# Conserved spin operator of Dirac’s theory in spatially flat FLRW spacetimes

Ion I. Cotăescu
West University of Timişoara,
V. Parvan Ave. 4, RO-300223 Timişoara Corresponding author E-mail:
<EMAIL_ADDRESS>
###### Abstract

New conserved spin and orbital angular momentum operators of Dirac’s theory on
spatially flat FLRW spacetimes are proposed generalizing thus recent results
concerning the role of Pryce’s spin operator in the flat case [I. I. Cotăescu,
Eur. Phys. J. C (2022) 82:1073]. These operators split the conserved total
angular momentum generating the new spin and orbital symmetries that form the
rotations of the isometry groups. The new spin operator is defined and studied
in active mode with the help of a suitable spectral representation giving its
Fourier transfor. Moreover, in the same manner is defined the operator of the
fermion polarization. The orbital angular momentum is derived in passive mode
using a new method, inspired by Wigner’s theory of induced representations,
but working properly only for global rotations. In this approach the
quantization is performed finding that the one-particle spin and orbital
angular momentum operators have the same form in any FLRW spacetime regardless
their concrete geometries given by various scale factors.

##  1 Introduction

The spin operator is defined naturally in the non-relativistic quantum
mechanics where this is conserved commuting with the Hamiltonian operator. In
contrast, the would-be spin operator of Dirac’s theory in the relativistic
quantum mechanics (RQM) whose components are the rotation generators of the
$SL(2,\mathbb{C})$ group does not commute with the the Dirac Hamiltonian
neither in Minkowski flat spacetime nor in spatially flat Friedman-Lemaître-
Robertson Walker (FLRW) manifolds. For this reason many authors studied the
possibility of defining new conserved spin and orbital angular momentum
operators splitting properly the conserved operator of total angular momentum.

This problem was solved in Minkowski spacetime by Pryce which proposed the
desired conserved spin operator in momentum representation (MR) giving its
Fourier transform many time ago. As this result was written in the active mode
in which the operators act on the mode spinors of MR it was less relevant
being ignored for more than seven decades. Recently, by using a new spectral
representation of the Fourier operators we re-defined this operator in
configuration representation (CR) showing that this is just the spin operator
we need in Dirac’s theory on Mikowski spacetime [2]. Moreover, in the same
manner we defined the operator of fermion polarization in the most general
case when the direction of spin projections depends on momentum. In other
respects, for avoiding the difficulties in manipulating operators in active
mode we proposed the method of associated operators in passive mode relating
any operator acting on the Dirac field in CR to a pair of operators acting
directly on the wave spinors in MR [3]. In this framework we studied the
entire operator algebra of Dirac’s theory in special relativity performing
then the quantization giving the conserved one-particle operators as well as
the operators having oscillating terms as those producing zitterbewegung [3].

In the present paper we would like to extend this study to any spatially flat
FLRW spacetime but restricting ourselves only to the conserved operators
generating the spin and orbital symmetries, i. e. the new conserved spin and
orbital angular momentum operators we have to study here. For this purpose we
use both the spectral representations in active mode and the method of
associated operators in passive mode. In Minkowski spacetime these methods are
based on the special structure of the mode spinors constructed according to
Wigner’s theory of induced representations [4, 5, 6, 7, 8]. As this theory
cannot be applied in the case of FLRW spacetimes where we do not have boost
transformations, we are forced to use an alternative approach but which works
properly only for global rotations. In this manner we derive the associated
isometry generators pointing out the associated spin and orbital angular
momentum operators we need for applying the Bogolyubov method of quantization
[9]. Finally we find that the conserved one-particle operators of the quantum
field theory (QFT) have the same forms in any FLRW spacetime as being
independent on time and implicitly on the scale factors. In this paper we use
exclusively conformal frames with conformal coordinates and diagonal tetrad
gauge.

We start in the next section presenting the Lagrangian theory of Dirac’s free
field on FLRW spacetimes defining the new spin and orbital symmetries
generated by the conserved spin and orbital angular momentum operators. The
third section is devoted to the structure of the mode spinors in MR separating
the time modulation functions governing their time evolution from the spin
parts constructed by using a new method inspired by Wigner’s one applied in
the flat case. We specify that these mode spinors are defined in rest frames
only if we set the rest frame vacuum we proposed recently [10]. In the next
section we adopt the active mode in which the operators act on the mode
spinors in CR. Consequently, for deriving the action of the components of the
new spin operator we must resort to our spectral representation for deriving
their Fourier transforms finding how these depend on the time modulation
functions. However, in spite of this apparent time dependence the spin
operator is conserved via Noether’s theorem. Moreover, in Minkowski spacetime
the Fourier transforms of the spin components become just those proposed by
Pryce in MR [1].

Observing then the difficulties in deriving the above results we abandon the
active mode developing in Sec. 5 the mentioned method of associated operators
in passive mode. This gives us the opportunity of pointing out that the
covariant representation of the Dirac field in CR is equivalent with a direct
product of representations carried by the wave spinors in MR, just as in
Wigner’s theory in Minkowski spacetime. Hereby we derive the associated
operators to the isometry generators preparing the quantization procedure
performed in Sec. 6 where we derive the corresponding one-particle operators
of QFT which have the same expressions in all the FRLW spacetimes including
the Minkowski one. Finally we present our concluding remarks. Some technical
details related to the $SL(2,\mathbb{C})$ are presented in an Appendix.

##  2 Covariant Dirac field on spatially flat FLRW spacetimes

The Dirac free field on $(1+3)$-dimensional local Minkowskian spacetimes $M$
may be defined in frames $\\{x;e\\}$ formed by local chats, $\\{x\\}$, and
orthogonal frames defined by the terades, $e$. The coordinates $x^{\mu}$ are
labeled by natural inidices, $\alpha,...\mu,\nu,...=0,1,2,3$ while the vector
fields $e_{\hat{\alpha}}=e_{\hat{\alpha}}^{\mu}\partial_{\mu}$ defining the
local orthogonal frames, and the 1-forms
$\omega^{\hat{\alpha}}=\hat{e}_{\mu}^{\hat{\alpha}}dx^{\mu}$ of the dual
coframes are labeled by local indices, $\hat{\mu},\hat{\nu},...=0,1,2,3$. The
natural indices are raised or lowered by the metric tensor of $M$,
$g_{\mu\nu}=\eta_{\hat{\alpha}\hat{\beta}}\hat{e}^{\hat{\alpha}}_{\mu}\hat{e}^{\hat{\beta}}_{\nu}$,
while for the local ones we have to use the Minkowski metric $\eta={\rm
diag}(1,-1,-1,-1)$. In an arbitrary frame $\\{x;e\\}$ of $M$ the tetrad-gauge
invariant action of the Dirac field $\psi:M\to{\cal V}_{D}$, of mass $m$,
minimally coupled to background’s gravity, reads,

${\cal
S}[e,\psi]=\int\,d^{4}x\sqrt{g}\left\\{\frac{i}{2}[\overline{\psi}\gamma^{\hat{\alpha}}\nabla_{\hat{\alpha}}\psi-(\overline{\nabla_{\hat{\alpha}}\psi})\gamma^{\hat{\alpha}}\psi]-m\overline{\psi}\psi\right\\}$
(1)

where $\bar{\psi}=\psi^{+}\gamma^{0}$ is the Dirac adjoint of $\psi$ and
$g=|\det(g_{\mu\nu})|$. The field $\psi$ takes values in the space of Dirac
spinors ${\cal V}_{D}$ which carries the Dirac representation
$\rho_{D}=(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ of the $SL(2,{\mathbb{C}})$
group. In this representation one can define the Dirac matrices
$\gamma^{\hat{\alpha}}$ (with local indices) which are Dirac self-adjoint,
$\overline{\gamma^{\hat{\mu}}}=\gamma^{0}{\gamma^{\hat{\mu}}}^{+}\gamma^{0}=\gamma^{\hat{\mu}}$,
and satisfy the anti-commutation rules (A.1). These matrices may extend the
$sl(2,{\mathbb{C}})$ Lie algebra to a $su(2,2)$ one such that $\rho_{D}$ is in
fact much larger than a representation of the $SL(2,{\mathbb{C}})$ group. The
gauge invariant field theories use covariant derivatives of the form

$\nabla_{\hat{\alpha}}=e^{\mu}_{\hat{\alpha}}\nabla_{\mu}=e_{\hat{\alpha}}+\frac{i}{2}\hat{\Gamma}^{\hat{\gamma}}_{\hat{\alpha}\hat{\beta}}s^{\hat{\beta}\,\cdot}_{\cdot\,\hat{\gamma}}\,,$
(2)

where $s^{\hat{\alpha}\hat{\beta}}$ are the $SL(2,{\mathbb{C}})$ generators
(A.2) while
$\hat{\Gamma}^{\hat{\sigma}}_{\hat{\mu}\hat{\nu}}=e_{\hat{\mu}}^{\alpha}e_{\hat{\nu}}^{\beta}(\hat{e}_{\gamma}^{\hat{\sigma}}\Gamma^{\gamma}_{\alpha\beta}-\hat{e}^{\hat{\sigma}}_{\beta,\alpha})$
are the connection components in local frames expressed in terms of tetrads
and Christoffel symbols, $\Gamma^{\gamma}_{\alpha\beta}$. Note that these
connections are known as the spin connections and denoted often by $\Omega$.
The covariant derivatives of the Dirac operator
$E_{x}=i\gamma^{\hat{\alpha}}\nabla_{\hat{\alpha}}-m$ guarantee that the Dirac
equation $E_{x}\psi=0$ is gauge invariant in the sense that this does not
change its form when we change the positions of local frames performing gauge
transformations. The particular solutions of the Dirac equation form a vector
space that may be equipped with the relativistic scalar product [10]

$\langle\psi,\psi^{\prime}\rangle_{D}=\int_{\Sigma}d\sigma_{\mu}\sqrt{g}\,e^{\mu}_{\hat{\alpha}}\,\bar{\psi}(x)\gamma^{\hat{\alpha}}\psi^{\prime}(x)\,,$
(3)

whose integral is performed on a space-like section $\Sigma\subset M$.

In what follows we study the Dirac free field on $(1+3)$-dimensional spatially
flat FLRW spacetimes $M(a)$ having scale factors $a$. We work exclusively in
conformal frames $\\{x_{c};e\\}$ formed by conformal chats,
$\\{t_{c},{\vec{x}}_{c}\\}$, and orthogonal frames defined by terades, $e$.
The conformal coordinates are the conformal time $t_{c}$ and the co-moving
Cartesian space coordinates, $x_{c}^{i}$ ($i,j,k...=1,2,3$), giving the line
element

$\displaystyle
ds^{2}=g_{\mu\nu}(x_{c})dx_{c}^{\mu}dx_{c}^{\nu}=a(t_{c})^{2}(dt_{c}^{2}-d{\vec{x}}_{c}\cdot
d{\vec{x}}_{c})\,.$ (4)

Here we adopt the diagonal tetrad-gauge in which the local frames are defined
by

$\displaystyle e_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{a(t_{c})}\,\partial_{t_{c}}\,,\qquad\omega^{0}=a(t_{c})dt_{c}\,,$
(5) $\displaystyle e_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{a(t_{c})}\,\partial_{i}\,,\qquad~{}~{}\omega^{i}=a(t_{c})dx^{i}\,,$
(6)

such that we preserve the global $SO(3)$ symmetry allowing us to use
systematically the $SO(3)$ vectors. In these frames the Dirac equation can be
put in Hamiltonian form, $i\partial_{t_{x}}\psi=H_{c}\psi$, with the help of
the Hamiltonian operator

$H_{c}({x_{c}})=ma(t_{c})\gamma^{0}-i\gamma^{0}\gamma^{i}\partial_{x^{i}_{c}}-\frac{3i}{2}\frac{\dot{a}(t_{c})}{a(t_{c})}\,,\quad\dot{a}(t_{c})=\frac{da(t_{c})}{dt_{c}}\,.$
(7)

while the scalar product takes the form

$\langle\psi,\psi^{\prime}\rangle_{D}=\int_{\Sigma}d^{3}x_{c}\,a(t_{c})^{3}\bar{\psi}(x_{c})\gamma^{0}\psi(x_{c})=\int_{\Sigma}d^{3}x_{c}\,a(t_{c})^{3}{\psi}^{+}(x_{c})\psi(x_{c})\,,$
(8)

where the integration is performed over a flat space section,
$\Sigma=\mathbb{R}^{3}$.

