††institutetext: Mathematics Department, King’s College London,
The Strand, London, WC2R 2LS, UK

#  Spinors in (Anti-)de Sitter Space

Vladimir Schaub<EMAIL_ADDRESS>
###### Abstract

We explore analytical aspects of correlators involving Dirac spinors in
$d+1$-dimensional de Sitter space. Adapting the formalism of Sleight and
Taronna, we show how to relate processes involving fermions in the in-in
formalism to equivalent Witten diagrams in (complexified) Euclidean anti-de
Sitter space. We exemplify the method for a fermion-exchange diagram. We
establish a positive spectral decomposition over the principal series of the
Wightman function of two spinors, showing the consequences of unitarity.

##  1 Introduction

The experimental facts motivating this theoretical work are the observations
that neutrinos exist and space expands. It is of crucial importance then to
have a well developed understanding of the physics of relativistic field of
spin $1/2$ in spacetimes which are asymptotically de Sitter (dS). The goal of
the present paper is to contribute to this edifice, by answering the question
: how does one computes cosmological correlators involving Dirac fermions in
pure dSd+1 ?

This endeavour is an incremental piece to a vast existing amount of
literature. Inflationary cosmologists have studied for a long time the
properties of cosmological correlators, which are the expectation values of
field insertions at the late time surface of de Sitter, as probes of the early
universeArkani-Hamed:2015bza , for a review see Baumann:2009ds ; Green:2022hhj
. Because of the difficulty in analytically evaluating these processes,
studies has turned toward more axiomatic approaches. Some focused on
bootstrapping the correlators, in works such as Arkani-Hamed:2018kmz ;
Baumann2018 ; Baumann:2020ksv ; Baumann:2020dch ; Baumann:2019oyu , reviewed
generally in Baumann:2022jpr . Others have followed an holographic perspective
McFadden:2009fg ; McFadden:2010na ; McFadden:2010vh ; McFadden:2011kk ;
Bzowski:2012ih . These are of course not the sole interesting objects in
expanding spaces. Many studies have investigated the rich physics of quantum
fields living in de Sitter, using an array of tools beyond late time
correlators only, such as Bros:1994dn ; Strominger:2001pn ; Anninos:2011ui ;
Gorbenko:2019rza ; Hogervorst:2021uvp . For a more thorough review, see
Flauger:2022hie .

The present work is motivated through the recent renewed activity on the
analytical front, started through the work of Sleight and Taronna
Sleight:2019hfp ; Sleight:2019mgd ; Sleight:2020obc ; Sleight:2021plv , which
have offered a new analytical approach to computing correlators. The crucial
insight they offered is that one is able to rewrite Lorentzian processes in dS
as equivalent processes in Euclidean Anti-de Sitter (AdS) space, which then
allows for an evaluation using standard methods. The Wick rotation between
these spaces Maldacena:2002vr  is there explicitly realised. This has led to
various interesting exact computations and results, and more generally to the
import of AdS knowledge into the properties of perturbative dS systems
Fichet:2021xfn ; DiPietro:2021sjt ; Heckelbacher:2022hbq ; Sleight:2021iix .

The goal of this article is to close the current gap in these development
relative to spinorial insertions. Our first main result, presented in sec.
3.4, is a new set of rules to rewrite cosmological diagrams involving fermions
using an equivalent Witten diagram, expanding the ones of Sleight:2021plv .
Regardless of any inflationary applications, from a physical and formal
standpoint these are perfectly legitimate objects to understand. This work is
a continuation of our previous study on fermions in de Sitter space
Pethybridge:2021rwf , which investigated the embedding formalism and showed in
detail how to uplift and project spinors, thereby streamlining their
manipulations and giving a simple and efficient derivation of their Wightman
functions. Work on spinors in expanding spaces goes back to Dirac Dirac:1935zz
, who attempted to formulate them in the embedding space already. Other
studies such as Candelas1975 ; Fang:1979hq ; Allen:1986qj ; Cotaescu2002 ;
Cotaescu2018  have considered them, and many more focused on the relation to
field theory defined on a sphere, such as Camporesi1992 ; Camporesi1996 ;
Letsios:2020twa . Our work is framed as a bridge between these considerations
and the computation of cosmological correlators involving Dirac particles. As
this endeavour makes us consider in detail the analytical structure of fermion
correlators, it is natural to wonder as well about the more general properties
of these objects, such as unitarity Bros:1990cu ; Bros:1994dn ; Bros:1995js ;
Bros:1998ik ; Bros:2010rku ; Sengor:2019mbz ; DiPietro:2021sjt ;
Hogervorst:2021uvp . To this aim, we also prove that spinors admit a positive
Källen-Lehman decomposition in de Sitter space. This forms our second main
result, presented in sec. 4.3

The plan of the paper is as follows. We first review the existing results for
the analytical continuation of bosonic cosmological correlators. Our approach
focuses on the differential relations defining the different propagators,
while working in the embedding picture. The analytical continuation rules are
specified in sec. 2.2. The second section discusses the analogue for fermions.
As fermions are fickle creatures very sensitive to signature, we setup in
detail the conventions in both the embedding and real spaces. A careful
analytical continuation follows, from which Feynman rules are extracted. As a
proof of concept, we compute a sample cosmological diagram in sec. 3.5 . The
final section is independent from the rest of the paper, and focuses on the
spectral decomposition. We review the canonical quantisation of scalar and
Dirac fields in dS, resumming the propagator from the mode functions. We
consider generic on-shell states, and show that symmetries fix the overlap of
the Dirac field with them to be proportional to the free-field modes. A
positive decomposition then ensues. Our conventions follow the first appendix
of Pethybridge:2021rwf . Our appendix is dedicated to a detailed rederivation
of the properties of spinorial harmonic functions.

##  2 Euclidean AdS Method for Bosonic Cosmological Correlator

In this section, we are concerned with how to compute cosmological correlation
functions by relating them to Witten diagrams. The content is not new, forming
an alternative exposition of some of the existing literature Sleight:2020obc .
However, unlike some of these works, we do not use Mellin space. Our goal is
to set the conventions and procedure in a way that makes the transition to the
spinor case smooth. The observables of interest are $n$-point functions of
field inserted on a fixed late-time slice, $\eta_{c}\approx 0^{-}$. Because of
the $SO(1,d+1)$ symmetry present at the late-time boundary, this must take the
form of a CFT $n$-point function. In principle, for a given Lagrangian QFT in
dS, this correlator can be derived from perturbation theory. The relevant
scheme is the Schwinger-Keldysh, or in-in formalism, which uses different type
of propagators Weinberg2005 ; Adshead2009 .

We first review the embedding approach to study field insertions in (EA)dS
Costa:2014kfa ; Pethybridge:2021rwf , and its result for the general Wightman
function of symmetric traceless tensor (STT) operators in (EA)dS. We then
review the in-in formalism, and its realisation using $i\epsilon$
prescription. We discuss how to wick rotate field insertions from dS to EAdS
Sleight:2019hfp ; Sleight:2020obc ; DiPietro:2021sjt , and check how the
analytically continued propagators decompose in term of AdS propagators.
Finally, we sum up the resulting procedure in a series of Feynman rules, like
in Sleight:2021plv .

###  2.1 Fields in the Embedding Space

Figure 1: The embedding of dSd+1 (red) and EAdSd+1 (blue) into
$\mathbb{R}^{1,d+1}$. Constant $\eta$ and $z$ coordinate lines are drawn in
black. These get deformed smoothly one into another through analytical
continuation, bypassing the light-cone.

The building blocks of perturbation theory in (EA)dS are the Wightman
functions of elementary fields. These objects are determined by the isometry
group and physical requirements. To take advantage of this fact, it is
convenient to consider them in a formalism that fully exploit the available
symmetries, the embedding picture.

Consider dSd+1 as the subset $X^{A}\in\mathbb{R}^{1,d+1}$ satisfying
$X^{A}X_{A}=+1$, where we measure in unit of the radius of de Sitter. A
coordinate system which covers part of the hyperboloid is given through the
Poincaré coordinates

$\displaystyle X^{A}$ $\displaystyle=(X^{+},X^{-},X^{a})$ (1)
$\displaystyle=\frac{1}{\eta}(1,x^{2}-\eta^{2},x^{a})\,,$

with $\eta<0$, and $X^{\pm}=X^{0}\pm X^{d+1}$. One can study EAdSd+1
insertions in this embedding space as well, by looking at points satisfying
$X_{H}\cdot X_{H}=-1$, which can be parametrised through

$\displaystyle X_{H}^{A}=\frac{1}{z}(1,x^{2}+z^{2},x^{a})\,,$ (2)

with $z>0$. The situation is represented in fig.1, with coordinate lines
drawn.

In this augmented space, the isometries are restated in term of Lorentz-
invariance, which is simple to impose. One can define the uplift and
projection of tensorial fields from $dS_{d+1}$ to and from the embedding space
$\mathbb{R}^{1,d+1}$, using the usual rules of differential geometry
bruhat:1982 . As before, the analogous statement hold for EAdS. A central
element of this analysis is the covariant derivative in dS, which in the
embedding becomes

$\displaystyle\nabla_{A}=\partial_{A}-X_{A}X\cdot\partial\,,$ (3)

And similarly for EAdS by switching the sign. The canonical example to
showcase the mileage brought by this formalism is that of the free massive
scalar. Consider the Wightman function of scalar fields in dSd+1, which obey
the equations of motions

$\displaystyle(\nabla_{A}\nabla^{A}-m^{2})\expectationvalue{\phi(X)\phi(Y)}=0\,.$
(4)

From Lorentzian symmetry, this is only a function of the scalar cross-ratio

$\displaystyle u=\frac{1+X\cdot
Y}{2}=\frac{(\eta_{1}+\eta_{2})^{2}-(x-y)^{2}}{4\eta_{1}\eta_{2}}\,.$ (5)

For points in AdS, the cross-ratio is defined identically in the embedding. We
can apply the equation of motions on this ansatz and solve the resulting
differential equation,

$\displaystyle\expectationvalue{\phi(X)\phi(Y)}$
$\displaystyle=W_{0,\Delta}(u)=\frac{\Gamma(\Delta)\Gamma(d-\Delta)}{(4\pi)^{\frac{d+1}{2}}\Gamma(\frac{d+1}{2})}\,_{2}F_{1}\left(d-\Delta,\Delta;\frac{d+1}{2};\frac{1+X\cdot
Y}{2}\right)\,,$ (6)

where we used the previously mentioned physical requirements, namely the
Bunch-Davies and Hadamard conditions. The Bunch-Davies condition selects a
vacuum state that maps to the Minkowski one in the far past limit, while the
Hadamard condition relates the normalisation of this correlator to the one in
flat space using the coincident point divergence. One can in principle
consider more general $\alpha$-vacua as well Allen:1985ux ; Sasaki:1994yt ;
Bousso:2001mw , but for simplicity we will not consider that complication in
the present work. The range of $\Delta$ allowed by unitarity is an interesting
matter, which depends non-trivially on the spin of the particle, and is made
of different series. The details are beyond the scope of our work, and we
refer the reader to the abundant literature touching on the subject, such as
Joung2006 ; Joung2007 ; Anous2020 ; Sun2021 ; Penedones:2023uqc .

This Wightman function can be used to construct the different propagators
which are needed to compute diagrams in the Schwinger-Keldysh framework.

The previous computation can be generalised to the case of tensorial operators
Costa:2011wa ; Costa:2014kfa . Consider the index-free field
$F_{J,\Delta}(X,W)=W^{A}\ldots W^{A_{J}}F_{A_{1}\ldots A_{J}}(X)$. This
homogeneous polynomial in the polarisation variables $W^{A}$ encodes a
spin-$J$ symmetric tensor in the embedding. Requiring $W^{A}W_{A}=0$ selects
out traceless tensors. Further requiring $W\cdot X=0$, we restrict ourselves
to transverse tensors in the embedding, which under the pull-back map to the
dS slice, these are in one-to-one map with STT fields. As in the scalar case,
the ambient symmetry fixes the ansatz for the Wightman function, on which one
can solve the equations of motions

$\displaystyle\expectationvalue{F_{J,\Delta}(X_{1},W_{1})F_{J,\Delta}(X_{2},W_{2})}$
$\displaystyle=W_{J,\Delta}\left(u\right)$ (7)
$\displaystyle=\sum_{k=0}^{J}(W_{1}\cdot W_{2})^{J-k}(W_{1}\cdot
X_{2}W_{2}\cdot X_{1})^{k}g_{k}(u)\,.$

The Wightman function $W_{J,\Delta}$ is determined by the functions
$g_{k}(u)$. The situation is the same in EAdS, and we call the analogous
propagator $\Pi_{J,\Delta}(u)$ Costa:2014kfa . The functions $g_{k}$ are
specified through the relation

$g_{k}(u)=\sum_{l=k}^{J}\left(\frac{l!}{k!}\right)^{2}\frac{\partial_{u}^{k}h_{k}(u)}{2^{l+k}(l-k)!}\,.$
(8)

with the $h_{k}$ recursively computed through

$\displaystyle h_{k}(u)$ $\displaystyle=\begin{aligned}
c_{J,\Delta,k}\bigg{(}&2(d-2k+2J-1)\bigg{(}(d+J-2)h_{k-1}(u)\\\
&-\left(\frac{1}{2}-u\right)\partial_{u}h_{k-1}(u)\bigg{)}-4(2-k+J)h_{k-2}(u)\bigg{)}\,,\end{aligned}$
(9) $\displaystyle c_{J,\Delta,k}$
$\displaystyle=\frac{J-k+1}{k(d+2J-k-2)(\Delta+J-k-1)(d-\Delta+J-k-1)}\,.$

Given a seed $h_{0}(u)$, these relations fix completely the 2-point function.
Surprisingly, they hold identically in both EAdS and dS, with only the seed
differing. In both cases, it is given by the Wightman function of scalars. For
fields in dS, we have

$\displaystyle h^{dS}_{0}(u)$
$\displaystyle=\frac{\Gamma(\Delta)\Gamma(d-\Delta)}{(4\pi)^{\frac{d+1}{2}}\Gamma(\frac{d+1}{2})}\,_{2}F_{1}\left(\Delta,d-\Delta;\frac{d+1}{2};u\right)\,,$
(10)

while for EAdS fields, it is the analogous result,

$\displaystyle h^{H}_{0}(u)$
$\displaystyle=\frac{\Gamma(\Delta)}{2\pi^{d/2}\Gamma(\Delta+1-\frac{d}{2})}\frac{1}{(-4u)^{\Delta}}\,_{2}F_{1}\left(\Delta,\Delta-\frac{d-1}{2};2\Delta-d+1;\frac{1}{u}\right)\,.$
(11)

These functions are related in that, as we are gonna see, the first one can be
written as a linear combination of the second, with conformal weight $\Delta$
and $d-\Delta$. They are found by solving the Casimir equation on the
appropriate spaces and imposing singular behaviour.

With these kinematical objects fixed, one is in principle able to write
diagrams in both spaces, in a given perturbative framework.

###  2.2 In-In Formalism and Analytical Continuation to AdS

The rules to compute correlators in de Sitter are rather intricate. The
objects of interest are correlation functions of fields inserted on a given
time-slice $\eta_{c}\sim 0^{-}$. Having settled on a set of interaction
vertices, we write all possible diagrams by inserting vertices decorated with
$l$ and $r$ labels. The propagators to be used depend on the type of
insertions, bulk or boundary, and on the label of the vertices it is joining
in the bulk. The different labels indicate the relative time ordering of the
operators, which is implemented through an $i\epsilon$ ($\epsilon>0$)
prescription Weinberg2005 . The prescription can be phrased in term of the
cross-ratio on which the 2-point function depends DiPietro:2021sjt

$\displaystyle\begin{aligned} (l,l)&\Rightarrow u+i\epsilon\,,\\\
(r,r)&\Rightarrow u-i\epsilon\,,\end{aligned}$ $\displaystyle\begin{aligned}
(l,r)&\Rightarrow u+i\epsilon\ \text{sgn}(\eta_{1}-\eta_{2})\,,\\\
(r,l)&\Rightarrow u-i\epsilon\ \text{sgn}(\eta_{1}-\eta_{2})\,.\end{aligned}$
(12)

The different propagators $W^{ab}(u)$ are given by evaluating the Wightman
function we previously computed, at the respective arguments $u_{ab}$. This
differs from EAdS. The theory is Euclidean and one works directly in
perturbation theory using the Wightman function. The integrals over vertices
encountered there are also under better analytical control.

This situation is not unlike usual quantum field theory, where the Lorentzian
diagrams are better understood as the analytical continuation of Euclidean
diagrams, which are better behaved. The procedure we now explain is the
analogue Wick rotation for de Sitter space, whereby one recasts cosmological
correlators in term of perturbative Euclidean AdS process Sleight:2019hfp .
This procedure hinges on a contour deformation, which must be done on a
continuous path that does not cross any branch cut or pole.

Figure 2: Analytical Continuation. The branch-cut should not be crossed as it
separates the different time orderings.

