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Article: Gielen, S. orcid.org/0000-0002-8653-5430, Oriti, D. and Sindoni, L. (2014) Homogeneous cosmologies as group field theory condensates. Journal of High Energy Physics, 2014 (6). 13. ISSN 1126-6708 https://doi.org/10.1007/jhep06(2014)013

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[email protected] https://eprints.whiterose.ac.uk/ JHEP06(2014)013 Springer ty, Lattice May 1, 2014 June 3, 2014 rmalism for : : January 27, 2014 : Accepted Published , atural extension of the 10.1007/JHEP06(2014)013 t quantum gravity mod- Received act an effective dynamics to be (approximate) solu- ondensate states to include aining the correct coupling signature to Lorentzian sig- doi: osmological Wheeler-DeWitt ns under which the dynamics uantum theory. The resulting in Bose-Einstein condensates. atial curvature. We show how states used in the description general programme for extract- opic non-perturbative theory of n the case of a kinetic term that roscopic, spatially homogeneous s, this approximation reproduces ons for the ‘condensate wavefunc- stein Institute), b Published for SISSA by [email protected] , and Lorenzo Sindoni b . 3 1311.1238 Daniele Oriti The Authors. a c Cosmology of Theories beyond the SM, Models of Quantum Gravi

We give a general procedure, in the group field theory (GFT) fo , [email protected] Am 1, M¨uhlenberg 14476 Golm, Germany E-mail: Max Planck Institute for Gravitational Physics (Albert Ein Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada [email protected] b a Open Access Article funded by SCOAP Abstract: Steffen Gielen, Homogeneous cosmologies as groupcondensates field theory quantum gravity, for constructing states thatuniverses. describe mac These statesof are close Bose-Einstein to condensates. coherenttions (condensate) to The the condition quantum equationsfor on of homogeneous such motion cosmologies of states directly GFT fromdescription is in the used general underlying to gives q extr nonlineartion’ and nonlocal which equati are analogousWe to show the the Gross-Pitaevskii general equation els, form as well of as thebecomes some effective concrete linear, equations examples. admitting for We anequation, identify curren and interpretation conditio give as its semiclassical a (WKB)includes quantum-c approximation a i Laplace-Beltrami operator.the For classical isotropic Friedmann state equation inthe vacuum formalism with can positive sp benature consistently extended models, from and Riemannian discussof the a addition massless of scalarGFT matter action. in fields, We the also obt Friedmann outlinecosmological the perturbations. equation procedure from Our for results the extending forming our the most effective basis c cosmological n of dynamics a directlyquantum from gravity. a microsc Keywords: Models of Gravity ArXiv ePrint: JHEP06(2014)013 ] 1 4 1 51 53 23 26 27 30 36 42 48 55 57 59 61 14 19 31 51 or existing e ctive large-scale description, hematically consistent. quantum gravity is the deriva- to be relevant for observation eedom presumably relevant at om the similarly fundamental are not directly relevant, must ith the standard model of par- ose done recently in WMAP [ SU(2) – 1 – / ) C , ]. It should also suggest new phenomena or new explanations f 2 5.3 Effective modified Friedmann equation: Lorentzian case 6.1 Adding matter:6.2 a scalar Perturbations/inhomogeneities field 4.3 Correlation functions 4.4 Condensate states as exact GFT vacua? 5.1 Simplicity constraints 5.2 Effective modified Friedmann equation: a concrete exampl 4.1 Self-consistency conditions4.2 GFT condensates vs. coherent and squeezed states observations. This challenge,challenge of of showing course, that is the proposed independent theory fr is in itself mat and Planck [ tion, from a fundamental theorythe describing Planck the scale, degrees of ofand effective fr for physics the at connection scales todescribing large other physical areas enough regimes of physics. wherebe quantum-gravity Any such effects consistent effe with theticle predictions physics of and General with Relativity, cosmological w observations such as th C The homogeneous space SL(2 1 Introduction One of the central challenges faced by any proposed theory of 7 Conclusions A Regularisation of Lorentzian models B Dynamics of the Bianchi IX model 6 Beyond vacuum and homogeneity: matter and perturbations 5 Effective cosmological dynamics 3 Approximate geometries and homogeneity 4 GFT condensates as continuum homogeneous geometries Contents 1 Introduction 2 Group field theory JHEP06(2014)013 el, in ation, spacetime d ], at least if one s create and anni- 12 – 3 etries, one faces two riate states within the his task is particularly ith what we know, this reconstruction procedure ) formalism for quantum s in the quantum theory, f a generic procedure that ions [ 1)-simplices in pin network states of loop al description of spacetime ng discrete geometric data. imately smooth geometries amental degrees of freedom, ld. While a finite number of p to possible corrections at he GFT Fock space in terms undamental vacuum state is ut the reconstructed metric, gths (which are necessary to semiclassical to be associated − s spacetime. This picture can c quantum dynamics of atoms a quantum fluid. The picture pproximate sense. inuum field on a differentiable as in the covariant formulation d st two kinds of approximations, ssue is at least as important as ( ral Relativity are thought of as at least in an approximate sense) ing the same data in loop quan- s are concerned, as shown already in connection) or simple bivectors of elementary GFT quanta which make up a – 2 – such building blocks to an approximate continuum N condensate ]. We use the physical intuition from Bose-Einstein condens ], and their dynamics is encoded, at the perturbative GFT lev 18 , 16 – 17 14 ]. We work in a Fock space picture in which the quantum GFT field The first issue is the definition and interpretation of approp In understanding the predictions of the theory for such geom In this paper, we describe in some detail the different steps o In background-independent approaches to quantum gravity t The second issue is the dynamical description of these state 13 basic issues. can address both ofgravity, at these least issues as far in as thein spatially [ homogeneous group geometrie field theory (GFT continuum, approximately smooth and spatiallybe homogeneou put on firm footingof elementary using parallel the transports geometric (giving interpretation(giving a of a discretised t discretised sp metric)tum which gravity derives or from spin interpret which foam maps models. a given We configuration are of able to give a general where one faces a similarto problem an of effective relating large-scale thewe microscopi description propose of is the indeed condensate that as of a hilate elementary building blocks of space (interpreted as challenging: in such theories,one the that most describes natural nocorresponding notion spacetime to of physically at a interesting all.states f solutions with Macroscopic, of a Gene approx veryallows large for number fluctuations of beyondbe quantum-geometric those able excitat with to very speak large of wavelen a smooth geometry at all). structures of the given theory, such thata one spacetime can metric associate description ( to them.a priori This independent involves from at one lea another:often one thought has of to map as themanifold, discrete fund and spacetime one structures, has to toto a describe a cont states classical that geometric are configuration, sufficiently again at least inat an some a effective continuumshould level. reproduce, at Inshort least order distances, in to the some dynamics be low-energythe of compatible General first, regime, w Relativity. lest i.e. This one u within i would quantum be gravity. constrained to a purely kinematic such building blocks can only contain finite information abo geometry given in terms of a metric on a differentiable manifo dimensions) with a finiteThese number of states degrees have of anquantum freedom gravity equivalent encodi [ description in terms of the s spin foam amplitudes and simplicialof the gravity same path theory integrals [ JHEP06(2014)013 to 4 consistent tructed metric ]. A natural possibility 19 mplest states one consid- ly unheard of; in fact, a sibly of the interpretation at can be interpreted as a choice of 3-dimensional Lie n spacetime; instead a field ematical Hilbert space and etric data: ‘single-particle’ collective wavefunction will try is defined on the config- rs of quantum-gravitational ase of spatial homogeneity, d, for example, in the loop vity and spin foam models, ovides the kind of effective vity. The states we consider property of being consistent eory. We consider two types ground-independent context: cribed by states in the GFT is then used in section given by the Gross-Pitaevskii rmations (or rotations in Rie- to directly define condensate relevant case for cosmology. unction directly from the un- space) extension of the usual ow in detail, this construction for a rethinking of the relation ion. We give a criterion for the ator. This property allows the nd ‘dipole’ condensates which ition of a condensate state as a his analogy is the adaptation of queezed states used in quantum Lorentz) rotations). We discuss etation as exact vacuum states of , we outline the general procedure 3 – 3 – , we give an introduction to the group field theory , the space of geometries [ 2 ]. 20 superspace Wheeler-DeWitt quantum cosmology. This interpretation is `ala acting on the space manifold with respect to which the recons G We proceed as follows. In section In the quantum physics of Bose-Einstein condensates, the si can be homogeneous. This essentially classical discussion its interpretation in termsis of a discrete reformulation geometries. of Asintroducing the we a basic sh second-quantised structure language ofstates that loop describing will quantum macroscopic allow gra universes. us In section (GFT) formalism, emphasising a Fock construction of the kin is to view the collectivewavefunction wavefunction appearing in the defin for associating to aFock given space, discrete a geometry, reconstructed ofspatial metric the hypersurface geometry type in on des canonical ain manifold which gravity. our th We procedure focusgroup requires no on additional the input c beyond a motivate the definition ofof condensate states condensates, in both groupcondensates, possessing field which th the are right particularlyare type simple automatically of gauge-invariant to (pre-)geom (with construct,properties respect of a to these states, local comparing ( themoptics, to looking coherent and at s correlation functions and their interpr with the geometric contentsatisfy of the in states general we aequations consider, of non-linear but quantum and the cosmology. non-local Thisbetween general (on quantum feature cosmology minisuper calls and fulland quantum use gravity, and of pos quantumnonlinear extension cosmology of itself. quantumquantum cosmology However, cosmology context has it in been is [ suggeste not total in the quantum theory wequanta can of consider space, to arbitrarily improve high arbitrarilyreconstructed numbe the metric same to approximat be spatially homogeneous, the most ers are coherent states thatderivation of are an eigenstates of effective dynamics thederlying microscopic field for dynamics oper the of condensate theequation, wavef atoms. has This dynamics, amacroscopic direct physics hydrodynamic that we interpretation, seekare and to such derive pr coherent for states, thewith but case the gauge of have invariance gra of to GFTmannian satisfy under signature). local the A Lorentz additional central transfo conceptualthe issue very in drawing notion t of awe hydrodynamic cannot interpretation expect to to the obtaincapturing back a some description effective degrees in of termsuration freedom of space of a for quantum gravity, ‘fluid’ i.e. geome o JHEP06(2014)013 propriate to be exact , not interpreted d G del incorporating data which could be used to tracting cosmology from fold nditions on the collective lassical analysis done for a generalisation of matrix etric degrees of freedom. ising for Lorentzian models o various models discussed kinetic operator and show deriving effective dynamics ding matter fields and per- p, the classical dynamics of ar differential equation. We equations that are ‘analytic y of the homogeneous space n space leads directly to the pectation values must vanish is incorporated into the GFT (which require extending the city constraints are imposed. erturbation field with coordi- nal dimension in superspace), otropic case. We then extend general form of this equation, ons over the exact condensate. ve dynamical equation for such ign changes corresponding to a les show that it can give the cor- the isotropic and WKB approx- ry of gravity, which is very promis- quantum gravity in the GFT or spin d – 4 – , deals with extensions of this formalism beyond 6 -dimensional gravity (where we will be only interested in d is the Lorentz group, its Riemannian counterpart, or some ap , we look at the dynamics of these condensate states. In order 5 G SU(2) in the appendix. / ) C , To summarise, in this paper we give a general procedure for ex For the convenience of the reader, we discuss divergences ar The last part of the paper, section = 4), where that are associated withBianchi the IX infinite universes, volume as of wellSL(2 the as gauge some grou facts about the geometr as spacetime, for models of d subgroup (usually SU(2)). In this, they can be viewed both as quantum gravity, that can beinterpretable applied as a to discrete any metric or GFTrect connection. or semiclassical Our spin limit examp foam corresponding mo toing, a but classical clearly theo morein work the is literature. needed This tofor paper apply an is this emergent a procedure spacetime first t geometry step from in a a theory programme of for pre-geom 2 Group field theory Group field theories (GFT) are field theories over a group mani the simple case ofturbations. homogeneous In universes a without simple matter: examplefield where as ad a an massless additional scalar argumenta field (corresponding natural to choice an of additio right kinetic coupling term of on a the masslessimation. scalar extended field configuratio We to gravity, present again ideasclass in of for states introducing we have inhomogeneities We been discuss considering) by the adding possibility fluctuati nates of in identifying the arguments background ofdevelop geometry the a defined p systematic by cosmological a perturbation GFT theory. condensate, Then, we specify some conditionsgive in its which this semiclassical becomesthat (WKB) a it approximation line reduces for tothe a the classical formalism Laplacian Friedmann to equation LorentzianRiemannian in signature. signature the can is We be findcontinuations’ done of analogously that and the the results previous semic in change ones, in i.e. the they metric contain signature. the s foam formulation, with special care in ensuring that simpli solutions to the equations offor motion, these an states. infinite We number focuswavefunction of defining on ex the two condensate. of This them definescollective, which an cosmological we effecti wavefunctions. express as Wewhich co write holds down the for