In the spacetimes $M(a)$ the action (1) is invariant at least under global
isometries, $(R,a):x_{c}^{i}\to
x_{c}^{i\,\prime}=R^{i\,\cdot}_{\cdot\,j}x_{c}^{j}+a^{i}$ formed by rotations
$R\in SO(3)$ and three-dimensional translations, ${\vec{a}}\in T(3)$. The
isometry group $E(3)=T(3)\circledS SO(3)$ is a semidirect product where $T(3)$
is the invariant subgroup. The Dirac field transforms under isometries
according to the covariant representation
${T}\,:\,(r,{\vec{a}})\to{T}_{r,{\vec{a}}}$ of the group
$\tilde{E}(3)=T(3)\circledS SU(2)$ which is the universal covering group of
the group $E(3)$ [8]. We have thus the global transformations

$({T}_{r,{\vec{a}}}\psi)(t_{c},{\vec{x}}_{c})=r\psi\left(t_{c},R(\hat{r})^{-1}({\vec{x}}_{c}-{\vec{a}})\right)\,,\quad\forall
r\in\rho_{D}[SU(2)]\,,\,{\vec{a}}\in T(3).$ (9)

generated by the basis-generators of the corresponding representation of the
algebra ${\rm Lie}({T})$ that read

$\displaystyle P^{i}=\left.i\frac{\partial{T}_{1,a}}{\partial
a^{i}}\right|_{a=0}\,,\quad
J_{i}=\left.i\frac{\partial{T}_{r(\theta),0}}{\partial\theta^{i}}\right|_{\theta=0}\,,$
(10)

where $r(\theta)$ is defined in Eq. (A.8). We obtain thus the momentum
components, $P^{i}=-i\partial_{x_{c}^{i}}$, and those of the total angular
momentum,

$\displaystyle J_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\varepsilon_{ijk}J_{jk}=-i\varepsilon_{ijk}\underline{x}^{j}\partial_{x_{c}^{k}}+s_{i}\,,$
(11)

where the components $\underline{x}^{i}$ of the position (or coordinate)
vector-operator $\underline{\vec{x}}$ act as
$(\underline{x}^{i}\psi)(x_{c})=x_{c}^{i}\psi(x_{c})$ while the reducible
matrices $s_{i}$ are given by Eq. (A.7). The operators $P^{i}$ and $J_{i}$ are
the standard basis-generators of the algebra ${\rm Lie}(T)$. As the action (1)
is invariant under isometries the scalar product (8) is also invariant,

$\langle{T}_{\lambda,a}\psi,{T}_{\lambda,a}\psi^{\prime}\rangle_{D}=\langle\psi,\psi^{\prime}\rangle_{D}\,.$
(12)

The generators $X\in{\rm Lie}(T)$ are self-adjoint operators,
$\langle\psi,X\psi^{\prime}\rangle_{D}=\langle
X\psi,\psi^{\prime}\rangle_{D}$, conserved via Noetrher’s theorem. Therefore,
we may conclude that in this framework the covariant representation ${T}$
behaves as a unitary one with respect to the relativistic scalar product (8).

The subgroup $SU(2)\subset\tilde{E}(3)$ is of a special interest such that we
denote by $T_{r,0}=T^{r}_{\hat{r}}$ the restriction of the covariant
representation ${T}$ to this subgroup. The basis-generators of the
representation $T^{r}$ are the components of total angular momentum operator,
${\vec{J}}=\underline{\vec{x}}\land{\vec{P}}+{\vec{s}}$, defined by Eq. (11),
which is formed by the orbital term $\underline{\vec{x}}\land{\vec{P}}$ and
the spin matrix ${\vec{s}}\in sl(2,\mathbb{C})$. As these operators are not
conserved separately via Noetrher’s theorem we must look for a new conserved
spin operator ${\vec{S}}$ associated to a suitable new position operator,
${\vec{X}}=\underline{\vec{x}}+\delta{\vec{X}}$, allowing the splitting

${\vec{J}}=\underline{\vec{x}}\land{\vec{P}}+{\vec{s}}={\vec{L}}+{\vec{S}}\,,\qquad{\vec{L}}={\vec{X}}\land{\vec{P}}\,,$
(13)

which impose the correction $\delta{\vec{X}}$ to satisfy
$\delta{\vec{X}}\land{\vec{P}}={\vec{s}}-{\vec{S}}$. This new splitting gives
rise to a pair of new $su(2)\sim so(3)$ symmetries, namely the orbital
symmetry generated by $\\{L_{1},L_{2},L_{3}\\}$ and the spin one generated by
$\\{S_{1},S_{2},S_{3}\\}$.

The plane wave solutions of the Dirac equation depend on arbitrary Pauli
spinors $\xi=\\{\xi_{\sigma}|\sigma=\pm\frac{1}{2}\\}$ determining the fermion
polarization. These spinors form similar bases in both the spaces ${\cal
V}_{P}$ of Pauli spinors carrying the irreducible representations
$(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ of $\rho_{D}$. The basis of
polarization spinors can be changed at any time, $\xi\to\hat{r}\xi$, by
applying a $SU(2)$ rotation $\hat{r}$. For this reason, when we study this
symmetry it is convenient to denote the Dirac field by $\psi_{\xi}$ instead of
$\psi$ pointing out explicitly its dependence on the polarization spinors. We
define now the transformations of spin symmetry with the help of a
representation ${T}^{s}:\hat{r}\to{T}^{s}_{\hat{r}}$ of the group $SU(2)$
acting as,

$\left(T^{s}_{\hat{r}(\theta)}\psi_{\xi}\right)(x_{c})=\psi_{\hat{r}(\theta)\xi}(x_{c})\,,$
(14)

where $\hat{r}(\theta)$ are the rotations (A.8) with Cayley-Klein parameters.
Then the components of the spin operator can be defined as the generators of
this representation, [2]

$S_{i}=\left.i\frac{\partial
T^{s}_{\hat{r}(\theta)}}{\partial\theta^{i}}\right|_{\theta^{i}=0}~{}~{}\Rightarrow~{}~{}S_{i}\psi_{\xi}=\psi_{\hat{s}_{i}\xi}\,,$
(15)

whose action is obvious. Similarly we define the orbital representation
$T^{o}:\hat{r}\to T^{o}_{\hat{r}}$ as

$\left(T^{o}_{\hat{r}(\theta)}\psi_{\xi}\right)(t_{c},{\vec{x}}_{c})=r(\theta)\psi_{\hat{r}(\theta)^{-1}\xi}\left(t_{c},R[\hat{r}(\theta)]^{-1}{\vec{x}}_{c}\right)\,,$
(16)

for accomplishing the factorization ${T}^{r}=T^{o}\otimes T^{s}$. The basis-
generators of the orbital representation,

$L_{i}=\left.i\frac{\partial
T^{o}_{\hat{r}(\theta)}}{\partial\theta^{i}}\right|_{\theta^{i}=0}\,,$ (17)

are the components of the new conserved orbital angular momentum operator. In
what follows we shall pay a special attention to the new operators ${\vec{S}}$
and ${\vec{L}}$.

##  3 Mode spinors in momentum representation

The Dirac equation can be solved formally on $M(a)$ in conformal frames
$\\{x_{c};e\\}$ allowing solutions of the general form [10],

$\displaystyle\psi({x}_{c})$ $\displaystyle=$
$\displaystyle\psi^{(+)}(x_{c})+\psi^{(-)}(x_{c})$ (18) $\displaystyle=$
$\displaystyle\int
d^{3}p\sum_{\sigma}[U_{{\vec{p}},\sigma}(x_{c}){\alpha}({\vec{p}},\sigma)+V_{{\vec{p}},\sigma}(x_{c}){\beta}^{*}({\vec{p}},\sigma)]\,,$

expressed in terms of particle, $\alpha$, and antiparticle, $\beta$, wave
spinors and mode spinors $U_{{\vec{p}},\sigma}$ and $V_{{\vec{p}},\sigma}$, of
positive and respectively negative frequencies. These spinors satisfy the
eigenvalues problems
$P^{i}U_{\vec{p},\sigma}(t_{c},{\vec{x}}_{c})=p^{i}U_{\vec{p},\sigma}(t_{c},{\vec{x}}_{c})$
and
$P^{i}V_{\vec{p},\sigma}(t_{c},{\vec{x}}_{c})=-p^{i}V_{\vec{p},\sigma}(t_{c},{\vec{x}}_{c})$
and form an orthonormal basis being related through the charge conjugation,

$V_{{\vec{p}},\sigma}(t_{c},{\vec{x}}_{c})=U^{c}_{{\vec{p}},\sigma}(t_{c},{\vec{x}}_{c})=C{U}^{*}_{{\vec{p}},\sigma}(t_{c},{\vec{x}}_{c})\,,\quad
C=i\gamma^{2}\,,$ (19)

satisfying the orthogonality relations

$\displaystyle\langle
U_{{\vec{p}},\sigma},U_{{{\vec{p}}\,}^{\prime},\sigma^{\prime}}\rangle$
$\displaystyle=$ $\displaystyle\langle
V_{{\vec{p}},\sigma},V_{{{\vec{p}}\,}^{\prime},\sigma^{\prime}}\rangle=\delta_{\sigma\sigma^{\prime}}\delta^{3}({\vec{p}}-{\vec{p}}\,^{\prime})$
(20) $\displaystyle\langle
U_{{\vec{p}},\sigma},V_{{{\vec{p}}\,}^{\prime},\sigma^{\prime}}\rangle$
$\displaystyle=$ $\displaystyle\langle
V_{{\vec{p}},\sigma},U_{{{\vec{p}}\,}^{\prime},\sigma^{\prime}}\rangle=0\,,$
(21)

with respect to the relativistic scalar product (8). Moreover, this basis is
supposed to be complete accomplishing the condition [10]

$\displaystyle\int
d^{3}p\sum_{\sigma}\left[U_{{\vec{p}},\,\sigma}(t_{c},{\vec{x}}_{c})U^{+}_{{\vec{p}},\sigma}(t_{c},{\vec{x}}_{c}^{\,\prime})+V_{{\vec{p}},\sigma}(t_{c},{\vec{x}}_{c})V^{+}_{{\vec{p}},\sigma}(t_{c},{\vec{x}}_{c}^{\,\prime})\right]$
$\displaystyle\hskip
91.04881pt=a(t_{c})^{-3}\delta^{3}({\vec{x}}_{c}-{\vec{x}}_{c}^{\,\prime})\,.~{}~{}~{}~{}~{}$
(22)

The space of the solutions of Dirac’s equation ${\cal
F}=\\{\psi|E_{x_{c}}\psi=0\\}={\cal F}^{+}\oplus{\cal F}^{-}$ is formed by the
subspaces of solutions of positive, ${\cal F}^{+}$, and negative, ${\cal
F}^{-}$, frequencies which are orthogonal with respect to the scalar product
(8).

