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 different ways, namely either as an invariant field defined on \(M^5\) or as a covariant one defined on M. The first approach, often referred to as the de Sitter ambient, has been developed by many authors considering a spinor field transforming under isometries according to linear representations of the universal covering group \({\textrm{Spin}}SO(1,4)\) 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 introduced by Nachtmann [5] and then developed 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 \(SL(2,{\mathbb {C}})={\textrm{Spin}}L_+^{\uparrow }\) group, which is the universal covering group of the proper orthochronous Lorentz group \(L_+^{\uparrow }\), that is, the 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 rise 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 opportunities for studying physical effects on M either focusing on the fermion propagators [16] in cosmological scenarios [17, 18] or deriving scattering amplitudes [19,20,21,22,23]. Under such circumstances, the study of the discrete de Sitter isometries generating the discrete symmetries of the Dirac theory has so far been neglected. Therefore, 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 phase convention, are the parity, \({{\textsf{P}}}\psi (t,{\textbf{x}})=\gamma ^0\psi (t,-{\textbf{x}})\), charge conjugation, \({\textsf{C}}\psi (t,{\textbf{x}})=i\gamma ^2\psi ^*(t,{\textbf{x}})\), and Wigner’s time reversal defined as \({\textsf{T}}\psi (t,{\textbf{x}})=-\gamma ^1\gamma ^3\psi ^*(-t,{\textbf{x}})\) [24] for closing the PCT theorem, \(\textsf{ PCT}\psi =\gamma ^5 \psi (-t,-{\textbf{x}})\) [25]. As \({\textsf{T}}\) is not related to an isometry, one often uses the transformation \({{\textsf{C}}}{{\textsf{T}}}\psi (t,{\textbf{x}})=\gamma ^0\gamma ^5\psi (-t,{\textbf{x}})\) as an alternative time reversal [26]. The transformations \({{\textsf{P}}}\), \({{\textsf{T}}}\) and \(\textsf{PT}\) are produced by the global discrete O(1, 3) isometries P, T and PT transforming the Cartesian coordinates \((t,{\textbf{x}})\) into \((t,-{\textbf{x}})\), \((-t,{\textbf{x}})\) and respectively \((-t,-{\textbf{x}})\).

The discrete transformations are important in constructing gauge models because of the parity violation in electroweak interactions. For this reason, the gauge models start with a spinor sector formed by left-handed (L) and right-handed (R) components, \(\psi _{L/R}=P_{L/R}\psi _0\), defined by applying the chiral projection operators \(P_{L/R}=\frac{1}{2}(1\mp \gamma ^5)\) on a massless Dirac field \(\psi _0\). In general, the parity violation is introduced assuming the sets of left-handed and right-handed components transforming according to different representations of the gauge group that one has chosen. Remarkably, the charge conjugation changes the chirality without affecting the coordinates such that a left-handed component can be seen as a charge-conjugated right-handed one and vice versa. This ensures the flexibility one needs for constructing new models with higher gauge symmetries as extensions of the Standard Model (see for instance Ref. [27]).

To obtain similar results for the Dirac field on the de Sitter manifold, M, we must focus on the discrete transformations 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 thus have 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 mere mathematical interest. We must thus study three proper discrete isometries and four improper ones, extending the method of induced representations [6] to the local discrete transformations of the group \({\textrm{Pin}}O(1,3)\) we expect to encounter here.

The frames we need here are formed by local charts and orthogonal non-holonomic frames defined by tetrads. The discrete isometries of the de Sitter manifold were studied long ago in a local chart on M with special coordinates [28], but unfortunately, we cannot use these results as we must work here with other types of coordinates able to cover the physical portions \(M_+\) and \(M_-\) separately. We first consider the conformal coordinates which are helpful in concrete calculations, and then we introduce the physical ones of Painlevé–Gullstrand type [29, 30] which allow us to understand the physical meaning of the transformations studied here. We must thus work with conformal or physical frames in which we hope to obtain a complete image of the discrete transformations of the covariant Dirac field produced by the de Sitter discrete isometries combined with the charge conjugation.

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 the covariant Dirac field presented is Sect. 3. Our study of the discrete de Sitter symmetries starts in Sect. 4, where we present the method for constructing the induced representations that can be extended to the discrete de Sitter isometries. We first derive these discrete transformations in conformal frames separating the proper ones. To understand their physical meaning, in the next section we study the discrete symmetries generated by the proper isometries in physical frames, adding the charge conjugation for completing the group of discrete symmetries. We first find that there exists a new local isometry reversing the time which helps us to define the analogy of the Wigner time reversal. In this way we obtain the discrete group of order 16 of the de Sitter expanding universe for which we derive the multiplication table. Furthermore, to obtain a global image of the discrete de Sitter isometries, in Sect. 6 we return to the conformal frames, where we write down all isometries including the improper ones. Finally, in Sect. 7, we discuss some physical consequences.

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 \(A,\,B,\ldots = 0,1,2,3,4\)) and metric \(\eta ^5={{\textrm{diag}}}(1,-1,-1,-1,-1)\) [31], denoting by \(\omega _{H}\) the de Sitter–Hubble constant (frequency). A local chart of coordinates \(x^{\mu }\) \((\mu ,\nu ,\ldots =0,1,2,3)\) 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, \(\{x_c\}=\{t_c, {\textbf{x}}_c\}\), with the conformal time \(t_c\) and co-moving Cartesian space coordinates, \({\textbf{x}}_c=(x_c^1,x_c^2,x_c^3)\), defined by the functions

$$\begin{aligned} z_c^0(x_c)= & {} -\frac{1}{2 \omega _{H}^2 t_c} \left[ 1- \omega _{H}^2\left( t_c^2-{{\textbf{x}}_c\,}^2\right) \right] , \end{aligned}$$
(1)
$$\begin{aligned} z_c^i(x_c)= & {} -\frac{x^i_c}{ \omega _{H} t_c}, \quad i,j,\ldots =1,2,3, \end{aligned}$$
(2)
$$\begin{aligned} z_c^4(x_c)= & {} -\frac{1}{2 \omega _{H}^2 t_c}\left[ 1+ \omega _{H}^2\left( t_c^2-{{\textbf{x}}_c\,}^2\right) \right] , \end{aligned}$$
(3)

giving rise to the conformal-flat line element

$$\begin{aligned} {\textrm{d}}s^2=\eta ^5_{AB}{\textrm{d}}z^A {\textrm{d}}z^B=\frac{1}{ \omega _{H}^2 t_c^2}\,({\textrm{d}}t_c^2-{\textrm{d}}{\textbf{x}}_c\cdot {\textrm{d}}{\textbf{x}}_c). \end{aligned}$$
(4)

