Table Of Content

A spin foam model for general Lorentzian 4–geometries Florian Conrady∗ Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada Jeff Hnybida† Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada and Department of Physics, University of Waterloo, Waterloo, Ontario, Canada 1 1 0 We derive simplicity constraints for the quantization of general Lorentzian 4– 2 geometries. Our method is based on the correspondence between coherent states n and classical bivectors and the minimization of associated uncertainties. For tri- a J angulations with spacelike triangles, this scheme agrees with the master constraint 8 method of the model by Engle, Pereira, Rovelli and Livine (EPRL). When it is ap- 1 plied to general triangulations of Lorentzian geometries, we obtain new constraints ] thatincludetheEPRLconstraintsasaspecialcase. Theyimplyadiscreteareaspec- c q trum for both spacelike and timelike surfaces. We use these constraints to define a - spin foam model for general Lorentzian 4–geometries. r g [ 4 v 9 I. INTRODUCTION 5 9 What happens if one describes geometry as a degree of freedom of quantum theory? Does 1 . geometry remain continuous or does it come in quanta? Are the singularities of classical 2 0 general relativity resolved? Loop quantum gravity originates from the attempt to answer 0 such questions1. In the Hamiltonian framework, this led to canonical loop quantum gravity, 1 and, in the path integral picture, it brought forth the notion of spin foam models [4]. A : v central result of both approaches is the discreteness of the area spectrum: it suggests that i X there is a minimal unit of area and that quantum geometry is indeed discrete. r The basic idea behind spin foam models is to divide spacetime into 4–simplices and to a study how the geometry of these simplices can be quantized. More precisely, bivectors B are used to describe triangles and constraints are imposed, so that four bivectors are equivalent to a tetrahedron. The bivectors and constraints are then translated in a suitable way to the quantum theory. The main elements of this procedure were introduced by Barrett and Crane [5]. In the language of field theory, this corresponds to the transition from topological BF theory to gravity: the B in BF is constrained to be simple, so that it becomes the wedge product of two tetrads. For this reason, the constraints are called simplicity constraints. In recent years, considerable progress was made in improving and clarifying this quantization process. The key to this progress were two new developments: firstly, Engle, Pereira, Rovelli and Livine (EPRL) defined a new model that resolves certain longstanding problems ∗Electronic address: [email protected] †Electronic address: [email protected] 1 See [1], [2] and [3] for reviews. 2 with the Barrett–Crane model and establishes a link with canonical loop quantum gravity [6]. The quantization is based on a so–called master constraint, which is the sum of the squares of all simplicity constraints. The second important innovation was the coherent state technique introduced by Livine and Speziale [7]. It provides a better geometric under- standing of quantum states, and led to the construction of the Freidel–Krasnov (FK) model [8, 9]. In this model, simplicity is imposed on expectation values of coherent states. Both of these developments spurred further results: The FK model was reexpressed as a path integral with a simple action [10]. The semiclassical limit of the new models was analyzed [11–13], likewise the graviton propagator [14]. It was found that intertwiner states can be understood in geometric terms [15]. Recently, coherent states were constructed for entire 3–geometries [16, 17]. The present paper is motivated by two questions. The first question concerns the relation between the EPRL and coherent state approach. What is the connection between these two lines of thought? Is there a way to understand the master constraint in terms of coherent states? In the Riemannian case, we know from explicit comparison that the EPRL and FK model are closely related [10]. To this extent, the coherent state ideas apply also to the EPRL model. In the Lorentzian case, however, it is not clear how a derivation from coherent states