# Discrete symmetries of Dirac’s theory in the de Sitter manifold

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

The discrete symmetries of the Dirac field on the de Sitter manifold are
studied taking into account that this has two portions that can play the role
of physical space-times, namely the expanding and a collapsing universes. The
proper discrete isometries which preserve the portion have a physical meaning
in contrast to the improper ones which change the portion remaining thus of a
mere mathematical interest. The discrete symmetries generated by the proper
isometries and charge conjugation are studied in physical frames on the
expanding portion shoving that all the discrete transformations reversing the
cosmic time are local depending on a local boost matrix. Thus the discrete
group of Dirac’s theory in the de Sitter expanding universe is obtained
showing that this is of the order 16 having a multiplication table similar to
that of Dirac’s theory in special relativity. Moreover, all the discrete de
Sitter isometries, including the improper ones, are studied in conformal
frames for obtaining a global image about the de Sitter isometries in spite of
the fact that these cannot be gathered in a larger discrete group with
physical meaning.

Pacs: 04.62.+v

Keywords: de Sitter space-time; discrete symmetries; proper isometries;
improper isometries; discrete symmetry group.

##  1 Introduction

The Dirac field on the $(3+1)$-dimensional de Sitter manifold, $M$, embedded
in a $(1+4)$-dimensional pseudo-Euclidean manifold, $M^{5}$, was considered in
two dissimilar manners, namely either as an invariant field defined on $M^{5}$
or as a covariant one defined on $M$. The first approach referred often as the
de Sitter ambient was developed by many authors considering a spinor field
transforming under isometries according to linear representations of the
universal covering group ${\rm Spin}SO(1,4)$ of the stable group $SO(1,4)$ of
the embedding space $M^{5}$ (see for instance [1, 2, 3, 4]).

The theory of the covariant Dirac field defined on $M$ was initiated by
Nachtmann [5] and developed then by one of us [6, 7, 8, 9, 10, 11]. In this
approach the Dirac field transforms under continuous $O(1,4)$ isometries
according to the Dirac representation $\rho_{D}$ of the $SL(2,\mathbb{C})={\rm
Spin}L_{+}^{\uparrow}$ group which is the universal covering group of the
proper orthochronus Lorentz group $L_{+}^{\uparrow}$ that is gauge group of
$M$ [12]. The $SL(2,\mathbb{C})$ transformations are associated through the
canonical homomorphism to the gauge ones which have to correct the tetrad
gauge after each isometry in order to leave the Dirac equation invariant [6]
but giving rice to the associated conserved quantities predicted by Carter and
McLenaghan [13, 14, 15]. For this reason we say that the covariant
representation of the isometry group $SO(1,4)$ is induced by $\rho_{D}$. In
other words this approach extends the Lorentz covariance of special relativity
to general relativity. This framework offers many attractive applications [16,
17, 18, 19, 20, 21] such that the study of the discrete de Sitter isometries
generating the discrete symmetries of the Dirac theory was neglected so far.
For this reason we devote this paper to the discrete transformations of the
covariant Dirac field on the de Sitter manifold.

In special relativity the discrete transformations of the Dirac field $\psi$,
written in our phases convention, are the parity, ${\cal P}\psi(t,{\bf
x})=\gamma^{0}\psi(t,-{\bf x})$, charge conjugation ${\cal C}\psi(t,{\bf
x})=i\gamma^{2}\psi^{*}(t,{\bf x})$, and Wigner’s time reversal, ${\cal
T}\psi(t,{\bf x})=-\gamma^{1}\gamma^{3}\psi^{*}(-t,{\bf x})$ [22] that closes
the famous PCT theorem, ${\cal PCT}\psi=\gamma^{5}\psi(-t,-{\bf x})$ [23]. As
${\cal T}$ is not related to an isometry, one uses often the transformation
${\cal CT}\psi(t,{\bf x})=\gamma^{0}\gamma^{5}\psi(-t,{\bf x})$ as an
alternative time reversal [24]. The transformations ${\cal P}$, ${\cal CT}$
and ${\cal PCT}$ are produced by the global discrete $O(1,3)$ isometries $P$,
$T$ and $PT$ transforming the Cartesian coordinates $(t,{\bf x})$ into
$(t,-{\bf x})$, $(-t,{\bf x})$ and respectively $(-t,-{\bf x})$.

For obtaining similar results for the Dirac field on the de Sitter manifold,
$M$, we must focus on the discrete isometries of the $O(1,4)$ isometry group.
On the other hand, we know that only a portion of $M=M_{+}\cup M_{-}$ can play
the role of physical space-time, either as an expanding ($M_{+}$) or as a
collapsing universe ($M_{-}$). We have thus two types of discrete isometries,
namely the proper ones that preserve the portion and the improper isometries
which are changing these portions among themselves. In what follows we intend
to study all these isometries even though the improper ones are of a mere
mathematical interest. Thus we have to study three proper discrete isometries
and four improper ones extending the method of induced representations to the
local discrete transformations of ${\rm Pin}O(1,3)$ we expect to meet here as
in this framework we can construct only three global discrete transformations,
as in the flat case.

The frames we need in our study are formed by a local chart and an orthogonal
non-holonomic frame defined by tetrads. The discrete isometries of the de
Sitter manifold were studied in a special coordonatization many time ago [25]
but unfortunately we cannot use these results as we need to work here with
other types of coordinates. We consider the conformal coordinates which are
helpful in concrete calculations and the physical ones of Painlevé -
Gullstrand type [27, 28] which allow us to understand the physical meaning of
the transformations studied here. Thus we have to work with conformal or
physical frames in which we hope to obtain a complete image about the discrete
transformations of the covariant Dirac field produced by the de Sitter
discrete isometries.

We start in the next section introducing these frames in which we have to
write down the Dirac equation after a brief review of the theory of covariant
Dirac field presented is Sec. 3. Our study of the discrete de Sitter
symmetries starts in Sec. 4 where we present the method of constructing the
induced representations that can be extended to the discrete de Sitter
isometries we derive first in conformal frames separating the proper ones. For
understanding their physical meaning we study in the next section the discrete
symmetries generated by the proper isometries in physical frames composed with
the charge conjugation. We find here a new local  isometry reversing the time
helping us to define the analogous of the Wigner time reversal. In this manner
we obtain the discrete group of order 16 of the de Sitter expanding universe
for which we derive the multiplication table. Furthermore, for getting a
global image about the discrete de Sitter isometries, in Sec. 6 we come back
to the conformal frames where we write down all of them including the improper
ones. Finally we present our concluding remarks.

##  2 Frames

Let us start with the $(1+3)$-dimensional de Sitter manifold, $M$, defined as
a hyperboloid of radius $\omega_{H}^{-1}$ embedded in the five-dimensional
flat space-time $M^{5}$ of coordinates $z^{A}$ (labeled by the indices
$A,\,B,...=0,1,2,3,4$) and metric $\eta^{5}={\rm diag}(1,-1,-1,-1,-1)$ [29],
denoting by $\omega_{H}$ the de Sitter-Hubble constant (frequency). A local
chart of coordinates $x^{\mu}$ ($\mu,\nu,...=0,1,2,3$) can be introduced on
$M$ giving the set of functions $z^{A}(x)$ which solve the hyperboloid
equation, $\eta^{5}_{AB}z^{A}(x)z^{B}(x)=-\omega^{-2}_{H}$.

