





\chapter{Properties of Hankel functions}\label{app:hankel}
This appendix is intended to provide a reference for the properties of the 
Hankel functions required for the development of this paper.
\begin{section}{Differential equation}
In this section we present the construction of the Hankel functions as well
as some important relations between them. The Hankel functions are solutions 
to the Bessel equation \cite{bookpart:smirnov_bess}:
\begin{equation}
  \left(z^2 \frac{d^2}{dz^2} + z \frac{d}{dz} + (z^2 - \nu^2)\right) 
  Z_p(z) = 0. \label{eq:bessel_equation}
\end{equation}
The series solutions to the above equation are the Bessel functions of order
$\pm \nu$:
\begin{equation}
  J_{\pm \nu}(z) = \left(\frac{z}{2}\right)^{\pm\nu}
  \sum_{k = 0}^{\infty} \frac{(-1)^k}{k! \Gamma{k \pm \nu + 1}}
  \left(\frac{z}{2}\right)^{2k}. \label{eq:bessel_function}
\end{equation}	
The complex conjugate passes onto the order and the argument:
\begin{equation}
	J_{\nu}(z)^* = J_{\nu^*}(z^*). \label{eq:bessel_hc}
\end{equation}
The wronskian \cite{bookpart:watson_wronsk} of the Bessel functions of opposite 
order is:
\begin{equation}
  W(J_{-\nu}, J_{\nu}) = \frac{\mathcal{C}}{z} = \frac{2 \sin{\nu\pi}}{\pi z}.
  \label{eq:bessel_wronskian}
\end{equation} 
The two solutions of order $\nu$ and $-\nu$ are linearly independent when 
$\nu$ is not an integer (because the wronskian is non-zero). When 
$\nu = n$ is an integer, the two functions are linearly dependent, since
they are related by
\begin{equation}
	J_{-n}(z) = (-1)^n J_{n}(z). \label{eq:bessel_minusn}
\end{equation}
Bessel functions of the second kind, or Neumann functions, can be 
constructed using $J_{\pm \nu}$:
\begin{equation}
	N_{\nu}(z) = \frac{J_\nu(z) \cos{\nu\pi} - J_{-\nu}(z)}{\sin{\nu\pi}}.
	\label{eq:neumann_function}
\end{equation}
Using \eqref{eq:bessel_hc}, we see that the complex conjugate passes onto the 
order and the argument:
\begin{equation}
	N_{\nu}(z)^* = N_{\nu^*}(z^*). \label{eq:neumann_hc}
\end{equation}
The wronskian of Neumann functions of opposite order follows:
\begin{equation}
  W(N_{-\nu}, N_{\nu}) = \frac{-2 \cos^2{\pi\nu} + 2}{\pi z \sin{\pi\nu}}
  = \frac{2 \sin{\nu\pi}}{\pi z}. \label{eq:neumann_wronskian}
\end{equation}
The wronskian with the bessel function of the same order is
\begin{equation}	
	W(J_{\nu}, N_{\nu}) = W(J_{-\nu}, J_{\nu})/\sin{\nu\pi} = \frac{2}{\pi z}.
  \label{eq:neumann_bessel_wronskian}
\end{equation}
Therefore, the two functions are always linearly independent.

The wronskian with the bessel function of opposite order is
\begin{equation}
	W(J_{-\nu}, N_{\nu}) = W(J_{-\nu}, J_{\nu}) \cos{\nu\pi}/\sin{\nu\pi} = 
	\frac{2 \cos{\nu\pi}}{\pi z}. \label{eq:neumann_besseln_wronskian}
\end{equation}	
Bessel functions of the third kind \cite{book:jackson}, or Hankel functions 
are constructed with these two linear independent solutions:
\begin{subequations} \label{eq:hankel_functions}
\begin{align}
  \Hank{1}{\nu}(z) &= J_\nu(z) + i N_\nu(z) & 
  &\textrm{Hankel function of the first kind}\label{eq:hankel1_def}\\
  \Hank{2}{\nu}(z) &= J_\nu(z) - i N_\nu(z) &
  &\textrm{Hankel function of the second kind}\label{eq:hankel2_def}
\end{align}
\end{subequations}
	The wronskian of the two kinds of Hankel functions is
\begin{equation}
	W(\Hank{1}{\nu}, \Hank{2}{\nu}) = -2i W(J_{\nu}, N_{\nu}) 
	= -\frac{4i}{\pi z}.	\label{eq:hankel12_wronskian}
\end{equation} 
Complex conjugation passes to the order and argument and switches between
the two kinds of Hankel functions:
\begin{equation}
  \Hank{1}{\nu}(z)^{*} = \Hank{2}{\nu^*}(z^*),\qquad
  \Hank{2}{\nu}(z)^{*} = \Hank{1}{\nu^*}(z^*).
