Pietro Donà
email: dona@cpt.univ-mrs.fr

Francesco Gozzini
email: gozzini@cpt.univ-mrs.fr

Alessandro Nicotra
email: alessandro.nicotra5@studio.unibo.it

A Wick rotation for EPRL spin foam models

published or not. The documents may come

Wick rotation for spin foam quantum gravity

Introduction

Spin foam theory attempts to define the Loop Quantum Gravity dynamics with a regularized, backgroundindependent, and Lorentz covariant quantum gravity path integral on a fixed triangulation. The most popular spin foam model is the EPRL-FK model defined in [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF][START_REF] Freidel | A New Spin Foam Model for 4d Gravity[END_REF] (see [START_REF] Perez | The Spin Foam Approach to Quantum Gravity[END_REF] or [START_REF] Rovelli | Covariant loop quantum gravity: an elementary introduction to quantum gravity and spinfoam theory[END_REF] for more pedagogical reviews). Both models naturally have as boundary states the spin network states from canonical Loop Quantum Gravity, which is why spin foam theory is considered "covariant Loop Quantum Gravity" [START_REF] Rovelli | A new look at loop quantum gravity[END_REF]. The theory assigns transition amplitudes to spin network states living on the boundary of four-dimensional triangulations. A version of the model exists for spacetimes with both Euclidean and Lorentzian signatures.

The two models differ by their gauge group structures: Spin(4) (the double covering of SO(4)) for the Euclidean signature and SL(2, C) (the double covering of SO + (3, 1)) for the Lorentzian one. The models are built upon unitary irreducible representations of the groups (the principal series ones in the Lorentzian case). The linear simplicity constraints, responsible for reducing the BF topological theory to General Relativity, are implemented weakly as a restriction on the representation labels.

Performing calculations with the Euclidean model is much simpler. The gauge group is compact, and it has finite-dimensional unitary irreducible representations. Moreover, Spin(4) SU (2) × SU (2), so its irreducible representations can be written as the tensor product of SU (2) ones. Those are very well studied, and the amplitude is the contraction of SU (2) invariants. The computations within the Lorentzian EPRL model are notably more complicated than in the Euclidean one. The group is non-compact, therefore its unitary representations are infinite-dimensional and much less studied. It is not surprising that the Euclidean model is the preferred first choice to perform complex calculations. Given the similarities between the two models, it has been so far assumed, as a strong hypothesis, that the results obtained in the simpler Euclidean model also hold for the Lorentzian one. The alternative is to repeat the specific calculation from scratch in the Lorentzian setting.

This work provides a map between the constrained irreducible representations and the spin foam vertex amplitudes of the two models. The map is realized through the "rotation", or analytic continuation, of the real Immirzi parameter γ to purely imaginary values. We show that the Euclidean vertex amplitude can be analytically continued to the Lorentzian amplitude, up to a multiplicative factor, sending γ → iγ. Conversely, the Lorentzian amplitude continues to the Euclidean one with the inverse map iγ → γ. We refer to this map as "Wick rotation" in analogy to the signature changing transformation in Quantum Field Theory. We remark that our transformation is a map between the internal gauge groups Spin(4) and SL(2, C) or equivalently a rotation between Euclidean and Lorentzian Ashtekar-Barbero variables [START_REF] Immirzi | Real and complex connections for canonical gravity[END_REF][START_REF] Ashtekar | A Generalized wick transform for gravity[END_REF][START_REF] Barbero | From Euclidean to Lorentzian General Relativity: The Real Way[END_REF], not on the spacetime manifold. This one has to be appropriately reconstructed in the semiclassical limit.

The paper is organized as follows. In Section 2, we review the SL(2, C) and Spin(4) algebras, group representations, useful decompositions, and the matrix elements of unitary irreducible representations. This is a well-known subject in the LQG community. However, the presentation for the two groups is traditionally very different. We take this occasion to fix our notations while treating the two groups along similar lines. In particular, we introduce the canonical basis for Spin [START_REF] Rovelli | Covariant loop quantum gravity: an elementary introduction to quantum gravity and spinfoam theory[END_REF]. In Section 3, we review the definition and implementation of the Y γ map, the main ingredient in the construction of the EPRL model. We also review the decomposition of the vertex amplitude in terms of booster functions. The EPRL models with different signatures differ only by the definition of their booster functions. The original contribution of this work is in Section 4. From a map between the algebras of SL(2, C) and Spin [START_REF] Rovelli | Covariant loop quantum gravity: an elementary introduction to quantum gravity and spinfoam theory[END_REF] we derive a map between unitary irreducible representations and group elements. We show that the matrix elements of the (ρ, k) representation of SL(2, C), with ρ ∈ R and k ∈ Z/2, can be obtained by analytic continuation from the matrix elements of the (p, k) representation of Spin(4), with p ∈ Z/2 and k ∈ Z/2. The analytic continuation is performed simultaneously on the representation labels (p, k) ↔ (iρ, k) and on the group elements. We define the analytic continuation of the group elements using the Cartan decomposition of both groups. Using these maps between group elements and representations, we show that the Euclidean and Lorentzian booster functions are related through analytic continuation when they are defined as line integrals along a particular complex contour. We conclude by applying this analytic continuation to the constrained representations of the vertex amplitudes with both signatures.

Mathematical preliminaries 2.1 Representations of SL(2, C)

The algebra of SL(2, C) is generated by L i , the generators of the spatial rotation subgroup, and K i , the generators of the corresponding boosts. They satisfy the commutation relations

[L i , L j ] = i ijk L k , [L i , K j ] = i ijk K k , [K i , K j ] = -i ijk L k . (1)
The two Casimir operators are K 2 -L 2 and K • L. The unitary irreducible representations in the principal series are labeled by the couple (ρ, k) where ρ is a real number and k is a half-integer. The Casimirs in this representations take the values

K 2 -L 2 |ρ, k = (ρ 2 -k 2 + 1) |ρ, k , K • L |ρ, k = ρk |ρ, k . (2)
The representation (-ρ, -k) is unitarily equivalent to the representation (ρ, k) [START_REF] Ruhl | The Lorentz group and harmonic analysis[END_REF]. This property is manifest in the expression for the matrix elements we derive in Section 4.2. Following the literature on the topic [START_REF] Ruhl | The Lorentz group and harmonic analysis[END_REF] we restrict ourselves to only positive values of ρ and k.

The group SL(2, C) is non-compact so the generic unitary representation (ρ, k) is infinite dimensional. The Hilbert space H (ρ,k) of the representation (ρ, k) decomposes in an infinite number of SU (2) representations with different values j of L 2

H (ρ,k) = ∞ j=k H j . (3)
The canonical basis of (ρ, k), given by |ρ, k; jm with j ≥ k and m = -j, . . . , j, diagonalizes L 2 and L 3

L 2 |ρ, k; j, m = j(j + 1) |ρ, k; j, m , L 3 |ρ, k; j, m = m |ρ, k; j, m . (4)
The canonical basis plays a central role in the construction of the EPRL model.

A useful decomposition of the group SL(2, C)is given by the map

SU (2) × A + × SU (2) -→ SL(2, C) (u, e r 2 σ3 , v) -→ ue r 2 σ3 v † .
(

) 5
where A + is the diagonal subgroup

A + : e r 2 σ3 = e r/2 0 0 e -r/2 , r ≥ 0 . (6)
This is usually called Cartan decomposition in the physics literature [START_REF] Ruhl | The Lorentz group and harmonic analysis[END_REF][START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF] 1 . The Haar measure with respect to this decomposition is [START_REF] Ruhl | The Lorentz group and harmonic analysis[END_REF][START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF] dµ SL(2,C) = 1 π sinh 2 r dr du dv = dµ(r) du dv .

The normalization factor 1 π we use in this paper differs from the one used in the literature by a factor 4. This choice allows us to write our formulas more cleanly. The matrix elements in the canonical basis of a group element g in this decomposition also decompose accordingly

D (ρ,k) jmln (g) ≡ ρ, k; j, m| g |ρ, k; l, n = D (ρ,k) jmln (ue r 2 σ3 v † ) = a,a D j ma (u)D (ρ,k) jala (e r 2 σ3 )D l a n (v † ) . (8)
In the expression above we used that the SU (2) subgroup is generated by L and its matrix elements are given by SU (2) Wigner matrices (see Appendix A)

ρ, k; j, m| u |ρ, k; l, n = δ jl D j mn (u) . (9)
Moreover, e

r 2 σ3 is diagonal, therefore D (ρ,k) jmln (e rσ3 ) = δ aa D (ρ,k) jala (e rσ3 ) ≡ δ aa d (ρ,k) jla (r) where d (ρ,k) jla are the reduced matrix elements of SL(2, C). Summarizing D (ρ,k) jmln (g) = a D j ma (u) d (ρ,k) jla (r) D j an (v † ) . (10)
The expression for d (ρ,k) jlm (r) was given in [START_REF] Ruhl | The Lorentz group and harmonic analysis[END_REF][START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF][START_REF] Vong | On the theory of unitary representations of the SL(2,C) group[END_REF][START_REF] Rashid | Boost matrix elements of the homogeneous Lorentz group[END_REF][START_REF] Basu | A Unified Treatment of the Groups SO(4) and SO(3,1)[END_REF] d

(ρ,k) jlm (r) = (-1) j-l (iρ -j -1)! (j + iρ)! (iρ -l -1)! (l + iρ)! (2j + 1)(2l + 1) (j + l + 1)! e (iρ-k-m-1)r (j + k)!(j -k)!(j + m)!(j -m)!(l + k)!(l -k)!(l + m)!(l -m)! s,t (-1) s+t e -2tr (k + s + m + t)!(j + l -k -m -s -t)! t!s!(j -k -s)!(j -m -s)!(k + m + s)!(l -k -t)!(l -m -t)!(k + m + t)! 2 F 1 {l -iρ + 1, k + m + s + t + 1}, {j + l + 2}; 1 -e -2r (11)
where 2 F 1 is the Gauss hypergeometric function, and the factorials of complex numbers in this formula and the rest of this paper have to be intended as Gamma functions using x! = Γ(x + 1).