The simplest local charts are the conformal ones, $\\{x_{c}\\}=\\{t_{c},{\bf
x}_{c}\\}$, with the conformal time $t_{c}$ and co-moving Cartesian space
coordinates, ${\bf x}_{c}=(x_{c}^{1},x_{c}^{2},x_{c}^{3})$, defined by the
functions

$\displaystyle z_{c}^{0}(x_{c})$ $\displaystyle=$
$\displaystyle-\frac{1}{2\omega_{H}^{2}t_{c}}\left[1-\omega_{H}^{2}\left(t_{c}^{2}-{{\bf
x}_{c}\,}^{2}\right)\right]\,,$ (1) $\displaystyle z_{c}^{i}(x_{c})$
$\displaystyle=$ $\displaystyle-\frac{x^{i}_{c}}{\omega_{H}t_{c}}\,,\quad
i,j,...=1,2,3\,,$ (2) $\displaystyle z_{c}^{4}(x_{c})$ $\displaystyle=$
$\displaystyle-\frac{1}{2\omega_{H}^{2}t_{c}}\left[1+\omega_{H}^{2}\left(t_{c}^{2}-{{\bf
x}_{c}\,}^{2}\right)\right]\,,$ (3)

giving rise to the conformal-flat line element

$ds^{2}=\eta^{5}_{AB}dz^{A}dz^{B}=\frac{1}{\omega_{H}^{2}t_{c}^{2}}\,(dt_{c}^{2}-d{\bf
x}_{c}\cdot d{\bf x}_{c})\,.$ (4)

These charts, defined up to an isometry, cover the expanding portion, $M_{+}$,
for $t_{c}\in(-\infty,0)$ and ${\bf x}_{c}\in{\mathbb{R}}^{3}$ while the
collapsing one, $M_{-}$, is covered by similar charts but with $t_{c}>0$ [29].
Note that the conformal coordinates for FLRW space-times were introduced by
Lemaître [26].

However, the coordinates with the obvious physical meaning are those of
Painlevé - Gullstrand type [27, 28] defining the physical local chart
$\\{x\\}=\\{t,{\bf x}\\}$. These coordinates are the cosmic time $t$ and the
physical space coordinates ${\bf x}=(x^{1},x^{2},x^{3})$ that can be
introduced on the expanding portion $M_{+}$ by substituting $x_{c}=\chi(x)$
where

$t_{c}=\chi^{0}(x)=-\frac{1}{\omega_{H}}e^{-\omega_{H}t}\,,\quad{x}^{i}_{c}=\chi^{i}(x)=x^{i}e^{-\omega_{H}t}\,,$
(5)

and obtaining the new line element

$ds^{2}=\left(1-\omega_{H}^{2}{{\bf x}}^{2}\right)dt^{2}+2\omega_{H}{\bf
x}\cdot d{\bf x}\,dt-d{\bf x}\cdot d{\bf x}\,.$ (6)

Similar coordinates can be defined on the collapsing portion $M_{-}$ following
the same procedure but with $\omega_{H}\to-\omega_{H}$.

For writing down the Dirac equation we need to set the tetrad gauge giving 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 co-
frames (labelled by the local indices, $\hat{\mu},\hat{\nu},...=0,1,2,3$).
Here we restrict ourselves to the Cartesian diagonal tetrad gauge defined by
the vector fields

$\displaystyle e_{0}$ $\displaystyle=$
$\displaystyle-\omega_{H}\,t_{c}\partial_{t_{c}}=\partial_{t}+\omega_{H}\,x^{i}\partial_{x^{i}}\,,$
$\displaystyle e_{i}$ $\displaystyle=$
$\displaystyle-\omega_{H}\,t_{c}\partial_{x^{i}_{c}}=\partial_{x^{i}}\,,$ (7)

and the corresponding dual 1-forms

$\displaystyle\omega^{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{\omega_{H}\,t_{c}}dt_{c}=dt\,,$
$\displaystyle\omega^{i}$ $\displaystyle=$
$\displaystyle-\frac{1}{\omega_{H}\,t_{c}}dx_{c}^{i}=d{x}^{i}-\omega_{H}\,{x}^{i}dt\,,$
(8)

which preserve the global $SO(3)$ symmetry allowing us to use systematically
the $SO(3)$ vectors. We obtain thus the conformal frames $\\{x_{c};e\\}$ and
the physical ones $\\{x;e\\}$ formed by a local chart, $\\{x_{c}\\}$ or
$\\{x\\}$, and a local orthogonal frame and co-frame given by the tetrads $e$
and $\hat{e}$, as defined by Eqs. (7) and (8) respectively. We remind the
reader that each physical frame is the proper frames of an observer staying at
rest in origin measuring the events inside the event horizon, for $|{\bf
x}|<\omega_{H}^{-1}$. Therefore, the physical interpretation must be done in
physical frames while the conformal ones are precious tools in our
investigations.

These frames, say the physical ones, can be transformed,
$\\{x;e\\}\to\\{x^{\prime};e^{\prime}\\}$, with the help of diffeomorphisms,
$x\to x^{\prime}=\phi(x)$, transforming the coordinates, and by using local
transformations $\Lambda(x)\in L^{\uparrow}_{+}$, for changing the tetrad
gauge as

$\displaystyle e_{\hat{\alpha}}(x)$ $\displaystyle\to$ $\displaystyle
e_{\hat{\alpha}}^{\prime}(x)=\Lambda_{\hat{\alpha}\,\cdot}^{\cdot\,\hat{\beta}}(x)\,e_{\hat{\beta}}(x)\,,$
(9) $\displaystyle{\omega}^{\hat{\alpha}}(x)$ $\displaystyle\to$
$\displaystyle{\omega}^{\prime\,\hat{\alpha}}(x)=\Lambda^{\hat{\alpha}\,\cdot}_{\cdot\,\hat{\beta}}(x)\,\omega_{\hat{\beta}}(x)\,.$
(10)

In general relativity any theory of fields with spin must be gauge invariant
in the sense that the above gauge transformations have to do not affect the
physical meaning of the theory. However, as here we intend to study discrete
transformations we add them to the gauge group which will be denoted from now
by $G=O(1,3)$.

##  3 Covariant Dirac field

In a 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 the
background gravity, reads,

${\cal
S}[e,\psi]=\int\,d^{4}x\sqrt{g}\left\\{\frac{i}{2}[\overline{\psi}\gamma^{\hat{\alpha}}\nabla_{\hat{\alpha}}\psi-(\overline{\nabla_{\hat{\alpha}}\psi})\gamma^{\hat{\alpha}}\psi]-m\overline{\psi}\psi\right\\}$
(11)

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
$\\{\gamma^{\hat{\alpha}},\,\gamma^{\hat{\beta}}\\}=2\eta^{\hat{\alpha}\hat{\beta}}$.
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 theory releases on the covariant derivatives defined as

$\nabla_{\hat{\alpha}}=e^{\mu}_{\hat{\alpha}}\nabla_{\mu}=e_{\hat{\alpha}}+\frac{i}{2}\hat{\Gamma}^{\hat{\gamma}}_{\hat{\alpha}\hat{\beta}}s^{\hat{\beta}\,\cdot}_{\cdot\,\hat{\gamma}}\,,\quad
s^{\hat{\alpha}\hat{\beta}}=\frac{i}{4}\,[\gamma^{\hat{\alpha}},\gamma^{\hat{\beta}}]\,,$
(12)

where $s^{\hat{\alpha}\hat{\beta}}$ are the $SL(2,{\mathbb{C}})$ generators of
$\rho_{D}$ 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 (known as the spin connections
and denoted often by $\Omega$) expressed in terms of tetrads and Christoffel
symbols, $\Gamma^{\gamma}_{\alpha\beta}$. Thanks to these derivatives, the
Dirac equation

$\left(i\gamma^{\hat{\alpha}}\nabla_{\hat{\alpha}}-m\right)\psi(x)=0\,,$ (13)

is gauge invariant in the sense that this does not change its form when we
perform the transformation