  \label{eq:hankel_hc}
\end{equation}
Replacing the Neumann function \eqref{eq:neumann_function} in 
\eqref{eq:hankel_functions} we obtain an alternative form for
the Hankel functions:
\begin{equation*}
  \Hank{1}{\nu}(z) = \frac{i}{\sin{\nu\pi}}\left(J_\nu(z) e^{-i \nu\pi} - 
  J_{-\nu}(z)\right) ,\qquad
  \Hank{2}{\nu}(z) = \frac{-i}{\sin{\nu\pi}}\left(J_\nu(z) e^{i \nu\pi} - 
  J_{-\nu}(z)\right).
\end{equation*}
From this form we can read the connection between $\Hank{1,2}{-\nu}$ 
and $\Hank{1,2}{\nu}$:	
\begin{equation}
  \Hank{1}{-\nu}(z) = e^{i\nu\pi} \Hank{1}{\nu}(z),\qquad
  \Hank{2}{-\nu}(z) = e^{-i\nu\pi} \Hank{2}{\nu}(z).
  \label{eq:hank_minusnu}
\end{equation}	
Denoting by $\Omega_{\nu}$ any of the Bessel functions introduced so far,
the following relations stand:
\begin{subequations}
\begin{gather}
	\Omega_{\nu - 1}(z) + \Omega_{\nu + 1}(z) = \frac{2\nu}{z} \Omega_{\nu}(z)
	\label{eq:bessel_rec_nu},\\
	\Omega_{\nu - 1}(z) - \Omega_{\nu + 1}(z) = 2 \Omega^\prime_{\nu}(z)		
	\label{eq:bessel_rec_nuprime},\\
	\Omega^\prime_{\nu}(z) = \Omega_{\nu - 1}(z) - \frac{\nu}{z} \Omega_{\nu}(z)
	\label{eq:bessel_rec_nuprime_minus},\\
	\Omega^\prime_{\nu}(z) = -\Omega_{\nu + 1}(z) + \frac{\nu}{z} \Omega_{\nu}(z)		
	\label{eq:bessel_rec_nuprime_plus}.
\end{gather}
\end{subequations}
A common example of Hankel functions is that of order $\nu = 1/2$:
\begin{subequations}\label{eq:hankel_1_2}
\begin{align}
	\Hank{1}{1/2} &= -i \sqrt{\frac{2}{\pi z}} e^{i z}\label{eq:hankel1_1_2},\\
	\Hank{2}{1/2} &= \phantom{-}i \sqrt{\frac{2}{\pi z}} e^{-i z}.
	\label{eq:hankel2_1_2}
\end{align}
\end{subequations}
\end{section}
\begin{section}{Asymptotic forms}
In this section we give the asymptotic forms for small and large values of the 
argument, corresponding to the Hankel functions and their derivatives.

For small values of the argument we have \cite{bookpart:zwillinger_bessel}:
\begin{subequations}\label{eq:bessel_all_as_smallz}
\begin{align}







	\Hank{1}{\nu}(z) &=
	\phantom{-}\left(\frac{2}{z}\right)^{\nu} \frac{\Gamma(\nu)}{i\pi} + 		
	\left(\frac{z}{2}\right)^{\nu} \frac{1+i\cot(\pi\nu)}{\Gamma(1+\nu)}+
	\mathcal{O}(z^{2 \pm \nu}), 
	\label{eq:hankel1_as_smallz}\\		
	\Hank{2}{\nu}(z) &=
	-\left(\frac{2}{z}\right)^{\nu} \frac{\Gamma(\nu)}{i\pi} + 
	\left(\frac{z}{2}\right)^{\nu} \frac{1-i\cot(\pi\nu)}{\Gamma(1+\nu)} +
	\mathcal{O}(z^{2\pm \nu}).
	\label{eq:hankel2_as_smallz}
\end{align}
\end{subequations}
The derivatives of the Hankel functions can be approximated by:
\begin{subequations}\label{eq:hankelp_as_smallz}
\begin{align}
	\Hank{1}{\nu}(z)^{\prime} &=
	- \frac{\nu}{z} \left(\phantom{-}
	\left(\frac{2}{z}\right)^{\nu} \frac{\Gamma(\nu)}{i\pi} -
	\left(\frac{z}{2}\right)^{\nu} \frac{1+i\cot(\pi\nu)}{\Gamma(1+\nu)}
	\right) + \mathcal{O}(z^{2\pm \nu}), \label{eq:hankel1prime_as_smallz}\\		
	\Hank{2}{\nu}(z)^{\prime} &=
	- \frac{\nu}{z} \left(
	-\left(\frac{2}{z}\right)^{\nu} \frac{\Gamma(\nu)}{i\pi} - 
	\left(\frac{z}{2}\right)^{\nu} \frac{1-i\cot(\pi\nu)}{\Gamma(1+\nu)}
	\right) + \mathcal{O}(z^{2\pm \nu}). \label{eq:hankel2prime_as_smallz}	
\end{align}
\end{subequations}
For large values of the argument we can use \cite{book:abramowitzstegun}:
\begin{subequations}\label{eq:bessel_all_as_largez}
\begin{align}
	\Hank{1}{\nu}(z) &=
	\sqrt{\frac{2}{\pi z}} e^{i(z - \nu\pi/2 - \pi/4)}
	\left(1 + \mathcal{O}(z^{-1})\right),
	\label{eq:hankel1_as_largez}\\			
	\Hank{2}{\nu}(z) &=
	\sqrt{\frac{2}{\pi z}} e^{-i(z - \nu\pi/2 - \pi/4)}
	\left(1 + \mathcal{O}(z^{-1})\right).
