Two Dimensional de Sitter Spinors and Their SL(2, R) Covariance

Abstract

We give a geometrical description of the double covering of the two-dimensional de Sitter universe as a coset space of the group SL(2, R). This identification is helpful in characterizing the de Sitter covariance of Dirac fields on the two-dimensional de Sitter spacetime or its double covering and opens the possibility to study CFT on that manifolds.

13.1 Introduction

In the general construction of systems of coherent states an important role is played by coset spaces G / H, where G is a (locally compact) Lie group and H a subgroup of G [1,2,3]. One particular example that has been studied and has played a distinguished role from the very beginning [1] is the coset space SL(2, R) / K identified with the upper complex plane (K is the compact subgroup of SL(2, R)). In this note we provide a simple geometric interpretation of another coset space of SL(2, R) that we identify with the double covering of the two-dimensional de Sitter spacetime $dS_2$. This identification may be useful for the study of conformal QFT on $dS_2$ and its double covering; we provide here some preliminary steps in that direction by studying the free quantum Dirac field. We study in particular the relations between Dirac fields living on the 2-dimensional Lorentzian cylinder and the ones living on the double-covering of the 2-dimensional de Sitter manifold, identified with a certain coset space of the group SL(2, R). We show that there is an extended notion of de Sitter covariance only for Dirac fields having the Neveu-Schwarz anti-periodicity and construct the relevant cocycle. Finally, we show that the de Sitter symmetry is naturally inherited by the Neveu-Schwarz massless Dirac field on the cylinder. This paper is an account of works in collaboration with Henri Epstein [4, 5].

13.2 The de Sitter Group $SO_0(1,2)$ and the Two-Dimensional de Sitter Manifold as a Coset Space

Let us consider the three-dimensional Minkowski spacetime ${M}_{3}$ whose metric tensor is

$$\begin{aligned} \eta _{\alpha \beta } = \mathrm{diag}(1,-1,-1) ,\quad \alpha ,\beta = 0,1,2. \end{aligned}$$

(13.1)

The Clifford anti-commutation relations relative to the above metric are

$$\begin{aligned} \gamma ^{\alpha }\gamma ^{\beta }+\gamma ^{\beta }\gamma ^{\alpha }=2\eta ^{\alpha \beta } \end{aligned}$$

(13.2)

and may be concretely realized by the following choice of matrices:

$$\begin{aligned} \gamma ^{0}= \begin{pmatrix}0&{}1\\ 1&{}0\end{pmatrix} ,\quad \gamma ^{1}=\begin{pmatrix}0&{}1\\ -1&{}0\end{pmatrix} ,\quad \gamma ^{2}=\begin{pmatrix}i&{}0\\ 0&{}-i\end{pmatrix} . \end{aligned}$$

(13.3)

The two dimensional de Sitter spacetime may be represented as the one-sheeted hyperboloid

$$\begin{aligned} dS_{2}=\left\{ X \in {M}_{3}:\ (X^{0})^{2}-(X^1)^{2}-(X^2)^{2}=- 1 \right\} \end{aligned}$$

(13.4)

embedded in ${M}_{3}$.

The de Sitter relativity group is the connected component $SO_0(1,2)$ of the (Lorentz) pseudo-orthogonal group of the three dimensional ambient spacetime $M_3$. We distinguish three one-parameter subgroups and write the Iwasawa decomposition of KNA of a generic element G of $SO_0(1,2)$ as follows:

$$\begin{aligned} G= & {} K(\theta ) N(\lambda ) A(u) \\= & {} \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} \cos \theta &{} \sin \theta \\ 0 &{} -\sin \theta &{} \cos \theta \\ \end{array} \right) \left( \begin{array}{ccc} 1+ \frac{\lambda ^2}{2} &{} \frac{\lambda ^2}{2} &{} -\lambda \\ -\frac{\lambda ^2}{2} &{} 1-\frac{\lambda ^2}{2} &{} \lambda \\ -\lambda &{} -\lambda &{} 1 \\ \end{array} \right) \left( \begin{array}{ccc} \cosh u &{} -\sinh u &{} 0 \\ -\sinh u &{} \cosh u &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{array} \right) . \end{aligned}$$

(13.5)

The above decomposition provides a natural parametrization $(\lambda ,\theta )$ of the quotient space $SO_0(1,2)/A$, which is seen to be topologically a cylinder.