Representations of Spin(4)

The algebra of Spin(4) = SU (2) × SU (2) is generated by two commuting SU (2) algebras

J i L , J j L = i ijk J k L , J i R , J j R = i ijk J k R , J i L , J j R = 0 . ( 12
)
The two Casimir operators are J 2 L and J 2 R and the unitary irreducible representations are labeled by the couples (j L , j R ) of half-integers such that

J 2 L |j L , j R = j L (j L + 1) |j L , j R , J 2 R |j L , j R = j R (j R + 1) |j L , j R . (13)
The group Spin( 4) is compact and the unitary representation (j L , j R ) has dimension (2j L + 1)(2j R + 1). The algebra of Spin( 4) is isomorphic to the algebra of SO(4). It is convenient to parametrize the algebra in terms of L i , the generators of the spatial rotation subgroup, and A i , the generators of time rotations or (Euclidean) boosts, as we will call them with a slight abuse of language. Defining the rotations and boost generators as

L = J L + J R , and A = J L -J R , (14)
they satisfy the algebra

[L i , L j ] = i ijk L k , [L i , A j ] = i ijk A k , [A i , A j ] = i ijk L k . (15)
We can sum and subtract the Casimirs (13) to obtain an equivalent set of two Spin(4) invariant operators

A 2 + L 2 = 2 J 2 L + J 2 R , L • A = J 2 L -J 2 R . (16)
We parametrize the representation (j L , j R ) in terms of two other half integer quantum numbers p ≡ j L +j R +1 and k ≡ j L -j R [START_REF] Biedenharn | Wigner Coefficients for the R4 Group and Some Applications[END_REF] 2 . In this work, without any loss of generality we will assume that j L ≥ j R such that p > k ≥ 0. In this representation, the Casimirs assume the values

A 2 + L 2 |p, k = (p 2 + k 2 -1) |p, k , L • A |p, k = pk |p, k . (17)
The representation space

H (p,k) decomposes in SU (2) representations with different values of L 2 = ( J L + J R ) 2
given by the usual sum of L(eft) and R(ight) angular momentum

H (p,k) = p-1 j=k H j . ( 18
)
The canonical basis of (p, k), given by |p, k; jm with p -1 ≥ j ≥ k and m = -j, . . . , j, diagonalizes L 2 and

L 3 L 2 |ρ, k; j, m = j(j + 1) |ρ, k; j, m , L 3 |ρ, k; j, m = m |ρ, k; j, m . (19)
Note that, in the spin foam literature, the Euclidean EPRL model is often formulated using the standard basis that diagonalizes J L3 , J R3 while J 2 L and J 2 R are taken as Casimirs. To highlight the similarities between the models with the two signatures, we formulate the Euclidean EPRL model in the canonical basis instead. We stress that our construction uses a different language from the traditional formulation, but it is entirely equivalent to it.

We want to find a decomposition analogue to the Cartan decomposition (5) for the group Spin(4). Inspired by the Euler parametrization of the rotation group, we look at the action

(g L , g R ) • h = g L hg † R (20)
of Spin( 4) SU (2) × SU (2) on the group H of quaternions. This action realizes the double-covering Spin(4) → SO(4) and in particular it is transitive on SU (2) S 3 in H. The diagonal subgroup (a, a) = D SU (2) ⊂ Spin(4) is the stabilizer of the identity. We parametrize an arbitrary element (g L , g R ) ∈ Spin(4) using two copies of this subgroup. We define the map

SU (2) × T × SU (2) -→ Spin(4) (u, e -i t 2 σ3 , v) -→ (ue -i t 2 σ3 v † , ue i t 2 σ3 v † ) ( 21
)
where

T = {exp -i t 2 σ 3 | t ∈ [0 4π
)} is the torus subgroup of SU (2). We show that the map is surjective. Let (g L , g R ) be a generic element of Spin(4). The equations

g L = ue -i t 2 σ3 v † g R = ue i t 2 σ3 v † imply g L g † R = ue -itσ3 u † g † R g L = ve -itσ3 v † .
The elements g L g † R and g † R g L are conjugate, and every element of SU (2) is conjugate to a diagonal matrix of the form exp(itσ 3 ). Hence we can solve the last equations for u, v. Notice, importantly, that it is enough to require t ∈ [0 2π) to get a unique solution. Therefore the proper Cartan decomposition for Spin(4) is

SU (2) × T + × SU (2) -→ Spin(4) (u, e -i t 2 σ3 , v) -→ (ue -i t 2 σ3 v † , ue i t 2 σ3 v † ) (22)
where

T + = {exp -i t 2 σ 3 | t ∈ [0 2π)}.
In the following we will also use the notation

E + = {(g, g † ) | g ∈ T + }.
The Haar measure with respect to the decomposition described in [START_REF] Dona | Infrared divergences in the EPRL-FK Spin Foam model[END_REF]

The matrix elements of g = (g L , g R ) in the standard basis are given by the tensor product of two SU (2) Wigner matrices

D (j L ,j R ) m L m R n L n R (g) ≡ j L , m L ; j R , m R | g |j L , n L ; j R , n R = = j L , m L | g L |j L , n L j R , m R | g R |j R , n R = D j L m L n L (g L )D j R m R n R (g R ) . (25)
This is the main reason why the |j L , j R basis is traditionally preferred for the construction of the EPRL model. The Cartan decomposition for Spin(4) [START_REF] Dona | Infrared divergences in the EPRL-FK Spin Foam model[END_REF] allows us to decompose the matrix elements in the canonical basis similarly to SL(2, C)

D (p,k) jmln (g) = o D j mo (u)d (p,k) jlo (t)D l on (v † ) . (26)
In the expression above we used that the D SU (2) is generated by L, the matrix elements of u ∈ SU (2) are given by

p, k; j, m| u |p, k; l, n = p, k; j, m|j L m L j R m R D j L m L n L (u)D j R m R n R (u) j L n L j R n R |p, k; l, n = δ jl D j mn (u) . ( 27
)
The reduced matrix elements d (p,k) jlo (t) are defined in terms of SU(2) Clebsch-Gordan coefficients intertwining between the (j L , j R ) and j or l SU (2) representation [START_REF] Biedenharn | Wigner Coefficients for the R4 Group and Some Applications[END_REF][START_REF] Lorente | Tensor and Spin Representations of SO(4) and Discrete Quantum Gravity[END_REF]

d (p,k) jlm (t) = D (p,k) jmlm (e i t 2 σ3 , e -i t 2 σ3 ) = m L ,m R j L , m L , j R , m R |jm e it(m L -m R ) j L , m L , j R , m R |lm (28) = 2p-2q-1 n=0 e it(p-q-1+m-2n) p + q -1 2 , p -q -1 2 + m -n, p -q -1 2 , n - p -q -1 2 jm p + q -1 2 , p -q -1 2 + m -n, p -q -1 2 , n - p -q -1 2 lm .

The Lorentzian and the Euclidean EPRL model

The starting point for constructing the EPRL model is the spin foam quantization of a topological BF theory leading to a well-defined state sum model. The classical simplicity constraints reduces the topological BF theory to gravity. The path integral for quantum gravity is obtained implementing the quantum simplicity constraints on the BF partition function. The simplicity constraints are expressed by an equation involving non-commuting operators and cannot be imposed strongly. The solution is to implement them weakly using master constraints techniques involving Casimir operators. The result is a restriction on the unitary representations that contribute to the state sum model. We will very briefly review the implementation of the linear simplicity constraints via the Y γ map in both the Lorentzian and Euclidean EPRL model. For an exhaustive discussion, we refer to the original paper [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] or the reviews [START_REF] Perez | The Spin Foam Approach to Quantum Gravity[END_REF][START_REF] Rovelli | Covariant loop quantum gravity: an elementary introduction to quantum gravity and spinfoam theory[END_REF]. The goal of the following sections is to fix the notation used in this work and to show the similarities in the implementation of the Y γ if we use the canonical basis for the groups with both signatures.