	\label{eq:hankel2_as_largez}
\end{align}	
\end{subequations}
The derivatives of the Hankel functions are simply
\begin{subequations}\label{eq:hankelprime_as_largez}
\begin{align}
	\Hank{1}{\nu}(z)^{\prime} &=
	i \sqrt{\frac{2}{\pi z}} e^{i(z - \nu\pi/2 - \pi/4)}
	\left(1 + \mathcal{O}(z^{-1})\right),
	\label{eq:hankel1prime_as_largez}\\		
	\Hank{2}{\nu}(z)^{\prime} &=
	-i \sqrt{\frac{2}{\pi z}} e^{i(z - \nu\pi/2 - \pi/4)}
	\left(1 + \mathcal{O}(z^{-1})\right).
	\label{eq:hankel2prime_as_largez}
\end{align}
\end{subequations}
Hankel's expansion is given by:
\begin{subequations}\label{eq:bessel_all_as_hankel_largez}
\begin{align}
	\Hank{1}{\nu}(z) &=
	\sqrt{\frac{2}{\pi z}}(P(\nu, z) + i Q(\nu, z)) e^{i\chi},
	\label{eq:hankel1_as_hankel}\\
	\Hank{2}{\nu}(z) &=
	\sqrt{\frac{2}{\pi z}}(P(\nu, z) - i Q(\nu, z)) e^{-i \chi},
	\label{eq:hankel2_as_hankel}
\end{align}		
with
\begin{align}
	P(\nu, z) &\xrightarrow[z\rightarrow \infty]{} 
	\sum_{k = 0}^{\infty} (-1)^k \frac{(\nu, 2k)}{(2z)^{2k}}\nonumber\\
	&= 1 - \frac{(\mu - 1)(\mu - 9)}{2!(8z)^2} + 
	\frac{(\mu - 1)(\mu - 9)(\mu - 25)(\mu - 49)}{4!(8z)^4} 
	+ \mathcal{O}(z^{-6}),\label{eq:P_hankel}\\
	Q(\nu, z) &\xrightarrow[z\rightarrow \infty]{} 
	\sum_{k = 0}^{\infty} (-1)^k \frac{(\nu, 2k + 1)}{(2z)^{2k + 1}}\nonumber\\
	&= \frac{\mu - 1}{8z} - 
	\frac{(\mu - 1)(\mu - 9)(\mu - 25)}{3!(8z)^3} +
	+ \mathcal{O}(z^{-5}),\label{eq:Q_hankel}\\
	(\alpha, n) &=
	- \frac{(n - 1/2)^2 - \alpha^2}{n} (\alpha, n - 1) ,\qquad
	(\alpha, 0) = 1,\nonumber\\
	\chi &= z - \frac{\pi\nu}{2} - \frac{\pi}{4},\nonumber\\
	\mu &= 4\nu^2. \nonumber
\end{align}
\end{subequations}	
Hankel's expansion for the derivatives is:
\begin{subequations}\label{eq:besselp_all_as_hankel_largez}
\begin{align}
	\Hank{1}{\nu}^\prime(z) &=
	i \sqrt{\frac{2}{\pi z}}(R(\nu, z) + i S(\nu, z)) e^{i\chi},
	\label{eq:hankel1p_as_hankel}\\
	\Hank{2}{\nu}^\prime(z) &=
	-i \sqrt{\frac{2}{\pi z}}(R(\nu, z) - i S(\nu, z)) e^{-i \chi},
	\label{eq:hankel2p_as_hankel}
\end{align}		
with
\begin{align}
	R(\nu, z) &\xrightarrow[z\rightarrow \infty]{} 
	\sum_{k = 0}^{\infty} (-1)^k 
	\frac{4\nu^2+16k^2-1}{4\nu^2-(4k-1)^2} \frac{(\nu, 2k)}{(2z)^{2k}}\nonumber\\
	&= 1 - \frac{(\mu - 1)(\mu + 15)}{2!(8z)^2} + \mathcal{O}(z^{-4}),
	\label{eq:R_hankel}\\
	S(\nu, z) &\xrightarrow[z\rightarrow \infty]{} 
	\sum_{k = 0}^{\infty} (-1)^k 
	\frac{4\nu^2+4(2k+1)^2-1}{4\nu^2-(4k+1)^2} \frac{(\nu, 2k + 1)}{(2z)^{2k + 1}}\nonumber\\
	&= \frac{\mu + 3}{8z} - 
	\frac{(\mu - 1)(\mu - 9)(\mu + 35)}{3!(8z)^3} + \mathcal{O}(z^{-5}).
	\label{eq:S_hankel}
\end{align}
\end{subequations}
\end{section}
\begin{section}{Hankel functions for the de Sitter scalar field}\label{sec:hank_kg}
Here we give some useful insight on the Hankel functions employed in the 
development of the scalar theory on de Sitter space.