These charts, defined up to an isometry, cover the expanding portion, \(M_+\), for \(t_c \in (-\infty ,0)\) and \({\textbf{x}}_c\in {\mathbb {R}}^3\), while the collapsing one, \(M_-\), is covered by similar charts but with \(t_c >0\) [31]. Note that the conformal coordinates for Friedmann–Lemaître–Robertson–Walker (FLRW) space-times were introduced for the first time by Lemaître [32].

However, the coordinates with the obvious physical meaning are those of Painlevé–Gullstrand type [29, 30] defining the physical local chart \(\{x\}=\{ t,{\textbf{x}}\}\). These coordinates are the cosmic time t and the physical space coordinates \({\textbf{x}}=(x^1,x^2,x^3)\) that can be introduced on the expanding portion \(M_+\) by substituting \(x_c=\chi (x)\), where

$$\begin{aligned} t_c=\chi ^0(x)=-\frac{1}{\omega _{H}}{\textrm{e}}^{-\omega _{H} t}, \quad {x}^i_c=\chi ^i(x)=x^i{\textrm{e}}^{-\omega _{H} t}, \end{aligned}$$
(5)

thus obtaining the new line element

$$\begin{aligned} {\textrm{d}}s^2=\left( 1-\omega _{H}^2 {{\textbf{x}}}^2\right) {\textrm{d}}t^2+2\omega _{H} {\textbf{x}}\cdot {\textrm{d}}{\textbf{x}}\,{\textrm{d}}t -{\textrm{d}}{\textbf{x}}\cdot {\textrm{d}}{\textbf{x}}. \end{aligned}$$
(6)

Similar coordinates can be defined on the collapsing portion \(M_-\) following the same procedure but with \(\omega _{H}\rightarrow -\omega _{H}\).

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 (labeled by the local indices \({\hat{\mu }},{\hat{\nu }},\ldots =0,1,2,3\)). Here we restrict ourselves to the Cartesian diagonal tetrad gauge defined by the vector fields

$$\begin{aligned} e_{0}= & {} -\omega _{H}\,t_c\partial _{t_c} =\partial _t+\omega _{H}\,x^i\partial _{x^i},\nonumber \\ e_{i}= & {} -\omega _{H}\,t_c\partial _{x^i_c}=\partial _{x^i}, \end{aligned}$$
(7)

and the corresponding dual 1-forms

$$\begin{aligned} \omega ^{0}= & {} -\frac{1}{\omega _{H}\,t_c}{\textrm{d}}t_c={\textrm{d}}t,\nonumber \\ \omega ^{i}= & {} -\frac{1}{\omega _{H}\,t_c}{\textrm{d}}x_c^i={\textrm{d}}{x}^i-\omega _{H}\,{x}^i {\textrm{d}}t, \end{aligned}$$
(8)

which preserve the global SO(3) symmetry, allowing us to systematically use the SO(3) vectors. We thus obtain 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 represents the proper frames of an observer staying at rest in origin measuring the events inside the event horizon, for \(|{\textbf{x}}|<\omega _H^{-1}\). Therefore, the physical interpretation must be done in physical frames, while the conformal ones are valuable tools in our investigations.

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

$$\begin{aligned} e_{{\hat{\alpha }}}(x)\rightarrow & {} e_{{\hat{\alpha }}}'(x)=\varLambda _{{\hat{\alpha }}\,\cdot }^{\cdot \,{\hat{\beta }}}(x)\, e_{{\hat{\beta }}}(x), \end{aligned}$$
(9)
$$\begin{aligned} {\omega }^{{\hat{\alpha }}}(x)\rightarrow & {} { \omega }^{\prime \,{\hat{\alpha }}}(x)=\varLambda ^{{\hat{\alpha }}\,\cdot }_{\cdot \,{\hat{\beta }}}(x) \,\omega _{{\hat{\beta }}}(x). \end{aligned}$$
(10)

In general relativity, any theory of fields with half-integer spin must be gauge-invariant in the sense that the above gauge transformation 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 \(\psi : M_+\rightarrow {{{\mathcal {V}}}}_D\), of mass m, minimally coupled to the background gravity, reads

$$\begin{aligned} {{{\mathcal {S}}}}[e,\psi ]=\int \, {\textrm{d}}^{4}x\sqrt{g}\Bigg \{ \frac{i}{2}[\overline{\psi }\gamma ^{{\hat{\alpha }}}\nabla _{{\hat{\alpha }}}\psi - (\overline{\nabla _{{\hat{\alpha }}}\psi })\gamma ^{{\hat{\alpha }}}\psi ] -m\overline{\psi }\psi \Bigg \} \end{aligned}$$
(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 \({{{\mathcal {V}}}}_D\) 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.