The Lorentzian Y γ map

The linear simplicity constraints in the Lorentzian model impose a linear dependence between the rotation and boost generators of the SL(2, C) algebra ( 1)

K = γ L . (29)
However, as rotation and boost generators do not commute, such equation cannot be imposed strongly at the quantum level. We apply it weakly using two master constraints by imposing strongly two commuting quadratic operators derived from [START_REF][END_REF]. We square [START_REF][END_REF] and project it onto L to obtain two constraints written in terms of the Casimirs of SL(2, C) (2)

K 2 -L 2 = (γ 2 -1)L 2 , L • K = γL 2 . ( 30
)
On the canonical basis of the unitary irreducible representation of SL(2, C) in the principal series the constraints [START_REF] Bianchi | LQG propagator from the new spin foams[END_REF] translate to an equation for the representation labels

ρ 2 -k 2 + 1 = (γ 2 -1)j(j + 1) , ρk = γj(j + 1) . (31)
For large quantum numbers this equation is solved by

ρ = γj and k = j . (32)
The constraints [START_REF] Bianchi | LQG propagator from the new spin foams[END_REF] select special representations of SL(2, C) (with representation labels proportional by a factor γ) and project to the lowest SU (2) subgroup of (3). Equations ( 32) define a map from the SU (2) representation of spin j to a subspace of the SL(2, C) representation (γj, j)

Y γ : |j, m → |γj, j; j, m . (33)
Therefore, Y γ is a map from SU (2) spin networks, the kinematical states of LQG, to SL(2, C) spin networks, the states at the boundary of a spin foam vertex amplitude.

The Euclidean Y γ

The linear simplicity constraints in the Euclidean model are very similar to the Lorentzian case. The generators of the Euclidean boosts A take the place of their Lorentzian counterpart

K A = γ L . (34)
The master constraints are

L 2 + A 2 = (γ 2 + 1)L 2 , L • A = γL 2 , (35)
that on the canonical basis reduce to an equation between representation labels

p 2 + k 2 -1 = (γ 2 + 1)j(j + 1) , pk = γj(j + 1) , (36)
that for large quantum numbers is solved by p = γj 3 and k = j. To be consistent with the convention of taking p > k we restrict to the case of γ > 1. Notice that this is not a limitation. The case with γ ≤ 1 can be studied by considering the case p < k and have as solution of (36) p = j and k = γj. We refrain to consider this case just for convenience, all the formulas in that case can be obtained by exchanging p with k.

The Euclidean Y γ map is defined as

Y γ : |j, m → |γj, j; j, m , (37)
and also in this case it provides a map between the LQG kinematical states to the states at the boundary of the spin foam vertex amplitude. If we insist that the map is valid for any SU (2) irrep j we also have an accidental quantization condition over the Immirzi parameter γ as the labels p, k and j are half-integers [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF][START_REF] Freidel | A New Spin Foam Model for 4d Gravity[END_REF]. At this point, the reader experienced with the presentation of the Euclidean EPRL model found in the literature could feel disoriented. Usually, the Y γ map in the Euclidean model is not defined on the canonical basis but on the standard one. In the standard basis the representation labels [START_REF] Basu | A Unified Treatment of the Groups SO(4) and SO(3,1)[END_REF]

are j L = p+k-1 2 and j R = p-k-1 2
. If we consider representations with p = γj and k = j we find the familiar restriction on the representations of Spin(4) with γ > 1 4

j L = (γ + 1) j 2 - 1 2 and j R = (γ -1) j 2 - 1 2 . ( 38
)
On the standard basis, we have the advantage of a simpler form of the matrix elements [START_REF] Helgason | Groups & Geometric Analysis: Radon Transforms, Invariant Differential Operators and Spherical Functions[END_REF]. The price to pay is to hide the parallelism with the Lorentzian version of the model, which becomes evident if we use the same (canonical) basis for both.

Decomposition of the vertex amplitude in terms of booster functions

The Cartan decomposition of the SL(2, C) group elements (5) allows us to recast the EPRL Lorentzian vertex amplitude as a superposition of SU (2) {15j} symbols weighted by the product of four booster functions

A L v (j f , i e ) = l f ,ke e d ke B L 4 (j f , l f , i e , k e ) {15j}(l f , k e , i ) (39)
This decomposition was first introduced in [START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF] and is one of the fundamental ingredients of the numerical calculations performed within the model [START_REF] Dona | SU(2) graph invariants, Regge actions and polytopes[END_REF][START_REF] Dona | Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude[END_REF][START_REF] Dona | Searching for classical geometries in spin foam amplitudes: a numerical method[END_REF][START_REF] Donà | Numerical analysis of spin foam dynamics and the flatness problem[END_REF]. The Lorentzian booster functions B L 4 are the integral over A + of the product of four SL(2, C) reduced matrix elements in the (γj f , j f ) representation contracted with 4jm-symbols (99) and have been extensively studied numerically [START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF][START_REF] Dona | Numerical methods for EPRL spin foam transition amplitudes and Lorentzian recoupling theory[END_REF][START_REF] Gozzini | High-performance spinfoam numerics[END_REF][START_REF] Dona | Infrared divergences in the EPRL-FK Spin Foam model[END_REF] and analytically [START_REF] Dona | Asymptotics of SL(2, C) coherent invariant tensors[END_REF]:

B L 4 (j a , l a , i, k) ≡ ma j a m a (i) ∞ 0 dµ(r) a d (γja,ja) jalama (r) l a m a (k) , (40)
where d (γja,ja) are the reduced matrix elements [START_REF] Vong | On the theory of unitary representations of the SL(2,C) group[END_REF] and dµ(r) is the part over A + of the SL(2, C) Haar measure [START_REF] Ashtekar | A Generalized wick transform for gravity[END_REF].

It is possible to write a similar decomposition also for the Euclidean EPRL vertex amplitude. The derivation is the same as in the Lorentzian case, and it is based on the Cartan decomposition of Spin(4) 3 As a side comment, since we are interested in solving [START_REF] Naimark | Linear representations of the Lorentz group[END_REF] for large quantum numbers, we could equivalently take p = γj + 1 and k = j as solution of [START_REF] Naimark | Linear representations of the Lorentz group[END_REF]. The advantage is that the matrix element D (γj+1,j) jnjm (g) = 1 for j = 0 for all group elements. This allows to define a cylindrically consistent Euclidean EPRL model. Unfortunately, there are no obvious analog choices for the Lorentzian model. A possibility is to define the Lorentzian Yγ map as an analytic continuation of the representations ρ = γj + i and k = j. We leave this as a speculative comment. 4 If we take the solution of [START_REF] Naimark | Linear representations of the Lorentz group[END_REF] for large quantum number to be p = γ(j + 1) and k = j instead as in [START_REF] Perez | The Spin Foam Approach to Quantum Gravity[END_REF], we have to add an extra γ 2 to both j L and j R . [START_REF] Anderson | Recursion and symmetry relations for the clebsch-gordan coefficients of the homogeneous lorentz group[END_REF]. The EPRL Euclidean vertex amplitude can be expressed as a superposition of SU (2) {15j} symbols weighted by the product of four Euclidean booster functions

A E v (j f , i e ) = l f ,ke e d ke B E 4 (j f , l f , i e , k e ) {15j}(l f , k e , i ) . (41)
The Euclidean booster functions B E 4 are defined in terms of the product of four Spin(4) reduced matrix elements in the (γj f , j f ) representation contracted with 4jm-symbols:

B E 4 (j a , l a , i, k) ≡ ma j a m a (i) 2π 0 dµ(t) a d (γja,ja) jalama (t) l a m a (k) , (42)
where d (γja,ja) are the reduced matrix elements [START_REF] Nicotra | Analytical map between EPRL spin foam models in loop quantum gravity[END_REF] and dµ(t) is the part over E + part of the Spin(4) Haar measure [START_REF] Hall | Lie Groups, Lie Algebras, and Representations[END_REF].

4 The map between the spin foam models

Mapping algebras and groups

The Lie algebra su( 2) is a compact real form of SL(2, C) [START_REF] Hall | Lie Groups, Lie Algebras, and Representations[END_REF]. Therefore, we get the (complex) algebra

sl(2, C) of SL(2, C) by complexification sl(2, C) = su(2) ⊕ isu(2) . (43)
If we consider the (real) algebra of Spin(4), spin(4) su(2) ⊕ su(2), we get the (realification of the) algebra of SL(2, C) by "rotating" half of the algebra to purely imaginary generators:

spin(4) su(2) ⊕ su(2) → su(2) ⊕ isu(2) sl(2, C) . ( 44
)
Equivalently, rotating the generator of boosts in SL(2, C) (considering its Lie algebra as a real algebra) we obtain the algebra of Spin(4). Using the canonical bases for Spin(4) and SL(2, C) introduced in the previous sections, the rotation maps the generators of Euclidean boosts to the generators of Lorentzian boosts and vice versa: ( L, i K) spin( 4) and ( L,

-i A) sl(2, C) . (45)
We write these isomorphisms of (real) Lie algebras as A ↔ i K and K ↔ -i A. The map [START_REF] Yutsis | Theory of Angular Momentum in Quantum Mechanics[END_REF] induces a map between group elements as follows. The Cartan decomposition Spin(4)

= D • E + • D we introduced in Section 2.2, is analogous to the Cartan decomposition SL(2, C) = SU (2) • A + • SU (2)
where by • we mean the group product. Therefore, the map (45) induces a map between the compact subgroup E + in ( 22) and the non-compact subgroup A + in (5) seen as subgroups of the complexified groups Spin(4

) C SL(2, C) C .
For example, the map from E + to A + can be achieved by sending

t -→ ir ( 46
)
where t ∈ [0 2π) parametrizes E + and r ∈ R + parametrizes A + . This relation between the compact subgroup E + and the non-compact subgroup A + can be given an interesting geometrical interpretation. Since the action of Spin( 4) is transitive on SU (2) S 3 ⊂ H and D stabilizes the identity, the 3-sphere S 3 is a homogeneous space for Spin(4), and we can identify the quotient subgroup Spin(4)/D with the 3-sphere. There is a similar result for SL(2, C) (a well known construction of geometric analysis [START_REF] Helgason | Groups & Geometric Analysis: Radon Transforms, Invariant Differential Operators and Spherical Functions[END_REF]): the quotient group SL(2, C)/SU (2) can be identified with hyperbolic 3-space H 3 .