The group $SO_0(1,2)$ acts on the coset space $SO_0(1,2)/A$ by left multiplication. Its action may be written as a nonlinear transformation of the parameters $(\lambda ,\theta )$:

$$\begin{aligned} G\, : \, (\lambda ,\theta ) \rightarrow (\lambda ',\theta '). \end{aligned}$$

(13.6)

It is useful to describe the action of the subgroups separately. The case of a rotation $K(\alpha ) \in K$ is of course the easiest one and amounts simply to a shift of the angle $\theta$:

$$\begin{aligned} \begin{array}{l} \lambda ' (\alpha ) = \lambda ,\ \ \ \ \theta '(\alpha ) = \theta +\alpha . \end{array} \end{aligned}$$

(13.7)

The two other subgroups give rise to slightly more involved transformation rules; an element $A(\kappa )$ of the abelian subgroup A gives

$$\begin{aligned} \left\{ \begin{array}{l} \lambda '(\kappa ) = \lambda \cosh \kappa +\sinh \kappa (\lambda \cos \theta +\sin \theta ), \\ \sin \theta ' (\kappa )={\sin \theta }/({\cos \theta \sinh \kappa +\cosh \kappa }),\\ \cos \theta ' (\kappa )=({\cos \theta \cosh \kappa +\sinh \kappa })/({\cos \theta \sinh \kappa +\cosh \kappa }). \end{array}\right. \end{aligned}$$

(13.8)

A short computation shows that

$$\begin{aligned} \cos \theta ' (\kappa )- \lambda ' (\kappa )\sin \theta ' (\kappa )= \cos \theta - \lambda \sin \theta . \end{aligned}$$

(13.9)

An element $N(\mu )\in N$ gives

$$\begin{aligned} \left\{ \begin{array}{l} \lambda ' (\mu )= {\lambda } \left( 1 +\frac{1}{2} {\mu ^2}\right) - \mu \left( \lambda +\frac{\mu }{2} \right) \sin \theta +{\mu } \left( 1- \frac{1}{2} \lambda \mu \right) \cos \theta , \\ \cos \theta '(\mu ) = \displaystyle { \frac{-2 \cos \theta +\mu ^2 \cos \theta -\mu ^2+2 \mu \sin \theta }{\mu ^2 \cos \theta -\mu ^2+2 \mu \sin \theta -2}}, \\ \sin \theta '(\mu )= \displaystyle {-\frac{2 (\sin \theta +\mu (\cos \theta -1))}{\mu ^2 \cos \theta -\mu ^2+2 \mu \sin \theta -2}}. \end{array}\right. \end{aligned}$$

(13.10)

The transformation rules (13.7), (13.8) and (13.10) have a simple geometrical interpretation that may be unveiled by introducing the following parametrization of $dS_2$ related to the Iwasawa decomposition (13.5):

$$\begin{aligned} X(\lambda ,\theta )= \left\{ \begin{array}{l} X^0= -\lambda , \\ X^1= \lambda \cos \theta +\sin \theta ,\\ X^2= \cos \theta -\lambda \sin \theta .\\ \end{array} \right. \end{aligned}$$

(13.11)

The coordinate system $(\lambda ,\theta )$ is not orthogonal (see Fig 13.1):

$$\begin{aligned} {\mathrm {d}}s^2 = \left. \left( {\mathrm {d}X^0}^2-{\mathrm {d}X^1}^2-{\mathrm {d}X^2}^2\right) \right| _{dS_2} = -2 \mathrm {d}\lambda \mathrm {d}\theta -\left( \lambda ^2+1\right) \mathrm {d}\theta ^2. \end{aligned}$$

(13.12)

One verifies easily that $X(\lambda '(\alpha ),\theta '(\alpha )) = K(\alpha ) X(\lambda ,\theta )$, that $X(\lambda '(\kappa ),\theta '(\kappa )) = A(\kappa ) X(\lambda ,\theta )$ and that $X(\lambda '(\mu )\theta '(\mu )) = N(\mu ) X(\lambda ,\theta )$. Therefore the coset space $SO_0(1,2)/A$ is identical to the two-dimensional de Sitter manifold and the left action of $SO_0(1,2)$ on the cosets coincides with the linear action of $SO_0(1,2)$ restricted to the manifold $dS_2$.

Fig. 13.1

The two-dimensional de Sitter manifold with the coordinate system $(\lambda ,\theta )$. Circles have constant values of $X^0=- \lambda$. Light rays have constant values of $\theta$

13.3 The Spin Group Sp(1, 2) and the Double Covering of $dS_2$

In our context the spin group (i.e. the double covering of $SO_0(1,2)$) is most usefully realized as the matrix group

$$\begin{aligned} Sp(1,2)=\bigl \{g\in SL(2,C) :\, \,\gamma ^{0} g^{\dagger }\gamma ^{0}= g^{-1}\bigr \}. \end{aligned}$$

(13.13)

An element of Sp(1, 2) may be parametrised in terms of four real numbers a, b, c, d subject to the condition $a d+bc=1$ as follows:

$$\begin{aligned} g= \left( \begin{array}{cc} a &{} i \,b \\ i \,c &{} d \\ \end{array} \right) . \end{aligned}$$

(13.14)

As a subgroup of SL(2, C), the group Sp(1, 2) is conjugated to SL(2, R):

$$\begin{aligned} h g h^{-1} = \left( \begin{array}{cc} a &{} -b \\ c &{} d \\ \end{array} \right) . \end{aligned}$$