In RQM the physical meaning of the free field $\psi$ is encapsulated in its
wave spinors

$\alpha=\left(\begin{array}[]{l}\alpha_{\frac{1}{2}}\\\
\alpha_{-\frac{1}{2}}\end{array}\right)\in\tilde{\cal
F}^{+}\,,\quad\beta=\left(\begin{array}[]{llc}\beta_{\frac{1}{2}}\\\
\beta_{-\frac{1}{2}}\end{array}\right)\in\tilde{\cal F}^{-}\,,$ (23)

whose values can be obtained applying the inversion formulas

$\alpha_{\sigma}({\vec{p}})=\langle
U_{{\vec{p}},\sigma},\psi\rangle_{D}\,,\quad\beta_{\sigma}({\vec{p}})=\langle\psi,V_{{\vec{p}},\sigma}\rangle_{D}\,,$
(24)

resulted from Eqs. (20) and (21). We assume now that the spaces $\tilde{\cal
F}^{+}\sim\tilde{\cal F}^{-}\sim{\cal
L}^{2}(\Omega_{\mathring{p}},d^{3}p,{\cal V}_{P})$ are equipped with the same
scalar product,

$\displaystyle\langle\alpha,\alpha^{\prime}\rangle=\int
d^{3}p\,\alpha^{+}({\vec{p}})\alpha^{\prime}({\vec{p}})=\int
d^{3}p\sum_{\sigma}\alpha_{\sigma}^{*}({\vec{p}})\alpha_{\sigma}^{\prime}({\vec{p}})\,,$
(25)

and similarly for the spinors $\beta$. Then after using Eqs. (20) and (21) we
obtain the important identity

$\langle\psi,\psi^{\prime}\rangle_{D}=\langle\alpha,\alpha^{\prime}\rangle+\langle\beta,\beta^{\prime}\rangle\,,$
(26)

showing that the spaces ${\cal F}$ and $\tilde{\cal F}=\tilde{\cal
F}^{+}\oplus\tilde{\cal F}^{-}$ related through the expansion (18) are
isometric.

The general form of the mode spinors can be studied in any manifold $M(a)$,
exploiting the Dirac equation in MR. As here we cannot apply the Wigner method
of constructing mode spinors in Minkowski spacetime, we must look for an
alternative approach inspired by the identity (A.12). For our further
purposes, it is convenient to separate from the beginning the orbital part
from the spin terms assuming that the mode spinors have the general form

$\displaystyle U_{\vec{p},\sigma}(t_{c},{\vec{x}}_{c})$ $\displaystyle=$
$\displaystyle[2\pi
a(t_{c})]^{-\frac{3}{2}}{e^{i{\vec{p}}\cdot{\vec{x}}_{c}}}{\cal
U}_{p}(t_{c})d({\vec{p}})u_{\sigma}({\vec{p}})$ (27) $\displaystyle
V_{\vec{p},\sigma}(t_{c},{\vec{x}}_{c})$ $\displaystyle=$ $\displaystyle[2\pi
a(t_{c})]^{-\frac{3}{2}}{e^{-i{\vec{p}}\cdot{\vec{x}}_{c}}}{\cal
V}_{p}(t_{c})d({\vec{p}})v_{\sigma}({\vec{p}})$ (28)

where we introduce the diagonal matrix-functions ${\cal U}_{p}(t_{c})$ and
${\cal V}_{p}(t_{c})$ which depend only on $t$ and $p=|{\vec{p}}\,|$,
determining the time modulation of the fundamental spinors.

The spin part is separated with the help of the Hermitian singular matrix

$d({\vec{p}})=d^{+}({\vec{p}})=1+\frac{\gamma^{0}{\vec{\gamma}}\cdot{\vec{p}}}{p}\,,$
(29)

acting on the Dirac spinors that in the standard representation of the Dirac
matrices (with diagonal $\gamma^{0}$) have the form [TH, 2]

$u_{\sigma}({\vec{p}})=\left(\begin{array}[]{c}\xi_{\sigma}({\vec{p}})\\\
0\end{array}\right)\,,\quad
v_{\sigma}({\vec{p}})=Cu^{*}_{\sigma}({\vec{p}})=\left(\begin{array}[]{c}0\\\
-\eta_{\sigma}({\vec{p}})\end{array}\right)\,,$ (30)

depending on the Pauli spinors, $\xi_{\sigma}({\vec{p}})$ and
$\eta_{\sigma}({\vec{p}})=i\sigma_{2}\xi^{*}_{\sigma}({\vec{p}})$ which are
normalized,
$\xi^{+}_{\sigma}({\vec{p}})\xi_{\sigma^{\prime}}({\vec{p}})=\eta^{+}_{\sigma}({\vec{p}})\eta_{\sigma^{\prime}}({\vec{p}})=\delta_{\sigma\sigma^{\prime}}$.
satisfying the completeness condition,

$\sum_{\sigma}\xi_{\sigma}({\vec{p}})\xi_{\sigma}^{+}({\vec{p}})=\sum_{\sigma}\eta_{\sigma}({\vec{p}})\eta_{\sigma}^{+}({\vec{p}})={\bf
1}_{2\times 2}\,.$ (31)

The form of these spinors depends on the direction of the spin projection
which, in general, may depend on ${\vec{p}}$ as in the case of the helicity
basis. Here we say that the polarization depending on momentum is a peculiar
polarization. Otherwise, the plarization independent on momentum will be
referred as common polarization. The corresponding Dirac spinors have the
completeness properties [2]

$\displaystyle\sum_{\sigma}u_{\sigma}({\vec{p}}){u}^{+}_{\sigma}({\vec{p}})=\frac{1+\gamma^{0}}{2}\equiv\pi_{+}\,,$
(32)
$\displaystyle\sum_{\sigma}v_{\sigma}({\vec{p}}){v}^{+}_{\sigma}({\vec{p}})=\frac{1-\gamma^{0}}{2}\equiv\pi_{-}\,,$
(33)

laying out the Hermitian matrices $\pi_{\pm}=(\pi_{\pm})^{+}$ that form a
complete system of projection operators as $\pi_{+}\pi_{-}=0$ and
$\pi_{+}+\pi_{-}=1\in\rho_{D}$. All the above auxiliary quantities will be
useful in the further calculations having simple calculation rules as, for
example, $d({\vec{p}})^{2}=2d({\vec{p}})$, $~{}d({\vec{p}})d(-{\vec{p}})=0$,
etc..

The principal pieces are the diagonal matrix-functions determining the time
modulation of the mode spinors which can be represented as

$\displaystyle{\cal U}_{p}(t_{c})$ $\displaystyle=$
$\displaystyle\pi_{+}u^{+}(t_{c},p)+\pi_{-}u^{-}(t_{c},p)\,,$ (34)
$\displaystyle{\cal V}_{p}(t_{c})$ $\displaystyle=$
$\displaystyle\pi_{+}v^{+}(t_{c},p)+\pi_{-}v^{-}(t_{c},p)\,,$ (35)

in terms of the time modulation functions $u^{\pm}(t_{c},p)$ and
$v^{\pm}(t_{c},p)$ whose differential equations in the general case of
$m\not=0$ may be obtained by substituting Eqs. (27) and (28) in Dirac’s
equation. Then, after a few manipulation, we find the systems of the first
order differential equations [10]

$\displaystyle\left[i\partial_{t_{c}}\mp m\,a(t_{c})\right]u^{\pm}(t_{c},p)$
$\displaystyle=$ $\displaystyle{p}\,u^{\mp}(t_{c},p)\,,$
$\displaystyle\left[i\partial_{t_{c}}\mp m\,a(t_{c})\right]v^{\pm}(t_{c},p)$
$\displaystyle=$ $\displaystyle-{p}\,v^{\mp}(t_{c},p)\,,$ (36)

which govern the time modulation of the free Dirac field on any spatially flat
FLRW manifold. The solutions of this system depend on integration constants
that must be selected according to the charge conjugation condition (19) which
requires to have ${\cal V}_{p}={\cal U}^{c}_{p}=C{\cal
U}_{p}^{*}C^{-1}=\gamma^{5}{\cal U}_{p}^{*}\gamma^{5}$ leading to the
mandatory condition

$v^{\pm}(t_{c},p)=\left[u^{\mp}(t_{c},p)\right]^{*}\,.$ (37)

The remaining normalization constants can be restricted as we have the prime
integrals of the system (36),
$\partial_{t_{c}}(|u^{+}(t_{c},p)|^{2}+|u^{-}(t_{c},p)|^{2})=\partial_{t_{c}}(|v^{+}(t_{c},p)|^{2}+|v^{-}(t_{c},p)|^{2})=0$,
allowing us to impose the normalization conditions

$|u^{+}(t_{c},p)|^{2}+|u^{-}(t_{c},p)|^{2}=|v^{+}(t_{c},p)|^{2}+|v^{-}(t_{c},p)|^{2}=1\,,$
(38)

which guarantee Eqs. (20) and (21) to be accomplished. Hereby we find the
calculation rules ${\rm Tr}({\cal U}_{p}{\cal U}_{p}^{*})={\rm Tr}({\cal
V}_{p}{\cal V}_{p}^{*})=2$ resulted from Eqs. (38) as ${\rm Tr}(\pi_{\pm})=2$.

Gathering all the above elements we obtain the intuitive forms of the mode
spinors in conformal frames and standard representation of the
$\gamma$-matrices [10],

$\displaystyle U_{{\vec{p}},\sigma}(t_{c},{\vec{x}}_{c})$ $\displaystyle=$
$\displaystyle\frac{e^{i{\vec{p}}\cdot{\vec{x}}_{c}}}{[2\pi
a(t_{c})]^{\frac{3}{2}}}\left(\begin{array}[]{c}u^{+}(t_{c},p)\,\xi_{\sigma}({\vec{p}})\\\
u^{-}(t_{c},p)\,\frac{\vec{\sigma}\cdot{\vec{p}}}{p}\,\xi_{\sigma}({\vec{p}})\end{array}\right)\,,$
(41) $\displaystyle V_{{\vec{p}},\sigma}(t_{c},{\vec{x}}_{c})$
$\displaystyle=$ $\displaystyle-\frac{e^{i{\vec{p}}\cdot{\vec{x}}_{c}}}{[2\pi
a(t_{c})]^{\frac{3}{2}}}\left(\begin{array}[]{c}v^{+}(t_{c},p)\,\frac{\vec{\sigma}\cdot{\vec{p}}}{p}\,\eta_{\sigma}({\vec{p}})\\\
v^{-}(t_{c},p)\,\eta_{\sigma}({\vec{p}})\end{array}\right)\,.$ (44)

A special problem comes from the fact that these spinors are not defined in
the rest frame where ${\vec{p}}=0$ because of the matrix
$\frac{\vec{\sigma}\cdot{\vec{p}}}{p}$ which is undefined in this limit. For
this reason we proposed the rest frame vacuum defined by the following
supplementary conditions [10]

$u^{-}(t_{c},0)=v^{+}(t_{c},0)=0\,,\quad|u^{+}(t_{c},0)|=|v^{-}(t_{c},0)|=1\,.$
(45)

In this vacuum the mode spinors in the rest frame are well-defined,

$\displaystyle U_{0,\sigma}(t_{c},{\vec{x}})$ $\displaystyle=$
$\displaystyle[2\pi a(t_{c})]^{-\frac{3}{2}}e^{-im\,t(t_{c})}u_{\sigma}(0)\,,$
(46) $\displaystyle V_{0,\sigma}(t_{c},{\vec{x}})$ $\displaystyle=$
$\displaystyle[2\pi a(t_{c})]^{-\frac{3}{2}}e^{im\,t(t_{c})}v_{\sigma}(0)\,,$
(47)

depending on the cosmic time

$t(t_{c})=\int_{t_{c\,0}}^{t_{c}}dt^{\prime}_{c}a(t^{\prime}_{c})\,,$ (48)

the rest energy of special relativity, $E_{0}=m$, and the rest frame spinors
$u_{\sigma}(0)$ and $v_{\sigma}(0)$. Note that in Minkowski spacetime the time
modulation functions (A.11) satisfy naturally the conditions (45) defining the
rest frame vacuum. In contrast, on the de Sitter expanding universe one
prefers the adiabatic vacuum [11] but which cannot be defined in FLRW
spacetimes with finite big bang times. In our opinion, the de Sitter adiabatic
vacuum can be seen as the partner of the rest frame one in the process of
cosmological particle creation under inflation [12].