In light of the Cartan decompositions, write an element of the quotient Spin(4)/D as de(t)D, e(t) ∈ E + , d ∈ D, and an element of SL(2, C)/SU (2) as ka(r)K, a(r) ∈ A + , k ∈ K = SU (2). The parameters r, t act as radial coordinates in the corresponding 3-manifolds. Hence the inverse map r → -it from A + to E + can be interpreted geometrically as mapping hyperbolic 3-space to spherical 3-space, similarly to the usual rotation t → iτ from physical time to Euclidean time that transforms Lorentzian metrics into Euclidean ones (and in particular flat Minkowski space to flat Euclidean space). This becomes manifest if we consider the metric of hyperbolic 3-space in radial coordinates

dH 2 = dr 2 + sinh 2 r dΩ 2 2 (47)
where dΩ 2 is the metric on the 2-sphere. The map r → -it maps this metric to

dS 2 = dt 2 + sin 2 t dΩ 2 2 (48)
which is exactly the metric of the 3-sphere (up to an innocuous global minus sign).

From the metrics (47) and (48) we can also read the Jacobians that enter the Haar measures of Spin(4) and SL(2, C) given in Sections 2.1 and 2.2. The measure on E + gets mapped to -the measure on

A + dµ(t) = 1 π sin 2 t -→ - 1 π sinh 2 r dr = -dµ(r) . ( 49
)
and the Haar measure of Spin(4)( 24) gets mapped to the Haar measure of SL(2, C) [START_REF] Ashtekar | A Generalized wick transform for gravity[END_REF] dµ

Spin(4) = 1 π sin 2 t dt du dv -→ 1 π sin 2 ir dr du dv = -dµ SL(2,C) . (50)
.

Mapping representations and matrix elements

The isomorphisms of real Lie algebras [START_REF] Yutsis | Theory of Angular Momentum in Quantum Mechanics[END_REF] can be used to find a correspondence between the unitary irreducible representations of Spin(4) and SL(2, C). We need the following facts:

(i) the complexification of spin( 4) is isomorphic to the complexification of sl(2, C) (as a real algebra)

spin(4) C su(2) C ⊕ su(2) C sl(2, C) ⊕ sl(2, C) sl(2, C) C , (51)
(ii) for any Lie algebra g, (real linear) representations of g on a complex vector space extend uniquely to holomorphic (i.e. complex linear) representations of g C on the same vector space.

These two results imply that we can map the (p, k) and (ρ, k) representations working with the complexified algebras. Concretely, we can compute the action of the Casimirs in the complexified algebras and find the map between representations looking at their eigenvalues on the respective canonical bases. From A ↔ i K we get

A 2 + L 2 ↔ -( K 2 -L 2 ) p 2 + k 2 -1 ↔ -ρ 2 + k 2 -1 , (52)
A • L ↔ i K • L pk ↔ iρk . (53)
Looking at the second Casimir we read the map from SL(2, C) to Spin(4) representations:

(ρ → -ip, k) (p, k) . ( 54
)
The converse isomorphism K ↔ -i A provides the map from Spin(4) to SL(2, C):

(p → iρ, k) (ρ, k) . (55)
In the following we also write (p, k) ↔ (iρ, k) to denote both (54) and (55). These correspondences are defined up to a global minus sign, which is irrelevant since the irreps (ρ, k) and (-ρ, -k) are unitarily equivalent. These maps between representations can be realized explicitly in terms of matrix elements, as follows. Using analytic continuation of the representation labels, the SL(2, C) matrix elements in the (ρ, k) representation can be obtained from the Spin(4) matrix elements in the (p, k) representation using (55) and [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF]. The converse from Spin(4) to SL(2, C) is also possible. This result can be found in the group theory literature [START_REF] Anderson | Recursion and symmetry relations for the clebsch-gordan coefficients of the homogeneous lorentz group[END_REF][START_REF] Basu | A Unified Treatment of the Groups SO(4) and SO(3,1)[END_REF][START_REF] Rashid | Boost matrix elements of the homogeneous Lorentz group[END_REF][START_REF] Wong | Boost matrix elements and Clebsch-Gordan coefficients of the homogeneous Lorentz group[END_REF]. Each paper in this list use a different technique and a set of different conventions, making challenging to compare them. The proof we present in Appendix B is original and it is based only on elementary properties of the hypergeometric functions. Deriving a new proof helps us to be immune to the numerous different conventions that plague the literature on the subject. We summarize the result here and refer to Appendix B and [START_REF] Nicotra | Analytical map between EPRL spin foam models in loop quantum gravity[END_REF] for more details for the interested reader. Because of the Cartan decompositions [START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF] and [START_REF] Anderson | Recursion and symmetry relations for the clebsch-gordan coefficients of the homogeneous lorentz group[END_REF] it is sufficient to show that the reduced matrix elements of SL(2, C) given by ( 11) can be analytically continued in the Spin(4) ones [START_REF] Nicotra | Analytical map between EPRL spin foam models in loop quantum gravity[END_REF]. We find that

d (ρ,k) jlm (r) = n e -(iρ-k-1+m-2n)r iρ + k -1 2 , iρ -k -1 2 + m -n ; iρ -k -1 2 , n - iρ -k -1 2 j, m iρ + k -1 2 , iρ -k -1 2 + m -n ; iρ -k -1 2 , n - iρ -k -1 2 l, m + n e -(-iρ+k-1+m-2n)r -iρ -k -1 2 , -iρ + k -1 2 + m -n ; -iρ + k -1 2 , n - -iρ + k -1 2 j, m -iρ -k -1 2 , -iρ + k -1 2 + m -n ; -iρ + k -1 2 , n - -iρ + k -1 2 l, m (56)
where j 1 , m 1 , j 2 , m 2 |j, m are the analytic continuation to complex spins of SU (2) Clebsch-Gordan coefficients (111). Notice that, under the change of sign of the representation labels (ρ, k) → (-ρ, -k) the first term becomes the second and vice-versa. This is an explicit sign of the unitary equivalence of the representations (ρ, k) and (-ρ, -k).

If we perform the analytic continuation in both the representation label ρ → -ip, equivalently iρ → p, and the group element r → -it, equivalently ir → t, we obtain

d (-ip,k) jlm (it) = n e (p-k-1+m-2n)it p + k -1 2 , p -k -1 2 + m -n , p -k -1 2 , n - p -k -1 2 j, m p + k -1 2 , p -k -1 2 + m -n , p -k -1 2 , n - p -k -1 2 l, m + n e (-p+k-1+m-2n)it -p -k -1 2 , -p + k -1 2 + m -n , -p + k -1 2 , n - -p + k -1 2 j, m -p -k -1 2 , -p + k -1 2 + m -n , -p + k -1 2 , n - -p + k -1 2 l, m . (57)
The second term vanishes identically since k ≤ j, l ≤ p -1 while the Clebsch-Gordan coefficients vanishes if j, l ≤ -p -1. If we shift the first summation n → n + p-k-1

2
we obtain the expression for the reduced matrix elements of Spin(4) as in [START_REF] Nicotra | Analytical map between EPRL spin foam models in loop quantum gravity[END_REF].

For later convenience, we introduce the generalized matrix element function on the complex plane

d (a,k) jlm (z) =(-1) j-l (a -j -1)! (j + a)! (a -l -1)! (l + a)! (2j + 1)(2l + 1) (j + l + 1)! z -(a-k-m-1) (j + k)!(j -k)!(j + m)!(j -m)!(l + k)!(l -k)!(l + m)!(l -m)! s,t (-1) s+t z 2t (k + s + m + t)!(j + l -k -m -s -t)! t!s!(j -k -s)!(j -m -s)!(k + m + s)!(l -k -t)!(l -m -t)!(k + m + t)!
where z ∈ C, k ∈ Z/2 and a can be either a half integer greater than 1 or a purely imaginary number. We use for (58) a notation similar to the reduced matrix elements of SL(2, C) and Spin(4) and we distinguish it from them by the use of the complex argument. The slight abuse of notation is justified since, if a = iρ and z = e -r then (58) turns into the SL(2, C) reduced matrix elements d

Mapping the vertex amplitude

We compare the expressions of the Lorentzian (39) and the Euclidean (41) vertex amplitude. To find a relation between the amplitudes, it is sufficient to study the relation between the Lorentzian booster function [START_REF] Han | Analytic Continuation of Spin foam Models[END_REF] and the Euclidean booster function [START_REF] Frodden | Black Hole Entropy from complex Ashtekar variables[END_REF].

Booster functions as integrals

The generalized matrix element d (a,k) jlm (z) (58) and its connection with the reduced matrix elements of Spin(4) and SL(2, C) induces an embedding of E + in the complex z-plane to the unit circle e it ∈ S 1 and an embedding of A + to the unit interval e -r ∈ [0 1].