should look like2. It could be very useful to have one, since coherent states provide a particularly transparent quantization of simplicity constraints. The second motivation for this paper comes from the fact that the EPRL model is only defined for spacelike area bivectors. That is, it can only describe geometries in which all surfaces are spacelike. The obvious question is therefore: how can one specify a model that covers realistic Lorentzian geometries, where bivectors can be both spacelike and timelike? In this article, we obtain an answer to our first question, and it turns out that we can also resolve the second question with this knowledge. What we find is a coherent state method that reproduces the EPRL constraints and applies at the same time to the cases which were not yet covered in this model. On the one hand, we recover the EPRL constraints for tetrahedra with timelike normals. Thus, the method gives the desired coherent state derivation of the EPRL spin foams. The same scheme, however, works also for tetrahedra with spacelike normals. In this case, it results in two new sets of constraints that pertain to spacelike and timelike triangles within such tetrahedra. Our method is inspired by the Riemannian FK model and extends its logic by an addi- tional condition. We demand that there are quantum states for which 1. The expectation value of the bivector operator is simple. 2. The uncertainty in the bivector is minimal3. With this, we enforce the existence of coherent states that correspond to classical simple bivectors. Such states can only exist in certain irreducible representations of SL(2,C) and its subgroups. This determines constraints on irreps and we interpret them as the quantum version of the simplicity constraints. Based on this, we define a new spin foam model that gives a quantization of tetrahedra with spacelike and timelike normals, and hence a quantization of general Lorentzian geometries. In addition, a coherent state vertex amplitude is specified. We should remark 2 A proposal was made in ref. [8]. 3 The precise meaning of “minimal” is stated in sec. II and sec. III. 3 that coherent states are not necessary, and, in fact, not used, when defining the spin foam sum. In this work, the coherent states are only essential in the derivation of the simplicity constraints. The paper is organized as follows: in sec. II we introduce our coherent state method. It is applied to tetrahedra with a timelike normal and the constraints of the EPRL model are reproduced. In sec. III, we treat tetrahedra with a spacelike normal and obtain two new constraints that refer to spacelike and timelike triangles respectively. Section IV summarizes the constraints for the different cases. In sec. V, we use these constraints to define a spin foam model for general Lorentzian 4–geometries. Conventions At the outset, we make a few remarks on notation to avoid confusion due to differing conventions in the literature. Our sign convention for the spacetime metric is (+, , , ), and (+, , ) for 3d − − − − − Minkowski spacetime. Moreover, ǫ0123 = +1. The Immirzi parameter is γ and we assume that γ > 0. Unit normal vectors of tetrahedra are denoted by U. The letter N stands for unit normal vectors of triangles. We have the Hodge dual operator ⋆, and ⋆B is the area bivector of triangles. Unitary irreducible representations of SL(2,C) are labelled by pairs (ρ,n), where ρ R and n Z . With regard to irreducible representations of subgroups, we follow the nota∈tion + ∈ in [18]: stands for SU(2) irreps. The discrete and continuous series of SU(1,1) are j D designated by ± and ǫ respectively. Representation matrices are symbolized by D(ρ,n)(g) Dj Cs in the SL(2,C) case, and by Dj(g) for subgroups. II. SIMPLICITY CONSTRAINTS FOR THE SU(2) REDUCTION In this section, we introduce our procedure for deriving the simplicity constraints on representations of the Lorentz group. We do so by applying it to a situation that has been already treated in the EPRL model [6]. As explained below, the simplicity constraints split into two main categories, depending on whether normal vectors U of tetrahedra are timelike or spacelike. The case considered in [6] is the one for timelike U. At the quantum level, this choice of