In the gauge-invariant theory, one uses covariant derivatives of the form

$$\begin{aligned} \nabla _{{\hat{\alpha }}}={\textrm{e}}^{\mu }_{{\hat{\alpha }}}\nabla _{\mu }= e_{{\hat{\alpha }}}+\frac{i}{2}{\hat{\varGamma }}^{{\hat{\gamma }}}_{{\hat{\alpha }} {\hat{\beta }}}s^{{\hat{\beta }}, \cdot }_{\cdot \, {\hat{\gamma }}},\quad s^{{\hat{\alpha }} {\hat{\beta }}}=\frac{i}{4}\,[\gamma ^{{\hat{\alpha }}}, \gamma ^{{\hat{\beta }}} ], \end{aligned}$$
(12)

where \(s^{{\hat{\alpha }} {\hat{\beta }}}\) are the \(SL(2,{{\mathbb {C}}})\) generators of \(\rho _D\) while \({\hat{\varGamma }}^{{\hat{\sigma }}}_{{\hat{\mu }} {\hat{\nu }}}=e_{{\hat{\mu }}}^{\alpha } e_{{\hat{\nu }}}^{\beta } ({\hat{e}}_{\gamma }^{{\hat{\sigma }}}\varGamma ^{\gamma }_{\alpha \beta } -{\hat{e}}^{{\hat{\sigma }}}_{\beta , \alpha })\) are the connection components in local frames (known as the spin connections and often denoted by \(\varOmega \)) expressed in terms of tetrads and Christoffel symbols, \(\varGamma ^{\gamma }_{\alpha \beta }\). Thanks to these derivatives, the Dirac equation

$$\begin{aligned} \left( i\gamma ^{{\hat{\alpha }}}\nabla _{{\hat{\alpha }}}-m\right) \psi (x)=0, \end{aligned}$$
(13)

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

$$\begin{aligned} \psi (x)\rightarrow \psi '(x)= \lambda (x)\psi (x), \end{aligned}$$
(14)

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

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

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

$$\begin{aligned} \lambda ^{-1}({\hat{\omega }})\gamma ^{{\hat{\alpha }}}\lambda ({\hat{\omega }}) =\varLambda ({\hat{\omega }})^{{\hat{\alpha }},\cdot }_{\cdot \,{\hat{\beta }}}\gamma ^{{\hat{\beta }}}, \end{aligned}$$
(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. To shorten the notation, we denote by \({\tilde{G}}={\textrm{Pin}}\,O(1,3)\) the group associated to the gauge group \(G=O(1,3)\) through canonical homomorphism, 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]

$$\begin{aligned} \left[ -i\omega _{H}t_c\left( \gamma ^0\partial _{t_c}+\gamma ^i\partial _{x^i_c}\right) +\frac{3i}{2}\omega _H\gamma ^{0}-m\right] \psi (t,{\textbf{x}})=0,\nonumber \\ \end{aligned}$$
(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]

$$\begin{aligned} \left[ i\gamma ^0\partial _{t}+i{\gamma ^i}{\partial _{x^i}} -m +i\gamma ^{0}\omega _{H} \left( {x}^i{\partial }_{x^i}+\frac{3}{2}\right) \right] \psi (t,{\textbf{x}})=0,\nonumber \\ \end{aligned}$$
(18)

is different from that on \(M_-\), which has a similar form but with \(\omega _H\rightarrow -\omega _H\) thus changing 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],

$$\begin{aligned} \psi ({x}_c\,)= \int {\textrm{d}}^{3}p \sum _{\sigma }[U_{{\textbf{p}},\sigma }(x_c){\alpha }({\textbf{p}},\sigma ) +V_{{\textbf{p}},\sigma }(x_c){\beta }^{*}({\textbf{p}},\sigma )], \end{aligned}$$
(19)

in terms of wave spinors in momentum representation of particle \(\alpha \) and antiparticle \(\beta \). The mode spinors \(U_{{\textbf{p}},\sigma }\) and \(V_{{\textbf{p}},\sigma }\), of positive and negative frequencies, respectively, are plane-wave solutions of the Dirac equation depending on the conserved momentum \({\textbf{p}}\) and an arbitrary polarization \(\sigma \) as presented in the Appendix. These spinors form an orthonormal basis 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

$$\begin{aligned} {\textsf{C}}\psi =i\gamma ^2 \psi ^*, \end{aligned}$$
(20)

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

4 Continuous and discrete isometries

The de Sitter manifold \(M=M_+\cup M_-\) has isometries generated by the group O(1, 4) that leaves the metric \(\eta ^5\) invariant. In general, these isometries, \(x\rightarrow x'=\phi _{\mathfrak {g}} (x)\), 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 \(\phi _{\mathfrak {g}}\circ \phi _{{\mathfrak {g}}'}=\phi _{\mathfrak {gg}'}\), \(\forall \mathfrak {g,g}'\,\in I(M)\). The identity element of I(M) is the identity function \({\textrm{id}}=\phi _{\mathfrak {e}}\), corresponding to the identity element \({\mathfrak {e}}\) of the group O(1, 4). Therefore, the calculation rules \(\phi \circ {\textrm{id}}={\textrm{id}}\circ \phi =\phi \) and \(\phi \circ \phi ^{-1}=\phi ^{-1}\circ \phi ={\textrm{id}}\) hold for all \(\phi \in I(M)\).

The problem here is that the isometries may change the tetrad gauge, thus modifying the form of the Dirac equation. To ensure 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 \(\lambda _{\mathfrak {g}} : x\rightarrow \lambda _{\mathfrak {g}}(x)\in \rho _D\) and \(\phi _{\mathfrak {g}} : x\rightarrow \phi _{\mathfrak {g}}(x)\), which have to correct in each point the positions of the local frames after an isometry. In other words, these transformations must simultaneously preserve the metric and the tetrad gauge as \(\omega (x')=\varLambda [\lambda _{\mathfrak {g}}(x)]\omega (x)\) having the form [6],

$$\begin{aligned} \varLambda ^{{\hat{\alpha }},\cdot }_{\cdot \,{\hat{\beta }}}[\lambda _{\mathfrak {g}}(x)]= {\hat{e}}_{\mu }^{{\hat{\alpha }}}[\phi _{\mathfrak {g}}(x)]\frac{\partial \phi ^{\mu }_{\mathfrak {g}}(x)}{\partial x^{\nu }}\,{\textrm{e}}^{\nu }_{{\hat{\beta }}}(x), \end{aligned}$$
(21)

which define the mapping \(\lambda _{\mathfrak {g}}\), assuming in addition that

$$\begin{aligned} \lambda _{{\mathfrak {g}}={\mathfrak {e}}}(x) =1\in \rho _D. \end{aligned}$$
(22)