(13.15)

where

$$\begin{aligned} h=\left( \begin{array}{cc} e^{\frac{i \pi }{4}} &{} 0 \\ 0 &{} e^{-\frac{i \pi }{4}} \\ \end{array} \right) \end{aligned}$$

(13.16)

Sp(1, 2) acts on $dS_2$ by similarity transformations:

(13.17)

where as usual

(13.18)

The covering projection $g \rightarrow \varLambda (g)$ of Sp(1, 2) onto $SO_0(1,2)$ coherent with the above action is expressed as follows:

$$\begin{aligned} g\rightarrow {\varLambda (g)^\alpha }_\beta = {1\over 2}\mathrm{tr}(\gamma ^{\alpha }g\gamma _{\beta }g^{-1}). \end{aligned}$$

(13.19)

$\varLambda (g)$ is the (real) Lorentz transformation that directly relates $X'$ and X

(13.20)

Let us now, as before, write the Iwasawa decomposition of Sp(1, 2):

$$\begin{aligned} g=k(\zeta )\,n(\lambda )\,a(\chi ) = \left( \begin{array}{cc} \cos \frac{\zeta }{2} &{} i \sin \frac{\zeta }{2} \\ i \sin \frac{\zeta }{2} &{} \cos \frac{\zeta }{2} \\ \end{array} \right) \left( \begin{array}{cc} 1 &{} i \lambda \\ 0 &{} 1 \\ \end{array} \right) \left( \begin{array}{cc} e^{\frac{\chi }{2}}&{} 0 \\ 0 &{}e^{-\frac{\chi }{2}}\\ \end{array} \right) ; \end{aligned}$$

(13.21)

the parameters $\zeta ,\lambda$ and $\chi$ are related to a, b, c and d by the following relations:

$$\begin{aligned} \cos \frac{\zeta }{2} = \frac{a}{\sqrt{a^2+c^2}}, \ \ \ \sin \frac{\zeta }{2} = \frac{c}{\sqrt{a^2+c^2}}, \ \ \ \lambda = a b-c d, \ \ \ e^{\frac{\chi }{2}} = {\sqrt{a^2+c^2}} \end{aligned}$$

(13.22)

where $0\le \zeta <4 \pi$ and $\lambda$ and $\chi$ are real. Note that a and c cannot be both 0 since $ad +bc = 1$. The above Iwasawa decomposition provides a natural parametrization of the coset space Sp(1, 2) / A

$$\begin{aligned} \tilde{X} (\lambda , \zeta )=k(\zeta )\,n(\lambda )= \left( \begin{array}{cc} \cos \frac{\zeta }{2} &{} i \lambda \cos \frac{\zeta }{2} +i \sin \frac{\zeta }{2} \\ i \sin \frac{\zeta }{2} &{} \cos \frac{\zeta }{2} -\lambda \sin \frac{\zeta }{2} \\ \end{array} \right) \end{aligned}$$

(13.23)

which again is topologically a cylinder. The coset space Sp(1, 2) / A is a symmetric space. The group Sp(1, 2) acts on the coset space by left multiplication:

$$\begin{aligned} g \, \tilde{X} (\lambda , \zeta )\rightarrow \tilde{X} (\lambda ', \zeta ') \end{aligned}$$

(13.24)

Let us describe the action of the subgroups separately. The case of a rotation $k(\alpha ) \in K$ amounts again to a shift of the angle $\zeta$:

$$\begin{aligned} \begin{array}{l} \lambda ' (\alpha ) = \lambda ,\ \ \ \ \zeta '(\alpha ) = \zeta +\alpha . \end{array} \end{aligned}$$

(13.25)

An element $a(\kappa )$ of the abelian subgroup A gives

$$\begin{aligned} \left\{ \begin{array}{l} \lambda '(\kappa ) = \lambda \cosh \kappa +\sinh \kappa (\lambda \cos \zeta +\sin \zeta ), \\ \cot \frac{\zeta '(\kappa )}{2}= e^{\kappa }\cot \frac{\zeta }{2} . \end{array}\right. \end{aligned}$$

(13.26)

An element $n(\mu )\in N$ gives

$$\begin{aligned} \left\{ \begin{array}{l} \lambda ' (\mu )= {\lambda } \left( 1 +\frac{1}{2} {\mu ^2}\right) - \mu \left( \lambda +\frac{\mu }{2} \right) \sin \zeta +{\mu } \left( 1- \frac{1}{2} \lambda \mu \right) \cos \zeta , \\ \cot \frac{\zeta '(\mu )}{2} = \cot \frac{\zeta }{2} - \mu . \end{array}\right. \end{aligned}$$

(13.27)