The solution to the scalar field equation \eqref{eq:kg_Z_def} is chosen
such that in the conformally coupled massless case ($\mu = 1/4$) it 
approaches the Minkowski plane-wave solution. Indeed, the Hankel function
of order $\nu = 1/2$, given by \eqref{eq:hankel1_1_2}, has the exponential
form we are looking for:
\begin{equation*}
	\varphi_p(\eta) \sim_{\substack{m = 0\\\xi = 1/6}} e^{-i p \eta}.
\end{equation*}
The complex conjugate of $Z_\nu(z)$ is
\begin{equation*}
	Z^{*}_\nu(z) = \begin{cases}
	\Hank{2}{\nu}(z) & \mu > 0,\\
	e^{-\frac{\pi\nu}{2}} \Hank{2}{-i \nu}(z) & \mu < 0.
\end{cases}
\end{equation*}
The imaginary argument flipped sign. We can use \eqref{eq:hank_minusnu} to 
get the sign back:
\begin{equation*}
	Z^{*}_\nu(z) = \begin{cases}
		\Hank{2}{\nu}(z) & \mu > 0,\\
		e^{\frac{\pi\nu}{2}} \Hank{2}{i \nu}(z) & \mu < 0.
	\end{cases}
\end{equation*}
which is just what \eqref{eq:kg_Z_def} states. It is important to note
that the Wronskian of $Z_{\nu}$ and $Z^*_\nu$ is the same in both branches, 
irrespective of the value of $M^2$, and is given by 
\eqref{eq:hankel12_wronskian}:
\begin{equation}
	W(Z_\nu(z), Z^*_\nu(z)) = - \frac{4 i}{\pi z}.
	\label{eq:kg_wronski}
\end{equation}
Although the normalization factor is dependent on the value of $\mu$,
it cancels in products of the form $Z_\nu Z^*_\nu$.
\end{section}
\begin{section}{Hankel functions for the de Sitter spinorial field}\label{sec:hank_dirac}
In this section we briefly discuss the Hankel functions used for the 
construction of the solutions to the Dirac equation on de Sitter space 
\eqref{eq:dirac_fieldeq}, which are written in terms of Hankel functions
$\Hank{1/2}{\nu_\pm}(z)$, with $\nu_{\pm} = \nicefrac{1}{2} \pm k$ and 
$z = -p\eta$. In the massless limit, the Dirac equation field is conformal 
to the Minkowski case and the solutions \eqref{eq:dirac_Usol_frw} reduce to
\begin{equation}
	U_{{\Vec{p}}, \lambda}(\eta,\Vec{x}) \xrightarrow[m\rightarrow 0]{} \frac{1}{(2\pi)^{3/2} \sqrt{2}} 
	(-\omega\eta)^{3/2} 
	\begin{pmatrix}
		\xi_{\lambda}({\Vec{p}})\\2\lambda \xi_{\lambda}({\Vec{p}})
	\end{pmatrix} 
	e^{-i p \eta + i {\Vec{p}} \Vec{x}}.
	\label{eq:dirac_Usol_m0}
\end{equation}
Up to a factor $(-\omega \eta)^{3/2}$, this is the polarized plane wave 
solution on the Minkowski space.

Under complex conjugation, the Hankel functions change order and kind:
\begin{equation}
	\Hank{1}{\nu_\pm}^*(z) = \Hank{2}{\nu_\mp}(z).
	\label{eq:hank_nupms}
\end{equation}
Using \eqref{eq:bessel_rec_nuprime_minus}, \eqref{eq:hank_minusnu} and 
$-\nu_\pm = 1 - \nu_\mp$ we find:
\begin{subequations}\label{hank_nupm_prime}
\begin{align}
	\Hank{1}{\nu_\pm}^\prime(z) &= \phantom{-}
	i e^{\pm\pi k} \Hank{1}{\nu_\mp}(z)-\frac{\nu_{\pm}}{z}\Hank{1}{\nu_\pm}(z),
	\label{eq:hank1_nupm_prime}\\
	\Hank{2}{\nu_\pm}^\prime(z) &= 
	-i e^{\mp\pi k}\Hank{2}{\nu_\mp}(z)-\frac{\nu_{\pm}}{z}\Hank{2}{\nu_\pm}(z).
	\label{eq:hank2_nupm_prime}
\end{align}
\end{subequations}
Replacing this in the wronskian \eqref{eq:hankel12_wronskian} we find
\begin{equation*}
	iW(\Hank{1}{\nu_-}, \Hank{2}{\nu_-}) = 
	e^{\pi k} \Hank{2}{\nu_+}(z) \Hank{1}{\nu_-}(z) + 
	e^{- \pi k} \Hank{2}{\nu_-}(z) \Hank{1}{\nu_+}(z),
\end{equation*}
and we arrive at the identity \cite{art:cota_polarized_fermions}:
\begin{equation}
	e^{\pi k} \Hank{2}{\nu_+}(z) \Hank{1}{\nu_-}(z) + 
	e^{- \pi k} \Hank{2}{\nu_-}(z) \Hank{1}{\nu_+}(z) = \frac{4}{\pi z}.