Consider a generic term arising from perturbation theory. The expressions are
all analytic in the variable $\eta_{l/r}$ in the domain which preserve the
ordering of the operator. Consider the path that rotates the time coordinate
depending on its $l/r$ label along

$\displaystyle\eta_{r}$ $\displaystyle\rightarrow e^{-i\pi/2}z\,,$ (13)
$\displaystyle\eta_{l}$ $\displaystyle\rightarrow e^{+i\pi/2}z\,.$

From the $i\epsilon$ prescriptions spelled out before, these do not cross any
singularity, hence are valid contour deformations of the different integrals
one encounters Visser:2017atf . The situation is illustrated in fig.2. Through
this transformation, the propagators and integration over coordinates are
analytically continued. From the Poincaré parametrisation of the EAdS and dS
slices, these translate also into an analytical continuation of the embedding
coordinates. For instance, consider the integral of a point in the left
contour, over all of dS,

$\displaystyle i\int_{dS}DX_{l}=i\int d^{d+2}X\,2\delta(X^{2}-1)$
$\displaystyle\rightarrow e^{i\frac{\pi}{2}(d+3)}\int
d^{d+2}X\,2\delta(X^{2}+1)$ (14)
$\displaystyle=e^{i\frac{\pi}{2}(d-1)}\int_{H}DX\,,$

Where we notice the last integral becomes an integral over the EAdS slice. One
might complain that this is an integral over both the future and past EAdS
slices. However, the integrals which appear in perturbative computation of
cosmological diagrams are not integrals over the full de Sitter space, but
only the causal past of the late time slice, which are the coordinate covered
by the Poincaré parametrisation (1). As such, it is sensible to invert the
logic and think of the integral over a single EAdS hyperboloid as defining the
relevant integral over dS. The analogous computation for $X_{r}$ gives the
expected result

$\displaystyle-i\int_{dS}DX_{r}$ $\displaystyle\rightarrow
e^{-i\frac{\pi}{2}(d-1)}\int_{H}DX\,,$ (15)

and allows us to rewrite generic vertices integrals in dS as vertices in EAdS.
The job is done, as we have yet to consider the behaviour of the propagators,
which is the topic of the next section.

###  2.3 Prolonging the In-In Propagators

The goal of this section is to demonstrate that a given STT dS field can be
rewritten as a (shadow)-pair of AdS fields of the same type, up to phase
factors, effectively transcribing perturbation theory from one space to
another. The starting point is the previously highlighted fact that the
analytic continuation acts on the embedding coordinate by sending

$\displaystyle X_{a}\cdot Y_{b}$ $\displaystyle\rightarrow
e^{-i\frac{\pi}{2}(a+b)}X\cdot Y\,,$ (16)

under which $u$ behaves as

$\displaystyle u_{ll}$ $\displaystyle\rightarrow 1-(u-i\epsilon)\,,$
$\displaystyle u_{rr}$ $\displaystyle\rightarrow 1-(u+i\epsilon)\,,$
$\displaystyle u_{lr}$ $\displaystyle\rightarrow u\,,$ $\displaystyle u_{rl}$
$\displaystyle\rightarrow u\,.$ (17)

In the $ll$ and $rr$ scenarii, the phases associated to terms such as
$u^{\Delta}$, since $u<0$ for points in AdS, are resolved by considering in
which direction in the complex plane one approaches, as we illustrated by
keeping the factors of $\epsilon$. The propagators of tensors also contain
structures, which are built using product of polarisation and position
vectors. These transform like (16) as well.

We now take the propagator along the paths (13) in the complex plane. Going
one step further, these analytically continued dS propagators are decomposed
into EAdS propagators. The matter is essentially solved by the application of
the hypergeometric identities gradshtein

$\,{}_{2}F_{1}\left(a,b;c;z\right)$ $\displaystyle=\begin{aligned}
&\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}\frac{1}{(-z)^{a}}\,_{2}F_{1}\left(a,a+1-c;a+1-b;\frac{1}{z}\right)\\\
&+\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}\frac{1}{(-z)^{b}}\,_{2}F_{1}\left(b,b+1-c;b+1-a;\frac{1}{z}\right)\,,\end{aligned}$
(18) $\,{}_{2}F_{1}\left(a,b;c;z\right)$ $\displaystyle=\begin{aligned}
&\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}\frac{1}{(1-z)^{a}}\,_{2}F_{1}\left(a,c-b;a+1-b;\frac{1}{1-z}\right)\\\
&+\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}\frac{1}{(1-z)^{b}}\,_{2}F_{1}\left(b,c-a;b+1-a;\frac{1}{1-z}\right)\,,\end{aligned}$
(19)

to the seed of the spinning propagators, as we now illustrate. Consider
$W_{J,\Delta}^{lr}(u)$; it is unchanged when performing the analytic
continuation, and one can reduce by linearity its decomposition to that of the
seed, which is readily done using (LABEL:eq:id1). This only works thanks to
the relations (9) being unchanged for $\Delta$ and $d-\Delta$. The analytic
continuation of $h_{0}$ implies the analytic continuation of the whole
propagator,

$\displaystyle
W_{J,\Delta}^{lr}(u)\rightarrow\frac{1}{2\sin(\frac{\pi(d-2\Delta)}{2})}\bigg{(}\Pi_{J,\Delta}(u)-\Pi_{J,d-\Delta}(u)\bigg{)}\,.$
(20)

Which is a way of saying that $W^{lr}$ is proportional to the AdS harmonic
function. $W^{rl}$ is identical. The cases $ll$ and $rr$ are more interesting.
Consider the scalar propagator, or seed, using the second identity
(LABEL:eq:id2),

$\displaystyle W_{0,\Delta}^{ll}(u)$ $\displaystyle\rightarrow
W_{0,\Delta}(1-u)$ (21)
$\displaystyle=\frac{1}{2\sin(\frac{\pi(d-2\Delta)}{2})}\bigg{(}\Pi_{0,\Delta}(u)e^{i\pi\Delta}-\Pi_{0,d-\Delta}(u)e^{i\pi(d-\Delta)}\bigg{)}\,,$

and likewise for $W_{0,\Delta}^{rr}$, with opposite phases. Having seen how
the relation holds for $J=0$, we can work our way through (9), noticing that
the analytical continuation leaves the relations relating the different
$h_{k}$ invariant, and allows us to extrapolate the decomposition of $h_{0}$
into the one of $W_{J}$. Explicitly,

$\displaystyle W_{J,\Delta}^{ll}(u)\rightarrow\sum_{k=0}^{J}(W_{1}\cdot
W_{2})^{J-k}(W_{1}\cdot X_{2}W_{2}\cdot
X_{1})^{k}\underbrace{(-1)^{J-k}g_{k}(1-u)}_{\bar{g}_{k}(u)}\,,$ (22)

and since $\partial_{(1-u)}f(u)=-\partial_{u}f(u)$, we find

$\displaystyle\bar{g}_{k}(u)=\sum_{l=k}^{J}\left(\frac{l!}{k!}\right)^{2}\frac{1}{2^{l+k}(l-k)!}\partial_{u}^{k}\underbrace{\left(h_{k}(1-u)(-1)^{J}\right)}_{\bar{h}_{k}(u)}\,.$
(23)

The crucial property is now that the recursive relation (9) for $h_{k}(u)$ is
invariant under the change to $\bar{h}_{k}(u)=(-1)^{k}h_{k}(1-u)$, as well as
under $\Delta\rightarrow d-\Delta$. It then follows that the analytical
continuation of $W_{0,\Delta}^{ll}$ is indeed sufficient to find the relation
for the general spin, as claimed,

$\displaystyle
W_{J,\Delta}^{ll}(u)\rightarrow\frac{1}{2\sin(\frac{\pi(d-2\Delta)}{2})}\bigg{(}\Pi_{J,\Delta}(u)e^{i\pi(J+\Delta)}-\Pi_{J,d-\Delta}(u)e^{i\pi(J+d-\Delta)}\bigg{)}\,.$
(24)

For the $rr$ propagator one proceeds identically, finding opposite phases.

From this result, we can rewrite all propagators of tensorial operators in the
in-in formalism. Notice that the four different decompositions found for the
de Sitter propagator can be restated as a change of basis to new AdS fields

$\displaystyle F_{J,\Delta}^{dS,l}$
$\displaystyle=e^{i\frac{\pi}{2}(J+\Delta)}F^{+}_{J}+e^{i\frac{\pi}{2}(J+d-\Delta)}F^{-}_{J}\,,$
$\displaystyle F_{J,\Delta}^{dS,r}$
$\displaystyle=e^{-i\frac{\pi}{2}(J+\Delta)}F^{+}_{J}+e^{-i\frac{\pi}{2}(J+d-\Delta)}F^{-}_{J}\,,$
(25)

having a non-standard kinetic term normalisation

$\displaystyle\expectationvalue{F_{J,\Delta}^{+}F_{J,\Delta}^{+}}$
$\displaystyle=\frac{\Pi_{J,\Delta}}{2\sin(\frac{\pi(d-2\Delta)}{2})}\,,$
$\displaystyle\expectationvalue{F_{J,\Delta}^{-}F_{J,\Delta}^{-}}=-\frac{\Pi_{J,d-\Delta}}{2\sin(\frac{\pi(d-2\Delta)}{2})}\,.$
(26)

Effectively, this gives a procedure to compute a generic diagram in dS. One
first draws the diagram, then assign weights $\Delta_{+}=\Delta$ or
$\Delta_{-}=d-\Delta$ to each line whose weight is not fixed. For each
internal line, one divides by the normalising factor
$\frac{1}{2\sin(\frac{\pi}{2}(d-2\Delta_{\pm}))}$. For each vertex, one must
fix the label $l/r$, multiply by the factors of $e^{\pm
i\frac{\pi}{2}(J+\Delta)}$ for each spin $J$, dimension $\Delta$ line attached
to it, multiply by $e^{\pm i\frac{\pi(d-1)}{2}}$, and finally sum. This is a
straightforward set of rules to relate cosmological diagrams to Witten
diagrams.

It is important to match the analytical continuation of the external points as
well. In general in AdS, to take the boundary limit, one simply rescales a
point and the field insertion there to infinity Harlow:2011ke . From the
operator perspective, we define

$\displaystyle\lim_{\lambda\to\infty}\lambda^{\Delta}\phi_{H}\left(\lambda
P+\frac{1}{\lambda}\ldots\right)$ $\displaystyle=\widehat{\phi}(P)\,.$ (27)

The boundary field $\widehat{\phi}$ is non-canonically normalised and obey a
CFT transformation law for a primary of dimension $\Delta$. We define the
bulk-to boundary-propagator in AdS

$\displaystyle\lim_{\lambda\to\infty}\lambda^{-\Delta}\Pi_{\Delta}\left(\lambda
P+\frac{1}{\lambda}\ldots,Y\right)=K_{\Delta}(P,Y)\,.$ (28)

For the scalar field

$\displaystyle K_{\Delta}(P,Y)$
$\displaystyle=\frac{\Gamma(\Delta)}{2\sqrt{\pi}^{d}\Gamma\left(\Delta-\frac{d}{2}+1\right)}\frac{1}{(-2P\cdot
Y)^{\Delta}}\,,$ (29)

and similarly for the spinning operators. When comparing this to the dS
result, one has to be careful of phases. Although the boundary point $P^{2}=0$
is the same, the path through which it is reached is different. In the
Poincaré patch in AdS, we follow $\lambda\sim\frac{1}{z}$, and in dS,
$\lambda\sim\frac{1}{(-\eta)}$, yielding a phase difference. This is precisely
the phase factor we accounted for previously when rewriting the field
insertions of dS in terms of a mixture of EAdS fields. The rules previously
explained suffice to fix the continuation of the external lines as well.

###  2.4 Feynman Rules for dS Amplitudes

Through the careful study of the propagators in de Sitter space, we have re-
derived the rules of Sleight:2021plv  to write tensorial amplitudes using
effective AdS processes.111Our formulation differs slightly as we do not
include possible phase factors due to derivative interactions, and we do not
write the rules in Mellin Space. This set of rules will be completed by the
equivalent ones for fermion lines in sec. 3.4 . The transcription scheme for
tensorial processes is :

  * •

First, draw the diagram in dS, specifying the dimensions of each external
line.

  * •

For each internal line with $\Delta,J$, sum over two diagrams:

$\displaystyle\includegraphics[valign={c},width=85.35826pt]{diagram/prop1}=\frac{1}{c_{\Delta_{+}}}\includegraphics[valign={c},width=85.35826pt]{diagram/prop2}+\frac{1}{c_{\Delta_{-}}}\includegraphics[valign={c},width=85.35826pt]{diagram/prop3}\,.$
(30)

  * •

For each vertex, we have to fix a choice of contour. Then, we multiply by the
required phases and normalisation factor to transform it into an AdS vertex.
We then sum over each contour. Doing this sum explicitly for a general vertex
we find :

$\displaystyle\includegraphics[valign={c},width=128.0374pt]{diagram/int1}=\lambda_{(\Delta_{i},J_{i})}\includegraphics[valign={c},width=113.81102pt]{diagram/int2}\,.$
(31)

  * •

Finally, evaluate the diagrams using whichever methods one prefers.

For each internal lines joining two vertices, the factors of $c_{\Delta}$ get
repeated on both vertices, and so we must divide the internal lines by it to
avoid overcounting. The different constant we defined are

$\displaystyle\lambda_{(\Delta_{i},J_{i})}$
$\displaystyle=2\sin(\frac{\pi}{2}(d+\sum(\Delta_{i}+J_{i})))\prod_{i}c_{\Delta_{i}}\,,$
$\displaystyle c_{\Delta}$
$\displaystyle=\frac{1}{2\sin(\frac{\pi}{2}(d-2\Delta))}\,.$ (32)

##  3 Fermionic Cosmological Correlator

We have reviewed how to rewrite a generic perturbative diagram involving
tensorial operators in dS as an effective EAdS theory. The present section
contains the main new result of this paper, which is an extension of the
previous analysis to fermions. Since continuation of the integrals is left
unchanged, the only issue left is to find how to transform the propagators.

This is a subtle matter, owing to fermion’s sensitivity to signature. There
are also multiple angles of attacks, as one can consider the problem in
embedding and real space, and in both cases one must heed the effect of the
Wick rotation. As continuation of the scalar functions found in the
propagators themselves is straightforward from the method previously used for
tensors, the technicality lies in the kinematical part.

We start by giving a parallel exposition of spinors in both dS
Pethybridge:2021rwf  and EAdS Henningson:1998cd ; Kawano:1999au ;
Nishida:2018opl , spelling out the embedding and real-space picture in both.
This sets out our convention and clarifies the different objects to be
related. We then turn to the analytical continuation proper. As in the
tensorial case, we use expression of the embedding object in terms of real-
space one to transform them from one space to the other. We give explicit
expressions for the structure transformations, which with the change of the
the scalar functions in the propagators, gives the final relations. The output
of this computation is presented as a set of Feynman rules incorporating
fermion lines. We finish with an illustrative computation using the formalism
outlined. As we make use of spinorial harmonic functions, we give back some
details regarding their properties in appendix A.

###  3.1 Fermions in de Sitter

Figure 3: Uplift and Projection from the Poincaré patch of dSd+1 into the
embedding picture for the spinor bundle

We look at Dirac fermions in $\mathbb{R}^{1,d+1}$. If $d+2$ is odd, pick a
direct sum of two Dirac fermions transforming in representation with the
reflected gamma matrices. This allows to work in a dimension agnostic way in
the embedding. We define the gamma matrices,

$\displaystyle\begin{aligned} \Gamma_{0}&=\begin{pmatrix}0&&1\\\
-1&&0\end{pmatrix}\otimes\mathds{1}\,,\\\
\Gamma_{\star}&=\begin{pmatrix}1&&0\\\
0&&-1\end{pmatrix}\otimes-i\gamma_{0}\,,\end{aligned}$
$\displaystyle\begin{aligned} \Gamma_{a}&=\begin{pmatrix}1&&0\\\
0&&-1\end{pmatrix}\otimes\gamma_{0}\gamma_{a}\,,\\\
\Gamma_{+}&=\begin{pmatrix}0&&1\\\
0&&0\end{pmatrix}\otimes\mathds{1}\,,\end{aligned}$
$\displaystyle\begin{aligned} \Gamma_{d+1}&=\begin{pmatrix}0&&1\\\
1&&0\end{pmatrix}\otimes\mathds{1}\,,\\\ \Gamma_{-}&=\begin{pmatrix}0&&0\\\
-1&&0\end{pmatrix}\otimes\mathds{1}\,.\end{aligned}$ (33)

Note that even if $d+2$ is odd, $\Gamma_{\star}$ exists still as the
representation is reducible. The real-space gamma matrices obey
$\gamma_{\mu}^{\dagger}=\gamma_{0}\gamma_{\mu}\gamma_{0}$ and
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\eta_{\mu\nu}$ with mostly plus metric. In
the embedding we define the conjugate of $\Psi$ to be
$\overline{\Psi}=\Psi^{\dagger}(-i\Gamma_{0})$. In real space, we define
$\overline{\psi}=i\psi^{\dagger}\gamma_{0}$. The minus-sign difference is
there for convenience, and does not influence the reality property of
bilinears VanProeyen:1999ni .

