SPIN-FOAM DYNAMICS of GENERAL RELATIVITY by Atousa Chaharsough Shirazi

SPIN-FOAM DYNAMICS OF GENERAL RELATIVITY by Atousa Chaharsough Shirazi

A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science In Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

ACKNOWLEDGEMENTS

I am sincerely grateful and indebted to my advisor Dr. Jonathan Engle for his dedication, help and guidance through all my PhD years. I appreciate everything you taught me. Thank you for being patient. Besides my advisor, I would like to acknowledge the rest of my dissertation committee members for their valuable insight in preparing this manuscript . To all my professors at Florida Atlantic University, thank you for sharing your expertise with me. A special thank you to Dr. Warner Miller whose invaluable and crucial support during these years helped me to achieve my full potential. My heartfelt gratitute goes to my family. Words can not express my gratitute to my late father who inspired and encouraged me to become a scientist. Thank you for all the support, love and confidence you gave me. To my beloved mother whose ongoing love and support through all these years made this journey possible for me. Thank you for your patience during the years I was away from home pursuing my passion. Finally, I would like to thank my friends for being beside me through the good times and the bad. Thank you for all the fun times and memories that we made together during these years. You are definitely one of the significant parts of this journey.

iv ABSTRACT

Author: AtousaChaharsoughShirazi Title: Spin-foamDynamicsofGeneralRelativity Institution: Florida AtlanticUniversity Dissertation Advisor: Dr. Jonathan S. Engle Degree: Doctor of Philosophy Year: 2015

In this dissertation the dynamics of general relativity is studied via the spin- foam approach to quantum gravity. Spin-foams are a proposal to compute a transition amplitude from a triangulated space-time manifold for the evolution of quantum 3d geometry via path integral. Any path integral formulation of a quantum theory has two important parts, the measure factor and a phase part. The correct measure factor is obtained by careful canonical analysis at the continuum level. The basic variables in the Plebanski-Holst formulation of gravity from which spin-foam is derived are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the ﬁnal spin-foam sum, one usually sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski- Holst path integral in which only the Plebanski two-form appears, and in which the connection degrees of freedom have been integrated out. Calculating the measure factor for Plebanksi Holst formulation without the connection degrees of freedom is one of the aims of this dissertation. This analysis is at the continuum level and in order to be implemented in spin-foams one needs to properly discretize and quantize

v this measure factor. The correct phase is determined by semi-classical behavior. In asymptotic analysis of the Engle-Pereira-Rovelli-Livine spin-foam model, due to the inclusion of more than the usual gravitational sector, more than the usual Regge term appears in the asymptotics of the vertex amplitude. As a consequence, solutions to the classical equations of motion of GR fail to dominate in the semi-classical limit. One solution to this problem has been proposed in which one quantum mechanically imposes restriction to a single gravitational sector, yielding what has been called the “proper” spin-foam model. However, this revised model of quantum gravity, like any proposal for a theory of quantum gravity, must pass certain tests. In the regime of small curvature, one expects a given model of quantum gravity to reproduce the predictions of the linearized theory. As a consistency check we calculate the graviton two-point function predicted by the Lorentzian proper vertex and examine its semiclassical limit.

vi SPIN-FOAM DYNAMICS OF GENERAL RELATIVITY

1 Introduction ...... 1

2 Background ...... 7 2.1 Plebanski-Holstformulationofgravity ...... 7 2.2 Discrete Geometry ...... 9 2.2.1 Bivector geometry ...... 10 2.3 Quantization ...... 12 2.4 EPRL model ...... 14 2.4.1 Imposingtheconstraints ...... 14 2.4.2 Boundary states ...... 15 2.4.3 The amplitude ...... 16 2.4.4 Asymptotics...... 16 2.5 “Proper” vertex amplitude ...... 18

3 “Proper” Graviton propagator ...... 21 3.1 Introduction ...... 21 3.2 Gravitonpropagatorinspin-foamformalism ...... 22 3.2.1 The metric operator ...... 23 3.3 EPRL graviton propagator ...... 24 3.4 Graviton propagator for the proper vertex ...... 27 3.4.1 Asymptoticlimit ...... 29 3.5 Hessianofthepropervertex: Calculation ...... 32

3.5.1 Derivatives of S˜EPRL with respect to ⌘ and one other variable 33 ˜o 3.5.2 Derivatives of S⇧ ...... 35 vii 3.5.3 Hessian ...... 36 3.6 Graviton propagator and its 0limit...... 36 ! 3.7 Coecient in the asymptotics of the proper vertex ...... 37 3.8 Discussion ...... 40

4 Canonical measure factor ...... 43 4.1 Introduction ...... 43 4.2 Reduced phase-space method ...... 44 4.3 Plebanski-Holstformulation ...... 46 4.4 Purely geometric path integral ...... 47 4.4.1 Integrating out the connection ...... 48 4.4.2 The determinant of a ...... 49 4.4.3 Finalcontinuumpathintegral ...... 50 4.5 ADMpathintegral ...... 51 4.5.1 Integrating out the momentum ...... 51 4.5.2 The determinant of A ...... 52 4.5.3 Space-time covariant variables ...... 53 4.5.4 Comparison with the result of the last section ...... 54 4.6 Results and conclusion ...... 55

Appendices ...... 57 ACoherentstates...... 58 A.1 SU(2)coherentstates...... 58 A.2 Extrinsiccoherentstates ...... 61 BIntegrationofapathintegralwithquadraticaction...... 63 C Equivalence of gauge-ﬁxed and non-gauge-ﬁxed path integrals . . . . 65 C.3 The argument ...... 65

Bibliography ...... 71

viii CHAPTER 1 INTRODUCTION

One of the main open problems in theoretical physics is to unify general relativity (GR) and quantum mechanics (QM). General relativity is valid for large scales and high energies and quantum mechanics governs physics at planck scale l Planck ⇠ 35 10 m.Bothofthesetheorieshavestrongempiricalsupportattheirregimeof validity. At some extreme cases where the space-time region is small and at high energies like black holes and the big bang, these two theories overlap. Therefore, having a consistent theory of “quantum gravity”(QG)thatreproducesgeneralrela- tivity and quantum mechanics in appropriate limit is of high importance. However, these two theories are not compatible and the problem is rooted in the fact that in GR, gravitational field modifies the notion of space-time. In quantum mechanics, the causal structure is defined by a fix background space-time and any dynamical field is quantized on this background. On the other hand, GR is a background independent theory meaning that there is no background needed for the field to live in. Gravi- tational field defines geometry and metric is a smooth dynamical field. Therefore, the metric is quantized in a quantum theory of GR and the standard QM approach which requires a fixed smooth background fails in this case. Di↵erent approaches to quantum theory of general relativity tackle the problems di↵erently. Among the most popular approaches are string theory, loop quantum gravity (LQG), spin-foams, causal dynamical triangulation and causal sets. Each of these approaches have their pros and cons but so far there is no experiment favoring any of these approaches. In this dissertation for the reasons given below we choose the spin-foam approach which is a path integral formulation of loop quantum gravity, to study the dynamics 1 of general relativity. Loop quantum gravity [1–5] is a canonical quantization of general relativity that preserves the background independence principle from GR. In GR background independence is realized as di↵eomorphism invariance of the action. After quantization space-time is expected to have quantum properties. Since interaction with gravitational field is through geometry, the quantities involved in the interaction are geometric quantities such as area, volume and length. Therefore, in LQG area and volume become operators and the eigenstates of these operators form a basis of the Hilbert space of states of our quantum system. One of the key results of LQG is that the area and volume operators have discrete eigenvalues [2,6]. Therefore at small scales geometry is discretized and space is built out of quanta. This discreteness is the result of compactness of the group SU(2) which is the local gauge group. The consequence of background independence is that the quanta of gravitational field does not live on aspace-timebuttheybuildspace-time. Spin-foams are a proposal for defining the dynamics of LQG via path integral [7– 10]. The spin-foam program has a di↵erent starting point from LQG, but nevertheless uses the loop quantum states which are eigenstates of the 3d boundary geometry as the boundary states and computes a transition amplitude between those states, yielding an evolution of quantum 3d geometry. It started in 3d with the work of Ponzano and Regge [11] by realizing that GR in 3d is a topological field theory (with no local degrees of freedom) and a quantum theory of GR can be obtained from a triangulated manifold by a topological (triangulation independent) state-sum. Since 4d GR has local degrees of freedom, the extension of the state-sum method to 4d is not trivial. However, the Plebanski formulation [12–15] of 4d gravity is in the form of a BF theory [16], which is a topological field theory, plus a so-called simplicity constraint. Using this, Barrett and Crane [17] extended the work of Ponzano and Regge to 4d and derived a quantum version of the simplicity constraint. The quantum constraint

2 becomes a restriction on the representations (of the local gauge group) summed over. Despite its remarkable success, the Barrett-Crane model fails to give correct tenso- rial structure for the graviton propagator in the low-energy limit [18] and its boundary states do not fully match the states of LQG. The reason for these is that the constraints are imposed too strongly and in [19,20] a new way of imposing the constraint is suggested in 4d Euclidean case that ﬁxes the problems with the Barett-Crane model. Later the new model was extended to the Lorentzian theory [21]. The classical continuum action that the new spin-foam models are derived from is the Plebanski-Holst action [19,22–25]. This action is obtained from the Plebanski action by adding an extra topological term that doesn’t change the equations of

1 motion. The coupling constant for this extra term introduces a new parameter R+ called the Barbero-Immirzi parameter [26, 27]. One of the most used spin- 2 foam models is the Engle-Pereira-Rovelli-Livine (EPRL) model [25] that extends the prior models [19, 20] to the case with ﬁnite Barbero-Immirzi parameter [26, 27] for both Euclidean and Lorentzian theories. The remarkable results of this model are that the space of boundary states matches the one from LQG for all values of and the area spectrum turns out to be discrete and the same as LQG area spectrum in both Euclidean and Lorentzian cases. In the path integral formulation of quantum mechanics, the transition amplitude between two points is obtained by restricting the integration in the partition function:

iScl(x) = (x)µ(x)e ~ Z D Z to all the possible paths between these points. The integrand of the path integral has two important parts: a phase part given by the exponential of i times the classical action Scl,andameasurefactorµ(x). The phase part,determinestheclassicalbehaviorusingthestationaryphase method [28]. In the classical limit, ~ 0, the phase oscillates fast and destruc- ! tive phase interference happens except in the vicinity of the stationary points of the 3 action x0 where Scl(x0) = 0, and the transition amplitude becomes

k µ(x )eiScl(x0m) Z⇠ 0m m=0 X where the sum is over the critical points m =1,...,k.Therefore,theformofthe phase part in terms of the classical action ensures that the solutions to the classical equations of motion dominate the path integral in the classical limit. The measure factor, µ(x)isdeterminedbycanonicalanalysisandisimportantfor the path integral to be equivalent to the corresponding canonical quantum theory. In order to establish this equivalence the measure factor for a constrained theory must be the Liouville measure on the reduced phase-space where all the first-class constraints are gauge-fixed [29,30]. This measure can then be extended to the full phase-space. In spin-foams the transition amplitudes are defined between loop quantum states that are eigenstates of the 3d geometry. In LQG eigenvalues of the 3-geometry have adiscretestructureandarelabeledbyspinnetworks.Therefore,inspin-foamsthe integral over paths is replaced by a discrete sum over histories of spin networks

= A() Z X where A is the contributed amplitude from each of these histories. In this dissertation, in studying the dynamics of general relativity via spin-foam approach we have paid attention to both the measure factor and the phase part. As mentioned above the correct measure factor is obtained by careful canonical analysis at the continuum level. For the Plebanski-Holst formulation of gravity from which spin-foams are derived this measure factor is calculated in [31]. The basic variables in this formulation are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the ﬁnal spin-foam sum, one usually sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski- Holst path integral in which only the Plebanski two-form appears, and in which the 4 connection degrees of freedom have been integrated out. Calculating the measure factor for Plebanksi Holst formulation without the connection degrees of freedom is one of the aims of this dissertation and it is done in chapter 4 and the results are reported in [32]. This analysis is at the continuum level and in order to be implemented in spin-foams one needs to properly discretize and quantize this measure factor. On the other hand, the correct phase is determined by requiring correct behavior in the classical limit. The asymptotics of the EPRL vertex amplitude in the Euclidean and Lorentzian case has been extensively studied in [33, 34] to check that the phase part has the correct semiclassical limit. One diculty with this and all spin-foam models before it is that, due to the inclusion of more than the usual gravitational sector [35], more than one term appears in the asymptotics of the vertex amplitude. As a consequence, solutions to the classical equations of motion of GR fail to dominate in the semi-classical limit [36,37]. One solution to this problem has been proposed in which one quantum mechanically imposes restriction to a single gravitational sector, yielding what has been called the ‘proper’ spin-foam model of quantum gravity [36,37]. However, this revised model of quantum gravity, like any proposal for a theory of quantum gravity, must pass certain tests. For example, in cases where the space- time curvature is small, one expects linearized quantum gravity to be correct. As aconsequence,inthisregime,oneexpectsagivenmodelofquantumgravityto reproduce the predictions of the linearized theory. One of the aims of this dissertation is to calculate the graviton two-point function predicted by the ‘proper’ vertex and examine its semiclassical limit to see if it matches the propagator calculated from the linearized theory. This dissertation is organized as follows: In chapter 2, the necessary preliminaries and background for the spin-foams formalism is given and the Lorentzian EPRL model and its asymptotics are reviewed . In chapter 3, as a consistency check of the ‘proper’

5 spinfoam model we calculate the gravitational two-point function predicted by the ‘proper’ spin-foam vertex to the lowest order in the vertex expansion to verify that it reproduces the predictions of the linearized theory in the appropriate limit. The analysis is restricted to the Lorentzian signature, as this is the physically relevant one. In chapter 4, in order to find the proper measure factor for spin-foams we calculate the measure factor for Plebanski-Holst formulation of gravity with the connection degrees of freedom integrated out. To confirm the results this analysis is done in two independent ways. The final results are summarized and discussed in chapter 5.