	\label{eq:hank_idcota}
\end{equation}
Because the order is $\nu_{\pm} = 1/2 \pm ik$, the small argument 
approximation \eqref{eq:bessel_all_as_smallz} is
\begin{subequations}\label{eq:hankelnu_as_smallz}
\begin{align}
	\Hank{1}{\nu_\pm}\left(\frac{p}{\omega}\right) &\xrightarrow[z\rightarrow 0]{} \phantom{-}
	\sqrt{\frac{2\omega}{p}} \frac{1}{i\pi} \Gamma_\pm(p,p) + 
	\sqrt{\frac{p}{2\omega}}\frac{e^{\pm \pi k}}{\pi\nu_\pm}
	\Gamma_\mp(p,p),
	\label{eq:hankelnu1_as_smallz}\\		
	\Hank{2}{\nu_\pm}\left(\frac{p}{\omega}\right) &\xrightarrow[z\rightarrow 0]{}
	-\sqrt{\frac{2\omega}{p}} \frac{1}{i\pi} \Gamma_\pm(p,p) + 
	\sqrt{\frac{p}{2\omega}}\frac{e^{\mp \pi k}}{\pi\nu_\pm}
	\Gamma_\mp(p,p).
	\label{eq:hankelnu2_as_smallz}
\end{align}
\end{subequations}
The shorthand notation $\Gamma_{\pm}(p, q)$ stands for
\begin{equation}
	\Gamma_{\pm}(p, q) = \left(\frac{2\omega}{\sqrt{pq}}\right)^{\pm ik}
	\Gamma(\nu_{\pm}),\qquad
	\Gamma_{+}^*(p, q) = \Gamma_{-}(p, q) \quad,\quad
	\abs{\Gamma_{\pm}(p, q)}^2 = \frac{\pi}{\cosh{\pi k}}.
	\label{eq:def_gammapm}
\end{equation}
We have used the identity
\begin{equation}
	\Gamma(x) \Gamma(1-x) = \frac{\pi}{\sin{\pi x}}.
	\label{eq:gamma_id}
\end{equation}
\end{section}

\chapter{Pauli spinors} \label{chap:pauli_spinors}
\begin{section}{Spinor construction}
	The Pauli spinors are two-component eigenvectors of the Pauli matrices:
	\begin{align}
		\sigma_1 = \begin{pmatrix} 
		0 & 1\\
		1 & 0 \end{pmatrix}
		,\qquad
		\sigma_2 = \begin{pmatrix} 
		0 & -i\\
		i & 0 \end{pmatrix}
		,\qquad
		\sigma_3 = \begin{pmatrix} 
		1 & 0\\
		0 & -1 \end{pmatrix}.
		\label{eq:pauli_matrices}
	\end{align}
	These matrices have the following (anti)-commutation rules:
	\begin{equation}
		\acomm{\sigma_i}{\sigma_j} = 2 \delta_{ij},\qquad
		\comm{\sigma_i}{\sigma_j} = 2i \varepsilon_{ijk} \sigma_k.
		\label{eq:sigma_comm_anticomm}
	\end{equation}	
	The goal of this section is to construct the Pauli spinors
	$\xi_\lambda({\Vec{p}})$ and $\eta_\lambda({\Vec{p}})$ defined by \eqref{eq:dirac_xietadef}. 
	Let $\Vec{n} = {\Vec{p}}/p$ be the 
	unit vector along ${\Vec{p}}$. The eigenvalue equations for $\xi$ and $\eta$ are
	\begin{equation}
		\Vec{n} \cdot \boldsymbol\sigma \xi_{\lambda}(\Vec{n}) = 2 \lambda \xi_{\lambda}(\Vec{n}),\qquad
		\Vec{n} \cdot \boldsymbol\sigma \eta_{\lambda}(\Vec{n}) = -2 \lambda \eta_{\lambda}(\Vec{n}).
		\label{eq:xieta_eigeneq}
	\end{equation}
	First, we solve the equation for $\Vec{n}=(0,0,1)$. If we denote 
	$\xi_\lambda(\Vec{e}_3) = (\xi^1_\lambda, \xi^2_\lambda)^{\rm T}$, 
	we arrive at the equation
	\begin{equation*}
		(1 - 2\lambda) \xi^1_\lambda = 0,\qquad 
		(1 + 2\lambda) \xi^2_\lambda = 0.
	\end{equation*}
	From this we conclude that $\lambda = \pm 1/2$, so
	$1 - 4\lambda^2 = (1 - 2\lambda)(1 + 2\lambda) = 0$. This gives the 
	natural solution for $\xi$ (and $\eta$ through (\ref{eq:dirac_etaxicc})):
	\begin{equation}
		\xi_{\lambda}(\Vec{e}_3) = \begin{pmatrix}
			\frac{1}{2} + \lambda \\
			\frac{1}{2} - \lambda
		\end{pmatrix},\qquad
		\eta_{\lambda}(\Vec{e}_3) = \begin{pmatrix}
			\frac{1}{2} - \lambda \\
			-\frac{1}{2} - \lambda
		\end{pmatrix}.