The Maureer-Cartan form $dg \, g^{-1}$ gives to the symmetric space Sp(1, 2) / A a natural Lorentzian metric that may be constructed as follows (see e.g. [6]). There exists a inner automorphism of Sp(1, 2)

$$\begin{aligned} g\rightarrow \mu (g) = - \gamma ^2 g \gamma ^2 \end{aligned}$$

(13.28)

that leaves invariant the elements of the subgroup A. It may be used to construct a map from the coset space Sp(1, 2) / A into the group Sp(1, 2):

$$\begin{aligned} g({\tilde{X}}) = g\mu (g)^{-1}= -{\tilde{X}} \gamma ^2 {\tilde{X}}^{-1} \gamma ^2. \end{aligned}$$

(13.29)

This map in turn allows to introduce a left invariant Lorentzian metric on the coset space as follows:

$$\begin{aligned} \mathrm {d}s^2 = \frac{1}{2} \text{ Tr } (\mathrm {d}g \, g^{-1})^2= -2 \mathrm {d}\lambda \mathrm {d}\zeta -\left( \lambda ^2+1\right) \mathrm {d}\zeta ^2 \end{aligned}$$

(13.30)

The following properties hold:

• 1. The metric (13.30) is invariant under the transformations (13.25), (13.26) and (13.27).

• 2. The curvature is constant ($R=-2$) and the Ricci tensor is proportional to the metric:

  $$\begin{aligned} R_{\mu \nu } - \frac{1}{2} R g_{\mu \nu } = R_{\mu \nu } + g_{\mu \nu } =0 \end{aligned}$$

  (13.31)

• 3. The map $p: Sp(1,2)/A \rightarrow dS_2$

  $$\begin{aligned} p : \tilde{X} (\lambda , \zeta ) \rightarrow X(\lambda ,\zeta )= \left\{ \begin{array}{l} X^0= -\lambda \\ X^1= \lambda \cos \zeta +\sin \zeta \\ X^2= \cos \zeta -\lambda \sin \zeta \\ \end{array} \right. \end{aligned}$$

  (13.32)

  is a covering map.

In conclusion: the symmetric space ${Sp(1,2)/A = \widetilde{dS_2}}$ may be identified with the double covering of the two-dimensional de Sitter universe. The spin group Sp(1, 2) acts directly on the covering space $\widetilde{dS_2}$ as a group of spacetime transformations:

$$\begin{aligned} {\tilde{X}}\rightarrow g{\tilde{X}}. \end{aligned}$$

(13.33)

We were not able to find the above identification in the (enormous) literature on the group SL(2, R).

13.4 Dirac Fields on the Cylinder and on $dS_2$

Let us consider a Minkowskian cylinder with metric:

$$\begin{aligned} \mathrm {d}s^2 = \mathrm {d}t^2- \mathrm {d}\theta ^2 \end{aligned}$$

(13.34)

(with $-\pi<\theta <\pi$). The Clifford anti-commutation relations $\gamma ^{a}\gamma ^{b}+\gamma ^{b}\gamma ^{a}=2\eta ^{ab},$ with $a,b=0,1,$ may be realised by $\gamma ^{0}$ and $\gamma ^{1}$ (see (13.3)). There are two inequivalent spin structures on the cylinder which correspond to two different monodromies for the spinor fields [7]. Spinors that are periodic:

$$(Ramond) \,\, \psi (t, \theta +2\pi ) = \psi (t, \theta );$$

and anti-periodic spinors:

$$(Neveu-Schwarz) \,\, \psi (t, \theta +2\pi ) = -\psi (t, \theta ). $$

Let us consider for simplicity the massless Dirac equation $\ i \gamma ^a \partial _a \psi = 0\$ on the cylinder. There are at least two canonical quantum fields that solve this equation and that correspond to the above two monodromies (having of course the same anticommutation relations). They are completely characterized by the corresponding Wightman functions that are most simply written by using the light-cone variables $u = t+\theta , \ \ \ v = t-\theta$. In the periodic (Ramond) case we have

$$\begin{aligned} (\varOmega ,\ \psi ^{\text{ R }}(x){\overline{\psi }^{\text{ R }}}(y) \varOmega ) = -\frac{i}{4\pi } \left( \begin{array}{cc} 0 &{} \cot \left( \frac{1}{2} (u-u' )\right) \\ \cot \left( \frac{1}{2} (v-v' )\right) &{} 0 \\ \end{array} \right) . \end{aligned}$$

(13.35)

in the usual matrix form, where $\overline{\psi } = {\psi }^+\gamma ^0$. $\psi ^{ \text{ R }}$ is a univalued distribution on the cylinder.

Neveu-Schwarz Dirac fields are obtained by summing over half-integer momenta. The 2-pt functions is now given by

$$\begin{aligned} (\varOmega ,\ \psi ^{ \text{ NS }}(x){\overline{\psi }^{\text{ NS }}}(y) \varOmega ) = - \frac{ i }{4\pi }\left( \begin{array}{cc} 0 &{}\displaystyle {\frac{1}{\sin \left( \frac{1}{2}({u-u'}) \right) }}\\ \displaystyle {\frac{1}{\sin \left( \frac{1}{2}({v-v'}) \right) }}&{} 0 \\ \end{array} \right) . \end{aligned}$$

(13.36)

Here $\psi ^{ \text{ NS }}$ is a bivalued distribution on the cylinder, i.e. is a distribution on the double covering of the cylinder.