The booster functions ( 40) and ( 42) depend on the integration of the product of four reduced matrix elements over the appropriate subgroup. This translates into the integration of the generalized matrix element d (a,k) jlm (z) (58) in the complex plane along the unit circle in the Euclidean case, or the unit interval in the Lorentzian case. In the following, we relate the integrals of products of d (iρ,k) (z) along these two paths.

For simplicity, we will focus on the minimal case j = l = k. This simplification allows us to avoid the clutter of the additional sums over the indices s, t present in the reduce matrix elements in the nonminimal case. However, the calculation in the general case follows the same steps and we can apply the same arguments of the simplified case. It is just more cumbersome and confusing to keep track of all the terms. In the simplified case the matrix elements become

d (a,k) kkm (z) = z (k+m+1)-a 2 F 1 {k -a + 1, k + m + 1}, {2k + 2}; 1 -z 2 . ( 59
)
The integrand of ( 40) and ( 42) with general a and omitting the intertwiner index for brevity is

f (a i , k i ; z) = mi k i m i 4 i=1 d (ai,ki) kikimi (z) k i m i = mi z K-A+4+M k i m i 4 i=1 2 F 1 {k i -a i + 1, k i + m i + 1}, {2k i + 2}; 1 -z 2 k i m i (60) with K = i k i , A = i a i , M = i m i .
Notice that the 4jm-symbols vanish if M = 0, therefore in the following we will assume M = 0 if needed.

From Euclidean to Lorentzian integrals

The Euclidean booster functions are defined as the integral of (60) with a i = p i = γj i half integer greater than 1. Our results are valid independently from the imposition of the Y γ map that constrains p i to assume specific values. In this section, we will keep p (and ρ in the Lorentzian case) generic until the very end.

To avoid unnecessary confusion, we will call the integral of (60) I E (p i , k i ), emphasizing its dependence on Spin(4) labels p i and k i , and discuss the connection to the booster function later. The Spin(4) Haar measure induces on the complex plane the integration measure dt sin 2 (t)

π -→ i (1 -z 2 ) 2 4πz 2 dz z = dµ(z) . (61)
The Euclidean integral I E expressed as a contour integral over the unit circle is

I E (p i , k i ) = i 4π S 1 dz (1 -z 2 ) 2 mi k i m i z K-A+M +1 × 4 i=1 2 F 1 {k i -p i + 1, k i + m i + 1}, {2k i + 2}; 1 -z 2 k i m i . (62)
The first argument of all the hypergeometric functions k i -p i is a strictly negative integer since k i < p i (see Section 2.2). It follows that the hypergeometric functions reduce to polynomials in 1 -z 2 . However, from k i -p i + 1 = -2J Ri we find that the prefactor z K-A+M +1 introduces a pole singularity in z = 0 and the complete integrand is meromorphic. The integral can be evaluated using Cauchy residue theorem, and the result is 2πi times the residue at z = 0 which depends non-trivially on all the parameters through the product of the hypergeometric functions.

As an example, we work out explicitly the degenerate case j Ri = 0 where it is particularly simple to compute the integral I E with Cauchy's closed curve theorem. Here k i = p i -1 and all the hypergeometric functions are equal to 1. We can take M = 0 otherwise the 4jm-symbols vanish, so that K -A = -4 in this case. The function (1 -z 2 ) 2 z -3 has a pole with residue -2 in the origin. The integral is immediately calculable, reintroducing the intertwiner indices (a) and (b) explicitly

mi k i m i (a) i 4π 2πi(-2) k i m i (b) = δ ab 2a + 1 . ( 63
)
This result is exactly what one would expect by performing the integral in the canonical basis remembering that

D J Ri (g R ) = 1 if J Ri = 0 for any g R ∈ SU (2).
The integrand is analytic in the punctured plane. Therefore, it is actually irrelevant which contour one uses, as long as it contains z = 0. Let us consider the contour C ε represented in red in Figure 1. The horizontal segments have small distance ε from the real axis. The semicircles around 0 and 1 have small radius ε. We can deform the contour integral in (64) from the unit circle to the contour C ε obtaining a completely equivalent definition of the Euclidean integral. Furthermore, taking the limit ε → 0 does not change the result. We will base the analytic continuation of the Euclidean integral on this expression:

I E (p i , k i ) = lim →0 i 4π Cε dz (1 -z 2 ) 2 mi j i m i z K-A+M +1 × 4 i=1 2 F 1 {k i -a i + 1, k i + m i + 1}, {2k i + 2}; 1 -z 2 j i m i . ( 64
)
The Euclidean booster function is obtained from (64) by imposing a restriction on p = γk coming from the Y γ -map.

The Lorentzian booster functions are defined as the integral of (60) with a i = iρ i with ρ i a positive real number. Once again we postpone the imposition of the Y γ -map to the very end and to avoid unnecessary confusion we will call the integral I L (ρ i , k i ), emphasizing its dependence on SL(2, C) labels ρ i and k i . The SL(2, C) Haar measure induces on the complex plane the integration measure

dr sinh 2 (r) π -→ - (1 -z 2 ) 2 4πz 2 dz z = idµ(z) . ( 65
)
The Lorentzian integral I L can be expressed as a line integral over the unit interval as

I L (ρ i , k i ) = 1 4π [0 1] dz mi j i m i (1 -z 2 ) 2 z K-A+M +1 × 4 i=1 2 F 1 {k i -iρ i + 1, k i + m i + 1}, {2k i + 2}; 1 -z 2 j i m i . ( 66
)
The integrand of (66) differs from the Euclidean one (64) only by a factor of i and the different values of a i and A = i a i = i i ρ i . This however implies that the Lorentzian integrand is not meromorphic anymore.

The hypergeometric functions develop a branch point singularity in 1 -z 2 = 1, i.e. z = 0. The analytic continuation of the hypergeometric series outside of the unit disc using Euler's formula has a branch cut discontinuity along real numbers x ≥ 1 [START_REF][END_REF]. In our case, the hypergeometric function is computed in 1 -z 2 . The branch cut discontinuity is along the whole imaginary axis, represented as a punctured line in Figure 1, and there are two disconnected domains of analyticity. We assign the principal branch | arg(z)| < π/2 on both sides of the imaginary axis and we define the value on the imaginary axis minus the origin by continuity from the left.

At the origin each one of the hypergeometric functions in (60) is in general divergent. This happens when Re(δ i -α i -β i ) < 0 where α i , β i , δ i are the three parameters of 2 F 1 {α i , β i }, {δ i }; 1 -z 2 . In our case Re(δ i -α i -β i ) = -m i , therefore for some m i the hypergeometric function is divergent at most of order

d i = 2 max(0, m i ), i.e. lim z→0 - z di+σ 2 F 1 {k i -a i + 1, k i + m i + 1}, {2k i + 2}; 1 -z 2 = 0 ( 67
)
for any real σ > 0 6 . The product of four hypergeometric functions in ( 60) is divergent at most logarithmically in the origin, i.e. i d i = 2 max(0, i m i ) = 2 max(0, M ) = 0, since we can always take M = 0. This implies that the prefactor z K-A+1 cures any potential divergence in the origin of (60). 7 6 The hypergeometric function has the property [START_REF][END_REF] lim

w→1 - (1 -w) α+β-δ 2 F 1 [{α, β}, {δ}, w] = Γ(δ)Γ(α + β -δ) Γ(α)Γ(β) , ( 68
) therefore lim w→1 - (1 -w) α+β-δ+σ 2 F 1 [{α, β}, {δ}, w] = 0 , (69)
for any real σ > 0. Substituting w → 1 -z 2 we obtain (67). 7 Notice that this is not peculiar for the minimal case j i = l i = k i . In the general case the order of divergence of the product of the four hypergeometric functions is given by 2 max(0, m i + t i -(j i -k i -s i )). The summation over s i is such that j i -k i -s i is always positive, the z 2t i factor counterbalances the possible divergence of order 2t i and i m i = 0. We conclude that also in the general case the divergence (at most logarithmic) of the product of hypergeometric functions is cured by the prefactor z K-A+1 .

The branch cut discontinuity along the imaginary axis is not the only one. The prefactor z K-A+1 also has branch point singularities in 0 and ∞ since A is purely imaginary. For this term we consider the branch | arg(-z)| < π so that the discontinuity is along the positive real axis (represented in Figure 1 as a punctured line). We note, however, that the hypergeometric functions are continuous across the positive real axis.

We conclude our analysis by relating the integral on the unit interval in (66) to the contour integral over C ε of the same function in the limit of ε → 0. The latter is i times the analytic continuation of the Euclidean integral (64) to purely imaginary a i .