$\psi(x)\to\psi^{\prime}(x)=\lambda(x)\psi(x)\,,$ (14)

simultaneously with the gauge transformation (9) and (10) with $\Lambda(x)$
depending on $\lambda(x)$ through the canonical homomorphism [12]. We remind
the reader that in the covariant parametrization with skew-symmetric real
valued parameters,
$\hat{\omega}_{\hat{\alpha}\hat{\beta}}=-\hat{\omega}_{\hat{\beta}\hat{\alpha}}$,
the $SL(2,{\mathbb{C}})$ transformations

$\lambda(\hat{\omega})=\exp\left(-\frac{i}{2}\,\hat{\omega}_{\hat{\alpha}\hat{\beta}}s^{\hat{\alpha}\hat{\beta}}\right)\in\rho_{D}[SL(2,{\mathbb{C}})]\,.$
(15)

correspond through the canonical homomorphism to the transformation matrices
$\Lambda(\hat{\omega})\in L^{\uparrow}_{+}$ having the matrix elements
$\Lambda(\hat{\omega})^{\hat{\alpha}\,\cdot}_{\cdot\,\hat{\beta}}=\delta^{\hat{\alpha}}_{\hat{\beta}}+\hat{\omega}^{\hat{\alpha}\,\cdot}_{\cdot\,\hat{\beta}}+...$.
Note that this property is encapsulated in the identity

$\lambda^{-1}(\hat{\omega})\gamma^{\hat{\alpha}}\lambda(\hat{\omega})=\Lambda(\hat{\omega})^{\hat{\alpha}\,\cdot}_{\cdot\,\hat{\beta}}\gamma^{\hat{\beta}}\,,$
(16)

that holds for any $\lambda(\hat{\omega})\in\rho_{D}[SL(2,{\mathbb{C}})]$.
Moreover, we assume that this identity can be used even for relating the
discrete transformations of $\rho_{D}$, constructed with the help of the
$\gamma$-matrices, to those of $G$. For shortening the notation we denote by
$\tilde{G}={\rm Pin}\,O(1,3)$ the group associated to the gauge group
$G=O(1,3)$ through canonical hamomorphism, understanding that $\rho_{D}$
includes a representation of $\tilde{G}$.

According to this general formalism the Dirac equation in the conformal frame
$\\{x_{c};e\\}$ of $M_{+}$ reads [7]

$\left[-i\omega_{H}t_{c}\left(\gamma^{0}\partial_{t_{c}}+\gamma^{i}\partial_{x^{i}_{c}}\right)+\frac{3i}{2}\omega_{H}\gamma^{0}-m\right]\psi(t,{\bf
x})=0\,,$ (17)

keeping the same form on $M_{-}$ where $t_{c}>0$. In contrast, the Dirac
equation in the physical frames $\\{x;e\\}$ of $M_{+}$, [8]

$\left[i\gamma^{0}\partial_{t}+i{\gamma^{i}}{\partial_{x^{i}}}-m+i\gamma^{0}\omega_{H}\left({x}^{i}{\partial}_{x^{i}}+\frac{3}{2}\right)\right]\psi(t,{\bf
x})=0\,,$ (18)

is different from that on $M_{-}$ which has a similar form but with
$\omega_{H}\to-\omega_{H}$ changing thus the sign of the last term due to the
de Sitter gravity. Both these equations can be solved analytically. For
example, in conformal frame the general solution may be written as a mode
integral [7],

$\displaystyle\psi({x}_{c}\,)=\int d^{3}p\sum_{\sigma}[U_{{\bf
p},\sigma}(x_{c}){\alpha}({\bf p},\sigma)+V_{{\bf
p},\sigma}(x_{c}){\beta}^{*}({\bf p},\sigma)]\,,$ (19)

in terms of wave functions in momentum representation of particle, $\alpha$,
and antiparticle, $\beta$. The mode spinors $U_{{\bf p},\sigma}$ and $V_{{\bf
p},\sigma}$, of positive and respectively negative frequencies, are plane
waves solutions of the Dirac equation depending on the conserved momentum
${\bf p}$ and an arbitrary polarization $\sigma$. These spinors form an
orthonormal basis being related through the usual charge conjugation,

$V_{{\bf p},\sigma}({x}_{c})=i\gamma^{2}\left[{U}_{{\bf
p},\sigma}({x}_{c})\right]^{*}\,,$ (20)

and satisfying orthogonality relations and a completeness condition with
respect to a suitable relativistic scalar product [7]. Similar properties hold
in physical frames such that we may consider the charge conjugation

${\cal C}\psi=i\gamma^{2}\psi^{*}\,,$ (21)

on the whole manifold $M$ where this has the same properties as in the flat
case being independent on geometry.

##  4 Continuous and discrete isometries

The de Sitter manifold $M=M_{+}\cup M_{-}$ has the isometries generated by the
group $O(1,4)$ that leaves the metric $\eta^{5}$ invariant. In general, these
isometries, $x\to x^{\prime}=\phi_{\mathfrak{g}}(x)$, are defined for any
${\mathfrak{g}}\in O(1,4)$ such that
$z[\phi_{\mathfrak{g}}(x)]={\mathfrak{g}}z(x)$ forming the isometry group
$I(M)\sim O(1,4)$ with respect to the composition rule
$\phi_{\mathfrak{g}}\circ\phi_{{\mathfrak{g}}^{\prime}}=\phi_{\mathfrak{gg}^{\prime}}$,
$\forall\mathfrak{g,g}^{\prime}\,\in I(M)$. The identity element of $I(M)$ is
the identity function ${\rm id}=\phi_{\mathfrak{e}}$, coresponding to the
identity element ${\mathfrak{e}}$ of the group $O(1,4)$. Therefore, the
calculation rules $\phi\circ{\rm id}={\rm id}\circ\phi=\phi$ and
$\phi\circ\phi^{-1}=\phi^{-1}\circ\phi={\rm id}$ hold for all $\phi\in I(M)$.

The problem here is that the isometries may change the tetrad gauge modifying
thus the form of the Dirac equation. For assuring the invariance under
isometries of the Dirac or other field equations one of us introduced the
combined transformations $(\lambda_{\mathfrak{g}},\phi_{\mathfrak{g}})$,
formed by the mappings
$\lambda_{\mathfrak{g}}:x\to\lambda_{\mathfrak{g}}(x)\in\rho_{D}$ and
$\phi_{\mathfrak{g}}:x\to\phi_{\mathfrak{g}}(x)$, which have to correct in
each point the positions of the local frames after an isometry. In other
words, these transformations must preserve simultaneously the metric and the
tetrad gauge as
$\omega(x^{\prime})=\Lambda[\lambda_{\mathfrak{g}}(x)]\omega(x)$ having the
form [6],

$\Lambda^{\hat{\alpha}\,\cdot}_{\cdot\,\hat{\beta}}[\lambda_{\mathfrak{g}}(x)]=\hat{e}_{\mu}^{\hat{\alpha}}[\phi_{\mathfrak{g}}(x)]\frac{\partial\phi^{\mu}_{\mathfrak{g}}(x)}{\partial
x^{\nu}}\,e^{\nu}_{\hat{\beta}}(x)\,,$ (22)

which define the mapping $\lambda_{\mathfrak{g}}$ assuming, in addition, that

$\lambda_{{\mathfrak{g}}={\mathfrak{e}}}(x)=1\in\rho_{D}\,.$ (23)

The resulted combined transformations
$(\lambda_{\mathfrak{g}},\phi_{\mathfrak{g}})$ preserve the gauge,
$e^{\prime}=e$ and $\omega^{\prime}=\omega$, transforming the Dirac field
according to the covariant representation
$\mathsf{T}:(\lambda_{\mathfrak{g}},\phi_{\mathfrak{g}})\to\mathsf{T}_{\mathfrak{g}}$
whose operators act as