any of the current models of 4 GFT. In section JHEP06(2014)013 . ]. )) G A by ] = C 21 [ I ¯ , ϕ (2.1) (2.2) (2.5) (2.3) g ; they g ilation , usually ~χ ϕ, [ c ) I S C ) (indexed g ( -connection , = SL(2 ∗ ~χ → /G G ψ d d G ~χ † G = 0 G ˆ c lds. For compact ( 2 i imposing the basic ) te them from loop ), . L ′ ~χ I i ) to make the theory g X n ); for non-compact ( 2.1 1 † , or more precisely the itational) ition of group-theoretic − 2.1 ˆ + 1 } ϕ ) I ) = , ′ ~χ I I g ) g annihilated by all ˆ n = 0 (2.4) g ). These operators can be G, I { ( | t multiplication of all ( I g . This particle is interpreted cuss in detail in appendix i i is a function h † ( g ∈ i tions on its Lie algebra ′ 1)-simplex with the group ele- 0 I ˆ † ( sformations for ϕ reedom. The basic variable in | 0 ~χ † and interaction (higher order) g ϕ + 1 | ˆ ~χ h − ˆ ϕ c ( ∀ , h ~χ ≡ ψ d δ K ~χ † stent with ( n ) ˆ c = d ) =1 h p d I  h ) d = ′ I Q = g  i h ( ′ ~χ , . . . , g ~χ ˆ d ϕ 1 ˆ c n , | g G , ) ( R ~χ ~χ † I † h, . . . , g ˆ ˆ c c g 2)-subsimplices). Local gauge transformations ˆ 1  ϕ ( – 5 – g ˆ − ϕ , when acting on ( ) = ,  } ′ I d ϕ , , i ′ G ) 1 , g I , there is a well-defined notion of non-commutative I g ∈ ~χ,~χ − g ) = ) and the dynamics is encoded in an action, G ( ( δ , d d ~χ ~χ ) G n , h ′ ψ I I = | } ~χ ~χ i , g h ′ ˆ c n I I ~χ † faces (i.e. ( , . . . , g g , . . . , g g ˆ c √ 1 ( 1 , d ~χ g g {{ G , which is the motivation for requiring ( ~χ ( X ( = I ˆ c ) itself creates a “particle” with data G ϕ ϕ i h I ~χ ], for a kinetic (quadratic) term ] = = g ) and promote the expansion coefficients to creation and annih ¯ of ) = ( n ϕ } ~χ | I † i I ~χ g h ) g ϕ ϕ, ( ˆ c ′ I [ { ˆ g ϕ i ( V weighted by appropriate coupling constants. † i ) is the identity operator on the space of gauge-invariant fie ≡ ˆ i ϕ ′ I λ ) , V i d ) , g I I P g g corresponding to elementary parallel transports of a (grav ( ( , this can be defined by ˆ ϕ I G h g G I ] + , . . . , g ¯ 1 ϕ The quantum field theory can be defined in operator language by At least for finite-dimensional Let us first define the basic structures of GFT, and then motiva One can now proceed to expand the field in a basis of functions o g ( ϕ, ˆ [ ϕ (non-relativistic) commutation relations which are consi in gravity. Fourier transform which takes functions on the group to func endowed with the invariance corresponding to local gauge transformations (Lorentz tran quantum gravity and spin foam models. The classical field gauge-invariant. as an elementary building block of simplicial geometry, a ( models, and as andata interpreted enrichment as of pre-geometric tensorGFT or models is geometric a through degrees complex the of field add f K polynomials where The field operator ˆ by some set of labels equivalence class [ one has to be more careful to avoid divergences, as we will dis will act as a common element ments group used to define a Fock space starting from a vacuum state act on the vertex where these links meet as simultaneous righ along the links dual to the operators, for an appropriate normalisation of the basis functions which will satisfy JHEP06(2014)013 ) ]. e- σ ( ) is 30 i σ (2.6) N . i V , and Aut( T model. Their , equivalently, as σ ynamics of quantum d spin foam models, Unless a fundamental , resentation of the same lar complex (a possible σ f gravitational models is n loop quantum gravity. of a quantum field theory, ter. A he GFT approach, let us s of freedom, a spin foam ails on this are given in a ] to loop quantum gravity -over-histories formulation o-one correspondence [ ) ynman diagram dual to this constructions in the rest of σ etwork graphs (appearing as to those that can be written 28 ( ) i ated to the cellular complexes ction in terms of kinetic (free) – nction also defines a spin foam up (and of the Lie algebra) is I n the different terms σ N y, specified by a choice of field x. A complete definition should ay in which the Hilbert space of ) ], which has in fact provided the B 22 nising the amplitudes associated i λ 18 ( i Aut( , ]). Q 17 31 σ X = ] ¯ ϕ – 6 – is the Feynman amplitude that the GFT model ϕ, [ , the possible interaction vertices are determined σ in the case of gravitational models. A basic issue in the Feynman diagram dual to i S V i A − e i . ¯ σ ϕ e P -product in this dual representation. The dual variables D ⋆ + -bein ϕ k d S D , are interpreted as bivectors associated to the faces, corr g = Z for a S = ] but we illuminate the most important points here. e 29 ∧ Z ... ∧ e ] and specifically its spin foam corner [ R and, generically, can be represented as a spin foam model (or 16 – σ are cellular complexes dual to the Feynman diagrams of the GF 14 σ What we have described so far is a self-contained formulation There is a very close relation between group field theories an Thus we see that group field theories not only encode the same d , which are elements of I combinatorial structure depends on thepart model; and writing interactions the as a the order of the automorphisms of where in this form, at the quantum level. We will come back to that la assigns to which are a proposalfor for quantum defining gravity, a basedIn discrete fact, and on spin algebraic the foam sum variables models and and group states field used theories i are in one-t by the combinatorial pattern of pairings of field arguments i B a non-commutative discrete gravity path integral [ geometry as spin foamtheory models, of but quantum that gravityformulation they possesses of it do a cannot more be finite based thaninvolve number on an that. a infinite of single class degree cellular of comple cellularthe complexes, canonical in theory the is same w boundary defined states), over an and in infinite particular class a of prescription spin for n orga (LQG) [ impetus for the developmentseparate of publication group [ field theories. More det To any spin foam‘history’ model of assigning spin an networks),and there amplitude action, exists that to reproduces a a the group samecellular given field amplitude complex. cellu for theor Conversely, the any GFT quantum Fe GFTmodel partition by fu specifying uniquely theappearing Feynman amplitudes in associ its perturbative expansion. In a formula, sponding to This can be usedtheory. to The define non-commutativity the of equivalentreflected multiplication “momentum in of space” the the rep introduction gro of a is then the number of vertices of type of the spin foamconcerned, and is GFT to approach, restrict as the far generic as Lie the algebra construction variables o and in fact containsthe everything paper. that will However,explain be to more required understand closely for better the the the relation of motivation group of field t theories [ JHEP06(2014)013 - j d β , as i = 4. (2.7) V d λ d -particle V = SU(2) in elements 2 G V as it generates a faces. 1 ) for esults. For a more d j ism, and on the link ted as covariant and β itly with complicated j i g , n turn a correspondence ( ) coupling constants leads however directly to φ ysics in condensed matter ch vertex is a node with of the spin foam approach. V d n with ols to study the physics of ned combinatorial weights. am dynamics. One advan- le ones) can be turned into equals the spacetime dimension, escription sm. that we encounter the same irit of lattice gauge theory, with tions for collective variables e of the GFT formalism: we of states associated to e cosmological dynamics for , provide both a definition of a group element of the group ll these are reasons for using d r exact correspondence is not ) = d j i pact case. ˜ on of spin networks (both their g H , . . . , g ( sociated to both open and closed 1)-simplex, i.e. a tetrahedron in φ V 2 m states of the theory, interpreted − inuum physics. For this direction of inves- : , g d V completion V 1 g ; ... vertices or their dual polyhedra, of the type ; 1 d V – 7 – , . . . , g 1 2 , g ) in quantum gravity GFT models, and 1 1 C g , ( φ ) by defining the inner product via the Haar measure on the V ]. ) = j i /G 29 g V ( · d φ G ( 2 L ]. 33 , 32 = SU(2), Spin(4), or SL(2 . The set of such functions (restricting to square-integrab Spin foam models and loop quantum gravity are usually presen In first-quantised language, one has a Hilbert space We now give some more details on this second quantised formal G Another prescription could be some refinement procedure in the sp In this paper, we restrict attention to the simplicial case, in which G ( 1 2 group, or some right/left-invariant measure in the non-com the Hilbert space standard LQG), with gauge invariance at vertices in of a Feynman diagram expansion andThus one thus can with say that canonically group assig field theory is actually a sum over complexes, weighted by spin foam amplitudes and the states are given by wavefunctions describing valent graph vertices (which includesgraphs, particular of states the as type defining the Hilbert space of LQG). Each su associated coarse graining methodstigation, used to see extract [ effective cont where each open link outgoingG from each vertex is associated and each GFT quantum (or spin network vertex) is dual to a ( to all such complexes. Group field theory provides one such pr canonical formulations of the sameyet quantum fully theory, understood. but thei Akinematics straightforward and second dynamics), quantisati andthe GFT thus formalism, of and loop thisbetween quantum GFT/LQG the correspondence gravity, canonical defines LQG i tage formulation of and the covariant GFT spinmany reformulation fo LQG is degrees that it ofspin provides networks freedom, and the to spin rightand bypass foams, to features the and of to need the derivequantum to non-perturbative effective field deal sector descrip theory of explic reformulations the oftheory theory. many-body and particle quantum A ph physics,advantages so in it quantum should gravity. come Ourwill paper as indeed exemplifies no bypass this surprise the us spininteresting, foam albeit formulation very of simple, the non-perturbativeas dynamics quantu cosmological quantum spacetimes,them, and using extract the second an quantised effectiv features of the GFTbetween formali LQG and GFT,extensive and treatment, see thus [ the direct LQG relevance of our r outgoing open links, and can be thought as dual to a polyhedro JHEP06(2014)013 , ⊂ φ (2.8) (2.9) . The (2.10) (2.11) of the (Γ) tion : a state G E . Γ φ ∈ ! I b j .e. elements , associated to B ) , g -th link at the } i a i , b j a β I g g I X , 1 , d d

− Γ. In the GFT setting i ) ⋆ efunction \ are assigned to each ...J ...n a p representations, or i V (Γ)), a wavefunction ˜ Γ 1 . LQG wavefunctions 1 g ~χ J j n E ( | G ⋆ δ ements { C V I ( # sentation and in the spin epresented by the closed ∈ g ) Γ # cylindrical consistency I ) ab ij I There is a relation B g ( G ( I i · · · h 3 I group elements g 1 given either by the spin network n e ) = Ψ I ~χ and target I } V | b j J m d 1 I i =1 g Y I g D h " ab ij (Γ)) which specifies the connectivity of V constrained by the (non-commutative) α d =1 = ; Y I E I a i i ...~χ " g 1 g , includes as a special class of states the ~χ e 6∈ ~χ | d ), for any = ab ij I ) connected, while the second pair identifies vertices (specified by φ )] ˜ } g i H α 1 ij h { ~χ   i a − V j | ( – 8 – i I -valent graphs Γ. β Γ )( ~χ g d φ ) = X ab ij h I Γ that do not depend on the edges i a G ab ij g V i =1 ( ⊃ i Y α β ~χ ) = d ˜ Γ { I   ψ ( of LQG, where there is a notion of g -particle state can be decomposed into products of ele- -th node, with source G ( Γ j = Z V ~χ ), where the group elements (Γ), there is a directed edge connecting the → i } ψ E φ (Γ) | ab ij E i ! I Y ∈ ) = Ψ ∈ G g (Γ) of the graph. These are labelled by two pairs of indices: (satisfying [( } e h { the subspace of single-particle (single-vertex) states, i → )] ( E 2 = 0 ab ij v ) Γ ) of each vertex glued together to form the link. We assume the j b I } G ∈ ) = H  -valent graph Γ with { ab ) = i B I ’s glue open spin network vertices corresponding to the func )( ( d I } g )] Γ α ( I ab ij i a -th link at the φ X j b b G , . . . , d cylindrical functions | { = 1. A general 1 ~ )( J, ~m, ( g  Γ , V i a ∈ v Ψ = } × { I H ) associated to the vertices of Γ, label a basis in the intertwiner space between the given grou ~χ B := [( with I 2.7

e d Given a closed Let us denote by The Hilbert space for these functions, associated to Γ can be obtained by group-averaging of any wav = These are the ˜ ,...,V H 3 1 Γ ~χ { -th node to the the vertices. the outgoing edges ( gauge invariance Ψ the first pair identifies the pair of vertices ( i link Ψ form ( such a graph: if [( usual LQG states associated to closed in such a way that each edge in Γ is associated with two group el integrals over the pairwise along commongraph links, Γ. thus The forming samerepresentation. the construction can spin be network phrased r in the flux repre ( are then of the form Ψ on a graph Γ is identified with states on used here, cylindrical consistency is not imposed. where ments of by a product of non-commutative plane waves where the complete basis ofwavefunctions single-vertex wave for functions individual is spin network vertices, closure condition for the fluxes, of JHEP06(2014)013 ), V LQG (2.16) (2.14) (2.15) (2.12) (2.13) /G in the V · φ d i   i , which is G j i ( ~χ 0 2 ,... . | | a i L , , and one only v V =1 annihilate  j O + 1 H V I ~χ V , . . . , n and n on numbers”, i.e. to | = 0 ; 1 ...~χ n 1 s all the states of the ce in second quantised i e group representation i | ′ ˜ ~χ i Γ , . . . , g + 1 ~χ † of spin network vertices, ~χ i φ I ! the one on ˆ , i.e. many-particle states c  | | ~χ ion/annihilation operators } ) , } i I create a h no degree of freedom of Fock vacuum n ~χ a V † V g n V ( v =1 ˆ c h n i O rs p h ), we can then rewrite { }|{ } s of the type we are considering X v i , . . . , g V vertices, and thus the symmetry =1 a i j = = I ~χ O ) are the coefficients of n H { | i  ( i

′ X ~χ ~χ   ~χ F { ,... | n   ˆ c | i ! I a , V ~χ † ⊗ · · · ⊗ H g , . . . , g are included. The inner product on this ~χ ∞ ˆ c h 2 I ˆ c v ...~χ 
(2) ! v 1 , g V i =1 I i ˜ ~χ Γ 1 I O g V . . . n , . . . , n g φ } ! 1 a  } } ⊗ H 1 n n a a X { φ n ( n – 9 – n { (1) v ⊗· · ·⊗H ˜ C = = v r }|{ ψ X i H i ) ~χ  i  = { V ~χ I ⊗H i ! , . For elements of | =0 v = SU(2)) to the LQG inner product of states on a i I ,... ∞ a i ∞ , and M V g a 1 H i h ,... G ! a − V =1 , . . . , g V i Y ) = . . . n ~χ ,... j I v ! a n , 1 | V , . . . , n H ′ ~χ n 1 ( n ...~χ n , . . . , n F ~χ,~χ 1 ( 1 r , . . . , g ˜ √ δ ~χ Γ n ˜ i I , . . . , n | C   φ 1 = = } } } i n a a i | i ~χ n n runs over a basis of the single-particle Hilbert space ′ i I ~χ X X X { { { ~χ † g n ˆ a c h , . . . , g | , 2 I ~χ = = =: ~χ ˆ c ˆ c , g
h 1 I i ) = I i g g ~g particles are in the state  ( ˜ Γ a φ } a φ n n { Being quantum field theories, GFTs describe this Hilbert spa One moves to a labelling of quantum states by their “occupati ψ interpreted as the “no-space” (or “emptiest”) state in whic It is clear from this algebra that these fundamental operato where the label network vertices. One can constructby arbitrary spin acting network multiple times on the special state given by the sums over those configurations compatible with the labels where we denote theby coefficients of the new basis elements in th occupation number basis.Fock space The of states the theory,defined of can by the be obtained new in basis, terms and of the thu creat where the corresponding Fock space is language. Assuming bosonic statistics for the spin network where only symmetric elements of Fock space descends, for each summand in the direct sum, from and is equivalentfixed (at graph. least for a new basis of the Hilbert space of a given (finite) number defined by JHEP06(2014)013 olv- spin (2.20) (2.19) (2.21) (2.17) (2.18) (2.22) m . , ) ) j I I h g ( ( , ˆ ∗ ~χ ϕ ds. Hence we