a timelike normal is reflected by the fact that unitary irreps of SL(2,C) are analyzed in terms of irreps of the subgroup SU(2). For this case, we will find that our method produces the same simplicity constraints as the master constraint employed in the EPRL paper. Encouraged by this agreement, we will then proceed to section III, where we apply our technique to the case when U is spacelike. Then, irreps of SL(2,C) will be decomposed into irreps of SU(1,1) and we will obtain a new set of simplicity constraints. A. Classcial variables Our starting point is the SO(1,3) bivector 1 J = B + ⋆B, (1) γ 4 which is used when defining gravity as a constrained BF theory with Immirzi parameter γ. The idea is to constrain B in such a way that B becomes B = ⋆(E E), where E is a ∧ co–tetrad. Under this constraint, the action 1 S = J F = B F + ⋆B F (2) ∧ ∧ γ ∧ Z Z (cid:18) (cid:19) reduces to the Hilbert–Palatini action with an Immirzi term. Like in [6], we denote the total bivector by the letter J. This choice is convenient, since J will be closely related to the generators of the Lorentz group. After quantization, the classical bivector JIJ will arise from expectation values of SO(1,3) generators JIJ. At the classical level, the derivation of the simplicity constraints proceeds as follows: first we state the simplicity constraints for B, which ensure that B = ⋆(E E). These constraints ∧ involve a vector U which is normal to the dual bivector ⋆B. Since U can be either timelike or spacelike, one obtains two classes of simplicity constraints. In the present section, we treat the case where U is timelike, as in the EPRL model. These constraints are then expressed as constraints on J—ready to be translated to the quantum theory. In full generality, the simplicity constraint reads U ⋆B = 0, (3) · where U is a Lorentz vector of unit norm, i.e. U2 = 1. This constraint implies that the ± dual ⋆B is simple and of the form4 ⋆B = E E , (4) 1 2 ∧ where E and E are two 4–vectors orthogonal to U (for a derivation see e.g. [8]). Moreover, 1 2 B is simple and given by B = AU N , (5) ∧ where N is a unit norm 4–vector such that U N = 0 and N E = N E = 0. The 1 2 · · · coefficient A in (5) is equal to the area A = E2E2 (E E )2 (6) | 1 2 − 1 · 2 | q of the parallelogram spanned by E and E . 1 2 Equations (4) and (5) elucidate the geometric meaning of the 4–vectors U and N. E 1 and E correspond to the tetrad and in a discrete setting they can be regarded as the two 2 edges of a triangle. The dual ⋆B is the so–called area bivector of this triangle, and it is orthogonal to both U and N. To obtain a tetrahedron, one starts from four bivectors B , a a = 1,...,4, and imposes the simplicity constraint (3) on each of them: U ⋆B = 0, a = 1,...,4. (7) a · This means that all four bivectors are simple and that they span a 3d subspace orthogonal to U. When the closure constraint 4 B = 0, (8) a a=1 X 4 V W stands for the bivector (V W)IJ =VIWJ WIVJ. ∧ ∧ − 5 is supplemented, it follows that the bivectors B are equivalent to a tetrahedron. The vector a U is the normal of this tetrahedron, the four vectors N (associated to the bivectors B ) a a are the normals to the four triangles, and the E’s are the edges of the triangles. In this way, geometry results from constraints on bivectors. In spin foam models the analog of (7) is imposed on representations of the gauge group, while the closure constraint arises dynamically. Let us assume now that U is timelike and gauge–fixed to U = (1,0,0,0). Then, the simplicity constraint becomes (⋆B)0i = 0 and ⋆B has to be spacelike. Next we express this constraint in terms of the bivector J. Solving for J in (1) gives γ2 1 B = J ⋆J . (9) γ2 +1 − γ (cid:18) (cid:19) It follows that the gauge–fixed simplicity constraint is equivalent to 1 Ji + Ki = 0 (10) γ where we use the usual definitions 1 Ji = ǫ0i Jjk and Ki = J0i. (11) jk 2 Eq. (10) will be the central equation for the derivation of the simplicity constraints in the quantum theory. It is also useful to express the normal vector N in terms of Ji and Ki. On the one hand, we have that Bij = 0, B0i = ANi. (12) On the other hand, γ2 1 B0i = J0i ǫ0i Jjk (13) γ2 +1 − 2γ jk (cid:18) (cid:19) γ2 1 = Ki Ji (14) γ2 +1 − γ (cid:18) (cid:19) Using the simplicity constraint (10) this yields ANi = γJi (15) − Therefore, N = (0,N~), where AN~ = γJ~. Observe also that the vector N~ is a point in the − 2–sphere S2 SU(2)/U(1). (16) ≃ and hence a point in the coadjoint orbit (or phase space) of spin j = 1. B. Quantum states Next we will describe our way of translating the simplicity constraint to quantum states. As in the EPRL model, we will not do this in a manifestly covariant way. We will start from eq. (10), which is the simplicity constraint after gauge–fixing U to (1,0,0,0). 