There are two equivalent pictures one can use. We can work with constrained
embedding spinors, which are in one-to-one map with real-space spinors, as is
illustrated in fig.3. They obey the equation

$\displaystyle X^{A}\Gamma_{A}\Psi(X)$ $\displaystyle=\Psi(X)\,,$ (34)

and their upper components transform like $\psi/\sqrt{-\eta}$, with $\psi$
transforming as a Dirac spinor field on $dS_{d+1}$, at the point $x^{\mu}$
associated to $X^{A}$, as in (1) Weinberg:2010ws ; Pethybridge:2021rwf . The
second perspective is to work with an unconstrained spinors $\Psi$, and
contract it with a (commuting) polarisation spinor $\overline{S}$ which
satisfies the constraint, projecting out the supplementary degrees of freedom.
This mimics the approach used for tensors, where we restricted the
polarisations to be transverse. We will proceed with this perspective now, as
it gives the more succint expressions. The polarisation spinors can be solved
for in terms of real-space objects

$\displaystyle X^{A}\Gamma_{A}S=-S$ $\displaystyle\implies\hskip
28.45274ptS=\begin{pmatrix}\gamma_{0}\\\
-\not{x}\end{pmatrix}\frac{s}{\sqrt{-\eta}}\,,$ (35)
$\displaystyle\overline{S}X^{A}\Gamma_{A}=\overline{S}$
$\displaystyle\implies\hskip
28.45274pt\overline{S}=\frac{\overline{s}}{\sqrt{-\eta}}\begin{pmatrix}-\not{x}\gamma_{0}&&1\end{pmatrix}\,.$

Like the covariant derivative, one can uplift the Dirac operator to the
embedding, becoming

$\displaystyle\not{\nabla}$
$\displaystyle=\not{\partial}-\not{X}\left(X\cdot\partial+\frac{d+1}{2}\right)\,.$
(36)

This expression can be derived through a formal study of spinor bundle
Trautman:1992 ; Trautman:1995fr , or by noticing it is fixed by its
anticommutation with the constraint. There is a slight subtlety in embedding,
which is understood by looking at the following object for constrained spinors

$\displaystyle\overline{\Psi}\not{\nabla}\Psi$
$\displaystyle=-2\overline{\psi}\not{\nabla}\psi\,,$
$\displaystyle\overline{\Psi}i\Gamma_{\star}\Psi$
$\displaystyle=-2\overline{\psi}\psi\,.$ (37)

From which it follows that one can equivalently write the Dirac equation as

$\displaystyle(\not{\nabla}+m)\psi=0\,\Leftrightarrow\,(i\Gamma_{\star}\not{\nabla}-m)\Psi=0\,.$
(38)

This machinery allows to solve the Dirac equation on the two-point function
easily. First, note that the Wightman function is necessarily of the form

$\displaystyle\expectationvalue{\Psi(X,\overline{S}_{1})\overline{\Psi}(Y,S_{2})}$
$\displaystyle=\overline{S}_{1}S_{2}g_{+}(u)+\overline{S}_{1}\Gamma_{\star}S_{2}g_{-}(u)$
(39) $\displaystyle=W_{1/2,\Delta}(X,\overline{S}_{1};Y,S_{2})$
$\displaystyle=\overline{s}_{1}W_{1/2,\Delta}(x;y)s_{2}\,.$

The different structures take the real-space form

$\displaystyle\overline{S}_{1}S_{2}$
$\displaystyle=\overline{s}_{1}\frac{\gamma^{\mu}(x-y)_{\mu}}{\sqrt{\eta_{x}\eta_{y}}}s_{2}\,,$
$\displaystyle\overline{S}_{1}\Gamma_{\star}S_{2}$
$\displaystyle=\overline{s}_{1}\frac{i\gamma_{0}\gamma^{\mu}(\tilde{x}-y)_{\mu}}{\sqrt{\eta_{x}\eta_{y}}}s_{2}\,,$
(40)

with $\tilde{x}^{\mu}=(-x^{0},x^{a})$ the time-reversal of $x^{\mu}$, $s$ and
$\overline{s}$ the real-space polarisation spinors. The Wightman function then
solves the Dirac equation with Bunch-Davies and Hadamard conditions.

As before, we restrict the singularities to coincident point. For the
normalisation, we take the short-distance limit of this singularity compared
with the flat-space one. In dS, the geodesic distance for nearly coincident
point is given by $D^{2}\sim 4(1-u)$, and we should have a coincident limit
$u\rightarrow 1$ of the propagator with behaviour
$-\kappa_{f}\frac{\gamma_{\mu}\partial^{\mu}(D^{2})}{2(D^{2})^{\frac{d+1}{2}}}$,
with $\kappa_{f}$ the usual fermion 2pt. function normalisation. An equivalent
computation is to note that the equations of motions are satisfied everywhere
except at the coincident limit $x\sim y$. Hence we can approximate the
correlator by its leading behaviour in the coincident limit, and act on it
with the Dirac operator. We recognise the resulting expression as a
specialisation of

$\displaystyle\delta^{d+1}(x-y)=\frac{1}{S_{d+1}}\partial_{\mu}\left(\frac{x^{\mu}-y^{\mu}}{\absolutevalue{x-y}^{d+1}}\right)\,,$
(41)

whose normalisation we match with the one for a Lorentzian Green function.

The output of this computation is

$\displaystyle g_{+}(u)$
$\displaystyle=\kappa\,_{2}F_{1}\left(\Delta+\frac{1}{2},d-\Delta+\frac{1}{2};\frac{d+1}{2};u\right)\,,$
(42) $\displaystyle g_{-}(u)$
$\displaystyle=\kappa\frac{(d-2\Delta)}{d+1}\,_{2}F_{1}\left(\Delta+\frac{1}{2},d-\Delta+\frac{1}{2};\frac{d+1}{2}+1;u\right)\,,$
$\displaystyle\kappa$
$\displaystyle=\frac{\Gamma(\Delta+\frac{1}{2})\Gamma(d-\Delta+\frac{1}{2})}{2(4\pi)^{\frac{d+1}{2}}\Gamma(\frac{d+1}{2})}\,,$

with the interplay between the Dirac and Casimir equation fixing
$\Delta=\frac{d}{2}+im$. This result can also be derived from a more involved
mode resummation in canonical quantisation, as in sec. 4.2 . The different
propagators follow from the previous prescription.

We will write $W_{1/2,\Delta}$ to represent interchangeably the propagators in
the real or embedding space picture. The embedding space picture is the most
natural one, we find, to express the kinematic transformation induced by Wick
rotation. Because of this we will first perform the continuation of each
element in the embedding picture independently, but then write out the result
in real-space. The position space picture is still practical, as one
encounters in diagrams objects of the form

$\displaystyle\ldots W_{1/2,\Delta}(x;y)W_{1/2,\Delta}(y;z)\ldots\,,$ (43)

Which are to be understood as a matrix multiplication. But in embedding space,
the equivalent expression is more convoluted, as one must project out the
unphysical component of the spinor indices by inserting $1+i\not{X}$ for
contractions at position $X^{A}$. Which object we are manipulating should be
clear from context.

Finally, we can take the boundary limit of the correlator if needed. One has
to be careful, as the polarisation spinors have a non-trivial scaling as one
goes towards the boundary, making the boundary spinor scale like
$\Delta+\frac{1}{2}$ instead of $\Delta$, as is usual in the embedding picture
for CFT Weinberg:2010ws . We define the boundary polarisation spinors to be

$\displaystyle P^{A}\Gamma_{A}S_{\partial}=0$ $\displaystyle\implies\hskip
28.45274ptS_{\partial}=\begin{pmatrix}\gamma_{0}\\\
-\gamma_{i}x^{i}\end{pmatrix}s\,,$ (44)
$\displaystyle\overline{S}_{\partial}P^{A}\Gamma_{A}=0$
$\displaystyle\implies\hskip
28.45274pt\overline{S}_{\partial}=\overline{s}\begin{pmatrix}-\gamma_{i}x^{i}\gamma_{0}&&1\end{pmatrix}\,.$

And when we take the boundary limit of the bulk to bulk correlator, one
obtains two different types of bulk-to-boundary correlator. They can be
written using

$\displaystyle K_{1/2,\Delta}^{dS,\pm}(X,\overline{S}_{1};Y,S_{2})$
$\displaystyle=\frac{N_{\Delta}}{2\cos(\frac{\pi(d-2\Delta)}{2})}\frac{\overline{S}_{1}(1\pm\Gamma_{\star})S_{2}}{(-2X\cdot
Y)^{\Delta+\frac{1}{2}}}\,,$ (45) $\displaystyle N_{\Delta}$
$\displaystyle=\frac{\Gamma\left(\Delta+\frac{1}{2}\right)}{2\sqrt{\pi}^{d}\Gamma\left(\Delta-\frac{d-1}{2}\right)}\,.$

Where either $X$ or $Y$ are on the lightcone, and respectively
$\overline{S}_{1}$ or $S_{2}$ are of the boundary type. The two specific
propagators encountered are $K^{dS,\pm}_{1/2,\Delta_{\pm}}$, with
$\Delta_{\pm}=\frac{d}{2}\pm im$.

###  3.2 Fermions in Anti-de Sitter

This section reproduces the previous rigmarole but in Euclidean anti-de Sitter
space this time. Our approach is inspired by previous exposition of
Pethybridge:2021rwf , and draws on Nishida:2018opl , but our conventions
differ vastly from the latter. Since we are in an Euclidean space, there is no
preferred reality structure to be imposed a-priori on the fermionic fields. In
effect, we will use the one inherited from the de Sitter space, which in the
usual picture gives a non-unitary, non-reflection positive theory in AdS.

Figure 4: Uplift and Projection from the Poincaré patch of EAdSd+1 into the
embedding picture for the spinor bundle

We reuse the previous choice of gamma matrices (33), but now define
$\overline{\gamma}_{0}=-i\gamma_{0}$. Because in AdS $X^{A}X_{A}=-1$, the
eigenvalue equation satisfied by the constrained embedding spinors is
different. Namely, one finds

$\displaystyle X^{A}\Gamma_{A}\Psi=-i\Psi\,,$ (46)

and because now the eigenvalue is complex, it follows that the conjugate
satisfies the same equation with eigenvalue $-i$ as well. Such a spinor has an
upper component which transforms like $\psi/\sqrt{z}$, with $\psi$ a Dirac
spinor field in EAdSd+1. This is illustrated in fig.4. As we did previously,
we can use the associated unconstrained picture instead. We introduce
polarisation spinors, which satisfy

$\displaystyle X_{H}^{A}\Gamma_{A}S_{H}=-iS_{H}$ $\displaystyle\implies\hskip
28.45274ptS_{H}=\begin{pmatrix}1\\\
-i\overline{\gamma}_{0}\not{x}\end{pmatrix}\frac{s}{\sqrt{z}}\,,$ (47)
$\displaystyle\overline{S}_{H}X_{H}^{A}\Gamma_{A}=-i\overline{S}_{H}$
$\displaystyle\implies\hskip
28.45274pt\overline{S}_{H}=\frac{\overline{s}}{\sqrt{z}}\begin{pmatrix}\not{x}\overline{\gamma}_{0}&&i\end{pmatrix}\,.$

Note that by writing $\gamma^{\mu}x_{\mu}=\not{x}$ we now mean
$\gamma^{i}x_{i}+\overline{\gamma}_{0}z$. Which one is meant should be clear
from context. Because these spinors are Euclidean, formally $s$ and
$\overline{s}$ are independent quantities. We are effectively free to specify
the reality structure encoded by the operation $\bullet^{\dagger}$ to make
sense of it. $\overline{S}$ and $S$ are still related by
$\overline{S}=S^{\dagger}(-i\Gamma_{0})$, but we also have to specify how to
make sense of the conjugation of the EAdS real-space spinors. We take the
reality conventions imported from dSd+1, which is not the usual choice of
using the ones of Lorentzian AdSd+1.

To find the Wightman function, we proceed as before. The Dirac operator takes
the form

$\displaystyle\not{\nabla}_{H}$
$\displaystyle=\not{\partial}+\not{X}_{H}\bigg{(}X\cdot\partial+\frac{d+1}{2}\bigg{)}\,.$
(48)

There is now a stark difference in the uplift of spinors bilinears, so we must
be careful when trying to write the Dirac equation properly in the embedding.
For constrained embedding spinors, one has

$\displaystyle\overline{\Psi}_{H}\Gamma_{\star}\not{\nabla}_{H}\Psi_{H}$
$\displaystyle=2i\overline{\psi}\not{\nabla}\psi\,,$
$\displaystyle\overline{\Psi}_{H}\Psi_{H}$
$\displaystyle=2\overline{\psi}\psi\,.$ (49)

It follows that the Dirac equation takes exactly the same form as in de
Sitter,

$\displaystyle(\not{\nabla}+m)\psi=0\,\Leftrightarrow\,(i\Gamma_{\star}\not{\nabla}_{H}-m)\Psi_{H}=0\,.$
(50)

Note however that because the polarisation spinors are quite different, the
structures appearing in the propagator, although seemingly identical in the
embedding, differ quite a lot once projected in real space

$\displaystyle\overline{S}_{H}^{1}\Gamma_{\star}S_{H}^{2}$
$\displaystyle=\overline{s}_{1}\frac{\gamma^{\mu}(x-y)_{\mu}}{\sqrt{z_{1}z_{2}}}s_{2}\,,$
$\displaystyle\overline{S}^{1}_{H}S_{H}^{2}$
$\displaystyle=-\overline{s}_{1}\frac{\overline{\gamma}_{0}\gamma^{\mu}(\tilde{x}-y)_{\mu}}{\sqrt{z_{1}z_{2}}}s_{2}\,.$
(51)

We have all the tools needed. Acting on the ansatz with the Dirac equation,
one finds two independent solutions. We write the general solution as

$\displaystyle\expectationvalue{\Psi_{H}(X,\overline{S}_{1})\overline{\Psi}_{H}(Y,S_{2})}$
$\displaystyle=\Pi_{1/2,\Delta}^{\pm}(u)=\overline{s}_{1}\Pi^{\pm}_{1/2,\Delta}(x;y)s_{2}$
(52)
$\displaystyle=\overline{S}_{H}^{1}\Gamma_{\star}S_{H}^{2}f_{+}(u)\pm\overline{S}_{H}^{1}S_{H}^{2}f_{-}(u)\,.$

The functions $f_{\pm}$ are specified through

$\displaystyle f_{+}(u)$
$\displaystyle=-\frac{\Gamma\left(\Delta+\frac{1}{2}\right)}{2\sqrt{\pi}^{d}\Gamma\left(\Delta-\frac{d-1}{2}\right)}\frac{1}{(-4z)^{\Delta+\frac{1}{2}}}\,_{2}F_{1}\left(\Delta+\frac{1}{2},\Delta-\frac{d}{2};2\Delta-d+1;\frac{1}{u}\right)\,,$
(53) $\displaystyle f_{-}(u)$
$\displaystyle=-\frac{\Gamma\left(\Delta+\frac{1}{2}\right)}{2\sqrt{\pi}^{d}\Gamma\left(\Delta-\frac{d-1}{2}\right)}\frac{1}{(-4z)^{\Delta+\frac{1}{2}}}\,_{2}F_{1}\left(\Delta+\frac{1}{2},1+\Delta-\frac{d}{2};2\Delta-d+1;\frac{1}{u}\right)\,.$

These functions are normalised by looking at the coincident limit and
requiring them to be Euclidean Green function which have a $-\delta(\ldots)$
source term.

The Dirac equation has two independent solutions given by
$\Pi^{\pm}_{1/2,\frac{d}{2}\pm m}(u)$ for $m>0$. $m\in\mathbb{R}^{+}$ is
equivalent to a unitary field in AdS in the so-called normal and alternate
quantisation Henningson:1998cd ; Henneaux:1998ch . These functions have an
interpretation in term of boundary conformal blocks of a fermionic bCFT, and
the structures are likewise related under a Weyl transformation Herzog:2022jlx
. The principal series, though unitary in de Sitter space, can be realised in
AdS through an imaginary mass for the fermion, which must be approached from
the positive real part part of the complex plane. Note that this is also the
type of mass which is consistent with our reality assignment for
$\overline{\psi}\psi$, although unusual from an AdS perspective. This is not
an issue whatsoever, because the AdS space we consider is an abstract object,
a useful computational tool.

As before, one can take the boundary limit of this object. The bulk-to-
boundary correlator is easily found to be given by

$\displaystyle K_{1/2,\Delta}^{\pm}(X,\overline{S}_{1};Y,S_{2})$
$\displaystyle=\overline{s}_{1}K_{1/2,\Delta}^{\pm}(x;y)s_{2}=-N_{\Delta}\frac{\overline{S}_{1}(\Gamma_{\star}\pm
1)S_{2}}{(-2X\cdot Y)^{\Delta+\frac{1}{2}}}\,,$ (54)

where either $X$ or $Y$ are on the lightcone, and one must replace the
corresponding polarisation by the boundary one. The structures takes the
explicit form

$\displaystyle\overline{S}_{1}(\Gamma_{\star}\pm
1)S_{2,\partial}=\overline{s}_{1}\frac{\gamma\cdot(x-y)(1\pm\overline{\gamma}_{0})}{\sqrt{z_{1}}}s_{2}\,,$
(55) $\displaystyle\overline{S}_{1,\partial}(\Gamma_{\star}\pm
1)S_{2}=\overline{s}_{1}\frac{(1\mp\overline{\gamma}_{0})\gamma\cdot(x-y)}{\sqrt{z_{2}}}s_{2}\,.$

To compute exchange diagrams, it is convenient to define the harmonic function
Costa:2014kfa ; Nishida:2018opl , which is bilocal in the bulk and is
constructed using an integrated product of bulk-to-boundary propagators with
specific dimensions,

$\displaystyle\Omega_{1/2,\nu}(X,\overline{S}_{1};Y,S_{2})$
$\displaystyle=\frac{i}{2\pi}\int_{\partial
H}[dP]\,K_{1/2,\frac{d}{2}+i\nu}^{+}(X,\overline{S}_{1};P)\not{P}K_{1/2,\frac{d}{2}-i\nu}^{-}(P;Y,S_{2})$
(56) $\displaystyle=\overline{s}_{1}\Omega_{1/2,\nu}(x;y)s_{2}\,.$

Where by $[dP]$ we denote the conformally invariant integral over the light-
cone Simmons-Duffin:2014wb . Through an explicit computation, easily done in
the embedding using the formula from appendix A of Herzog:2022jlx , one can
relate it to the functions previously found

$\Omega_{1/2,\nu}(x;y)=\frac{1}{2\pi}\left(\Pi^{+}_{1/2,h+i\nu}(x;y)-\Pi^{-}_{1/2,h-i\nu}(x;y)\right)\,.$
(57)

The harmonic function satisfies the differential equation

$\displaystyle(\not{\nabla}_{x}+i\nu)\Omega_{1/2,\nu}(x;y)=0=\Omega_{1/2,\nu}(x;y)(\overset{\leftarrow}{\not{\nabla}}_{y}-i\nu)\,,$
(58)

making them an orthonormal basis. We give in appendix A a detailed derivation
of the orthogonality relation

$\displaystyle\int_{AdS}d^{d+1}y\,\Omega_{1/2,\nu}(x;y)\Omega_{1/2,\nu^{\prime}}(y;z)=\delta(\nu-\nu^{\prime})\Omega_{1/2,\nu}(x;z)\,,$
(59)

which explains the specific normalisation chosen for the integral. This type
of relation allows us to rewrite the propagators we found in terms of a
harmonic decomposition

$\displaystyle\Pi_{1/2,h\pm
i\nu}^{\pm}(x;y)=-i\int_{-\infty}^{\infty}\frac{d\nu^{\prime}}{\nu^{\prime}-\nu\pm
i\epsilon}\Omega_{1/2,\nu^{\prime}}(x;y)\,.$ (60)

These expressions are particularly symmetric because one has to analytically
continue the result with real $m>0$ to $i\nu$, and the $\epsilon$ part is
necessary to pick up the right contour. Away from the principal series, the
two propagators have more distinct expressions.