ψ ′ n n =1 ~χ = 0 † Y j )-body operators, ˆ c ) i i I m ) g des either via Peter- ′ I . ~χ ( + g jection’ operator onto X † ly from the canonical t it is annihilated by , . . . , ~χ ( n ˆ ′ 1 † ϕ ˆ phys ϕ ,( i , ~χ on operators) and single- ) = , m =1 ) I i Y Ψ itable normalisation of the m ... I | g
ing constants, operators, it is then standard n on physical states to be of ( g n I † ( geometry). = † ˆ ϕ ˆ ϕ ell), which encodes the action of h ≡ , . . . , ~χ phys 1 ) i . = ) spin network vertices, and possibly , . . . , h d ~χ , that is an operator acting on spin 1 I  Ψ m | ) m ′ I , h i m + phys + g i + n m I n ( n ... ˆ Ψ , . . . , g \ ϕ O | ,( 1 , O + g ) ( = I ... 3 = † g b , . . . , g P ˆ ( i ϕ – 10 – 1 I 3 , and so all of its occupation numbers are zero, ˆ ′ n ϕ g phys λ ~χ  ) from the perspective of spin networks in LQG. i , ∀ + ) Ψ m ) or via the non-commutative Fourier transform for | 2.2 I 2 + I g b b n P P ( , . . . , ~χ , 2 ′ 1 ~χ O ) = 0 λ ~χ ′ I ψ h | n i · ~ J, ~m, ~χ d 0 , g spin network vertices and resulting in states with m | )-body operator’ = ˆ c ) I + ~χ g h n n m c = ( ( ~χ \ (d O G | phys X ~χ + I i m m · n Ψ d = = 0). In the second case the formula has to be understood as inv | field operators )) ) ) = b I I I g P i g ) g ~ B ( ′ ( I (d ~χ ˆ . g , . . . , ~χ ϕ will in general decompose into 2-body, 3-body, i ( Z 1 =1 ψ ) is the identity operator on the space of gauge invariant fiel † 0 d I ~χ ′ I b ≡ ˆ = P ϕ h i ) , P , g † ) d | I ˆ I ϕ g g ,..., g ( ( 0 ˆ ϕ, ∈ G , ˆ -multiplication between field modes (creation/annihilati ϕ h I 0 ⋆ h ~ | B , . . . , g m 1 The quantum dynamics of spin networks can be encoded in a ‘pro These are the fundamental GFT field operators, expanded in mo The kinematical operators of GFTs are also obtained natural Given a general ‘( From the linear superposition of creation and annihilation + = g n ( = ( i ˆ ϕ \ 0 O ~χ physical states (other possibilitiessome can Hamiltonian be constraint considered operator. asthe w We form take the conditio quantum geometry isall present. annihilation operators, It ˆ is defined by the property tha network states formed by and a corresponding operator on the GFT Fock space, network vertices, one can define its matrix elements LQG kinematics (or, equivalently, from quantum simplicial ing a to define the bosonic The operator i.e. into operators whose action involves 2, 3, vertex wave functions. | recover and motivate the definition ( Weyl decomposition for satisfying the commutation relationsbasis functions (here we are using a su where an infinite number of components, weighted by suitable coupl JHEP06(2014)013   ) j I h ( (2.24) (2.23) (2.25) (2.26) (2.29) (2.28) (2.27) acting ˆ ϕ b P n =1 ′ n Y j ~χ − . ) ˆ c I i I ~χ g ϕ ( ... † ~χ ′ 1 ˆ ϕ ~χ ϕ ˆ c , satisfying also encoded in the
m =1 i i Y ~χ ′ n 0 ] , | X |  ϕ
j I ϕ ϕ, ) [ h I . , h ) ession into the GFT path g g from the above quantum eff i i rections to be absorbed in  ) = I . I ( the Fock space is then given S ) s , , . . . , ~χ I g i † g responding resolution of the | es, | ′ 1 g − V (  ) nstates of the GFT quantum ϕ ϕ ( v ( equires some assumption. We one obtains ϕ | mary in many-body quantum b ) ˆ ϕ N h , ~χ ϕ m I ) µ ϕ e ) ~χ g + b m N ( I − n D ϕ µ ) = g b ϕ P F I ( − ϕ ( = g d n ϕ b · ( ) F − D d ~χ † † ( d g e ) , . . . , ~χ ˆ ⊗ · · · ⊗ H c ˆ | ) ϕ | − 1 h | Z s g (d selects quantum states with many or few e h ϕ ~χ ϕ | (2) v R h (d h (d µ = s ϕ m h X m · Z i 2 d + s | ⊗ H ) | = n ≡ ϕ ) – 11 – , g , i P b 2 i (1) v −| N i = exp s | (d m | e µ ϕ ϕ i H ϕ ~χ | g † | | Z − ˆ ˆ 0 c ρ  ~χ | ) | ≡ b   F I s ϕ  ( ] h g m =0 ... ~χ † -point functions. We have chosen here the normal or- ( ϕ − + 1 ∞ V = ˆ c s n , e ϕ ϕ, n ~χ † ~χ | [ i X | ⊕ ˆ c λ s ϕ ϕ = ϕ eff h } | ′ = S ~χ i ~χ ih ∞ s are indeed the classical GFT fields, and the measure over them − g ) = ˆ n,m c X ϕ ensemble ϕ ~χ,~χ X X e v | P | Z { ) ϕ  − 2 H I | = m ( ) g ϕ I g ( + F g ˆ −| n and ϕ Z ( λ ϕ ϕ on the GFT Fock space that corresponds to the operator = exp ) ˆ ϕ e i I ∞ b n,m D X F g ϕ ( | † ϕ − ˆ ϕ ~χ D d ˆ c grandcanonical ) ~χ † Z g ˆ c (d = ~χ I X Z where the sign of the ‘chemical potential’ The identification of the corresponding GFT action, startin These are the GFT dynamical operator equations, which can be The functions ≡ = b F spin network vertices asintegral, dominant. we introduce In then orderfield a operator, to second turn quantised this basis expr of eige dynamics expressed in canonicalconsider (second a quantised) form, r and acts on states in Schwinger-Dyson equations for dering for creation andphysics. annihilation Different operators, operatorthe as orderings interaction is would kernels custo give of quantum the theory. cor or equivalently on first quantised spin-network states. It is defined by and a completeness relation as is usual for such coherent stat in terms of an effective action is the (formal) GFTidentity path in integral the measure. formula for Inserting the the quantum cor partition function, The second quantised counterpart of theby quantum an dynamics in operator JHEP06(2014)013 . y µ − ,   ) = 1 j I 2 h red by the ( field of BF m ϕ B n =1 Y j ) by ) b F i I g ( ds itself to different ¯ ϕ gral. Later on, we will m a different operator m =1 iscussion of the effective tion, rescaling the term the local gauge group of i Y classical GFT action heory, such as the Ooguri , one can just define the  sion of the Euler-Lagrange the basis of (products of) pose suitable conditions on G j I nection (the needed for explicit manipu- perator in ) are thus nothing else than he formulation of gravity as . To give proper meaning to ce them to be a function of egy, only some of which have = Spin(4) in the Riemannian , h he most studied ones) aiming ld have anyway to be properly i I uantum field theory. on, one has then to go through ry, also known as the Plebanski ues of general functionals of the g G  m ; (2.30) ) (2.31) + i d n i ϕ | V ϕ | b F n · | ϕ d h ϕ ) , . . . , g h 1 h g ( (d ϕ ] = – 12 – m · ) ϕ d d ) g ϕ, [ . Because of this, and because both sides of the two (d 0 . b S F , . . . , g Z  1 j I   g ( m , h ¯ ϕ i I + g n d . ) in Lorentzian signature and )  λ = 0 on physical states. The relation between the two is given b 5 g C is the quantum corrected version of the ¯ m , ϕ ∞ (d + n,m X n eff Z − P S δS/δ = = SL(2 = quantum gravity. These have been defined via a strategy inspi ] =  d j I ¯ G ϕ , , h ϕ, d i I [ g δS/δϕ S  m + n The corresponding classical GFT action is then of the form The GFT formalism as outlined above is quite general, and len In this section we define the quantum GFT through the path inte The same type of quantum corrections would have appeared fro V possible definitions of thedefined, quantum partition with functions wou aquantum careful GFT theory handling starting of fromthe the such corresponding above path quantum classical integral action corrections andthe to usual the renormalisation partition and functi constructive procedures of q the chemical potential becomescoming from a the identity operator mass (which term became the in number o the effective ac ordering in the very definition of theory) which enforce the geometric nature of them, i.e. for The effective action The GFT interaction kernels (or spin foam vertex amplitudes the matrix elementssingle-vertex of states. the canonical projection operator in case; 2) to startmodel, from and a thus GFT quantising model onlythe describing flat Lie topological connections, algebra-valued BF and t variables 3) conjugate to im to such a flat con formulation of gravity as aformulation. constrained topological We BF will theo givelations. more details Now at we only aa recapitulate later corresponding some stage, justification features when of ina this the constrained strat canonical BF LQG theory theory.gravity suggests: T in 1) 4 to choose for the group constructions. In theat following, describing we 4 will consider models (t use an operator formalism inequations which we impose the operator ver the Schwinger-Dyson equations, which give expectationfield val from the pathGFT integral. dynamics We in will section come back to those in the d JHEP06(2014)013 tor e is (2.33) (2.32) where a G ace construction . i ) m ( ) are normalisable. spatially homogeneous I ) (or Spin(4)) variables for compact ipulate in the quantum ). For such a state, the B dual links. Unlike in the C | I ction of these tetrahedra, , , however, that after such 2.32 ing five of these tetrahedra e full group encoded in the g ither metric or connection ⋆ me variables used above to ( s incorporating the Immirzi ) /or interaction terms of the spin foam construction takes e spin foam/GFT literature, s ( ality principle’ in GFT. One t of such constraining is the ϕ ) lling the faces of the tetrahe- lst formulation of gravity, the , entation of GFT, as we have d associated to the 4-simplex, ontext. The basic objects of bosons to have general wave- m i an be constructed out of basis screte counterpart of BF fields: ( 0 ions in the continuum. How to ted to the faces of the tetrahedra d uch algebraic data, intended as | N ) ) m ( I ,...,B B ) ( † m ˆ ˜ ϕ 1( B =1 N ( Y m m ! – 13 – Ψ 1 N =1 N := Y m i ) N 4 m ( ) I B B particles that could be interpreted purely in terms of bivec | (d N Z specify the bivectors to be attached to each face, while ther := B i m Ψ | ) is the Fourier transform of the field operator ˆ I B ( ˆ ˜ ϕ Our strategy for the rest of the paper will be to use the Fock sp A slightly more general construction allows for the GFT Fock states with functions, to SU(2) ones, with thedynamics detailed of embedding of the SU(2) theory,GFT data i.e. action. in in th the details of the kinetic and where Lie algebra variables non-commutative Fourier transform can be defined, the state a tetrad field.Palatini In formulation of the gravity. continuum, Theplace classical corresponding at GFT theory, and the the discrete resul the theory level, are more combinatorial precisely tetrahedra,group elements labelled in or by (conjugate) a the Lie algebra simplicial di (or elements c links associa of thelabel dual states spin and network amplitudes vertices).encoded in in These the the GFT are GFT vertex, thein formalism. is the sa given boundary. in The terms This intera ofthen corresponds a to imposes 4-simplex a hav restrictions specificdiscrete (“simplicity choice counterpart of constraints”) ‘loc of on the s do Plebanski this geometricity imposition condit correctlyand is different a models main haveconditions focus been have of proposed. been activity imposed, The indron the general th can Lie point be algebra is interpreted elements aswith labe derived edge from vectors a discrete associatedparameter, tetrad to and fiel aiming tetrahedron at edges. asame quantisation In constraints of imply model the Plebanski-Ho that one can move from covariant SL(2 no information about theanalogous group case elements to of be scalar attached field to theory the on Minkowski space, and data, with completely undeterminedstates parallel of the transports, form c of GFT tomacroscopic define geometries, states and thattheory. that we are can For the comparably interpretvariables, geometric geometrically easy corresponding interpretation, as to to one the man described. can Lie focus algebra on or e group repres JHEP06(2014)013 4 R ta, (2.34) linearly , keeping N =1 for three N m } B k e m A j Ψ . For states of the e { i in the group at the jk ra formulation, in a i ǫ ducts of the functions -particle states in which e need an excited state, and which does not factorise = is entails, in this section o-space state, containing geometries corresponding N N 4 of the Lie algebra elements. unctions bra ). Our goal is to select those AB i les, but this not essential for . around the peak). In this way, there is no sense in which in- class of states that is naturally B (4) eral functions Ψ tric data to each tetrahedron. for the gravitational connection te ambiguities in their solution; ). Furthermore, we imagine we ly with a high occupation num- ∞ 2.32 so ally homogeneous metric geome- sed by the data appearing in the ) is parametrized by 3 < ) ) on I ) 2.32 B ,...,N N ( j 4( Ψ ], the assignment of geometric data can be = 1 9 ⋆ = 0, ( – ) 3 I I , m ,...,B – 14 – tetrahedra with geometric data attached to its B B 3 ( , i I 2 1(1) N Ψ ), and consider, for example, coherent states such as , P B 4 ) = 1 2.33 B i (d ( building blocks (simplices) with certain pre-geometric da } Z ) N m ( i B { ), requiring normalisability means that pairwise star-pro . A i e 2.33 If we use a smooth state like ( The gauge invariance of the GFT field results, in the Lie algeb We next turn to a classical discussion of the type of discrete have to be integrable, m there would be a general function Ψ( as a product. multiplication by a (noncommutative) DiracBecause delta of of the this sum (closure) constraint to GFT states, consisting of form ( in mind that thesedividual are tetrahedra indistinguishable can be particles, associated so with that particular wave Of course, such states still form a rather small subset of gen independent bivectors discrete geometries that correspond totries. macroscopic, Here spati we focusthe on criterion metric of homogeneity rather which than oneinstead connection could of similarly variab the give metric. visualised in terms of the position of the peak in the Lie alge the ones introduced in the LQG literature [ can impose the simplicity constraintswe and then ignore take the discre the independent bivectors to be of the form faces and dual links according to the properties of the wavef Such a state corresponds to a set of same time, in a controlled waywe (i.e. can with a attach given simultaneously finite intrinsic spread and extrinsic geome vectors 3 Approximate geometries and homogeneity As said, the Fock vacuum of group field theory represents the n to identify the ones that are useful for cosmology. no geometry at all.i.e. In a order state to obtainedber. model from Among a superpositions the macroscopic of various geometry possibilities, states,adapted we w possib to need to the construct concept a we of first homogeneity. focus on ToGFT understand classical Fock what discrete space th (parallel geometries, transports characteri and bivectors), like ( Ψ JHEP06(2014)013 , B N N and (3.1) (3.3) (3.2) ≤ e ed in in m ≤ , for 1 . } ) m . ( uous geometry at j AB ) edding of the discrete ij , there is an action of B g m ) ( { m h tetrahedron is six-dimen- ( AB i ) A i ontinuum geometries from um geometry is possible. e cannot be interpreted as ke metric, a spacetime met- ansforms covariantly under e with a homogeneous spatial m B espond to three-dimensional ( n discrete geometry given by f the bivectors alone: es that we are manipulating s specifying a choice of local i see below, they are naturally he conjugate connection vari- , corresponding to local frame e tify further this interpretation. en by meaning of the variables d remains, in its full generality, := nformation about the extrinsic the geometric data, this geome- he two are not obviously related. ng integrals of a 2-form over the ). 7→ ij sible to associate to a finite set of ˜ ) B ) 3.1 m ( i m ( , compatible ln Aj e ˜ , e B ) ) m m km have been constructed from the bivectors ( ( A i ˜ B e h ij ) – 15 – g = m mn ( j i ) ǫ m B ( kl 1 i ij ǫ − are the gauge-invariant content of states for a single g )