		\label{eq:xieta_e3}
	\end{equation}
	Explicitly, the spinors have the following form:
	\begin{equation}
	\begin{array}{c|cc}
		& \xi_{\lambda}(\Vec{e}_3) & \eta_{\lambda}(\Vec{e}_3)\\\hline
		\lambda=\frac{1}{2} & \begin{pmatrix}	1\\ 0	\end{pmatrix} & 
		\begin{pmatrix}	0\\ -1	\end{pmatrix}\smallskip\\
		\lambda=-\frac{1}{2} & \begin{pmatrix}	0\\ 1	\end{pmatrix} & 
		\begin{pmatrix}	1\\ 0	\end{pmatrix}		
	\end{array}
	\end{equation}		
	These spinors are already orthonormal in the sense that
	\begin{equation}
		\xi_{\lambda}^{\dagger}(\Vec{e}_3)\xi_{\lambda^{\prime}}(\Vec{e}_3) = \delta_{\lambda\lambda^{\prime}},\qquad
		\eta_{\lambda}^{\dagger}(\Vec{e}_3)\eta_{\lambda^{\prime}}(\Vec{e}_3) = \delta_{\lambda\lambda^{\prime}}.
		\label{eq:xieta_ortho_e3}
	\end{equation}
	The spinors corresponding to arbitrary orientations $\Vec{n}$ can be 
	constructed from the above by using the spinorial representation
	of the $SU(2)$ group, $D_{\Vec{n}}(\theta) = \exp(-i \Vec{n} \boldsymbol\sigma/2)$.
	We shall use the {\em Euler angles} parametrization of the rotation
	group:
	\begin{equation*}
		R(\alpha, \beta, \gamma) = R_3(\alpha)R_2(\beta)R_3(\gamma).
	\end{equation*}
	$R$ stands for the standard $3$-dimensional representation of the rotation
	group, and $D(R)$ is the spinorial representation of the rotation $R$.
	The subscripts of the form $i$ denote rotations about the $i$ coordinate
	axis.	The rotation which brings a vector on the third axis given by
	spherical coordinates $\theta=0, \varphi=0$ to an arbitrary position
	$\Vec{n}(\theta, \varphi)$ is $R(\varphi, \theta, 0)$, and thus we expect 
	that the eigenvector of the operator $\Vec{n} \cdot \boldsymbol\sigma$ is given by
	\begin{equation*}
		\Vec{n} \boldsymbol\sigma \xi_{\lambda}(\Vec{n}) = 2\lambda \xi_{\lambda}(\Vec{n}),\qquad
		\xi_{\lambda}(\Vec{n}) = D_3(\varphi) D_2(\theta) \xi_{\lambda}(\Vec{e}_3).
	\end{equation*}
	We can prove the validity of this construction by considering the action of
	the rotation $D(\varphi,\theta,0)$ over the eigenvalue equation 
	\eqref{eq:xieta_eigeneq} for $\Vec{n} = \Vec{e}_3$. The following identity 
	($A$ and $B$ are matri\-ces) 
	is useful:
	\begin{equation}
	e^{A} B e^{-A} = B + \comm{A}{B} + \frac{1}{2!} \comm{A}{\comm{A}{B}} + \dots
	= B + \sum_{n} \frac{1}{n!} \ad{A}^n(B).\label{eq:id_eA_B_eA}
	\end{equation}
	With \ad{X} given by
	\begin{equation}
		\ad{X}(Y) = \comm{X}{Y},\qquad \ad{X}^{n+1}(Y) 
		= \comm{X}{\ad{X}^n(Y)}. \label{eq:ad_def}
	\end{equation}
	By using the commutation relations \eqref{eq:sigma_comm_anticomm} between 
	the $\sigma$ matrices, we can evaluate
	\begin{equation}
		D_j(\theta) \sigma_i D_j^{\dagger}(\theta) 
		= \sigma_j \cos{\theta} + \varepsilon_{ijk} \sigma_k \sin{\theta},
		\qquad i\neq j.
		\label{eq:sigmai_rotj}
	\end{equation}
	Succesively applying relation \eqref{eq:sigmai_rotj} for the two rotations
	$D(\Vec{n}) = D_3(\varphi) D_2(\theta)$ yields
	\begin{equation*}
		D(\varphi,\theta,0) \sigma_3 \xi_{\lambda}(\Vec{e}_3) = 
		(\sigma_1 \sin{\theta} \cos{\varphi} + \sigma_2 \sin{\theta} \sin{\varphi}
		+ \sigma_3 \cos{\theta}) D_3(\varphi) D_2(\theta) \xi_{\lambda}(\Vec{e}_3),
	\end{equation*}
	which is equivalent to the eigenvalue equation \eqref{eq:xieta_eigeneq}.