###  3.3 Analytic Continuation

Having properly defined the different players, we can consider their relation
under the analytic continuation 13 . The fermions have a more involved
kinematic behaviour under this transformation, as we can illustrate quickly
through a simple argument.

We have seen that the embedding coordinates are deformed along the path
$X\rightarrow e^{\mp i\frac{\pi}{2}}X$. At the level of the eigenvalue
equation satisfied by the constrained spinors, this gives a are mapping from
one to the other, up to a sign. This sign is meaningful, since we pick spinors
in EAdS to obey the specific eigenvalue equation
$X^{A}_{H}\Gamma_{A}\Psi_{H}(X)=-i\Psi_{H}(X)$, from which it follows that
$\Psi^{r}\rightarrow\Psi_{H}$, while
$\Psi^{l}\rightarrow\Gamma_{\star}\Psi_{H}$. Likewise, since $\overline{\Psi}$
has opposite eigenvalue, it follows that one has
$\overline{\Psi}^{l}\rightarrow\overline{\Psi}_{H}$ and
$\overline{\Psi}^{r}\rightarrow\Gamma_{\star}\overline{\Psi}_{H}$. This
implies the Wick rotation of the spinors act non-trivially on the kinematical
structures entering the Wightman function.

To be more quantitative about this behaviour, we can turn to the
unconstrained-spinor contracted with polarisation picture. We use the explicit
expressions (35),(47) and perform the Wick rotation of (13), to obtain the
relations

$\displaystyle\begin{aligned}
\overline{S}^{l}[\overline{s}]&\rightarrow+e^{+i\frac{\pi}{4}}\overline{S}_{H}[\overline{s}\overline{\gamma}_{0}]\Gamma_{\star}\,,\\\
S^{l}[s]&\rightarrow-e^{+i\frac{\pi}{4}}S_{H}[\overline{\gamma}_{0}s]\,,\end{aligned}$
$\displaystyle\begin{aligned}
\overline{S}^{r}[\overline{s}]&\rightarrow-e^{-i\frac{\pi}{4}}\overline{S}_{H}[\overline{s}]\,,\\\
S^{r}[s]&\rightarrow-e^{-i\frac{\pi}{4}}\Gamma_{\star}S_{H}[s]\,.\end{aligned}$
(61)

By combining this result with the hypergeometric identities (LABEL:eq:id1),
(LABEL:eq:id2), we can analytically continue the propagators. Replacing the
structures and the functions multiplying them by their analytically continued
expressions, we decompose the result by inspection into a sum of AdS
propagators.

Consider first the simplest case of fields in the $lr$ and $rl$ configuration,

$\displaystyle W_{1/2,\Delta}^{lr}(x;y)$
$\displaystyle\rightarrow\frac{-\overline{\gamma}_{0}}{2\cos(\frac{\pi(d-2\Delta)}{2})}\left(\Pi^{+}_{1/2,\Delta}(x;y)-\Pi^{-}_{1/2,d-\Delta}(x;y)\right)\,,$
(62) $\displaystyle W_{1/2,\Delta}^{rl}(x;y)$
$\displaystyle\rightarrow\left(\Pi^{+}_{1/2,\Delta}(x;y)-\Pi^{-}_{1/2,d-\Delta}(x;y)\right)\frac{\overline{\gamma}_{0}}{2\cos(\frac{\pi(d-2\Delta)}{2})}\,.$

The non-trivial factors are now due to the spinor polarisation. Looking at
fields in the same configuration,

$\displaystyle W_{1/2,\Delta}^{ll}(x;y)$
$\displaystyle\rightarrow\frac{-\overline{\gamma}_{0}}{2\cos(\frac{\pi(d-2\Delta)}{2})}\left(e^{i\pi\Delta}\Pi^{+}_{1/2,\Delta}(x;y)+e^{i\pi(d-\Delta)}\Pi^{-}_{1/2,\Delta}(x;y)\right)\overline{\gamma}_{0}\,.$
(63) $\displaystyle W_{1/2,\Delta}^{rr}(x;y)$
$\displaystyle\rightarrow\frac{1}{2\cos(\frac{\pi(d-2\Delta)}{2})}\left(e^{-i\pi\Delta}\Pi^{+}_{1/2,\Delta}(x;y)+e^{-i\pi(d-\Delta)}\Pi^{-}_{1/2,\Delta}(x;y)\right)\,.$

Like we did for tensors, these different propagators decomposition can be
reformulated in terms of a change of basis to a pair of shadow-dual AdS fields

$\displaystyle\begin{aligned}
\psi^{l}&=-\overline{\gamma}_{0}\left(e^{i\frac{\pi}{2}\Delta}\psi_{+}+e^{i\frac{\pi}{2}(d-\Delta)}\psi_{-}\right)\,,\\\
\psi^{r}&=\left(e^{-i\frac{\pi}{2}\Delta}\psi_{+}-e^{-i\frac{\pi}{2}(d-\Delta)}\psi_{-}\right)\,,\end{aligned}$
$\displaystyle\begin{aligned}
\overline{\psi}^{l}&=\left(e^{i\frac{\pi}{2}\Delta}\overline{\psi}_{+}+e^{i\frac{\pi}{2}(d-\Delta)}\overline{\psi}_{-}\right)\overline{\gamma}_{0}\,,\\\
\overline{\psi}^{r}&=\left(e^{-i\frac{\pi}{2}\Delta}\overline{\psi}_{+}-e^{-i\frac{\pi}{2}(d-\Delta)}\overline{\psi}_{-}\right)\,,\end{aligned}$
(64)

whose propagators are normalised in a unusual way

$\displaystyle\expectationvalue{\psi_{+}(x)\overline{\psi}_{+}(y)}$
$\displaystyle=\frac{\Pi_{1/2,\Delta}^{+}(x;y)}{2\cos(\frac{\pi(d-2\Delta)}{2})}\,,$
$\displaystyle\expectationvalue{\psi_{-}(x)\overline{\psi}_{-}(y)}$
$\displaystyle=\frac{\Pi_{1/2,d-\Delta}^{-}(x;y)}{2\cos(\frac{\pi(d-2\Delta)}{2})}\,.$
(65)

Note that these equations are consistent with the eigenvalue discussion. The
pair of AdS fields obtained sit in the two quantisation schemes of spinors
with imaginary masses $\pm im$. It is interesting to note how for tensors the
“spinning” part of the phase factor, $e^{\pm i\frac{\pi}{2}J}$, came about
because of the analytical continuation of the tensor structures, which was
quite simple, while here we have a more intricate phase structure appearing.
Note also how the bosonic correlators where normalised using a $\sin(\ldots)$,
while the fermionic ones uses a $\cos(\ldots)$ factor. Using the relations
(64) one can discuss the analytical continuation of general interaction terms.

As in the bosonic case, these replacement rules are enough to take into
account the change in the bulk-to-boundary propagators as well. These fully
specifies the rules to translate perturbation theory when fermions are
involved. They are enough to perform a practical computation involving spinors
in expanding space-time.

###  3.4 Feynman Rules With Spinors

Because of the slightly more involved phase in the fermion change of basis,
pairs of fermion lines alter the rules of sec.2.4. If a pair of fermion lines
in different quantisation are joined together at a vertex, they pick up a
further $-1$ difference between the left and right contour from the bosonic
case, turning the $\sin$ into a $\cos$.

Taking into account the previous results, we find that to evaluate a fermionic
correlator, one should proceed as follows

  * •

First, draw the diagram in dS, specifying the dimensions of each external
line.

  * •

For each internal spinor line, sum over two diagrams with weight
$\Delta_{\pm}$, and divide with factors of $c_{f,\Delta_{i}}$.

$\displaystyle\includegraphics[valign={c},width=85.35826pt]{diagram/propbis1}=\frac{1}{c_{f,\Delta_{+}}}\includegraphics[valign={c},width=85.35826pt]{diagram/propbis2}+\frac{1}{c_{f,\Delta_{-}}}\includegraphics[valign={c},width=85.35826pt]{diagram/propbis3}\,.$
(66)

  * •

For each vertex, we have to fix a choice of contour. Then, We multiply by the
required phases and normalisation factor to transform it into an AdS vertex.
We then sum over each contour. Doing this sum explicitly for a general vertex
with one pair of spinors, we find

$\displaystyle\includegraphics[valign={c},width=142.26378pt]{diagram/intbis1}=\lambda_{(\Delta_{k},J_{k})}^{ij}\includegraphics[valign={c},width=113.81102pt]{diagram/intbis2}\,.$
(67)

  * •

Finally, evaluate the AdS diagram using whichever method one prefers.

The constants are given by

$\displaystyle\lambda_{(\Delta_{k},J_{k})}^{(ij)}=\left(c_{f,\Delta_{i}}c_{f,\Delta_{j}}\prod_{i}c_{\Delta_{i}}\right)\bigg{(}$
$\displaystyle 2\delta_{ij}\cos(\frac{\pi}{2}(d+\sum(\Delta_{i}+J_{i})))$ (68)
$\displaystyle+2i\sigma_{ij}\sin(\frac{\pi}{2}(d+\sum(\Delta_{i}+J_{i})))\bigg{)}\,,$

Where in the sum one takes all the legs of the vertex, including fermions with
$J=1/2$. The normalisation and structure are specified by

$\displaystyle c_{f,\Delta}$
$\displaystyle=c_{\Delta+\frac{1}{2}}=\frac{1}{2\cos(\frac{\pi}{2}(d-2\Delta))}\,,$
$\displaystyle\sigma_{ij}$ $\displaystyle=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,.$ (69)

The generalisation to more pairs of spinors insertions follows through
straightforwardly using the relations (64) and choosing a specific vertex type
and summing over the ordering label. There is no new feature beyond the ones
already present here.

###  3.5 An Example of Cosmological Correlator

Using the formalism we develop, we can perform an analytical computation of a
cosmological correlator with fermions. To illustrate it, we will consider a
slightly contrived example which has the benefit of being relatively
tractable.

We will be looking at the fermion-exchange process between two scalars and two
spinors in different boundary conditions, interacting through a Yukawa term.
Methods to study such configurations are known in AdS Kawano:1999au ;
Nishida:2018opl ; Giombi:2021cnr . To lighten the notation, since all the
polarisation spinors which will appear are of the boundary type, we will drop
the $\partial$ label in this section. We also abbreviate $h=\frac{d}{2}$
throughout.

We use the rules previously derived to relate the tree-level process to a
linear combination of Witten diagrams. For each Witten diagram, we rewrite the
bulk-to-bulk spinor propagator using the harmonic representation in eq. (60).
We gather the expressions together,

$\displaystyle\includegraphics[valign={c},width=142.26378pt]{diagram/exchange1}=\int
d\nu\,r_{\nu}\int[dP]\,\includegraphics[valign={c},width=142.26378pt]{diagram/exchange2}\,.$
(70)

The precise configuration chosen is motivated by the ensueing selection rules
that simplify the computation to be performed. Like in boundary CFT, the
boundary spinors have a ”chirality”, and the structures to be sewn together by
the boundary integral over $P$ are simpler in this configuration. The spectral
parameter entering the integral is given by

$\displaystyle
r_{\nu}=i\frac{2\lambda^{2}}{\pi}c_{\Delta_{2}}c_{\Delta_{4}}c_{f,h+im}$
$\displaystyle
c_{f,h-im}\bigg{(}\frac{c_{f,h+im}\cos(\frac{\pi}{2}(2d+\Delta_{4}))\sin(\frac{\pi}{2}(2d+2im+\Delta_{2}))}{\nu-m+i\epsilon}$
(71)
$\displaystyle+\frac{c_{f,h-im}\cos(\frac{\pi}{2}(2d+\Delta_{2}))\sin(\frac{\pi}{2}(2d-2im+\Delta_{4}))}{\nu-
m-i\epsilon}\bigg{)}\,.$

By using the harmonic function, we have already split the difficulty. We have
two related bulk integrals to compute, and a final boundary one. Both bulk
integrals can be computed starting from the simpler diagrams

$\displaystyle\includegraphics[valign={c},width=85.35826pt]{diagram/intermediate}=$
$\displaystyle\int_{AdS}\frac{DX\,2C_{\Delta_{2}}N_{h+i\alpha
m_{1}}N_{h-i\alpha
m_{3}}\alpha\overline{S}_{1}(\Gamma_{\star}+\alpha)S_{3}}{(-2P_{1}\cdot
X)^{h+i\alpha m_{1}+\frac{1}{2}}(-2P_{2}\cdot X)^{\Delta_{2}}(-2P_{3}\cdot
X)^{h-i\alpha m_{3}+\frac{1}{2}}}$ (72)

The numerator of the integrand is found as follows. In the embedding, we free
the spinor indices of the bulk-to-boundary, multiply by $1+i\not{X}$ to
project out the unphysical components, and simplify. In real-space, we simply
multiply through the gamma matrices, finding the same result once uplifted
back. The configuration we chose for the 4-point function is tuned so that one
can get rid of the factors of $X^{A}$ on both 3-point functions.

We have written $C_{\Delta}=N_{\Delta-\frac{1}{2}}$, the normalisation of the
scalar propagator. As the remaining numerator is independent of $X^{A}$, it
reduces to the known result Freedman:1998tz ; Cornalba:2008qf ; Costa:2014kfa
of a scalar 3-point function vertex integration in AdS with some specific
weight combination. Collecting the factors, we find

$\displaystyle
n_{\Delta_{2},m_{1},m_{3},\alpha}\frac{\alpha\overline{S}_{1}(\Gamma_{\star}+\alpha)S_{3}}{(P_{12})^{\Delta_{12,3}}(P_{23})^{\Delta_{23,1}}(P_{31})^{\Delta_{31,2}}}\,.$
(73)

and we used the abbreviation $P_{ij}=-2P_{i}\cdot P_{j}$,
$\Delta_{ij,k}=\frac{\Delta_{i}+\Delta_{j}-\Delta_{k}}{2}$, and one should set
$\Delta_{1}=h+\frac{1}{2}+i\alpha m_{1}$ and $\Delta_{3}=h+\frac{1}{2}+i\alpha
m_{3}$. The coefficient is given by

$\displaystyle
n_{\Delta_{2},m_{1},m_{3},\alpha}=2C_{\Delta_{1}}^{2}C_{\Delta_{2}}^{2}C_{\Delta_{3}}^{2}\frac{\sqrt{\pi}^{d}\Gamma(\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}-d}{2})\Gamma(\Delta_{12,3})\Gamma(\Delta_{23,1})\Gamma(\Delta_{31,2})}{2\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\Delta_{3})}\,.$
(74)

With the abbreviations of earlier still applying. This factor, though
unwieldy, will simply become a part of the spectral integral left-over at the
end.

We can now rewrite the left-hand side of (70) as

$\displaystyle=\int
d\nu\mathcal{R}_{\nu}\mathcal{W}^{(h+im,\Delta_{2},h-im,\Delta_{4})}_{\nu}(P_{1},P_{2},P_{3},P_{4})\,.$
(75)

The spectral function $\mathcal{R}_{\nu}$ is given by

$\displaystyle\mathcal{R}_{\nu}$
$\displaystyle=-r_{\nu}n_{\Delta_{2},m,\nu,+}n_{\Delta_{4},\nu,m,-}\,,$ (76)

and the function $\mathcal{W}_{\nu}$ is the (non-properly normalised)
Euclidean conformal partial waves associated to the exchange of a primary of
spin $1/2$. There are multiple such partial waves in general, but this
corresponds to the single-one allowed by the selection rules of this
“chirality” configuration. It is found by recognising the leftover integral
over $P$ as a conformal integral Ferrara:1972kab ; Dolan:2001wg ; Dolan:2004up
; Dolan:2012wt  written in the embedding space Simmons-Duffin:2014wb . It can
be further expressed, using Mellin space for example, or by explicit
evaluation in some dimensions.