3 ) ij m B g ( 2 h B 1 1 B 7→ ) are precisely the gauge-invariant data that we are interest m in the Lorentzian case), ( 8 tr( ij i g N B = 1) ij , g ]. ) a unique smooth continuous geometry. 12 – 3.2 3 in such a way that it is clear whether it is (or SO(3 } N ) m The coefficients The resulting gauge-invariant configuration space for each We are interested here in those spatial geometries that corr Such discrete geometries can be seen as a sampling of a contin The next step is to relate classical discrete quantities giv On the space of bivectors, or alternatively the space of ( ij , as defining discrete metric coefficients. We will shortly jus g the construction of the states. Since the frame, removed by the condition of invariance under ( tetrahedron; the remaining information can be understood a which are invariant under coordinatefaces), transformations they (bei must bemetric scalars coefficients under in a diffeomorphisms,interpreted coordinate and as basis. henc giving Instead, the asdiffeomorphisms. metric we in will a Indeed, given the fixed frame that tr geometry or not.geometry, In such order that to a do comparison with this a we homogeneous need continu to construct an emb This group of transformations isrotations. a gauge symmetry of gravity SO(4) This formula bears some resemblance toric constructed the out well of known a Urbant triple of spacetime two-forms, but t homogeneous spaces. We must be{ able to characterise a certai sional and may be parametrised by the quantities with continuum geometries.discrete The data has problem been of discussedlargely several reconstructing times open. in c the In pasthere, an particular, see for [ LQG, which uses the variabl which can be interpreted, in light of the discrete geometric different points. Furthermore, giventry the is interpretation of only acurvature, spatial and slice hence of the embedding aables). of four-dimensional the Clearly, manifold slice, without (i is further givennumbers instructions, by ( t it is impos B These metric coefficients can be expressed directly in terms o JHEP06(2014)013 , 4 at M M m (3.4) ∈ M x T m x ∈ ) m ( i is defined with v provides a natural in its geometric M M . , pulled back on on ]. For our purposes, ret data of the form

G and hence the metric, , our spatial slice. The g to a GFT quantum) G 34 . In fact, homogeneous ry on [ M dron (bivectors, tetrads, t vectors pecify the point M m M x G ities as resulting from con- n the structure of the tetra- d parallel transports as act- requires however some extra set of all possible embeddings. T on on t can be represented in terms , e.g. for discrete geometries with e. Furthermore, being gauge function, mapping each point old ction of ⊂ which is then exponentiated. ] and references therein). , should be used to make statements self-consistent t at the given vertex. The expo- M arallel transports with the con- prinkling of points in a given manifold hedron has to be much smaller gs, while ensuring that our statements econstructed metric and connec- ij

kground structure: only the dynamical Once this is guaranteed, everything we v 12 g ) m 3( v , ) m 2( ]. Defining a Poisson process requires a choice of v 35 , is a discrete subgroup of a three-dimensional ) m – 16 – G 1( and define it everywhere else by the pull-back of the group v , acting transitively on ⊂  G , X ∈ M x ∈ M ), such as the determinant of m 3.3 x , where in terms of a discrete sampling of continuum geometric data  curvature is small over the size of the same tetrahedra, so th /X N by the embedding of the tetrahedra, we need to assume that the 7→ G ≤ m M ¸ m M ≃ ], or in the causal set approach [ ≤ 12 , in correspondence with the embedded tetrahedra . . This is a condition on our procedure to be reconstructed M G for 1 M } ) m We proceed as follows. Each of the tetrahedra (correspondin For example, in order to reconstruct an approximate tetrad, With the construction described above, we are able to interp Let us recall, first, that homogeneity of a Riemannian geomet The exact translation of the data associated to each tetrahe ( We stress that we do not have to make reference to any notion of s 4 ij g respect to a Lie group of isometries Lie group as it is induced on measure, associating a volume to a given region, and while the group a we assume that manifolds can be classified in terms of their isometry group geometric variables derived from ( action) that could be used, we regard this measure as a fiducial bac by using a PoissonLQG-type process, data as [ it is customarily done in these contexts measure (fix a volume form at one point about densities and volumes.about We hence spatial consider homogeneity arbitrary do embeddin say not will depend also on hold this if arbitrary one choice. chooses to restrict to sprinklings which are a sub or parallel transports) to continuum fields on the manifold interpretation. care. We want totinuum interpret geometric all fields the integratedvariant, above over such domains pre-geometric integrations of quant require finite thetinuum siz connection use (see of for appropriate example p the construction in [ associated we can approximate their interioring as trivially. flat and More regardthan precisely, the different the neede possible linear curvature radii sizetion inferred of on using each the tetra r can be embedded into a three-dimensional topological manif defined on emanating from it, corresponding to thenential three map edges naturally inciden defined by the Maurer-Cartan connecti embedding of each tetrahedron requiresof a sufficiently the smooth tetrahedron (boundary andhedron, bulk) the to embedding our can manifold.at be Give which completely one determined once of we the s vertices is embedded, and the three tangen allows us to embed the wholeof tetrahedron, a since linear any combination point of in the i three independent vectors { JHEP06(2014)013 : G vec- (3.6) (3.5) 4 vector . This , is the R ) m provides ]). x m as a local ( fields on 38 ij M – g discrete data 36 of approximation. , but with different the point m x unspecified, even two grated along the edges te the integral of the 1- ) we interpret the nding state could not be m ion which has to be taken hich is a statement about ( fficients would be constant hoice of this sort is always i oints neity? At this stage, any and the physical tetrad: he coefficients v action on space tric reconstruction procedure ) m cosmology [ ill have to be replaced with a tetrahedra are embedded with , d to the scale of the embedded mate sense where one looks at ess reliable, as the approxima- m riant vector fields. dynamics. Indeed, an effective eomorphism-variant quantities. ( )) A i e . m x )) obtained by push-forward of a basis ( j m e x , M ( ) i e m , avoiding such issues: the tetrahedra are ) x -invariant inner product in the Lie algebra )( ( m i G m ( i e x v ( )( – 17 – A m e x ( = g , this implies that ) ij = m g ( . Fixing a A i ) e G m ( ij in general cannot be undone by a diffeomorphism. We could g are the vector fields on ij g } i e { , would lead to physically distinct reconstructed metrics, ) m ( i are the metric components in the frame of the left-invariant , a natural choice for which is a basis of left-invariant vector ) v where ) m , ( ) is, the more confident we can be of the association between our m i ( v m ij N x g . ( associated to a tetrahedron as physical tetrad vectors inte i } representing the physical tetrad over an edge by its value at ) i e points in the manifold. This is indeed our second main source e m A , such a basis is unique up to a global action of O(3). ( { = A i e N G e Within the approximation of near-flatness, we can approxima This approximation is the first source of error in our discuss Now assuming that the approximation of near-flatness holds, If we did not make such a choice but left the vectors Clearly, if the spatial geometry was homogeneous, these coe ) m of ( i always oriented along the local frame given by the left-inva us with a natural, canonical way of fixing GL(3) transformation on have chosen a different set of vectorsuch fields tangent and assume vectors, that of all course, but the existence of a group embeddings in which all tetrahedra are embedded at the same p tors tangent vectors g form of left-invariant vector fields on and thus implies we assume thethe edges coordinate to system be we ofrequired are when unit expressing passing coordinate from the diffeomorphism-invariant length, metric toWe diff w in. then have Some the c following relation between the vectors For the gauge-invariant quantities v fields specified by into account, especially indynamics the predicting construction a regime of of an verytetrahedra high effective would curvature compare make the abovetion geometric on interpretation l which ittrusted hinges to breaks have such down. simplebetter guess, geometric In or interpretation, this be and reanalysed caseand w in the terms interpretation of (see correspo a also more the subtle geome related work on loop quantu underlying metric geometrypositive compatible answer to with this spatial questiononly can homoge only hold in the approxi and a continuum geometry. The larger in space. We are interested in the converse question: given t JHEP06(2014)013 , ij g (3.7) ) can be , we can M 2.33 homogeneity is clear. With on For a scalar field, G M ) or ( of the elementary 5 : we shall see later ). pretation as describ- nction’ for many ele- G is unspecified and its 6 T formalism one usu- 2.32 the intrinsic geometry configuration space as G tency conditions. First, epresenting an isotropic otivation for considering gauge-invariant quantities bitrary anisotropic homo- e group ike ( riate gauge-covariant com- he GFT dynamics, just like e with respect to the size of s were expressed was the same for e the same for all tetrahedra; ices for tive dynamics corresponds to trahedra as part of our embedding s the criterion of e frame of left-invariant vector group coming from a particular GFT es. It is a dynamical question n of wavefunctions for quantum ), i.e. the microscopic geometric near-flatness c curvature), or in fact for any , i.e. close to the identity in the s case). In the quantum theory e of discrete geometric data. if iscussion holds for the connection l criterion for homogeneity would have (see section 3.7 = SU(2) for consistency. ,...N. G = 1 m ∀ – 18 – ) results. In the next section, we will also lift this , ij 3.7 g = ¯ ) m ( ij g -point sampling of a homogeneous geometry N compatible with spatial homogeneity states, characterised by just a single macroscopic ‘wavefu As we have said, the procedure is subject to some self-consis Within this approximation, the criterion for which a state l Again we should stress that we focussed the discussion only on Let us summarise the conditions on GFT states to have an inter In general the reconstructed continuum geometries can be ar Note also that we had only to specify a homogeneity criterion for the 5 which at the classical level corresponds to the condition ( data must be thecondensate same for all tetrahedra. This is our primary m interpreted as a discrete the criterion of homogeneity wouldfields simply with be tensor that indices its wouldfields, value be so b interpreted that as a given criterion inhomogeneity analogous th criterion to to ( the quantum setting,states, as and a thus conditio to probability distributions over the spac mentary building blocks. Thetetrahedra: second the condition components is of thatbinations the of of elementary curvature, parallel given transports,gauge by group must approp be (SO(4) in small ally what we considers have the considered universal so covering far group, — Spin(4) in in the thi GF because we had previouslyand fixed reconstruction a procedure. unique referencebeen Had frame to we require for not exactly all done thatall our this, the tetrahedra. te local an frame additiona in which the metric quantitie (the three-dimensional metric) but a perfectly(which analogous d includes, in theother Ashtekar field, formulation, such the as extrinsi additional matter data characterising degrees the elementary of tetrahedra freedom added to the GFT the embedding discussed above, making use of the action of th ing macroscopic homogeneous spatial geometries. The first i say that the state is geneous metrics, corresponding towhether all an possible approximately isotropic Bianchiin typ geometry classical emerges general from relativity. t one has to ensurethe that tetrahedra) the is flatness satisfied. condition (small Second, curvatur while in principle the on that, for ahomogeneous special geometry, choice the of semiclassicala GFT regime positively of action curved the and 3-geometry, effec quantum which suggests state that r choice provides an additional input,model the can effective provide dynamics conditions on the possible consistent cho JHEP06(2014)013 , 3 r. (4.1) for all theory . We will ), one can ) of identical roximation of e given an in- 2.33 , 3.7 i eat kernel) or not 0 | N  ) he immediate quantum 4 antum spacetime degrees of semiclassicality given quantum state, the s valid. This is a standard a WKB approximation; the etries as GFT states, lifting ithin the more general class ensate wavefunction must be ce would be to use coherent and vious section. We work first ies emerging from a quantum he shape of the wavefunction se the semiclassical approximation. terms), as in ( frame) becomes, at the quan- l that these represent geomet- ,...,B l affect the result. In our procedure, The third condition is that the gical approximation, the semiclassical tain quantum cosmology-like effective 1 riables): at the end of section eneity criterion ( onding cosmological wavefunction, here lassical approximation can be taken at y approximate homogeneous (and B : at least in some regime, in order ( † ˆ ˜ ϕ ⋆ ) 4 homogeneity ]. In the context of this paper, we will extract 39 variables to label our states (assumed to satisfy ,...,B – 19 – 1 ) is easy to identify. Working in the Lie algebra semiclassicality B B 3.7 Ψ( 4 ) . B 4.1 (d identical distribution over the space of geometric data Z ], which we do not really tackle in its generality in this pape  ! 