We split the contour C ε in four pieces: let C 0 ε be the small semicircle around 0, C 1 ε be the small semicircle around 1, C + ε be the straight line above the real axis and C - ε be the straight line below the real axis. The integral along C 1 ε vanishes since Re(K -A + 1) > 0 and the hypergeometric function is regular there. The integral along C 0 ε vanishes for the same reason, since the eventual logarithmic divergence at z = 0 of the product of the four hypergeometric functions is more than canceled by the prefactor z K-A+1 . The integrals along C + ε , C - ε differ by a factor exp 2πi(K -A + 1) due to the presence of the branch cut of z K-A+1 while the hypergeometric function is continuous in the right half-plane Re z > 0. In the limit ε → 0 we have

lim ε→0 Cε dz f (a i ; z) = lim ε→0 C + ε dz f (a i ; z) + C - ε dz f (a i ; z) = e 2πi(K-A+1) -1 [0 1] dz f (a i ; z) (70)
taking into account the opposite directions induced by the contour C ε . From the definition (64) of the Euclidean integral we conclude that taking its analytic continuation p i → iρ i we obtain (up to a multiplicative factor) the Lorentzian integral

I E (p i , k i ) pi→iρi ------→ i e 2πi i (ki-iρi) -1 I L (ρ i , k i ) (71)
or equivalently

I L (ρ i , k i ) = i 1 -e 2πi i (ki-iρi) I E (iρ i , k i ) (72)
where on the right side we mean the analytic continuation of I E (p i , k i ) as a function of p i and k i to purely imaginary values of the first argument 8 .

In deriving (71) we considered the minimal case l i = j i = k i for simplicity. The generalization to any j , l i ≥ k i , and in particular to the half-minimal case relevant for the booster function k i = j i and l i ≥ j i , is straightforward. All the arguments we made are immediately extended. In particular, the prefactor z K-A+1 remains the same and all the considerations about the product of the hypergeometric functions with minimal arguments apply also to the more complicated sum over s i , t i of products of hypergeometric functions.

From Lorentzian to Euclidean integrals

Formulae (71) and (72) provide also the converse result for the rotation iρ i → p i :

I E (p i , k i ) = i lim qi→pi e 2πi i (ki-qi) -1 I L (-iq i , k i ) , (73)
where first we do the analytic continuation ρ i → -iq i of I L (ρ i , k i ) with q i ∈ R \ Z and then we take the limit q i → p i = k i + n i and n i ∈ N + to regularize the product of the vanishing prefactor with the divergent function I L (-iq i , k i ). In fact, the defining integral representation (66) of I L (ρ i , k i ) is divergent if we perform the substitution iρ i → p i , for any half-integer p i > k i ≥ 0. However, we can overcome this difficulty noticing that the same apparent obstruction appears for example in the Euler's integral representation of the standard Gamma and Beta functions [START_REF][END_REF]. In particular, after the substitution iρ i → p i the integral in (66) reduces to

1 0 dz (1 -z 2 ) 2 z 1+ i (ki-pi) P [1 -z 2 ] ( 74
)
where P [1 -z 2 ] stands for a generic polynomial in the variable 1 -z 2 and where we set M = 0. Changing variables z 2 → w we can write this integral as a finite sum of Beta functions

1 0 dw w -2+ 1 2 i (ki-pi+1) P [1 -w] ∼ j B -2 + 1 2 i (k i -p i + 1), n j (75)
with first argument always a negative integer and second argument a positive integer. The Beta function can be analytically continued to complex values of its arguments using for example the Pochammer contour, possibly with simple poles at the negative integers. Since I L (-iq i , k i ) tends to (75) continuously for q i → p i , this implies that it is possible to analytically extend I L (ρ i , k i ) to generic complex values of the first parameter, again possibly with simple poles at the negative integers, i.e. at the values k i -p i +1 relevant for our case. The simple form (75) holds however only in small a neighborhood of the poles since in general the hypergeometric functions won't be expressible as simple polynomials. Remarkably, the vanishing prefactor in (73) exactly cancels the divergence of the analytically continued I L (-iq i , k i ) at its simple poles. We show how this works in the simple case p i = k i + σ, σ → 1 where we expect to recover (63). In this case the hypergeometric functions are equal to unity and the integral in (66) reduces to

1 0 dz (1 -z 2 ) 2 z 1-4σ . ( 76
)
The integral is divergent for σ → 1. Changing variables z 2 → w we obtain the integral 1 2

1 0 dw (1 -w) 2 w -2σ = 1 2 B(1 -2σ, 3) (77)
which as a complex function of σ has a simple pole in σ = 1 with residue -1. Recalling the prefactor from (73) we get i 2 lim

σ→1 e -8πiσ -1 B(1 -2σ, 3) = 4π (78)
and inserting this in (66) we get exactly the Euclidean result (63).

Notice that since each ρ i span an open subset of C, the previous considerations support strongly the conjecture that the two functions I E (p i , k i ) and I L (ρ i , k i ) are particular integral representations of a unique function I(a i , k i ) defined on the whole space C 4 × Z 4 , which agrees with I E (p i , k i ) for a i = p i and with I L (ρ i , k i ) for a i = iρ i . Hence, we can speak unambiguously of the analytic continuation of I E and I L . We do not provide a rigorous proof of this interesting claim here, which would require a more careful treatment of the interplay between the analytic continuation of the hypergeometric functions, the prefactor z K-A+1 and the integration on the unit interval.

Analytic continuation of vertex amplitudes

Imposing the Y γ map to both the Lorentzian and Euclidean integrals we find the desired relation between the (analytic continuations of the) Lorentzian and Euclidean booster functions. The analytic continuation p i → iρ i reads γj i → iγj i and can be interpreted fascinatingly as the rotation of the Immirzi parameter γ → iγ

B L 4 (j a , l a , i, k) = i 1 -e 2π(i+γ) i ji B E 4 (j a , l a , i, k) (γ→iγ) . ( 79
)
The prefactor can be furthermore simplified remembering that i j i is always an integer, obtaining the simpler

B L 4 (j a , l a , i, k) = i 1 -e 2πγ i ji B E 4 (j a , l a , i, k) (γ→iγ) . ( 80
)
From the definition of the vertex amplitudes [START_REF] Varadarajan | From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier[END_REF] and [START_REF] Han | Spinfoam on a Lefschetz thimble: Markov chain Monte Carlo computation of a Lorentzian spinfoam propagator[END_REF] and using (80) we find

A L v (j f , i e ) = e i 1 -e 2πγ i∈e ji A E v (j f , i e ) (γ→iγ) . (81)
Vice versa, the analytic continuation iρ i → p i when the Y γ map is imposed reads iγj i → γj i and can be interpreted as the inverse rotation of the Immirzi parameter iγ → γ, equivalently γ → -iγ. The booster functions are related by

B E 4 (j a , l a , i, k) = i e 2πγ i ji -1 B L 4 (j a , l a , i, k) (iγ→γ) (82)
where the rotation iγ → γ must be regularized taking the limit defined in (73). A similar expression for the vertex amplitude is readily obtained

A E v (j f , i e ) = i e e 2πγ i∈e ji -1 A L v (j f , i e ) (iγ→γ)
.

This completes the derivation of the prescription for the analytic continuation of Euclidean and Lorentzian vertex amplitudes one into the other.

Conclusion

In the spin foam literature, the Euclidean and Lorentzian EPRL models are traditionally presented differently.

Although the guiding principle is the same, that is imposing the linear simplicity constraints weakly, the resulting implementations look dissimilar. Performing calculations in the Euclidean model is much more straightforward than in the Lorentzian one. Therefore, many results are derived within the first [START_REF] Bianchi | LQG propagator from the new spin foams[END_REF][START_REF] Barrett | Asymptotic analysis of the EPRL four-simplex amplitude[END_REF][START_REF] Hellmann | Holonomy spin foam models: Asymptotic geometry of the partition function[END_REF][START_REF] Bahr | Investigation of the Spinfoam Path integral with Quantum Cuboid Intertwiners[END_REF] and then inferred to be valid in the second or re-derived from scratch [START_REF] Bianchi | Lorentzian spinfoam propagator[END_REF][START_REF] Barrett | Lorentzian spin foam amplitudes: Graphical calculus and asymptotics[END_REF]. Moreover, the model with Euclidean signature carries the stigma of not being relevant or connected to physical calculations.

When formulated in the canonical basis and adopting the Cartan decomposition, the two models look very alike, differing only in the booster functions (66) and ( 64), thanks to a few key correspondences. First, the algebra of SL(2, C) maps to the algebra of Spin(4) if we rotate the generators of the Lorentzian boosts into i times the generators of the Euclidean "boosts" A ↔ i K and vice versa. Second, this map induces a correspondence between SL(2, C) and Spin(4) group elements that, using the Cartan decompositions [START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF] and [START_REF] Anderson | Recursion and symmetry relations for the clebsch-gordan coefficients of the homogeneous lorentz group[END_REF], reduces to a rotation of the non-compact coordinate ir → t. Third, the map between the algebras induces a correspondence between irreducible representation (p, k) of Spin(4) and the unitary irreducible representation (ρ, k) in the principal series of SL(2, C) as (p, k) ↔ (iρ, k). We obtain the matrix elements in the (p, k) representation of Spin(4) from the matrix elements in the (ρ, k) representation of SL(2, C) through the analytic continuation of the representation labels and group elements simultaneously. Finally, we show that the booster functions of the models with different signature, and, more in general, the vertex amplitudes, can be obtained from one another by rotation of the Immirzi parameter γ ↔ iγ.

This work bridges the gap between the spin foam EPRL models with different signatures and gives a prescription on how to map the results obtained in the Euclidean framework to the Lorentzian one and back.