##  4 Operators in active mode

In linear algebra, a linear operator may act changing the basis in the active
mode or transforming only the coefficients without affecting the basis if we
adopt the passive mode. Let us start with the operators in active mode.

The simplest operators of Dirac’s theory on $M(a)$ are the multiplicative and
differential operators in CR. The differential operators are $4\times 4$
matrices depending on space derivatives, $f(i\partial_{i})\in\rho_{D}$, whose
action on the mode spinors

$\displaystyle\left[f(i\partial_{i})\psi\right](x)$ $\displaystyle=$
$\displaystyle\int
d^{3}p\sum_{\sigma}\left[f({p^{i}})U_{{\vec{p}},{\sigma}}(x)\alpha_{\sigma}({\vec{p}})+f(-{p^{i}})V_{{\vec{p}},{\sigma}}(x)\beta^{*}_{\sigma}({\vec{p}})\right]\,,$
(49)

is given by the momentum-dependent matrices $f(p^{i})$ called the Fourier
transforms of the operators $f(i\partial_{i})$. The principal differential
operators are the translation generators $P_{i}=i\partial_{i}$ and the time-
dependent Hamiltonian operator (7) whose Fourier transform

$H_{c}(t_{c},{\vec{p}})=ma(t_{c})\gamma^{0}+\gamma^{0}\vec{\gamma}\cdot{\vec{p}}-\frac{3i}{2}\frac{\dot{a}(t_{c})}{a(t_{c})}\,,$
(50)

allows us to write the formal actions

$\left({H}_{c}U_{{\vec{p}},\sigma}\right)(x_{c})=H_{c}(t_{c},{\vec{p}})U_{{\vec{p}},\sigma}(x_{c})\,,\quad\left({H}_{c}V_{{\vec{p}},\sigma}\right)(x_{c})=H_{c}(t_{c},-{\vec{p}})\,,V_{{\vec{p}},\sigma}(x_{c})\,,$
(51)

which will be useful for identifying conserved quantities.

Most interesting are the integral operators whose kernels may depend on time.
Here we restrict ourselves to the equal-time operators acting in conformal
frames as

$(A\psi)(t_{c},{\vec{x}}_{c})=\int
d^{3}x_{c}^{\prime}{\mathfrak{A}}(t_{c},{\vec{x}}_{c},{\vec{x}}_{c}^{\,\prime})\psi(t_{c},{\vec{x}}_{c}^{\,\prime})\,,$
(52)

preserving the conformal time $t_{c}$. The multiplication takes over this
property,

$A=A_{1}A_{2}~{}~{}\Rightarrow~{}~{}{\mathfrak{A}}(t_{c},{\vec{x}}_{c},{\vec{x}}_{c}^{\,\prime})=\int
d^{3}x_{c}^{\prime\prime}\,{\mathfrak{A}}_{1}(t,{\vec{x}}_{c},{{\vec{x}}_{c}}^{\prime\prime}){\mathfrak{A}}_{2}(t_{c},{{\vec{x}}_{c}}^{\prime\prime},{\vec{x}}_{c}^{\,\prime})\,,$
(53)

such that the set of equal-time operators forms an algebra
$E[t_{c}]\subset{\rm Aut}({\cal F})$ at any fixed time $t_{c}$. A special
subalgebra $F[t_{c}]\subset E[t_{c}]$ is formed by operators with local
kernels,
${\mathfrak{A}}(t_{c},{\vec{x}}_{c},{\vec{x}}_{c}^{\prime})={\mathfrak{A}}(t_{c},{\vec{x}}_{c}-{\vec{x}}_{c}^{\prime})$,
allowing three-dimensional Fourier representations,

${\mathfrak{A}}(t_{c},{\vec{x}}_{c})=\int
d^{3}p\,\frac{e^{i{\vec{p}}\cdot{\vec{x}}_{c}}}{(2\pi)^{3}}{A}(t_{c},{\vec{p}})\,,$
(54)

depending on the matrices $A(t_{c},{\vec{p}})\in{r}_{D}$ we call here the
Fourier transforms of the operators $A$. Then the action (52) on a field (18)
can be written as

$\displaystyle(A\psi)(t_{c},{\vec{x}}_{c})=\int
d^{3}x_{c}^{\prime}\,{\mathfrak{A}}(t,{\vec{x}}_{c}-{\vec{x}}_{c}^{\,\prime})\psi(t_{c},{\vec{x}}_{c}^{\,\prime})$
$\displaystyle~{}~{}=\int
d^{3}p\sum_{\sigma}\left[A(t_{c},{\vec{p}})U_{{\vec{p}},{\sigma}}(t_{c},{\vec{x}})\alpha_{\sigma}({\vec{p}})+A(t_{c},-{\vec{p}})V_{{\vec{p}},{\sigma}}(t,{\vec{x}})\beta^{*}_{\sigma}({\vec{p}})\right]\,.$
(55)

Remarkably, the operator multiplication, $A=A_{1}A_{2}$, in $F[t]$ algebra
leads to the convolution of the corresponding kernels
${\mathfrak{A}}={\mathfrak{A}}_{1}*{\mathfrak{A}}_{2}$ defined as

${\mathfrak{A}}(t_{c},{\vec{x}}_{c}-{\vec{x}}_{c}^{\,\prime})=\int
d^{3}p^{\prime\prime}{\mathfrak{A}}_{1}(t,{\vec{x}}-{{\vec{x}}\,}^{\prime\prime}){\mathfrak{A}}_{2}(t,{{\vec{x}}\,}^{\prime\prime}-{\vec{x}}^{\,\prime})\,,$
(56)

and, consequently, to the multiplication of their Fourier transforms,
$A(t_{c},{\vec{p}})=A_{1}(t_{c},{\vec{p}})A_{2}(t_{c},{\vec{p}})$. One obtains
thus the new algebra $\bar{F}[t_{c}]$ in MR formed by the Fourier transforms
of the Fourier operators in which the identity is $I({\vec{p}})=1\in\rho_{D}$.
Obviously, the operator $A\in F[t_{c}]$ is invertible if its Fourier transform
is invertible in $\bar{F}[t_{c}]$.

We say that a Fourier operator $A$ is reducible if its Fourier transform
satisfies

$\left.\langle
V_{{\vec{p}},{\sigma}},A(t_{c},{\vec{p}})U_{{\vec{p}},{\sigma}}\rangle_{D}\right|_{t_{c}}=\left.\langle
U_{{\vec{p}},{\sigma}},A(t_{c},-{\vec{p}})V_{{\vec{p}},{\sigma}}\rangle_{D}\right|_{t_{c}}=0\,,$
(57)

at any instant $t_{c}$. These operators have simple expectation values

$\left.\langle\psi,A\psi\rangle_{D}\right|_{t_{c}}=\left\langle
U_{{\vec{p}},{\sigma}},A(t_{c},{\vec{p}})U_{{\vec{p}},{\sigma}}\rangle_{D}\right|_{t_{c}}+\left\langle
V_{{\vec{p}},{\sigma}},A(t_{c},-{\vec{p}})V_{{\vec{p}},{\sigma}}\rangle_{D}\right|_{t_{c}}$
(58)

that become conserved quantities when the Fourier transform
$A(t_{c},{\vec{p}})$ accomplishes the condition

$\frac{d}{dt_{c}}\left.\langle\psi,A\psi\rangle_{D}\right|_{t_{c}}=0~{}~{}\Rightarrow~{}~{}i\partial_{t_{c}}A(t_{c},{\vec{p}})=\left[H_{c}(t_{c},{\vec{p}}),A(t_{c},{\vec{p}})\right]\,,$
(59)

allowing us to identify the conserved operators directly without resorting to
Noether’s theorem.

In this framework we may define the transformations of the spin symmetry (14)
as Fourier operators constructed with the help of the spectral representations
we proposed recently. We assume that the operators $T^{s}_{\hat{r}}$ are
Fourier operators whose kernels allow the spectral representation

$\displaystyle{\mathfrak{T}}^{s}_{\hat{r}}(t_{c},{\vec{x}}_{c}-{\vec{x}}_{c}^{\,\prime})$
$\displaystyle=$ $\displaystyle\int
d^{3}p\,a(t_{c})^{3}\sum_{\sigma,\sigma^{\prime}}\left[U_{{\vec{p}},\xi_{\sigma}}(t_{c},{\vec{x}}_{c})D_{\sigma\sigma^{\prime}}(\hat{r},{\vec{p}})U^{+}_{{\vec{p}},\xi_{\sigma^{\prime}}}(t_{c},{\vec{x}}_{c}^{\,\prime})\right.$
(60) $\displaystyle\hskip
56.9055pt\left.+V_{{\vec{p}},\eta_{\sigma}}(t_{c},{\vec{x}}_{c})D^{*}_{\sigma\sigma^{\prime}}(\hat{r},{\vec{p}})V^{+}_{{\vec{p}},\eta_{\sigma^{\prime}}}(t_{c},{\vec{x}}_{c}^{\,\prime})\right]\,,$

where

$D_{\sigma\sigma^{\prime}}(\hat{r},{\vec{p}})=\xi^{+}_{\sigma}({\vec{p}})\hat{r}\xi_{\sigma^{\prime}}({\vec{p}})$
(61)

are the usual matrix elements of the fundamental representation of the little
group $SU(2)$ in the basis of polarization spinors $\\{\xi\\}$. The rotations
$\hat{r}\in SU(2)$ transform this basis and implicitly the mode spinors as

$\displaystyle\hat{r}\,\xi_{\sigma}({\vec{p}})$ $\displaystyle=$
$\displaystyle\sum_{\sigma^{\prime}}\xi_{\sigma^{\prime}}({\vec{p}})D_{\sigma^{\prime}\sigma}(\hat{r},{\vec{p}})\Rightarrow
U_{{\vec{p}},\hat{r}\xi_{\sigma}}(x_{c})=\sum_{\sigma^{\prime}}U_{{\vec{p}},\xi_{\sigma^{\prime}}}(x_{c})D_{\sigma^{\prime}\sigma}(\hat{r},{\vec{p}})\,,$
(62) $\displaystyle\hat{r}\,\eta_{\sigma}({\vec{p}})$ $\displaystyle=$
$\displaystyle\sum_{\sigma^{\prime}}\eta_{\sigma^{\prime}}({\vec{p}})D^{*}_{\sigma^{\prime}\sigma}(\hat{r},{\vec{p}})\Rightarrow
V_{{\vec{p}},\hat{r}\eta_{\sigma}}(x_{c})=\sum_{\sigma^{\prime}}V_{{\vec{p}},\eta_{\sigma^{\prime}}}(x_{c})D^{*}_{\sigma^{\prime}\sigma}(\hat{r},{\vec{p}})\,.~{}~{}~{}~{}$
(63)

Therefore, we obtain the desired action of the operators of the spin symmetry

$\displaystyle[{T}^{s}_{\hat{r}}\psi_{\xi}](t_{c},{\vec{x}}_{c})=\int
d^{3}x_{c}^{\prime}{\mathfrak{T}}^{s}_{\hat{r}}(t_{c},{\vec{x}}_{c}-{\vec{x}}_{c}^{\,\prime})\psi_{\xi}(t_{c},{\vec{x}}_{c}^{\,\prime})=\psi_{\hat{r}\xi}(t_{c},{\vec{x}}_{c})\,,$
(64)

in accordance with the definition (14). The next step is to substitute the
mode spinors (27) and (28) the integral (60) for deriving the Fourier
transforms

$\displaystyle T^{s}_{\hat{r}}(t_{c},{\vec{p}})={\cal
U}_{p}(t_{c})d({\vec{p}})\,r\,\pi_{+}d({\vec{p}}){\cal U}^{*}_{p}(t_{c})+{\cal
V}_{p}(t_{c})d(-{\vec{p}})\,r\,\pi_{-}d(-{\vec{p}}){\cal V}^{*}_{p}(t_{c})\,,$
(65)

of the operators $T^{s}_{\hat{r}}$ in terms of projection operators, (32) and
(33), and rotations (A.5).