41 1 N The problem of emergence of classical physics from a quantum := 6 i ]. ], as done for example in [ N should be reasonably large, and the larger the better the app Ψ 40 12 | – quanta each associated to the same wavefunction. These can b N ), or to whether it represents a semiclassical state (e.g. a h 4 10 N , 8 – 3 ,...,B 1 Notice that, along the way from the microscopic description of the qu B 6 to be able tostate speak must about have a semiclassicalstates classical properties. [ universe emerging The from standard a choi the continuum. The fourth condition is discuss specific conditions onof the condensate choice states of in wavefunction section Ψ w (e.g. a Dirac-delta-like distribution in the Lie algebra va semiclassical physics from the quantumcondition dynamics is by means then of toprocedure be in in quantum a cosmology regime tostate, where discuss see this e.g. spacetime approximation histor [ i consider special states of the product type geometric data for alltum tetrahedra level, (in the requirement the of tetrahedra. appropriate local Notice that thisΨ( requirement does not refer to t we distinguished the independent conditions of is of course a major one [ representation, for example, thus using containing freedom to the effectivevarious points. macroscopic cosmological The ones, specificsince stage the the notion at semic of which quantum oneapproximation condensate cannot takes is be it needed taken to will until obtainequations, in the a and genera very cosmolo it end. is Inarising on fact, as these we the final will collective ob equations variable (better, for the on GFT the condensate) corresp that we will u simplicity conditions, so to be interpretable in geometric this must be phrased inpeaked terms around of small expectation values: valuesric of the cond quantities elementary curvatures integrated (recal oversampling the size size of the tetrahedra). terpretation in terms of discretepossibly geometries anisotropic) that naturall spatial slices. The classical homog 4 GFT condensates asIn continuum this homogeneous section geometries we willto describe the continuum quantum homogeneous level geom in the the classical Riemannian considerations contextversion of where of the the the pre homogeneity gauge criterion group ( is Spin(4). T JHEP06(2014)013 ), re emi- 4.2 (4.3) (4.2) . in ( i accuracy. 0 N | c N to infinity, we .  i ) ′ 0 I N | B ( N †  ˆ ˜ ϕ ) sibly depending on a to this region, e.g. by ) 4 try describes a compact I the radius of curvature. L atness condition, i.e. the . Indeed, there are many r superpositions of states B forward way to go beyond ( used in condensed matter hat are built out of larger † ogoliubov approximation of ˆ ˜ terpreted as a condensate of ϕ e the coefficients r given modes, controlled by ,...,B correlations between particles. ⋆ 1 uous discrete geometries, with s and the discrete ones, we can mpling to zero. In this way, a nce this condition refers only to xactly homogeneous geometries orrelations among a larger and of the Fock space allows us also ) tum dynamics first. B ′ 4 ( † ˆ ˜ ϕ ⋆ ) ,...,B 4 ]. ′ 1 is compact, as will be the case later on 44 ,B – , since they correspond to particular second- M 4 42 ,...,B 1 mentioned before). Then, probing the geometry – 20 – B ,...,B M Ψ( 1 4 different points can be understood as a restriction to B ), but could be more physically appropriate to describe ) B N Ψ( 3.7 in the reconstructed geometry. Sending 4 (d ) 3 ′ / Z 1 B GFT condensates  (d ! is the total volume with respect to some arbitrary fixed measu 4 N ) L/N c N V B , giving the size of the sampling. We ignore the condition of s =0 ∞ (d N X N to infinity will allow us to recover homogeneity to arbitrary tetrahedra at Z :=  where N N ! i 3 N / Ψ c N 1 ]. | V is a three-sphere). One can associate a length scale 44 =0 ∞ – X = N 42 M L := i We will call these states The above definition of condensate states can be generalised To summarise, we assume that the underlying continuum geome In principle, nothing prevents us from considering states t Again, there are two sources of approximations. First, the fl Having a Fock space structure for our states gives a straight Ψ | (such as the left-invariant measure on of this region by setting the wavefunction Ψ, thustheory following [ the standard terminology quantised states having macroscopic occupation numbers fo ways to construct states that,a as given many-body building states, block. can Besides be the in obvious freedom to choos (where region in space, which can be all of space if wavelengths longer than take this approximation scalesecond-quantised associated state to will the allow discretethe us only sa to source of approximate error contin encodedthe in discrepancy the between flatness the condition. continuumsay Si geometric that quantitie sending one can imagine to consider moreOne general possibility states is that to include consider states of the form elementary building blocks, i.e.larger states number that are of encodingslightly quanta, larger c set or of pre-geometric even data.according combinations These to may our of not simple be criterion them, e ( and pos smallness of theSecond, ratio the between value of the size of the tetrahedra and classicality for this discussion, focussing on the GFT quan a formalism in which thewith different number particle of numbers. quanta In isto addition, fixed, take the states and structure with conside an infinite number of particles, e.g. Such states are used,the for dynamics instance, of Bose-Einstein in condensates the [ discussion of the B JHEP06(2014)013 (4.5) is often Spin(4), ∈ ) (4.4) ′ 4 k → ∞ ∀ ) 4 N ) is a normalisa- is taken. , . . . , g σ 1 ( g ( N † , . . . , g ϕ . 1 → ∞ ) ˆ way. In this sense they rmation about the best g ite expectation value for 4 ( t correspond to a system es. These are the simplest  N σ y used to describe weakly 2 tries or correspond to the | re. For instance, a better le modes and for pairs, con- umber of GFT quanta. It is ) besides the gauge invariance er ate, with states of arbitrarily high e previous section. Therefore, icle number, acting on them ontain the idea of sampling a 4 t our system, a macroscopic f atoms will be finite for any g to condensation of the GFT pose is that the state contains e system has a large but finite nd removing a relatively small ) = ng the use of the original Fock , . . . , g ′ 1 e freedom in the definition of the k ). ) to define exponential operators g 4 ment of normalisability, i.e. of ob- ( Spin(4), and , . . . , g σ 3.1 4.3 1 ∈ effectively depends on less arguments. 4 g ) ( σ k g , . . . , g σ ′ ∀ | (d k ) 4 1 4 ) ) and ( g Z g ( σ 4.2 (d := – 21 – Z σ , . . . , g 1 1 2 g ( − σ  with ˆ ) = 4 i 0 | appearing in ( ) ) := exp σ N σ c ( , . . . , kg N 1 bosons), although the formal thermodynamic limit ) exp (ˆ kg 3 ( σ ( σ N ) imposes automatically that := i 2.1 σ | The simplest class of states is a ‘single-particle’ condens Despite the fact that, strictly speaking, these states do no In this paper, we will focus on just two of these possible stat Since ( We model the states after coherent states for single-partic where we require tion factor, with infinite number ofwith creation particles, and for annihilation sufficientlynumber operators large of (that quanta) part is, adding doescapture a not part of change the theirspace. thermodynamic shape limit, in while This an still isinteracting allowi essential Bose-Einstein the condensates, reason where whyparticular the condensate coherent number state states (as o aresize, dictated by typically extensivel the of fact 10 that th possible choices that will allowcontinuous homogeneous us geometry to in work the sense withfor specified states our in purposes, that th the c onlyexactly restriction the that we geometric are data goingof of to GFT im a fields, tetrahedron. it has Consequently, to include the gauge symmetry ( straining the coefficients (giving the desired coherence properties),state and to reducing the th choice of a single function. approximately homogeneous (andoutcome thus of more a realistic)state more geome that refined encodes reconstructionhomogeneous the procedure. universe, appropriate possesses, physical Thecontrol properties should on info the tha come properties from ofquanta the might elsewhe GFT give phase hints. transition leadin useful as an approximation. we have two restrictions telling us that the function At least for thetaining ‘single-particle’ an condensate, element the of require the the number GFT operator. Fock Hence, space, whileparticle is involving number, a equivalent the superposition to condensate aonly always fin has in a the finite first sense average n that the limit of infinite particle numb JHEP06(2014)013 ) 4 ) we (4.9) (4.6) ation (4.11) (4.10) k 4 satisfies ) (4.12) g ,...,B 1 ... 1 , +4 − p ) B ]. + 4 ( 4 g i ˜ σ 31 1 4 p | ⋆ , − 4 ξ ) I h| 21 k, . . . , k 4 1 , . . . , h 1 B ) 1 ) g 4 I m [ 4 h 1 + , , . . . , h ( ) automatically − 2 † P 4 ) (4.7) i k ( +1 4 ϕ 2 ξ ( | p k ) ˆ 4 σ ξ 4 e , . . . , g ) is a normalisation factor g , . . . , g 1 ξ h| 3 1 ( g e identified as corresponding 1 8 , . . . , g g ( − 1 ( ) = 1 p N σ ,...,B 4 − 4 ) = g + σ ) ) (4.8) ). This is indeed the simplest 1 ( 4 , . . . , g ) h 1 i 1 1 ( I σ 2 B − − | g ξ ) ( 3.3 B k ( ξ k I ) ˜ ( σ I † 4 p h| k B ), and 4 ϕ ⋆ ,...,B I ( 2 1 , . . . , g 1 ) ˆ g h 1 I kB 1 4 1 − 1 I g ! ( p g B h − e g I I − 4 ( ( 1 k g ( B σ e σ e ξ − 4 4 =1 Y , . . . , kB I = 1 + I 4 4 1 =1 =1 4 Y Y X I I ) − )] = 0, the function ) =   , . . . , g 4 4 – 22 – I g k i

I 3 ) ) 1 k h g , . . . , g ⋆ − g g 2 ( (d ( 1 δ | p † ξ 4 h ξ (d (d kB ˆ Z ϕ 1 ( h h| , 3 − 1 , with stores exactly and only the gauge-invariant data needed Z Z σ ) ) = k g − 1 I 4 1 σ 2 ( p σ ) = g = = ˜ − 4 ξ 4 ( i 1 g † 0 ) = ≥ 4 ( | 4 ϕ ) k X ξ  h ˆ   ξ ,...,B k =1 (d 1 we obtain  Y p Spin(4) and ,...,B 4 ). Here we have used some elementary properties of the plane B 1 ) k k ( 4 ∈ g ,...,B B ) ˜ ) and [ ˆ σ 1 ′ ( 3.1 = exp h has unit norm in the Fock space. Using the fundamental commut (d ) exp ˜ σ B ξ i i ( 2.1 (d ( Z 0 ξ k, k ˜ σ | k | ) 4 ∀ 2 1 N ˆ ) ξ ) g ′ k := := (d I ˆ i ξ ξ Z ) exp( kg | † ( and of the corresponding non-commutative Fourier transfor ˆ ξ ξ := g i e k 2 The second class of states that we are considering is We see that the wavefunction exp( ) = | | I ξ 0 g h| h ( where, thanks to ( ensuring the state ξ waves to reconstruct thechoice metric of from quantum the statesto that bivectors, continuum possesses quantum as homogeneous all in geometries. the ( properties w relations and a bit of combinatorics, one can show that Defining the Fourier transform of where the invariances that we impose imply that first ˜ obtain which takes care of ( so that integrating over and the closure constraint is satisfied; from JHEP06(2014)013 . In ξ (4.13) encode i , possibly ) is finite, ξ | 3 atisfied, we 4.11 and i ] aims at deriving σ | 48 tial in ( , es logies, the above states ondensate states whose 47 ), provides two kinds of states of the full theory. in the arguments of dent of a chosen reference acuum state of the system, 4.4 ty [ i se states are not states of a cal dynamics, let us make a s at this point that there is A condensate of cubes would ght of as composed of several g e necessary for the geometric the extraction of an effective utlined in section ss of states, while possessing reconstruction procedure. A . setting, and could be used to ticle states in the GFT Fock ry one often thinks of cubical imple two-particle correlations, m geometries. The distribution = ft for future work. This should   articular they support arbitrary ould be imposed on our quantum i i k + 2 k | 4 ξ g , which can be seen as functions over h| ξ k 1 4 or 1 a ‘dipole’ function on the gauge-invariant ] in more detail. ≥ σ k X ξ 48 , − – 23 – ) is imposed in a natural way, without any further   47 3.1 ], we call ) := exp 46 ξ ( , to be in the Fock space, these moments of the profile function N 45 i 0 | ) ˆ ξ , which is expected to be a rather strong constraint. If it is s in the product it is understood that ∞ k < i = k 2 p | ξ h| It would be straightforward to define, along the same lines, c According to our previous analysis, the GFT condensate stat Despite being general enough to encode all the Bianchi cosmo Following recent work [ k 1 2 must go to zero fast enough so that the argument of the exponen improvements: first, invariance under ( external restriction. Second, thea state feature encodes that some should very bedue s expected to its to highly be interacting necessary nature. in the true v P cosmological dynamics from LQG usingrepresent cubulations of the space. closest analoguecompare to our such approach a with construction the in work our of [ are simple enough tocosmological lead dynamics, to as explicit we calculations will and see allow in the4.1 following. Self-consistency conditions Before moving on tobit the more extraction precise of the the approximationsinterpretation mentioned effective of above cosmologi that the ar abovedetailed analysis states, of according the geometric to conditionsstates that our can to simple or ensure sh theentail correct a geometric refined interpretation version is of le the reconstruction procedure we o and for elementary building blocks arespace. more For complicated instance, multi-par graphs, in where similar the elementary contexts cell in istetrahedra. a discrete “cube” Most geomet that notably, can recent be thou work in loop quantum gravi can set the same gauge invariance and the same geometric data as ( configuration space of a single tetrahedron. This second cla the minisuperspace of homogeneousa geometries. difference with Let the ussymmetry standard stres reduced minisuperspace theory. approach.Furthermore, The Rather, given they the are sum symmetrylattice over reduced structure, samplings, or they fixedperturbations discretisation are over of indepen homogeneous space, geometries. and in p continuous homogeneous (but possibly anisotropic)of quantu geometric data is encoded in the functions order for the state exp( ξ JHEP06(2014)013 and 4 (4.15) (4.17) (4.18) (4.14) ˜ g i σ | ) ,..., 4 1 g to the familiar ) (4.16) 4 , . . . , g 1 g ( ed as the expectation etric observables. For ϕ , . . . , g cified by ˜ ) ˆ 1 . 4 , g his gas of tetrahedra. At 2 ( ) | we are probing geometry, 4 ) σ he spatial slice captured by e quantum state, and is not mics of the theory. Here, we ) 4 in its GFT formulation), the el, i.e. as a choice of quantum ven state. For simplicity, we 4 e have to ask that the volume evious results on semiclassical ld in terms of (a large number istent with the intuition that to consist in the discretisation mulation. ometric condition should take, ed, this error can be expressed , . . . , g 1 g , . . . , g ; 1 , . . . , g 4 g , . . . , g 1 ˜ g ( 1 g . ]. The same reconstruction procedure ( g ϕ 1 ; ) ˆ σ . 