##  4 Källen-Lehman Decomposition For Spinors

Our previous result has probed the analytical structure of spinorial
insertions in de Sitter. The goal of the present section is to showcase a
different perspective on this issue, by investigating the notion of unitarity
Bros:1990cu ; Bros:1994dn ; Bros:1995js ; Bros:1998ik ; Bros:2010rku . The
central result of this section is a proof that the Wightman function of
generic spinor field in dS admits a positive spectral decomposition over the
principal series. The idea is the same as in flat space Weinberg2005 , that
one can insert in the correlator a complete set of state transforming
irreducibly under the isometry group. The contribution to the correlator of a
specific family of state then takes the form of an integral over a product of
wavefunctions. These are entirely fixed by symmetry to obey the Dirac
equation, and are precisely the mode functions one finds by canonically
quantising the free Dirac field. Finding the explicit spectral decomposition
becomes equivalent to performing the mode resummation for the Wightman
function of free fields.

Our exposition proceeds in reverse order. We first review the canonical
quantisation of the free scalar field in de Sitter space, to introduce a few
useful functions and tools. We then proceed to do the same for the free Dirac
field, rederiving (42). Finally, using this knowledge, we use the methods of
Hogervorst:2021uvp  to fix the wavefunctions of a generic Dirac spinor over an
on-shell state to be the free field modes, completing our proof.

###  4.1 Free Scalar Field, a Brief Reminder

The quantisation of the scalar field is a standard procedure Tagirov:1972vv ;
Chernikov:1968zm ; Schomblond:1976xc ; Mottola:1984ar ; Allen:1985ux ;
Spradlin:2001pw ; Anninos:2012qw , which we review here to set our conventions
and give a springboard for the more complicated case of the Dirac spinor.

Consider the action for a free massive scalar field,

$\displaystyle S$
$\displaystyle=\frac{1}{2}\int_{dS}d^{d+1}x\,\phi\left(\Box-m^{2}\right)\phi=-\frac{1}{2}\int\frac{d^{d}x\,d\eta}{(-\eta)^{d+1}}\left(\partial_{\eta}\phi\partial_{\eta}\phi-\partial_{i}\phi\partial^{i}\phi+m^{2}\phi^{2}\right)\,,$
(77)

with equations of motions

$\displaystyle\eta^{2}\partial_{\eta}^{2}\phi-(d-1)\eta\partial_{\eta}\phi+\left(\frac{d^{2}}{4}+\nu^{2}-\eta^{2}\nabla^{2}\right)\phi$
$\displaystyle=0\,.$ $\displaystyle m^{2}$
$\displaystyle=\frac{d^{2}}{4}+\nu^{2}\,,$ (78)

This equation will reappear in another form when we solve the Dirac equation.
Naturally, it simplifies in Fourier space, where one finds that the solutions
to (78) are linear combinations of Hankel functions multiplied by power-laws

$\displaystyle\phi(x,\eta)$ $\displaystyle=\int\frac{d^{d}k}{(2\pi)^{d}}\
e^{ik\cdot x}\phi_{k}(\eta)\,,$
$\displaystyle\phi_{k}(\eta)\sim(-\eta)^{\frac{d}{2}}H_{i\nu}^{(1/2)}(-k\eta)\,.$
(79)

These functions have many interesting properties, among which that they appear
generically in the modes of particles in de Sitter space. We pick the vacuum
of the theory, i.e. some specific behaviour for our solution. In the Bunch-
Davies vacuum, as $\eta\rightarrow-\infty$, we recover the polarisations of a
massless particle of the same type in flat-spacetime, i.e.

$\displaystyle\lim_{k}\phi_{k}(\eta)\propto\frac{e^{-ik\eta}}{\sqrt{2k}}a_{k}+\ldots\,.$
(80)

It is useful to define functions which have well normalised limits already

$\displaystyle h_{\alpha}(k\eta)$
$\displaystyle=\frac{\sqrt{\pi}e^{\frac{-i\alpha\pi}{2}-i\frac{\pi}{4}}}{2}H^{(2)}_{\alpha}(-k\eta)\,,$
$\displaystyle\overline{h}_{\alpha}(k\eta)$
$\displaystyle=\frac{\sqrt{\pi}e^{\frac{i\alpha\pi}{2}+i\frac{\pi}{4}}}{2}H^{(1)}_{\alpha}(-k\eta)\,.$
(81)

These satisfy

$\displaystyle\overline{h}_{\overline{\alpha}}(k\eta)$
$\displaystyle=\left(h_{\alpha}(k\eta)\right)^{\dagger}\,,$ $\displaystyle
h_{-\alpha}(k\eta)$ $\displaystyle=h_{\alpha}(k\eta)\,,$
$\displaystyle\lim_{\eta\rightarrow-\infty}h_{\alpha}(k\eta)$
$\displaystyle=\frac{1}{\sqrt{-\eta}}\frac{e^{ik\eta}}{\sqrt{2k}}\,,$ (82)

The most general solution for the field with the given boundary conditions is
then

$\displaystyle\phi(x,\eta)=\int\frac{d^{d}p}{(2\pi)^{d}}\
\underbrace{(-\eta)^{\frac{d}{2}}e^{ip\cdot
x}\overline{h}_{i\nu}(p\eta)}_{f_{p}(\eta)}a_{p}+(-\eta)^{\frac{d}{2}}e^{-ip\cdot
x}h_{i\nu}(p\eta)a_{p}^{\dagger}\,,$ (83)

where we also added the negative energy modes. $f_{p}(\eta)$ is the
wavefunction overlap between the field $\phi(x,\eta)$ and a one-particle state
in the Fock space of the $a_{p}$ oscillator.

We can check that this decomposition and normalisation reproduce the
commutation relations for the fields and the creation-annihilation operator,
by normalising using the Klein-Gordon inner product. Consider a cconstant
$\eta$ slice, the inner product is given by

$\displaystyle(f,g)=i\int\frac{d^{d}x}{(-\eta)^{d-1}}(f\nabla_{\eta}g^{\star}-g^{\star}\nabla_{\eta}f)$
(84)

The orthonormality of the wavefunctions follows directly

$\displaystyle(f_{p},f_{k})$
$\displaystyle=-i\int\frac{d^{d}x}{(-\eta)^{d-1}}e^{i(p-k)\cdot
x}(-\eta)^{d}\left(h_{i\nu}(k\eta)\partial_{\eta}\overline{h}_{i\nu}-\overline{h}_{i\nu}(k\eta)\partial_{\eta}h_{i\nu}\right)$
(85)
$\displaystyle=i(2\pi)^{d}\delta^{d}(p-k)\left(h_{i\nu}(k\eta)\eta\partial_{\eta}\overline{h}_{i\nu}-\overline{h}_{i\nu}(k\eta)\eta\partial_{\eta}h_{i\nu}\right)$
$\displaystyle=(2\pi)^{d}\delta^{d}(p-k)$

With this mode decomposition in hand, one can check that the canonical
commutation relations impose that the operators $a$ and $a^{\dagger}$ be of
Fock type. This is all that is needed to turn to the question of the Wightman
function. By definition, it is given through the evaluation of the 2-point
function on the Fock-vacuum $\ket{0}$ of the creation operators,

$\displaystyle\expectationvalue{\phi(x,\eta_{1})\phi(y,\eta_{2})}$
$\displaystyle=\bra{0}\phi(x,\eta_{1})\phi(y,\eta_{2})\ket{0}$ (86)
$\displaystyle=(\eta_{1}\eta_{2})^{\frac{d}{2}}\int\frac{d^{d}p\,d^{d}k}{(2\pi)^{2d}}e^{i(p\cdot
x-k\cdot
y)}\overline{h}_{i\nu}(p\eta_{1})h_{i\nu}(k\eta_{2})\bra{0}a_{p}a^{\dagger}_{k}\ket{0}$
$\displaystyle=(\eta_{1}\eta_{2})^{\frac{d}{2}}\int\frac{d^{d}p}{(2\pi)^{d}}\
e^{ip\cdot(x-y)}\ \overline{h}_{i\nu}(p\eta_{1})h_{i\nu}(p\eta_{2})$
$\displaystyle=\frac{\Gamma(\frac{d}{2}+i\nu)\Gamma(\frac{d}{2}-i\nu)}{(4\pi)^{\frac{d+1}{2}}\Gamma(\frac{d+1}{2})}\,_{2}F_{1}\left(\frac{d}{2}+i\nu,\frac{d}{2}-i\nu;\frac{d+1}{2};u\right)=I_{\nu}(u)\,.$

This does reproduce the result (6), with the variable $u$ given as in (5).
Performing this Fourier transform is quite involved. The interested reader can
follow the steps outlined in appendix A of Bunch:1978yq . These are the
analytical continuation of an integral representation for the product of
Bessel-K functions, followed by a chain of change of variables to massage the
expression into the Euler parametrisation of the hypergeometric function.

###  4.2 Quantising the Free Dirac Spinor

Armed with the results obtained for the scalar, we can turn to the more
complex Dirac field. The modes in Poincaré coordinates have been considered in
Cotaescu:2016sof . Consider a free massive Dirac fermion in dSd+1. Its action
is given through

$\displaystyle
S=-\int\frac{d^{d}x\,d\eta}{\eta^{d+1}}\overline{\psi}\left(\not{\nabla}+m\right)\psi\,,$
(87)

and noting the classic result
$\not{\nabla}=\eta\gamma^{\mu}\partial_{\mu}+\frac{d}{2}\gamma_{0}$
Henningson:1998cd  for the Dirac operator, the equation of motions are

$\displaystyle(\eta\partial_{\eta}-\frac{d}{2}+m\gamma_{0}+\eta\gamma_{0}\gamma_{i}\partial_{i})\psi=0\,.$
(88)

We divide $\psi$ in $\gamma_{0}$ eigenvectors $\psi_{+}$ and $\psi_{-}$, with
$\gamma_{0}\psi_{\pm}=\pm i\psi_{\pm}$, and obtain a system of coupled
differential equations

$\displaystyle\left(\eta\partial_{\eta}-\frac{d}{2}+im\right)\psi_{+}(\eta,k)$
$\displaystyle=+\eta k^{i}\gamma_{i}\psi_{-}(\eta,k)\,,$ (89)
$\displaystyle\left(\eta\partial_{\eta}-\frac{d}{2}-im\right)\psi_{-}(\eta,k)$
$\displaystyle=-\eta k^{i}\gamma_{i}\psi_{+}(\eta,k)\,.$

At this point, it is useful to name these operators and note some of of their
properties. Defining $f_{\nu}=(-\eta)^{\frac{d}{2}}h_{\nu}(k\eta)$, we have

$\displaystyle L_{\pm}$
$\displaystyle=\eta\partial_{\eta}-\left(\frac{d}{2}\pm\nu\right)\,,$
$\displaystyle L_{\pm}f_{\nu}$ $\displaystyle=ik\eta f_{\nu\pm 1}\,.$ (90)

To solve (89), we can iterate the equations to obtain a decoupled set of 2nd
order equations, for each individual eigenvector

$\displaystyle\left(\eta^{2}\partial_{\eta}^{2}-d\eta\partial_{\eta}+k^{2}\eta^{2}+\frac{(d+1)^{2}}{4}+\left(m\mp\frac{i}{2}\right)^{2}\right)\psi_{\pm}(\eta,k)$
$\displaystyle=0\,.$ (91)

These are equivalent to the equation for scalars (78), with the modification
$d\rightarrow d+1$, and identification $i\nu_{\pm}=im\pm\frac{1}{2}$. The
solutions can now be written using the functions (81), some $k$ dependant
constants, and an orthonormal basis of eigenspinors of $\gamma_{0}$, $\xi_{r}$
and $\chi_{r}$. This choice is particularly convenient to analyse physically
the mode expansion when $d=2k$, because one can take $\gamma_{0}$ diagonal and
all other matrices off-diagonal, giving a simple structure. When $d=2k+1$, it
is more convenient to diagonalise $\gamma_{\star}$, giving a chiral
representations. In that case, it is then interesting to redecompose the basis
into chiral eigenmodes. This is not necessary for our computation however. We
will pick $\xi_{r}$ to be an orthonormal basis of the half spinor, with
$\gamma_{0}\xi=+i\xi$. It spans a space with dimension
$\frac{1}{2}2^{[\frac{d+1}{2}]}$, which corresponds to the physical
polarisations state of a spinor.

From here, it is straightforward to work out the positive energy modes, which
correspond to asymptotic behaviour $e^{-ik\eta}$, selecting out the
$\overline{h}$ solution. Further imposing the first order equation relating
$\psi_{+}$ and $\psi_{-}$, fixes the relative coefficients, and expresses
$\chi$ modes in term of $\frac{k\cdot\gamma}{k}\xi_{r}$. To find the negative
energy mode, we simply have to consider repeating this operation while sending
$k^{i}\rightarrow-k^{i}$, and solving the analogous equation for $h$. We now
instead relate the $+i$ $\gamma_{0}$-eigenvectors to the $-i$ ones.

The end result of this computation is the mode decomposition for the spinor
field

$\displaystyle\psi(x,\eta)$
$\displaystyle=\int\frac{d^{d}p}{(2\pi)^{d}}\sum_{r}e^{ip\cdot
x}u_{r,p}(\eta)a_{r,p}+e^{-ip\cdot x}v_{r,p}(\eta)b^{\dagger}_{r,p}\,,$ (92)
$\displaystyle u_{r,p}(\eta)$
$\displaystyle=\frac{1}{\sqrt{2}}(-\eta)^{\frac{d+1}{2}}\left(\sqrt{p}\
\overline{h}_{im+\frac{1}{2}}(p\eta)-i\frac{p\cdot\gamma}{\sqrt{p}}\
\overline{h}_{im-\frac{1}{2}}(p\eta)\right)\xi_{r}\,,$ $\displaystyle
v_{r,p}(\eta)$
$\displaystyle=\frac{1}{\sqrt{2}}(-\eta)^{\frac{d+1}{2}}\left(i\frac{p\cdot\gamma}{\sqrt{p}}\
h_{im+\frac{1}{2}}(p\eta)+\sqrt{p}\
h_{im-\frac{1}{2}}(p\eta)\right)\chi_{r}\,.$

The results for the conjugate field are similarly found

$\displaystyle\overline{\psi}(x,\eta)$
$\displaystyle=\int\frac{d^{d}p}{(2\pi)^{d}}\sum_{r}e^{-ip\cdot
x}\overline{u}_{r,p}(\eta)a_{r,p}^{\dagger}+e^{ip\cdot
x}\overline{v}_{r,p}(\eta)b_{r,p}\,,$ (93)
$\displaystyle\overline{u}_{r,p}(\eta)$
$\displaystyle=\frac{1}{\sqrt{2}}(-\eta)^{\frac{d+1}{2}}\overline{\xi}_{r}\left(\sqrt{p}\
h_{im-\frac{1}{2}}(p\eta)-i\frac{p\cdot\gamma}{\sqrt{p}}\
h_{im+\frac{1}{2}}(p\eta)\right)\,,$ $\displaystyle\overline{v}_{r,p}(\eta)$
$\displaystyle=\frac{1}{\sqrt{2}}(-\eta)^{\frac{d+1}{2}}\overline{\chi}_{r}\left(i\frac{p\cdot\gamma}{\sqrt{p}}\
\overline{h}_{im-\frac{1}{2}}(p\eta)+\sqrt{p}\
\overline{h}_{im+\frac{1}{2}}(p\eta)\right)\,.$

Note $\overline{\xi}_{r}\gamma_{0}=i\overline{\xi}_{r}$ satisfy the same
eigenvalue equation as its conjugate. We remind the reader interested in
reproducing the expression for the conjugate that in our conventions
$\psi\rightarrow\overline{\psi}=i\psi^{\dagger}\gamma_{0}$,
$\gamma_{0}^{\dagger}=-\gamma_{0}$ VanProeyen:1999ni . For future convenience,
we will sometimes write these modes as $u_{r,p}^{+m}(\eta)$, to specify the
sign of the mass in the Dirac equation.

The normalisation we picked for the mode can be understood through the
following argument. Consider a free massless spinor in Minkowski spacetime.
Its polarisations spinors will have a generic shape
$\sqrt{p^{\mu}\sigma_{\mu}}\xi_{r}\sim\frac{p+\sigma_{i}p^{i}}{\sqrt{2p}}\xi_{r}$,
which is the one reproduced by (92) in the early-time limit. One does not have
to rely on this argument however, since these modes can be orthonormalised
with respect to the Dirac inner product, like in bosonic case, yielding back
this normalisation.

The Dirac inner product is given through the integral over a space-like slice
of the conserved Dirac current. A quick way to rederive it is as follows. We
write a manifestly vanishing integral over a volume $\Omega$ between two times
slices $\Sigma_{i}$ and $\Sigma_{f}$,

$\displaystyle 0$
$\displaystyle=i\int_{\Omega}\frac{d^{d}x\,d\eta}{(-\eta)^{d+1}}\left(\overline{\psi}_{1}(\not{\nabla}-m)\psi_{2}-\overline{\psi}_{1}(-\overset{\leftarrow}{\not{\nabla}}-m)\psi_{2}\right)$
(94) $\displaystyle=i\int
d^{d}x\,d\eta\,\partial_{\eta}\left(\frac{\overline{\psi}_{1}\gamma^{0}\psi_{2}}{(-\eta)^{d}}\right)=i\int_{\Sigma_{i}-\Sigma_{f}}\frac{d^{d}x}{(-\eta)^{d}}\overline{\psi_{1}}\gamma^{0}\psi_{2}\,,$

which allows to extract an invariant scalar product between Dirac spinors,
which we use to orthonormalise the modes.