The resulting combined transformations \((\lambda _{\mathfrak {g}},\phi _{\mathfrak {g}})\) preserve the gauge, \(e'=e\) and \(\omega '=\omega \), transforming the Dirac field according to the covariant representation \({\mathfrak {T}} : (\lambda _{\mathfrak {g}},\phi _{\mathfrak {g}})\rightarrow {\mathfrak {T}}_{\mathfrak {g}}\) whose operators act as

$$\begin{aligned} \left( {\mathfrak {T}}_{\mathfrak {g}}\psi \right) [\phi _{\mathfrak {g}}(x)]=\lambda (x)\psi (x)~~\Rightarrow ~~ {\mathfrak {T}}_{\mathfrak {g}}\psi =(\lambda _{\mathfrak {g}} \psi ) \circ \phi _{\mathfrak {g}}^{-1}.\nonumber \\ \end{aligned}$$
(23)

The frame transformation \((\lambda _{\mathfrak {g}},\phi _{\mathfrak {g}}) : (x;e)\rightarrow \left( \phi _{\mathfrak {g}}(x);\varLambda (\lambda _{\mathfrak {g}})\, e \right) \) and the covariant one (23) 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 \((\lambda _{\mathfrak {g}}, \phi _{\mathfrak {g}})\), respectively, 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 \(\lambda :M\rightarrow {\tilde{G}}\) are sections on the principal fiber bundle \(M\times {\tilde{G}}\), while the mappings associated through canonical homomorphism, \(\varLambda (\lambda ):M\rightarrow G\), are sections on \(M\times G\). These mappings can be composed as \(\lambda _{{\mathfrak {g}}'}\times \lambda _{\mathfrak {g}}\) such that (\(\lambda _{{\mathfrak {g}}'}\times \lambda _{\mathfrak {g}})(x)=\lambda _{{\mathfrak {g}}'}(x) \lambda _{\mathfrak {g}}(x)\) and similarly for the mappings \(\varLambda \). Moreover, observing that \((\lambda _{{\mathfrak {g}}'}\circ \phi _{\mathfrak {g}})\times \lambda _{\mathfrak {g}}=\lambda _{{\mathfrak {g}}'{\mathfrak {g}}}\), 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]

$$\begin{aligned}{} & {} (\lambda _{{\mathfrak {g}}'},\phi _{{\mathfrak {g}}'})*(\lambda _{\mathfrak {g}}, \phi _{\mathfrak {g}})=((\lambda _{{\mathfrak {g}}'}\circ \phi _{\mathfrak {g}})\times \lambda _{\mathfrak {g}}, \phi _{{\mathfrak {g}}'}\circ \phi _{\mathfrak {g}})\nonumber \\{} & {} \quad =(\lambda _{{\mathfrak {g}}'{\mathfrak {g}}},\phi _{{\mathfrak {g}}'{\mathfrak {g}}}). \end{aligned}$$
(24)

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 \(\lambda _{\mathfrak {d}} :x \rightarrow \lambda _{\mathfrak {d}}(x)\) give rise to local discrete transformation \(\lambda _{\mathfrak {d}}(x)\in \rho _D\) 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 \(\varLambda (\lambda _{\mathfrak {d}})\varLambda (\lambda _{\mathfrak {d}})=I\), 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 of 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,

$$\begin{aligned} \varLambda (\gamma ^0)= & {} P={\textrm{diag}}(1,-1,-1,-1), \end{aligned}$$
(25)
$$\begin{aligned} \varLambda (\gamma ^0\gamma ^5)= & {} T={\textrm{diag}}(-1,1,1,1), \end{aligned}$$
(26)
$$\begin{aligned} \varLambda (\gamma ^5)= & {} PT=-I, \end{aligned}$$
(27)

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

The de Sitter discrete isometries \(\phi ^c_{\mathfrak {d}}\) in conformal frames defined by the functions (1–3) satisfy \(z_c[\phi ^c_{\mathfrak {d}}(x_c)]={\mathfrak {d}}z_c(x_c)\). The simplest discrete transformations on \(M^5\) are the mirror reflections, denoted by [A], which change the sign of a single coordinate, \(z_c^A\rightarrow -z_c^A\), giving rise to the discrete isometries \(x_c\rightarrow x_c'=\phi ^c_{[A]}(x_c)\). A rapid inspection indicates that there are two interesting local isometries produced by \(\phi ^c_{[0]}\) as

$$\begin{aligned} t_c'(x_c)= & {} \phi _{[0]}^{c\,0}(x_c)=\frac{t_c}{\omega _H^2 (t_c^2-{\textbf{x}}_c^2)}, \end{aligned}$$
(28)
$$\begin{aligned} x_c^{\prime \, i}(x_c)= & {} \phi _{[0]}^{c\,i}(x_c)=\frac{x^i_c}{\omega _H^2 (t_c^2-{\textbf{x}}_c^2)}, \end{aligned}$$
(29)

and by \(\phi ^c_{[4]}\) which gives

$$\begin{aligned} \phi _{[4]}^{c\,0}(x_c)=- t_c'(x_c), \quad \phi _{[4]}^{c\,i}(x_c)=-x_c^{\prime \, i}(x_c). \end{aligned}$$
(30)

The space isometries, \(\phi ^c_{[i]}\), are simple mirror transformations of the space co-moving coordinates, \(x_c^i\rightarrow -x_c^i\), 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, \(x_c^{\mu }\rightarrow - x_c^{\mu }\). The antipodal transformation, \({\mathfrak {a}} : z_c\rightarrow -z_c\), 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, \(t_c\rightarrow -t_c\).