Hereby it results that the components of the spin operator (15) are Fourier
operators whose Fourier transforms read

$\displaystyle S_{i}(t_{c},{\vec{p}})={\cal
U}_{p}(t_{c})d({\vec{p}})\,s_{i}\,\pi_{+}d({\vec{p}}){\cal
U}^{*}_{p}(t_{c})+{\cal
V}_{p}(t_{c})d(-{\vec{p}})\,s_{i}\,\pi_{-}d(-{\vec{p}}){\cal
V}^{*}_{p}(t_{c})\,,$ (66)

where $s_{i}$ are the diagonal matrices (A.7) which commutes with $\pi_{\pm}$.
Furthermore, taking into account that the time modulation functions satisfy
the conditions (37) and(38) we obtain the definitive form of these Fourier
transforms as

$\displaystyle S_{i}(t_{c},{\vec{p}})$ $\displaystyle=$
$\displaystyle\left[|u^{+}(t_{c},p)|^{2}-|u^{-}(t_{c},p)|^{2}\right]s_{i}+\frac{2|u^{-}(t_{c},p)|^{2}}{p^{2}}p^{i}{\vec{p}}\cdot{\vec{s}}$
(67) $\displaystyle+$
$\displaystyle\frac{i}{p}\left[u^{+}(t_{c},p)u^{-}(t_{c},p)^{*}\pi_{+}+u^{+}(t_{c},p)^{*}u^{-}(t_{c},p)\pi_{-}\right]\epsilon_{ijk}p^{j}\gamma^{k}\,.~{}~{}~{}~{}$

If, in addition, we set the rest frame vacuum imposing the limit (45) then we
obtain the correct limit in rest frames

$\lim_{{\vec{p}}\to 0}S_{i}(t_{c},{\vec{p}})=s_{i}\,.$ (68)

In the particular case of the Minkowski spacetime, after substituting the time
modulation functions (A.11) in Eq. (67), we obtain the components of Pryce’s
spin operator [1],

$S_{i}({\vec{p}})_{\rm
Pryce}=\frac{m}{E(p)}s_{i}+\frac{p^{i}({\vec{s}}\cdot{\vec{p}})}{E(p)(E(p)+m)}+\frac{i}{2E(p)}\epsilon_{ijk}p^{j}\gamma^{k}\,,$
(69)

which are the Fourier transforms of the components of the conserved spin
operator of Dirac’s theory in special relativity.

The components of the spin operator ${\vec{S}}$ act in CR through their
Fourier transforms (67) according to the general rule (55). The action of
these operators on the field $\psi$ can be derived as in Eq. (55) applying the
operators (66) on the mode spinors (27) and (28) and using the obvious
identities

$\displaystyle\pi_{+}d(\pm{\vec{p}}){\cal U}^{*}_{p}(t_{c}){\cal
U}_{p}(t_{c})d(\pm{\vec{p}})\pi_{+}$ $\displaystyle=$
$\displaystyle\pi_{+}\,,$ (70) $\displaystyle\pi_{-}d({\pm\vec{p}}){\cal
V}^{*}_{p}(t_{c}){\cal V}_{p}(t_{c})d(\pm{\vec{p}})\pi_{-}$ $\displaystyle=$
$\displaystyle\pi_{-}\,,$ (71)

resulted from the condition (38). We obtain thus

$\displaystyle(S_{i}U_{{\vec{p}},\xi_{\sigma}})(x_{c})$ $\displaystyle=$
$\displaystyle
S_{i}(t_{c},{\vec{p}})U_{{\vec{p}},\xi_{\sigma}}(x_{c})=U_{{\vec{p}},\hat{s}_{i}\xi_{\sigma}}(x_{c})\,,$
(72) $\displaystyle(S_{i}V_{{\vec{p}},\eta_{\sigma}})(x_{c})$ $\displaystyle=$
$\displaystyle
S_{i}(t_{c},-{\vec{p}})V_{{\vec{p}},\eta_{\sigma}}(x_{c})=V_{{\vec{p}},\hat{s}_{i}\eta_{\sigma}}(x_{c})\,,$
(73)

recovering the desired action (15). Moreover, we verify that the operators
$S_{i}$ are conserved as their Fourier transforms (67) satisfy the condition
(59) in spite of their complicated dependence on time. In addition, these
operators are translation invariant obeying $\left[P^{j},S_{i}\right]=0$ and
have the expected algebraic properties

$\displaystyle\left[S_{i}(t_{c},{\vec{p}}),S_{j}(t_{c},{\vec{p}})\right]=i\epsilon_{ijk}S_{k}(t_{c},{\vec{p}})$
$\displaystyle~{}\Rightarrow~{}$
$\displaystyle\left[S_{i},S_{j}\right]=i\epsilon_{ijk}S_{k}\,,$ (74)
$\displaystyle{\vec{S}}^{\,2}(t_{c},{\vec{p}})=\frac{3}{4}\cdot 1\in\rho_{D}$
$\displaystyle~{}\Rightarrow~{}$ $\displaystyle\vec{S}^{\,2}=\frac{3}{4}\cdot
1\in\rho_{D}\,,$ (75)

showing that they generate a fundamental representation of the $SU(2)$ group.

With the above elements we may define the operator of fermion polarization for
any peculiar polarization given by a pair of related spinors
$\xi_{\sigma}({\vec{p}})$ and $\eta_{\sigma}({\vec{p}})$ assumed to satisfy
the eigenvalues problems

$\hat{s}_{i}{n}^{i}({\vec{p}})\xi_{\sigma}({\vec{p}})=\sigma\,\xi_{\sigma}({\vec{p}})\Rightarrow\hat{s}_{i}{n}^{i}({\vec{p}})\eta_{\sigma}({\vec{p}})=-\sigma\,\eta_{\sigma}({\vec{p}}),$
(76)

where the unit vector ${\vec{n}}({\vec{p}})$ gives the peculiar direction with
respect to which the peculiar polarization is measured. We define the
polarization operator as the Fourier operator $W_{s}$ having the Fourier
transform

$\displaystyle W_{s}(t_{c},{\vec{p}})={\cal
U}_{p}(t_{c})d({\vec{p}})\,w({\vec{p}})\,\pi_{+}d({\vec{p}}){\cal
U}^{*}_{p}(t_{c})+{\cal
V}_{p}(t_{c})d(-{\vec{p}})\,w(-{\vec{p}})\,\pi_{-}d(-{\vec{p}}){\cal
V}^{*}_{p}(t_{c})\,\,,$ (77)

where $w({\vec{p}})=s_{i}n^{i}({\vec{p}})$. This operator is conserved and
translation invariant acting as

$\displaystyle(W_{s}U_{{\vec{p}},\xi_{\sigma}({\vec{p}})})(x)=W_{s}({\vec{p}})U_{{\vec{p}},\xi_{\sigma}({\vec{p}})}(x)$
$\displaystyle=$ $\displaystyle
U_{{\vec{p}},w({\vec{p}})\xi_{\sigma}({\vec{p}})}(x)=\sigma
U_{{\vec{p}},\xi_{\sigma}({\vec{p}})}(x)\,,$ (78)
$\displaystyle(W_{s}V_{{\vec{p}},\eta_{\sigma}({\vec{p}})})(x)=W_{s}(-{\vec{p}})V_{{\vec{p}},\eta_{\sigma}({\vec{p}})}(x)$
$\displaystyle=$ $\displaystyle
V_{{\vec{p}},w({\vec{p}})\eta_{\sigma}({\vec{p}})}(x)=-\sigma
V_{{\vec{p}},\eta_{\sigma}({\vec{p}})}(x)\,.~{}~{}~{}~{}~{}$ (79)

on the mode spinors depending on the polarization spinors satisfying Eqs.
(76).

The above calculations in active mode are tedious and less intuitive because
of the complicated Fourier transforms of the principal operators. This explain
why the Pryce spin operator proposed long time ago was ignored for more than
seven decades. The solution we proposed recently is to consider the passive
mode in which we focus on the operators acting on the wave spinors (23).

##  5 Operators in passive mode

In the passive mode we relate the operators acting on the free field (18) in
CR to corresponding operators acting on the wave spinors (23) we call the
associated operators. We associate thus to each operator $A:{\cal F}\to{\cal
F}$ the pair of operators, $\tilde{A}:\tilde{\cal F}^{+}\to\tilde{\cal F}$ and
$\tilde{A}^{c}:\tilde{\cal F}^{-}\to\tilde{\cal F}$, obeying

$\displaystyle(A\psi)(x)=\int
d^{3}p\sum_{\sigma}\left[U_{{\vec{p}},\sigma}(x)(\tilde{A}\alpha)_{\sigma}({\vec{p}})+V_{{\vec{p}},\sigma}(x)(\tilde{A}^{c}\beta)^{*}_{\sigma}({\vec{p}})\right]\,,$
(80)

such that the brackets of $A$ for two different fields, $\psi$ and
$\psi^{\prime}$, can be calculated as

$\langle\psi,A\psi^{\prime}\rangle_{D}=\langle\alpha,\tilde{A}\alpha^{\prime}\rangle+\langle\beta,\tilde{A}^{c\,+}\beta^{\prime}\rangle\,.$
(81)

Hereby we deduce that if $A=A^{+}$ is Hermitian with respect to the Dirac
scalar product (LABEL:sp) then the associated operators are Hermitian with
respect to the scalar product (25), $\tilde{A}={\tilde{A}}^{+}$ and
$\tilde{A}^{c}={\tilde{A}^{c}\,}^{+}$. For simplicity we denote the Hermitian
conjugation of the operators acting on ${\cal F}$ and $\tilde{\cal F}$ with
the same symbol but bearing in mind that the scalar products of these spaces
are different. Note that, the operators $A\in E[t_{c}]$ and their associated
operators, $(\tilde{A},\tilde{A}^{c})$, may depend on time such that we must
be careful considering the entire algebra we manipulate as frozen at a fixed
time $t_{c}$.