38 | 4 4 – ˜ g 3 ,..., 4 ≪ / 1 ) 1 36 g g ) ¸ (˜ 1 ). ¸ ) is the matrix element of the volume operator (d N ,..., V , . . . , g V 4 1 ) 1 ( 4.4 Z 4 g g – 24 – (˜ ˜ g := ( , we expect one of the first sources of error for the ∼ † = 3 V ˆ ϕ ) i ¸ L tot 4 , . . . , g 4 V σ 1 ˜ V g | ) ,..., g 1 g ¸ ; g ˆ 4 (˜ N (d ˜ | g † ]. Also, the geometric conditions so identified should then ,..., ˆ σ ϕ Z 1 h | g 12 (˜ σ = – = h ,..., σ 3 1 ¸ ¸ 4 4 g ˆ ) ) (˜ N N g g V (d˜ (d˜ 4 4 ) ) volume encoded by the state is g g (d (d Z Z total = SU(2), = = G . Its form can be obtained from a Peter-Weyl decomposition in 4 tot V Similar one-body operators can be used to extract other geom The number of tetrahedra contained in the state can be obtain Following the arguments of section , . . . , g 1 instance, the quantum geometry states [ error associated to the approximationof) of a discrete continuum constituents. manifo Oncein the terms states of have expectation beenwill specifi values consider of just certain the simple operators condensates in ( the gi value of the one-body operator should be generalised beyond the homogeneous case, using pr effective theory that we are going to derive from these states be incorporated into the quantum theoryensemble at being the considered, dynamical and lev thuslimit in ourselves the to fundamental a dyna basicfor discussion the of class the of form that states such we ge consider, in a second quantised for The volume of a tetrahedron defines the typical scale at which reproducing and extending to thekinematical set-up full of quantum loop theory quantum (here, cosmology [ where, for in LQG between two spin network nodes with geometric data spe This scale uses onlyfixed the externally. geometric information These encodedthe very in continuum simple th limit considerations can are be cons seen as a thermodynamic limit for t representations in terms of spins.of As each we tetrahedron have is already smallthe said, compared w state, to the total volume of t g which is JHEP06(2014)013 by L (4.19) (4.20) i σ nts. In the case | that are invariant ) 4 4 representation, the ). The interpretation 1 2 ) for all tetrahedra in χ = t is self-consistent comes tes, with the appropriate ctation values of suitable , . . . , g 4.19 1 out the expectation values also try to constrain nection are too large, the efore, the curvature of the j emerges, one must use the e stated as a condition on g tions are nothing else than t the relevant states should ( ahedra of the type proposed in ( . orrelations between different hedron from their values at L tain functionals of the wave- ϕ ogether with a more detailed ing a thermodynamic variable 1 ) ˆ ng the precise nature of these essarily semiclassical in the Lie , applied to our quantum states. 4 e fluctuations around the mean the microscopic GFT dynamics ) ture) has to be small on the scale ≪ 4 1 − , . . . , g ] 1 , . . . , g g σ , which can be derived from the theory, ( [ 1 ¸ g L χ ( N ) tot 4 χ χ 2 ) | should be much larger than the typical length e ) 1 ( c 4 – 25 – χ L , . . . , g to be the trace in the 1 g χ ( 1 = † 1. , . . . , g ˆ 1 ϕ − g | should be, or how it is related to fundamental (Planck- ≪ ( ] σ ) σ c , which motivates their identification with components of σ h | L e [ I ( g ) give a complete set of functions on SU(2) 4 4 ¸ ′ χ ) ) 1 χ k L/L g g − 4 , g (d (d j 7→ L g 1 Z Z I − g 4 g = i ] = g ( σ and [ χ k tot I := χ g ij χ 7→ can be seen as a character of a suitable product of group eleme ] is that of a sum of curvature components (defined by I σ χ g [ = SU(2), for instance, taking Information about the connection can be extracted from expe Another condition that is needed to ensure that the treatmen In fact, this condition might involve not only a statement ab This reasoning can be exported to the case of more general sta tot G χ operators, for instance from the fact thatconnection the that tetrahedra they have encode to (including beof the extrinsic the close curva tetrahedra to themselves. flat. Ther of where changing the type of ensembleconjugate one is to considering, the e.g. extensive byconstraints add quantity is ‘volume’. crucial Understandi forand clarifying the the resulting relation effectiveanalysis cosmological between of scenario the kinematical and reconstruction goes procedure t to be GFT dynamics. One possibilityonly is occurs that within a a condensationinstead of certain of tetr range being of put values for into the definition of the states. One can functions under of tetrahedra, one has to go beyond using a one-body operator as the curvature for a single tetrahedron. In order to measure c ian) units. To understand physically how a particular scale scale of the tetrahedron this stage it is not clear what In terms of thethe scales observation defined that the by curvature the scale tetrahedron, these condi modifications. They will be expressed as restrictions on cer the condensate. Inthe general, deviation our of flatnessthe the condition identity, then curvature can expectation b values per tetra of connection-related operators, butvalues. also conditions It on is th flatness clear requirement that is not ifbe satisfied. the rather One fluctuations peaked can in around the thenalgebra the connection expect variables. variables, flat tha albeit con not nec JHEP06(2014)013 ), ) is tion w these 4.10 ( (4.23) (4.22) (4.25) (4.21) (4.24) ˆ S ) and ( . 4.4 ! 2 | z in the state cannot | z − | ¸ |  ]. 1 ˆ a N ˆ a t states only in a rough 49 ion. In particular, since formation and define the p he condensate as a whole 2 w on in which the GFT field

goliubov transformation can edra 1 − ly nontrivial constraints to be ahedra alone. in quantum optics and in the antum states ( − . † -point correlation function, the es that um geometric interpretation to ons; it gives rise to correlation tions that ˆ e, they represent a natural class a † ˆ a sinh = 0 . 2 | w i i is annihilated by it, and hence it is z z 0 | | σ  , | i ) 2 −  0 | I † | † z g = 1 the two would-be ladder operators commute, ˆ a ˆ a
( † | † z ) = z ˆ σ a ˆ a | − | z † 2 z − ( ) = exp 1 ˆ a =  ˆ a 2 z w i – 26 – p ( σ ˆ | S ) = I ) exp b , f ˆ g † ( i are ladder operators and the unitary operator ˆ a ) ˆ ϕ z , † z i ˆ a ( 0 − f | a, ) a ) are coherent states, i.e. eigenstates of the field annihila (ˆ w are more similar to squeezed states [ ( 4.4 i ∝ | ˆ i S 0 ξ | | =  1. The state exp i † ˆ a ) that are used in the parametrisation of the states. In turn, w † | ξ < ˆ a 2 2 z | or  z | σ 7 1 the role of annihilation and creation operators is exchanged. exp > 2 | is a complex number, ˆ z | w The states of the form ( The second class of states, dipole condensates, are coheren For a single mode, a squeezed state is defined as We mention, for completeness, that in the case of 7 the so-called squeezing operator. It follows from the defini Indeed, using the properties of the ladder operators, one se sense. In fact, the states where be tuned by hand.imposed Therefore, on these the conditions resulting represent effectivebe tru dynamics, trusted. for its They continu that contain cannot information be about captured the by properties dynamics of of4.2 the t individual tetr GFT condensatesNext, vs. it coherent is and worth stressing squeezed some states further properties of the qu especially in their relationshipphysics with of quantum states fluids. commonly used operator, as a straightforward calculationof shows. states Therefor for aacquires sort a of nontrivial vacuum Hartree-Fock expectation or value, mean field approximati while for provided that functions (e.g. As said, this statefunctions does that are not products encode andmean convolutions multiparticle field. of correlati a single one functions are determinedthese dynamically equations by are the nonlinear, equations the of average mot number of tetrah With a simple rescalingladder operator we can complete the Bogoliubov trans proportional to the corresponding Fock vacuum. A general Bo JHEP06(2014)013 , . ξ ) j I h (4.30) (4.27) (4.31) (4.28) (4.29) (4.32) ( σ =1 m Y j . ) i ) I , , I g ) I ( h I g 1 σ ˆ h ξ and their Hermi- 1 − I . n k − =1 I I i Y ( ) g ˆ ξ ξ ˆ ξ ) =: kg ; (4.26) = ) 4 ( † I i 4 − a k σ )ˆ ics. | 1 ) exp( | e of our quantum gravity ) I k δ − I I y to see that w g g g d | m I ( ˆ ( ξ , . . . , h ation operators, and to show ormation of the ladder oper- h ξ 1 ϕ ( Z Dirac delta distribution on the ions of the group field theory, ombination of ladder operators h pated. This is just the Hartree is to use the characterisation of 4 ˆ ϕ ( ) . A proof of such a statement, . Evaluating the commutator we ≡ † sinh( has to be the infinite-dimensional )] = . As a consequence of this, k ξ ξ ˆ | † gebra of creation and annihilation ϕ ) ξ , ... ξ ) ˆ I (d ϕ w w ) 4 | h 1 I ). h 1 Z h = 0 1 exp( − + I ( , − i 4 g ) ϕ a ) are squeezed states, i.e. to write down ξ ( 4.23 is itself proportional to a group-averaged ) + ) ˆ )ˆ I | 3 | to be a squeezed state. I g n I  ξ ( h g w i I | ( 4.10 1 ξ ϕ g † | [ ˆ ˆ , . . . , g ξ − I ˆ 1 ϕ g Spin(4) ), the correlation functions are simply factorised – 27 – ( h − δ 3 1 ) ... 4.4 − 1 I ∝ , ) g g = cosh( ] 1 I 1 ( ( I g † ξ ˆ − † ϕ h ( ) Spin(4) n ˆ † ξ 4  δ ) ), but more generally w ) ˆ ˆ ϕ I ξ ( − h | ( g I ( ) σ ] = g aS (d 3 I h ˆ I )ˆ ξ h † h Z n ( ˆ w ) we get the desired ( ξ ) can be interpreted as squeezed states only if the function † )= ( ˆ ϕ S − m I in the creation operator field ˆ , Spin(4) ] = ] = ) 4.26 I δ ˆ ξ 4.10 n I = are squeezed states if the operators ˆ g ) , ˆ ξ h ˆ ) b ˆ ξ ∝ i . . . h ( I ( ξ † ) 1 I − , g | ˆ I linear ) ( ϕ h ) g I , ; I ϕ ) and ( ( I g [ ˆ is n g I ξ g ( ( ˆ ξ I ϕ g ϕ [ ˆ [ ˆ ˆ ξ 4.25 − ) , . . . g I 1 I g g ( ( ) ϕ [ ˆ , corresponding to another annihilation operator. It is eas For the single-particle condensate ( We would like to show that our states ( † n,m ˆ ( ϕ G ˆ which will in general only be true for specific choices of 4.3 Correlation functions The particular form of the state chosen as a trial vacuum stat system implies specificwhich properties are of the the true encoding correlation of funct the fundamental quantum dynam tian conjugates satisfy theoperators, (suitably gauge-invariant) al and so the states find that Therefore, the states ( analogue of a unitary symmetric matrix for Notice that in terms of theapproximation, one-point correlation function, as we antici ators, presumably requiring appropriate conditions on squeezing operators that correspond to a Bogoliubov transf comparing ( delta function, convoluted with its complex conjugate,group is manifold. proportional to A a trivial case of this is that however, is not straightforward atsqueezed all. states An as alternative path Fockthat vacua these of states Bogoliubov are rotated annihilated annihil by an appropriate linear c be expressed as ϕ, JHEP06(2014)013 i 0 | (4.38) (4.33) (4.34) (4.35) (4.36) ]]]] D , e ) (4.37) ), it is easy ) which enter ′ I . d I ) g , k 4) I , ( , I 4.28 ) (0 ϕ k , k [ ˆ ′ I ( ′ I , G 2 k k ) 1 ( c I G ≡ − 2 I g ) ( k I 4 G ( . ϕ ) ξ [ ˆ , k y provided by the very I ) ,G , I dynamics. k ) is just provided by the I ) k . , with the straightforward h 1 2) , this simple class of states b I k , in the given state. Instead, tive cosmological dynamics ( n-zero correlation functions σ − I g 1 ) = 0 2 or more kernels. its expectation value on such 4.4 ed on the classical field con- (0 ( ′ I h − I functions are written in terms d = 2 htly more involved, but shows ( G ϕ k G h [ ˆ s among the different quanta. ) ξ 1 ted, and we just limit ourselves ( ′ , I ) m n that any equation or condition ξ − ) I ≡ ′ I ) a I k , k k ′ 2 I + ( g I 1 k ( ξ G g − 1 I ) ( ) ϕ g I [ ˆ − I 2 I ( † k g , n ξ G 1 ( D , k ) ξ e 4 − I I 4 ) | ) m I g h 4 ′ ′ 0 ( ( ) k , unless this function satisfies the condition ′ k . In this case, all correlation functions con- ξ 1) ξ k ) , ξ (d (d ′ I . . . h (1 4 4 i∝h (d k ) – 28 – ) 1 I ) satisfies 4 1 G k d I ) k h ) − i I encodes the geometric data that we need, it does g ; k I ξ g ( (d (d n I | k ( ξ ) (d ϕ 1 ξ I Z ) ˆ Z − I c I 4 h Z h ) g ( − ′ ( ( , . . . g ϕ ) + k ) ξ 1 I ϕ ) ˆ I I 4 ) + g ) ˆ I ) (d h I ( b I g 4 1 ) k , h h g ( ) I − ( 1 I ˆ ϕ k GFT models we are most interested in. Using ( (d g g | − I ϕ ( n,m ( ξ ) ˆ ( g d 2 (d Z ξ a ( I G ξ G g Z ( ≡ h = ˆ ϕ ≈ ≈ ) ) = h I I . ) ) I I σ , h , h )= h I I d I 1 , h g → g I ( − I ( , g g 2 g 2 ϕ c ( I ( 2 G G ξ does not in general coincide with , g G b I 2 G , g a I g The analysis of the four-point correlation function is slig This is a crucial point, because such equations are obviousl In the case of dipole condensate states, all the correlation Therefore, the mean field theory encoded in the state ( ( 4 G the general pattern of the calculations. By definition, definition of the fundamental GFToffers model an to immediate be and used. straightforwardfrom Therefore way any to given obtain GFT an definition effec of fundamental quantum gravity of the two-point function, parametrised by are of the form taining an odd number of arguments simply vanish; the only no the calculations for the 4 Closed expressions for the generalto case the are cases rather of complica the two-point and four-point functions classical GFT equations, withfigurations. an additional symmetry impos to see that As said, thisFurthermore, particular the result class immediately of leads toimposed states the upon conclusio ignores the field correlation operators,a when considered state, in would terms lead of to the corresponding equation for the fiel where we are neglecting terms built with convolutions of five replacement ˆ not immediately correspond to the two-point function of GFT we have Hence, This means that, while the function JHEP06(2014)013 ) b I × g D]] ) 1 , d ) I − (4.40) (4.41) b I g ) 1 g a I ( -particle − g ) ϕ [ ˆ ′ (( I N , ξ k ) a I (( = ]]] (4.39) the four-point g ξ . ( ) D b I ) ξ c I ) + permut. e g ϕ 6 g I d a I I ) 1 g g 2 , k = [ ˆ − D ] still try to estimate G I ) e b I ,D nt function, the four- D ,D ′ I h g ) ((  a e I ( h a c I I ate is quadratic in the ) g d I 2 g O g ˆ g ) all higher commutators d ξ onals of I (( ( c I g b G I ξ + g c I ) of the two-point function, ϕ g case of general ) b = I g e recursion relations among } [ ˆ d I b 1 I g b I , g g tion functions to only a few g a a I I − ) D], D 1 1 g g ) b . I ence of further conditions, do equation: , a − I I − I g ) g ( k k a I + D ( ( g ϕ (( ξ [ ˆ ξ d ( I eed, following standard procedures, ξ ) , ) g + permut. ϕ c I ) a I D a g I = d I g ) + permut. c I 1 g g 1 d I g b = [ ˆ I ( − I g − I D D a ϕ I a ]] I h , g c I h b g I g ( c g I ( D {z g ξ g ξ e D ( ) + permut. + D 4 b I 4 2 d I ) g ) b I g ′ a I Gaussian-like k G g ]]]] = [ ˆ g k ) c I and D D – 29 – D b g ) + permut. I (d c I ˆ b I ξ d I (d g 4 . However, it is also clear that all the correlation g , e 4 ) , g g ξ ) . ) 1 a I h D d I ) k g a − I g ′ + D I ( g ) (d in a highly nonlinear way. ( 2 c (d I to emphasise that the calculation that follows is totally d I ϕ 4 , k g g ξ Z [ ˆ | G ˆ ) c ′ I ξ I + permut. + D ′ , ) (( g + D ) b d I h I ξ , h c I d I g g ) I c g I ) = (D g b 1 I (d g ( d , I 4 g + permut. + D − D ) , k ) 1 ϕ ) D c I b I d I I [ ˆ , g b g I h a I − g g , c I h g ) c g I ( ) ( D g (d a I b I b 4 I , g ϕ (( g [ ˆ (D g g , the last two contributions vanish, D b I D ξ ( ˆ Z G , , ξ b I (( D ) ) g , g ϕ × + + ξ a I a a I I [ ˆ a I a I g g , g g g ) ( ( ( D a I ) = 4 ϕ ϕ  g +D d I ( G , g ϕ = [ ˆ = = [ ˆ [ ˆ c I , g b I , g a I As in the case of the two-point functions, regarded as functi In absence of an accurate analysis of the critical limit of th It is convenient to introduce the notation D Therefore, at least at leading order in the expansion, the st To the same order of approximation used above for the two-poi g ( 4 G etc. In the case of the dipole where D = in general. For D = vanish. Then where we are using D instead of and the four-point function is the solution to the following general and is valid (with appropriate modifications) in the sense that we canitself determined express by the the function correlation functions in terms correlation functions are givennot only correspond simply implicitly to and, bilinears in in abs coherent states. functions are given in terms of the two-point functionpoint alone function is GFT correlators, itthe is theoretical hard error to ofrepresentatives say in a terms much of truncation a more. Ginzburg-like of criterion. Ind the However, tower one of can correla JHEP06(2014)013 exact lation (4.43) (4.44) (4.42) (4.45) , ) c I g ( φ )+permut. ) d b I I . ates. We show g ( , g c I φ ) , g rise in the large N a I b I g le the approximation g ( cific condensate states i efined can at most be e the theoretical error ( φ ) )) c ) + permut. b I in the resulting effective such a true vacuum state ynamics of Bose-Einstein d I 3( g duced from the Schwinger- ( rephrased in an equivalent G , g d terms. For instance, in the φ ) antum state considered. om the point of view of the Consequently, the validity of c I ) that conditions on them to ed in the following) are ty system, even if one believes s and fluctuations, a I g . While less clear in terms of g ( ( ) ) + on is signalled not necessarily by ce, in the case of matrix models c s (and their relations encoded in φ b 4.10 I 2( g ( by the large value of the fluctuation ) is. G )+permut.