Let us now consider the following coordinate system for $dS_{2}$

$$\begin{aligned} X(t,\theta ) = \left\{ \begin{array}{l} X^0= \tan t ,\\ X^1 = \sin \theta /\cos t, \\ X^2 = \cos \theta /\cos t. \end{array}\right. \ \end{aligned}$$

(13.37)

Here the de Sitter metric is conformal to the metric of the Minkowskian cylinder:

$$\begin{aligned}&{\mathrm {d}}s^2 = \left. \left( {{\mathrm {d}}X^0}^2-{{\mathrm {d}}X^1}^2-{{\mathrm {d}}X^2}^2\right) \right| _{{\mathrm {d}}S_2}=\frac{1}{\cos ^2 t} ({\mathrm {d}}t^2- {\mathrm {d}}\theta ^2), \end{aligned}$$

(13.38)

and the interval of the ambient spacetime restricted to the de Sitter manifold has the following expression:

$$\begin{aligned}&(X-Y)^{2}= \frac{2\cos (\theta -\theta ' )-2\cos (t-t')}{ \cos {t}\cos t' } . \end{aligned}$$

(13.39)

The curved-space matrices $\alpha ^i = e^i_a \gamma ^a$ in the above coordinates are simply proportional to the flat space gamma matrices $\gamma ^0$ and $\gamma ^1$:

$$\begin{aligned} \alpha ^t = \left( \begin{array}{cc} 0 &{} \cos t\\ \cos t &{} 0 \\ \end{array} \right) = (\cos t) \,\gamma ^0, \ \ \ \ \ \alpha ^\theta = \left( \begin{array}{cc} 0 &{} \cos t\\ -\cos t &{} 0 \\ \end{array} \right) = (\cos t) \,\gamma ^1, \end{aligned}$$

(13.40)

where the components of the natural zweibein are $e^t_0 = \cos t$, $e^t_1 = 0$, $e^\theta _0 = 0$ and $e^\theta _1=\cos t$. In two-dimensions there is only one non-vanishing component of the spin connection $\omega _{\mu \, \!a b} = e_a^\nu \, \nabla _\mu e_{b\nu }.$ namely $\omega _{\theta 01} = -\omega _{\theta 10} = \tan t.$ Correspondingly

$$\begin{aligned} \varGamma _t=0, \ \ \ \varGamma _{\theta } = \frac{1}{4} [\gamma ^0,\gamma ^1] \omega _{\theta 01} = \tan {t}\left( \begin{array}{cc} -\frac{1}{2} &{} 0 \\ 0 &{} \frac{1}{2} \\ \end{array} \right) . \end{aligned}$$

(13.41)

Putting everything together, in the above coordinates the Fock-Ivanenko-Dirac equation [8] on the de Sitter manifold is finally written as follows:

$$\begin{aligned} i \alpha ^i (\partial _i + \varGamma _i) \phi -m\phi = i \cos t (\gamma ^a \partial _a \phi + \frac{i}{2} \tan t\, \gamma ^0 \phi ) -m\phi = 0. \end{aligned}$$

(13.42)

13.5 Another Equation by Dirac

In this section we elaborate on another first order equation for a spinor on the de Sitter manifold which is due to Dirac himself [9]. The generators $L_{\alpha \beta }$ of the Lorentz group $SO_0(1,2)$ are given by

$$\begin{aligned} L_{\alpha \beta }=M_{\alpha \beta }+S_{\alpha \beta }, \end{aligned}$$

(13.43)

where $M_{\alpha \beta }=-i(X_{\alpha }\partial _{\beta }-X_{\beta }\partial _{\alpha })$ and $S_{\alpha \beta }=-{i\over 4}[\gamma _{\alpha },\gamma _{\beta }]$ are respectively the ‘orbital’ and the ‘spinorial’ parts of $L_{\alpha \beta }$. In the case under study ($s= 1/2$) the first Casimir operator takes the form

$$\begin{aligned} Q_{1} = - {1\over 2} L^{\alpha \beta }L_{\alpha \beta } ={ \left( {1\over 2}\gamma _{\alpha }\gamma _{\beta }M^{\alpha \beta }\right) }^{2}+ i\gamma _{\alpha }\gamma _{\beta }M^{\alpha \beta }-{{3}\over {4}} \end{aligned}$$

(13.44)

and its eigenvalues ${1\over 4}+\nu ^{2}$ are parametrized by a nonzero real number $\nu$ as described in Bargmann’s classic paper [10]. Since

$$\begin{aligned} Q_{1} -\frac{1}{4} ={ \left( {1\over 2}\gamma _{\alpha }\gamma _{\beta }M^{\alpha \beta }+i\right) }^{2} \end{aligned}$$