6 CHAPTER 2 BACKGROUND

2.1 PLEBANSKI-HOLST FORMULATION OF GRAVITY

Consider space-time as a 4d manifold .Thespace-timemanifoldindices,are M denoted here by lower case Greek letters ↵,, µ, ⌫, 0, 1, 2, 3 and ‘internal’ ··· 2{ } indices by I,J,K, 0, 1, 2, 3 which are raised and lowered with a fixed ‘internal’ ···2{ } metric ⌘ := diag(s, 1, 1, 1). s := +1 and s := 1 corresponds to Euclidean and IJ Lorentzian gravity respectively. The starting point for spin-foams is not General Relativity but BF theory [16] which is a topological field theory whose analysis is simpler than GR. Specifically, the Plebanski formulation of GR can be written in the form of a BF theory plus a constraint. The BF action is defined by:

S [B,!]= B F IJ[!](2.1) BF IJ ^ ZM where BIJ = 1 BIJdxµ dx⌫ is a two-form with values in the Lie algebra of so(⌘)1, 2 µ⌫ ^ IJ IJ IJ µ F is the curvature two-form of the so(⌘)-valued connection ! = !µ dx and is deﬁned by: F IJ = d ! = d!IJ + !I !KJ ! K ^ where d and d! are exterior derivative and covariant exterior derivative respectively.

IJ Consider BIJ and ! as independent variables. The Euler-Lagrange equations of motion for this action, obtained by varying these variables, are (1) No curvature:

IJ F =0and(2)d!B=0.

1For Euclidean case s = 1, so(⌘)=so(4) and for the Lorentzian case where s = 1, so(⌘)= so(3, 1). 7 IJ In order to reproduce GR’s action the Bµ⌫ ﬁeld is constrained to satisfy the simplicity constraint

s C := ✏ BIJBKL ✏ ✏↵✏ BIJBKL 0(2.2) µ⌫⇢ IJKL µ⌫ ⇢ 4! µ⌫⇢ IJKL ↵ ⇡

↵ where ✏IJKL denotes the internal Levi-Civita tensor with ✏0123 =1,and✏µ⌫⇢, ✏ denote the Levi-Civita tensor densities of weight 1and1,respectively.Thiscon- IJ straint has 20 independent components per point and restricts Bµ⌫ to belong to one of the following ﬁve sectors [4]. The ﬁrst, is the degenerate sector, in which

µ⌫⇢ IJ KL IJ ✏ ✏IJKLBµ⌫ B⇢ =0.TheotherfoursectorseachcorrespondtoBµ⌫ taking one of the following four forms

1 eI eJ (I ) BIJ = ± ^ ± 8⇡G 8 eK eL = 1 ✏IJ eK eL (II ) < ±⇤ ^ ± 2 KL ^ ± for some non-degenerate tetrad: eI . Here is the exterior product or wedge product µ ^ and is the Hodge dual operator with respect to the internal metric. However, just ⇤ aspecialcombinationofthesectors(II+) and (II ), namely the Einstein-Hilbert sector [35,38], gives the usual Einstein-Hilbert action for General Relativity. This will be relevant later on in this thesis. It is convenient to use units where 8⇡G =1, which we will do from now on. There is another term ( B ) F IJ,thatcanbeaddedtotheactionthatpreserves ⇤ IJ ^ the symmetries of the theory and doesn’t change the equations of motions. The total action now reads 1 S (B + B ) F IJ (2.3) BF ⌘ IJ ⇤ IJ ^ Z IJ 1 IJ KL 1 + where ( B) := 2 ✏ KLB , is the coupling constant and R is called the ⇤ () 2 Barbero-Immirzi parameter [26, 27]. Deﬁne, BIJ := (B + 1 B)IJ.Theaction(2.3) ⇤ constrained to the simplicity constraint (2.2) is called Plebanski-Holst action and is the starting point for spin-foams.

8 Restriction to the (II )sectorsreducesthisactiontotheHolst action of General ± Relativity [39]

1 1 s S (B + B ) F IJ = ( e e + eK eL) F IJ S . (2.4) BF ⌘ IJ ⇤ IJ ^ ±2 ⇤ I ^ J 2 ^ ^ ⌘± Holst Z Z The equations of motion derived by varying this action are

() d BIJ =0 d BIJ =0 d eI =0 ! , ! , ! and ✏ eI F JK =0 IJKL ^ the ﬁrst is the torsion-free condition on !,andthesecondisequivalenttotheEinstein equations. For ﬁnite non-trivial Immirzi parameter, the sector (I )alsogivesGR, ± s but the values of the e↵ective Newton constant and immirzi parameter are G and —di↵erentthansection(II ), where they are simply G and . ±

2.2 DISCRETE GEOMETRY

Spin-foams compute a transition amplitude for evolution of quantum 3-geometries from a triangulated space-time manifold .Speciﬁcally,weconsiderasimplicial M decomposition of the manifold which we assume to be oriented. For the rest M of this chapter we restrict to the Lorentzian manifold which is more physical and for simplicity the calculation is restricted to the case of just one 4-simplex .The generalization of the results to the case of multiple simplices can be done by using group ﬁeld theory techniques [40–42].

ALorentziangeometric4-simplexisaconvexhullofﬁvepointsinR3,1 called vertices. If all the vertices do not lie on a hyperplane the 4-simplex is called non- degenerate. The 4-simplex has ﬁve tetrahedra ⌧a on the boundary @ where a = 0, 1,...,4 and the tetrahedra are labeled by the point that the tetrahedron does not include. The numbering of the vertices imposes an orientation. Each tetraheron has 9 four triangles so there are twenty triangles associated to the boundary of a 4-simplex. But the boundary is closed and each triangle is shared between two tetrahedra. Thus, in total there will be ten triangles, tab,ontheboundarywheretheindicesdenotethat the triangle is shared between tetrahedron ⌧a and tetrahedron ⌧b.

2.2.1 Bivector geometry

Suppose V and W are two vectors in R3,1.Asimplebivectorcanbeconsideredasan antisymmetric tensor that can be expressed as a wedge product of two one-forms

V W = V W W V. ^ ⌦ ⌦

GR restricts the BIJ ﬁelds to be of the form eI eJ .ForeachchoiceofI and J, the ⇤ ^ bivector BIJ is simple.

Each oriented triangle t in R3,1 deﬁnes a simple bivector

1 X = e e . t 2 t01 ^ t02 by the exterior product of its two edges e = x x , e = x x where t01 t0 t1 t02 t0 t2 3,1 xt0 ,xt1 ,xt2 R are position coordinates for its vertices. Equivalently this bivector 2 can be deﬁned by the 4d outward normals Na, Nb to the tetrahedra sharing it

N () N () X ():=X = A(t ) a ^ b ab tab ab ⇤ N () N () | a ^ b | where A(tab)istheareaofthetriangletab.In[17,33,34]thesetofbivectorsXab as defined above are chosen to describe the geometry of a given 4-simplex. Here we choose B = X to be our set of bivectors to be consistent with [19,25]. According ab ⇤ ab to bivector geometry theorem, a set of ten bivectors satisfy certain properties com- pletely define the geometry of a 4-simplex. Bivector geometry theorem [17, 34]: Each geometric 4-simplex determines a set of bivectors satisfying the bivector geometry constraints stated below and each set of bivectors satisfying these conditions determines a geometric 4-simplex unique up to 10 parallel translation and inversion through the origin. Namely, there is a parameter µ = 1andageometric4-simplex unique up to translation and inversion, such ± that Bab()=µBab. Bivector geometry constraints: 1) Orientation condition: The bivector changes sign if the orientation of the triangle is changed, B = B . ab ab 2) Closure: The sum of the four bivectors corresponding to the faces of each tetrahedron is zero: a B =0. 8 ab b:b=a X6 3) Diagonal simplicity: Each bivector must define a geometric plane, i.e., is simple,

B B =0. (2.5) ab ^ ab

4) Cross simplicity: Each set of bivectors belonging to the same tetrahedron must span a 3D hyperplane

(3,1) IJ a, Na R , such that NaI ( Bab )=0, b = a. (2.6) 8 9 2 ⇤ 8 6

5) Tetrahedron non-degeneracy: For three triangles meeting at a vertex e of the ath tetrahedron (bcde): Tr(B [B ,B ]) =0. ab ac ad 6 6) Nondegeneracy: The assignment of bivectors is non-degenerate. This means that for six triangles sharing a common vertex, the six bivectors are linearly independent. I refer to [17] for the proof of this theorem. The Lorentzian geometrical 4-simplex is the classical geometry that we start with. Therefore, a correct spinfoam theory should be peaked on such classical geometries. In the asymptotic analysis of the spinfoam vertex amplitude that will be discussed in section 2.4.4 it becomes apparent how the Lorentzian geometrical 4-simplex emerges in the semiclassical limit.

11 2.3 QUANTIZATION

In the formulation of quantum theory which we use, a Hilbert space of states is associated to the boundary of a 2-complex ⇤ that is dual to a simplicial decomposition

2 of the 4d manifold .Theverticesv,edges e and faces f of ⇤ are dual to M 4-simplices ,tetrahedra⌧ and triangles t respectively. The points where edges intersect the boundary are called nodes n and the lines where faces intersect the boundary are called links l, which together form the graph which is the boundary of the

2-complex ⇤. The calculations in this dissertation is restricted to one 4-simplex, which has ﬁve tetrahedra and ten triangles on the boundary @,sothatthegraph has ﬁve nodes and ten links. Before quantization one needs to properly discretize the continuum variables of the theory. As was seen in section 2.1 the basic variables in Plebanski-Holst formulation

IJ are the two-form B and the curvature two-form FIJ. In quantum theory the groups are replaced by their universal coverings. Therefore, we replace SO(3, 1) by SL(2, C).

IJ Now, the B ﬁelds are so(3, 1) ⇠= sl(2, C)valuedtwo-formsandcanbeintegratedon triangles to give an element of so(3, 1) ⇠= sl(2, C),

IJ IJ Btab := B . Ztab The only assumption in the integration above is that the B ﬁeld is constant on the surface of integration, i.e., each triangle. In the discrete theory the continuum BIJ

IJ ﬁelds are replaced by their discrete version Btab which are elements of sl(2, C)and therefore can be identiﬁed with the generators of the group SL(2, C). The faces of ⇤ are dual to triangles of the 4-simplex . Therefore, by assigning an algebra element BIJ := BIJ to a face f dual to t the boundary variables of a 4-simplex will be a fab tab ab ab set of ten algebra elements B . { f } The curvature F is now replaced by holonomies Ue SL(2, C)alongedgese.On 2 2The reason for the font change is so that the edges e are not confused with tetrads e.

12 the boundary the holonomies Ul are deﬁned along the links. The relation between this discrete variable and the continuous sl(2, C)connection,!IJ,is

! Ul = Pe l , R where P stands for the path-ordered and the connection one-form is integrated along the link l.Sothediscreteboundaryvariablesareasetoften(Bf ,Ul). Before imposing the simplicity constraints in the quantum theory it is helpful to do a change of variables. From (2.3) the variable conjugate to Ul reads

1 J = B + B . f f ⇤ f ✓ ◆ Inverting this equation gives

2 1 B = J J . f 2 s f ⇤ f ✓ ◆✓ ◆ The diagonal simplicity constraint (2.5) and the cross simplicity constraint (2.6) in terms of new variables respectively become:

1 2 B .B = J .J 1+s s J .J =0, (2.7) ⇤ f f ⇤ f f 2 f f ✓ ◆ 1 N ( B )IJ = N ( J )IJ s J IJ =0 (2.8) I ⇤ f I ⇤ f f ✓ ◆ IJ where B.B = B BIJ. Also, the closure for Bf implies the closure for Jf . Note that imposing (2.6) selects sector (II ), i.e., B = e e and it eliminates sector (I ). ± ±⇤ ^ ± These constraints need to be imposed in the quantum theory. The quantization steps then consist in (1.) associating a Hilbert space of states to the boundary graph , (2.) imposing the constraints and (3.) computing a transition amplitude for these boundary states. In the next section we will review one of the most used spin-foam models namely Engle-Pereira-Rovelli-Livine (EPRL) model [25].

13 2.4 EPRL MODEL

2.4.1 Imposing the constraints

EPRL model [25] treats the simplicity constraints by ﬁrst imposing them properly in aﬁxedSO(3, 1) gauge, and then projecting on the gauge invariant spaces.

0 Lets ﬁx NI = I .Thegeneralcasewillberecoveredbygaugeinvariance.Inthe Lorentzian case this choice restricts all tetrahedra to be spacelike. With this choice (2.8) becomes 1 1 1 Cj = ✏j J kl s J 0j = Lj s Kj 0(2.9) f 2 kl f f f f ⇡

j 0j j 1 j kl where ✏ kl := ✏ kl,andLf := 2 ✏ klJf are the generators of the SU(2) subgroup that j 0j leaves NI invariant, and Kf := Jf are the generators of the corresponding boosts.

Since we are considering the Lorentzian case, Jf are the generators of SL(2, C)so C = J .J =2(L2 + sK2)andC = J .J =4sL.K are the casimir and pseudo- 1 f f 2 ⇤ f f casimir operators of sl(2, C). In terms of these operators, the constraint (2.7) can be written as s 2s C 1+ C =0 (2.10) 2 2 1 ✓ ◆ which is a ﬁrst class constraint and in the EPRL model it is imposed strongly. On the other hand, the equation (2.9) is a second class constraint. In the EPRL model the “master” constraint technique [4,43] is used to implement this constraint:

s 2 M := (Ci)2 = Li Ki 0 f ⇡ i i X X ✓ ◆ that leads to

2 C2 =4L . (2.11)

The Hilbert space of L2 functions on SL(2, C)canbeexpandedintoirreducible unitary representations of SL(2, C). The principal series of irreducible representations of SL(2, C)arelabelledbypositiverealnumbersp and non-negative half-integers k.

Each SL(2, C) irreducible representation p,k can be decomposed into a direct sum H 14 of SU(2) irreducibles which are labeled by a non-negative integer or half-integer Hj j. The casimir operators for the representation (p, k), are given by

C = p2 k2 +1 1

C2 = pk.