$\left(\mathsf{T}_{\mathfrak{g}}\psi\right)[\phi_{\mathfrak{g}}(x)]=\lambda(x)\psi(x)~{}~{}\Rightarrow~{}~{}\mathsf{T}_{\mathfrak{g}}\psi=(\lambda_{\mathfrak{g}}\psi)\circ\phi_{\mathfrak{g}}^{-1}\,.$
(24)

Thus the frame transformation
$(\lambda_{\mathfrak{g}},\phi_{\mathfrak{g}}):(x;e)\to\left(\phi_{\mathfrak{g}}(x);\Lambda(\lambda_{\mathfrak{g}})\,e\right)$
and the covariant one (24) leave the Dirac equation invariant preserving the
physical meaning of the entire theory. In the de Sitter frames we consider
here, $\\{x_{c};e\\}$ and $\\{x;e\\}$, we denote the combined transformations
by $(\lambda_{\mathfrak{g}}^{c},\phi_{\mathfrak{g}}^{c})$ and respectively
$(\lambda_{\mathfrak{g}},\phi_{\mathfrak{g}})$ bearing in mind that these are
related as $\phi_{\mathfrak{g}}=\phi_{\mathfrak{g}}^{c}\circ\chi$ and
$\lambda_{\mathfrak{g}}=\lambda_{\mathfrak{g}}^{c}\circ\chi$ where the
function $\chi$ is defined in Eq. (5).

Mathematically speaking, we must specify that the mappings
$\lambda:M\to\tilde{G}$ are sections on the principal fiber bundle
$M\times\tilde{G}$ while the mapping associated through canonical
homomorphism, $\Lambda(\lambda):M\to G$ are sections on $M\times G$. These
mappings can be composed as
$\lambda_{{\mathfrak{g}}^{\prime}}\times\lambda_{\mathfrak{g}}$ such that
($\lambda_{{\mathfrak{g}}^{\prime}}\times\lambda_{\mathfrak{g}})(x)=\lambda_{{\mathfrak{g}}^{\prime}}(x)\lambda_{\mathfrak{g}}(x)$
and similarly for the mappings $\Lambda$. Moreover, observing that
$(\lambda_{{\mathfrak{g}}^{\prime}}\circ\phi_{\mathfrak{g}})\times\lambda_{\mathfrak{g}}=\lambda_{{\mathfrak{g}}^{\prime}{\mathfrak{g}}}$
we may conclude that the pairs $(\lambda_{\mathfrak{g}},\phi_{\mathfrak{g}})$
form a well-defined Lie group with respect to the new operation [6]

$(\lambda_{{\mathfrak{g}}^{\prime}},\phi_{{\mathfrak{g}}^{\prime}})*(\lambda_{\mathfrak{g}},\phi_{\mathfrak{g}})=((\lambda_{{\mathfrak{g}}^{\prime}}\circ\phi_{\mathfrak{g}})\times\lambda_{\mathfrak{g}},\phi_{{\mathfrak{g}}^{\prime}}\circ\phi_{\mathfrak{g}})=(\lambda_{{\mathfrak{g}}^{\prime}{\mathfrak{g}}},\phi_{{\mathfrak{g}}^{\prime}{\mathfrak{g}}})\,.$
(25)

This group can be seen as a representation of the universal covering group of
$I(M)$. Therefore, we may say that the covariant representations defined in
Ref. [6] transfer the Lorentz covariance from special to general relativity.

The discrete isometries are generated by the discrete transformations
${\mathfrak{d}}$ of the group $O(1,4)$ that satisfy
${\mathfrak{d}}{\mathfrak{d}}={\mathfrak{e}}$. Each discrete transformation
${\mathfrak{d}}$ defines the combined transformation
$(\lambda_{\mathfrak{d}},\phi_{\mathfrak{d}})$ whose mappings
$\lambda_{\mathfrak{d}}:x\to\lambda_{\mathfrak{d}}(x)$ give rise to local
discrete transformations $\lambda_{\mathfrak{d}}(x)\in\rho_{D}$ which must
satisfy $\lambda_{\mathfrak{d}}(x)\lambda_{\mathfrak{d}}(x)=\pm 1\in\rho_{D}$.
However, the corresponding transformations through the canonical homomorphism
have to obey
$\Lambda(\lambda_{\mathfrak{d}})\Lambda(\lambda_{\mathfrak{d}})=I$ where $I$
is the identity of the group $G$. In general, the discrete transformations
$\lambda_{\mathfrak{d}}(x)$ are local depending on $x$ but there are global
discrete transformations which reduce to simple matrices independent on $x$.
For example, the matrices $\gamma^{0}$, $\gamma^{0}\gamma^{5}$ and
$\gamma^{5}$ of $\rho_{D}$ are associated through the canonical homomorphism
to the global discrete transformations,

$\displaystyle\Lambda(\gamma^{0})$ $\displaystyle=$ $\displaystyle P={\rm
diag}(1,-1,-1,-1)\,,$ (26) $\displaystyle\Lambda(\gamma^{0}\gamma^{5})$
$\displaystyle=$ $\displaystyle T={\rm diag}(-1,1,1,1)\,,$ (27)
$\displaystyle\Lambda(\gamma^{5})$ $\displaystyle=$ $\displaystyle PT=-I,,$
(28)

giving rice to the subsets of the Lorentz group $G$ [12].

The de Sitter discrete isometries $\phi^{c}_{\mathfrak{d}}$ in the conformal
frames given by the functions (1-3) solve the equations
$z_{c}[\phi^{c}_{\mathfrak{d}}(x_{c})]={\mathfrak{d}}z_{c}(x_{c})$. The
simplest discrete transformations on $M^{5}$ are the mirror reflections,
denoted by $[A]$, which change the sign of a single coordinate, $z_{c}^{A}\to-
z_{c}^{A}$, giving rise to the discrete isometries $x_{c}\to
x_{c}^{\prime}=\phi^{c}_{[A]}(x_{c})$. A rapid inspection indicates that there
are two interesting local isometries produced by $\phi^{c}_{[0]}$ as,

$t_{c}^{\prime}(x_{c})=\phi_{[0]}^{c\,0}(x_{c})=\frac{t_{c}}{\omega^{2}(t_{c}^{2}-{\bf
x}_{c}^{2})}\,,\quad
x_{c}^{\prime\,i}(x_{c})=\phi_{[0]}^{c\,i}(x_{c})=\frac{x^{i}_{c}}{\omega^{2}(t_{c}^{2}-{\bf
x}_{c}^{2})}\,,$ (29)

and by $\phi^{c}_{[4]}$ which gives

$\phi_{[4]}^{c\,0}(x_{c})=-t_{c}^{\prime}(x_{c})\,,\qquad\phi_{[4]}^{c\,i}(x_{c})=-x_{c}^{\prime\,i}(x_{c})\,.$
(30)

The space isometries, $\phi^{c}_{[i]}$, are simple mirror transformations of
the space co-moving coordinates, $x_{c}^{i}\to-x_{c}^{i}$ such that the space
parity can be defined globally as
$\phi^{c}_{\mathfrak{p}}=\phi^{c}_{[1]}\circ\phi^{c}_{[2]}\circ\phi^{c}_{[3]}$.
Moreover, we can construct the isometries
$\phi^{c}_{{\mathfrak{p}}[0]}=\phi^{c}_{\mathfrak{p}}\circ\phi^{c}_{[0]}$,
$\phi^{c}_{{\mathfrak{p}}[4]}=\phi^{c}_{\mathfrak{p}}\circ\phi^{c}_{[4]}$ and
$\phi^{c}_{[0][4]}=\phi^{c}_{[0]}\circ\phi^{c}_{[4]}$, the last one changing
the signs of all the conformal coordinates, $x_{c}^{\mu}\to-x_{c}^{\mu}$. The
antipodal transformation, ${\mathfrak{a}}:z_{c}\to-z_{c}$, gives rise to the
isometry
$\phi^{c}_{\mathfrak{a}}=\phi^{c}_{\mathfrak{p}}\circ\phi^{c}_{[0][4]}$ that
reverses the conformal time on $M$, $t_{c}\to-t_{c}$.

