}}(t_{\text{in}})=\rho_{in_{1}}e^{\eta_{in_{1}}}$ and
$\beta_{\text{in}}(t_{\text{in}})=\rho_{in_{2}}e^{\eta_{in_{2}}}$ will be

$\begin{split}&\text{Re}\left(\alpha_{\text{in}}(t)\bar{\beta}_{\text{in}}(t)\right)=\left(\rho_{in_{1}}^{2}-\rho_{in_{2}}^{2}\right)\cos{(2g(t))}\cos{(2F(t,t_{\text{in}}))}\\\
&+2\rho_{in_{1}}\rho_{in_{2}}\left[\sin{(2g(t))}\cos{(\eta_{in_{1}}-\eta_{in_{2}})}\right.\\\
&\left.-\cos{(2g(t))}\sin{(\eta_{in_{1}}-\eta_{in_{2}})}\sin{(2F(t,t_{\text{in}}))}\right]\end{split}$
(160)

and the imaginary part will be

$\begin{split}&\text{Im}\left(\alpha_{\text{in}}(t)\bar{\beta}_{\text{in}}(t)\right)=\left(\rho_{in_{2}}^{2}-\rho_{in_{1}}^{2}\right)\sin{(2F(t,t_{\text{in}}))}\\\
&+2\rho_{in_{1}}\rho_{in_{2}}\sin{(\eta_{in_{1}}-\eta_{in_{2}})}\cos{(2F(t,t_{\text{in}}))}.\end{split}$
(161)

In our case, $\vec{r}=(\begin{array}[]{ccc}0&0&1\end{array})$ to get
$\sigma^{z}$. Following Kofman in [33], the vectors $\vec{f}$ and $\vec{i}$
are computed as follows

$\vec{w_{f}}=\begin{pmatrix}\langle\xi_{\text{out}}(t)|\sigma^{1}|\xi_{\text{out}}(t)\rangle\\\
\langle\xi_{\text{out}}(t)|\sigma^{2}|\xi_{\text{out}}(t)\rangle\\\
\langle\xi_{\text{out}}(t)|\sigma^{3}|\xi_{\text{out}}(t)\rangle\end{pmatrix}=\left(\begin{array}[]{c}2\text{Re}\left(\alpha_{\text{out}}(t)\bar{\beta}_{\text{out}}(t)\right)\\\
2\text{Im}\left(\alpha_{\text{out}}(t)\bar{\beta}_{\text{out}}(t)\right)\\\
|\alpha_{\text{out}(t)}|^{2}-|\beta_{\text{out}(t)}|^{2}\end{array}\right),$

wherein $\xi_{\text{out}}(t)=\left(\begin{array}[]{c}\alpha_{\text{out}}(t)\\\
\beta_{\text{out}}(t)\end{array}\right)$. We obtain the following vector
$\vec{w_{f}}$ components that are given as:

$\displaystyle\left\\{\begin{array}[]{lll}{w_{f}}^{1}&=&-\Delta\rho^{2}_{out}\sin{(2g(t))}+2\rho_{out_{1}}\rho_{out_{2}}\cos{(2g(t))}\cos{(\Delta_{out})}\\\
{w_{f}}^{2}&=&2\rho_{out_{1}}\rho_{out_{2}}\sin{(\Delta_{out})}\\\
{w_{f}}^{3}&=&\Delta\rho^{2}_{out}\cos{(2g(t))}+2\rho_{out_{1}}\rho_{out_{2}}\sin{(2g(t))}\cos{(\Delta_{out})},\end{array}\right.$
(165)

where $(\eta_{out_{1}}-\eta_{out_{2}}+2F(t_{\text{out}},t))=\Delta_{out}$,
$\rho_{out_{1}}^{2}-\rho_{out_{2}}^{2}=\Delta\rho^{2}_{out}$.
Its norm is
$|\vec{w_{f}}|=\rho_{out_{1}}^{2}+\rho_{out_{2}}^{2}=\lambda^{2}-\frac{1}{4}$.
This implies that the unit vector $\vec{f}$ would be given by

$\vec{f}=\frac{1}{\lambda^{2}-\frac{1}{4}}\vec{w_{f}}.$

In the same way, the preselected state is given by:

$\vec{w_{i}}=\begin{pmatrix}\langle\xi_{\text{in}}(t)|\sigma^{1}|\xi_{\text{in}}(t)\rangle\\\
\langle\xi_{\text{in}}(t)|\sigma^{2}|\xi_{\text{in}}(t)\rangle\\\
\langle\xi_{\text{in}}(t)|\sigma^{3}|\xi_{\text{in}}(t)\rangle\end{pmatrix}=\left(\begin{array}[]{c}2\text{Re}\left(\alpha_{\text{in}}(t)\bar{\beta}_{\text{in}}(t)\right)\\\
2\text{Im}\left(\alpha_{\text{in}}(t)\bar{\beta}_{\text{in}}(t)\right)\\\
|\alpha_{\text{in}(t)}|^{2}-|\beta_{\text{in}(t)}|^{2}\end{array}\right).$

wherein $\xi_{\text{in}}(t)=\left(\begin{array}[]{c}\alpha_{\text{in}}(t)\\\
\beta_{\text{in}}(t)\end{array}\right)$.
The vector $\vec{f}$ is equal to $\frac{\vec{w_{f}}}{|\vec{w_{f}}|},\text{and
}\vec{i}=\frac{\vec{w_{i}}}{|\vec{w_{i}}|}$. They represent the unit vectors
on the Bloch sphere of post-selected and pre-selected state. In terms of the
components, the $\sigma^{z}$ weak measurement is given by:

$\sigma_{r,w}=\frac{{w_{f}}^{3}+{w_{i}}^{3}+i(-{w_{f}}^{1}{w_{i}}^{2}+{w_{f}}^{2}{w_{i}}^{1})}{\frac{1}{2}(1+{w_{f}}^{1}{w_{i}}^{1}+{w_{f}}^{2}{w_{i}}^{2}+{w_{f}}^{3}{w_{i}}^{3})}$
(166)

We obtain the following vector $\vec{w_{i}}=$

$\displaystyle\left\\{\begin{array}[]{lll}{w_{i}}^{1}&=&\Delta\rho_{in}^{2}\cos{(2g(t))}\cos{(2F(t,t_{\text{in}}))}+2\rho_{in_{1}}\rho_{in_{2}}\left(\sin{(2g(t))}\cos{(\Delta_{in})}+\cos(2g)\sin(\Delta_{in})\sin{(2F(t,t_{\text{in}}))}\right)\\\
{w_{i}}^{2}&=&-\Delta\rho_{in}^{2}\sin{(2F(t,t_{\text{in}}))}+2\rho_{in_{1}}\rho_{in_{2}}\sin{(\Delta_{in})}\cos{(2F(t,t_{\text{in}}))}\\\
{w_{i}}^{3}&=&\Delta\rho_{in}^{2}\sin{(2g(t))}\cos{(2F(t,t_{\text{in}}))}+2\rho_{in_{1}}\rho_{in_{2}}\left(-\cos{(2g(t))}\cos{(\Delta_{in})}+\sin{(2g(t))}\sin{(\Delta_{in})}\sin{(2F(t,t_{\text{in}}))}\right),\end{array}\right.$
(170)

wherein $\eta_{in_{1}}-\eta_{in_{2}}=\Delta_{in}$,
$\rho_{in_{1}}^{2}-\rho_{in_{2}}^{2}=\Delta\rho^{2}_{in}$,
$|\vec{w_{f}}|=\lambda^{2}-\frac{1}{4}$

The real and the imaginary part of the numerator of the weak measurement of
$\sigma^{z}$ operator , $\sigma_{r,w}$, are given respectively by:

${w_{f}}^{3}+{w_{i}}^{3}=$

$\begin{split}&\Delta\rho_{\text{in}}^{2}\cos\left(2F\left(t,t_{\text{in}}\right)\right)\sin(2g(t))+\Delta\rho_{\text{out}}^{2}\cos(2g(t))+2\rho_{in_{1}}\rho_{in_{2}}\left(-\cos(2g(t))\cos\left(\Delta_{\text{in}}\right)+\sin\left(2F\left(t,t_{\text{in}}\right)\right)\sin(2g(t))\sin\left(\Delta_{\text{in}}\right)\right)+\\\
&2\rho_{out_{1}}\rho_{out_{2}}\sin(2g(t))\cos\left(\Delta_{\text{out}}\right),\end{split}$
(171)