Equations (2.10) and (2.11) impose p = k and k = j [25] , where j labels the sub- spaces diagonalizing L2.Therefore,impositionofthesimplicityconstraintsbecomes arestrictionontherepresentationssummedoverinthespin-foamsum.Speciﬁcally, states p, k; j, m in satisfying these constraints have the form j,j; j, m .These | i Hp,k | i states are in one-to-one correspondence with the states in the irreducible representations of SU(2). Lets deﬁne a map as I

: I Hj 7! Hj,j

j, m j,j; j, m . | i 7! | i The map encodes the constraints (2.10) and (2.11). I

2.4.2 Boundary states

The boundary states which we will use in our calculations are chosen to be coherent states of SL(2, C). These states can be obtained by the map from certain SU(2) I coherent states (see appendix A). To deﬁne these coherent states, we use the real-

1 ization of j and p,k in terms of homogeneous functions on CP [44]. The SU(2) H H coherent states corresponding to spinor ⇠ and spin j are then deﬁned as:

j dj ¯ 2j C⇠ (z)= ⇠,z r ⇡ h i where the Hermitian inner product is deﬁned by

w, z =¯w z +¯w z h i 0 0 1 1

15 and d =2j +1. Using the explicit form of the map in terms of this realization of j I j and p,k, we ﬁnd the SL(2, C)boundarystatestobe[34], H H j (j,j) j dj 1+ij j ¯ 2j C⇠ (z) = C⇠ (z)= z,z ⇠,z . I r ⇡ h i h i

2.4.3 The amplitude

The vertex amplitude for these boundary states for a single 4-simplex then is deﬁned by [34]:

Av =( 1) (g0) dga ab 5 P SL(2,C) a Z Y Ya

jab 1 jab ab = ↵( C ,g gb C ) P I ⇠ab a I ⇠ba is deﬁned for each triangle (ab). Here ↵ is the invariant bilinear form satisfying ↵( ,)=( 1)2j↵(, ). Then we can rewrite the amplitude as:

SEPRL Av = c( 1) (g0) dga dµzab e (2.12) 5 1 10 SL(2,C) a ! (CP ) Z Y Z Ya

2 2 J⇠ab,Zab Zba,⇠ba Zba,Zba SEPRL = jab log h i h i + ijab log h i (2.13) Zab,Zab Zba,Zba Zab,Zab Xa

2.4.4 Asymptotics

According to the correspondence principle quantum systems must reproduce classical physics in the limit of large quantum numbers. Therefore, the asymptotics of the

3 Here we are using [45] convention for defining zab. By a change of variable inz ãb = Jzab we get the convention used in [38]. 16 vertex amplitude (2.12) in the limit of large spins jab must be peaked on the discrete geometry described in section 2.2. To assure that all the spins grow at the same rate the asymptotics is studied by first rescaling j by j then taking the limit . ab ab !1

The EPRL action (2.13) is linear in jab.Thereforethevertexamplide(2.12)isan integral of the general form:

n S(x) n Re (S(x)) iIm (S(x)) Av = d xf(x)e = d xf(x)e e Z Z By using the extended stationary phase method [28], in the limit the dominant !1 terms come from critical points x0 of the action. Critical points are the stationary points of the action Stot(x0)=0forwhichtherealpartoftheactionisfurthermore maximum. Since the real part of (2.13) satisﬁes Re(S ) 0, for critical points we tot  have Re (Stot(x0)) = 0. If there are more than one critical point present, the large spin limit of the integral is approximated by the sum of individual contribituions from each critical point

2⇡ n 1 S(x ) A = ( ) 2 f(x )e i . v i i det( H) X The asymptotics of the EPRL vertexp amplitude has been studied by [33] in the Euclidean case and in [34] for the Lorentzian case. Here we state the results for the Lorentzian case which we are discussing here: Given a set of non-degenerate boundary data, in the limit ,andforp = j , !1 ab ab kab = jab, 1) if the bounday state is the Regge state [33, 34] of the boundary geometry of a Lorentzian 4-simplex

12 1 L L Av ( 1) N+ exp i SRegge + N exp i SRegge ⇠ ✓ ◆ " a

17 4-simplex E

12 1 E E E E Av ( 1) N+ exp i SRegge + N exp i SRegge (2.15) ⇠ " ! !# ✓ ◆ Xa

1 12 A N v ⇠ V ✓ ◆ 4) for a set of boundary data that is neither a non-degenerate Lorentzian 4-geometry nor admits a vector geometry solution, the amplitude is suppressed for large

K A = o( ) non-negative integer K v 8

E E where N+,N ,N+ ,N and NV are independent of . In the ﬁrst two cases two terms appear in the asymptotics. The ﬁrst one is exp(iSRegge)andcorrespondstothecorrectReggeactionofgravityforadiscrete geometry, the other one corresponds to the negative of the Regge action, which cor- reponds to the opposite orientation. Together they result in the cosine of the Regge action. The appearance of the second term prevents the correct classical equation of motions from dominating in the classical limit when more than one 4-simplex is

present. In this case the result is a product over simplices cos(SRegge)whichis neither of the form exp(iSRegge)noroftheformcos(SRegge). Therefore,Q in the case of multiple simplices the action is non-Regge like which results in a wrong classical limit. In order to eliminate the second term, the “proper” vertex amplitude was introduced in [36–38] which will be discussed in the next section.

2.5 “PROPER” VERTEX AMPLITUDE

According to the Bivector geometry theorem stated in section 2.2.1 a set of bivectors

Bab satisfying bivector geometry constraints determines a geometric 4-simplex up 18 to parallel translation and inversion via B ()=µB where µ = 1. It was shown ab ab ± that the second term in the asymptotics of the vertex amplitude disappears if one restrict to a particular combination of sectors (II+) and (II )whichgivesµ =1 called the Einstein-Hilbert sector [36–38]. This sign is a multiplication of two factors µ = !⌫. The ﬁrst is the sign of the orientation ✏ BIJ BKL determined by BIJ IJKL ^

↵ IJ KL !(Bµ⌫):=sgn(✏ ✏IJKLB↵B ) where ✏↵ is the ﬁxed orientation on . The second, ⌫(B )is+1whenBIJ is in M µ⌫ µ⌫ the Plebanski sector (II+), 1ifitisinthePlebanskisector(II )and⌫(B )=0 µ⌫ otherwise. In order to have a well-deﬁned operator in quantum theory to implement the projection to the Einstein-Hilbert sector, this operator must be written in terms of the boundary data and group variables. The projection µ =1isdonebyintroducing aprojector

i ⇧ba[gab]=⇧(0, ) ba[g]tr igbagba† L . 1 h ⇣ ⌘ i 1 i Here gba = gb ga, L is the rotation generator in the spin jab representation of SU(2) and

i j k l m n ba =sgn ✏ijknbcnbdnbe✏lmnnacnadnae where c, d, e = 0,...,4 a, b and { } { }\{ }

i 1 i n [g]= tr( g g† ). ba 2 ba ba

The EPRL vertex amplitude restricted to the Einstein-Hilbert sector is now called “proper” vertex amplitude and for a general boundary state is given by:

(+) 1 Av =( 1) (g0) dga ↵( ab,ga gb ⇧ba( gab ) ba). (2.16) 5 I I { } SL(2,C) a ! Z Y Ya

19 Using Thm. 6 in [38] we can rewrite this amplitude for coherent states as:

(+) jab 1 jab A =( 1) (g ) dg ↵( ⇧ ( g )C ),g g C ). v 0 a ab ab ⇠ab a b ⇠ba 5 I { } I SL(2,C) a ! Z Y Ya

⇧abC⇠ab = dµ˜⌘âb C⌘âb (C⌘âb , ⇧abC⇠ab )(2.17) Z yielding

(+) jab 1 jab jab jab A =( 1) (g ) dg dµ˜ ↵( C ,g g C )(C , ⇧ ( g )C ) v 0 a ⌘âb ⌘âb a b ⇠ba ⌘âb ab ⇠ab 5 I I { } SL(2,C) a ! Z Y Ya

djab tation of SL(2, C)anddµ˜⌘ˆab = ⌦⌘ˆab . Here⌘ ˆ are normalized spinors⌘ ˆ = ⌘/ ⌘ , ⇡ k k so that ⌦ =⌦/ ⌘ 4.Rewritingeachinnerproductintermsofanintegralovera ⌘ˆ ⌘ k k spinor zab as before, we obtain the integral representation

A(+) =( 1) (g ) dg dµ˜ dµ eSprop . (2.18) v 0 a ⌘ˆab zab a ! Z Y Z Ya

Sprop = SEPRL + S⇧ where the ﬁrst term is a modiﬁed EPRL action

2 2 ab J⌘ˆab,Zab Zba,⇠ba Zba,Zba SEPRL = SEPRL =: jab log h i h i + ijab log h i Zab,Zab Zba,Zba Zab,Zab Xa

S = Sab =: log(C , ⇧ ( g )C ) . ⇧ ⇧ ⌘ˆab ab { ab} ⇠ab jab Xalinearized gravity in the appropriate limit. 20 CHAPTER 3 “PROPER” GRAVITON PROPAGATOR

3.1 INTRODUCTION

As explained in the last chapter, the “proper” spin-foam vertex amplitude (2.18) was obtained from the EPRL vertex by projecting to Einstein-Hilbert sector, in order to enable correct semi-classical behavior [36,37]. However, this revised model of quantum gravity, like any proposal for a theory of quantum gravity, must pass certain tests. For example, one expects a given model of quantum gravity to reproduce the predictions of the linearized theory in the low energy limit. The graviton propagator for the EPRL model in the Euclidean and Lorentzian case and their asymptotics has been studied in the literature [18, 45–50]. In this chapter, as a consistency check for the “proper” vertex we calculate the connected two-point correlation function for the metric operator also called the graviton propagator using the “proper” spin-foam vertex. Then the semiclassical limit of the propagator is studied. The method for calculating the semiclassical limit of two-point function is to cast it in terms of an action integral and to use stationary phase method. Thus, the calculation of the Hessian matrix of the action plays a key role. However, the fact that the “proper” action (2.5) is not linear in spins makes using the standard stationary phase method not straightforward [51]. This chapter is organized as follows. In section 3.2 the general framework and technique for calculating the graviton propagator in spin-foam formalism is deﬁned. 3.3 gives a review of the prior propagator calcuation for the Lorentzian EPRL model. Then comes the calculation of the propagator with the proper vertex in 3.4 which is

21 the main aim of this chapter, and ﬁnally the results and conclusion are discussed in 3.8.

3.2 GRAVITON PROPAGATOR IN SPIN-FOAM FORMALISM

We follow the same setting and general framework for calculationg the propagator as in [45,46,52] and the consideration is restricted to the case of just one 4-simplex. Here we briefly review the framework. Consider a 4d manifold with the topology of a 4-ball. Its boundary is a 3d M manifold ⌃= @ with the topology of S3 which is homeomorphic to the boundary of M a 4-simplex. Consider a simplicial decomposition of the boundary. The 4-simplex has five tetrahedra ⌧a on the boundary where a =0, 1,,...4. Associate a node to the center of each tetrahedron and four links that comes from the node and puncture through the faces of the tetrahedron, one for each face. The graph consisting of these nodes and links is a graph dual to the triangulation . In total there are five tetrahedra and ten faces on the boundary. Respectively the graph has five nodes and ten links. Since the area spectrum in LQG is discrete and is labeled by SU(2) spins j:

A =8⇡~G j(j +1) p where j’s are half-integers, each of those links can be labeled by a spin jab that gives the area of the triangle tab that the link is puncturing through a, b =0, 1,...4. As discussed in the preliminaries section, we associate a Hilbert space of states to this graph : .Thedynamicalexpectationvalueofanoperator | i2H⌃ O W h |O| i is deﬁned via: := W where W is the spin-foam map and W is the hOi h | i h | h | i spinfoam amplitude. The operator can be a geometric operator. Since we would O like to calculate the graviton propagator in this chapter, the geometric operator we are interested in is the densitized inverse metric operator qab. The connected two-point correlation function of the metric operator, also called 22 graviton propagator, is deﬁned by

W qab(x)qcd(y) W qab(x) W qcd(y) Gabcd(x, y)= | | | | | | W W W ⌦ h | i ↵ ⌦ h | i ↵ ⌦ h | i ↵ where x and y are associated to distinct nodes of the boundary spin network or equivalently to the center of each tetrahedron. Following [45] we choose the Lorentzian semiclassical states deﬁned in ap- | 0i pendix A as the boundary states. These states are a superposition of the Livine- Speziale coherent states [53], and are peaked on both intrinsic and extrinsic geometry. With this choice the spin-foam amplitude W will be the same as EPRL except h | 0i that there appear some extra terms in the action A. The only remaining element to deﬁne is the metric operator qab.

3.2.1 The metric operator

The metric operator that we are interested in is the operator associated to the densi-

1 a b a b 2 tized inverse 3d metricg ˜ 0 0 (x)=(detg)g 0 0 (x)wherea0,b0 =1, 2, 3 are 3d tangent indices. This metric can be written in terms of inverse triad

a0b0 a0b0 2 a0 b0 ij a0 b0 ij g˜ (x)=(detg)g (x)=(dete) ei (x)ej (x) = Ei (x)Ej (x) (3.1) where Ea0i := (det e)ea0i and i, j =1, 2, 3are3dinternalindicesinR3. Associate a normal nnm to each triangle t .Thefluxoperatorisdefinedby a0 nm Ei := Ea0i(x)nnm.InLQG,thefluxoperatorEi is a well-defined operator. The nm a0 nm m i n i two operators (En ) and (Em) at both endes of the link lnm of the graph dual to tnm are related by the parallel transport of the link. Our convention is that the lower

a index shows the frame, i.e., En is the flux operator corresponding to the face fna dual to the triangle tna parallel transported in tetrahedron n frame. The action of the 1Propagator of the linearized metric is the same as the propagator of the densitized metric if one uses the trace-free condition. 2Here we use the prime indices for the 3d space-time indices to distinguish them from labels of tetrahedra and triangles a, b =1,...5. Therefore, ga0b0 is the smooth metric and qab is the metric operator. 23 flux operator on the link lnm is identified with either right or left multiplication of the generator J i of SU(2) on that link depending on which end of the link the flux is located [2, 46]. Therefore, the (densitized) inverse metric operator associated to the classical quantity (3.1) is defined by3

ab ij a b qn (x):= (En)i(En)j where point x is identiﬁed with node n of the boundary spin network or equivalently with tetrahedron n of the triangulation.