We hereby see that the set of discrete isometries \({\mathfrak D}={\mathfrak D}_+\cup {\mathfrak D}_-\) can be split into two subsets, \({\mathfrak D}_+=\{\phi _{\mathfrak {d}} | \phi _{\mathfrak {d}} : M_{\pm }\rightarrow M_{\pm }\}\), of proper discrete isometries which preserve the portions of M and \({\mathfrak D}_-=\{\phi _{\mathfrak {d}} | \phi _{\mathfrak {d}} : M_{\pm }\rightarrow M_{\mp }\}\) of improper discrete isometries that change these portions among themselves. As the physical measurements can be performed only inside the light-cone where \(|t_c|>|{\textbf{x}}_c|\), we deduce that the isometry (28) preserves the sign of conformal time while the isometry (30) changes it. Therefore, we may conclude that

$$\begin{aligned}{} & {} \phi _{\mathfrak {e}}={\textrm{id}},\,\phi ^c_{\mathfrak {p}},\,\phi ^c_{[0]},\,\phi ^c_{{\mathfrak {p}}[0]}\in {\mathfrak D}_+ \end{aligned}$$
(31)
$$\begin{aligned}{} & {} \phi ^c_{\mathfrak {a}},\,\phi ^c_{[4]},\,\phi ^c_{[0][4]},\,\phi ^c_{{\mathfrak {p}}[4]}\in {\mathfrak D}_- \end{aligned}$$
(32)

and similarly for the isometries \(\phi _{\mathfrak {d}}\) of the physical frames. To understand the physical meaning of these isometries, we first discuss the proper discrete isometries in physical frames.

5 Discrete symmetries in physical frames

Let us now consider 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

$$\begin{aligned} z^0(x)= & {} \frac{1}{2 \omega _{H}}\left[ {\textrm{e}}^{\omega _H t} -{\textrm{e}}^{-\omega _H t} \left( 1-\omega _{H}^2{{\textbf{x}}}^2\right) \right] , \end{aligned}$$
(33)
$$\begin{aligned} z^i(x)= & {} x^i, \end{aligned}$$
(34)
$$\begin{aligned} z^4(x)= & {} \frac{1}{2 \omega _{H}}\left[ {\textrm{e}}^{\omega _H t} +{\textrm{e}}^{-\omega _H t} \left( 1-\omega _{H}^2{{\textbf{x}}}^2\right) \right] . \end{aligned}$$
(35)

Similar functions can be obtained that define the physical coordinates on \(M_-\) changing \(\omega _H\rightarrow -\omega _H\) in Eqs. (33–35).

Table 1 Transformations induced by the nontrivial proper discrete isometries in physical frames \(M_+\). The function \(t'(t, |{\textbf{x}}|)\) is defined in Eq. (37) while the matrices \(B({\textbf{x}})\) and \(\beta ({\textbf{x}})\) are given by Eqs. (40) and (44), respectively
Full size table

We first observe that the trivial isometry \({\textrm{id}}\) is needed for closing a discrete group. Denoting \({\mathfrak {T}}_{\mathfrak {e}}= {\textsf{I}}\), we relax the condition (22) for recovering the kernel \({\mathbb {Z}}_2=\{ {\textsf{I}},\, - {\textsf{I}}\}\) of the canonical homomorphism we need when we study discrete transformations. The simplest nontrivial 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 \(\lambda _{\mathfrak {p}}=\gamma ^0\). Then, denoting \({\mathfrak {T}}_{\mathfrak {p}}= {{\textsf{P}}}\), we obtain the parity transformation of the Dirac field,

$$\begin{aligned} \left( {{\textsf{P}}}\psi \right) (t,{\textbf{x}})=\gamma ^0 \psi (t,-{\textbf{x}}), \end{aligned}$$
(36)

acting just as in the flat case but in addition transforming the tetrad fields with \(\varLambda _{\mathfrak {p}}=P\).

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

$$\begin{aligned} t'(t, |{\textbf{x}}|)= & {} \phi _{[0]}^0(x)=-t +\frac{1}{\omega _H}\ln \left( 1-\omega _H^2 {\textbf{x}}^2\right) ,\end{aligned}$$
(37)
$$\begin{aligned} x^{\prime \,i}= & {} \phi _{[0]}^i(x)=x^i, \end{aligned}$$
(38)

giving the new local time \(t'(x)\le -t\) but without affecting the space coordinates. The problem now is to find the associated matrices \(\lambda _{[0]}\) and \(\varLambda _{[0]}\equiv \varLambda (\lambda _{[0]})\) transforming the Dirac and the tetrad fields, respectively. We first apply the definition (21) for deriving the local matrix \(\varLambda _{[0]}(x)\), finding that this is time-independent, having the matrix elements

$$\begin{aligned} \left<\varLambda _{[0]}({\textbf{x}})\right>^{0\,\cdot }_{\cdot \,0}= & {} -\frac{1+\omega ^2_H{\textbf{x}}^2}{1-\omega ^2_H{\textbf{x}}^2}<0, \nonumber \\ \left<\varLambda _{[0]}({\textbf{x}})\right>^{0\,\cdot }_{\cdot \,i}= & {} -\left<\varLambda _{[0]}(x)\right>^{i\,\cdot }_{\cdot \,0}=-\frac{2\omega _H x^i}{1-\omega ^2_H{\textbf{x}}^2},\\ \left<\varLambda _{[0]}({\textbf{x}})\right>^{i\,\cdot }_{\cdot \,j}= & {} \delta ^i_j+\frac{2\omega ^2_H x^i x^j}{1-\omega ^2_H{\textbf{x}}^2}.\nonumber \end{aligned}$$
(39)

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

$$\begin{aligned}{} & {} B({\textbf{x}})=\exp \left( -iK_i\,\frac{x^i}{|{\textbf{x}}|}\, \alpha ( |{\textbf{x}}|)\right) \end{aligned}$$
(40)
$$\begin{aligned}{} & {} \alpha ( |{\textbf{x}}|)=\textrm{tanh}^{-1}\left( \frac{2 \omega _H |{\textbf{x}}|}{1+\omega ^2_H{\textbf{x}}^2}\right) , \end{aligned}$$
(41)

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

$$\begin{aligned} \varLambda _{[0]}({\textbf{x}})[\varLambda _{[0]}({\textbf{x}})=I~ ~\Longleftrightarrow ~~ B({\textbf{x}})TB({\textbf{x}})=T, \end{aligned}$$
(42)