$-{w_{f}}^{1}{w_{i}}^{2}+{w_{f}}^{2}{w_{i}}^{1}=$

$\begin{split}&2\rho_{in_{1}}\rho_{in_{2}}\left(\Delta\rho_{\text{out}}^{2}\cos\left(2F\left(t,t_{\text{in}}\right)\right)\sin(2g(t))\sin\left(\Delta_{\text{in}}\right)+2\rho_{out_{1}}\rho_{out_{2}}\left(-\cos(2g(t))\cos\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)\sin\left(\Delta_{\text{in}}\right)+\cos\left(\Delta_{\text{in}}\right)\sin(2g(t))\sin\left(\Delta_{\text{out}}\right)\right)\right)+\\\
&\Delta\rho_{\text{in}}^{2}\left(-\Delta\rho_{\text{out}}^{2}\sin\left(2F\left(t,t_{\text{in}}\right)\right)\sin(2g(t))+2\rho_{out_{1}}\rho_{out_{2}}\cos(2g(t))\sin\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)\right)\end{split},$
(172)

and $1+{w_{f}}^{1}{w_{i}}^{1}+{w_{f}}^{2}{w_{i}}^{2}+{w_{f}}^{3}{w_{i}}^{3}$
is given by:

$\begin{split}1-2\Delta\rho_{\text{out}}^{2}\rho_{in_{1}}\rho_{in_{2}}\cos\left(\Delta_{\text{in}}\right)+2\rho_{out_{1}}\rho_{out_{2}}\left(\Delta\rho_{\text{in}}^{2}\cos\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)+2\rho_{in_{1}}\rho_{in_{2}}\sin\left(\Delta_{\text{in}}\right)\sin\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)\right).\end{split}$
(173)

##  Appendix C Berry phase

The weak value of a pure is given by:

$\hat{\Pi}_{r,w}=\frac{1+\vec{f}.\vec{r}+\vec{r}.\vec{i}+\vec{f}.\vec{i}+i\vec{f}.(\vec{r}\wedge\vec{i})}{\frac{1}{2}(1+\vec{f}.\vec{i})}.$
(174)

We obtain from $\vec{f}=\frac{\vec{w_{f}}}{|\vec{w_{f}}|},\text{and
}\vec{i}=\frac{\vec{w_{i}}}{|\vec{w_{i}}|}$ already computed in general terms:

$1+\vec{f}.\vec{r}+\vec{r}.\vec{i}+\vec{f}.\vec{i}=$

$\begin{split}&1+\cos(\varphi)\sin(\theta)\left(\Delta\rho_{\text{in}}^{2}\cos(2g(t))\cos\left(2F\left(t,t_{\text{in}}\right)\right)-\Delta\rho_{\text{out}}^{2}\sin(2g(t))\right.\\\
&\left.+2\rho_{in_{1}}\rho_{in_{2}}\left(\cos(2g(t))\sin\left(\Delta_{\text{in}}\right)\sin\left(2F\left(t,t_{\text{in}}\right)\right)+\sin(2g(t))\cos\left(\Delta_{\text{in}}\right)\right)+2\cos(2g(t))\cos\left(\Delta_{\text{out}}\right)\rho_{out_{1}}\rho_{out_{2}}\right)\\\
&+\cos(\theta)\left(\Delta\rho_{\text{in}}^{2}\sin(2g(t))\cos\left(2F\left(t,t_{\text{in}}\right)\right)+\Delta\rho_{\text{out}}^{2}\cos(2g(t))+2\rho_{in_{1}}\rho_{in_{2}}\left(-\cos(2g(t))\cos\left(\Delta_{\text{in}}\right)\right.\right.\\\
&\left.\left.\sin(2g(t))\sin\left(\Delta_{\text{in}}\right)+\sin\left(2F\left(t,t_{\text{in}}\right)\right)\right)+2\sin(2g(t))\cos\left(\Delta_{\text{out}}\right)\rho_{out_{1}}\rho_{out_{2}}\right)+\\\
&\sin(\theta)\sin(\varphi)\left(2\rho_{in_{1}}\rho_{in_{2}}\sin\left(\Delta_{\text{in}}\right)\cos\left(2F\left(t,t_{\text{in}}\right)\right)+\left(-\Delta\rho_{\text{in}}^{2}\right)\sin\left(2F\left(t,t_{\text{in}}\right)\right)+2\sin\left(\Delta_{\text{out}}\right)\rho_{out_{1}}\rho_{out_{2}}\right)\end{split}.$
(175)