6 The quantum analog of the bivectors are states in unitary irreducible representations of SL(2,C). The latter are labelled by pairs (ρ,n), where ρ R and n Z . Since + U = (1,0,0,0) singles out SU(2) as a little group of SL(2,C), i∈t is convenie∈nt to express everything in an “SU(2) friendly” way. This can be done by using the decomposition of SL(2,C) irreps into SU(2) irreps: namely, ∞ (17) (ρ,n) j H ≃ D j=n/2 M where denotes the Hilbert space of the irrep (ρ,n) and stands for the spin j irrep (ρ,n) j H D of SU(2) (see e.g. [23]). The corresponding completeness relation reads ∞ j = Ψ Ψ . (18) (ρ,n) jm jm 1 | ih | j=n/2m=−j X X The states Ψ , m = j,...,j, span a subspace of that is isomorphic to , so we jm (ρ,n) j | i − H D identify them with states jm of . j | i D Our recipe for quantizing the simplicity constraints is formulated as follows. We require the existence of quantum states for which the expectation value of the bivector JIJ satisfies the simplicity constraints. Moreover, the quantum uncertainty in this bivector should be small. Here, we work with a gauge–fixing, so the components of JIJ are organized in terms ~ ~ ~ ~ ~ ~ of J and K. Let us define associated lengths J J and K K , where stands | | ≡ |h i| | | ≡ |h i| h i for the expectation value w.r.t. the quantum state. We then demand that ∆J 1 = O , (19) J~  ~  J | | | | q  1 J~ + K~ = O(1), (20) h i γh i ∆K 1 = O . (21) K~  K~  | | | | q  Through these three conditions we establish a correspondence between classical variables and semiclassical states: the first requirement says that the states should be peaked around classical values of J~. The second condition states that their expectation values fulfill the ~ simplicity constraint. The last point adds that the states should not only be peaked in J, but also in the remaining components K~. The first condition is easily met by using SU(2) coherent states [19]: these states have the form jg Dj(g) jj . (22) | i ≡ | i and arise from SU(2) rotations of the “reference” coherent state jj . From such states we | i get ∆J √j 1 1 = = = O . (23) J~ j √j  J~  | | | | q  7 Equation (20) is more subtle, as it involves SL(2,C) generators outside of SU(2). Since both J~ and K~ transform as vectors under SU(2), it is sufficient to impose (20) on the reference state jj . If it is satisfied for jj , then it will be true for all coherent states jg . | i | i | i Therefore, we require5 1 jj J~ jj = jj K~ jj . (24) h | | i −γh | | i It is clear from commutation relations that J1, J2, K1 and K2 change the eigenvalue of J3, so their expectation values will be zero. We therefore only need to consider J3 and K3. The action of K3 is given by6 K3 jm = (j +m+1)(j m+1)C j +1m j+1 | i − − | i pmA jm j − | i + (j m)(j +m)C j 1m , (25) j − | − i p where ρn A = , (26) j 4j(j +1) and i (j2 n2)(j2 + ρ2) C = − 4 4 . (27) j js 4j2 1 − Hence eq. (24) leads to 1 ρn j = ( jA ) or γ = A = . (28) j j −γ − 4j(j +1) In order to deal with the variance in K~, we recall the Casimirs of SL(2,C): 1 C = 2 J~2 K~2 = (n2 ρ2 4) (29) 1 − 2 − − (cid:16) (cid:17) C = 4J~ K~ = nρ (30) 2 − · As a result, one gets (∆K)2 = K~2 K~ 2 (31) h i−h i 1 = J~2 (n2 ρ2 4) K~ 2 (32) h − 4 − − i−h i 1 = j(j +1) (n2 ρ2 4) j2A2. (33) − 4 − − − j 5 For the next couple of lines we omit the order symbol O(1). 6 See, for instance, [20]. 