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

The above parameterization helps us to find the local matrix \(\lambda _{[0]}({\textbf{x}})\) which transforms the Dirac field. According to Eq. (26), this can be written as

$$\begin{aligned} \lambda _{[0]}({\textbf{x}})=\gamma ^0\gamma ^5 \beta ({\textbf{x}}), \end{aligned}$$
(43)

where the matrix

$$\begin{aligned} \beta ({\textbf{x}})=\exp \left( -is_{i0}\,\frac{x^i}{|{\textbf{x}}|}\, \alpha ( |{\textbf{x}}|)\right) =\frac{1+\omega _H \gamma ^0\gamma ^i x^i}{\sqrt{1-\omega ^2_H{\textbf{x}}^2}}\in \rho _D,\nonumber \\ \end{aligned}$$
(44)

is calculated by using the boost generators \(s_{i0} \in \rho _D\) defined in Eq. (12). This matrix has the obvious properties \(\beta ({\textbf{x}})^{-1}=\beta (-{\textbf{x}})\) and \(\gamma ^0\beta ({\textbf{x}})=\beta (-{\textbf{x}})\gamma ^0\) such that \(\lambda _{[0]}({\textbf{x}})\lambda _{[0]}({\textbf{x}})=-1\in \rho _D\). Moreover, we verify that the matrices \(\lambda _{[0]}({\textbf{x}})\) and \(\varLambda _{[0]}({\textbf{x}})\) are related through the canonical homomorphism satisfying the identities

$$\begin{aligned} \lambda _{[0]}({\textbf{x}})^{-1}\gamma ^{{\hat{\alpha }}} \lambda _{[0]}({\textbf{x}})= \left<\varLambda _{[0]}({\textbf{x}})\right>^{{\hat{\alpha }},\cdot }_{\cdot \,{\hat{\beta }}} \gamma ^{{\hat{\beta }}}, \end{aligned}$$
(45)

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

Collecting the above results and denoting \({\mathfrak {T}}_{[0]}= {\textsf{B}}\), we define the discrete transformation of the Dirac field

$$\begin{aligned} \left( {\textsf{B}}\psi \right) (t,{\textbf{x}})= & {} \lambda _{[0]}({\textbf{x}})\psi (t'(x), {\textbf{x}})\nonumber \\= & {} \gamma ^0\gamma ^5\beta ({\textbf{x}})\psi (t'(t, |{\textbf{x}}|)), {\textbf{x}}), \end{aligned}$$
(46)

where \(t'(x)\) is given by Eq. (37). Moreover, the transformation of the Dirac field associated to the isometry \(\phi ^c_{{\mathfrak {p}}[0]}\) reads

$$\begin{aligned} \left( {{{\textsf{P}}}}{{\textsf{B}}}\psi \right) (t,{\textbf{x}})= \gamma ^5 \beta (-{\textbf{x}})\psi (t'(t, |{\textbf{x}}|), -{\textbf{x}} ), \end{aligned}$$
(47)

while the tetrads have to be transformed by \(\varLambda _{{\mathfrak {p}}[0]}=PTB(-{\textbf{x}}) \) \(=-B(-{\textbf{x}})\). Obviously, the transformations \( {\textsf{B}}\) and \( {{{\textsf{P}}}}{{\textsf{B}}}\) take over the role of \( {{\textsf{C}}}{{\textsf{T}}}\) and \({\textsf{PCT}}\) ones, respectively, of the Dirac theory in Minkowski space-time.

Table 2 Multiplication table of the subset \({{{\mathcal {G}}}}/{{\mathbb {Z}}}_2\) of the group \({{{\mathcal {G}}}}\)
Full size table

We thus see that the Dirac theory on \(M_+\) has three nontrivial proper discrete isometries, \( {{\textsf{P}}},\, {\textsf{B}}\) and \( {{{\textsf{P}}}}{{\textsf{B}}}=- {{\textsf{B}}}{{{\textsf{P}}}}\) whose actions are summarized in Table 1. These isometries can be composed at any time with the charge conjugation \({\textsf{C}}\) which is independent of geometry. Under such circumstances, we may define the Wigner time reversal on \(M_+\) as \( {\textsf{T}}_+= {\textsf{C}} {\textsf{B}}= {{\textsf{B}}}{{\textsf{C}}}\), again obtaining a local transformation which in our phase convention reads

$$\begin{aligned} \left( {\textsf{T}}_+\psi \right) (t,{\textbf{x}})= & {} i\gamma ^2\left( {\textsf{B}}\psi (t,{\textbf{x}})\right) ^*\nonumber \\= & {} -\beta (-{\textbf{x}})\gamma ^1\gamma ^3\psi ^*(t'(t, |{\textbf{x}}|),{\textbf{x}}), \end{aligned}$$
(48)

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

$$\begin{aligned} {{\textsf{P}}} {\textsf{T}}_+\psi (t,{\textbf{x}})=-\beta ({\textbf{x}}) \gamma ^0\gamma ^1\gamma ^3\psi ^*(t'(t, |{\textbf{x}}|),{-\textbf{x}}), \end{aligned}$$
(49)

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

Finally, we substitute \( {\textsf{B}}= {\textsf{C}} {\textsf{T}}_+\), obtaining the standard components

$$\begin{aligned} {{{\mathcal {G}}}}_+=\{ \pm {\textsf{I}},\,\pm {\textsf{C}},\,\pm {{\textsf{P}}},\, \pm {{{\textsf{P}}}}{{\textsf{C}}},\,\pm {\textsf{T}}_+,\,\pm {{\textsf{P}}} {\textsf{T}}_+,\,\pm {\textsf{C}} {\textsf{T}}_+,\,\pm {{{\textsf{P}}}}{{\textsf{C}}} {\textsf{T}}_+\}\nonumber \\ \end{aligned}$$
(50)

of the discrete group of Dirac’s theory on the de Sitter expanding universe, \(M_+\). This group is of the order 16, with the multiplication table (given in Table 2) which is similar to that of the Dirac theory in Minkowski space-time. In this group, we find the usual parity \({{\textsf{P}}}\) and charge conjugation \({\textsf{C}}\) which are global while all the transformations involving the Wigner time reversal \({\textsf{T}}_+\) are local, depending on the matrix \(\beta ({\textbf{x}})\).