The imaginary part, $\vec{f}.(\vec{r}\wedge\vec{i})$ is given by

$\begin{split}&\sin(\theta)\sin(\varphi)\left(\left(-\Delta\rho_{\text{in}}^{2}\right)\Delta\rho_{\text{out}}^{2}\cos\left(2F\left(t,t_{\text{in}}\right)\right)-2\rho_{in_{1}}\rho_{in_{2}}\left(\Delta\rho_{\text{out}}^{2}\sin\left(\Delta_{\text{in}}\right)\sin\left(2F\left(t,t_{\text{in}}\right)\right)+2\cos\left(\Delta_{\text{in}}\right)\cos\left(\Delta_{\text{out}}\right)\rho_{out_{1}}\rho_{out_{2}}\right)\right)\\\
&+\cos(\theta)\left(2\rho_{in_{1}}\rho_{in_{2}}\Delta\rho_{\text{out}}^{2}\left(\sin(2g(t))\sin\left(\Delta_{\text{in}}\right)\cos\left(2F\left(t,t_{\text{in}}\right)\right)+2\rho_{out_{1}}\rho_{out_{2}}\left(-\cos(2g(t))\sin\left(\Delta_{\text{in}}\right)\cos\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)\right.\right.\right.\\\
&+\left.\left.\left.\sin(2g(t))\cos\left(\Delta_{\text{in}}\right)\sin\left(\Delta_{\text{out}}\right)\right)\right)+\Delta\rho_{\text{in}}^{2}\left(2\rho_{out_{1}}\rho_{out_{2}}\cos(2g(t))\sin\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)-\Delta\rho_{\text{out}}^{2}\sin(2g(t))\sin\left(2F\left(t,t_{\text{in}}\right)\right)\right)\right)\\\
&+\cos(\varphi)\sin(\theta)\left(2\rho_{in_{1}}\rho_{in_{2}}\Delta\rho_{\text{out}}^{2}\left(\cos(2g(t))\sin\left(\Delta_{\text{in}}\right)\cos\left(2F\left(t,t_{\text{in}}\right)\right)+2\rho_{out_{1}}\rho_{out_{2}}\left(\sin(2g(t))\sin\left(\Delta_{\text{in}}\right)\cos\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)\right.\right.\right.\\\
&+\left.\left.\left.\cos(2g(t))\cos\left(\Delta_{\text{in}}\right)\sin\left(\Delta_{\text{out}}\right)\right)\right)-\Delta\rho_{\text{in}}^{2}\left(2\rho_{out_{1}}\rho_{out_{2}}\sin(2g(t))\sin\left(2F\left(t,t_{\text{in}}\right)+\Delta_{\text{out}}\right)+\Delta\rho_{\text{out}}^{2}\cos(2g(t))\sin\left(2F\left(t,t_{\text{in}}\right)\right)\right)\right).\end{split}$
(176)

The denominator is the same as computed for the weak measurements of
$\sigma^{z}$.