3.3 EPRL GRAVITON PROPAGATOR

In [45] the connected two-point correlation function of the (density-two inverse) metric operator

ab cd ab cd W q (x)q (y) 0 W q (x) 0 W q (y) 0 Gabcd = | n m | | n | | m | nm W W W ⌦ h | 0i ↵ ⌦ h | 0i ↵ ⌦ h | 0i ↵ is calculated for a Lorentzian EPRL spinfoam model with a single spinfoam vertex and where the semiclassical state is a Lorentzian semi-classical boundary state | oi peaked both on intrinsic and extrinsic geometry and is constructed as follows. The Lorentzian coherent spin network states j , ⌥ ( ~n ) (see appendix A), | ab a { ab} i peaked on a choice of intrinsic boundary geometry, are labeled by a set of spins jab and Lorentzian coherent intertwiners ⌥a Inv b:b=a j .EachLorentziancoherent 2 ⌦ 6 H ab intertwiner ⌥a( ~n ab )isdeterminedbyasetoffourunit3-vectors ~n ab b:b=a via { } { } 6

⌥ ( ~n )=exp i ⇥ j dh h j ,~n (3.2) a { ab} ab ab | ab abi a

i ⇥abjab phase condition [34]. e a

= (j) j , ⌥ (~n ) (3.3) | oi jo,o | ab a i j Xab with coecients given by a Gaussian times a phase,

(ab)(cd) jab (j0)ab jcd (j0)cd ab jo,o (j)=exp ↵ i 0 (jab (j0)ab) (j0)ab (j0)cd ! ab,cdX Xab p p (3.4) ab where o is the dihedral angle between tetrahedron a and b,representingthesim- plicial extrinsic curvature, (jo)ab are the background spins on which the Gaussian is peaked, and ↵(ab)(cd) is a 10 10 matrix assumed to be complex with positive deﬁnite ⇥ real part. Using these states as boundary states adds three extra terms to the EPRL action and the spin-foam vertex amplitude becomes (compare to (2.12))

tot SEPRL Av0 = W 0 = c( 1) (g0) dga dµzab e (3.5) h | i 5 1 10 SL(2,C) a ! (CP ) Z Y Z Ya

tot 1 (ab)(cd) jab (j0)ab jcd (j0)cd ab SEPRL = ↵ i 0 (jab (j0)ab) i⇥abjab+SEPRL 2 (j0)ab (j0)cd Xab p p and SEPRL is deﬁned by (2.13).

Semiclassical limit is obtained by ﬁrst rescaling jab to jab and (j0)ab to (j0)ab then taking . Also, in the Lorentzian case we can’t deﬁne a regular 4-simplex !1 4All tetrahedra are assumed to be space-like so their normals are time-like. A wedge composed by two tetrahedra a and b that share the triangle (ab) is called thick wedge if their outward timelike normals are both future-pointing or both past-pointing otherwise called thin wedge.

25 where all jab are equal. But we can still factorize each spin into jab = j✏ab where ✏ab is ﬁnite and let j .Inthislimitthepropagatorbecomes !1 ab cd abcd BD 1 @qn @qm 4 3 ✏ G = (Q ) + j (X + ()) (3.6) nm (pq)(rs) @j @j O p,q,r,s pq rs X ✏ BD 5 ab where X is independent from j and and Q(pq)(rs) and qn are deﬁned as

(ab)(cd) BD ↵ Q(pq)(rs) = jab jcd Stot crit = + jab jcd critSRegge, (j0)ab (j0)cd ab a b qn =pAn.Anp, (3.7) i i Zna,⇠na Aan jan h i. ⌘ Zna,⇠na h i ab See chapter 2 for deﬁnition of Z and ⇠ .Theterm@qn appearing in (3.6) scales na na @jpq as 2j2 [45]. Thus, the ﬁrst term of the propagator (3.6)

ab cd BD 1 @qn @qm 3 3 abcd (Q ) = j R (↵) (pq)(rs) @j @j nm p,q,r,s pq rs X scales as 3j3 where Rabcd(↵)isindependentofj and .Therefore,inthelimit 0 nm ! and j keeping the size of the quantum geometry !1

A =8⇡~G j(j +1) j ⇠ p ﬁnite and ﬁxed 6 the leading order term of the propagator (3.6), is

Gabcd 3j3Rabcd(↵). (3.8) nm ⇠ nm

a b The object En.En has dimension area squared and in the large spin limit scales as Ea.Eb (j)2 + (j). Therefore, the leading order term of the propagator predicted n n ⇠ O by Regge calculus is

( Ea.Eb Ea .Eb Ea.Eb Ea .Eb ) (j)3. h n n m mih n nih m mi ⇠ This shows that in the asymptotic limit the scaling behavior of the propagator (3.8) matches the results from the Regge calculus. However, the limit j is not enough !1 by itself to ensure the correct asymptotic limit. 5BD refers to the authors of [45], E. Bianchi and Y. Ding. 6This limit corresponds to the case where the distance between x and y is ﬁnite and ﬁxed. 26 3.4 GRAVITON PROPAGATOR FOR THE PROPER VERTEX

Following the same procedure in [45] we want to calculate the propagator for the proper vertex reviewed in section 2.5. In this case the propagator becomes

a b c d a b + j W+ E .E E .E j, ⌥(~n ) j W+ E .E j, ⌥(~n ) Gabcd = j h | n n m m| i j h | n n| i nm W j, ⌥(~n ) W j, ⌥(~n ) · P j jh +| i P j jh +| i W Ec .Ed j, ⌥(~n ) P j Pjh +| m m| i · W j, ⌥(~n ) P j jh +| i where (3.3) are used as boundary states and W : P W is the proper h +| | i 7! h +| i vertex amplitude deﬁned by (2.16). The indices are restricted to a, b, c, d = n, m and 6 n = m.Thenumberingofthetetrahedraisarbitraryuptotheorientationitdeﬁnes 6 on the 4-simplex. Using this arbitrariness, we can, without loss of generality, assume

a i a, b < n and c, d < m.(En) is in the frame of tetrahedron n and for a

a b i ab ⇥abjab W+ En.En j, ⌥(⇠) = e dga (g0) h | | i 5 SL(2,C) P a=0 ! Z Y

jcd 1 jpq jan 1 a i jna ↵( ⇧pq( g )C ,g gq C ) ↵( ⇧an( g )C ,g gn (E ) C ) 0 I { } ⇠pq p I ⇠qp 1 I { } ⇠an a I n ⇠na p

In order to separate the projector in the above formula we insert the resolution of the identity (2.17) in terms of new spinor variables ⌘cd:

jpq 1 jpq jpq 1 jpq jpq jpq ↵( ⇧pq( g )C ,g gq C )= dµ˜⌘ ↵( C ,g gq C )(C , ⇧pq( g )C ) I { } ⇠pq p I ⇠qp pq I ⌘ˆpq p I ⇠qp ⌘ˆpq { } ⇠pq Z and

jan 1 a i jna jan 1 a i jna jan jan ↵( ⇧an( g )C ,g gn(E ) C )= dµ˜⌘ ↵( C ,g gn(E ) C )(C , ⇧an( g )C ). I { } ⇠an a n I ⇠na an I ⌘ˆan a n I ⇠na ⌘ˆan { } ⇠an Z 27 Therefore, (3.9) becomes

a b i ab ⇥abjab W+ En.En j, ⌥(⇠) = e dga (g0)ij h | | i 5 · SL(2,C) P a=0 ! Z Y jpq 1 jpq jpq jpq dµ˜⌘ ↵( C ,g gq C )(C , ⇧pq( g )C ) · pq I ⌘ˆpq p I ⇠qp ⌘ˆpq { } ⇠pq · p

4 a b W+ En.En j, ⌥(⇠) = c dga (g0) h | | i 5 · SL(2,C) a=0 ! Z Y dj a0b0 ab Sprop0 dµza b dµ˜⌘a b qn e · ⇡ 0 0 0 0 ! Z aY0

4 a b c d W+ En.EnEm.Em j, ⌥(⇠) = c dga (g0) h | | i 5 · SL(2,C) a=0 ! Z Y dj a0b0 ab cd Sprop0 dµza b dµ˜⌘a b qn qn e · ⇡ 0 0 0 0 ! Z aY0

S0 := S i ⇥ j = S + S i ⇥ j prop prop ab ab EPRL ⇧ ab ab Xa

ab The qn ’s turn out to be the same as in the EPRL case [45], but here the EPRL action is replaced by the proper action. So using (3.3), the ﬁnal expression for the propagator becomes

tot tot 4 10 10 ab cd Sprop 4 10 10 ab Sprop abcd+ j d gd µzd µ˜⌘qn qm e j d gd µzd µ˜⌘qn e Gnm = tot tot (3.11) 4 10 10 Sprop 4 10 10 Sprop · P R j d gd µzd µ˜⌘e P Rj d gd µzd µ˜⌘e tot 4 10 10 cd Sprop P R P j R d gd µzd µ˜⌘qm e tot · 4 10 10 Sprop P Rj d gd µzd µ˜⌘e where P R

tot 1 (ab)(cd) jab (j0)ab jcd (j0)cd ab Sprop(j, g, z, ⌘)= ↵ i 0 (jab (j0)ab) 2 (j0)ab (j0)cd ab,cdX Xab p p i ⇥ j + S + S . ab ab EPRL ⇧ Xa

Notice that the coecients j are absorbed in the total action.

3.4.1 Asymptotic limit

As in [45], to pose the asymptotic limit, we scale all of the spins (jo)ab uniformly, setting (j ) =: (j ) ,with(j ) ﬁxed, and consider the limit .Inaddition, o ab o ab o ab !1 tot in each sum in (3.11) we perform a change of variables from jab to jab := jab/. Sprop then becomes

tot ˜ ˜ ⇧ Sprop = S + SEPRL + Sab h i Xa

˜ 1 (ab)(cd) jab (j0)ab jcd (j0)cd ab S := ↵ i o (jab (jo)ab) i ⇥abjab 2 (j0)ab (j0)cd ab,cdX Xa

ab projector part ofP the total action (3.12), a

tr(gbagba† ) n⌫ba = ⌫ba,⌫ba = ab[X] , h i tr(g g† ) | ba ba | and let

x := ⌘ ,⌫ ⌫ ,⇠ y := ⌘ ,J⌫ J⌫ ,⇠ . ab h ba baih ba bai ab h ba baih ba bai

Then we have [51]

exp(Sab) f˜ exp(S˜ab) ⇧ ⇠ ab ⇧ where

0ifjab = jab is integer ↵ = ab 8 < 1/2ifjab = jab is half integer.

ab : ab ab 2 ab Lastly, letq ñ be the same as qn ,butwithjab replaced by jab,sothatqn = qñ . With the above definitions and asymptotic limit, (3.11) becomes

1 d4gd10µ d10µ˜ f˜qãbq˜cdeS˜ Gabcd+() j z ⌘ n m (3.13) 4 nm ⇠ 4 10 10 ˜ S˜ P R j d gd µzd µ˜⌘fe 4 10 10 ˜ ab S˜ 4 10 10 ˜ cd S˜ P R j d gd µzd µ˜⌘fqñ e j d gd µzd µ˜⌘fq˜m e 4 10 10 ˜ S˜ 4 10 10 ˜ S˜ P Rj d gd µzd µ˜⌘fe P Rj d gd µzd µ˜⌘fe 8 Due to this expression for ↵ab, strictlyP R speaking, in order to haveP a measureR factor in (3.11) which is independent of , the asymptotic limits for when jab is integer or half-integer should be considered separately. However, as in [51], the final answer is the same in either case because there are no critical points in the region where the asymptotic integrand depends on ↵ab. 30 where

f˜ := f˜ab and S˜ := S˜ + S˜EPRL + S˜⇧ (3.14) Ya

4 10 10 ˜ ab cd S˜ 1 d gd µzd µ˜⌘fq˜n q˜m e Gabcd+() j (3.15) 4 nm ⇠ 4 10 10 ˜ S˜ R R j d gd µzd µ˜⌘fe 4 10 10 ˜ ab S˜ 4 10 10 ˜ cd S˜ R R d gd µzd µ˜⌘fq˜n e d d µzd µ˜⌘fq˜m e j j . 4 10 10 ˜ S˜ 4 10 10 ˜ S˜ R Rj d gd µzd µ˜⌘fe R jR d gd µzd µ˜⌘fe

Critical point equations obtainedR R from maximality, andR byR varying zab, ⌘ab,and ga are [51]: eiba eiba ⇠ = g†z =: Z (3.16) ba Z b ab Z ba k bak k bak eiab eiab J⌘ˆ = g†z =: Z (3.17) ab Z a ab Z ab k abk k abk i✓ ⌘ˆab = e ⇠ab (3.18)

i ab( g )n tr gabg† i > 0(3.19) { } ⇠ab ab ⇣ ⌘ for all a

1 10 These equations determine [zab], [⌘ab] CP ,andga uniquely [51]. Furthermore, 2 the ﬁnal critical point equation, from varying jab,gives

jab =(jo)ab (3.20)

9In the asymptotic limit, each sum over j’s in the expression (3.13) becomes an integral with almost everywhere smooth integrand, so that stationary phase can later be applied. Note, this is precisely where the phase convention for the coherent intertwiners ⌥( ~n ab ) becomes important: Without this phase convention, the sum over j’s would not approach a{ continuous} integral, so that stationary phase would not be applicable [54]. 1 10Here [z] denotes the equivalence class in CP , that is, modulo rescaling by C 0 . Also note \{ } that we have gauge-ﬁxed the symmetry in the ga’s with the factor (g0), as usual. 31 so that there is exactly one critical point of this action.Itfollowsthattheasymptotic expression for each integral in (3.15) will consist in only a single term. The asymptotics can then be evaluated using the extended stationary phase method. As there is only one critical point, the asymptotic expression becomes [45,52]

ab cd 1 abcd+ 1 1 ij @q˜n (xo) @q˜m (xo) G ()= (H (x )) +higherorderterms (3.21) 4 mn o @xi @xj where H is the Hessian of the asymptotic action S˜, xi = j ,g ,z ,⌘ and x is { ab a ab ab} o the one critical point of S˜ (3.16-3.20). Note that, if we deﬁne

S˜o := 2 j log(x + y )=2 j log ⌘ˆ ,⇠ , ⇧ ab ab ab ab h ab abi Xa

S˜ = S˜o := S˜ i ⇥ j + S˜ +2 j log ⌘ˆ ,⇠ (3.22) ab ab EPRL ab h ab abi Xa

3.5 HESSIAN OF THE PROPER VERTEX: CALCULATION

Calculating the Hessian of the asymptotic action is crucial in studying the asymptotics of the spinfoam propagator and also the asymptotics of the spinfoam vertex amplitude. In this section we evaluate the Hessian for the proper action at the critical point satisfying (3.16-3.20).

All second derivatives of the ﬁrst three terms in (3.22) with respect to jab,ga,zab were calculated in [45]. To complete the calculation of the Hessian, it remains to calculate all second derivatives of S˜EPRL involving ⌘ab, and all second derivatives of ˜o S⇧.Wedothisinthissection.

32 3.5.1 Derivatives of S˜EPRL with respect to ⌘ and one other variable

To calculate the derivatives in the Hessian for use in the stationary phase theorem,

1 it is necessary to choose a section of CP (and coordinates thereon) for each variable

⌘ab and zab,aswellasacoordinatesystemonSL(2, C) for each variable ga. To this end, for the variables ⌘ we we introduce complex coordinates ⇣ s.t. ⇣ 1 and ab ab | ab| deﬁne

⌘ = 1 ⇣ 2⇠ + ⇣ J⇠ . (3.23) ab | ab| ab ab ab p The calculation of derivatives with respect to the ⌘’s then reduces to taking the

@ @ holomorphic and anti-holomorphic derivatives @⇣ and @⇣¯.