Having fixed the mode decomposition of the field, we can use it to compute the
Wightman function of the Dirac spinors over the Bunch-Davies vaccuum. Consider
then

$\displaystyle W_{1/2,\frac{d}{2}+im}$
$\displaystyle=\bra{0}\psi(x,\eta_{1})\overline{\psi}(y,\eta_{2})\ket{0}$ (95)
$\displaystyle=\int\frac{d^{d}p}{(2\pi)^{d}}e^{ip\cdot(x-y)}\sum_{r}u_{r,p}(\eta_{1})\overline{u}_{r,p}(\eta_{2})\,.$

Replacing the explicit expressions for the polarisations, and using the
orthogonality relations for the eigenspinors $\xi$, we find the rather
complicated expression

$\displaystyle W_{1/2,\frac{d}{2}+im}$
$\displaystyle=\frac{1}{2}(\eta_{1}\eta_{2})^{\frac{d+1}{2}}\int\frac{d^{d}p}{(2\pi)^{d}}e^{ip\cdot(x-y)}\biggm{(}$
(96)
$\displaystyle-i\gamma_{0}p\frac{\overline{h}_{im+\frac{1}{2}}(\eta_{1}p)h_{im-\frac{1}{2}}(\eta_{2}p)+\overline{h}_{im-\frac{1}{2}}(\eta_{1}p)h_{im+\frac{1}{2}}(\eta_{2}p)}{2}$
$\displaystyle-
ip\cdot\gamma\frac{\overline{h}_{im+\frac{1}{2}}(\eta_{1}p)h_{im+\frac{1}{2}}(\eta_{2}p)+\overline{h}_{im-\frac{1}{2}}(\eta_{1}p)h_{im-\frac{1}{2}}(\eta_{2}p)}{2}$
$\displaystyle+p\frac{\overline{h}_{im+\frac{1}{2}}(\eta_{1}p)h_{im-\frac{1}{2}}(\eta_{2}p)-\overline{h}_{im-\frac{1}{2}}(\eta_{1}p)h_{im+\frac{1}{2}}(\eta_{2}p)}{2}$
$\displaystyle-\gamma_{0}p\cdot\gamma\frac{\overline{h}_{im+\frac{1}{2}}(\eta_{1}p)h_{im+\frac{1}{2}}(\eta_{2}p)-\overline{h}_{im-\frac{1}{2}}(\eta_{1}p)h_{im-\frac{1}{2}}(\eta_{2}p)}{2}\biggm{)}\,,$

and we will refer to the contributions associated to each line as
$I^{(1/2/3/4)}$. These are not as daunting as it may seem, as their evaluation
follows from the one of the scalar $I_{\nu}$, (78). We can extrapolate these
integrals using differential relations. Note, from our knowledge of the
embedding space, that the results must conspire into the shape

$\displaystyle
W_{1/2,\frac{d}{2}+im}=\frac{\gamma_{a}(x-y)^{a}+\gamma_{0}\left(\eta_{1}-\eta_{2}\right)}{\sqrt{\eta_{1}\eta_{2}}}g_{+}(u)+\frac{i\gamma_{0}\gamma_{a}(x-y)^{a}+i\left(\eta_{1}+\eta_{2}\right)}{\sqrt{\eta_{1}\eta_{2}}}g_{-}(u)\,,$
(97)

with $x^{0}=\eta_{1}$, and $\gamma_{0}x^{0}=\gamma_{0}\eta_{1}$. The terms
with factors of $p\cdot\gamma$ are already of the right form to be evaluated
using $I_{im\pm\frac{1}{2}}$. One must simply trade
$p\cdot\gamma\equiv-i\gamma\cdot\partialderivative{(x-y)}$, which further
simplifies when acting on a function of $u$. This gives

$\displaystyle I^{(2)}$
$\displaystyle=\frac{\gamma_{a}(x-y)^{a}}{\sqrt{\eta_{1}\eta_{2}}}\frac{\partial_{u}}{8}\left(I_{im-\frac{1}{2}}(u)+I_{im+\frac{1}{2}}(u)\right)\,,$
(98) $\displaystyle I^{(4)}$
$\displaystyle=\frac{i\gamma_{0}\gamma_{a}(x-y)^{a}}{\sqrt{\eta_{1}\eta_{2}}}\frac{\partial_{u}}{8}\left(I_{im-\frac{1}{2}}(u)-I_{im+\frac{1}{2}}(u)\right)\,.$

By itself, these are enough to extrapolate the functions $g_{\pm}$. We can
however also evaluate the remaining integrals. The trick is to use the raising
operators (90), noting

$\displaystyle
L_{\pm,\nu}^{(\eta)}\left[(-\eta)^{\frac{d}{2}}h_{\nu}(k\eta)\right]$
$\displaystyle=\pm i\eta k(-\eta)^{\frac{d}{2}}h_{\nu\pm 1}(k\eta)\,,$ (99)

and likewise for $\overline{h}$, we can write

$\displaystyle
I^{(1)}=\frac{\gamma_{0}}{4}\sqrt{\eta_{1}\eta_{2}}\left(\frac{1}{\eta_{1}}L_{+,im-\frac{1}{2}}^{(\eta_{1})}-\frac{1}{\eta_{2}}L_{+,im-\frac{1}{2}}^{(\eta_{2})}\right)\left[I_{im-\frac{1}{2}}(u)\right]\,,$
(100) $\displaystyle
I^{(3)}=i\frac{1}{4}\sqrt{\eta_{1}\eta_{2}}\left(\frac{1}{\eta_{1}}L_{+,im-\frac{1}{2}}^{(\eta_{1})}+\frac{1}{\eta_{2}}L_{+,im-\frac{1}{2}}^{(\eta_{2})}\right)\left[I_{im-\frac{1}{2}}(u)\right]\,.$

It is rather non-trivial that these different expressions combine back to form
the precise structure needed, forming a strong check of our computation.

These derivatives give linear combinations of hypergeometric functions with
shifted arguments. These can be further simplified by using a variety of
hypergeometric identities, or more prosaically by taking the Taylor series,
simplifying each terms at a given order, and resumming back. The endpoint of
this discussion is that one lands back on the result (42) specialised to
$\Delta=\frac{d}{2}+im$,

$\displaystyle g_{+}(z)$
$\displaystyle=\kappa\,_{2}F_{1}\left(\frac{d}{2}+im+\frac{1}{2},\frac{d}{2}-im+\frac{1}{2};\frac{d+1}{2};u\right)\,,$
(101) $\displaystyle g_{-}(z)$
$\displaystyle=-\frac{2im}{d+1}\kappa\,_{2}F_{1}\left(\frac{d}{2}+im+\frac{1}{2},\frac{d}{2}-im+\frac{1}{2};\frac{d+1}{2};u\right)\,,$
$\displaystyle\kappa$
$\displaystyle=\frac{\Gamma(\frac{d+1}{2}+im)\Gamma(\frac{d+1}{2}-im)}{2(4\pi)^{\frac{d+1}{2}}\Gamma(\frac{d+1}{2})}\,,$

in perfect agreement. The reader should note that our derivations are
completely independent on the value of $d$, and that further requiring reality
conditions such as for Majorana particles does not alter the result in any
meaningful way. The rederivation of the structures from a mode resummation is
an interesting output of this exercise.

Guided by these results, we can consider the harder problem of a generic
spinor 2-point function from purely group-theoretical arguments.

###  4.3 Deriving the Spectral Decomposition

Having studied in details the structure of free fields, we know enough to
discuss general insertions in a non-perturbative fashion. Our derivation
follows the strategy of Hogervorst:2021uvp . We consider bulk fields $\psi$
which transforms as a Dirac spinor representation of $SO(1,d+1)$,
$\psi(x^{\mu})$, with the $SO(1,d)$ subgroup made manifest. Our goal is to fix
the Wightman function
$\bra{\Omega}\psi(\eta_{1},x)\overline{\psi}(\eta_{2},y)\ket{\Omega}$ of the
fields $\psi$, over the interacting vacuum $\ket{\Omega}$, using only the
symmetry properties of the fields and states. The fields are taken to
transform under the isometry group as

$\displaystyle=x^{\mu}\partial_{\mu}\psi\,,$
$\displaystyle[K_{a},\psi(x^{\mu})]$
$\displaystyle=(2x_{a}x^{\mu}\partial_{\mu}-x^{\mu}x_{\mu}\partial_{a}+2\Sigma_{a\mu}x^{\mu})\psi\,,$
(102) $\displaystyle[P_{a},\psi(x^{\mu})]$ $\displaystyle=\partial_{a}\psi\,,$
$\displaystyle[M_{ab},\psi(x^{\mu})]$
$\displaystyle=-(x_{a}\partial_{b}-x_{b}\partial_{a}+\Sigma_{ab})\psi\,.$

The Casimir of the conformal group, the operator
$\mathcal{C}=D^{2}-\frac{1}{2}(P\cdot K+K\cdot P)-\frac{1}{2}M_{ab}M^{ab}$,
acts on it as

$\displaystyle C_{1/2,\Delta}$
$\displaystyle=\Delta(\Delta-d)+\frac{d(d-1)}{8}\,.$ (103)

The isometry group also fixes the structure of the Hilbert space. We use this
to work in a basis of on-shell states. We pick the states to realise
manifestly the $SO(1,1)\times SO(d)$ subgroup. They will be labelled
$\ket{\Delta,k}^{r}$, with $r$ an index of an irreducible representation of
$SO(d)$ and $k^{a}$ a momentum label. Without loss of generality, we can pick
states which diagonalise the action of the generators $P_{a}$, with
eigenvalues $-ik_{a}$. The action of all the other generators on the states
follow directly through dimensional analysis and the commutation relations
Hogervorst:2021uvp ,

$\displaystyle D\ket{\Delta,k}$
$\displaystyle=-\left(k\cdot\partialderivative{k}+\frac{d}{2}\right)\ket{\Delta,k}\,,$
(104) $\displaystyle P_{a}\ket{\Delta,k}$
$\displaystyle=-ik_{a}\ket{\Delta,k}\,,$ $\displaystyle M_{ab}\ket{\Delta,k}$
$\displaystyle=\left(k_{b}\partialderivative{k^{a}}-k_{a}\partialderivative{k^{b}}-\Sigma_{ab}\right)\ket{\Delta,k}\,,$

and finally,

$\displaystyle K_{a}\ket{\Delta,k}=$
$\displaystyle-i\left(k_{a}\partialderivative{k}\cdot\partialderivative{k}-2k\cdot\partialderivative{k}\partialderivative{k^{a}}-d\partialderivative{k^{a}}+\left(\Delta-\frac{d}{2}\right)^{2}\frac{k_{a}}{k^{2}}\right)\ket{\Delta,k}$
(105)
$\displaystyle+2i\Sigma_{ab}\left(\partialderivative{k_{b}}\pm\left(\Delta-\frac{d}{2}\right)\frac{k^{b}}{k^{2}}\right)\ket{\Delta,k}\,.$

The action of the operator $K_{a}$ on the state is not fully fixed by the
algebra. There is an ambiguity, associated to the shadow transform
automorphism, as indicated by the $\pm$ sign. We will now pick the definite
$+$ sign, thereby fixing what we mean by the label $\Delta$ and settling on a
definite realisation of the group. This peculiarity will turn out to be
important later on.

Going back to the object of interest, we now insert a complete set of state in
the middle of the correlator. By Wigner-Eckart, only states sitting in a
spin-$1/2$ representation can have non-zero overlap with the state created by
$\bra{\Omega}\psi$. Our task is to fix by symmetry the wavefunction $\Psi^{r}$
of the field over the state

$\displaystyle\bra{\Omega}\psi(\eta,x)\ket{\Delta,k}^{r}=\Psi^{r}(\eta,x,k)\,.$
(106)

The result for the conjugate wavefunction follows directly from this one and
does not carry new information. We proceed with the commutation relations of
$P_{a}$ with the fields to get rid of all position dependance through a
Fourier-transform term

$\displaystyle\Psi^{r}(\eta,x,p)=e^{ik\cdot x}\Psi^{r}(\eta,k)$ (107)

The next step is rotational symmetry. Picture using an induced representation
for the ket $\ket{\Delta,k}$ to bring ourselves to a given standard momenta
$\overline{k}_{a}=\delta_{a,d}k$. In this frame, we can use rotational
symmetry to fix spinning contributions up to a few functions of $\eta$ and
$\absolutevalue{k}=k$. Once this is done, we can rotate back to obtain an
expression valid in any frame.

From the oscillator construction of the spinor representations, see for
example Polchinski:1998rr , it is clear that in a Dirac spinor $\psi$ of
$Spin(1,d)$ contains two independent pieces transforming as Dirac spinors
under $Spin(d)$. Without loss of generality, these can be taken through
definite $\gamma_{0}$ eigenvalue sectors, as they do not mix under spatial
rotations. We decompose $\Psi^{r}$ using a basis of $+i,\xi^{r}$, and
$-i,\chi^{r}$ eigenvectors of $\gamma_{0}$. This basis is left unchanged as
one goes back from the standard momenta to the generic one. Note furthermore
that in the standard frame, we can trade of
$\chi\rightarrow\frac{k\cdot\gamma}{k}\xi$, since this is just a basis
redefinition. This expression is already covariant, hence is valid in all
frames. All in all, we aruged that rotational symmetry allows us to write

$\displaystyle\Psi^{r}(\eta,k)=\left(\sqrt{k}f_{1}(\eta,k)+\frac{ik\cdot\gamma}{\sqrt{k}}f_{2}(\eta,k)\right)\xi^{r}\,,$
(108)

where we already wrote the ansatz in a suggestive way, but without loss of
generality.

Our remaining task is to fix the functions $f_{1}$ and $f_{2}$. We have not
yet made any further assumption apart from translational and rotational
symmetry. We will now use more group-theoretic properties of the state. By
definition, it must be an eigenvector of the Casimir operator, as is the bulk
field. This also means that the wave-function is a solution of the Casimir
acting on $\psi$. Knowing the transformation law of $\psi$, we can rewrite the
eigenvalue equation for $\ket{\Delta,k}$ as a differential equation for
$f_{1}$ and $f_{2}$. The output is a set of 2nd order coupled differential
equation. The equations can be decoupled when written in terms of the
variables

$\displaystyle g_{1}$ $\displaystyle=f_{1}+f_{2}\,,$ $\displaystyle g_{2}$
$\displaystyle=f_{1}-f_{2}\,,$ (109)

where they take the form

$\displaystyle\left(\eta^{2}\partial_{\eta}^{2}-(d-1)\eta\partial_{\eta}+(k^{2}\eta^{2}+\frac{(d+1)^{2}}{4}+\left(\Delta(d-\Delta)+k\eta(k\eta-i)\right)\right)g_{1}(\eta,k)$
$\displaystyle=0\,,$ (110)
$\displaystyle\left(\eta^{2}\partial_{\eta}^{2}-(d-1)\eta\partial_{\eta}+(k^{2}\eta^{2}+\frac{(d+1)^{2}}{4}+\left(\Delta(d-\Delta)+k\eta(k\eta+i)\right)\right)g_{2}(\eta,k)$
$\displaystyle=0\,.$

This system is similar to (89) we had found when canonically quantising the
fermion. It can be solved using specific linear combinations of the type of
Hankel functions previously used. A peculiar result thus far is that these
equations span a four-dimensional parameter space, not two. The solutions for
$\Psi$ which are equivalent to the modes $u$ and $v$ previously found, as well
as the equivalent modes with $m\rightarrow-m$. All in all, we have so far
showed that translations, rotations and the Casimir equation have fixed

$\displaystyle\Psi^{r}(\eta,k,x)=e^{ik\cdot
x}\left(c(i\nu)u^{+\nu}_{r,k}(\eta)+\tilde{c}(i\nu)u^{-\nu}_{r,k}(\eta)+c^{\sharp}(i\nu)v^{+\nu}_{r,-k}(\eta)+\tilde{c}^{\sharp}(i\nu)v^{+\nu}_{r,-k}(\eta)\right)\,.$
(111)

The situation is now rather strange. The negative energy modes are not an
issue, we can rule them out by the Wightman axiom positive energy condition,
like in the scalar case Hogervorst:2021uvp . But the doubling of positive
energy mode is problematic. From the perspective of massive Dirac fermions, it
looks like the bulk field interpolates for twice too many degrees of freedom
on-shell. This counting clashes obviously with what one expects from a flat-
space limit as well. This is a strong indication that this second set of mode
should simply not be there as well.

To understand why, let us take a step back. These solutions are compatible
with the Casimir equation, as well as rotation and translations. However, as
is clear from their action on states, all of these actions are invariant under
the shadow transform $m\leftrightarrow-m$. It is then expected that this far,
both solutions should be allowed.

However, there is a supplementary, first order, constraint to specify that the
state $\ket{\Delta,k}^{r}$ transforms as a state with definite weight
$\Delta=\frac{d}{2}+im$, and not its shadow. This is of course precisely the
issue that arises from the realisation of $K_{a}$ on states. Hence the cure is
to impose on our wave-function that it realises special conformal
transformations correctly, through the obvious identity

$\displaystyle\bra{\Omega}\psi(x,\eta)K_{a}\ket{\frac{d}{2}+im,k}+\bra{\Omega}\big{[}K_{a},\psi(x,\eta)\big{]}\ket{\frac{d}{2}+im,k}$
$\displaystyle=0\,.$ (112)

Using the relations (102) and (105), we can impose this condition onto a
generic polarisation $e^{ik\cdot x}u_{r,k}^{\nu}(\eta)$. The correct
realisation of the special-conformal transformations then requires $m=\nu$,
thereby ruling out the second set of modes.

Figure 5: Once the functional form of the correlator in a specific channel is
fixed, we can prove its positivity by considering it as two glued copies of de
Sitter, whose far-past are regulated by the Bunch-Davies condition to allow
for the analytical continuation to the sphere Schlingemann:1998cw ;
Schlingemann:1999mk ; Higuchi:2010xt . Glueing the spaces together, we can
arrange the points in a symmetric, manifestly positive, configuration, proving
the positivity of the coefficient in the expansion.