An important particular case is the Majorana neutral fields, \(\psi _M^{\pm }\), defined as eigenspinors of the charge conjugation satisfying \({\textsf{C}}\psi _M^{\pm }=\pm \psi _M^{\pm }\). These fields have the form (19) with the same mode spinors as those of the charged field but with only one set of wave spinors for particle and antiparticle restricting \(\beta =\pm \alpha \). In this case, the discrete group is simpler, of the order 8, having the components

$$\begin{aligned} {{{\mathcal {G}}}}_{M\,+}=\{ \pm {\textsf{I}},\,\pm {{\textsf{P}}},\,\pm {\textsf{T}}_+,\,\pm {{\textsf{P}}} {\textsf{T}}_+,\} , \end{aligned}$$
(51)

as is natural after eliminating the charge conjugation.

Similar results can be obtained for the collapsing portion, where we have a similar discrete group \({{{\mathcal {G}}}}_-\) but with the Wigner time reversal \({\textsf{T}}_-\) that can be obtained from \({\textsf{T}}_+\) substituting \(\omega _H\rightarrow -\omega _H\). An important opportunity with this approach is the correct flat limit when \(\omega _H\rightarrow 0\). Indeed, in Eq. (37) we have

$$\begin{aligned} \lim _{\omega _H\rightarrow 0}\frac{1}{\omega _H}\ln (1-\omega _H^2{\textbf{x}}^2)=0 ~~\Rightarrow ~~ \lim _{\omega _H\rightarrow 0}{\textsf{T}}_+=\lim _{\omega _H\rightarrow 0}{\textsf{T}}_-={\textsf{T}},\nonumber \\ \end{aligned}$$
(52)

as in this limit, \(B({\textbf{x}})\rightarrow I\) and \(\beta ({\textbf{x}})\rightarrow 1\in \rho _D\). Therefore, \({{{\mathcal {G}}}}_+\sim {{{\mathcal {G}}}}_-\) tend to the usual discrete group of the Dirac theory in special relativity.

Then, trying to continue this study with the improper isometries in physical frames, we face difficulties, as we must consider the explicit transformation \(\omega _H\rightarrow -\omega _H\) 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, returning to the conformal frames where we have technical opportunities.

6 Discrete isometries in conformal frames

Table 3 Transformations induced by the discrete de Sitter isometries in conformal frames. The functions \(t'_c(x_c)\) and \({x}^{\prime \, i}_c(x_c)\) are defined in Eq. (28), while the matrices \(B^c(x_c)\) and \(\beta ^c(x_c)\) are given by Eqs. (55) and (56)
Full size table

In the conformal frame, we have the technical advantage of the Dirac equation and metric that do not change their forms when we change the portion by changing the sign of \(t_c\). For this reason, these frames are suitable for studying the discrete transformations on the whole manifold M. Starting with the proper isometries (31), we first observe that the parity remains the same,

$$\begin{aligned} \left( {{\textsf{P}}}\psi \right) (t_c,{\textbf{x}}_c)=\gamma ^0 \psi (t_c,-{\textbf{x}}_c), \end{aligned}$$
(53)

as the three-vector \({\textbf{x}}_c\) and \({\textbf{x}}\) are parallel to \(\textbf{z}\). The isometry \(\phi ^c_{[0]}\) defined by Eq. (28) mixes the time and Cartesian coordinates such that now we may write

$$\begin{aligned} \left( {\textsf{B}}\psi \right) (t_c,{\textbf{x}}_c)= & {} \lambda ^c_{[0]}({x}_c)\psi (t_c'({x}_c), {\textbf{x}}_c'(x_c))\nonumber \\= & {} \gamma ^0\gamma ^5\beta ^c(x_c)\psi (t'_c(x_c), {\textbf{x}}'_c(x_c)) , \end{aligned}$$
(54)

where the functions \(t'_c(x_c)\) and \({\textbf{x}}'_c(x_c)\) are defined in Eq. (28). The matrix \(\beta ^c=\beta \circ \chi ^{-1}\) and the matrix transforming tetrads, \(\varLambda ^c_{[0]}=TB^c=TB\circ \chi ^{-1}\), are obtained after substituting the physical coordinates according to Eq. (5). A few manipulations allow us to carry out the matrix elements of \(B^c(x_c)\) that read

$$\begin{aligned} \left<B^c({x}_c)\right>^{0\,\cdot }_{\cdot \,0}= & {} \frac{t_c^2+{\textbf{x}}_c^2}{t_c^2-{\textbf{x}}_c^2}, \nonumber \\ \left<B^c({x}_c)\right>^{0\,\cdot }_{\cdot \,i}= & {} \left<B^c({x}_c)\right>^{i\,\cdot }_{\cdot \,0}=-\frac{2 x_c^i}{t_c^2-{\textbf{x}}_c^2},\\ \left<B^c({x}_c)\right>^{i\,\cdot }_{\cdot \,j}= & {} \delta ^i_j+\frac{2 x_c^i x_c^j}{t_c^2-{\textbf{x}}_c^2},\nonumber \end{aligned}$$
(55)

and the new form

$$\begin{aligned} \beta ^c(t_c,{\textbf{x}}_c)=\frac{t_c- \gamma ^0\gamma ^i x_c^i}{\sqrt{t_c^2-{\textbf{x}}_c^2}}\in \rho _D, \end{aligned}$$
(56)

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

$$\begin{aligned} \left( {{{\textsf{P}}}}{{\textsf{B}}}\psi \right) (t_c,{\textbf{x}}_c)= \gamma ^5\beta ^c(t_c,-{\textbf{x}}_c)\psi (t'_c(x_c), -{\textbf{x}}'_c(x_c)) , \end{aligned}$$
(57)

and \(\varLambda ^c_{{\mathfrak {p}}[0]}(x_c)=-B^c(t_c,-{\textbf{x}}_c)\). Thus, all the results concerning the proper isometries are in accord with those obtained in physical frames.