The speciﬁc form of the section for each zab and the coordinate system for each ga turns out not to be needed for any of the results of this paper, so we leave them

i general. Derivatives with respect to these general coordinates we denote by zab ,with i i =1, 2, and ga ,withi =1,...6andtheindexi sometimes suppressed. We begin by noting

@ ⇣¯ab J⌘ab,Zab = J⇠ab,Zab ⇠ab,Zab @⇣ab h i 2 1 ⇣ ⇣¯ h ih i ab ab @ p ⇣ab ¯ J⌘ab,Zab = J⇠ab,Zab @⇣ab h i 2 1 ⇣ ⇣¯ h i ab ab so that p

¯ @ ãb ⇣ab J⇠ab,Zab ⇠ab,Zab SEPRL = jab h i 2jab h i (3.24) @⇣ab 1 ⇣ ⇣¯ J⌘ab,Zab J⌘ab,Zab ab ab h i h i @ ãb p ⇣ab J⇠ab,Zab SEPRL = jab h i (3.25) @⇣ 1 ⇣ ⇣¯ J⌘ab,Zab ab ab ab h i Noting that at the critical pointp⇠ ,Z =0and⇣ =0,weget: h ab abi ab @ @ @ @ Sãb = Sãb =0 @j @⇣ EPRL @j @⇣¯ EPRL ab ab crit. pt. ab ab crit. pt.

33 Next, from (3.24-3.25), we calculate the mixed g⌘ derivatives:

@ ˜ab ⇣ab J⇠ab,Zab ga SEPRL = jab ga h i @⇣ab J⌘ab,Zab 1 ⇣ab⇣ab ✓h i◆ q (g )⇠ ,z ⇠ ,Z 2j h a ab abi h ab abi (g )J⌘ ,Z ab J⌘ ,Z J⌘ ,Z 2 h a ab abi ✓ h ab abi h ab abi ◆ @ ˜ab ⇣ab J⇠ab,Zab ga SEPRL = jab ga h i @⇣¯ab J⌘ab,Zab 1 ⇣ab⇣ab ✓h i◆ q Evaluating at the critical point gives

@ ab 1 S˜ = 2j (g g )⇠ ,J⇠ @⇣ ga EPRL abh a a ab abi ab crit. pt. @ ˜ab ga S =0 @⇣¯ EPRL ab crit. pt. ˜ab The second derivative ⌘ab gSEPRL for variations g of any group elements other than ga is zero. The mixed derivatives ⌘z are

@ ˜ab ⇣ab J⇠ab,Zab zab SEPRL = jab zab h i @⇣ab J⌘ab,Zab 1 ⇣ab⇣ab ✓h i◆ q g ⇠ ,z ⇠ ,Z 2j h a ab abi h ab abi g J⌘ ,z ab J⌘ ,Z J⌘ ,Z 2 h a ab abi ✓ h ab abi h ab abi ◆ @ ˜ab ⇣ab J⇠ab,Zab zab SEPRL = jab zab h i @⇣¯ab J⌘ab,Zab 1 ⇣ab⇣ab ✓h i◆ q Evaluating it at the critical point:

@ g ⇠ ,z S˜ab = 2(j ) h a ab abi zab @⇣ EPRL o ab J⇠ ,Z ab crit. pt. ab ab h i @ ˜ab zab S =0 @⇣¯ EPRL ab crit. pt. The last to evaluate are the derivatives ⌘⌘: 2 @ @ ab J⇠ab,Zab ⇣ab S˜ = jab h i EPRL 3/2 @⇣ab @⇣ab J⌘ab,Zab 2(1 ⇣ ⇣ ) h i ab ab @ @ ab J⇠ab,Zab 1 ⇣ab⇣ab S˜ = jab h i + EPRL 1/2 3/2 @⇣ab @⇣ J⌘ab,Zab (1 ⇣ ⇣ ) 2(1 ⇣ ⇣ ) ab ✓ ab ab ab ab ◆ h i 2 @ @ ab J⇠ab,Zab ⇣ab S˜ = jab h i EPRL 3/2 @⇣ @⇣ J⌘ab,Zab 2(1 ⇣ ⇣ ) ab ab h i ab ab 34 At the critical point ⇣ =0, J⌘ ,Z = J⇠ ,Z =1andj =(j ) ,sotheonly ab h ab abi h ab abi ab o ab non-zero term is:

@ @ S˜ab = (j ) @⇣¯ @⇣ EPRL o ab ab ab crit. pt. ˜o 3.5.2 Derivatives of S⇧

˜o We begin by writing S⇧ in terms of the coordinates ⇣ab:

S˜o =2 j log ⌘ˆ ,⇠ = j log(1 ⇣ ⇣ ) ⇧ ab h ab abi ab ab ab Xa

@ @ õ ⇣ab S⇧ = @⇣ab @jab 1 ⇣ ⇣ ab ab @ @ õ ⇣ab S⇧ = @⇣ab @jab 1 ⇣ab⇣ab 2 @ @ j ⇣ Sõ = ab ab ⇧ 2 @⇣ab @⇣ab (1 ⇣ ⇣ ) ab ab @ @ j Sõ = ab ⇧ 2 @⇣ @⇣ab (1 ⇣ ⇣ ) ab ab ab @ @ j ⇣2 Sõ = ab ab @⇣ @⇣ ⇧ (1 ⇣ ⇣ )2 ab ab ab ab õ for all a, b,withallothersecondderivativesofS⇧ zero. Evaluating these using the critical point equations ⇣ab =0andjab =(jo)ab,weseethatonlyonesecondderivative õ of S⇧ is non-zero:

@ @ ˜o @ @ ˜o S⇧ = S⇧ = (jo)ab. @⇣ @⇣ab @⇣ @⇣ab ab crit. pt. ab crit. pt.

35 3.5.3 Hessian

Bringing these results together, we can write the Hessian as a (10 + 24 + 20 + 10 + 10) (10 + 24 + 20 + 10 + 10) matrix: ⇥

Qjj 010 24 010 20 010 10 010 10 ⇥ ⇥ ⇥ ⇥ 0 1 024 10 Hgg Hgz Hg⇣ 024 10 ⇥ ⇥ B C H(xo):=S00(xo)=B0 H H H 0 C B 20 10 zg zz z⇣ 20 10C B ⇥ ⇥ C B C B010 10 H⇣g H⇣z 010 10 H⇣⇣¯ C B ⇥ ⇥ C B C B010 10 010 24 010 20 H⇣⇣¯ 010 10C B ⇥ ⇥ ⇥ ⇥ C @ A BD Qjj = Qjj

BD Hgg = Hgg

BD Hgz = Hgz

BD Hzz = Hzz

BD BD BD Here Qjj , Hgg , Hgz are the Hessian components calculated by Bianchi and Ding

[45], and Hg⇣, Hz⇣,andH⇣⇣¯ are respectively given by

g⇣ 1 i H = 2(j ) (g g )⇠ ,J⇠ (ai)(ab) o abh a a ab abi g ⇠ ,z Hz⇣ = 2(j ) h a ab abi (ab)(cd) o ab J⇠ ,Z ac bd h ab abi ¯ H⇣⇣ = 2(j ) . (ab)(cd) o ab ac bd

3.6 GRAVITON PROPAGATOR AND ITS 0LIMIT. !

According to (3.21), calculation of the graviton propagator requires us to invert the Hessian matrix calculated in the last section. Doing this exactly is a non-trivial task. However, as was done in the work [45], we consider the 0limit,inwhichcasethe ! Hessian matrix simpliﬁes greatly. Such a limit can be motivated by stipulating that, as becomes large, the eigenvalues of areas, which are proportional to ,staythe same, by letting 0atthesametime.Thisisthelimitusuallyconsidered[45]. ! 36 According to the results in [45] and the results in the Hessian section 3.5, all the nonvanishing elements of the Hessian at critical points are independent of except

BD Qjj ,whichislinearin. If we write the Hessian in the form:

BD Qjj 010 64 ⇥ H(xo)= 0 R 1 064 10 H ⇥ @ A then its inverse is 1 BD Qjj 010 64 1 ⇥ (H(xo)) = . (3.26) 0 R 1 1 064 10 (H ) ⇥ ab @ A Theq ñ calculated using the proper action in this paper, equation (3.10), turn out to be the same as the qab obtained in [45]. Consequently their derivatives qãb , n z¯an n |crit qãb , qãb evaluated at critical points (3.16-3.20) are the same and they ga n |crit jcd n |crit 2 ab ¯ ab scale as and zan qn = 0. Here we have two more sets of variables ⇣ and ⇣,butqn

ab ab does not depend on these variables: ⇣ qn = ⇣¯qn = 0. Using the derivatives above and the matrix (3.26) in the asymptotic formula (3.21) yields the same results as [45]

ab cd 1 @q˜ @q˜ Gabcd = 3 QBD n m + o(3)+o(3)(3.27) nm (pq)(rs) @j @j p,q,r,s pq rs X where f = o(g)iff/g vanishes in the limit and 0. In this limit the !1 ! propagator is asymptotic to

ab cd 1 @q˜ @q˜ Gabcd 3 QBD n m . nm ⇠ (pq)(rs) @j @j p,q,r,s pq rs X 3.7 COEFFICIENT IN THE ASYMPTOTICS OF THE PROPER VER- TEX

In this section we want to ﬁnd the large spin limit of the proper vertex amplitude, including the overall coecient. From (2.18) we have

4 + Sprop A⌫ =( 1) c dga (g0) dµzab dµ˜⌘ab e . 1 10 SL(2,C) a=0 ! (CP ) Z Y Z Ya

4 ˜ + ˜ Sprop A⌫ =( 1) c dga (g0) dµzab dµ˜⌘ab fe 1 10 SL(2,C) a=0 ! (CP ) ! Z Y Z Ya

32 4 2⇡ 1/2 ˜ A+ det( HR(x )) dµHaar [dµ ][dµ˜ ] eSprop(xo) v ⇠ o ga zab ⌘ab · ✓ ◆ a=1 a

BD BD BD Hgg Hgz Hgz¯

EPRL 0 BD BD BD 1 H = Hzg Hzz Hzz¯ B C B BD BD BD C B Hzg¯ Hzz¯ Hz¯z¯ . C B C @ A Applying (3.29) to the usual formula [55] for the determinant of block matrices, together with the skew symmetry of the determinant under exchange of columns, 38 yields

R 2 EPRL det( H (x )) = (det( H ¯)) det( H (x )) (3.30) o ⇣⇣ o with

10 det( H ¯)= 2(j ) =2 (j ) . ⇣⇣ o ab o ab Ya

det( HR(x )) = 210 j det( HEPRL(x )). o ab o a

The projector part of the action vanishes at critical points so S˜prop(xo)=S˜EPRL(x0).

Furthermore, in terms of the coordinates ⇣ab, ⇣ab, dµ˜⌘ab is given by

d i dµ˜ = jab ✏ ⌘A d⌘B ✏ ⌘A d⌘B ⌘ab ⇡ 2 AB ab ab ^ AB ab ab

djab i = d⇣ab d⇣ab. xo ⇡ 2 ^

Hence

i(2jab +1) 10 10 [dµ˜⌘ab ]= ⇡ jab = ⇡ (jo)ab. xo 2⇡ ⇠ ⇡ xo ⇡ Ya

39 Therefore (3.28) becomes

22 4 + 2⇡ 1/2 Haar S˜ (x ) A (det( H (x ))) dµ [dµ ] e EPRL o . v ⇠ EPRL o ga zab · ✓ ◆ a=1 a

3.8 DISCUSSION

In this chapter, we calculated the connected two-point function of the metric operator predicted by the Lorentzian “proper” vertex amplitude [36–38] at the lowest order in the vertex expansion and studied its asymptotic behavior. To perform the analysis, we defined the correlation function in the boundary amplitude formalism [56–58] extended to a background-independent theory [46]. In this formalism, the gravitational two-point correlation function between two points depends on the state of the gravitational field on the boundary of a closed surface (3-sphere for simplicity) on which the points lie [46]. We use the Lorentzian semi-classical states defined in [45] and appendix A as the boundary states. These semi-classical states are obtained by asuperpositionofLorentziancoherentspinnetworkstateswhicharepeakedonboth intrinsic and extrinsic geometry. The asymptotics of the propagator is studied by first taking the limit of large spins up to the leading order. In order to make sure that all the spins increase at the same rate we first set j to j then take the limit of 0. The proper ! action turns out to be asymptotically linear in j [51]. Therefore, the large spin limit of the propagator was investigated using extended stationary phase methods which necessitated calculation of the Hessian of the proper action. This calculation comprised the main technical challenge of this chapter. As in the previous propagator calculation for the EPRL model [45, 46, 52] in addition to the limit above one needs 40 to take 0. This limit corresponds to the fixed area spectrum A j therefore ! ⇠ keeps the size of the quantum geometry fixed, meaning that the distance between the two points for which we calculated the propagator is kept constant. We found that in this limit, the leading order term of the Lorentzian proper propagator (??)andthe Lorentzian EPRL propagator match. The EPRL model was shown to reproduce the same two-point correlation function as linearized gravity [45] at the limit considered above. Thus we have confirmed the validity of the proper spin-foam model in the weak curvature regime to lowest order in the vertex expansion in this limit. Calculation of the Hessian also allowed us to complete the asymptotic analysis of the proper vertex [51]. Specifically, by calculating the determinant of the Hessian we were able to evaluate the factor arising in the semi-classical limit. Due to some unexpected cancellations this factor matches exactly the factor appearing in the semiclassical limit of the EPRL vertex in front of the term corresponding to the Einstein- Hilbert sector. In [59] we calcuated the two-point correlation function for a single spin-foam vertex. The obvious direction for future work is to extend the analysis to the case of multiple vertices. Since the proper vertex amplitude has the correct semi-classical limit, we believe this calculation will again be consistent with linearized quantum gravity in the appropriate regime whereas we suspect a calculation based on the EPRL vertex will not be consistent. Specifically, the proper vertex is expected to remove unphysical contributions to the semi-classical limit which, when multiple vertices are considered, would otherwise be generated by the EPRL vertex. In this calculation, as in prior graviton propagator calculations [18, 45–50], the boundary data was chosen to correspond to flat space-time. In the presence of matter and cosmological constant the space-time won’t be flat. A natural first choice for such acurvedspace-timewouldbedeSitterspace-timesinceitisthemaximallysymmetric solution of the vacuum Einstein field equation with positive cosmological constant.