## References

  * Finster and Hainzl [2011] F. Finster and C. Hainzl, A spatially homogeneous and isotropic einstein–dirac cosmology, Journal of mathematical physics 52 (2011).
  * Kipnis [1991] N. Kipnis,  _History of the Principle of Interference of Light_, Science networks historical studies (Springer, 1991).
  * Aharonov  _et al._ [2017] Y. Aharonov, E. Cohen, F. Colombo, T. Landsberger, I. Sabadini, D. C. Struppa, and J. Tollaksen, Finally making sense of the double-slit experiment, Proceedings of the National Academy of Sciences 114, 6480 (2017).
  * Tamir and Cohen [2013] B. Tamir and E. Cohen, Introduction to weak measurements and weak values, Quanta 2, 7 (2013).
  * Kocsis  _et al._ [2011] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Observing the average trajectories of single photons in a two-slit interferometer, Science 332, 1170 (2011).
  * Rozema  _et al._ [2012] L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, Violation of heisenberg’s measurement-disturbance relationship by weak measurements, Physical review letters 109, 100404 (2012).
  * Ferraz  _et al._ [2022] L. B. Ferraz, D. L. Lambert, and Y. Caudano, Geometrical interpretation of the argument of weak values of general observables in n-level quantum systems, Quantum Science and Technology 7, 045028 (2022).
  * Ellis and Rothman [2010] G. F. R. Ellis and T. Rothman, Time and spacetime: The crystallizing block universe, International Journal of Theoretical Physics 49, 988 (2010).
  * Aharonov and Gruss [2005] Y. Aharonov and E. Y. Gruss, Two-time interpretation of quantum mechanics, arXiv preprint quant-ph/0507269 (2005).
  * Davies [2014] P. Davies, Quantum weak measurements and cosmology, in  _Quantum Theory: A Two-Time Success Story: Yakir Aharonov Festschrift_ (Springer, 2014) pp. 101–112.
  * DeWitt [1965] B. S. DeWitt,  _Dynamical theory of groups and fields_ (Gordon and Breach, 1965).
  * Boulware [1975] D. G. Boulware, Quantum field theory in schwarzschild and rindler spaces, Physical Review D 11, 1404 (1975).
  * Hartle and Hu [1979] J. Hartle and B. Hu, Quantum effects in the early universe. ii. effective action for scalar fields in homogeneous cosmologies with small anisotropy, Physical Review D 20, 1772 (1979).
  * Cisowski  _et al._ [2022] C. Cisowski, J. Götte, and S. Franke-Arnold, Colloquium: Geometric phases of light: Insights from fiber bundle theory, Reviews of Modern Physics 94, 031001 (2022).
  * Anastopoulos [2024] C. Anastopoulos, Final states in quantum cosmology: Cosmic acceleration as a quantum post-selection effect, arXiv preprint arXiv:2401.07662 (2024).
  * Ribas  _et al._ [2005] M. Ribas, F. Devecchi, and G. Kremer, Fermions as sources of accelerated regimes in cosmology, Physical Review D 72, 123502 (2005).
  * Ochs and Sorg [1993] U. Ochs and M. Sorg, Fermions and expanding universe, International journal of theoretical physics 32, 1531 (1993).
  * Saha [2005] B. Saha, Spinor field and accelerated regimes in cosmology, arXiv preprint gr-qc/0512050 (2005).
  * Ribas  _et al._ [2007] M. Ribas, F. Devecchi, and G. Kremer, Cosmological model with non-minimally coupled fermionic field, Europhysics Letters 81, 19001 (2007).
  * Rakhi  _et al._ [2010] R. Rakhi, G. Vijayagovindan, and K. Indulekha, A cosmological model with fermionic field, International Journal of Modern Physics A 25, 2735 (2010).
  * Kremer and de Souza [2013] G. M. Kremer and R. C. de Souza, Cosmological models with spinor and scalar fields by noether symmetry approach, arXiv preprint arXiv:1301.5163 (2013).
  * Aharonov  _et al._ [1964] Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Time symmetry in the quantum process of measurement, Physical Review 134, B1410 (1964).
  * Aharonov and Vaidman [2008] Y. Aharonov and L. Vaidman, The two-state vector formalism: an updated review, Time in quantum mechanics , 399 (2008).
  * Dressel  _et al._ [2014] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Colloquium: Understanding quantum weak values: Basics and applications, Reviews of Modern Physics 86, 307 (2014).
  * Hartle and Hawking [1983] J. B. Hartle and S. W. Hawking, Wave function of the universe, Physical Review D 28, 2960 (1983).
  * Hosten and Kwiat [2008] O. Hosten and P. Kwiat, Observation of the spin hall effect of light via weak measurements, Science 319, 787 (2008).
  * Hu and Zhang [2017] M.-J. Hu and Y.-S. Zhang, Gravitational waves detection via weak measurements, arXiv preprint arXiv:1707.00886 (2017).
  * Dixon  _et al._ [2009] P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Ultrasensitive beam deflection measurement via interferometric weak value amplification, Physical review letters 102, 173601 (2009).
  * Finster and Reintjes [2009] F. Finster and M. Reintjes, The dirac equation and the normalization of its solutions in a closed friedmann–robertson–walker universe, Classical and Quantum Gravity 26, 105021 (2009).
  * Plebanski and Krasinski [2006] J. Plebanski and A. Krasinski,  _An introduction to general relativity and cosmology_ (Cambridge University Press, 2006).
  * Moore [2014] T. A. Moore,  _Relativité générale_ (De Boeck, 2014).
  * Cormann  _et al._ [2016] M. Cormann, M. Remy, B. Kolaric, and Y. Caudano, Revealing geometric phases in modular and weak values with a quantum eraser, Physical Review A 93, 042124 (2016).
  * Kofman  _et al._ [2012] A. G. Kofman, S. Ashhab, and F. Nori, Nonperturbative theory of weak pre-and post-selected measurements, Physics Reports 520, 43 (2012).
  * Cormann and Caudano [2017] M. Cormann and Y. Caudano, Geometric description of modular and weak values in discrete quantum systems using the majorana representation, Journal of Physics A: Mathematical and Theoretical 50, 305302 (2017).