We have shown that the de Sitter isometries require the wave-function of a
generic Dirac spinor with a spin-$1/2$ state to be proportional to that of a
free massive Dirac spinor mode,

$\displaystyle\bra{\Omega}\psi(\eta,x)\ket{\frac{d}{2}+i\nu,k}^{r}=c(i\nu)e^{ik\cdot
x}u_{r,k}^{\nu}(\eta)\,.$ (113)

This result is equivalent to the statement that on-shell states satisfy the
Dirac equation. A further consequence of which is that these states must all
lie on the principal series, with $m\in\mathbb{R}$. This is expected from a
group-theoretic perspective, as spin-$1/2$ representation of the de Sitter
group sits in the principal series Thieleker1973 ; Thieleker1974 ;
Dobrev:1977qv ; Joung2006 ; Joung2007 , but offers a physical derivation of
this fact. The result for the conjugate wavefunction follows directly.
Bringing these pieces together, we find that the 2-point function admits a
decomposition

$\displaystyle\bra{\Omega}\psi(\eta_{1},x)\overline{\psi}(\eta_{2},y)\ket{\Omega}$
$\displaystyle=\int_{-\infty}^{\infty}d\nu\rho(i\nu)\int\frac{d^{d}k}{(2\pi)^{d}}\sum_{r}u^{\nu}_{r,k}(\eta_{1})\overline{u}^{\nu}_{r,k}(\eta_{2})$
(114)
$\displaystyle=\int_{-\infty}^{\infty}d\nu\rho(i\nu)W_{1/2,\frac{d}{2}+i\nu}(u)\,.$

Where we used back the result previously derived from the mode resummation.
The spectral parameter given by

$\displaystyle\rho(i\nu)=\frac{\absolutevalue{c(i\nu)}^{2}}{\mathcal{N}}\geq
0\,,$ (115)

is manifestly positive.

An independent argument for the positivity of $\rho$ is obtained through the
analytical continuation to the sphere, as illustrated in fig.5. This concludes
our proof.

##  5 Discussion

The stated goal of this work was to tackle cosmological correlators involving
fermions. We have reviewed the current understanding for bosonic analytical
computations, which revolves around a Wick contraction to Euclidean AdS space.
Though they involve some technicalities, we have shown that the same framework
accommodates fermions, expanding the previously known transcription rules.
This allowed us to import the knowledge developed for Witten diagrams to the
evaluation of perturbative cosmological correlators, as we showed in an
example with fermions exchange. Our exposition aimed to show that fermions are
no more exotic than bosons in that regard, as the existing strategies for them
can be adapted successfully. In the same endeavour, we used the techniques
used to study unitarity of some bosonic observables, to prove the analogous
statement, at the level of the Källen-Lehman decomposition, for spinors. These
results are important as they enrich the observables under analytical control,
through which we better our understanding of phenomenologically relevant
spacetimes with standard matter content.

There are many ways through which one can push forward in the future. Much of
the literature in de Sitter, such as anninos2012 ; anninos2019Holo ; David2019
; Anninos:2020hfj ; Harris2021 ; Mirbabayi:2022gnl ; Muhlmann:2021aa ;
Anninos2021 , has focused on the properties of the cosmological horizon, often
through the angle of the euclidean sphere. We believe an embedding picture can
offer insights on this front as well. The associated study of quasinormal
modes Lopez-Ortega:2004vrh ; Du:2004jt ; Lopez-Ortega:2006aal ; Lopez-
Ortega:2006tjo ; Lopez-Ortega:2012xvr ; Jafferis:2013qia ; Sun:2020sgn  has
already touched on spinors and the embedding picture, but one can make more
general statements still. For example, the discussion of Mirbabayi:2022gnl
can clearly be applied to spinors as well, relating the spectral decomposition
we proved to the quasinormal spectrum. A more involved direction would be to
study whether the current Euclidean perspective for late-time correlators can
be exploited to study the scattering problem in de Sitter, perhaps along the
lines of Albrychiewicz:2020ruh . Another interesting direction is to make
contact with the literature on unitarity and on-shell amplitudes
Meltzer:2019nbs ; Goodhew:2020hob ; Albayrak:2020fyp ; Melville:2021lst ;
Jazayeri2021 ; Bonifacio2021 ; Benincasa:2022gtd ; Armstrong:2020woi ;
Gomez:2021qfd ; Armstrong:2022mfr ; Armstrong:2022vgl  or the studies of the
wavefunctional perspective Anninos2015 ; Arkani-Hamed:2017fdk ;
Cespedes:2020xqq ; Salcedo:2022aal . We believe that on these topics as well,
spinors would offer interesting extensions and help better appreciate the
structures under study.

###### Acknowledgements.

The author would like to thank A. Chalabi, N. Drukker, E. Harris, P. Kravchuk,
B. Pethybridge, and J. Stedman for discussions. The author would also like to
thank C. Herzog, D. Anninos for their comments on the drafts and
encouragements. Part of this work was motivated and initiated during the
school on “Black Holes, Cosmology and Holography” at the STAG Research Centre
in Southampton, and during the Corfu Summer Institute “Workshop on Quantum
Features in a de Sitter Universe”. The author is grateful to the organisers of
these events. This work was supported by the U.K. Science & Technology
Facilities Council Grant ST/P000258/1.

##  Appendix A Spinorial Harmonic Functions

Harmonic functions are useful tools for the evaluation of Witten diagrams. The
relevant results for bosonic propagators can be found in Costa:2014kfa , while
the spinorial variant was introduced in Nishida:2018opl . However, the latter
is not very detailed, and uses conventions which are very different from ours.
It is useful then to study these object in the setup most suited to our
concerns.

The goal of this section is to detail the decomposition of the propagator in
harmonics. We start by proving the orthonormality property of the spinorial
harmonics,

$\displaystyle\int_{AdS}dx\;\Omega_{1/2,\nu}(x_{1};x)\Omega_{1/2,\nu^{\prime}}(x;x_{2})=\delta(\nu-\nu^{\prime})\Omega_{1/2,\nu}(x_{1};x_{2})\,.$
(116)

We use the explicit real-space expression

$\displaystyle\Omega_{1/2,\nu}(x_{1};x_{2})$
$\displaystyle=N_{h+i\nu}N_{h-i\nu}\frac{i}{2\pi}\int
d^{d}y\,\frac{\gamma\cdot(x_{1}-y)2(1\pm\gamma_{0})\gamma\cdot(y-x_{2})}{\sqrt{z_{1}z_{2}}\left(\frac{(x_{1}-y)^{2}}{z_{1}}\right)^{h+i\nu+\frac{1}{2}}\left(\frac{(x_{2}-y)^{2}}{z_{2}}\right)^{h-i\nu+\frac{1}{2}}}\,,$
(117)

which is invariant under the replacement
$\nu,\gamma_{0}\rightarrow-\nu,-\gamma_{0}$, and we abbreviated
$h=\frac{d}{2}$. The integral we want to compute takes the unwieldy form

$\displaystyle\frac{i}{2\pi}\frac{N_{h+i\nu}N_{h-i\nu}}{\sqrt{z_{1}z_{2}}}\int
d^{d}y_{1}\,\frac{i}{2\pi}4N_{h+i\nu^{\prime}}N_{h-i\nu^{\prime}}\int\frac{d^{d}xdz}{z^{d+1}}\int
d^{d}y_{2}\times$ (118)
$\displaystyle\frac{\gamma\cdot(x_{1}-y_{1})(1+\gamma_{0})\gamma\cdot(y_{2}-x)\gamma\cdot(x-y_{2})(1-\gamma_{0})\gamma\cdot(y_{2}-x_{2})}{z\left(\frac{(x_{1}-y_{2})^{2}}{z_{1}}\right)^{h+i\nu+\frac{1}{2}}\left(\frac{(y_{2}-x)^{2}}{z}\right)^{h-i\nu+\frac{1}{2}}\left(\frac{(x-y_{2})^{2}}{z}\right)^{h-i\nu+\frac{1}{2}}\left(\frac{(y_{2}-x_{2})^{2}}{z_{2}}\right)^{h+i\nu+\frac{1}{2}}}\,.$

We will concentrate first on the middle integral over $AdS$, which sits
between two boundary integrals. After simplification, this integral can be
evaluated similarly to the ones encountered for bosonic harmonic functions,

$\displaystyle\int\frac{d^{d}x\,dz}{z^{d+1}}\frac{(1+\gamma_{0})(\gamma\cdot(y_{1}-x)\gamma\cdot(x-y_{2}))(1-\gamma_{0})}{z\left(\frac{(x-y_{1})^{2}}{z}\right)^{h+\frac{1}{2}-i\nu}\left(\frac{(x-y_{2})^{2}}{z}\right)^{h+\frac{1}{2}-i\nu^{\prime}}}$
(119) $\displaystyle=$ $\displaystyle
2(1+\gamma_{0})\gamma\cdot(y_{1}-y_{2})\int\frac{d^{d}x\,dz}{z^{d+1}}\frac{1}{\left(\frac{(x-y_{1})^{2}}{z}\right)^{h+\frac{1}{2}-i\nu}\left(\frac{(x-y_{2})^{2}}{z}\right)^{h+\frac{1}{2}-i\nu^{\prime}}}$
$\displaystyle=$
$\displaystyle(1+\gamma_{0})\gamma\cdot(y_{1}-y_{2})\left(\frac{4\pi\sqrt{\pi}^{d}\Gamma(\frac{1}{2}-i\nu)\delta(\nu-\nu^{\prime})}{\Gamma(\frac{d+1}{2}-i\nu)\left((y_{1}-y_{2})^{2}\right)^{h+\frac{1}{2}-i\nu}}+\delta^{d}(y_{1}-y_{2})\ldots\right)\,.$

In the last line we used the integral identity derived in the appendix of
Costa:2014kfa . The second piece in the last expression does not contribute
because of the spinorial structure. We are left with an integral over the
point $y_{2}$, which is effectively a shadow transform changing the last power
law from $h+\frac{1}{2}+i\nu$ to $h+\frac{1}{2}-i\nu$. This integral is
tedious to perform in position space, and so we rewrite the expression back to
the embedding space,

$\displaystyle\int
d^{d}y_{2}\frac{\gamma\cdot(y_{1}-y_{2})\gamma\cdot(y_{2}-x_{2})}{((y_{1}-y_{2})^{2})^{h+\frac{1}{2}-i\nu}\left(\frac{(y_{2}-x_{2})^{2}}{z_{2}}\right)^{h+\frac{1}{2}+i\nu}}$
(120) $\displaystyle=$
$\displaystyle\int[dP_{2}]\frac{\overline{S}_{1,\partial}[y_{1}]\not{P}_{2}S[x_{2}]}{(-2P_{1}\cdot
P_{2})^{h+\frac{1}{2}-i\nu}(-2P_{2}\cdot X)^{h+\frac{1}{2}+i\nu}}$
$\displaystyle=$
$\displaystyle\frac{\overline{S}_{1,\partial}[y_{1}]\Gamma_{A}X^{A}S[x_{2}]\sqrt{\pi}^{d}\Gamma(\frac{d}{2}+1)}{\Gamma(h+\frac{1}{2}+i\nu)\Gamma(h+\frac{1}{2}-i\nu)}\int_{0}^{\infty}d\lambda\frac{\lambda^{\frac{d-1}{2}+i\nu}}{\big{(}1+\lambda(-2X\cdot
P_{1})\big{)}^{\frac{d+2}{2}}}$ $\displaystyle=$
$\displaystyle(-i)\frac{\pi^{d/2}\Gamma\left(i\nu+\frac{1}{2}\right)}{\Gamma(\frac{d+1}{2}+i\nu)}\frac{\overline{S}_{1,\partial}[y_{1}]S[x_{2}]}{(-2X\cdot
P_{1})^{h-i\nu+\frac{1}{2}}}\,.$

The remaining integral then joins together the bulk-to-boundary propagator,
giving back $\Omega_{1/2,\nu}$ and the $\delta(\nu-\nu^{\prime})$ needed. One
must simply be careful to piece together the constants. Once the dust settles
we are left with the final result

$\displaystyle\int_{AdS}d^{d+1}x\,\Omega_{1/2,\nu}(x_{1};x)\Omega_{1/2,\nu^{\prime}}(x;x_{2})=\delta(\nu-\nu^{\prime})\Omega_{1/2,\nu}(x_{1};x_{2})\,.$
(121)

This computation implies the identity

$\displaystyle\int d\nu\;\Omega_{1/2,\nu}(x;y)=\delta_{H}^{d+1}(x-y)\,.$ (122)

This completeness relation implies one can write a generic propagators using a
superposition of harmonics i.e. the propagator admits a decomposition

$\displaystyle\Pi^{+}_{h+m}(x;y)=-i\int\frac{d\nu^{\prime}}{\nu^{\prime}+im}\Omega_{1/2,\nu^{\prime}}(x;y)\,.$
(123)

The integration kernel is fixed using the equations of motions. Since we know
that the difference of the two propagators is proportional to the harmonic
function, we can solve for $\Pi^{-}_{h-m}$. These expressions are valid
provided $Re[m]>0$, hence to obtain $\Pi^{+}_{h+i\nu}$ we consider
$m=\epsilon+i\nu$, $\epsilon>0$, and take the limit to the principal series

$\displaystyle\Pi^{+}_{h+i\nu}=-i\int\frac{d\nu^{\prime}}{\nu^{\prime}-\nu+i\epsilon}\Omega_{1/2,\nu^{\prime}}(x;y)\,.$
(124)

The analogous expression for $\Pi^{-}_{h-i\nu}$ follows through by using the
Sokhotski-Plemelj formula. The final result is particularly appealing

$\displaystyle\Pi^{\pm}_{h\pm i\nu}(x;y)$
$\displaystyle=-i\int\frac{d\nu^{\prime}}{\nu^{\prime}-\nu\pm
i\epsilon}\Omega_{1/2,\nu^{\prime}}(x;y)\,.$ (125)

From this, we gather that to evaluate a general diagram involving any type of
fermionic propagator, we simply have to replace each propagator by an harmonic
function with arbitrary weight, and integrate over it with the appropriate
kernel.