In addition, we found that the integral forms of the booster functions can be thought as special cases of a general integral defined for complex a i that analytically continues both the Euclidean and Lorentzian integrals. In terms of the Y γ maps that enforce a i = γj i , we can think of this as an extension of the booster functions to the case of an arbitrary complex Immirzi parameter9 . Then, it could be possible in principle to define a general spin foam model defined with complex Immirzi parameter γ that reduces to the Lorentzian EPRL model and the Euclidean EPRL model for purely imaginary or real integer values of γ. This idea is supported by the fact that the (ρ, k) representations of the principal series of SL(2, C) are defined for any ρ ∈ C, but they are unitary only when ρ ∈ R [START_REF] Naimark | Linear representations of the Lorentz group[END_REF]. However, the physical meaning of these hypothetical "complex EPRL models" is not clear to us and we leave the exploration of these ideas to future works.

It is interesting to relate our findings to the early formulation of Loop Quantum Gravity. The original canonical formulation of LQG was based on complex (self-dual) Ashtekar variables. In terms of these variables the constraints of the Hamiltonian formulation of General Relativity are low order polynomials. The major drawback is that one has to impose "reality conditions" on the canonical variables to recover real Lorentzian GR. Since the quantization of these reality conditions is problematic, the focus of the LQG community has shifted to the use of real variables (the Barbero-Immirzi variables) as soon as they were introduced. The price to pay is a more complicated form of the constraints and a less clear geometric interpretation of the real connection [START_REF] Samuel | Is Barbero's Hamiltonian formulation a gauge theory of Lorentzian gravity?[END_REF]. However, in the Euclidean signature these problems do not arise since the self-dual connection is real, and it is possible to show that a "Wick rotation" maps the Euclidean constraints of General Relativity to the Lorentzian ones [START_REF] Ashtekar | A Generalized wick transform for gravity[END_REF][START_REF] Thiemann | Reality conditions inducing transforms for quantum gauge field theory and quantum gravity[END_REF][START_REF] Varadarajan | From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier[END_REF]. In this work we define a similar "Wick rotation" in the quantum theory using the covariant formulation of Loop Quantum Gravity.

Our work opens the way to many interesting ideas that deserve future explorations. In the context of computer simulations, we expect our result to contribute to numerical codes for the EPRL model [START_REF] Dona | Numerical methods for EPRL spin foam transition amplitudes and Lorentzian recoupling theory[END_REF][START_REF] Gozzini | High-performance spinfoam numerics[END_REF] to speed up the calculation of the booster functions. Alternatively, we could rethink the entire numerical calculation scheme, avoiding the Cartan decomposition, and setting up the computation using the canonical basis of Spin(4) with the rotation γ → iγ. It would be interesting also to study a possible connection to other analytic continuations of the EPRL spin foam models based on the complexification of the integration domain [START_REF] Han | Analytic Continuation of Spin foam Models[END_REF] or using Markov Chain Monte-Carlo computations [START_REF] Han | Spinfoam on a Lefschetz thimble: Markov chain Monte Carlo computation of a Lorentzian spinfoam propagator[END_REF]. The computation of the Bekenstein-Hawking entropy of black holes in LQG can be derived using state counting after analytically continuing the formula to γ = ±i [START_REF] Frodden | Black Hole Entropy from complex Ashtekar variables[END_REF][START_REF] Ben Achour | Analytic continuation of the rotating black hole state counting[END_REF]. We can look for an interpretation of this analytic continuation using our results. The applications go beyond LQG and spin foam models. For example, the booster functions are related to the Clebsch-Gordan coefficients of the respective groups [START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF]. We can use our results to relate the Clebsch-Gordan coefficients of SL(2, C) in the principal series to the analytic continuation of the ones of Spin(4) (given by a {9j} symbol) [START_REF] Anderson | Recursion and symmetry relations for the clebsch-gordan coefficients of the homogeneous lorentz group[END_REF][START_REF] Wong | Boost matrix elements and Clebsch-Gordan coefficients of the homogeneous Lorentz group[END_REF]. One possibility is to use well known explicit formulae for the SL(2, C) Clebsch-Gordan coefficients [START_REF] Kerimov | Clebsch-Gordan Coefficients of the SL(2,c) Group[END_REF] and the {9j} symbol [START_REF] Yutsis | Theory of Angular Momentum in Quantum Mechanics[END_REF] in terms of sums of products of hypergeometric functions.

We conclude with the remark that our prescription for mapping through analytic continuation the unitary irreducible representations of Spin(4) and SL(2, C) can be immediately adapted to other spin foam models based on the same gauge groups. More generally, it would be interesting to study possible physical implications of this intriguing analytic continuation beyond the context of spin foam models.

The matrix elements of a group element g ∈ SU (2) in the representation of spin j is called Wigner matrix and in the basis |j, m is given by:

D j mn (g) ≡ j, m| g |j, n =e iφ m 2 e -iψ n 2 (j + m)!(j -m)!(j + n)!(j -n)!• (86) s (-1) s cos θ 2 2j+n-m-2s sin θ 2 m-n+2s (j + n -s)!s!(m -n + s)!(j -m -s)! . (87)
Which satisfies the orthogonality relation:

SU (2) dgD j mn (g)D j m n = 1 2j + 1 δ jj δ mm δ nn , (88)
and the symmetry property:

D j mn (g † ) = D j nm (g) = (-1) n-m D j -n,-m (g) . (89)
The tensor product of two SU(2) representations j 1 and j 2 can be decomposed in terms of a sum of SU (2)

representations j with j = |j 1 -j 2 |, • • • , j 1 + j 2 . The Clebsh-Gordan coefficients j 1 , m 1 , j 2 , m 2 |j, m (90)
relates the states of the three representations. The Clebsh-Gordan coefficients are real and non-zero if and only if:

|j 1 -j 2 | ≤ j ≤ j 1 + j 2 and m = m 1 + m 2 . (91)
They satisfy the orthogonality relation

m1,m2 j 1 , m 1 , j 2 , m 2 |j, m j 1 , m 1 , j 2 , m 2 |l, n = δ jl δ mn . (92)
There are many explicit expressions for the Clebsh-Gordan coefficients. In Section 4.2 we used the Van Der Waerden's formula [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF]:

j 1 , m 1 , j 2 , m 2 |j, m = δ m,m1+m2 2j + 1 (j 1 + j 2 -j)!(j 1 -j 2 + j)!(-j 1 + j 2 + j)! (j 1 + j 2 + j + 1)! (j 1 + m 1 )!(j 1 -m 1 )!(j 2 + m 2 )!(j 2 -m 2 )!(j + m)!(j -m)! t (-1) t 1 t!(j 1 + j 2 -j -t)!(j 1 -m 1 -t)!(j 2 + m 2 -t)!(j -j 2 + m 1 + t)!(j -j 1 -m 2 + t)! (93)
Where the range of summation is given by the existence conditions of the factorials. This expression can be also used to define an analytic continuation of the Clebsh-Gordan coefficients with complex spins [START_REF] Rashid | Boost matrix elements of the homogeneous Lorentz group[END_REF]. A symmetric equivalent of the Clebsh-Gordan coefficients are the Wigner 3jm-symbols

j 1 j 2 j 3 m 1 m 2 m 3 ≡ (-1) j1-j2-m3 √ 2j 3 + 1 C j3,-m3 j1m1j2m2 (94)
The 3jm-symbols we use are reals and non zero if and only if

|j 1 -j 2 | ≤ j 3 ≤ j 1 + j 2 and m 1 + m 2 + m 3 = 0 . (95)
They satisfy the orthogonality relations:

j,m (2j + 1) j 1 j 2 j m 1 m 2 m j 1 j 2 j n 1 n 2 m = δ m1n1 δ m2n2 , (96) m1,m2
(2j + 1)

j 1 j 2 j m 1 m 2 m j 1 j 2 l m 1 m 2 n = δ jl δ mn . (97)
The integral of thee matrix elements is given by the product of two 3jm-symbols

SU (2) dgD j1 m1n1 (g)D j2 m2n2 (g)D j3 m3n3 (g) = j 1 j 2 j 3 m 1 m 2 m 3 j 1 j 2 j 3 n 1 n 2 n 3 . (98)
We can couple four SU (2) representations j 1 , j 2 , j 3 , and j 4 in many (equivalent) ways. If choosing the recoupling basis [START_REF] Rashid | Boost matrix elements of the homogeneous Lorentz group[END_REF] we define the 4jm-symbols as are the orthogonal invariant states in the tensor product of the four representations j i . They obey the orthogonality relations:

m1,m2,m3,m4

j 1 j 2 j 3 j 4 m 1 m 2 m 3 m 4 (i) j 1 j 2 j 3 j 4 m 1 m 2 m 3 m 4 (i ) = δ ii d i . (101)
In this work we adopted the compact notation [START_REF] Speziale | Boosting Wigner's nj-symbols[END_REF]:

j i m i = j i m i (i) = j 1 j 2 j 3 j 4 m 1 m 2 m 3 m 4 (i)
.