(13.45)

following Dirac [9] we may introduce another first order spinorial equation:

$$\begin{aligned} \left( iD+i+{ \nu } \right) \varPsi = 0 \end{aligned}$$

(13.46)

where we have set

$$\begin{aligned} iD = {1\over 2} \gamma _{\alpha }\gamma _{\beta }M^{\alpha \beta } . \end{aligned}$$

(13.47)

To solve (13.46) we observe that the two-dimensional de Sitter - d’Alembert’s operator can be factorized as follows:

$$ \square = -i D(i D+i). $$

This relation implies that given a solution of the scalar Klein-Gordon equation

$$\begin{aligned} (\square +{ \nu }^{2}+i{ \nu })\varPsi = 0 \end{aligned}$$

(13.48)

by simply applying to it the operator $\left( -i{D}+{\nu } \right)$ to it we may construct a solution of (13.46) [11].

Given a nonzero light-like vector $\xi \in {M}_3$ and a complex number $\lambda \in \mathbf{C}$ we consider the homogeneous function [12, 13]:

$$\begin{aligned} (X\cdot \xi )_\pm ^\lambda = \lim _{Z\in \mathcal{T}_\pm , \, Z \rightarrow X} (Z\cdot \xi )^\lambda \ , \end{aligned}$$

(13.49)

where Z belongs to the tuboids ${\mathcal{T}}_\pm$ obtained by intersecting the ambient spacetime tubes $T_\pm = {M}_3 \pm i V^+$ ($V^+$ is the future cone of the origin) with the complex de Sitter manifold $dS_2^c$. The above boundary values are solutions of the de Sitter Klein-Gordon equation

$$\begin{aligned} \left( \Box +m_\lambda ^2\right) (X\cdot \xi )_\pm ^\lambda =0, \ \ \ \ \ \ \ \ \left( \Box +m_\lambda ^2\right) ( X\cdot \xi )_\pm ^{-1-\lambda } =0. \end{aligned}$$

(13.50)

The parameter $\lambda$ is here unrestricted, i.e. we consider a complex squared masses $m_\lambda ^2= -\lambda (\lambda +1)$. Spinorial plane waves can therefore be written in terms of scalar plane waves as follows [11,12,13,14]:

$$\begin{aligned} \left( -i{D}+{\nu } \right) (X\cdot \xi )_\pm ^\lambda { \upsilon }(\xi )\, \end{aligned}$$

(13.51)

where $\upsilon = \upsilon (\xi )$ is a two-component spinor and the complex number $\lambda$ may take either of the following two values:

$$\begin{aligned} {\lambda }_{1}=-i{ \nu },\ \ \ \ \ \ {\lambda }_{2}=-1+i{ \nu }. \end{aligned}$$

(13.52)

A straightforward calculation gives that

(13.53)

where we set . The spinor $w(\xi )$ satisfies the condition . In general, let us consider the linear equation

(13.54)

For any given lightlike vector $\xi$ the unique solution (apart from normalisation) of the above equation is ([15]—Chap. 3)

$$\begin{aligned} u(\xi )= \frac{1}{\sqrt{2(\xi ^0-\xi ^1)}} \begin{pmatrix}{\xi ^0}-{\xi ^1}\\ i{\xi _2}\end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{{\xi ^0}-{\xi ^1}} \\ i\sqrt{{\xi ^0}+{\xi ^1}}\end{pmatrix} \end{aligned}$$

(13.55)

Defining the adjoint spinor in the usual way ${\overline{u}}(\xi )=u^{\dagger }(\xi )\gamma ^{0}$ there follows the completeness relation

(13.56)

We obtain the following (complete, unnormalised) set of spinorial plane wave solutions, labeled by the wave vector $\xi$ and the mass parameter $\nu$:

$$\begin{aligned} (X\cdot \xi )_\pm ^{-1+i{ \nu }}u(\xi ). \end{aligned}$$

(13.57)

where u is the spinor given in (13.55). It is now possible to build the two-point function of the quantum Dirac field $\varPsi _\nu$ by superposing spinorial plane waves having the correct analyticity properties; these analyticity properties are there to replace of a true global spectral condition which is absent in de Sitter Quantum Field Theory [11,12,13,14]:

(13.58)

and the canonical anticommutation relations fix the value of the constant:

$$\begin{aligned} c_{\nu }=\frac{\varGamma (1-i{\nu })\varGamma (1+i{ \nu })}{(2\pi )^{2}e^{\pi {\nu }}}. \end{aligned}$$

(13.59)

Since we get (in the complex analyticity domain $\mathcal{T}_-\times \mathcal{T}_+$)

(13.60)

The integral at the RHS is a hypergeometric function of the invariant scalar product $\zeta = {Z_1}\cdot {Z_2}$ and all the spinorial content is carried by the operator acting on it:

(13.61)

Taking the massless limit $\nu \rightarrow 0$ we get

(13.62)

As a consequence of the above chain of identities we obtain a nice integral representation of the two-point function of the massless field $\varPsi _0$:

(13.63)

valid for $Z_1\in \mathcal{T^-}$ and $Z_2\in \mathcal{T^+}$. The transformation law for spinors under the action of Sp(1, 2) is the standard one:

$$\begin{aligned} \varPsi ^{\prime }(X)=g\varPsi \bigr (\varLambda ^{-1}(g)X\bigr ), \ \ \ \ \ {\overline{\varPsi }}^{\prime }(X)={\overline{\varPsi }}\bigr (\varLambda ^{-1}(g)z\bigr )g^{-1}. \end{aligned}$$

(13.64)

The covariance of the two-point function (13.58) under the transformation (13.64) can now be easily shown. In the massless case this is obvious:

(13.65)

In the general case, for any $g\in Sp(1,2)$ there holds the following chain of equalities:

(13.66)

The last step is a consequence of the Stokes theorem exactly as in the scalar case [13].

13.6 Cocyclic Covariance of the de Sitter Dirac-Fock-Ivanenko Field

Since there are two apparently distinct Dirac’s equations on the de Sitter manifold, namely (13.42) and (13.46), it is natural to ask whether there is a relation between them. In four dimensions, this question has been raised first by Gürsey and Lee [16] and they provided a way to build a bridge between the two equations. The two-dimensional case is trickier (also more interesting) because of its topological peculiarities [4].

Given a solution $\varPsi$ of the Dirac (13.46) the dressed spinor

(13.67)

solves the equation

$$\begin{aligned} i \alpha ^t \left( \partial _t +\varGamma _t \right) \phi + i \alpha ^\theta \left( \partial _\theta +\varGamma _\theta \right) \phi - i \alpha ^i (\partial _i \ln f) \phi - \nu \phi = 0 \end{aligned}$$

(13.68)

where $S(t,\theta )$ is the following particular element of the spin group Sp(1, 2):

$$\begin{aligned} S(t,\theta )=\frac{1}{\sqrt{\cos t}} \left( \begin{array}{cc} {\cos \frac{t-\theta }{2}} &{} {i \sin \frac{t-\theta }{2}}\\ -{i \sin \frac{t+\theta }{2}} &{} { \cos \frac{t+\theta }{2}} \\ \end{array} \right) , \end{aligned}$$

(13.69)

and $f(t,\theta )$ is an arbitrary function on the double-covering of the de Sitter spacetime $\widetilde{dS_2}$, Clearly, the arbitrary function $f(t,\theta )$ can be reabsorbed by a gauge transformation. We may therefore set¹ $f=1$ in (13.67) and observe the coincidence of (13.68) with the covariant Dirac (Fock-Ivanenko) (13.42).

The matrix S is anti-periodic $S(t,\theta +2\pi ) = -S(t,\theta )$ and therefore well-defined only on the double covering of the de Sitter hyperboloid. The map $(t,\theta )\rightarrow S(t,\theta )$ is thus a map from the double covering of the de Sitter spacetime $\widetilde{dS_2}$ with values in the spin group Sp(1, 2) (see (13.13)); when acting on a spinor field it changes its periodicity: periodic (R) fields become anti-periodic (NS) and viceversa.

The group element $S(t,\theta ) =S(\tilde{X}(t, \theta ))= S(\tilde{X})$ has a very simple geometrical interpretation that is made clear by examining the Lorentz transformation associated to it through the projection (13.19):

$$\begin{aligned} \varLambda (S(t,\theta )) = \left( \begin{array}{ccc} \sec t &{} -\sin \theta \tan t &{} -\cos \theta \tan t \\ 0 &{} \cos \theta &{} -\sin \theta \\ -\tan t &{} \sec t \sin \theta &{} \cos \theta \sec t \\ \end{array} \right) \end{aligned}$$

(13.70)

so that

$$\begin{aligned} \varLambda (S(t,\theta )) X(t,\theta ) =X(0,0)=\left( \begin{array}{c} 0 \\ 0 \\ 1\\ \end{array} \right) \end{aligned}$$

(13.71)

All the above features are not present in the original construction by Gürsey and Lee which was relative to the four-dimensional case.