41 The vertex amplitude with a cosmological constant has already been studied by [60– 68]. It would be very interesting and crucial to extend the propagator calculation to study the correlation functions for boundary data corresponding to a curved space- time. To test the theory, in the appropriate limit the results must match the results from the quantum ﬁeld theory of linearized gravity in a de Sitter background.

42 CHAPTER 4 CANONICAL MEASURE FACTOR FOR SPIN-FOAMS

4.1 INTRODUCTION

As mentioned in the introduction, the measure factor is important for the path integral to be equivalent to the corresponding canonical quantum theory. The covariant formulation of gravity from which spin-foams are derived is the Plebanski-Holst for-

IJ mulation, in which the basic variables are a Lorentz connection !µ and a Lorentz- algebra valued two-form XIJ = BIJ, called the Plebanski two-form. However, in µ⌫ ⇤ µ⌫ the ﬁnal spin-foam sum, one usually sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two- form appears, and in which the connection degrees of freedom have been integrated out. The measure factor for the Plebanski-Holst path integral with both connection and two-form variables was calculated in [31]. Before one discretizes this measure and incorporates it into a spin-foam sum, however, for the reason explained above one must integrate out the connection. This is the main purpose of this section. This is done in two independent ways, determining which powers of the three and four dimensional volumes appear in the measure factor. The results of this chapter is published in [32]. This chapter is organized as follow. In section 4.2 the reduced phase space method for calculating the measure factor in canonical analysis is introduced. In section 4.3 the results [31] of applying this method to Plebanski-Holst formulation of gravity are

43 given. Then in 4.4 comes the main calculation of this chapter in two di↵erent ways (1) starting from the Plebanski-Holst path integral calculated in [31] and integrating out the connection degrees of freedom (2) by starting from ADM path integral and integrating out the momentum degrees of freedom. Finally, comes results and conclusion in 4.6.

4.2 REDUCED PHASE-SPACE METHOD

For an unconstrained ﬁeld theory with canonically conjugate variables ',⇡,thephase space path integral takes the form [69]

( )= ' ⇡ (',⇡)expiS[',⇡]. (4.1) Z O D D O Z where the integration is over histories of pairs (',⇡), S[',⇡]isthe‘phasespaceaction’ d4x(˙'⇡ )with the Hamiltonian density, and ' ⇡ is a formal Cartesian H H D D productR of Lesbesgue measures at each point in or, equivalently, a Cartesian product of Liouville measures on phase space at each moment of time. For a system with second class constraints1,if(¯', ⇡¯)aretakentobecanonically conjugate coordinates on the constrained phase space, then (4.1) still applies. How- ever, if one uses coordinates (',⇡,Ci)ontheunconstrained (reduced) phase space, one obtains [30]

n 1 ( )= ' ⇡ (C ) det C ,C 2 exp iS[',⇡](4.2) Z O D D i | { i j}| O Z n 1 where the combined factor (C ) C ,C 2 is independent of the way the constraint i |{ i j}| surface is represented by constraint functions Ci.Forasystemwithﬁrstclasscon-

i i straints Fi in which the action takes the form S[',⇡,]=So[',⇡]+ Fi,with Lagrange multipliers, one can write the path integral as [70] R

( )= ' ⇡ exp iS[',⇡,]. (4.3) Z O D D D O Z 1A constraint is called first class if its Poisson bracket with every constraint weakly vanishes. Meaning that the Poisson bracket vanishes if all the constraints are satisfied. Otherwise, a constraint is second class. 44 Alternatively, one may introduce gauge fixing functions ⇠i so that ⇠i and Fi together form a second class set of constraints, in which case (4.2) again applies. If we assume that, e.g., the ⇠i are pure momentum or pure configuration, so that they all Poisson commute, the path integral then takes the form [71]

( )= ' ⇡ n(⇠) det F ,⇠ exp iS[',⇡,]. (4.4) Z O D D D | { i j}| O Z In fact, under very general assumptions (which, however, have yet to be fully proven in the case of gravity) (4.3) and (4.4) are equal upto an inﬁnite constant equal to the gauge volume (see [32,70,72,73], so that they determine the same physics in the manner to be reviewed below. In the rest of this chapter for simplicity we will use expression (4.3). However, all derivations in this chapter can just as well be done starting from expression (4.4) – one need only reinsert the omitted factor n(⇠) det F ,⇠ . | { i j}| For a general constrained theory with both ﬁrst and second class constraints the path integral takes the form:

n n 1 ( )= ' ⇡ (⇠) det F ,⇠ (C ) det C ,C 2 exp iS[',⇡,]. (4.5) Z O D D D | { i j}| i | { i j}| O Z The path integral was originally discovered as a way to write transition amplitudes between states in quantum mechanics. Let O denote a set of phase space functions { i} whose quantum analogues Oi form a complete commuting set. Let (O ,⌫ ) denote { } { i i } the simultaneous eigenstate of the operators O with eigenvalues ⌫ .Thenthe b { i} { i} path integral determines the transition amplitudeb between two such states via

(O0,⌫0) ,U(T 0 T ) (Oi,⌫i) = (Oi0((',⇡)(T 0)) ⌫i0)(Oi((',⇡)(T )) ⌫i) { i i } { } Z i ! ⌦ ↵ Y = ' ⇡ n(⇠) det C ,⇠ exp iS[',⇡,](4.6) D D D | { i j}| Z Oi((',⇡)(T ))=⌫i Oi0((',⇡)(T 0))=⌫i0 where U(T 0 T )isthetimeevolutionoperatorfromt = T to t = T 0.Foratheory with ﬁrst class constraints of which the Hamiltonian is a linear combination — i.e., atimereparametrizationinvarianttheory—thereisnotimeevolutionoperator, 45 and, instead of (4.6), the interpretation of the path integral involves a ‘rigging map’ or ‘group averaging map’ ⌘ [74] from kinematical (unconstrained) states to physical states (states annihilated by the constraints) [2,10,75]:

n ⌘ (O ,⌫ ) ,⌘ (O ,⌫ ) = ' ⇡ (⇠) det Ci,⇠j exp iS[',⇡,]. i0 i0 { i i } phys { } Oi((',⇡)(T ))=D⌫i D D | { }| ZO ((',⇡)(T ))=⌫ ⌦ ↵ i0 0 i0 (4.7) This is the case relevant for us. Note the physical inner product, and therefore ( ), Z O is only physically meaningful up to rescaling by a constant. In the following we will use the notation =todenote“equalityuptorescalingbyaconstant.” By setting =1,oneobtainstheso-calledpartitionfunction.Ofteninthe Ob literature one says that the partition function defines the quantum theory. Of course, what is meant is that the mathematical form or more precisely, the integrand of the partition function defines the quantum theory. One then uses the integrand to integrate against general functions ,theresultsofwhichareusedasabove.Inthis O dissertation, we are explicit about this fact for clarity. The transition amplitude (4.6) or (4.7) defines the dynamics of the theory. This, together with a kinematical quantum framework of states and relevant operators, is enough to allow one to calculate any quantity of physical interest. In the case of GR, canonical loop quantum gravity provides the kinematics. In this case, by choosing the observables Oi appropriately, the canonical states (O ,⌫ ) can be made to be spin- { i i } network states [6], generalized spin-network states [1], or Livine-Speziale coherent states [53]. In all of these cases, the relevant observables Oi are purely geometric,

IJ depending only on the pull-back of Xµ⌫ to the spatial slice.

4.3 PLEBANSKI-HOLST FORMULATION

We will consider both Euclidean and Lorentzian gravity simultaneously, deﬁning s := +1 and s := 1respectivelyinthesetwocases.Recallingthevariablesanddeﬁnitions

46 IJ from section 2.1 the basic variables are an so(⌘)-valued connection !µ and an so(⌘)-

IJ valued two-form Xµ⌫ , the Plebanski two-form. Using these variables and notation, we start from the Plebanski-Holst path integral derived in [31]

() IJ IJ 9 IJ ( )= !µ Xµ⌫ (C) Vs exp i XIJ F (4.8) Z O (II ) D D V O ^ Z ± Z

() where XIJ := (X + 1 ?X)IJ, R+ is the Barbero-Immirzi parameter and the action 2 is an integral over the space-time manifold ,andthe“(II )” subscript indicates M ± that we are restricting the integration to the (II+) and (II )sectors,inwhich 1 XIJ = ✏IJ eK eL. (4.9) ±2 KL ^

The appearing in (4.8) is the space-time volume density and V is the spatial V s volume density determined by eI . For brevity, from now on we will omit the “(II )” µ ± subscript on the path integral (4.8), leaving it understood. It should be noted that the path integral (4.8) is valid only when is purely O IJ geometric (i.e., a function of Xµ⌫ only). For, in the derivation in [31], when the

IJ Henneaux-Slavnov trick [76] is used, a change of variables involving !µ is performed. If the presence of is made explicit throughout the derivation, one sees that if O O IJ depends explicitly on !µ , then the argument leading to (4.8) will change, as will the ﬁnal path integral expression2 However, the restriction of to be purely geometric is O not a real restriction, as, on-shell, ! is uniquely determined by X.Furthermore,the principal application of (4.8) will be to computing the physical inner product between spin-network states, which are eigenstates of X only (see equation (4.7)).

4.4 PURELY GEOMETRIC PATH INTEGRAL

The goal of this section is to integrate out the connection in the expression (4.8). We do this by ﬁrst casting the integral explicitly in Gaussian form and integrating

2 Speciﬁcally, in the ﬁnal path integral, if 'HS denotes the Henneaux-Slavnov canonical transformation, then the ! argument of will need to be replaced by ' !. O HS · 47 out the connection, giving the result in terms of the determinant of the appropriate matrix. The determinant of this matrix is then calculated.

4.4.1 Integrating out the connection

The next step is to evaluate the Gaussian integral:

() () I(X):= !IJ exp i X F IJ = !IJ exp i X (d!IJ + !I !KJ). D µ IJ ^ D µ IJ ^ K ^ Z Z Z Z Because, in the path integral (4.8), X is constrained to be of the form (4.9), we assume X to be of this form throughout the section. We begin by noting that if we deﬁne () ↵ 1 ↵⇢ a IJ KL := ✏ X⇢[K [I ⌘J] L] 2 | | h ih i and 1 () b↵ := ✏↵⇢(dX ) . IJ 3! IJ ⇢ the action becomes

4 ↵ IJ KL ↵ IJ SBF [X,!]= d x a IJ KL !↵ ! + bIJ!↵ (4.10) Z ⇣ h ih i ⌘ so that I(X)= ! exp i d4x a ↵ !IJ!KL + b↵ !IJ . D IJ KL ↵ IJ ↵ Z Z ⇣ h ih i ⌘ This casts the integral in the explicit Gaussian form in equations (B.1-B.2) in appendix B. Using the result (B.4), one has that, modulo an overall X-independent constant,

1 I(X)=(det a) 2 exp iS[X](4.11) where S[X]:=SBF [X,![X]] with b![X] the unique connection determined by X via the equation of motion of BF theory found by varying !,

IJ IJ [I K J] d X =dX +2![X] X| | =0. ![X] K ^

48 Because X is of the form (4.9), ![X] is furthermore the unique Lorentz spin-connection determined by the tetrad e,i.e.,suchthat

I d!e =0, so that the action reduces to the Holst and Palatini actions

S[X] S [X,![X]] S [e, ![e]] = S [e, ![e]] ⌘ BF ⇡± Holst ± Palatini where denotes equality when X is of the form (4.9), and the last equality holds ⇡ because, when ! = ![X], the extra ‘topological’ term added to the Palatini action to give the Holst action vanishes due to the Bianchi identity [39]. In order to use equation (4.11) in the path integral, it remains to calculate the determinant of a.

4.4.2 The determinant of a

We have

() ↵ 1 ↵⇢ a IJ KL = ✏ X⇢[K [I ⌘J] L] 2 | | h ih i 1 ↵⇢ M N 2s = ✏ e e⇢ ✏MN[K [I + e[K e⇢[I ⌘J] L] 2 | | | ✓ ◆ ↵ PQMN 2s ↵ PQ = V eP eQ✏ ✏MN[K [I + eP eQ✏ [K [I ⌘J] L] 2 | | | ✓ ◆ ↵ [Q P ] s PQ = V eP 2 ⌘J][L + ✏ [K [I ⌘J] L] eQ  [I K] | | ✓ ◆ ↵ M N [Q P ] s PQ R S = V eP [I J] 2 ⌘N][S + ✏ [R [M ⌘N] S] eQ[K L]  [M R] | | ✓ ◆ ⇣ ⌘ so that if we deﬁne the 24 by 24 matrices

↵ MN ↵ M N E IJ P := eP [I J] h ih i and

P Q [Q P ] s PQ ˚a MN RS := 2 ⌘N][S + ✏ [R [M ⌘N] S], [M R] | | h ih i 49 the above becomes

a ↵ = V E ↵ MN ˚a P Q E RS . IJ KL  IJ P · MN RS · KL Q h ih i h ih i h ih i h ih i Thus,

24 det a = V (det E)2(det˚a)  ✓ ◆ = 24(det E)2 V where the fact that det˚a is constantb on the space of histories has been used3. Finally,

↵ because E is block diagonal with six copies of the block eP ,onehas

↵ 6 6 det E =(dete ) = P V whence det a= 12. V b 4.4.3 Final continuum path integral

Using this expression in (4.11) and substituting that into (4.8) yields the ﬁnal pure geometric continuum path integral

( )= XIJ(C) 3V exp i X F [![X]]IJ. (4.12) Z O D µ⌫ V sO IJ ^ Z Z We wish to note thatb this expression derives from the full canonical path integral, with primary as well as secondary simplicity constraints imposed, and with the full requisite determinant factors. The only assumption necessary in the derivation is that does not explicitly depend on !.Inspin-foamsums,eachspin-foamislabeledby O quantum numbers which are a discrete analogue of precisely the two-form X,making the above expression ideally suited for use with spin-foams.

3 I det˚a is furthermore non-zero. This is clear from the fact that when e↵ is non-degenerate, so is a, which is equivalent to the usual fact that the equations of motion obtained by varying the action (2.4),(4.10) with respect to ! determines ! uniquely.

50 4.5 ADM PATH INTEGRAL

As a secondary derivation of (4.12) we start from the ADM formalism. Because one has no second class constraints in the ADM formalism, one doesn’t need to use the Henneaux-Slavnov trick [76]. As a consequence, this derivation can also be thought of as a ‘check’ on the use of the Henneaux-Slavnov trick in this case.