+ -point functions of the theory. φ ) c I n b I − g , g 4.10 ( b I ) ) and ( φ b I g ) ( g ) a I 4.4 ( c g , ϕ ( 2( )+permut.+ ) φ ) and ( b d I I G ))( ˆ ) – 30 – a I , g a , g I g 4.4 c a I I ) + g , ( ( g d g I ) φ ( ( g φ d I ) ) ( c c φ , g 2( 2( ) ) + c I ) + c I a I G G c I g g ) , g ( ( b b I I , g φ φ b I ) + ) , g , g b b I I a a I I − , g g g g g a I ) ( ( ( ( g , a I ) ) φ φ ( c c ) might be directly verified once the relations between corre g ) ) ) ) ( a a a c I I I 2( 4( g g g ϕ 3( G G ( ( ( ( ˆ 4.10 h φ G φ φ + + =: =: ) =: ) =: i ) = ) or ( ) c I d I b I a I , g g , g ]. This behaviour would be matched by the simple condensates 4.4 , g ( b I c I a I ˆ ϕ to the microscopic quantum dynamics, and thus true vacuum st 51 g , g , g h ( , b a I I 2 gravity it has been shown that correlation function do facto g 50 ( G , g 3 d a I These considerations allow us to at least estimate how reliab It is clear from the analysis of the states ( In some cases, however, one can do even more, and show that spe g G ( 4 G limit [ functions are investigated infor the critical 2 limit. For instan 4.4 Condensate states as exactAs GFT we vacua? discussed above,approximations to GFT the condensates true vacuum state of ofthat the the something quantum simple gravi akin type to afor we GFT d our condensation quantum is universe. whatmade determines in We using have such also approximation seen by that analysing we the can estimat be good approximations toform in physically terms relevant of statesthe the can GFT properties condensate be of interpretations,the correlation the function Schwinger-Dyson correlation equations) functions mightanalysis be of more the perturbative accessible (spinthe fr foam) ansatz expansion ( of GFTs. and so on. ThenDyson the truncation equation, of the to towertheory a of that equations, can given as be order de estimatedcase by leads the of magnitude to a of a the Hartree-Fockcondensates, neglecte mean theoretical the field breakdown error of approximation the toa Gross-Pitaevskii singularity equati the of hydrod the particular solutionwith itself, respect but to rather the mean field associated to theencoded particular in qu the use of the simple states ( one can split the correlation functions in terms of mean field (slightly more involved than thesolutions ones presented above and us here one example. JHEP06(2014)013 (5.1) (5.2) is the (4.46) (4.47) , , ] N ¯ ϕ = 0 ϕ, ) [ 4 V ] λ ¯ ϕ ϕ, istribution in some , . . . , g [ ) + trivial kinetic term 1 ′ 4 V g such an equation. We δ ( antity, being the partition ¯ , ϕ amiltonian constraint in δ lass of states representing her models, obtained in a , . . . , g λ e creation operators, = 0 ′ 1 simplex. quadratic (kinetic) part and e another to form a 3-sphere s of such states, as well the g i ) and to insert an identity written ( , and they represent geomet- ction, they do not yet satisfy i ) + he Friedmann equation in the ψ , 0 ϕ i | ′ 4 ons. | = 1, it turns out that ) ) 0 s are kinematical. While we have 2 | ′ 4 ]) | . Assuming the action to be real, ! i † ) σ  ϕ E ] ϕ 4 [ ˆ | † −| ˆ V , . . . , g ϕ E [ ˆ ] ′ 1 λ h , . . . , g † and ¯ g ˆ ′ V 1 exp( ( ϕ | [ ˆ λ ϕ , . . . , g ϕ , g σ ˆ 1 ) V 4 − g ′ 4 δ i h  ( In the case of Boulatov-Ooguri theories, this σ † | ˆ ϕ 8 σ δ – 31 – exp D , . . . , g 1 , . . . , g σ λ 1 ′ 1 − g D ( , g N ) + Z 4 K 4 ) 0 = 1 4 Z i = E | I , . . . , g , . . . , g 1 1 , . . . , g g g 1 ( ( g K ˆ ( ϕ ¯ 4 ϕ

) ′ 4 ) g ′ g (d (d Z 4 = ) g ) 4 (d ] ¯ Z ϕ is in general a differential operator, but can also be a delta d , needed for the normalisation of the integral, is itself a divergent qu 0 ]= ϕ, K , . . . , g [ ¯ Z ϕ 1 The dynamics of a given GFT action provides us with precisely In the case in which the quantum equation of motion involves a It will be interesting to investigate further the propertie Applying Wick’s theorem to the each term in g δS To see this, it suffices to take the norm of the state exp( ( ϕ, 8 [ ¯ ϕ S δ start with a general actionan that interaction, we only assume to consist of a function for a Gaussian ensemble. where models which simply identifies the arguments of where state can be seen astopology, a in condensate of the five combinatorial tetrahedra pattern glued of to the on boundary ofexistence a of 4- other exactsimilar solutions fashion. of the GFT dynamics for ot 5 Effective cosmological dynamics In the previous sectionhomogeneous spatial we geometries. have At constructed thisensured stage, and these that discussed state they a are c ric invariant data under invariant local underany frame spatial form rotations diffeomorphisms of by dynamical constru geometrodynamics, equation or that to would an correspondcosmological appropriate to setting. generalisation the of H t and a (non-Hermitian) potential term that depends only on th there is one independent classical field equation, as a (formal) integral over single field coherent states, the special state square root of the GFT partition function. is an exact solution to the interacting operator field equati JHEP06(2014)013 . ) 4.3 5.3 (5.4) (5.3) (5.5) . i de the ξ | ) that all or 6= 0. The . -point func-  i ) 4.32 n ] σ 4 )] ) ¯ | ϕ ′ I I ] g † g ( ( ϕ, ˆ † ϕ [ ¯ ock space of the free ϕ ˆ ϕ δ , . . . , g , ϕ, δS [ ˆ 1 ) ] g I ˆ and its complex conju- V ¯ ϕ ( g act an equation for the δ † ( nd we also adopt this σ an operator equation on the nvolving all ˆ ϕ ϕ, undary conditions of the ϕ [ expectation value is to be δ at we cannot delve into a more nsate states, understood to implicitly assume ecessarily the Fock vacuum. l [ ˆ ese can be formally derived ed on any state in the Fock can be justified to an extent λ e quantum theory, that solutions to the simplest field operator is to be interpreted nd assuming that there is no solutions of the full quantum amical solution to all of them. − O hen be consistency conditions. , where this would be the case, rests nction , ] ld theories one would not expect the and its complex conjugate, ( ) + ¯ ) ′ 4 ϕ example, we saw in ( I ϕ . We are first looking for solutions g = 0 ϕ, ( σ [ ¯ i ϕ O δ ψ | δ , . . . , g ) ′ 1 I  g g ( ( = ϕ ˆ C ) ˆ | ′ 4  ] ψ ¯ ϕ h – 32 – ϕ, [ := S , . . . , g − , the connection of operator equations of motion and ψ ′ 1 e 2 i ) = 1 in which we obtain the requirement that ] , g I ¯ 4 ϕ g O ( ˆ ϕ, C [ h O , . . . , g  1 ) g I ( -point functions of the theory, as we have outlined in section g n appearing in the definition of these states, which would enco δ K ( ¯ ϕ 4 ξ ) δ ′ g or ¯ ϕ (d σ D 9 Z ϕ D is one of the condensate states we are considering. or its functional derivative to be mathematically well-defined on the F ) := Z I i ˆ V g ψ ( | ˆ . This implies that the tower of Schwinger-Dyson equations i C 0 = σ The task will be to use the Schwinger-Dyson equations to extr The simplest case occurs for As we have mentioned in section This does not suffice, of course, to make the equation well defined as 9 -point functions are just products of the condensate wavefu requirement that the correspondingdynamics. states In are a systematic approximate treatment, oneSchwinger-Dyson would equations have already to prove approximate aFor fully our dyn present purposes, thisfrom is an a analysis working of assumption the which profile functions for any functional ofinterpreted the as field taken and in itspath complex the integral. conjugate. “vacuum Hence, state” The thespace specified resulting that by is equations assumed are the to toIn bo play our be the setting, impos role we of will “ground choose state”, this not state n to be one of our conde gate ¯ requires a choice of operator ordering, given that in genera the path integral isby given using by the Schwinger-Dyson “fundamental equations. theoremboundary of term, Th functional so that calculus” a In the simplest case ofn the single-particle condensate, for usual procedure isstandard choice. to adopt a normal ordering prescription a Fock space, from theas rigorous an functional operator-valued analytic distribution,operator point in of usual view. interacting If quantum the fie which we can associate with the corresponding operator in th For a general classical potential term depending both on tions just reduces to a set of (nonlinear) equations for to the simplest ones; all the higher-order equations would t where theory without regularisation. Weon note Poincar´einvariance however and that causality relativisticdetailed whose QFT mathematical role analysis in here. GFTthat is an Our appropriate unclear, discussions regularisation so in has th this been section chosen. are JHEP06(2014)013 le (5.7) (5.6) . neous spa- , this does σ ould have to = 0 ¯ σ → ). It bears close ¯ ϕ , and nonlocal on 3 σ σ, are to the left of all , etc. are interpreted → † ξ ϕ ϕ ,

) and with the estimate ) σ 4 -Einstein condensates, it 0) 5.6 s nonlinear in selves with the approxima- ] ¯ , ation for real Bose-Einstein is not an inconsistency, but ϕ oss-Pitaevskii-like equations ill of course break the super- > antisation of minisuperspace n reduces to the classical field n ( . This is the direct analogue ) takes a particularly simple ]. nlinear in ted if ψ i nterpretation of the domain of σ ϕ, h particle density and momen- s a form of hydrodynamics for time evolution’ under which an -point functions in the case of , . . . , g [ σ ion of loop quantum cosmology quation as a standard quantum 52 + | 5.5 1 n ] V g σ † ) δ o Bose-Einstein condensates where ( h ˆ I in which all ˆ ϕ ¯ ), ( are eigenstates of the field operator ϕ g ( δ ˆ V i ϕ, † [ ˆ 4.4 λ σ ˆ ϕ | ˆ has the interpretation of an initial-value O δ δ ˆ is clearly just one condition to be satisfied C ) + * ′ 4 ˆ C = ψ – 33 – i (and specifically for the type of potentials typi- ) , . . . , g I ′ 1 V , the states g g ( ( ˆ C σ 4.2 ] ) † ′ 4 ˆ ϕ ϕ, [ ˆ ˆ = 0 reduces to (using that O , . . . , g , could be equivalently used to derive conditions on the profi h i ′ 1 ˆ O σ | , g ) 4 I g ( ˆ C as defining a probability distribution on the space of homoge | σ , . . . , g σ h 1 . Clearly, since there is an infinity of such conditions, one w as implied by the reconstruction procedure of section g ξ ( σ K or 4 ) σ ′ g (d ] and in the simplified ‘group field theory’ model of [ We interpret For the single-particle condensate defined in ( The vanishing of the expectation value of Z 20 , the condition . Then, using the normal ordering prescription for ˆ ˆ show that not all oftion them to are the independent. full Here quantum we dynamics content represented our by the equatio functions for an arbitrary operator Hence the expectation value of theequation, quantum equation to of be motio satisfied by the ‘condensate wavefunction’ of the theoretical error obtained from the study of the similarity to the equationsin studied [ in the nonlinear extens form. As we have noted in section simple condensates. not lead to any immediate issue with unitarity; constraint, not an evolution equation giving any notion of ‘ inner product would have toposition be principle preserved. of The quantum mechanics nonlinearity that w would be expec in the group fieldcondensates. theory context For a of the general Gross-Pitaevskii potential equ cally considered in the GFT literature), this equation is no the minisuperspace of homogeneousdefinition geometries (recall of the i by a genuine physical state. Any other condition of the form ϕ as wavefunctions. Linearof combination motion of will solutions not ofit be the does solutions prevent Gr any themselves, straightforwardcosmology in interpretation equation, general. of as the it This e geometries. would follow from Rather, the canonical againsuggests qu a in re-interpretation analogy ofquantum with quantum spacetime. the cosmology theory itself a of Bose ϕ tial geometries, as anticipated. Again,the this is condensate analogous wavefunction t can directlytum be density associated as wit functions on space. Even though our equation i JHEP06(2014)013 , i i . . ) ξ ϕ ξ | I h + (5.8) (5.9) ( ] = 0 ) (5.11) (5.14) (5.12) (5.13) (5.10) ¯ ϕ ϕ I ξ ) ˆ g I ( + ϕ, . g † [ ) ( ˆ 4 ϕ ˆ V , ˆ ϕ δ ] δ ξ h † = 0 i ) ˆ ) ′′ I ϕ ξ ′ I g . ( + h . Again, we start , . . . , g ϕ, ) ( ˆ ) [ ˆ ξ 1 ϕ I 4 g ϕ ˆ V * ( . ) ˆ + 1)-point functions ] δ , h † ′ I † λ it does not contribute ′ I ˆ g ϕ n ˆ ϕ g ( quantum gravity. Then δ = 0 , obtaining ˆ ]( ϕ ) taken to be the field ˆ i , . . . , g ) + d ϕ, h ξ ξ ′′ 4 ′ [ ˆ 1 I ξ satisfies ) i | g g ˆ ˆ ′ I ) V ξ, O i ( 1 ′′ 4 h δ † ξ le function − | 1 ˆ ϕ ) − I ) Ξ[ ′ I δ , . . . , g ′ h I h . ′′ 1 ( g * g ξ 1 (( , . . . , g ( ) ) with ξ λ ′′ − 1 I ′ ˆ I ) ϕ g = 0 g g ′ I ( ( + 5.7 1 * ξ ξ ϕ ξ , g − I 4 ) ˆ , the two-point function in the state i + λ I ) g is invertible, no condensation of ‘dipoles’ ′ 4 ) ′ and its complex conjugate: ) g ( K ′ 4 4 g ( + ξ ξ K 4 ] ξ K ) (d † i ′ ξ ) ˆ ), here in the state ϕ h i ′ 4 Z -point functions, this equation cannot directly , . . . , g ) – 34 – ′ 1 , . . . , g n , . . . , g ′ I (d ϕ, 5.5 ′ 1 g [ ˆ 1 4 h ( g ) ) + g ˆ must itself vanish. ( V ′ ( ˆ ( I ϕ δ g ϕ ˆ † h ξ ϕ h , . . . , g ) ˆ ) ˆ in deriving the analogue of the Gross-Pitaevskii equa- h ϕ 1 ′ 1 I ′ I (d ) δ g i vanishes, leading us to conclude that − I ′ 4 h vanishes since it only contains (2 ( ( g σ , g Z i * | ) ( ϕ ˆ I ϕ ξ V ′′ ) ˆ ξ I g | h . A simplification occurs if we assume that the interaction ′′ ( g 4 ξ ) ) + 1 = K ′′ I I , . . . , g − i g ′ g 1 ) 4 ) 1 1 ′ I ) has a nontrivial kernel. Then the equation to be satisfied is I ′ , g − I g − I g h , . . . , g 4 ) states that = 0) ), by recursion, one finds that the two-point function h , the two-point function in the state h ( K (( ′′ 1 ( ( ) and the invertibility of (d ¯ ξ g ϕ ϕ ξ ξ ) ( ) ˆ 4.3 ′ I 5.12 4 I ˆ Z 5.11 ϕ = ) g h ′ 5.10 , g ( , . . . , g ) ξ h I 1 δϕ/δ ˆ ′ i I ϕ g d ) g h ( ( I , g h I h K K d ) becomes ( g ′ 4 ( 4 ϕ g ) ) ) ˆ ′ ′ K I g g 5.10 (d 4 g ), the operator ) ( (d (d ′ Z ˆ ϕ g + h Z Z Let us assume that while the interaction is of odd order so that The philosophy followed for . If then, in addition, the kinetic operator 5.12 (d ξ is of odd order, as is indeed the case in many GFT models of 4 Z 0 = is possible; from ( so that ( off by computing the expectation value ( can be expressed in terms of a power series in of to ( Without explicit expressions for general yielding (we use be written as a condition on But the one-point function for the last term that depends on V vanishes, but then ( Using the relation ( As shown in section tion can also be applied to the dipole condensate and its profi Similarly, we can compute an expectation value ( JHEP06(2014)013 ) ) ξ , in  ξ ) 5.11 5.15 ′′ I (5.15) (5.17) (5.16) (5.19) (5.20) , in the and the , g i ′ I ) ) ξ h I I h ]( ξ ( i , h i ′ I ˆ ) ϕ ϕ would change, ) ˆ h ′ I ˆ ϕ I h h i ]( g ( i , ˆ ( ϕ i goes to zero. This ˆ ϕ ϕ ˆ ˆ s of convergence of ϕ to include different ϕ ˆ ) ˆ ϕ ˆ h h ϕ ξ ′ I , ˆ linear term in ( ˆ h ϕ ξ g ) h , ( ) (5.18) i 