Hereby we see that the set of discrete isometries
${\mathfrak{D}}={\mathfrak{D}}_{+}\cup{\mathfrak{D}}_{-}$ can be split in two
subsets,
${\mathfrak{D}}_{+}=\\{\phi_{\mathfrak{d}}|\phi_{\mathfrak{d}}:M_{\pm}\to
M_{\pm}\\}$ of proper discrete isometries which preserve the portions of $M$
and ${\mathfrak{D}}_{-}=\\{\phi_{\mathfrak{d}}|\phi_{\mathfrak{d}}:M_{\pm}\to
M_{\mp}\\}$ of improper discrete isometries that change these portions among
themselves. As the physical measurements can be performed only inside the
light-cone where $|t_{c}|>|{\bf x}_{c}|$ we deduce that the isometry (29)
preserve the sign of conformal time while the isometry (30) changes it.
Therefore, we may conclude that

$\displaystyle\phi_{\mathfrak{e}}={\rm
id},\,\phi^{c}_{\mathfrak{p}},\,\phi^{c}_{[0]},\,\phi^{c}_{{\mathfrak{p}}[0]}$
$\displaystyle\in$ $\displaystyle{\mathfrak{D}}_{+}$ (31)
$\displaystyle\phi^{c}_{\mathfrak{a}},\,\phi^{c}_{[4]},\,\phi^{c}_{[0][4]},\,\phi^{c}_{{\mathfrak{p}}[4]}$
$\displaystyle\in$ $\displaystyle{\mathfrak{D}}_{-}$ (32)

and similarly for the isometries $\phi_{\mathfrak{d}}$ of the physical frames.
For understanding the physical meaning of these isometries we discuss first
the proper discrete isometries in physical frames.

##  5 Discrete symmetries in physical frames

Let us consider now the physical frames of the expanding portion $M_{+}$ known
as the de Sitter expanding universe. The physical coordinates on $M_{+}$ can
be introduced by the new functions $z=z_{c}\circ\chi$ that read

$\displaystyle z^{0}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\omega_{H}}\left[e^{\omega_{H}t}-e^{-\omega_{H}t}\left(1-\omega_{H}^{2}{{\bf
x}}^{2}\right)\right]\,,$ (33) $\displaystyle z^{i}(x)$ $\displaystyle=$
$\displaystyle x^{i}\,,$ (34) $\displaystyle z^{4}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\omega_{H}}\left[e^{\omega_{H}t}+e^{-\omega_{H}t}\left(1-\omega_{H}^{2}{{\bf
x}}^{2}\right)\right]\,.$ (35)

Similar functions that define the physical coordinates on $M_{-}$ can be
obtained changing $\omega_{H}\to-\omega_{H}$ in Eqs. (33-35).

We observe first that the trivial isometry id is needed for closing a discrete
group. Denoting $\mathsf{T}_{\mathfrak{e}}={\cal I}$ we relax the condition
(23) for recovering the kernel $\mathbb{Z}_{2}=\\{{\cal I},\,-{\cal I}\\}$ of
the canonical homomorphism we need when we study discrete transformations. The
simplest non-trivial discrete isometry is the parity ${\mathfrak{p}}$ which is
global on $M^{5}$ such that $\phi^{0}_{\mathfrak{p}}(x)=t$,
$\phi^{i}_{\mathfrak{p}}(x)=-x^{i}$ and $\lambda_{\mathfrak{p}}=\gamma^{0}$.
Denoting then $\mathsf{T}_{\mathfrak{p}}={\cal P}$ we obtain the parity
transformation of the Dirac field,

${\cal P}\psi(t,{\bf x})=\gamma^{0}\psi(t,-{\bf x})\,,$ (36)

acting just as in the flat case but transforming, in addition, the tetrads
fields with $\Lambda_{\mathfrak{p}}=P$.

$~{}{\mathfrak{d}}$ |  $T_{\mathfrak{d}}$ |  $\phi^{0}_{\mathfrak{d}}(x)~{}~{}$ |  $\phi^{i}_{\mathfrak{d}}(x)$ | $\lambda_{\mathfrak{d}}(x)$ | $\Lambda_{\mathfrak{d}}(x)$ |  Eq.
---|---|---|---|---|---|---
${\mathfrak{p}}$ | ${\cal P}$ | $t$ | $-{x}^{i}$ | $\gamma^{0}$ | $P$ | (36)
$[0]$ | ${\cal B}$ | $t^{\prime}(t,|{\bf x}|)$ | ${x}^{i}$ | $\gamma^{0}\gamma^{5}\beta({\bf x})$ | $TB({\bf x})$ | (45)
${\mathfrak{p}}[0]$ |  ${\cal PB}$ | $t^{\prime}(t,|{\bf x}|)$ | $-{x}^{i}$ | $\gamma^{5}\beta(-{\bf x})$ | $-B(-{\bf x})$ | (46)
Table 1: Transformations induced by the non-trivial proper discrete isometries
in physical frames of the de Sitter expanding universe, $M_{+}$. The function
$t^{\prime}(t,|{\bf x}|)$ is defined in Eq. (37) while the matrices $B({\bf
x})$ and $\beta({\bf x})$ are given by Eqs. (40) and respectively (43).

Furthermore, we consider the isometry $x^{\prime}=\phi_{[0]}(x)$ generated by
the mirror transformation $[0]$ which changes $z^{0}\to-z^{0}$. Remarkably,
this is a local time reversal,

$\displaystyle t^{\prime}(t,|{\bf x}|)$ $\displaystyle=$
$\displaystyle\phi_{[0]}^{0}(x)=-t+\frac{1}{\omega_{H}}\ln\left(1-\omega_{H}^{2}{\bf
x}^{2}\right)\,,$ (37) $\displaystyle x^{\prime\,i}$ $\displaystyle=$
$\displaystyle\phi_{[0]}^{i}(x)=x^{i}\,,$ (38)

giving the new local time $t^{\prime}(x)\leq-t$ but without affecting the
space coordinates. The problem now is to find the associated matrices
$\lambda_{[0]}$ and $\Lambda_{[0]}\equiv\Lambda(\lambda_{[0]})$ transforming
the Dirac and respectively the tetrad fields. We first apply the definition
(22) for deriving the local matrix $\Lambda_{[0]}(x)$ finding that this is
time-independent having the matrix elements

$\displaystyle\left<\Lambda_{[0]}({\bf x})\right>^{0\,\cdot}_{\cdot\,0}$
$\displaystyle=$ $\displaystyle-\frac{1+\omega^{2}_{H}{\bf
x}^{2}}{1-\omega^{2}_{H}{\bf x}^{2}}<0\,,$
$\displaystyle\left<\Lambda_{[0]}({\bf x})\right>^{0\,\cdot}_{\cdot\,i}$
$\displaystyle=$
$\displaystyle-\left<\Lambda_{[0]}(x)\right>^{i\,\cdot}_{\cdot\,0}=-\frac{2\omega_{H}x^{i}}{1-\omega^{2}_{H}{\bf
x}^{2}}\,,$ (39) $\displaystyle\left<\Lambda_{[0]}({\bf
x})\right>^{i\,\cdot}_{\cdot\,j}$ $\displaystyle=$
$\displaystyle\delta^{i}_{j}+\frac{2\omega^{2}_{H}x^{i}x^{j}}{1-\omega^{2}_{H}{\bf
x}^{2}}\,.$

Moreover, we find that $\det[\Lambda_{[0]}{(\bf x})]=-1$ which means that
$\Lambda_{[0]}\in TL^{\uparrow}_{+}$. Therefore, we can use the factorization
$\Lambda_{[0]}({\bf x})=TB({\bf x})$ where now $B({\bf x})\in
L^{\uparrow}_{+}$. It is not difficult to verify that $B({\bf x})$ is a
Lorentz boost that can be parametrized as