4.5.1 Integrating out the momentum

The non-gauge-ﬁxed canonical path integral in the ADM formalism in terms of the

ab canonical variables (hab,⇡ )is

( )= h ⇡ab N N a exp i d4x(⇡abh˙ [h, ⇡,N, N~ ]) (4.13) ZADM O D abD D D O ab HADM Z Z where recall a, b, c . . . denote spatial indices. Here N and N a are lapse and shift lagrange multipliers and is the Hamiltonian density given by [77] HADM 1 (3) 1 ab 1 1 2 ab = h 2 N( R+h ⇡ ⇡ h ⇡ )+2⇡ D N , HADM ab 2 a b a (3) where h := det hab, ⇡ := ⇡ a,and Rdenotesthescalarcurvatureofhab.The integral (4.13) can be written

( )= h ⇡ab N N a exp i d4x(A ⇡ab⇡cd + B ⇡ab + C)(4.14) ZADM O D abD D D O ab,cd ab Z Z where A, B and C are 1 Nh 2 A := (h h + h h h h ), ab,cd 2 ac bd ad bc ab cd B := h˙ 2D N ab ab (a b) and

1 (3) C := Nh2 R.

We assume does not depend explicitly on ⇡,sothat(4.14)isagainoftheform O (B.1-B.2) in appendix B, and the integration over ⇡ab yields

a 1/2 4 ab ( )= h N N det A exp i d x(⇡ [h, N, N~ ]h˙ [h, ⇡[h, N, N~ ]]) ZADM O D abD D | | O ab HADM Z Z (4.15) b 51 ab ab a where ⇡ [h, N, N~ ]isthevalueof⇡ determined by hab, N,andN via the appropriate equation of motion of the Hamiltonian theory. Substituting the explicit expression for ⇡ab[h, N, N~ ]into(4.15)yields

a 1 4 ( )= h N N det A 2 exp i d x . (4.16) ZADM O D abD D | | O LADM Z Z where b := Nph(K Kab K2 +(3) R) LADM ab is the usual ADM Lagrangian with

1 1 K := h = (h˙ D N D N ) ab 2Ln ab 2N ab a b b a the usual extrinsic curvature. It remains to ﬁnd the determinant of A.

4.5.2 The determinant of A

e e f By the symmetry of hab,thereexistsanorthogonalmatrixOa such that Oa Ob hef =

aab for some a ,sothat

e f g h N A0 := Oa Ob Oc Od Aef,gh = [abacbd + abadbc acabcd]. ab,cd 2ph

Now because rows (ab)=(12),(13)and(23)inA0 each have just one non zero element

(A120 ,12, A130 ,13 and A230 ,23, respectively) the determinant of the above equation yields

A120 ,12A130 ,13A230 ,23 det Q =detA where Q := A . Using A = N , A = N and A = N ab aa,bb0 120 ,12 2ph 1 2 130 ,13 2ph 1 3 230 ,23 2ph 2 3 gives N 3 det A = ( )2 det Q. p 1 2 3 ✓2 h◆

52 Furthermore,

2 A11,11 A11,22 A11,33 1 12 13 3 0 1 N 0 1 det Q =det A A A = det 2 22,11 22,22 22,33 p 1 2 2 2 3 B C ✓2 h◆ B C B C B 2 C B A33,11 A33,22 A33,33 C B 13 23 3 C B C B C @ N 3 A @ A = 4 ( )2 p 1 2 3 ✓2 h◆ So that, 6 6 6 N 4 N 4 N det A = 4(123) = h = h p 4 p 6 24 ✓2 h◆ 2 ( h) where 123 =dethab has been used. Substituting this into (4.16) gives

a 3 1 4 ( )= h N N N h 2 exp i d x . (4.17) ZADM O D abD D O LADM Z Z b 4.5.3 Space-time covariant variables

a It remains to change from the variables (hab,N,N )tothespace-timecovariantvari- able gµ⌫,wehave

g = g g g = det J h N N a D µ⌫ D abD 00D 0a | |D abD D where

@(gab,g00,g0a) J := c . @(hcd,N,N ) Using the relations

g = h g = N 2 + h N aN b g = h N b ab ab 00 ab 0a ab it is easy to check that det J = 2hN so that g =2hN h N N a. D µ⌫ D abD D

53 Equation (4.17) thus yields4

4 3 ( )= g N h 2 exp iS . (4.18) ZADM O D µ⌫ O ADM Z b 4.5.4 Comparison with the result of the last section

Our goal in this section is to compare the ADM measure in (4.18) to the ﬁnal continuum path integral (4.12) derived in section 4.4.1

( )= XIJ(C) 3V exp i X F [![X]]IJ. Z O D µ⌫ V sO IJ ^ Z Z Since these two expressionsb have di↵erent variables, again a change of variables is necessary. As discussed in [31], one can perform a change of variables XIJ (eI ,C), µ⌫ ! µ 5 IJ I from Xµ⌫ to the tetrad eµ together with the simplicity constraints. The resulting change in measure is given by [31]

IJ 6 I X = e C D µ⌫ V D µD so that b I 3 IJ ( )= e C(C) V exp i X F [![X]] . Z O D µD V sO IJ ^ Z Z Next, we change fromb tetrad variables to metric variables. This can be done by ﬁxing I an arbitrary reference tetrad ˚eµ[g]foreachmetricgµ⌫,anddeﬁningalocalgauge

I transformation ⇤ J via

I I I eµ =⇤ J˚eµ[g].

4Superficially the measure in (4.18) seems to di↵er from that in equations (1.3), (2.23) of [78] by a single power of the lapse. However, in order to compare (4.18) with the path-integral (1.3) in [78], one must include gauge fixing in (4.18). In doing this, one inserts the canonical [71] Faddeev- Popov determinant, appearing in equation (4.4), appropriate for the ADM formulation (note that this determinant depends on the scaling of the constraints in the particular canonical formulation used). By contrast, the measure (2.23) in [78] is intended to be used in (1.3) of [78] with a di↵erent, covariantly defined Faddeev-Popov determinant J[g]. These two determinants di↵er by precisely that single power of the lapse needed to make the total measure here and in [78] the same, as can be deduced from the following equations in [78]:(1.5) and those between (2.28) and (2.31) inclusive. 5The indices for C are suppresed here for simplicity. Here C is defined as (2.2) whereas in [31] ab C is defined in terms of three parts (C , C˜, C˜⇡). The formal Lebesgue measure and Dirac delta functions of these two formulations of the simplicity constraint are related by an irrelevant constant Jacobian. 54 I At each space-time point, ⇤ J is an element of either the 4-dimensional Euclidean group or the Lorentz group, depending on s. The change of variables (eI ) (g , ⇤ J ) µ ! µ⌫ I yields the change of measure [79]

g ⇤ J =pg eI D µ⌫D I D µ

J I where ⇤I is the measure deﬁned in [79].b Because the integral ⇤ J is indepen- D D dent of the metric, it can be dropped from the path integral, leavingR

4 IJ ( )= Dg C(C)V V exp i X F [![X]] . Z O µ⌫D sO IJ ^ Z Z 1 1 Using the relations bV = h 2 , = g 2 = NV and the fact that C(C)=1gives s V s D R 4 3 ( )= g N h 2 exp iS Z O D µ⌫ O ADM Z which is the same as (4.18). b

4.6 RESULTS AND CONCLUSION

In this chapter, we calculated the path integral measure factor for the Plebanski-Holst formulation of gravity with the Lorentz connection degrees of freedom integrated out. The calculation is at the continuum level and it is done in two independent ways to double-check the results. First, starting from the measure factor calculated in [31] for the Plebanski-Holst formulation with the Plebanski two-form and Lorentz connection variables, we integrate out the connection degrees of freedom. The integration is a generalized gaussian integral and one needs to calculate the determinant of the coecient of the quadratic term. Second, we start from the ADM formulation of gravity. In this formulation the variables are the 3-metric and its conjugate momentum plus lapse and shift Lagrange multipliers. Then we integrate out the momentum degrees of freedom. Because these two formulations have di↵erent variables, in order to compare the results one needs to do a change of variables. After incorporating 55 the jacobian factors for these change of variables the results from the two methods match. Some powers of the 3 and 4 dimensional volumes appear in the measure factor. These factors were also present in the measure factor for Plebanski-Holst before the integration [31] but with a di↵erent power of 4-volume. Specially the appearance of the 3-volume is interesting since it breaks the manifest general covariance of the path integral o↵shell. For a discussion on this issue I refer you to [31]. The equivalence of the results from the two formulations was expected. It has been shown by careful analysis in [29] that if one uses the Liouville measure on the reduced phase-space, the path integral formulation is formally equivalent to the canonical theory regardless of what formulation of general relativity is used since they all have the same reduced phase-space. In order to implement this factor in spin-foams these factors need to be discretized and quantized. The next step is to discretize this measure on a spin-foam cell complex, expressing it directly in terms of spins and intertwiners.

56 APPENDICES

57 APPENDIX A COHERENT STATES

A.1 SU(2) COHERENT STATES

The irreducible representations of the group SU(2), are labeled by a non-negative Hj integer or half-integer spin j.Thebasisofstatesj, m are simultaneous | i2Hj eigenstates of a complete set of commuting operators J 2,J ,namelythesquare { z} 2 2 2 2 of the total angular momentum J = Jx + Jy + Jz and the angular momentum in the z-direction Jz,withtheeigenvalues:

J 2 j, m = j(j +1)j, m | i | i

J j, m = m j, m z| i | i with m = j, j +1,...,j 1,j. The components of the angular momentum operator satisfy the commutation relation

[Jx,Jy]=iJz and the dispersion or standard deviation in the angular momentum is given by

(J)2 = J 2 J~ 2 = j(j +1) m2 (A.1) h ih i where the expection values are deﬁned by J~ = j, m J~ j, m and J 2 = j, m J 2 j, m . h i h | | i h i h | | i Clearly, for m = j this dispersion is minimized. ± Coherent states are obtained by searching for the most classical-like states. In particular, one seeks to minimize the Heisenberg uncertainty relations

1 AB [A, B] 2|h i| for appropriate pairs of operators A and B.Inthepresentcase,itisnatural[80]to ﬁrst consider states that satisfy the minimum uncertainty relation

1 J J = J (A.2) x y 2|h zi| 58 of the states j, m ,anditscyclicpermutations.Thesearethestateswitheither | i maximum or minimum weight j, j .Thesealsominimizethedispersionrelation | ± i (A.1). Furthermore, from j, j J~ j, j = jzˆ h | | i one can see that j, j is a coherent state corresponding to J~ = jzˆ,thatis,theˆz | i direction. The states corresponding to an arbitrary direction deﬁned by a unit vector ~n on the sphere S2 can be obtained up to a phase from the coherent states in the z-direction by rotating the states by a group action

j, ~n = g(~n ) j, j . | i | i where g(~n ) SU(2). The state j, ~n minimizes the appropriately rotated version of 2 | i the uncertainty relations (A.2). The group variables can equivalently be labeled by a

2 normalized spinor ⇠ C associated to unit vectors ~n ⇠ by 2

⇠0 ⇠¯1 g(~n )= ⇠ 0 1 ⇠1 ⇠¯0 @ A where ⇠0,⇠1 C. 2 For calculation purposes it is easier to use the realization of in terms of homo- Hj 1 geneous functions on CP [44]. In terms of this realization, coherent state j, ~n⇠ is | i given by the function j dj ¯ 2j C⇠ (z)= ⇠,z r ⇡ h i where w, z =¯w z +¯w z is the Hermitian inner product and d =2j +1. The h i 0 0 1 1 j inner product between these states is deﬁned by

j j j j (C (z),C (z)) = C (z)C (z)⌦z ⇠ ⇠0 ⇠ ⇠0 1 ZCP with integration measure ⌦ = i (z0dz1 z1dz0) (¯z0dz¯1 z¯1dz¯0). z 2 ^ 59 A.1.1 Livine-Speziale Coherent intertwiner

In this dissertation we have used spin-foam models to compute a single 4-simplex transition amplitude (see chapter 2 for more details). A 4-simplex has ﬁve tetrahedra

⌧a (a =1,...,5) on the boundary. We want the coherent states to be peaked on the geometry of a classical Euclidean tetrahedron. Associate an SU(2) coherent state j ,~n in the spin representation j to each triangle t (a, b =1...5anda

jab~n b =0. b=a X6 Assign to each Euclidean tetrahedron a Livine-Speziale coherent intertwiner [53]

(~n ,...,~n )= dg g j ,~n j ,~n j ,~n j ,~n a 1 4 | 1 1i⌦| 2 2i⌦| 3 3i⌦| 4 4i ZSU(2) which is an SU(2) invariant tensor product of the representations of its triangles. Here g represents both an SU(2) group element as well as its action on representations.

A.1.2 Lorentzian case

The Lorentzian coherent spin network states j , ⌥ (~n ) are labeled by a Lorentzian | ab a i coherent intertwiner ⌥a for each tetrahedron ⌧a and a spin jab for each triangle tab on the boundary of a Lorentzian 4-simplex, and is deﬁned in terms of Euclidean coherent intertwiner and Euclidean coherent state j , (~n ) by [45]: a | ab a i j , ⌥ (~n ) =exp( i ⇥ j ) j , (~n ) . (A.3) | ab a i ab ab | ab a i Xa

In search for the most classical like states in quantum mechanics, extrinsic coherent states were derived. These semiclassical states are peaked on both conﬁguration variables q and their conjugate momenta p so they identify a point in phase space. Because these states have good semiclassical properties, they are useful in asymptotic analysis of the vertex amplitude and make the geometric picture of the theory more clear. Lets ﬁrst derive them for a general quantum mechanical system then extend the method to our case of interest, spin network states.