	This enables us to define the spinors of arbitrary orientation $\Vec{n}$ as
	\begin{subequations}\label{eq:xieta_vn_def}
	\begin{gather}
		\Vec{n} \cdot \boldsymbol\sigma \xi_{\lambda}(\Vec{n}) = 2\lambda \xi_{\lambda}(\Vec{n}),\qquad
		\xi_{\lambda}(\Vec{n}) = D(\Vec{n}) \xi_{\lambda}(\Vec{e}_3), \label{eq:xi_vn_def}\\
		\Vec{n} \cdot \boldsymbol\sigma \eta_{\lambda}(\Vec{n}) = -2\lambda \eta_{\lambda}(\Vec{n}),\qquad
		\eta_{\lambda}(\Vec{n}) = D(\Vec{n}) \xi_{\lambda}(\Vec{e}_3). \label{eq:eta_vn_def}
	\end{gather}
	\end{subequations}
	The explicit form of these spinors can be obtained by evaluating the rotation
	matrices $D_3(\varphi)$ and $D_2(\theta)$ and applying them following the 
	prescription \eqref{eq:xieta_vn_def}:
	\begin{subequations} \label{eq:xi_eta_vn}
	\begin{align}
		\xi_{\lambda}(\theta,\varphi) &= \begin{pmatrix}
			e^{-i\varphi/2} \left( 
			\cos{\frac{\theta}{2}} \left(\frac{1}{2} + \lambda\right) -
			\sin{\frac{\theta}{2}} \left(\frac{1}{2} - \lambda\right) \right)\\
			e^{i\varphi/2} \left( 
			\cos{\frac{\theta}{2}} \left(\frac{1}{2} - \lambda\right) +
			\sin{\frac{\theta}{2}} \left(\frac{1}{2} + \lambda\right) \right)
		\end{pmatrix}, \label{eq:xi_vn}\\
		\eta_{\lambda}(\theta,\varphi) &= \begin{pmatrix}
			e^{-i\varphi/2} \left( 
			\cos{\frac{\theta}{2}} \left(\frac{1}{2} - \lambda\right) +
			\sin{\frac{\theta}{2}} \left(\frac{1}{2} + \lambda\right) \right)\\
			e^{i\varphi/2} \left( 
			-\cos{\frac{\theta}{2}} \left(\frac{1}{2} + \lambda\right) +
			\sin{\frac{\theta}{2}} \left(\frac{1}{2} - \lambda\right) \right)
		\end{pmatrix}.\label{eq:eta_vn}
	\end{align}
	\end{subequations}
	Explicitly, these spinors are
	\begin{equation}
	\begin{array}{c|cc}
		& \xi_{\lambda}(\Vec{n}) & \eta_{\lambda}(\Vec{n})\\\hline
		%
		\lambda=\frac{1}{2} & 
		\begin{pmatrix}
			e^{-i\varphi/2} \cos{\frac{\theta}{2}}\\
			e^{\phantom{-}i\varphi/2} \sin{\frac{\theta}{2}}
		\end{pmatrix}
		& 
		\begin{pmatrix}
			\phantom{-}e^{-i \varphi/2} \sin{\frac{\theta}{2}}\\
			-e^{\phantom{-}i \varphi/2} \cos{\frac{\theta}{2}} 
		\end{pmatrix}
		\smallskip\\
		%
		\lambda=-\frac{1}{2} & 
		\begin{pmatrix}
			-e^{-i \varphi/2} \sin{\frac{\theta}{2}}\\
			\phantom{-}e^{\phantom{-}i \varphi/2} \cos{\frac{\theta}{2}}
		\end{pmatrix}
		&
		\begin{pmatrix}
			e^{-i\varphi/2} \cos{\frac{\theta}{2}}\\
			e^{\phantom{-}i\varphi/2} \sin{\frac{\theta}{2}} 
		\end{pmatrix}
	\end{array}
	\end{equation}				
	The appearance of half angles should not be a surprise, since we are working
	in a spinorial representation of $\mathcal{S}\mathcal{U}(2)$ (spin $\frac{1}{2}$). The
	interesting feature of this representation is that 
	$\xi_\lambda(\theta, \phi+2\pi) = -\xi_{\lambda(\theta,\phi)}$. This property
	will be very important in understanding the Bogoliubov coefficients defined
	in \autoref{chap:dirac_bogo}.
	
	In order to be convinced that the above definition for $\eta$ still obeys 
	the conjugation relation \eqref{eq:dirac_etaxicc}, we use the anticommutation 
	relations \eqref{eq:sigma_comm_anticomm} and the explicit forms of the 
	$\sigma$ matrices \eqref{eq:pauli_matrices} to obtain
	\begin{equation}
		\sigma_2 \sigma_i = - \sigma_i^* \sigma_2. \label{eq:sigma_cc}
	\end{equation}
	We apply the conjugation relation \eqref{eq:dirac_etaxicc} 
	to the equation defining $\xi_\lambda(\Vec{n})$ \eqref{eq:xieta_vn_def} 
	and we obtain
	\begin{equation}
		i \sigma_2 D^{*}(\Vec{n})\xi^*_{\lambda}(\Vec{e}_3) = 
		D(\Vec{n}) (i \sigma_2 \xi^*_{\lambda}(\Vec{e}_3)),
	\end{equation}
	since $D^{*}_k(\varphi) = \exp(i \varphi \sigma_k^* /2)$.