8 By inserting the simplicity constraint A = γ, we obtain furthermore j 1 1 (∆K)2 = ρn (n2 ρ2 4) j2γ2 (34) 4γ − 4 − − − 1 1 = ρn (n2 ρ2) γ2j(j +1)+γ2j +1 (35) 4γ − 4 − − 1 1 γ = ρn (n2 ρ2) ρn+γ2j +1 (36) 4γ − 4 − − 4 1 n = ρ γn ρ+ +γ2j +1. (37) 4 − γ (cid:16) (cid:17)(cid:16) (cid:17) If one sets ρ = γn (38) or n ρ = , (39) −γ the first term vanishes, and ∆K γ2j +1 1 = = O . (40) K~ p γj  K~  | | | | q  However, when plugging back (38) into the simplicity constraint (28), we obtain n2 = 4j(j +1). (41) Since this cannot be solved for generic values of n and j, we proceed like in [6] and adopt the approximate solution j = n/2. (42) One can check that (38) and (42) fulfill conditions (20) and (21). When (39) is inserted into (28), on the other hand, we get n2 = 4γ2j(j +1). (43) − This has no solution. Thus, our final result are the constraints ρ = γn and j = n/2, which are the same constraints as in the EPRL model! The area spectrum can be derived by squaring the classical equation (15) and setting the right–hand side equal to the expectation value of the coherent state. This gives us the quantum area A = γ J~2 = γ j(j +1). (44) h i Up to this point, we have just usqed a new pperspective to obtain something that was already known, i.e. the constraints of the EPRL model. This shows that the EPRL master constraint is equivalent to a set of semiclassical constraints, namely to the requirement that there exist quantum states that are peaked in J~ and K~, and that their expectation values satisfy the classical simplicity constraint. In the next section, we will apply our procedure to derive something new: we will provide a prescription for spacelike U, and hence for bivectors ⋆B that can be both spacelike and timelike. 9 III. SIMPLICITY CONSTRAINTS FOR THE SU(1,1) REDUCTION A. Classical variables In this section, the normal vector U is assumed to be spacelike, and we gauge–fix it to U = (0,0,0,1). Then, the classical simplicity constraint is (⋆B)3i = 0, where i = 0,1,2. A short calculation shows that this is equivalent to 1 K1 J1 = 0, (45) − γ 1 K2 J2 = 0, (46) − γ 1 J3 + K3 = 0. (47) γ By defining the quantities F0 = J3, F1 = K1, F2 = K2, G0 = K3, G1 = J1, G2 = J2, − − one can write these constraints in the more symmetric form 1 Fi + Gi = 0, i = 0,1,2. (48) γ Here and below we use the indices i,j to denote vectors in 3–dimensional Minkowski spacetime, in the same way that indices i,j were used for vectors of 3–dimensional Euclidean space in the previous section. Inspection of the commutation relations reveals, in fact, that F and G transform like 3d Minkowski vectors under SU(1,1) [21]: [Fi,Fj] = iCij Fk, (49) k [Fi,Gj] = iCij Gk, (50) k where C01 = C10 = C20 = C02 = C21 = C12 = 1. (51) 2 2 1 1 0 0 − − − Thus, eq. (48) has a similar structure as eq. (10): it relates two vectors that have the same transformation property under the little group, and one of the vectors is the generator of the little group. As before, we determine the relation between the vector N and the generators. On the one hand, Bij = 0, B3i = ANi, where i,j = 0,1,2. (52) On the other hand, we also have γ2 1 B3i = J3i ǫ3i Jjk (53) γ2 +1 − 2γ jk (cid:18) (cid:19) 10 For i = 0, this gives γ2 1 B30 = J30 ǫ30 J12 (54) γ2 +1 − γ 12 (cid:18) (cid:19) γ2 1 = K3 + J3 (55) γ2 +1 − γ (cid:18) (cid:19) On account of the simplicity constraint this reduces to AN0 = γJ3. (56) Similarly, we obtain AN1 = γK2 and AN2 = γK1 (57) − for the cases i = 1 and i = 2. Altogether we get N0 J3 F0 AN1  = γ K2  = γ F2  . (58) N2   K1   F1          −  −        Like in the SU(2) case, the vector N~ = (N0,N1,N2) is related to quotient spaces of SU(1,1): ~ timelike vectors N coordinatize the two–sheeted hyperboloid H H , H = N~ N~2 = 1,N0 ≷ 0 , (59) + − ± ∪ { | } and each sheet is isomorphic to the quotient SU(1,1)/U(1). Spacelike N~ parametrize the spacelike single–sheeted hyperboloid H = N~ N~2 = 1 . This hyperboloid is isomorphic sp to the quotient SU(1,1)/(G Z ), where{G |is the −one}–parameter subgroup of SU(1,1) 1 2 1 ⊗ generated by K [22]. Appendix A describes the details of these isomorphisms. 1 B. Quantum states Since we have set U equal to (0,0,0,1), the little group is SU(1,1). Accordingly, we should work with unitary irreducible representations of SU(1,1). These come in a discrete and a continuous series. In both cases, the irreps can be built from eigenstates jm of J3, | i satisfying jm jm′ = δmm′ , (60) h | i J3 jm = m jm . (61) | i | i Combinations of K1 and K2 act as raising and lowering operators. The Casimir is given by Q = (J3)2 (K1)2 (K2)2. − − In irreps of the discrete series, one has Q jm = j(j 1) jm , where j = 1, 1, 3, ... (62) | i − | i 2 2