A.2.1 General case

Consider a general quantum mechanical system with canonical variables (q, p)that

~ satisfy the Heisenberg uncertainty relation qp > 2 where the deviation is deﬁned by 2 = (ˆq qˆ )2 = qˆ2 qˆ 2. q h h i i! q h ih i p ~ The classical like states satisfy the minimum uncertainty condition qp = 2 .The states with minimal uncertainty satisfy

(ˆq qˆ ) ↵ = i(ˆp pˆ ) ↵ (ˆq + ipˆ) ↵ =(qˆ + i pˆ ) ↵ . (A.4) h i | i h i | i! | i h i h i | i

Consequently, the wave function = q ↵ takes the form q h | i 1 (q qˆ )2 i pˆ (q qˆ ) = exp[ h i + h i h i ] (A.5) q 2 2 2⇡q 4q ~ which is a Gaussian wavep packet peaked at ( qˆ , pˆ ). These extrinsic coherent states h i h i or semiclassical states ↵ are not eigenstates of conﬁguration variableq ˆ or their | i conjugate momentump ˆ.Instead,theyareeigenstatesoftheannihilationoperator

aˆ :=q ˆ+ ip.ˆ

61 A.2.2 Spin networks

The classical action that spin-foams are peaked on is the Regge action. From the boundary Regge action one concludes that the conﬁguration variables and their conjugate momenta are the spins jl and dihedral angles ⇥l:

@SR SR(jl,~nl,~nl0)= jl⇥l(jl,~nl,~nl0) =⇥l. ! @(jl) Xl The asymptotics of the spinfoam vertex amplitude has been studied in [33,34] using the spins and normals representation. In this representation each link of the graph is labeled with a spin jl and two unit normals ~n l and ~n l0 for the source/target nodes of the link. This representation diagonalizes the area operator;

ˆ 3 Al jl,~nl,~n0 =8⇡~Gc jl(jl +1) jl,~nl,~n0 | l i | l i p j 1 A l j l ⇡ l as a result the spread on its conjugate momentum ⇥l is maximum. To deﬁne spin network states that are peaked on both these variables, in analogy with (A.5) Rovelli proposed semiclassical boundary states that are a superposition of spin network states with coecients given by a gaussian times a phase [46]:

(ab)(cd) jab (j0)ab jcd (j0)cd ab j0,0 =exp ↵ i 0 (jab (j0)ab) 0 (j0)ab (j0)cd 1 (abX),(cd) Xab @ p p (A.6)A ab where (j0)ab and 0 are the background spins and the simplicial extrinsic curvature where is peaked, and ↵(ab)(cd) is a 10 10 matrix assumed to be complex j0,0 ⇥ ab with positive deﬁnite real part. The background values ((j0)ab,0 )correspondto the geometry of the boundary of a 4-simplex where the (j0)ab give the areas of the

ab triangles and 0 are the dihedral angles between the tetrahedra.

62 APPENDIX B INTEGRATION OF A PATH INTEGRAL WITH QUADRATIC ACTION

Let ( , )beasymmetric,non-degenerate,butnotnecessarilypositivedeﬁnite,inner · · product on some real vector space V .Letˆa be an invertible operator on V symmetric with respect to ( , ), b an element of V ,andc arealnumber.Considertheaction · ·

S[v]:=(v, avˆ )+(v, b)+c (B.1) and the path integral veiS[v] (B.2) D Z where v is any translation-invariant measure on V (unique up to rescaling). If one D ﬁxes a basis on V and uses components with respect to this basis as coordinates on V , v will just be a real number times the Lebesgue measure in the chosen coordinates. D From the complex analytic continuation of the usual formula for the Gaussian integral, one obtains

iS[v] 1/2 i 1 ve = C(deta ˆ) exp (b, aˆ b)+c (B.3) D 4 Z ✓ ◆ where the constant C depends exclusively on the relative scaling of the inner product ( , )andthemeasure v. Variation of the action gives · · D

S =(v,avˆ )+(v, aˆv)+(v, b)=(v,2ˆav + b). |v=v0

Setting this equal to zero for all v yields the following expression for the extremum v0 of S[v]:

1 1 v = aˆ b. 0 2

i 1 Substituting this into (B.1) directly gives S[v ]= (b, aˆ b)+c,sothat(B.3)can 0 4 be written

iS[v] 1/2 iS[v0] ve = C(deta ˆ) e (B.4) D Z 63 with v0 the unique extremum of S[v]andC depending exclusively on the relative scaling of ( , )and v. · · D

64 APPENDIX C EQUIVALENCE OF GAUGE-FIXED AND NON-GAUGE-FIXED PATH INTEGRALS

In this appendix, we address the equivalence of the path integrals (4.3) and (4.4). The argument used here is based on the proof for the Yang-Mills case given by Faddeev and Popov [70]. A version of this argument is also given in [73] and [72]. However, here we keep the argument more general and give more details.

C.3 THE ARGUMENT

Consider a system with ﬁrst class constraints C ,generatingagaugegroup on shell, i G i and an action of the form S[⇣,]=So[⇣]+ Ci,where⇣ is short hand for a set of canonically conjugate variables (',⇡), and Rare Lagrange multipliers. We start with (4.3)

( )= ⇣ [⇣]expiS[⇣,]= ⇣(C(⇣)) [⇣]expiS [⇣], (C.1) Z O D D O D O o Z Z where ⇣ := ' ⇡, and where, for this appendix, we assume is gauge invariant. D D D O Faddeev and Popov in their original paper [70] almost start with this same path integral, the only di↵erence being that here we use phase space variables, which is the more fundamental starting point from the canonical perspective [30,71]. The Faddeev-Popov strategy [70] is to factor out the divergences in the path integral due to the integration over the gauge group. We here adapt their argument to the general path integral (C.1) as follows. First choose gauge-ﬁxing functions ⇠j = ⇠j(⇣)whichare regular, that is, have non-vanishing gradient, at ⇠ 0. The ⇠ then form a good set j ⌘ j of coordinates on each gauge-orbit in a neighborhood of ⇠ 0. Furthermore, given j ⌘ any coordinates ↵ g(↵) on the gauge group ,andanyphasespacepoint i 7! 2G G ⇣ =(,⇡), one can deﬁne another set of coordinates on the gauge orbit containing ⇣

65 via ↵ g(↵) ⇣.Onethenhas i 7! · @⇠j(g(↵) ⇣) ↵(⇠(g(↵) ⇣)) det · = ⇠(⇠)=1 D · @↵i D Z Z which can be inserted into the path integral (C.1) to obtain @⇠ (g(↵) ⇣) ( )= ⇣ ↵(C(⇣))(⇠(g(↵) ⇣)) det j · [⇣]expiS [⇣]. (C.2) Z O D D · @↵i O o Z We next perform the change of variables ⇣ ⇣0 := g(↵) ⇣. As ⇣ g(↵) ⇣ is a 7! · 7! · canonical transformation, and d⇣ = d'd⇡ is the Liouville measure, we have d⇣ = d⇣0.

Furthermore, S [⇣]=S [⇣0]and [⇣]= [⇣0]astheaction(andhencetheaction o o O O with constraints satisfied, S ) and [⇣]aregauge-invariant.Itremainstoconsider o O @⇠j (g(↵) ⇣) · the constraint factor (C(⇣)), and the determinant factor det @↵i .Inorder i to facilitate calculation, we now make a specific choice for g(↵): For each ↵ ,we define g(↵): to be the Hamiltonian flow generated by ↵iC ,evaluatedatunit ! i parameter time. Equivalently, g(↵)maybedefinedbytheequations

d ↵ C, f ˜ = f(g( t↵) ⇣˜) , (C.3) { · }|⇣ dt · t=0 g(~0) = id

If we let X↵ C denote the derivative operator X↵ C f := ↵ C, f ,thentheequations · · { · } above implies the explicit expression

˜ 1 1 n f(g( ↵) ⇣)= (X↵ C ) f ⇣˜ exp (X↵ C ) f ˜ . · n! · | ⌘ · |⇣ n=0 X The constraint factor (C(⇣)) can be rewritten (g( ↵)⇤C(⇣0)), where ⇤ denotes pull- back. Because the ﬂow generated by ﬁrst class constraints is always tangent to the constraint surface, g( ↵)⇤C will again be a linear combination of the constraints, so i that

j g( ↵)⇤C =: µ(↵) C i i j j for some matrix µ(↵)i of functions on phase space, whence

1 (g( ↵)⇤C(⇣0)) = det µ(↵) (C(⇣0)). (C.4) | | 66 Turning now to the last remaining factor, we have

@⇠j(g(↵) ⇣) d d · = ⇠ (g(↵ + t ) ⇣) = ⇠ (g(↵ + t )g( ↵) ⇣0) @↵i dt j i · dt j i · t=0 t=0 1 d = ds ⇠j(g(s↵)g(ti)g( s↵) ⇣0) dt · Z0 t=0 1 1 d = ds (g(s↵)⇤⇠j)(g(ti)g( s↵) ⇣0) = ds Ci,g(s↵)⇤⇠j g( s↵) ⇣0 dt · { }| · Z0 t=0 Z0 1 1 k = ds g( s↵)⇤Ci,⇠j ⇣0 = dsµ(s↵)i Ck,⇠j ⇣0 (C.5) { }| { }| Z0 ✓Z0 ◆ j j where (i) := i . Here the definition of the partial derivative has been used in the first line, equation (C.6) from the next subsection has been used in the second line, equation (C.3) has been used in the third line, and the invariance of the Poisson bracket under the gauge transformation g(s↵)hasbeenusedinthelastline.Define

j 1 j M(↵)i := 0 dsµ(s↵)i .Takingthedeterminantof(C.5)thenyields R @⇠ (g(↵) ⇣) det j · =detM(↵) det C ,⇠ . @↵i |⇣0 { i j}|⇣0 ✓ ◆ Using this and equation (C.4) in equation (C.2) gives us

det M(↵) ( )= ⇣0 ↵ (C(⇣0))(⇠(⇣0)) det C ,⇠ [⇣0]expiS [⇣0] Z O D D det µ(↵) | { i j}|⇣0 O o Z Z ⇣0 ! det M(↵) = ⇣0 ↵ (⇠(⇣0)) det C ,⇠ [⇣0]expiS[⇣0,]. D D D det µ(↵) | { i j}|⇣0 O Z Z ⇣0 ! At this point, all dependence on ↵ is restricted to the inner integral in parentheses, det M(↵) which can be thought of as a “gauge orbit volume,” with det µ(↵) acting as a ⇣0 “volume element”. If we can show that this gauge orbit volume is independent of ⇣0, then we can drop it as an overall constant, thereby proving the equivalence of the non-gauge-fixed (C.1, 4.3) and gauge-fixed (4.4) path integrals. In the case of gravity, or any other theory with non-compact gauge orbits, this gauge orbit volume is infinite, so it is not clear what it means to be constant on phase space. In a moment, we will show that, in the case when the algebra of constraints under consideration has structure constants,thenatleastthegaugeorbitvolume 67 element is constant on phase space. Assuming that the ranges of the coordinates ↵i on each gauge orbit are then also constant, this implies that the fully integrated gauge orbit volume itself is also constant, as required for equivalence.

det M(↵) The proof of the constancy of the gauge orbit volume element, det µ(↵) ,when there are structure constants, is straight forward. We have

1 n j s µ(s↵) C := g( s↵)⇤C = ↵ C, ↵ C, . . . ↵ C, C ... , i j i n! { · { · { · i} } } n=0 X where in each term there are n nested Poisson brackets, with the n =0termbeing just Ci.Ifonehasstructureconstants,eachPoissonbracketintroducesmultiplication by a matrix which is constant on phase space. The product of these matrices, summed

j over n,isthenequaltothematrixµ(s↵)i on the left hand side, which is therefore also

j det M(↵) constant on phase space, leading to M(↵)i ,andhencealso det µ(↵) being constant on phase space. To handle the case of structure functions, which is in the case relevant for gravity, one must look not only at the ‘gauge orbit volume element’, but also at the fully integrated ‘gauge orbit volume’. As this volume is infinite, as already mentioned, it is not so clear what it means for it to be constant on the phase space. A better understanding of how to handle this infinite volume through an appropriate regulator, or experimentation with toy models with structure functions in which the gauge volume is finite, could shed light on how to extend the last step of this proof of equivalence to systems of interest with structure functions.

C.3.1 A result for non-Abelian gauge groups

In this subsection we prove equation (C.6) below, which has been key in the above subsection. We begin by proving a general result for linear operators, which we then apply to the relevant case at hand.

68 Lemma 1. For any two linear operators A and B on any vector space V ,

1 d A tB A sA sA e e = dse ( B)e . dt t=0 Z0 Proof. Let I := d [exp A exp( tB A)] .Wethenhave dt t=0 1 @2 1 @2 1 ( tB sA)n I = ds [s exp(sA)exp( tB sA)] = ds s exp(sA) . 0 @s@t t=0 0 @s@t " n! # Z Z n=0 t=0 X Using the Leibniz rule to evaluate the derivative with respect to t,oneobtains n 1 1 @ 1 n m m 1 I = ds s exp(sA) ( sA) ( B)( sA) @s n! Z0 " n=0 m=1 # X nX 1 1 n @ ( s) n m m 1 = ds exp(sA) A BA @s n! 0 " n=0 m=1 # Z X X 1 1 1 n n n 1 ( s) n m m 1 ( 1) s n m m 1 = ds exp(sA) AA BA + A BA . n! (n 1)! 0 " m=1 n=m # Z X X ✓ ◆ Notice that we have switched the order of summation in the last line and used the fact that A exp(sA)=exp(sA)A.Simplifyingfurther,

1 1 1 n n 1 ( s) n m+1 m 1 ( s) n m m 1 I = ds exp(sA) A BA + A BA . n! (n 1)! 0 " m=1 n=m # Z X X ✓ ◆ In this expression, the sum over n telescopes, so that all terms in the sum cancel except the second term in the n = m case. One is thus left with

1 1 m 1 1 ( s) m 1 I = ds exp(sA) BA = exp(sA)B exp( sA)ds. (m 1)! 0 " m=1 # 0 Z X Z ⌅

Proposition 2.

d 1 d f(g(↵ + t)g( ↵)⇣) = ds f(g(s↵)g(t)g( s↵)⇣) (C.6) dt dt t=0 Z0 t=0

69 Proof. d d f(g(↵ + t)g( ↵)⇣) = exp X( ↵ t) C f (g( ↵)⇣) dt dt · t=0 t=0 ⇥ ⇤ d = exp (X↵ C )exp X( ↵ t) C f (g(⇣) dt · · t=0 ⇥ ⇤ d = [exp (X↵ C )exp( X↵ C tX C )]t=0 f dt · · · ⇣ 1 = ds [exp (sX↵ C )( X C )exp( sX↵ C )] f · · · Z0 ⇣ 1 d = ds [exp (sX↵ C )exp( tX C )exp( sX ↵ C )] f dt · · · Z0 t=0 ⇣ 1 d = ds [exp (Xs↵ C )exp(X t C )exp(X s↵ C ) f](⇣) dt · · · Z0 t=0 1 d = ds f(g(s↵)g(t)g( s↵)⇣) dt Z0 t=0 where, in the fourth line, lemma 1 was used in the case where V is the space of functions on , A = X↵ C ,andB = X C . · · ⌅

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