$B({\bf x})=\exp\left(-iK_{i}\,\frac{x^{i}}{|{\bf x}|}\,\alpha(|{\bf
x}|)\right)\,,\quad\alpha(|{\bf x}|)={\rm
tanh}^{-1}\left(\frac{2\omega_{H}|{\bf x}|}{1+\omega^{2}_{H}{\bf
x}^{2}}\right)\,,$ (40)

using the boost generators $K_{i}$ of the $L^{\uparrow}_{+}$ group (having the
non-vanishing components
$\left<K_{i}\right>^{j}_{0}=\left<K_{i}\right>_{j}^{0}=i\delta_{ij}$ [12]).
Note that these boost matrices satisfy $B({\bf x})^{-1}=B(-{\bf x})$ and
$B({\bf x}=0)=I$. In addition, we can verify that

$\Lambda_{[0]}({\bf x})[\Lambda_{[0]}({\bf
x})=I~{}~{}\Longleftrightarrow~{}~{}B({\bf x})TB({\bf x})=T\,,$ (41)

as the pseudo-orthogonal boosts $B({\bf x})$ are symmetric, $B({\bf
x})^{T}=B({\bf x})$.

The above parametrization helps us to find the local matrices
$\lambda_{[0]}({\bf x})$ which transform the Dirac field. According to Eq.
(27) this can be written as

$\lambda_{[0]}({\bf x})=\gamma^{0}\gamma^{5}\beta({\bf x})\,,\quad\,,$ (42)

where the matrix

$\beta({\bf x})=\exp\left(-is_{i0}\,\frac{x^{i}}{|{\bf x}|}\,\alpha(|{\bf
x}|)\right)=\frac{1+\omega_{H}\gamma^{0}\gamma^{i}x^{i}}{\sqrt{1-\omega^{2}_{H}{\bf
x}^{2}}}\in\rho_{D}\,,$ (43)

is calculated by using the boost generators $s_{i0}\in\rho_{D}$ defined in Eq.
(12). This has the obvious properties, $\beta({\bf x})^{-1}=\beta(-{\bf x})$
and $\gamma^{0}\beta({\bf x})=\beta(-{\bf x})\gamma^{0}$ which guarantee that
$\lambda_{[0]}({\bf x})\lambda_{[0]}({\bf x})=-1\in\rho_{D}$. Moreover, we
verify that the matrices $\lambda_{[0]}({\bf x})$ and $\Lambda_{[0]}({\bf x})$
are related through the canonical homomorphism satisfying the identities

$\lambda_{[0]}({\bf x})^{-1}\gamma^{\hat{\alpha}}\lambda_{[0]}({\bf
x})=\left<\Lambda_{[0]}({\bf
x})\right>^{\hat{\alpha}\,\cdot}_{\cdot\,\hat{\beta}}\gamma^{\hat{\beta}}\,,$
(44)

which assure the invariance of the Dirac equation (18) under this isometry.

Collecting then the above results and denoting $\mathsf{T}_{[0]}={\cal B}$ we
define the discrete transformation of the Dirac field

${\cal B}\psi(t,{\bf x})=\lambda_{[0]}({\bf x})\psi(t^{\prime}(x),{\bf
x})=\gamma^{0}\gamma^{5}\beta({\bf x})\psi(t^{\prime}(t,|{\bf x}|)),{\bf
x})\,,$ (45)

where $t^{\prime}(x)$ is given by Eq. (37). The above results help us to write
the transformation of the Dirac field associated to the isometry
$\phi^{c}_{{\mathfrak{p}}[0]}$ as

${\cal PB}\psi(t,{\bf x})=\gamma^{5}\beta(-{\bf x})\psi(t^{\prime}(t,|{\bf
x}|),-{\bf x})\,,$ (46)

while the tetrads have to be transformed by
$\Lambda_{{\mathfrak{p}}[0]}=PTB(-{\bf x})=-B(-{\bf x})$. Obviously, the
transformations ${\cal B}$ and ${\cal PB}$ take over the role of ${\cal CT}$
and respectively ${\cal PCT}$ ones of the Dirac theory in Minkowski space-
time.

${\cal I}$ |  ${\cal C}$ |  ${\cal P}$ |  ${\cal PC}$ |  ${\cal T}_{+}$ |  ${\cal P}{\cal T}_{+}$ |  ${\cal C}{\cal T}_{+}$ |  ${\cal PC}{\cal T}_{+}$
---|---|---|---|---|---|---|---
${\cal C}$ | ${\cal I}$ | $-{\cal PC}$ | $-{\cal P}$ | ${\cal C}{\cal T}_{+}$ | $-{\cal PC}{\cal T}_{+}$ | ${\cal T}_{+}$ | $-{\cal P}{\cal T}_{+}$
${\cal P}$ | ${\cal PC}$ | ${\cal I}$ | ${\cal C}$ | ${\cal P}{\cal T}_{+}$ | ${\cal T}_{+}$ | ${\cal PC}{\cal T}_{+}$ | ${\cal C}{\cal T}_{+}$
${\cal PC}$ | ${\cal P}$ | $-{\cal C}$ | $-{\cal I}$ | ${\cal PC}{\cal T}_{+}$ | $-{\cal C}{\cal T}_{+}$ | ${\cal P}{\cal T}_{+}$ | $-{\cal T}_{+}$
${\cal T}_{+}$ | ${\cal C}{\cal T}_{+}$ | ${\cal P}{\cal T}_{+}$ | ${\cal PC}{\cal T}_{+}$ | $-{\cal I}~{}~{}$ | $-{\cal P}$ | $-{\cal C}$ | $-{\cal PC}$
${\cal P}{\cal T}_{+}$ | ${\cal PC}{\cal T}_{+}$ | ${\cal T}_{+}$ | ${\cal C}{\cal T}_{+}$ | $-{\cal P}$ | $-{\cal I}$ | $-{\cal PC}$ | $-{\cal C}$
${\cal C}{\cal T}_{+}$ | ${\cal T}_{+}$ | $-{\cal PC}{\cal T}_{+}$ | $-{\cal P}{\cal T}_{+}$ | $-{\cal C}$ | ${\cal PC}$ | $-{\cal I}$ | $-{\cal P}$
${\cal PC}{\cal T}_{+}$ | ${\cal P}{\cal T}_{+}$ | $-{\cal C}{\cal T}_{+}$ | $-{\cal T}_{+}$ | $-{\cal PC}$ | ${\cal C}$ | $-{\cal P}$ | $-{\cal I}$
Table 2: Multiplication table of the subset ${\cal G}/{\mathbb{Z}}_{2}$ of the
group ${\cal G}$ of the discrete transformations of the covariant Dirac field
in the de Sitter expanding universe $M_{+}$.

We see thus that the Dirac theory on $M_{+}$ has three non-trivial proper
discrete isometries, ${\cal P},\,{\cal B}$ and ${\cal PB}=-{\cal BP}$, that
can be composed at any time with the charge conjugation ${\cal C}$ which is
independent on geometry. Under such circumstances we may define of the Wigner
time reversal on $M_{+}$ as ${\cal T}_{+}={\cal C}{{\cal B}}=\cal{BC}$,
obtaining again a local transformation which in our phase convention reads,

$\displaystyle{\cal T}_{+}\psi(t,{\bf x})$ $\displaystyle=$ $\displaystyle
i\gamma^{2}\left({\cal B}\psi(t,{\bf x})\right)^{*}=-\beta(-{\bf
x})\gamma^{1}\gamma^{3}\psi^{*}(t^{\prime}(t,|{\bf x}|),{\bf x})\,,$ (47)

as the matrix $\beta({\bf x})$ satisfies $\gamma^{1}\gamma^{3}\beta^{*}({\bf
x})=\beta(-{\bf x})\gamma^{1}\gamma^{3}$. Another transformation can be
defined as

${\cal P}{\cal T}_{+}\psi(t,{\bf x})=-\beta({\bf
x})\gamma^{0}\gamma^{1}\gamma^{3}\psi^{*}(t^{\prime}(t,|{\bf x}|),{-\bf
x})\,,$ (48)

observing that ${\cal P}{\cal T}_{+}={\cal T}_{+}{\cal P}$. Note that under
these last two transformations the tetrad fields have to transform according
to the transformations $\Lambda_{[0]}$ and respectively
$\Lambda_{{\mathfrak{p}}[0]}$ as ${\cal C}$ does not produce geometric
effects.