	
	Finaly, these spinors obey the same orthonormalization relations as their 
	$\Vec{e}_3$ counterparts (\eqref{eq:xieta_ortho_e3}): 
	\begin{equation}
		\xi^\dagger_{\lambda^{\prime}}(\Vec{n}) \xi_{\lambda}(\Vec{n}) = \delta_{\lambda\lambda^{\prime}}
		,\qquad
		\eta^\dagger_{\lambda^{\prime}}(\Vec{n}) \eta_{\lambda}(\Vec{n}) = \delta_{\lambda\lambda^{\prime}}
		\label{eq:xieta_ortho_vn}
	\end{equation}
\end{section}
\begin{section}{Behaviour under parity transformations}
	It is instructive to consider the parity-transformed spinor corresponding 
	to $-\Vec{n}$. In spherical coordinates, the easiest way to define this 
	reflection about the origin of the coordinate axes is 
	\begin{equation}
		\Vec{n}(\theta, \varphi) \mapsto -\Vec{n}(\pi - \theta, \varphi + \pi)
		\mapsto -(-\Vec{n})(\theta, \varphi + 2\pi).
		\label{eq:sph_rev}
	\end{equation}
	We can see that two consecutive reversals (as defined above)
	are equivalent to a shift of $2\pi$ in the angle $\varphi$, which has the
	effect of introducing a $-$ sign in the corresponding spinor. To prove this,
	we must investigate the transformed spinor. Let's analyze first the spinor
	$\xi_{\lambda}(-\Vec{e}_3)$ (which we read from \eqref{eq:xi_vn}):
	\begin{equation}
		\xi_{\lambda}(-\Vec{e}_3) = i \xi_{-\lambda}(\Vec{e}_3).
		\label{eq:xi_me3}
	\end{equation}
	The $i$ factor is the hallmark of the spinorial representations
	of the rotation group. Next, in order to evaluate de general case, we
	must use
	\begin{equation}
		D_3(\pi) D_2(\theta) D_3^{\dagger}(\pi) = D_2(-\theta).
	\end{equation}	
	The above follows from the unitarity of the representation 
	matrices and relation \eqref{eq:sigmai_rotj}. We can now evaluate the $-\Vec{n}$
	spinor:
	\begin{subequations}\label{xi_eta_mvn}
	\begin{multline}
		\xi_{\lambda}(-\Vec{n}) = D_3(\pi + \varphi) D_2(\pi - \theta) \xi_{\lambda}(\Vec{e}_3) = 
		D_3(\varphi) D_2(\theta) D_3(\pi) D_2(\pi) \xi_{\lambda}(\Vec{e}_3)\\
		= i \xi_{-\lambda}(\Vec{n}).
		\label{eq:xi_mvn}
	\end{multline}
	The $\eta$ spinor has a similar behaviour, which we investigate by applying
	the charge conjugation \eqref{eq:dirac_etaxicc} to the above expression:
	\begin{equation}
		\eta_{\lambda}(-\Vec{n}) = -i \eta_{-\lambda}(\Vec{n}).
		\label{eq:eta_mvn}
	\end{equation}
	\end{subequations}
\end{section}

\begin{section}{Relation between \texorpdfstring{$\xi$ and $\eta$}{xi and eta}}
	If we take a closer look at the forms of the spinors $\xi$ and $\eta$ 
	\eqref{eq:xieta_e3}, we see that there is an easy relation between them:
	\begin{equation}
		\eta_{\lambda}(\Vec{e}_3) = -2 \lambda \xi_{-\lambda}(\Vec{e}_3).
	\end{equation}
	Rotating the above relation to an arbitrary vector $\Vec{n}$, we obtain 
	\begin{equation}
		\eta_\lambda(\Vec{n}) = - 2\lambda \xi_{-\lambda}(\Vec{n}).
	\end{equation}
	If we let $\Vec{n} \mapsto -\Vec{n}$ as defined in \eqref{eq:sph_rev}, we find the
	relevant relation
	\begin{equation}
		\eta_{\lambda}(-\Vec{n}) = -2 i \lambda \xi_{\lambda}(\Vec{n}).
		\label{eq:eta_mvn_xi}
	\end{equation} 
	With this we can evaluate the inner products
	\begin{equation}
		\eta^\dagger_{\lambda}(-\Vec{n}) \xi_{\lambda^{\prime}}(\Vec{n}) = 
		-2 i \lambda \delta_{\lambda\lambda^{\prime}},\qquad
		\xi^\dagger_{\lambda}(-\Vec{n}) \eta_{\lambda^{\prime}}(\Vec{n}) = 
		2 i \lambda \delta_{\lambda\lambda^{\prime}}\label{eq:eta_mvn_xi_vn_prod}
	\end{equation}
	These relations will be used in the derivation of the matching coefficients
	between de Sitter and Minkowski modes in \autoref{chap:dirac_bogo}.
\end{section}


\section{abc}\label{sec:aaa}... \nameref{sec:aaa} -> section abc
\def