The new operators $\tilde{A}$ and $\tilde{A}^{c}$ are well-defined as their
action can be derived by applying the inversion formulas (24) at any given
instant $t_{c}$. We find thus that $\tilde{A}$ and $\tilde{A}^{c}$ are
integral operators that may depend on time acting as

$\displaystyle\left.(\tilde{A}\alpha)_{\sigma}({\vec{p}})\right|_{t_{c}}$
$\displaystyle=$ $\displaystyle\int
d^{3}p^{\prime}\sum_{\sigma^{\prime}}\left.\langle
U_{{\vec{p}},\sigma},AU_{{\vec{p}}\,^{\prime},\sigma^{\prime}}\rangle_{D}\right|_{t_{c}}\alpha_{\sigma^{\prime}}({\vec{p}}\,^{\prime})$
(82) $\displaystyle+$ $\displaystyle\int
d^{3}p^{\prime}\sum_{\sigma^{\prime}}\left.\langle
U_{{\vec{p}},\sigma},AV_{{\vec{p}}\,^{\prime},\sigma^{\prime}}\rangle_{D}\right|_{t_{c}}\beta^{*}_{\sigma^{\prime}}({\vec{p}}\,^{\prime})\,,$
$\displaystyle\left.(\tilde{A}^{c}\beta)_{\sigma}({\vec{p}})\right|_{t}$
$\displaystyle=$ $\displaystyle\int
d^{3}p^{\prime}\sum_{\sigma^{\prime}}\left.\langle
U_{{\vec{p}}\,^{\prime},\sigma^{\prime}},AV_{{\vec{p}},\sigma}\rangle_{D}\right|_{t_{c}}\alpha^{*}_{\sigma^{\prime}}({\vec{p}}\,^{\prime})$
(83) $\displaystyle+$ $\displaystyle\int
d^{3}p^{\prime}\sum_{\sigma^{\prime}}\left.\langle
V_{{\vec{p}}\,^{\prime},\sigma^{\prime}},AV_{{\vec{p}},\sigma}\rangle_{D}\right|_{t}\beta_{\sigma^{\prime}}({\vec{p}}\,^{\prime})\,,$

through kernels which are the matrix elements of the operator $A$ in the basis
of mode spinors.

In what follows we restrict ourselves to reducible operators which do not mix
the particle and antiparticle subspaces complying with the condition (57) at
any time. In the particular case of reducible Fourier operators, $A\in\,F[t]$,
having time-dependent Fourier transforms $A(t,{\vec{p}})$, the non-vanishing
matrix elements can be calculated easier as

$\displaystyle\left.\langle
U_{{\vec{p}},\sigma},AU_{{\vec{p}}\,^{\prime},\sigma^{\prime}}\rangle_{D}\right|_{t_{c}}=\left.\langle
U_{{\vec{p}},\sigma},A(t_{c},{\vec{p}}\,^{\prime})U_{{\vec{p}}\,^{\prime},\sigma^{\prime}}\rangle_{D}\right|_{t_{c}}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\delta^{3}({\vec{p}}-{\vec{p}}\,^{\prime})\mathring{u}_{\sigma}^{+}({\vec{p}})\pi_{+}d({\vec{p}}){\cal
U}^{*}_{p}(t_{c})A(t_{c},{\vec{p}}){\cal
U}_{p}(t_{c})d({\vec{p}})\pi_{+}\,\mathring{u}_{\sigma^{\prime}}({\vec{p}})\,,$
(84) $\displaystyle\left.\langle
V_{{\vec{p}},\sigma},AV_{{\vec{p}}\,^{\prime},\sigma^{\prime}}\rangle_{D}\right|_{t_{c}}=\left.\langle
V_{{\vec{p}},\sigma},A(t_{c},-{\vec{p}}\,^{\prime})V_{{\vec{p}}\,^{\prime},\sigma^{\prime}}\rangle_{D}\right|_{t_{c}}$
$\displaystyle~{}~{}~{}~{}~{}~{}=\delta^{3}({\vec{p}}-{\vec{p}}\,^{\prime})\mathring{v}_{\sigma}^{+}({\vec{p}})\pi_{-}d({\vec{p}}){\cal
V}_{p}^{*}(t_{c})A(t_{c},-{\vec{p}}){\cal
V}_{p}(t_{c})d({\vec{p}})\pi_{-}\,\mathring{v}_{\sigma^{\prime}}({\vec{p}})\,,$
(85)

observing that in this case the associated operators are simple $2\times 2$
matrix-operators acting separately on the spaces $\tilde{\cal F}^{+}$ and
$\tilde{\cal F}^{-}$. We observe that when $A(t_{c},{\vec{p}})$ commutes with
${\cal U}_{p}(t_{c})$, ${\cal V}_{p}(t_{c})$, $d({\vec{p}})$ and $\pi_{\pm}$
then we must use directly the identities (70) and (71). Applying first this
rule to the components of our conserved spin operator we obtain the associated
operators

$\displaystyle
S_{i}~{}~{}\Rightarrow~{}~{}\tilde{S}_{i}=-\tilde{S}_{i}^{c}=\frac{1}{2}\Sigma_{i}({\vec{p}})\,,$
(86)

where the $2\times 2$ matrices $\Sigma_{i}({\vec{p}})$ have the matrix
elements

$\Sigma_{i\,\sigma\sigma^{\prime}}({\vec{p}})=2\mathring{u}_{\sigma}^{+}({\vec{p}})s_{i}\mathring{u}_{\sigma^{\prime}}({\vec{p}})=\xi^{+}_{\sigma}({\vec{p}})\sigma_{i}\,\xi_{\sigma^{\prime}}({\vec{p}})\,,$
(87)

depending on the polarization spinors and having the same algebraic properties
as the Pauli matrices. A similar procedure gives the simple associated
operators of the polarization operator (77),

$W_{s}~{}\Rightarrow~{}~{}\tilde{W}_{s}=-\tilde{W}_{s}^{c}=\frac{1}{2}\sigma_{3}\,,$
(88)

according to the definition (76) of the polarization spinors.

It is interesting now to see how the spin operator is related to the
generators of the $E(3)$ isometries of the spacetimes $M(a)$. As the covariant
representation ${T}$ defined by Eq. (9) is reducible this may be associated to
a pair of representations whose operators $\tilde{T}\in{\rm Aut}(\tilde{\cal
F}^{+})$ and $\tilde{T}^{c}\in{\rm Aut}(\tilde{\cal F}^{-})$ are related to
$T$ as,

$\displaystyle({T}_{r,{\vec{a}}}\,\psi)(x_{c})$ $\displaystyle\hskip
34.1433pt=\int
d^{3}p\sum_{\sigma}\left[U_{{\vec{p}},\sigma}(x_{c})(\tilde{T}_{r,{\vec{a}}}\,\alpha)_{\sigma}({\vec{p}})+V_{{\vec{p}},\sigma}(x_{c})(\tilde{T}^{c}_{r,{\vec{a}}}\,\beta)^{*}_{\sigma}({\vec{p}})\right]\,.$
(89)

In other respects, by using the identity
$(R{\vec{x}})\cdot{\vec{p}}={\vec{x}}\cdot(R^{-1}{\vec{p}})$ we expand Eq. (9)
changing the integration variable as

$\displaystyle({T}_{r,{\vec{a}}}\psi)(t_{c},{\vec{x}}_{c})=r\psi\left(t_{c},R(\hat{r})^{-1}({\vec{x}}_{c}-{\vec{a}})\right)$
$\displaystyle\hskip 28.45274pt=\int
d^{3}p\sum_{\sigma}\left[rU^{\prime}_{{\vec{p}},\sigma}(x_{c})\alpha_{\sigma}({\vec{p}}^{\,\prime})e^{i{\vec{a}}\cdot{\vec{p}}}+rV^{\prime}_{{\vec{p}},\sigma}(x_{c})\beta^{*}_{\sigma}({\vec{p}}^{\,\prime})e^{-i{\vec{a}}\cdot{\vec{p}}}\right]\,,$
(90)

where the new mode spinors

$\displaystyle U^{\prime}_{\vec{p},\sigma}({x}_{c})$ $\displaystyle=$
$\displaystyle[2\pi
a(t_{c})]^{-\frac{3}{2}}{e^{i{\vec{p}}\cdot{\vec{x}}_{c}}}{\cal
U}_{p}(t_{c})d({\vec{p}}\,^{\prime})u_{\sigma}({\vec{p}}\,^{\prime})\,,$ (91)
$\displaystyle V^{\prime}_{\vec{p},\sigma}({x}_{c})$ $\displaystyle=$
$\displaystyle[2\pi
a(t_{c})]^{-\frac{3}{2}}{e^{-i{\vec{p}}\cdot{\vec{x}}_{c}}}{\cal
V}_{p}(t_{c})d({\vec{p}}\,^{\prime})v_{\sigma}({\vec{p}}\,^{\prime})\,,$ (92)

depend on the transformed momentum

${{\vec{p}}\,}^{\prime}=R(\hat{r})^{-1}{\vec{p}}~{}~{}\Rightarrow~{}~{}|{\vec{p}}\,^{\prime}|=|{\vec{p}}|\,.$
(93)

Observing then that, according to Eq. (A.4), we have
$rd({\vec{p}}\,^{\prime})=d({\vec{p}})r$ we deduce that
$\tilde{T}_{r,{\vec{a}}}\simeq\tilde{T}^{c}_{r,{\vec{a}}}$ acting alike on the
subspaces $\tilde{\cal F}^{+}$ and $\tilde{\cal F}^{-}$ as ,

$\displaystyle(\tilde{T}_{r,{\vec{a}}}\,\alpha)_{\sigma}({\vec{p}})=e^{i{\vec{a}}\cdot{\vec{p}}}\sum_{\sigma^{\prime}}{D}_{\sigma\sigma^{\prime}}(\hat{r},{\vec{p}})\alpha_{\sigma^{\prime}}({{\vec{p}}\,}^{\prime})\,,$
(94)

and similarly for $\beta$, where ${D}(\hat{r},{\vec{p}})$ is just the matrix
(61).

The representation $\tilde{T}_{r,{\vec{a}}}$ is unitary with respect to the
scalar product (25),
$\langle\tilde{T}_{\lambda,a}\alpha,\tilde{T}_{\lambda,a}\alpha^{\prime}\rangle=\langle\alpha,\alpha^{\prime}\rangle$.
As the covariant representations are unitary with respect to the scalar
product (8) which can be decomposed as in Eq. (26) we conclude that the
expansion (18) establishes the unitary equivalence,
${T}=\tilde{T}\oplus\tilde{T}$. Consequently, the self-adjoint generators
$\tilde{X}\in{\rm Lie}(\tilde{T})$ defined as

$\displaystyle\tilde{P}^{i}=-\left.i\frac{\partial\tilde{T}_{1,{\vec{a}}}}{\partial
a^{i}}\right|_{{\vec{a}}=0}\,,\quad\tilde{J}_{i}=\left.i\frac{\partial\tilde{T}_{r(\theta),0}}{\partial\theta^{i}}\right|_{\theta=0}\,,$
(95)

are related to the corresponding ones, $X\in{\rm Lie}({T})$, such that

$\displaystyle(X\psi)(x_{c})$ $\displaystyle=$ $\displaystyle\int
d^{3}p\sum_{\sigma}\left[U_{{\vec{p}},\sigma}(x_{c})(\tilde{X}\,\alpha)_{\sigma}({\vec{p}})-\,V_{{\vec{p}},\sigma}(x_{c})(\tilde{X}\,\beta)^{*}_{\sigma}({\vec{p}})\right]\,,$
(96)

as we deduce deriving Eq. (89) with respect to the corresponding group
parameter $\zeta\in(\theta^{i},a^{i})$ in $\zeta=0$. We find thus that the
isometry generators are reducible on $\tilde{\cal F}$ obeying
$\tilde{X}^{c}=-\tilde{X}$ as a consequence of the fact that
$\tilde{T}^{c}\simeq\tilde{T}$.