Finally, we substitute ${\cal B}={\cal C}{\cal T}_{+}$ obtaining the standard
components

${\cal G}_{+}=\\{\pm{\cal I},\,\pm{\cal C},\,\pm{\cal P},\,\pm{\cal
PC},\,\pm{\cal T}_{+},\,\pm{\cal P}{\cal T}_{+},\,\pm{\cal C}{\cal
T}_{+},\,\pm{\cal PC}{\cal T_{+}}\\}$ (49)

of the discrete group of Dirac’s theory on the de Sitter expanding universe,
$M_{+}$. This group is of the order 16 having the multiplication table given
in Tab. 2 which is similar to that of the Dirac theory in Minkowski space-
time. In this group we meet the usual parity ${\cal P}$ and charge conjugation
${\cal C}$ which are global while all the transformations involving the Wigner
time reversal ${\cal T}_{+}$ are local, depending on the matrix $\beta({\bf
x})$. Similar results can be obtained for the collapsing portion where we meet
a similar discrete group ${\cal G}_{-}$ but with the Wigner time reversal
${\cal T}_{-}$ that can be obtained from ${\cal T}_{+}$ substituting
$\omega_{H}\to-\omega_{H}$. An important opportunity of this approach is the
correct flat limit when $\omega_{H}\to 0$. Indeed, in Eq. (37) we have

$\lim_{\omega_{H}\to 0}\frac{1}{\omega_{H}}\ln(1-\omega_{H}^{2}{\bf
x}^{2})=0~{}~{}\Rightarrow~{}~{}\lim_{\omega_{H}\to 0}{\cal
T}_{+}=\lim_{\omega_{H}\to 0}{\cal T}_{-}={\cal T}\,,$ (50)

as in this limit $B({\bf x})\to I$ and $\beta({\bf x})\to 1\in\rho_{D}$.
Therefore ${\cal G}_{+}\sim{\cal G}_{-}$ tend to the usual discrete group of
the Dirac theory in special relativity.

Trying then to continue this study with the improper isometries in physical
frames we meet difficulties because we must consider the explicite
transformation $\omega_{H}\to-\omega_{H}$ if we use the same coordinates on
$M_{+}$ and $M_{-}$. However, once we have investigated the physical meaning
of the proper isometries in physical frames we may look for a general picture
coming back to the conformal frames where we have technical opportunities.

##  6 Discrete isometries in conformal frames

The advantage of the conformal frames is that here the Dirac equation as well
the metric do not change their forms when we change the portion by changing
the sign of $t_{c}$. Starting with the proper isometries (31) we observe first
that the parity remains the same,

${\cal P}\psi(t_{c},{\bf x}_{c})=\gamma^{0}\psi(t_{c},-{\bf x}_{c})\,,$ (51)

as the three-vectors ${\bf x}_{c}$ and ${\bf x}$ are parallel to ${\bf z}$.
The isometry $\phi^{c}_{[0]}$ defined by Eq. (29) mixes the time and the
Cartesian coordinates such that now we may write

${\cal B}\psi(t_{c},{\bf
x}_{c})=\lambda^{c}_{[0]}({x}_{c})\psi(t_{c}^{\prime}({x}_{c}),{\bf
x}_{c}^{\prime}(x_{c}))=\gamma^{0}\gamma^{5}\beta^{c}(x_{c})\psi(t^{\prime}_{c}(x_{c}),{\bf
x}^{\prime}_{c}(x_{c}))\,,$ (52)

where the functions $t^{\prime}_{c}(x_{c})$ and ${\bf x}^{\prime}_{c}(x_{c})$
are defined in Eq. (29). The matrix $\beta^{c}=\beta\circ\chi^{-1}$ and the
matrix transforming tetrads, $\Lambda^{c}_{[0]}=TB^{c}=TB\circ\chi^{-1}$, are
obtained after substituting the physical coordinates according to Eq. (5). A
few manipulation carries out the matrix elements of $B^{c}(x_{c})$ that read

$\displaystyle\left<B^{c}({x}_{c})\right>^{0\,\cdot}_{\cdot\,0}$
$\displaystyle=$ $\displaystyle\frac{t_{c}^{2}+{\bf x}_{c}^{2}}{t_{c}^{2}-{\bf
x}_{c}^{2}}\,,$
$\displaystyle\left<B^{c}({x}_{c})\right>^{0\,\cdot}_{\cdot\,i}$
$\displaystyle=$
$\displaystyle\left<B^{c}({x}_{c})\right>^{i\,\cdot}_{\cdot\,0}=-\frac{2x_{c}^{i}}{t_{c}^{2}-{\bf
x}_{c}^{2}}\,,$ (53)
$\displaystyle\left<B^{c}({x}_{c})\right>^{i\,\cdot}_{\cdot\,j}$
$\displaystyle=$
$\displaystyle\delta^{i}_{j}+\frac{2x_{c}^{i}x_{c}^{j}}{t_{c}^{2}-{\bf
x}_{c}^{2}}\,,$

and the new form

$\beta^{c}(t_{c},{\bf
x}_{c})=\frac{t_{c}-\gamma^{0}\gamma^{i}x_{c}^{i}}{\sqrt{t_{c}^{2}-{\bf
x}_{c}^{2}}}\in\rho_{D}\,,$ (54)

of the associated matrix through the canonical homomorphism. Finally we derive
the transformations produced by the isometry $\phi^{c}_{{\mathfrak{p}}[0]}$,

${\cal PB}\psi(t_{c},{\bf x}_{c})=\gamma^{5}\beta^{c}(t_{c},-{\bf
x}_{c})\psi(t^{\prime}_{c}(x_{c}),-{\bf x}^{\prime}_{c}(x_{c}))\,,$ (55)

and $\Lambda^{c}_{{\mathfrak{p}}[0]}(x_{c})=-B^{c}(t_{c},-{\bf x}_{c})$. Thus
all the results concerning the proper isometries are in accordance to those
obtained in physical frames.