## References

  * (1) N. Arkani-Hamed and J. Maldacena, _Cosmological Collider Physics_ , 1503.08043.
  * (2) D. Baumann, _Inflation_ , in _Theoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small_ , pp. 523–686, 2011, DOI [0907.5424].
  * (3) D. Green et al., _Snowmass Theory Frontier: Astrophysics and Cosmology_ , 2209.06854.
  * (4) N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, _The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities_ , _JHEP_ 04 (2020) 105 [1811.00024].
  * (5) D. Baumann, G. Goon, H. Lee and G.L. Pimentel, _Partially massless fields during inflation_ , _Journal of High Energy Physics 2018 2018:4_ 2018 (2018) 1.
  * (6) D. Baumann, C. Duaso Pueyo and A. Joyce, _Bootstrapping Cosmological Correlations_ , _AAPPS Bull._ 30 (2020) 2.
  * (7) D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, _The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization_ , 2005.04234.
  * (8) D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, _The cosmological bootstrap: weight-shifting operators and scalar seeds_ , _JHEP_ 12 (2020) 204 [1910.14051].
  * (9) D. Baumann, D. Green, A. Joyce, E. Pajer, G.L. Pimentel, C. Sleight et al., _Snowmass White Paper: The Cosmological Bootstrap_ , in _2022 Snowmass Summer Study_ , 3, 2022 [2203.08121].
  * (10) P. McFadden and K. Skenderis, _Holography for Cosmology_ , _Phys. Rev. D_ 81 (2010) 021301 [0907.5542].
  * (11) P. McFadden and K. Skenderis, _The Holographic Universe_ , _J. Phys. Conf. Ser._ 222 (2010) 012007 [1001.2007].
  * (12) P. McFadden and K. Skenderis, _Holographic Non-Gaussianity_ , _JCAP_ 05 (2011) 013 [1011.0452].
  * (13) P. McFadden and K. Skenderis, _Cosmological 3-point correlators from holography_ , _JCAP_ 06 (2011) 030 [1104.3894].
  * (14) A. Bzowski, P. McFadden and K. Skenderis, _Holography for inflation using conformal perturbation theory_ , _JHEP_ 04 (2013) 047 [1211.4550].
  * (15) J. Bros, U. Moschella and J.P. Gazeau, _Quantum field theory in the de Sitter universe_ , _Phys. Rev. Lett._ 73 (1994) 1746.
  * (16) A. Strominger, _The dS / CFT correspondence_ , _JHEP_ 10 (2001) 034 [hep-th/0106113].
  * (17) D. Anninos, T. Hartman and A. Strominger, _Higher Spin Realization of the dS/CFT Correspondence_ , _Class. Quant. Grav._ 34 (2017) 015009 [1108.5735].
  * (18) V. Gorbenko and L. Senatore, _$\lambda\phi^{4}$ in dS_, 1911.00022.
  * (19) M. Hogervorst, J.a. Penedones and K.S. Vaziri, _Towards the non-perturbative cosmological bootstrap_ , 2107.13871.
  * (20) R. Flauger, V. Gorbenko, A. Joyce, L. McAllister, G. Shiu and E. Silverstein, _Snowmass White Paper: Cosmology at the Theory Frontier_ , in _2022 Snowmass Summer Study_ , 3, 2022 [2203.07629].
  * (21) C. Sleight and M. Taronna, _Bootstrapping Inflationary Correlators in Mellin Space_ , _JHEP_ 02 (2020) 098 [1907.01143].
  * (22) C. Sleight, _A Mellin Space Approach to Cosmological Correlators_ , _JHEP_ 01 (2020) 090 [1906.12302].
  * (23) C. Sleight and M. Taronna, _From AdS to dS Exchanges: Spectral Representation, Mellin Amplitudes and Crossing_ , 2007.09993.
  * (24) C. Sleight and M. Taronna, _From dS to AdS and back_ , 2109.02725.
  * (25) J.M. Maldacena, _Non-Gaussian features of primordial fluctuations in single field inflationary models_ , _JHEP_ 05 (2003) 013 [astro-ph/0210603].
  * (26) S. Fichet, _On holography in general background and the boundary effective action from AdS to dS_ , _JHEP_ 07 (2022) 113 [2112.00746].
  * (27) L. Di Pietro, V. Gorbenko and S. Komatsu, _Analyticity and Unitarity for Cosmological Correlators_ , 2108.01695.
  * (28) T. Heckelbacher, I. Sachs, E. Skvortsov and P. Vanhove, _Analytical evaluation of cosmological correlation functions_ , _JHEP_ 08 (2022) 139 [2204.07217].
  * (29) C. Sleight and M. Taronna, _On the consistency of (partially-)massless matter couplings in de Sitter space_ , _JHEP_ 10 (2021) 156 [2106.00366].
  * (30) B. Pethybridge and V. Schaub, _Tensors and spinors in de Sitter space_ , _JHEP_ 06 (2022) 123 [2111.14899].
  * (31) P.A.M. Dirac, _The Electron Wave Equation in De-Sitter Space_ , _Annals Math._ 36 (1935) 657.
  * (32) P. Candelas and D.J. Raine, _General-relativistic quantum field theory: An exactly soluble model_ , _Phys. Rev. D_ 12 (1975) 965.
  * (33) J. Fang and C. Fronsdal, _Massless, Half Integer Spin Fields in De Sitter Space_ , _Phys. Rev. D_ 22 (1980) 1361.
  * (34) B. Allen and C.A. Lutken, _Spinor Two Point Functions in Maximally Symmetric Spaces_ , _Commun. Math. Phys._ 106 (1986) 201.
  * (35) I.I. Cotăescu, _Polarized dirac fermions in de sitter spacetime_ , _Physical Review D_ 65 (2002) 084008.
  * (36) I.I. Cotaescu, _Integral representation of the feynman propagators of the dirac fermions on the de sitter expanding universe_ , _European Physical Journal C_ 78 (2018) .
  * (37) R. Camporesi, _The spinor heat kernel in maximally symmetric spaces_ , _Commun.Math.Phys._ 148 (1992) 283.
  * (38) R. Camporesi and A. Higuchi, _On the eigen functions of the dirac operator on spheres and real hyperbolic spaces_ , _J.Geom.Phys._ 20 (1996) 1.
  * (39) V.A. Letsios, _The eigenmodes for spinor quantum field theory in global de Sitter space–time_ , _J. Math. Phys._ 62 (2021) 032303 [2011.07875].
  * (40) J. Bros, _Complexified de Sitter space: Analytic causal kernels and Kallen-Lehmann type representation_ , _Nucl. Phys. B Proc. Suppl._ 18 (1991) 22.
  * (41) J. Bros and U. Moschella, _Two point functions and quantum fields in de Sitter universe_ , _Rev. Math. Phys._ 8 (1996) 327 [gr-qc/9511019].
  * (42) J. Bros, H. Epstein and U. Moschella, _Analyticity properties and thermal effects for general quantum field theory on de Sitter space-time_ , _Commun. Math. Phys._ 196 (1998) 535 [gr-qc/9801099].
  * (43) J. Bros, H. Epstein and U. Moschella, _Particle decays and stability on the de Sitter universe_ , _Annales Henri Poincare_ 11 (2010) 611 [0812.3513].
  * (44) G. Sengör and C. Skordis, _Unitarity at the Late time Boundary of de Sitter_ , _JHEP_ 06 (2020) 041 [1912.09885].
  * (45) S. Weinberg, _Quantum contributions to cosmological correlations_ , _Physical Review D - Particles, Fields, Gravitation and Cosmology_ 72 (2005) 1.
  * (46) P. Adshead, R. Easther and E.A. Lim, _“in-in” formalism and cosmological perturbations_ , _Physical Review D_ 80 (2009) .
  * (47) M.S. Costa, V. Gonçalves and J.a. Penedones, _Spinning AdS Propagators_ , _JHEP_ 09 (2014) 064 [1404.5625].
  * (48) Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick, _Analysis, manifolds, and physics_ , North-Holland Pub. Co. (1982).
  * (49) B. Allen, _Vacuum States in de Sitter Space_ , _Phys. Rev. D_ 32 (1985) 3136.
  * (50) M. Sasaki, T. Tanaka and K. Yamamoto, _Euclidean vacuum mode functions for a scalar field on open de Sitter space_ , _Phys. Rev. D_ 51 (1995) 2979 [gr-qc/9412025].
  * (51) R. Bousso, A. Maloney and A. Strominger, _Conformal vacua and entropy in de Sitter space_ , _Phys. Rev. D_ 65 (2002) 104039 [hep-th/0112218].
  * (52) E. Joung, J. Mourad and R. Parentani, _Group theoretical approach to quantum fields in de sitter space, i. the principal series_ , _Journal of High Energy Physics_ 2006 (2006) .
  * (53) E. Joung, J. Mourad and R. Parentani, _Group theoretical approach to quantum fields in de sitter space ii. the complementary and discrete series_ , _JHEP_ 2007 (2007) 030.
  * (54) T. Anous and J. Skulte, _An invitation to the principal series_ , _SciPost Physics_ 9 (2020) .
  * (55) Z. Sun, _A note on the representations of $\text{\\{}SO\\}(1,d+1)$_, _arXiv_ (2021) arXiv:2111.04591.
  * (56) J. Penedones, K. Salehi Vaziri and Z. Sun, _Hilbert space of Quantum Field Theory in de Sitter spacetime_ , 2301.04146.
  * (57) M.S. Costa, J. Penedones, D. Poland and S. Rychkov, _Spinning Conformal Correlators_ , _Journal of High Energy Physics_ 2011 (2011) 71.
  * (58) M. Visser, _How to Wick rotate generic curved spacetime_ , 1702.05572.
  * (59) I.S. Gradshteĭn and D. Zwillinger, _Table of integrals, series, and products_ , Elsevier, Academic Press is an imprint of Elsevier, Amsterdam ; Boston, eighth edition ed. (2015).
  * (60) D. Harlow and D. Stanford, _Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT_ , 1104.2621.
  * (61) M. Henningson and K. Sfetsos, _Spinors and the AdS / CFT correspondence_ , _Phys. Lett. B_ 431 (1998) 63 [hep-th/9803251].
  * (62) T. Kawano and K. Okuyama, _Spinor exchange in AdS(d+1)_ , _Nucl. Phys. B_ 565 (2000) 427 [hep-th/9905130].
  * (63) M. Nishida and K. Tamaoka, _Fermions in Geodesic Witten Diagrams_ , _JHEP_ 07 (2018) 149 [1805.00217].
  * (64) A. Van Proeyen, _Tools for supersymmetry_ , _Ann. U. Craiova Phys._ 9 (1999) 1 [hep-th/9910030].
  * (65) S. Weinberg, _Six-dimensional Methods for Four-dimensional Conformal Field Theories_ , _Physical Review D_ 82 (2010) 045031.
  * (66) A. Trautman, _Spinors and the dirac operator on hypersurfaces. i. general theory_ , _Journal of Mathematical Physics_ 33 (1992) 4011.
  * (67) A. Trautman, _The Dirac operator on hypersurfaces_ , _Acta Phys. Polon. B_ 26 (1995) 1283 [hep-th/9810018].
  * (68) M. Henneaux, _Boundary terms in the AdS / CFT correspondence for spinor fields_ , in _International Meeting on Mathematical Methods in Modern Theoretical Physics (ISPM 98)_ , pp. 161–170, 9, 1998 [hep-th/9902137].
  * (69) C.P. Herzog and V. Schaub, _Fermions in Boundary Conformal Field Theory : Crossing Symmetry and $\epsilon$-Expansion_, 2209.05511.
  * (70) D. Simmons-Duffin, _Projectors, Shadows, and Conformal Blocks_ , _Journal of High Energy Physics_ 2014 (2014) 146.
  * (71) S. Giombi, E. Helfenberger and H. Khanchandani, _Fermions in AdS and Gross-Neveu BCFT_ , 2110.04268.
  * (72) D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, _Correlation functions in the CFT(d) / AdS(d+1) correspondence_ , _Nucl. Phys. B_ 546 (1999) 96 [hep-th/9804058].
  * (73) L. Cornalba, M.S. Costa and J. Penedones, _Eikonal Methods in AdS/CFT: BFKL Pomeron at Weak Coupling_ , _JHEP_ 06 (2008) 048 [0801.3002].
  * (74) S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, _Covariant expansion of the conformal four-point function_ , _Nucl. Phys. B_ 49 (1972) 77.
  * (75) F.A. Dolan and H. Osborn, _Conformal Four Point Functions and the Operator Product Expansion_ , _Nuclear Physics B_ 599 (2001) 459.
  * (76) F.A. Dolan and H. Osborn, _Conformal Partial Waves and the Operator Product Expansion_ , _Nuclear Physics B_ 678 (2004) 491.
  * (77) F.A. Dolan and H. Osborn, _Conformal Partial Waves: Further Mathematical Results_ , _arXiv:1108.6194 [hep-th]_ (2012) .
  * (78) E.A. Tagirov, _Consequences of field quantization in de Sitter type cosmological models_ , _Annals Phys._ 76 (1973) 561.
  * (79) N.A. Chernikov and E.A. Tagirov, _Quantum theory of scalar fields in de Sitter space-time_ , _Ann. Inst. H. Poincare Phys. Theor. A_ 9 (1968) 109.
  * (80) C. Schomblond and P. Spindel, _Unicity Conditions of the Scalar Field Propagator Delta(1) (x,y) in de Sitter Universe_ , _Ann. Inst. H. Poincare Phys. Theor._ 25 (1976) 67.
  * (81) E. Mottola, _Particle Creation in de Sitter Space_ , _Phys. Rev. D_ 31 (1985) 754.
  * (82) M. Spradlin, A. Strominger and A. Volovich, _Les Houches lectures on de Sitter space_ , in _Les Houches Summer School: Session 76: Euro Summer School on Unity of Fundamental Physics: Gravity, Gauge Theory and Strings_ , pp. 423–453, 10, 2001 [hep-th/0110007].
  * (83) D. Anninos, _De Sitter Musings_ , _Int. J. Mod. Phys. A_ 27 (2012) 1230013 [1205.3855].
  * (84) T.S. Bunch and P.C.W. Davies, _Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting_ , _Proc. Roy. Soc. Lond. A_ 360 (1978) 117.
  * (85) I.I. Cotaescu, _Canonical quantization of the covariant fields on de Sitter space–times_ , _Int. J. Mod. Phys. A_ 33 (2018) 1830007 [1602.06810].
  * (86) J. Polchinski, _String theory. Vol. 2: Superstring theory and beyond_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (12, 2007), 10.1017/CBO9780511618123.
  * (87) D. Schlingemann, _From Euclidean field theory to quantum field theory_ , _Rev. Math. Phys._ 11 (1999) 1151 [hep-th/9802035].
  * (88) D. Schlingemann, _Euclidean field theory on a sphere_ , hep-th/9912235.
  * (89) A. Higuchi, D. Marolf and I.A. Morrison, _On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT_ , _Phys. Rev. D_ 83 (2011) 084029 [1012.3415].
  * (90) E. Thieleker, _On the quasi-simple irreducible representations of the lorentz groups_ , _Transactions of the American Mathematical Society_ 179 (1973) 465.
  * (91) E.A. Thieleker, _The unitary representations of the generalized lorentz groups_ , _Transactions of the American Mathematical Society_ 199 (1974) 327.
  * (92) V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, _Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory_ , vol. 63, Springer (1977), 10.1007/BFb0009678.
  * (93) D. Anninos, S.A. Hartnoll and D.M. Hofman, _Static patch solipsism: conformal symmetry of the de sitter worldline_ , _Classical and Quantum Gravity_ 29 (2012) 075002.
  * (94) D. Anninos, D.A. Galante and D.M. Hofman, _De sitter horizons & holographic liquids_, _Journal of High Energy Physics_ 2019 (2019) .
  * (95) A. David, N. Fischer and Y. Neiman, _Spinor-helicity variables for cosmological horizons in de sitter space_ , _Phys.Rev.D_ 100 (2019) .
  * (96) D. Anninos, F. Denef, Y.T.A. Law and Z. Sun, _Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions_ , _JHEP_ 01 (2022) 088 [2009.12464].
  * (97) D. Anninos and E. Harris, _Three-dimensional de sitter horizon thermodynamics_ , _Journal of High Energy Physics 2021 2021:10_ 2021 (2021) 1.
  * (98) M. Mirbabayi and F. Riccardi, _Probing de Sitter from the horizon_ , 2211.11672.
  * (99) B. Mühlmann, _The two-sphere partition function in two-dimensional quantum gravity at fixed area_ , _JHEP_ 09 (2021) .
  * (100) D. Anninos, T. Bautista and B. Mühlmann, _The two-sphere partition function in two-dimensional quantum gravity_ , _JHEP_ 09 (2021) 116.
  * (101) A. Lopez-Ortega, _Dirac fields in 3D de Sitter spacetime_ , _Gen. Rel. Grav._ 36 (2004) 1299.
  * (102) D.-P. Du, B. Wang and R.-K. Su, _Quasinormal modes in pure de Sitter space-times_ , _Phys. Rev. D_ 70 (2004) 064024 [hep-th/0404047].
  * (103) A. Lopez-Ortega, _Quasinormal modes of D-dimensional de Sitter spacetime_ , _Gen. Rel. Grav._ 38 (2006) 1565 [gr-qc/0605027].
  * (104) A. Lopez-Ortega, _Absorption and quasinormal modes of classical fields propagating on 3D and 4D de Sitter spacetime_ , _Gen. Rel. Grav._ 38 (2006) 743 [gr-qc/0605022].
  * (105) A. Lopez-Ortega, _On the quasinormal modes of the de Sitter spacetime_ , _Gen. Rel. Grav._ 44 (2012) 2387 [1207.6791].
  * (106) D.L. Jafferis, A. Lupsasca, V. Lysov, G.S. Ng and A. Strominger, _Quasinormal quantization in de Sitter spacetime_ , _JHEP_ 01 (2015) 004 [1305.5523].
  * (107) Z. Sun, _Higher spin de Sitter quasinormal modes_ , 2010.09684.
  * (108) E. Albrychiewicz and Y. Neiman, _Scattering in the static patch of de Sitter space_ , _Phys. Rev. D_ 103 (2021) 065014 [2012.13584].
  * (109) D. Meltzer, E. Perlmutter and A. Sivaramakrishnan, _Unitarity Methods in AdS/CFT_ , _JHEP_ 03 (2020) 061 [1912.09521].
  * (110) H. Goodhew, S. Jazayeri and E. Pajer, _The Cosmological Optical Theorem_ , _JCAP_ 04 (2021) 021 [2009.02898].
  * (111) S. Albayrak, S. Kharel and D. Meltzer, _On duality of color and kinematics in (A)dS momentum space_ , _JHEP_ 03 (2021) 249 [2012.10460].
  * (112) S. Melville and E. Pajer, _Cosmological Cutting Rules_ , _JHEP_ 05 (2021) 249 [2103.09832].
  * (113) S. Jazayeri, E. Pajer and D. Stefanyszyn, _From locality and unitarity to cosmological correlators_ , _JHEP_ 10 (2021) 065.
  * (114) J. Bonifacio, E. Pajer and D.-G. Wang, _From amplitudes to contact cosmological correlators_ , _arXiv_ 2021 (2021) arXiv:2106.15468.
  * (115) P. Benincasa, _Amplitudes meet Cosmology: A (Scalar) Primer_ , 2203.15330.
  * (116) C. Armstrong, A.E. Lipstein and J. Mei, _Color/kinematics duality in AdS 4_, _JHEP_ 02 (2021) 194 [2012.02059].
  * (117) H. Gomez, R.L. Jusinskas and A. Lipstein, _Cosmological Scattering Equations_ , _Phys. Rev. Lett._ 127 (2021) 251604 [2106.11903].
  * (118) C. Armstrong, H. Gomez, R. Lipinski Jusinskas, A. Lipstein and J. Mei, _New recursion relations for tree-level correlators in anti–de Sitter spacetime_ , _Phys. Rev. D_ 106 (2022) L121701 [2209.02709].
  * (119) C. Armstrong, A. Lipstein and J. Mei, _Enhanced soft limits in de Sitter space_ , _JHEP_ 12 (2022) 064 [2210.02285].
  * (120) D. Anninos, T. Anous, D.Z. Freedman and G. Konstantinidis, _Late-time structure of the bunch-davies de sitter wavefunction_ , _Journal of Cosmology and Astroparticle Physics_ 2015 (2015) .
  * (121) N. Arkani-Hamed, P. Benincasa and A. Postnikov, _Cosmological Polytopes and the Wavefunction of the Universe_ , 1709.02813.
  * (122) S. Céspedes, A.-C. Davis and S. Melville, _On the time evolution of cosmological correlators_ , _JHEP_ 02 (2021) 012 [2009.07874].
  * (123) S.A. Salcedo, M.H.G. Lee, S. Melville and E. Pajer, _The Analytic Wavefunction_ , 2212.08009.