(102)

The integration over the Haar measure of four Wigner matrices can be expressed in term of 4jm-symbols as:

SU (2)
duD j1 m1n1 (u)D j2 m2n2 (u)D j3 m3n3 (u)D j4 m4n4 (u) = j (2j + 1)

j i m i (i) j i n i (i) . ( 103
)
The vertex amplitude includes the 15j-symbols of the first kind [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF], which can be expressed as the contraction over their magnetic indices of the product of five 4jm-symbols:

B Explicit proof of (56)

The starting point is the expression for the SL(2, C) matrix elements [START_REF] Vong | On the theory of unitary representations of the SL(2,C) group[END_REF]. For better bookkeeping we will denote z = e -r . We use the properties of the 2 F 1 function to write it as the sum of two 2 F 1 functions evaluated at z -2 , obtaining: d (ρ,k) jlm (r) = (-1) j-l (iρ -j -1)! (j + iρ)! (iρ -l -1)! (l + iρ)! (2j + 1)(2l + 1) (j + l + 1)! z -(iρ-k-m-1)

(j + k)!(j -k)!(j + m)!(j -m)!(l + k)!(l -k)!(l + m)!(l -m)! s,t (-1) s+t z 2t (k + s + m + t)!(j + l -k -m -s -t)! t!s!(j -k -s)!(j -m -s)!(k + m + s)!(l -k -t)!(l -m -t)!(k + m + t)! (j + l + 1)!(l -m -iρ -k -s -t -1)! (l -iρ)!(j + l -m -k -s -t)! z -2(k+m+s+t+1)

2 F 1 {j + iρ + 1, k + m + s + t + 1}, {m + iρ + k + s + t -l + 1}; z -2 + (j + l + 1)!(m + iρ + k + s + t -l -1)! (j + iρ)!(k + m + s + t)! z -2(l-iρ+1)

2 F 1 {j + l -m -k -s -t + 1, l -iρ + 1}, {l -m -iρ -k -s -t + 1}; z -2 . ( 105
)
We can write the two 2 F 1 functions explicitly, and using the properties of the Pochhammer symbols, we obtain:

d (ρ,k)
jlm (z) = (-1) j-l (2j + 1)(2l + 1) (iρ -j -1)! (l + iρ)! (j + iρ)! (j + k)!(j -k)!(j + m)!(j -m)!(l + k)!(l -k)!(l + m)!(l -m)! 1 (l -iρ)! (iρ -l -1)! s,t,n (-1) s+t (-1) n z -(iρ+k+m+2s+2n+1) (j + iρ + n)!(k + m + s + t + n)!(l -m -iρ -k -s -t -1 -n)! t!s!n!(j -k -s)!(j -m -s)!(k + m + s)!(l -k -t)!(l -m -t)!(k + m + t)! + (iρ -l -1)! s,t,n (-1) s+t z 2t+iρ+k+m-2l-2n-1 (j + l -m -k -s -t + n)!(m + iρ + k + s + t -l -1 -n)! t!s!n!(j -k -s)!(j -m -s)!(k + m + s)!(l -k -t)!(l -m -t)!(k + m + t)!(iρ -l -1 -n)! .

(106)

The summations over s and t can be decoupled by shifting the index n → n -s -k -m in the first sum and n → n + t + k -l in the second. Notice that this change of variable is well defined since s, k + m and k -l are all integers. Moreover, since the expression is getting quite lengthy we split it in two pieces

d (ρ,k) jlm (z) = F 1 + F 2 (107)
Where we defined F 1 = (-1) j-l (2j + 1)(2l + 1) (iρ -l -1)! (iρ -j -1)! (l + iρ)! (j + iρ)! (j + k)!(j -k)!(j + m)!(j -m)!(l + k)!(l -k)!(l + m)!(l -m)! s,t,n (-1) s+t z iρ-k-1+m-2n

(j -m + n -s)!(iρ + m -n -1 + s)! s!(j -k -s)!(j -m -s)!(k

+ m + s)! 1 t!(l -k -t)!(l -m -t)!(k + m + t)!(iρ -k -1 -n -t)!(k -l + n + t)! , (108)
two sums as analytically continued Clebsch-Gordan coefficients10

F 1 = n z iρ-k-1+m-2n iρ + k -1 2 , m -n + iρ -k -1 2 , iρ -k -1 2 , n - iρ -k -1 2 j, m iρ + k -1 2 , m -n + iρ -k -1 2 , iρ -k -1 2 , n - iρ -k -1 2 l, m . (112)
Similarly, in F 2 we can perform the sums over s and t in terms of two 3 F 2 functions evaluated in 1 and we can identify the analytically continued Clebsch-Gordan coefficients

F 2 = n z -(iρ-k+1-m+2n) -iρ -k -1 2 , -iρ + k -1 2 + m -n , -iρ + k -1 2 , n - -iρ + k -1 2 j, m -iρ -k -1 2 , -iρ + k -1 2 + m -n , -iρ + k -1 2 , n - -iρ + k -1 2 l, m (113)
Inserting the expression of F 1 and F 2 in (107) we obtain an expression of the reduced matrix elements in the (ρ, k) representation in terms of complex Clebsch-Gordan coefficients

d (ρ,k) jlm (r) = n e -(iρ-k-1+m-2n)r iρ + k -1 2 , iρ -k -1 2 + m -n , iρ -k -1 2 , n - iρ -k -1 2 j, m iρ + k -1 2 , iρ -k -1 2 + m -n , iρ -k -1 2 , n - iρ -k -1 2 l, m + n e -(-iρ+k-1+m-2n)r -iρ -k -1 2 , -iρ + k -1 2 + m -n , -iρ + k -1 2 , n - -iρ + k -1 2 j, m -iρ -k -1 2 , -iρ + k -1 2 + m -n , -iρ + k -1 2 , n - -iρ + k -1 2 l, m . (114)

(

  iρ,k) jlm (e -r ) ≡ d (ρ,k) jlm (r) 5 . If a = p ≥ 1, p ∈ Z/2 and z = e it then (58) turns into the Spin(4) reduced matrix elements d (p,k) jlm (e it ) ≡ d (p,k) jlm (t).

Figure 1 :

 1 Figure 1: The contour Cε in red. The point z = 0 in red is a pole singularity in the Euclidean case and a branch point singularity in the Lorentzian case. The black point is z = 1. In the Lorentzian case the integrand is discontinuous on the punctured axes.

|j 1

 1 , m 1 , j 2 , m 2 , j 3 , m 3 , j 4 , m 4 (100)

{15j}(i k , j a ) ≡ ma j 1 j 2 j 3 j 4 m 1 m 2 m 3 m 4 (

 4

While in the mathematical literature this is referred to either as the KAK decomposition or as the polar decomposition.

We use a slightly different definition from[START_REF] Biedenharn | Wigner Coefficients for the R4 Group and Some Applications[END_REF], in which p ≡ j L + j R .

F 1 {l -a + 1, k + m + s + t + 1}, {j + l + 2}; 1 -z 2 ,(58)

This must not be confused with the analytic continuation ρ → -ip which could be equivalently written as iρ → p.

More rigorously, the "analytic continuation" of a function defined on the integers cannot be performed in the mathematical sense since Z is not an open subset of C. Hence one can just define I E (iρ i , k i ) to be the evaluation of I E (p i , k i ) with p i purely imaginary. Notice, however, that in light of the converse result (73) we can speak unambiguously of the unique function on the whole complex plane that extends both I E (p i , k i ) and I L (ρ i , k i ) at the same time.

Amusingly, the key contribution of Giorgio Immirzi (who extended an idea from Fernando Barbero) was to highlight how the complex phase space of Ashtekar variables could be canonically transformed to a real phase space using a real parameter, namely the Immirzi parameter, to avoid the imposition of the reality conditions.

F 2 (j -j 1 -j 2 , m 1 -j 1 , -j 2 -m 2 ; j -j 2 + m 1 + 1, j -j 1 -m 2 + 1; 1) . (111)The 3 F 2 admits a well-defined analytic continuation for complex parameters, and as a consequence a welldefined analytic continuation of the Clebsch-Gordan coefficients.We can perform the sums over s and t in (110) exactly in terms of 3 F 2 hypergeometric functions evaluated in 1. Using an identity of the 3 F 2 functions and manipulating the factorials in front we can recognize the

The precise manipulation are quite cumbersome to report here. We refer to the Master thesis of one of the autors[START_REF] Nicotra | Analytical map between EPRL spin foam models in loop quantum gravity[END_REF] for a step by step description. Note that the names F 1 and F 2 have been reversed.

Acknowledgments

The work of P.D. is partially supported by the grant 2018-190485 (5881) of the Foundational Questions Institute and the Fetzer Franklin Fund. A.N. acknowledges support from the Università di Bologna and its Erasmus+ program.

Appendices A SU (2) conventions

Here we summarize the SU (2) objects we use in the paper. A useful parametrization of a group element g ∈ SU (2) is the so called Euler angles parametrization

where σ i are the Pauli matrices and 0 ≤ φ < 2π, 0 ≤ θ < π, and 0 ≤ ψ < 4π. The Haar measure in this parametrization is given by

and

We focus on F 1 first. We shift one of the sum by t → t + l -k -n and we rearrange the terms:

The SU (2) Clebsch-Gordan coefficients can be expressed, using the Van der Waerden's formula (93) and [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF], in terms of the 3 F 2 hypergeometric function evaluated in 1

(j 1 -j 2 + j)!(-j 1 + j 2 + j)! (j 1 + j 2 + j + 1)!(j 1 + j 2 -j)! (j 1 + m 1 )!(j 2 -m 2 )!(j + m)!(j -m)! (j 1 -m 1 )!(j 2 + m 2 )! 1 (j -j 1 -m 2 )!(j -j 2 + m 1 )!