The associated Abelian generators are trivial being diagonal in momentum
basis,

$P^{i}~{}~{}\Rightarrow~{}~{}\tilde{P}^{i}=-\tilde{P}^{c\,i}=p^{i}\,.$ (97)

For rotations we use the Cayley-Klein parameters as in Eq. (A.8) recovering
the natural splitting (13),

$\displaystyle J_{i}=L_{i}+S_{i}~{}~{}$ $\displaystyle\Rightarrow$
$\displaystyle~{}~{}\tilde{J}_{i}=-\tilde{J}^{c}_{i}=\tilde{L}_{i}+\tilde{S}_{i}\,,$
(98)

laying out the components of the spin operator (86) and intuitive components
of the orbital angular momentum operator,

$L_{i}~{}~{}\Rightarrow~{}~{}\tilde{L}_{i}=-\tilde{L}_{i}^{c}=-i\epsilon_{ijk}p^{j}\tilde{\partial}_{k}\,,$
(99)

where $\tilde{\partial}_{k}$ are the covariant derivatives [2],

$\tilde{\partial}_{i}=\partial_{p^{i}}1_{2\times 2}+\Omega_{i}({\vec{p}})\,,$
(100)

defined such that
$\tilde{\partial}_{i}[\xi_{\sigma}({\vec{p}})\alpha_{\sigma}({\vec{p}})]=\xi_{\sigma}({\vec{p}})\tilde{\partial}_{i}\alpha_{\sigma}({\vec{p}}$.
Therefore, the connections,

$\displaystyle\Omega_{i\,\sigma\sigma^{\prime}}({\vec{p}})=\xi^{+}_{\sigma}({\vec{p}})\left[\partial_{p^{i}}\xi_{\sigma^{\prime}}({\vec{p}})\right]=\left\\{\eta^{+}_{\sigma}({\vec{p}})\left[\partial_{p^{i}}\eta_{\sigma^{\prime}}({\vec{p}})\right]\right\\}^{*}\,,$
(101)

are anti-Hermitian,
$\Omega_{i\,\sigma\sigma^{\prime}}({\vec{p}})=-\Omega_{i\,\sigma^{\prime}\sigma}^{*}({\vec{p}})$,
such that the operators $i\tilde{\partial}_{i}$ are Hermitian. The principal
property of the covariant derivatives is to commute with the spin components,
$[\tilde{\partial}_{i},\tilde{S}_{j}]=0$ thanks to the connections
$\Omega_{i}({\vec{p}})$. This becomes trivial in the case of common
polarization when $\Omega_{i}=0$ and $\tilde{S}_{i}$ are independent on
${\vec{p}}$. Note that we proposed these derivatives for the first time in
Dirac’s theory in Minkowski spacetime [2].

The sets of conserved operators
$\\{\tilde{L}_{1},\tilde{L}_{2},\tilde{L}_{3}\\}$ and
$\\{\tilde{S}_{1},\tilde{S}_{2},\tilde{S}_{3}\\}$ generate the representations
$\tilde{T}^{o}$ and $\tilde{T}^{s}$ of the associated factorization

${T}^{r}=T^{o}\otimes
T^{s}~{}~{}\Rightarrow~{}~{}\tilde{T}^{r}=\tilde{T}^{o}\otimes\tilde{T}^{s}\,,$
(102)

of the $SU(2)$ restriction $\tilde{T}^{r}\equiv\left.\tilde{T}\right|_{SU(2)}$
of the representation $\tilde{T}$. Thus we have found the generators of the
associated orbital representation studying the isometry generators without
resorting to new spectral representations. In this manner we cannot come back
to the active mode in CR but we have all we need for performing the
quantization.

##  6 Quantization

The association between the operator acting in CR and MR allows us to derive
at any time the expectation values of operators defined in MR according to the
general rule (81). We may apply thus the Bogolyubov method [9] for quantizing
the isometry generators of the massive Dirac fermions of arbitrary
polarization. According to this method, we replace first the wave spinors in
MR with field operators,
$(\alpha,\alpha^{*})\to({\mathfrak{a}},{\mathfrak{a}}^{{\dagger}})$ and
$(\beta,\beta^{*})\to({\mathfrak{b}},{\mathfrak{b}}^{{\dagger}})$, satisfying
canonical anti-commutation relations among them the non-vanishing ones are,

$\displaystyle\left\\{{\mathfrak{a}}_{\sigma}({\vec{p}}),{\mathfrak{a}}_{\sigma^{\prime}}^{{\dagger}}({\vec{p}}^{\,\prime})\right\\}=\left\\{{\mathfrak{b}}_{\sigma}({\vec{p}}),{\mathfrak{b}}_{\sigma^{\prime}}^{{\dagger}}({\vec{p}}^{\,\prime})\right\\}=\delta_{\sigma\sigma^{\prime}}\delta^{3}({\vec{p}}-{\vec{p}}^{\,\prime})\,.$
(103)

The Dirac free field becomes thus the field operator

$\displaystyle\psi(x)=\int
d^{3}p\sum_{\sigma}\left[U_{{\vec{p}},\sigma}(x){\mathfrak{a}}_{\sigma}({\vec{p}})+V_{{\vec{p}},\sigma}(x){\mathfrak{b}}^{{\dagger}}_{\sigma}({\vec{p}})\right]\,,$
(104)

denoted with the same symbol but acting on the Fock state space equipped with
the scalar product $\langle~{}~{}|~{}~{}\rangle$ and a normalized vacuum state
$|0\rangle$ accomplishing

${\mathfrak{a}}_{\sigma}({\vec{p}})|0\rangle={\mathfrak{b}}_{\sigma}({\vec{p}})|0\rangle=0\,,\quad\langle
0|{\mathfrak{a}}_{\sigma}^{\dagger}({\vec{p}})=\langle
0|{\mathfrak{b}}_{\sigma}^{\dagger}({\vec{p}})=0\,.$ (105)

The sectors with different number of particles have to be constructed applying
the standard method for constructing generalized momentum bases of various
polarizations.

Through quantization the expectation value of any time-dependent operator
$A(t)$ of RQM becomes a one-particle operator

$A(t_{c})~{}\to~{}\mathsf{A}=\left.:\langle\psi,A(t_{c})\psi\rangle_{D}:\right|_{t_{c}=t_{c\,0}}\,,$
(106)

calculated respecting the normal ordering of the operator products [13, 14] at
the initial time $t_{c\,0}$. This procedure allows us to write down any one-
particle operator $\mathsf{A}$ directly in terms of operators
$(\tilde{A},\tilde{A}^{c})$ associated to the operator
$A=A(t_{c})|_{t_{c\,0}}$. We consider here only the reducible operators for
which we obtain the general formula

$\mathsf{A}=\int
d^{3}{p}\left[{\mathfrak{a}}^{{\dagger}}({\vec{p}})(\tilde{A}{\mathfrak{a}})({\vec{p}})-{\mathfrak{b}}^{{\dagger}}({\vec{p}})(\tilde{A}^{c\,+}{\mathfrak{b}})({\vec{p}})\right]\,,$
(107)

written with the compact notation

${\mathfrak{a}}^{{\dagger}}({\vec{p}})(\tilde{A}{\mathfrak{a}})({\vec{p}})\equiv\sum_{\sigma}{\mathfrak{a}}^{{\dagger}}_{\sigma}({\vec{p}})(\tilde{A}{\mathfrak{a}})_{\sigma}({\vec{p}})\,,$
(108)

and similarly for the second term. We specify that the bracket (106) is
calculated according to Eq. (81) in which the last term changes its sign after
introducing the normal ordering of the operator products.

Given an arbitrary operator $A\in{\rm Aut}({\cal F})$ and its Hermitian
conjugated $A^{+}$ we define the adjoint operator of $\mathsf{A}$,

$\displaystyle
A^{+}(t_{c})~{}\to~{}\mathsf{A}^{\dagger}=\left.:\langle\psi,A(t)^{+}\psi\rangle_{D}:\right|_{t_{c\,0}}=\left.:\langle
A(t)\psi,\psi\rangle_{D}:\right|_{t_{c\,0}}\,,$ (109)

complying with the standard definition
$\langle\alpha|\mathsf{A}^{\dagger}\beta\rangle=\langle\mathsf{A}\alpha|\beta\rangle$
on the Fock space. In what follows we meet only self-adjoint one-particle
operators as all their corresponding operators of RQM are reducible and
Hermitian with respect to the scalar products of the spaces in which they act.
We obtain thus an operator algebra formed by fields and self-adjoint one-
particle operators which have the obvious properties

$\displaystyle\left[\mathsf{A},\psi(x)\right]=-(A\psi)(x)\,,\qquad\left[\mathsf{A},\mathsf{B}\right]=:\left<\psi,[A,B]\psi\right>_{D}:\,,$
(110)

preserving the structures of Lie algebras but without taking over other
algebraic properties of their parent operators from RQM as the product of two
one-particle operators is no longer an operator of the same type. Therefore,
we must restrict ourselves to the Lie algebras of symmetry generators and
unitary transformations whose actions reduce to sums of successive
commutations.

The simplest one-particle operator is the electric charge

$\mathsf{Q}=:\langle\psi,\psi\rangle_{D}:=\int
d^{3}p\,\left[{\mathfrak{a}}^{{\dagger}}({\vec{p}}){\mathfrak{a}}({\vec{p}})-{\mathfrak{b}}^{{\dagger}}({\vec{p}}){\mathfrak{b}}({\vec{p}})\right]\,,$
(111)

related to the internal gauge symmetry $U(1)_{\rm em}$ [13]. Other diagonal
operators in momentum basis are the momentum components

$\displaystyle\mathsf{P}^{i}$ $\displaystyle=$
$\displaystyle:\langle\psi,P^{i}\psi\rangle_{D}:=\int
d^{3}p\,p^{i}\left[{\mathfrak{a}}^{{\dagger}}({\vec{p}}){\mathfrak{a}}({\vec{p}})+{\mathfrak{b}}^{{\dagger}}({\vec{p}}){\mathfrak{b}}({\vec{p}})\right]\,,$
(112)

as well as our new operator of fermion polarization ,

$\displaystyle\mathsf{W}_{s}=:\langle\psi,W_{s}\psi\rangle_{D}:=\frac{1}{2}\int
d^{3}p\left[{\mathfrak{a}}^{{\dagger}}({\vec{p}}){\sigma}_{3}{\mathfrak{a}}({\vec{p}})+{\mathfrak{b}}^{{\dagger}}({\vec{p}}){\sigma}_{3}{\mathfrak{b}}({\vec{p}})\right]\,,$
(113)

which enters in the incomplete set of commuting operators
$\\{\mathsf{P}^{1},\mathsf{P}^{2},\mathsf{P}^{3},\mathsf{W}_{s},\mathsf{Q}\\}$
determining the momentum bases of the Fock state space up to an integration
constant that has to be fixed setting the vacuum as in the case of our rest
frame vacuum defined by Eqs. (45). In Minkowski spacetime we do not have this
inconvenience as there exists a conserved energy operator able to complete
this set commuting with all the other operators. In contrast, in the de Sitter
expanding universe we have a conserved energy operator but this does not
commute with the momentum components. Thus the problem of setting the vacuum
arises in all the FLRW spacetimes apart from the Minkowski one.