Let us apply the map (13.67) to the field $\varPsi _\nu$ defined in (13.58) and get a quantum field $\phi _\nu$ solving the standard Dirac (Fock-Ivanenko) equation. The field $\phi _\nu$ has the NS antiperiodicity and therefore is well-defined only on the manifold $\widetilde{dS_2}$. Equation (13.64) tells us how the field $\phi _\nu$ is transformed by the action of the de Sitter group:

$$\begin{aligned} \phi '({\tilde{X}})= & {} \varSigma (g,{\tilde{X}}) \, \phi (g^{-1} {\tilde{X}}), \end{aligned}$$

(13.72)

where the matrix

$$\begin{aligned} \varSigma (g,{\tilde{X}}) = S({\tilde{X}}) \, g\, S({g^{-1}{\tilde{X}}})^{-1} \end{aligned}$$

(13.73)

is also an element of the spin group Sp(1, 2) depending on the point ${\tilde{X}}\in \widetilde{dS_2}$ and the group element g; one immediately verifies that $\varSigma (g,{\tilde{X}})$ is a nontrivial cocyle of Sp(1, 2):

$$\begin{aligned} \varSigma (g_1,{\tilde{X}}) \varSigma (g_2,g_1^{-1}{\tilde{X}})= \varSigma (g_1g_2,{\tilde{X}}). \end{aligned}$$

(13.74)

The de Sitter covariance of the de Sitter Dirac NS field $\phi _\nu$ is thus expressed by (13.72). On the other hand there is no covariant Dirac field (in the above sense) in the Ramond sector. The following remarkable result play an important technical role in the construction of the de Sitter-Thirring model [5]. For any g in the spin group Sp(1, 2) the cocycle $\varSigma (g,{\tilde{X}})$ is diagonal.

13.7 Massless Fields: From the de Sitter Manifold to the Cylinder and Back

In this concluding section we examine the various incarnations of the massless Dirac field. In the massless case the dressing is simpler:

$$\begin{aligned}&\psi (t,\theta ) = \frac{1}{ \sqrt{\cos t}} S(t,\theta ) \varPsi _0(t,\theta ) \end{aligned}$$

(13.75)

The LHS has to be understood as a Dirac field on the double covering of the cylinder obtained from the massless field (13.63). Computing the two-point function we get

(13.76)

The two-point function completely characterizes the field: the remarkable result is that by the above construction the covariant massless de Sitter-Dirac field (13.62) is precisely mapped into the Neveu-Schwarz-Dirac field on the cylinder

$$\begin{aligned} \psi ^{ \text{ NS }}(t,\theta )= \frac{1}{ \sqrt{\cos t}} S(t,\theta ) \varPsi _0(t,\theta ) \end{aligned}$$

(13.77)

and viceversa

$$\begin{aligned}&\varPsi _0(t,\theta )= { \sqrt{\cos t}} \, S(t,\theta )^{-1} \psi ^{ \text{ NS }}(t,\theta ). \end{aligned}$$

(13.78)

It is also instructive to apply the inverse transform (13.78) to the two components of the field separately:

$$\begin{aligned}&\varPsi _{0,r(l)}(t,\theta )= { \sqrt{\cos t}} \, S(t,\theta )^{-1} \psi ^{ \text{ NS }}_{1(2)}(t,\theta ). \end{aligned}$$

(13.79)

where the index r refers to the right moving part and l to the left moving parts(i.e. the parts depending only on the u and v variables). We get in this way a splitting of the massless Dirac field $\varPsi _{0} = \varPsi _{0,r}+\varPsi _{0,l}$ into its right and left moving parts:

(13.80)

where

(13.81)

(13.82)

These expressions are useful in computing the image of the Ramond field under the same transformation:

$$\begin{aligned} \varPsi ^{\text{ R }}_0(t,\theta )= { \sqrt{\cos t}} \, S(t,\theta )^{-1} \psi ^{\text{ R }}(t,\theta ). \end{aligned}$$

(13.83)

The field $\varPsi ^{\text{ R }}_0(t,\theta )$ is defined on the double covering of the de Sitter spacetime and solves the massless Dirac (13.46) there. Its two-point function is written in the simplest way as follows in terms of the left and right part of the covariant two-point function as follows:

$$\begin{aligned} (\varOmega , \varPsi ^{ \text{ R }}_0(t,\theta ) \overline{\psi }^{ \text{ R }}_0(t',\theta ') \varOmega ) = \cos \left( \frac{1}{2} (u-u')\right) A(u,u')+ \cos \left( \frac{1}{2} (v-v')\right) B(v,v') \\\end{aligned}$$

(13.84)

On the other hand, expressing the above two-point function using the ambient space variables gives a very complicated expression, not particularly useful. The representation (13.84) allows to prove that the $\varPsi ^{ \text{ R }}_0(t,\theta )$ is covariant under rotations. On the other hand, the boosts are broken.

13.8 Conclusion

In conclusion we have displayed some interesting structures that emerge when spinor fields are considered on manifolds having the topology of a two-dimensional cylinder. In particular there is a natural map that transforms massless Neveu-Schwarz quantum Dirac fields on the Minkowskian cylinder into de Sitter covariant Dirac fields. This is a non trivial fact. One could say that Dirac NS massless fields have a hidden de Sitter symmetry. The result presented in this note may be useful in considering soluble models of two-dimensional QFT on the de Sitter universe like for instance the de Sitter-Thirring model [17].