To study the improper isometries (32), we start with the antipodal isometry \(\phi ^c_{\mathfrak {a}}\), for which we have \(\varLambda _{\mathfrak {a}}^c=P \rightarrow \lambda ^c_{\mathfrak {a}} =\gamma ^0\) such that by denoting \({\mathfrak {T}}_{\mathfrak {a}}={\textsf{A}}\), we obtain the antipodal transformation of the Dirac field,

$$\begin{aligned} \left( {\textsf{A}}\psi \right) (t_c,{\textbf{x}}_c)=\gamma ^0 \psi (-t_c,{\textbf{x}}_c), \end{aligned}$$
(58)

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

$$\begin{aligned} \left( {{{\textsf{P}}}}{{\textsf{A}}}\psi \right) (t_c,{\textbf{x}}_c)= \psi (-t_c,-{\textbf{x}}_c), \end{aligned}$$
(59)

and the associated matrix \(\varLambda _{[0][4]}=I\). 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

$$\begin{aligned} \left( {\textsf{PAB}}\psi \right) (t_c,{\textbf{x}}_c)= \gamma ^0\gamma ^5\beta ^c(-t_c,-{\textbf{x}}_c)\psi (-t'_c(x_c), -{\textbf{x}}'_c(x_c)) , \end{aligned}$$
(60)

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

$$\begin{aligned} \left( {{\textsf{A}}}{{\textsf{B}}}\psi \right) (t_c,{\textbf{x}}_c)= \gamma ^5\beta ^c(-t_c,{\textbf{x}}_c)\psi (-t'_c(x_c), {\textbf{x}}'_c(x_c)) , \end{aligned}$$
(61)

and \(\varLambda ^c_{{\mathfrak {p}}[4]}(x_c)=-B^c(-t_c,{\textbf{x}}_c)\).

In Table 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 are local, depending on the matrix (56). The improper isometries comprise two global and two local, depending on the same boost. All these transformations can be obtained using the basis formed by \({{\textsf{P}}}\), \({\textsf{B}}\) and \({\textsf{A}}\). Moreover, these transformations can be composed at any time with the charge conjugation \({\textsf{C}}\) giving rise to a large collection of discrete transformations. For example, the transformations (48) and (49) can be rewritten in conformal frames as

$$\begin{aligned} \left( {\textsf{T}}_c\psi \right) (t_c,{\textbf{x}}_c)&=-\beta ^c(t_c,-{\textbf{x}}_c)\gamma ^1\gamma ^3\psi ^*(t_c'({\textbf{x}}),{\textbf{x}}'_c({x_c})), \end{aligned}$$
(62)
$$\begin{aligned} \left( {{\textsf{P}}}{\textsf{T}}_c\psi \right) (t_c,{\textbf{x}}_c)&=-\beta ^c(t_c,{\textbf{x}}_c)\gamma ^0\gamma ^1\gamma ^3\psi ^*(t_c'({\textbf{x}}),-{\textbf{x}}'_c({x_c})), \end{aligned}$$
(63)

having the same form on \(M_+\) and \(M_-\). With their help we may construct the discrete group \({\mathcal {G}}_c\) in conformal frames which has a similar multiplication table as in Table 2 but with \({\textsf{T}}_c\) instead of \({\textsf{T}}_+\). We observe that Eqs. (62) and (63) are far from an intuitive interpretation, again confirming 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 Table 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 Some physical consequences

After the above mathematics, it is worth discussing some physical consequences from the perspective of an observer staying at rest in the origin of their proper frame with physical coordinates on the expanding portion \(M_+\). In this frame, the discrete group (50) is formed by global transformations and new local ones we do not meet in the flat case.

Remarkably, the global transformations \({{\textsf{P}}}\), \({\textsf{C}}\) and \({{{\textsf{P}}}}{{\textsf{C}}}\) have the same action and properties as in Minkowski space-time. This is important from the point of view of the gauge models that can be built just as in the flat case. More specifically, the charge conjugation changes the chirality of the components \(\psi _{L/R}=\left. P_{L/R}\psi \right| _{m=0}\) as

$$\begin{aligned} \left( {\textsf{C}}\psi \right) _{L/R}(t,{\textbf{x}})=i\gamma ^2\psi _{R/L}(t,{\textbf{x}})^*, \end{aligned}$$
(64)

ensuring the aforementioned flexibility in constructing the spinor sectors.

In contrast, the new local transformations, \({\textsf{B}}\), \({{{\textsf{P}}}}{{\textsf{B}}}\) and \({\textsf{T}}_+={{\textsf{C}}}{{\textsf{B}}}\), depend on the matrix (44) which is singular on the observer’s event horizon at \(|{\textbf{x}}|=\omega _H^{-1}\), and consequently, all the local discrete transformations are singular on the event horizon. This may pose some difficulties in interpreting their local effects such that our observer may be forced to consider the local transformations exclusively in the origin where \(\beta ({\textbf{x}})|_{{\textbf{x}}=0}=1\in \rho _D\).

We might hope to use the new discrete transformations studied here in interpreting dynamical effects or selecting new vacua. On \(M_+\), the propagators (or two-point functions), which play a crucial role in describing the field dynamics in de Sitter’s gravitational field, are maximal symmetric functions depending on the geodesic distance between the two points. This is proportional to the Euclidean distance in \(M^5\) which is invariant under isometries such that all the mentioned propagators are invariant under such transformations. On the other hand, the structure of the functions (A.6) and (A.7) shows that we cannot use discrete transformations for setting the vacuum. We conclude that on \(M_+\), the discrete transformation of the Dirac field does not play an important role in selecting the vacuum or interpreting dynamical processes, just as happens in Minkowski space-time.

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 [33].