Furthermore, applying the general rule (107) to the rotation generators we
find first the splitting of the total angular momentum

$\mathsf{J}_{i}=:\langle\psi,J_{i}\psi\rangle_{D}:=\mathsf{L}_{i}+\mathsf{S}_{i}\,,$
(114)

where the components of the orbital angular momentum, $\mathsf{L}_{i}$, and
spin operator, $\mathsf{S}_{i}$, can be written as

$\displaystyle\mathsf{L}_{i}$ $\displaystyle=$ $\displaystyle-\frac{i}{2}\int
d^{3}p\,\epsilon_{ijk}p^{j}\left[{a}^{{\dagger}}({\vec{p}}){\stackrel{{\scriptstyle\leftrightarrow}}{{\tilde{\partial}_{i}}}}{a}({\vec{p}})+{b}^{{\dagger}}({\vec{p}}){\stackrel{{\scriptstyle\leftrightarrow}}{{\tilde{\partial}_{i}}}}{b}({\vec{p}})\right]\,,$
(115) $\displaystyle\mathsf{S}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
d^{3}p\left[{a}^{{\dagger}}({\vec{p}})\Sigma_{i}({\vec{p}}){a}({\vec{p}})+{b}^{{\dagger}}({\vec{p}})\Sigma_{i}({\vec{p}}){b}({\vec{p}})\right]\,,$
(116)

according to Eqs. (99) and (86), using the special notation

$\alpha^{+}\stackrel{{\scriptstyle\leftrightarrow}}{{\tilde{\partial}_{i}}}\beta=\alpha^{+}(\partial_{p^{i}}\beta)-(\partial_{p^{i}}\alpha^{+})\beta+2\alpha^{+}\Omega({\vec{p}})\beta\,,$
(117)

inspired by Green’s theorem, which points out explicitly that $\mathsf{L}_{i}$
are self-adjoint operators. The components $\mathsf{L}_{i}$ and
$\mathsf{S}_{i}$ form the bases of two independent unitary representations of
the $su(2)\sim so(3)$ algebra, $\left[\mathsf{L}_{i},\mathsf{S}_{j}\right]=0$,
generating the orbital and respectively spin symmetries. Moreover, these
operators are conserved while the commutation relations

$\left[\mathsf{L}_{i},\mathsf{P}^{j}\right]=i\epsilon_{ijk}\mathsf{P}^{k}\,,\qquad\left[\mathsf{S}_{i},\mathsf{P}^{j}\right]=0\,,$
(118)

show that only the spin operator is, in addition, invariant under space
translations.

We obtained thus all the one-particle operators of QFT coming from the
conserved operators of RQM. All these operators have similar forms in any
spacetime $M(a)$, including the Minkowski one, being independent on the scale
factors. However, this universality is limited as the evolution of the time-
dependent operators is determined by the time modulation functions satisfying
the system (36). For example, if we apply the above procedures to the
Hamiltonian (7) at an arbitrary instant $t_{c}$ we obtain the time-dependent
operator

$\displaystyle\mathsf{H}_{c}(t_{c})=\left.:\langle\psi,H_{c}(t_{c})\psi\rangle_{D}:\right|_{t_{c}}=\int
d^{3}p\,\tilde{H}_{c}(t_{c},{p})\left[{\mathfrak{a}}^{{\dagger}}({\vec{p}}){\mathfrak{a}}({\vec{p}})+{\mathfrak{b}}^{{\dagger}}({\vec{p}}){\mathfrak{b}}({\vec{p}})\right]\,,$
(119)

where the quantity

$\displaystyle H_{c}(t_{c},{p})$ $\displaystyle=$ $\displaystyle
m\,a(t_{c})\left(|u^{+}(t_{c},p)|^{2}-|u^{-}(t_{c},p)|^{2}\right)$ (120)
$\displaystyle+$ $\displaystyle
p\left(u^{+}(t_{c},p)u^{-}(t_{c},p)^{*}+u^{+}(t_{c},p)^{*}u^{-}(t_{c},p)\right)-\frac{3i}{2}\frac{\dot{a}(t_{c})}{a(t_{c})}\,,~{}~{}~{}~{}$

plays the role of energy giving just the special relativistic energy $E(p)$ in
the flat limit when $a\to 1$, $\dot{a}\to 0$ and the time modulation functions
take the form (A.11). In other respects, the form of the orbital angular
momentum (115) suggests us that the operator of the initial position related
to the conserved spin as in Eq. (13) could have the same form as in the flat
case, [2, 3]

$\mathsf{X}^{i}=\frac{i}{2}\int
d^{3}p\left[{\mathfrak{a}}^{{\dagger}}({\vec{p}})\stackrel{{\scriptstyle\leftrightarrow}}{{\tilde{\partial}_{i}}}{\mathfrak{a}}({\vec{p}})-{\mathfrak{b}}^{{\dagger}}({\vec{p}})\stackrel{{\scriptstyle\leftrightarrow}}{{\tilde{\partial}_{i}}}{\mathfrak{b}}({\vec{p}})\right]\,.$
(121)

For studying the time evolution of this operator we need to calculate
commutators as $[\mathsf{H}_{c}(t_{c}),\mathsf{X}^{i}]$ which depends on the
derivatives of time modulation functions, $\partial_{p}u^{\pm}(t_{c},p)$.
Therefore we may conclude that the time evolution of the principal observables
must be studied in each particular case separately.

##  Appendix A  $SL(2,\mathbb{C})$ transformations

The Dirac field $\psi:M\to{\cal V}_{D}$ takes values in the space ${\cal
V}_{D}={\cal V}_{P}\oplus{\cal V}_{P}$ of the Dirac representation
$\rho_{D}=(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ of the $SL(2,\mathbb{C})$
group where one defines the Dirac $\gamma$-matrices with local indices and the
Hermitian form $\overline{\psi}\psi$ with the Dirac adjoint
$\overline{\psi}=\psi^{+}\gamma^{0}$ of $\psi$. These matrices satisfy the
anticommutation rules

$\\{\gamma^{\hat{\mu}},\gamma^{\hat{\nu}}\\}=2\eta^{\hat{\mu}\hat{\nu}}\,$
(A.1)

giving rise to the $SL(2,\mathbb{C})$ generators

$s^{\hat{\mu}\hat{\nu}}=\frac{i}{4}\left[\gamma^{\hat{\mu}},\gamma^{\hat{\nu}}\right]=\overline{s^{\hat{\mu}\hat{\nu}}}\in{\rho_{D}}[sl(2,\mathbb{C})]$
(A.2)

which are Dirac self-adjoint such that the transformations

$\lambda(\hat{\omega})=\exp\left(-\frac{i}{2}\hat{\omega}^{\hat{\alpha}\hat{\beta}}s_{\hat{\alpha}\hat{\beta}}\right)\in\rho_{D}[SL(2,\mathbb{C})]\,,$
(A.3)

having real-valued parameters,
$\hat{\omega}^{\hat{\alpha}\hat{\beta}}=-\hat{\omega}^{\hat{\beta}\hat{\alpha}}$,
leave the Hermitian form invariant as
$\overline{\lambda(\hat{\omega})}=\lambda^{-1}(\hat{\omega})=\lambda(-\hat{\omega})$.
The corresponding Lorentz transformations,
$\Lambda^{\hat{\mu}\,\cdot}_{\cdot\,\hat{\nu}}(\hat{\omega})\equiv\Lambda^{\hat{\mu}\,\cdot}_{\cdot\,\hat{\nu}}[\lambda(\hat{\omega})]=\delta^{\hat{\mu}}_{\hat{\nu}}+\hat{\omega}^{\hat{\mu}\,\cdot}_{\cdot\,\hat{\nu}}+\frac{1}{2}\,\hat{\omega}^{\hat{\mu}\,\cdot}_{\cdot\,\hat{\alpha}}\hat{\omega}^{\hat{\alpha},\cdot}_{\cdot\,\hat{\nu}}+\cdots$
satisfy the identities

$\lambda^{-1}(\hat{\omega})\gamma^{\hat{\alpha}}\lambda(\hat{\omega})=\Lambda(\hat{\omega})^{\hat{\alpha}\,\cdot}_{\cdot\,\hat{\beta}}\gamma^{\hat{\beta}}\,,$
(A.4)

which encapsulate the canonical homomorphism [8].

In the chiral representation of the Dirac matrices (with diagonal
$\gamma^{5}$) the transformations $\lambda(\hat{\omega})$ generated by the
matrices $s^{\mu\nu}$ are reducible to the subspaces of Pauli spinors ${\cal
V}_{P}$ carrying the irreducible representations $(\frac{1}{2},0)$ and
$(0,\frac{1}{2})$ of ${\rho_{D}}$ [8, 7]. We denote by

$r={\rm diag}(\hat{r},\hat{r})\in{\rho_{D}}\left[SU(2)\right]$ (A.5)

the transformations we call here simply rotations, and by

$l={\rm
diag}(\hat{l},\hat{l}^{-1})\in{\rho_{D}}\left[SL(2,\mathbb{C})/SU(2)\right]$
(A.6)

the Lorentz boosts. For rotations we use the generators

$s_{i}=\frac{1}{2}\epsilon_{ijk}s^{jk}={\rm
diag}(\hat{s}_{i},\hat{s}_{i})\,,\quad\hat{s}_{i}=\frac{1}{2}\sigma_{i}\,,$
(A.7)

and Cayley-Klein parameters
$\theta^{i}=\frac{1}{2}\epsilon_{ijk}\hat{\omega}^{jk}$ such that

$\displaystyle r(\theta)={\rm
diag}(\hat{r}(\theta),\hat{r}(\theta))\,,\quad~{}~{}\hat{r}(\theta)=e^{-i\theta^{i}\hat{s}_{i}}=e^{-\frac{i}{2}\theta^{i}\sigma_{i}}\,,$
(A.8)

where $\sigma_{i}$ are the Pauli matrices. Similarly, we chose the parameters
$\tau^{i}=\hat{\omega}^{0i}$ and the generators $s_{0i}=-s^{0i}={\rm
diag}(i\hat{s}_{i},-i\hat{s}_{i})$ for the Lorentz boosts, $l(\tau)={\rm
diag}(\hat{l}(\tau),\hat{l}^{-1}(\tau))$ where
$\hat{l}(\tau)=e^{\tau^{i}\hat{s}_{i}}=e^{\frac{1}{2}\tau^{i}\sigma_{i}}$. The
corresponding transformations of the group $L_{+}^{\uparrow}$ will be denoted
as $R(\hat{r})=\Lambda(r)$ and $L(\hat{l})=\Lambda(l)$.

Particularly, the boosts (A.6) of Wigner’s method of constructing the
covariant Dirac field in Minkowski spacetime have the parameters
$\tau^{i}=-\frac{p^{i}}{p}{\rm tanh}^{-1}\frac{p}{E(p)}$ taking the form [7]

$\displaystyle
l_{{\vec{p}}}=\frac{E(p)+m+\gamma^{0}{\vec{\gamma}}\cdot{\vec{p}}}{\sqrt{2m(E(p)+m)}}=l_{{\vec{p}}}^{+}\,,\quad
l^{-1}_{\vec{p}}=l_{-\vec{p}}=\bar{l}_{\vec{p}}\,,$ (A.9)

where $E(p)=\sqrt{m^{2}+{\vec{p}}^{2}}$ is the energy in special relativity.
These boosts which determine the form of the mode spinors in Minkowski
spacetime can be related as

$\sqrt{\frac{m}{E(p)}}l_{\vec{p}}\,e^{-iE(p)t}=u^{+}_{M}(t,p)+\frac{\gamma^{0}{\vec{\gamma}}\cdot{\vec{p}}}{p}u_{M}^{-}(t,p)$
(A.10)

to the time modulation functions in this manifold,

$u^{\pm}_{M}(t,p)=\sqrt{\frac{E(p)\pm m}{2E(p)}}e^{-iE(p)t}\,.$ (A.11)

Hereby we deduce the identity

$\left(u^{+}_{M}(t,p)+\frac{\gamma^{0}{\vec{\gamma}}\cdot{\vec{p}}}{p}u_{M}^{-}(t,p)\right)\pi_{+}=\left(u_{M}^{+}(t,p)\pi_{+}+u_{M}^{-}(t,p)\pi_{-}\right)\left(1+\frac{\gamma^{0}{\vec{\gamma}}\cdot{\vec{p}}}{p}\right)\pi_{+}$
(A.12)

that inspires our method of constructing Dirac mode spinors in any FLRW
spacetime with the help of the matrix (29) revealed here.

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