For studying the improper isometries (32) we start with the antipodal isometry
$\phi^{c}_{\mathfrak{a}}$ for which we have
$\Lambda_{\mathfrak{a}}^{c}=P\to\lambda^{c}_{\mathfrak{a}}=\gamma^{0}$ such
that by denoting $\mathsf{T}_{\mathfrak{a}}={\cal A}$ we obtain the antipodal
transformation of the Dirac field,

${\cal A}\psi(t_{c},{\bf x}_{c})=\gamma^{0}\psi(-t_{c},{\bf x}_{c})\,,$ (56)

with the help of which we may deduce the actions of the other improper
isometries. Indeed, observing that
$\phi^{c}_{[0][4]}=\phi^{c}_{\mathfrak{p}}\circ\phi^{c}_{\mathfrak{a}}$ we
obtain its action

${\cal PA}\psi(t_{c},{\bf x}_{c})=\psi(-t_{c},-{\bf x}_{c})\,,$ (57)

and the associated matrix $\Lambda_{[0][4]}=I$. Similarly we find the action
of the isometry (30) which can be decomposed as
$\phi^{c}_{[4]}=\phi_{\mathfrak{p}}\circ\phi^{c}_{\mathfrak{a}}\circ\phi^{c}_{[0]}$
such that we may write

${\cal PAB}\psi(t_{c},{\bf x}_{c})=\gamma^{0}\gamma^{5}\beta^{c}(-t_{c},-{\bf
x}_{c})\psi(-t^{\prime}_{c}(x_{c}),-{\bf x}^{\prime}_{c}(x_{c}))\,,$ (58)

deducing that the tetrad fields are transformed by
$\Lambda^{c}_{[4]}=\Lambda^{c}_{[0]}=TB^{c}$. Hereby we find that the isometry
$\phi^{c}_{{\mathfrak{p}}[4]}=\phi^{c}_{\mathfrak{a}}\circ\phi^{c}_{[0]}$
gives the transformations

${\cal AB}\psi(t_{c},{\bf x}_{c})=\gamma^{5}\beta^{c}(-t_{c},{\bf
x}_{c})\psi(-t^{\prime}_{c}(x_{c}),{\bf x}^{\prime}_{c}(x_{c}))\,,$ (59)

and $\Lambda^{c}_{{\mathfrak{p}}[4]}(x_{c})=-B^{c}(-t_{c},{\bf x}_{c})$.

${\mathfrak{d}}$ | $T_{\mathfrak{d}}$ | $\phi^{c\,0}_{\mathfrak{d}}(x_{c})$ | $\phi^{c\,i}_{\mathfrak{d}}(x_{c})$ | $\lambda^{c}_{\mathfrak{d}}(x_{c})$ | $\Lambda^{c}_{\mathfrak{d}}(x_{c})$ | Eq.
---|---|---|---|---|---|---
${\mathfrak{p}}$ | ${\cal P}$ | $t_{c}$ | $-{x}_{c}^{i}$ | $\gamma^{0}$ | $P$ | (51)
$[0]$ | ${\cal B}$ | $t^{\prime}_{c}(x_{c})$ | ${x}^{\prime\,i}_{c}(x_{c})$ | $\gamma^{0}\gamma^{5}\beta^{c}(t_{c},{\bf x}_{c})$ | $TB^{c}(t_{c},{\bf x}_{c})$ | (52)
${\mathfrak{p}}[0]$ | ${\cal PB}$ | $t^{\prime}_{c}(x_{c})$ | $-{x}^{\prime\,i}_{c}(x_{c})$ | $\gamma^{5}\beta^{c}(t_{c},-{\bf x}_{c})$ | $-B^{c}(t_{c},-{\bf x}_{c})$ | (55)
${\mathfrak{a}}$ | ${\cal A}$ | $-t_{c}$ | $x_{c}^{i}$ | $\gamma^{0}$ | $P$ | (56)
$[0][4]$ | ${\cal PA}$ | $-t_{c}$ | $-x_{c}^{i}$ | $1\in\rho_{D}$ | $I$ | (57)
$[4]$ | ${\cal PAB}$ | $-t^{\prime}_{c}(x_{c})$ | $-{x}^{\prime\,i}_{c}(x_{c})$ | $\gamma^{0}\gamma^{5}\beta^{c}(-t_{c},-{\bf x}_{c})$ | $TB^{c}(t_{c},{\bf x}_{c})$ | (58)
${\mathfrak{p}}[4]$ | ${\cal AB}$ | $-t^{\prime}_{c}(x_{c})$ | ${x}^{\prime\,i}_{c}(x_{c})$ | $\gamma^{5}\beta^{c}(-t_{c},{\bf x}_{c})$ | $-B^{c}(-t_{c},{\bf x}_{c})$ | (59)
Table 3: Transformations induced by the discrete de Sitter isometries in
conformal frames. The functions $t^{\prime}_{c}(x_{c})$ and
${x}^{\prime\,i}_{c}(x_{c})$ are defined in Eq. (29) while the matrices
$B^{c}(x_{c})$ and $\beta^{c}(x_{c})$ are given by Eqs. (53) and respectively
(54).

In Tab. 3 we summarize the results obtained in this section listing the
transformations of the Dirac field produced by the discrete isometries in
conformal frames. The first three are due to the proper ones among them only
the parity remains global as the last two ale local, depending on the matrix
(54). The improper isometries are two global and two local depending on the
same boost. All these transformations can be obtained by using the basis
formed by ${\cal P}$, ${\cal B}$ and ${\cal A}$. Moreover, these
transformations can be composed at any time with the charge conjugation ${\cal
C}$ giving rise to a large collection of discrete transformations. For
example, the transformations (47) and can (48) be rewritten in conformal
frames as

$\displaystyle{\cal T}_{c}\psi(t_{c},{\bf x}_{c})$ $\displaystyle=$
$\displaystyle-\beta^{c}(t_{c},-{\bf
x}_{c})\gamma^{1}\gamma^{3}\psi^{*}(t_{c}^{\prime}({\bf x}),{\bf
x}^{\prime}_{c}({x_{c}}))\,,$ (60) $\displaystyle{\cal P}{\cal
T}_{c}\psi(t_{c},{\bf x}_{c})$ $\displaystyle=$
$\displaystyle-\beta^{c}(t_{c},{\bf
x}_{c})\gamma^{0}\gamma^{1}\gamma^{3}\psi^{*}(t_{c}^{\prime}({\bf x}),-{\bf
x}^{\prime}_{c}({x_{c}}))\,,$ (61)

having the same form on $M_{+}$ and $M_{-}$. With their help we can construct
the discrete group ${\cal G}_{c}$ in conformal frames which has a similar
multiplication table as in Tab. 2 but with ${\cal T}_{c}$ instead of ${\cal
T}_{+}$. We observe that Eqs. (60) and (61) are far from an intuitive
interpretation confirming again that the conformal frames are useful for
calculating the action of the discrete isometries whose physical meaning can
be better understood in physical frames. On the other hand, we must stress
that the transformations listed in Tab. 3 cannot be gathered in the same
discrete group as the improper ones do not have a physical meaning if we
accept that one cannot perform measurements simultaneously in the expanding
and collapsing universes.

##  7 Concluding remarks

We presented for the first time the complete theory of the discrete
transformations of the covariant Dirac field on the de Sitter manifold whose
expanding and collapsing portions are models of universes. We focused on the
group of discrete symmetries in the physical frames of the expanding universe
which is of actual interest in cosmology pointing out that similar results on
the collapsing portion can be obtained simply by substituting
$\omega_{H}\to-\omega_{H}$. Moreover, we derived all the transformations
generated by the discrete isometries, including the improper ones, using
conformal frames where we have obvious technical advantages but obscure
physical meaning as the improper isometries do not have a physical meaning
being of a mere mathematical interest. Thus we may say that this paper
complete the previous studies concerning the de Sitter continuous isometries
[7, 10, 11] generating the principal invariants of the Dirac theory on the de
Sitter expanding universe [6, 14, 15].

However, our results differ from those obtained in the invariant approach of
the de Sitter ambient where only linear representations of the group ${\rm
Pin}O(1,4)$ are considered obtaining a discrete group formed exclusively by
global transformations [3, 4]. In contrast, in our approach only the parity,
antipodal isometry, charge conjugation and their compositions remain global
while all the symmetries containing the Wigner time reversal, ${\cal T}_{+}$,
lead to local transformations depending on the matrix (43). This indicates
that on the de Sitter manifold a global time reversal cannot be defined at
least in our covariant approach.

Finally we note that the groups of the discrete symmetries on the de Sitter
portions are of the same order as in the case of the Minkowski geometry
because both these manifolds are of maximal symmetry. Therefore, we expect to
obtain similar results on the Anti-de Sitter space-time which is the third
$(1+3)$-dimensional manifold of maximal symmetry [30].

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