1 e would be inclined to ˆ for I ϕ nt-function not to grow ˆ Γ[ − I ), the relation between ϕ h . ), but this constant drops ˆ ) ξ ϕ K ) , h ator ξ i ′ I h kh 4 I i ) ) describes non-perturbative I h ) g ′ I 5.18 se Γ[ e condensate, as can be seen 1 g ′′ 4.13 I h ]( ( umber − I ξ g ( i ndensate is diluted. A rescaling i ( h ˆ 5.18 ϕ ϕ ) ( ϕ k δ ) ˆ , . . . , g ′ I ˆ ξ ϕ ) ˆ ′ I d 1 ) h h ′ I g g ( ′ I , ) with the expression given by ( g ( ( i g ( Z ϕ ˆ ϕ ˆ 1 ϕ ) ˆ ˆ ), one can verify that ϕ ϕ h ) ˆ ) we see that I − I ˆ h k = ) ϕ 4 5.15 g ) g I ′ I h i ( ′ I ( ) ˆ ξ ϕ , g , h Φ[ 5.16 I , g 4 h ) as I I h ) I 4 g g ′ ( − ) g ( ( ′ , . . . , g h ( ) ϕ 1 . h K K ) ˆ I 5.11 g K – 35 – (d ) 4 ′ I ( ) 4 I (d g and there is a second branch of solutions for † ′ , h ) ( 4 ′ I ˆ h ) = ϕ ) ξ ˆ Z , h g g ϕ ′ I I ). ( h 4 g (d ) g ) + (d K , h ′ I g Z ]( (d ξ I − i i Z g , g 5.18 (d ) Z + ˆ ( ϕ I I − ˆ g K ϕ Z h ( h ) = ξ ( as a power series in the two-point function and its complex i , I ) = K = ϕ ) i ′ I 4 h ) ˆ I ξ ˆ ) g 1 ϕ I ′ ¸ h 1 ˆ − g I ϕ g ˆ ) and ( , leads to a different normalisation ( in the right-hand side of ( − ( N h . On the second branch of solutions ( g I ) ϕ ǫ ( ˆ ξ (d ϕ ) ˆ ǫ ξ ′′ I ξ h , g I ), we see that the relation between the dipole wavefunction g I Z g ( → h (( =: Φ[ ( ˆ ξ ϕ ξ ) 5.16 h K ′ I , ) = ξ I , g h for small I ) = , the structure of the equation relating the kernels to ) = 1 † g I I ǫ 1 ) if Φ does. Therefore, already without investigating issue − I ( h ϕ g is not analytic, and the two-point function blows up when 1 , h K ( ) is a quadratic equation for is rescaled in this way, the relative contribution of the non I − i I ξ 4 g 5.15 g ˆ ) ϕ ξ ( ′ ( ˆ ϕ ξ g K 5.11 h Focussing on the first branch of solutions ( (d Z powers of ˆ two-point functions is not one to one. The second branch ( branch of solutions isconsider not it as connected spurious: to the we Fock expect vacuum, that, and if w we deform the oper terms of the two-point function. Namely, if there is an inver becomes smaller and smaller,faster than but only if we assume the two poi to get a representation of condensate states that behave non-analytically whenof the the co function out of expectation values.from It looking at corresponds the to expectation a value dilution of of the th total particle n and leading to the disappearance of ( and replace each of the conjugate, sense that Following the same idea, one can rewrite ( solves ( with Eq. ( When the power series ( JHEP06(2014)013 (5.21) (5.22) ]; other 18 , must satisfy . 17 ξ ) for Lorentzian 6= 0 C , ) ′′ acting on the field, I , g S ′ I ntum gravity, one has he terms that admit a traints can be achieved g r quantum gravity, the cuss the last important ( ) to hold, s and assumptions made, tions of motion, and the amics. um level. The qualitative K (or SL(2 ere are several ways to do tion of general relativity as ) be found in [ ion of the partition function s (or spin foams). The sum scretised. The GFT actions ′ ates. However, before being I e continuum theory into the . 5.13 ). by the fact that the quantisa- ) becomes a Wheeler-DeWitt- , g I definition of current spin foam ue. A recent review, containing g variables, and only them. ( 5.21 5.21 ) = 0 K ′ I ably discretised, will be translated g 4 1 ) ′ − g ) ′′ I (d g constraint operator (( Z ξ ) − ′ I ] are indeed designed to provide a quantisation – 36 – , g I ) = 54 g ′ I ( g 1 K − 4 ) ) ′′ I ′ g g which can then be interpreted as a quantum cosmology ) this is no longer true, as we would in general have (( (d ξ ξ ) 5.18 Z ′ I ]. that has a nontrivial kernel, ( , g ] and Ooguri [ I 60 K g 53 – ( 55 K 4 ) ′ g (d Z In order to turn these models into candidate theories for qua The precise translation of the simplicity constraints of th The construction of spin foam models follows the interpreta For the second branch ( In the language of GFT, the imposition of the simplicity cons This is our key dynamical equation, under the approximation to find a waypicture to is impose that the the simplicityinto classical constraints restrictions simplicity at on constraints, the the suit labels quant over of Feynman the amplitudes GFT willgeometric Feynman then amplitude interpretation be in terms converted of to simplicial the geometric sumlanguage of of t discrete models likea spin detailed foams discussion is a of delicate the iss construction of some models can a topological BF theory withtion constraints. of BF This is theories motivated proposed is under by control, Boulatov and [ they can be easily di the linear differential equation models) and modify the action with a suitable For GFT models with a with a modification of the actioninvolves such the that the summation Feynman expans overthis. geometric One configuration way is only. to start Th with the GFT for BF theory for Spin(4) of BF theories. constructions are in [ type equation for a function 5.1 Simplicity constraints So far we haverelated examined problems, only when the restricted general toable the structure to case of do of the GFTingredient the equa condens in calculation the forgeometricity construction concrete or of models, simplicity spin constraints weand foam implemented GFT need in and models the to GFT for 4d dis models quantum fo gravity. wavefunction, encoding some part of the full GFT quantum dyn for a quantum universe described by our dipole condensate. In the following, for the reasons explained, we focus on ( and so it follows that for the quantum equation of motion ( JHEP06(2014)013 c ) I are and G ( ) can ± S (5.26) (5.23) (5.25) (5.24) i − I p , ) diag + I p − I diag − I ,j q + I + I j ( he coefficients q , in general, is SU(2) S / D 4 SU(2) − I . restricts the sum- j / , 4 ) d S ) I + I g j ]). Since we are using ( d ) in upper case. into representations of I diag (SU(2) tor 61 C he gauge invariant field 2 N 4 =1 – e described, it is possible , I D Y I L I hich can be given in terms m, only in the interaction ctive equations for cosmol- can be defined in terms of ) J scopic equations of motion M (2 55 in this section we will write − out the contributions to the 4 of constraint operator valent, since ) ver SU(2) j D ces SU(2) , we still have the freedom to − 4 I + mirzi parameter has a different 4 / re [ . S J j ,p 4 ( d S , translated in the language of + 4 ) ... ions in p features encoded by the model at ) ( − 4 4 =1 − 1 ,j Y I j ) ,..., + 4 4 ) 4 + 1 − 4 j j (SU(2) ( − 1 ,q ( 2 ...J ,p + 4 − ...N 1 ,..., i L . They are, in principle, four independent q 1 + 1 ) ( + p S JJ N i − 1 ( ι → I ,j ,..., 4 4 ) ) ) + 1 − 4 − j 1 ( ...J ,j ,q ...M is a four-valent Spin(4) intertwiner and 4 1 – 37 – ) 1 + diag 4 + 1 j q I − 4 JJ M ( ...J ( ,q 1 4 ψ J + 4 ,..., } q ) − I ( i ...M − 1 + 1 ,N ,j Spin(4) ,..., I M Ji ) + SU(2), / 1 X j 4 S M − 1 ( { × ,q − J, i can be written in terms of a Peter-Weyl decomposition as + 1 q } + i ( I S can be described by its action on a basis of functions. Generi } } J ϕ ) ) X { S − I − I (Spin(4) ,q ,p 2 + I + I X ) = q p L ( ( I { { : g ( − S i ψ + X i can be then specified in terms of its action on a basis, i.e. by t } ) S − I ]. Furthermore, for fixed form of the operator ,j + I X 60 j – ( { 55 )= I The general form of Let us consider first the Riemannian case. The (linear) opera As a result, different models will result also in different effe G . The choice of the precise form of the constraint operator (w ( ˆ ϕ ϕ required to mapfunctions between the in coefficients representations.representation of theory, determine the The the coefficients expansions simplicity of constraints t be decomposed as decide where to applyterm, it or in in both. the These action: three only choices are in not the completely kinetic equi ter functions in the domain of two different groups, to makeelements the of notation SU(2) more in lower transparent, case, and elements of Spin(4) or SL in the case of finite Immirzistructure, parameter (the but case can of be infinite Im easily obtained via a limiting procedu The operator where we are using the splitting of the representation matri SU(2) using Spin(4) = SU(2) the modifications to the effective dynamics. mation over representations by turning the field into a field o S ogy. Given the veryof simple GFT correspondence and between the the macroscopichowever to cosmological micro give dynamics a that specific we correspondence hav between the choice additional angular momenta labelling it. Similarly, funct not a projector. Insum addition, over one amplitudes might of still the have configurations to annihilated worry by ab constraints on the different arguments of the field, and hence of so-called fusion coefficients) distinguisheshand the [ specific JHEP06(2014)013 . , − I − I p j − I + I p + I ). p j (5.29) (5.27) (5.28) (5.30) + I I I p A J N − I q ) C , different + I I − I V q j J I ) + I , γ j N . Apart from 2 ( ) − I 3 I and − 4 (1 J M H j , S + − 4 I K j j y of Spin(4) to the ≃ ( , δ 4 representations into =1 I ... Y I − ) J − p ) ) j − 1 γ + j − 4 count gauge invariance, + 2 j ] to define other models, + 1 ) ) scuss in appendix j + 4 Jj rett-Crane model, is again (1+ Np SU(2) ( − j − 4 , 60 / ( 4 ,i C + ) ,p I hoices for – ... to slightly different theories, j + − ng homogeneous cosmologies, olds also in the case in which + 4 ) one can insert the constraint i C − ...J δ q etic term of the action, in the ( p j 55 1 − , 1 ), by contracting indices of the ( + J j J + 4 esponding explicit expression of =1 f + Y 1 I ,..., j Jj ) , but also where it is inserted) will Mq ) ! 5.23 )( ˆ − S 4 − 1 C j − − − I I J ,p ) i j + q 4 ) j + 1 − 4 + + I + γ I ( i p j 2 ( ,p ( − the Immirzi parameter. Plugging these I ... I + 4 ) J (1 Mq a four-valent Spin(4) intertwiner, and the 4 p , 4 − γ 1 ( C j − I j ...J + 1 ...N ,..., 1 δ j 4 =1 ) 1 J Y I )( − 1 JJ N ) − ι γ – 38 –

,p } i 2 ) + 1 + i − I p (1+ = imposing simplicity constraints can be written as ( ( , ,p ) I + S + I 4 j 4 − 4 p δ ( ,j ) a similar expression, with different weights for the rep- X { ...J + , 4 ...N 1 − 4 j = } 1 ), provided that one deals with the regularisation issues ( S I ,q } JJ N C − ) N + ι 4 p , ,..., { q − I ) ( + ,p = p − 1 + I ) − ,j ,..., ) p ) q − + 4 ( 1 − − j + 1 X { j are known as the fusion coefficients. Their expression is ,j , ( ) ,q } 4 + + 4 j I − f 4 j + 1 ( ( ) one obtains q ,q N ...J ( J MNq { 1 4 + 4 = ,..., S J q ) ) ( − 5.27 − 4 − i 1 determine how simplicity is imposed. ...M j ,j + 1 ,..., + 4 ) + 1 S j M Ji − j 1 ( ( S ,q ... 4 ) + 1 q − 1 from the space of square integrable functions on a single cop ...J ( j 1 4 is a four-valent SU(2) intertwiner and being Clebsch-Gordan coefficients and + 1 J ) S j ι ( − − i 4 C ...M ,i + 1 Looking at the construction for condensate states describi This means that the coefficients For instance, with the EPRL prescription of embedding SU(2) The case of infinite Immirzi parameter, associated to the Bar + i ...J M Ji ( 1 S J J f there are now severaloperators possible (in choices the in appropriateinteraction form) the term, in construction: the or states, inwith in combinations. different the Feynman kin rules. Each choice Therefore, even will with lead the same c in general change the effective dynamics. where the coefficients ways to implement the simplicity constraints (not only where this, this case can be treated in exactly the same way. Spin(4) ones we get coefficients into ( appearing due to the noncompactness of the group (which we di representation matrices with four-valent intertwiners. we obtain for the coefficients of slightly different, and one has to replace SU(2) with SL(2 with coefficients of resentations. It is thenthe straightforward effective to equations obtain for the otherSpin(4) corr models. is A replaced similar with formula SL(2 h If we impose the simplicity constraints in a different way [ a map space of functions oni.e. SU(2). the fact However, that we the must field also lives take on into a ac quotient space in ( JHEP06(2014)013 ) ), ′ I g ly in 1 5.34 , (5.31) (5.33) (5.34) − t least I σ g ( on ( δ . σ = 0 ; (5.32) ) = i ′ I ψ = 0 | general effective , g ) I ! 4 g ) ] ( 4 σ ] K ˆ S consider only the case cannot be interpreted ϕ ˆ S σ, m, σ ,...,G ˆ S , we find that avity defined so far, with 1 [ . ϕ, ,...,G S 1 ˆ V S G [ ( δ approximate) dynamics, and nstraints are imposed at the G ate. It is the ˆ . V σ licity constraints have not yet = 0 ( itting the function ] δ † δ ) GFT (thus spin foam) model. ˆ ϕ ϕ 4 λ . Consequently, it is applicable ˆ S δ d ] ll be allowed. ˆ S σ λ ϕ, ˆ S ) + ) simplifies to ˆ S ′ 4 [ σ, ) + V ,...,G ˆ S ′ 4 ) = 0 (5.35) 1 [ 5.33 λ 4 , we are using interactions for what would be an V G in the kernel of S ( δ ,...,G ] + σ σ as before, this means that in terms of ′ 1 : geometric and non-geometric configurations δ ϕ | σ ,...,G G λ – 39 – σ ′ 1 ( ,...,G ϕ, h [ 1 σ ) is a general result: it does not depend on the G ) ( K G ′ 4 ) + cannot be interpreted immediately as a distribution ( ϕ 4 ) ˆ K 5.33 σ ′ 4 σ ] = ¯ ϕ ,...,G ϕ, ′ 1 [ ,...,G 1 S ,...,G ′ 1 ,G G ( 4 Then, the arguments of the function ). A first possibility is to insert the constraint operator on σ ,G is the component of ) only depends on ˜ 4 4.4 extracted from the fundamental quantum gravity dynamics (a K 10 σ 5.34 ,...,G and multiplying with 1 i ,...,G G 1 ( σ | where K G ( = 4 σ ) K i ′ + ˜ 4 ψ G | ) ′ K (d σ G Z = (d As an example, consider GFT models with a trivial kinetic ter It is important to stress that ( To make the analysis more concrete and easy to follow, we will The quantum equation of motion is Notice that, for consistency with the definition of σ Z 10

cosmological dynamics in the approximate sense clarified above) for a generic 4 plus a general potential. The effective equation ( The insertion of thelevel constraint of operator the ensures dynamics. that the co to all the modelsthe for only (Riemannian restriction being and the Lorentzian) special quantum form gr of the quantum st decouple. as particular form of the simplicity constraint operator of simple condensates ( the interaction term of the GFT action, Non-geometric tetrahedra are not allowed dynamically. Spl SU(2) GFT. as the potential in ( choosing thus of the simplicity constraints, this interpretation wi over minisuperspace. Only after the implementation of the ( immediately in terms ofbeen geometric implemented. variables, since In the other simp words, JHEP06(2014)013 , / ) C can C , → (5.36) (5.37) ̟ × ) diag τ ( f the labels of × (or (SL(2 τ ) Y SU(2) 4 − 20 × / ) ,j 4 he correspondence − 12 + 20 ) , j ,j SU(2) − 20 )( − 17 } + 12 ) × j ,q j j − 19 )) BF theories modified − , 12 − )( 19 − + 17 17 + 20 ,j he net effect of reducing theory is defined directly j ariant theory for Spin(4) j q q ) ,q 15 : SU(2) C − − with the homogeneous space 19 11 generic fusion coefficients, + 19 etric ones or not. Second, , + { )( 19 − q + 16 r, and the correspondence 17 + 12 j ,j ve dynamics for geometric j } j q ϕ q − + 19 19 )( the GFT vertex. Once the hus a specific model chosen, (2 − + diag − 6 11 + 16 q − )( 16 j j ,q s in terms of which spin foam j − q 18 ,j ht be seen as the imposition of − 6 − + 11 6 )( , with no room for additional + 19 ,j q + 10 j + /G 16 q ,q cally, in the states used. Instead − q J 5 10 + 6 4 ,j + 18 ι )( J q j + ,j 11 − the four-dimensional gauge group. ι G q − − 13 )( 18 ,i + ( 10 − j 5 )( j ,q 2 + j − 13 − 17 has been constructed. In fact, X + q − 13 5 ,i )( − 10 + 5 ,j + L 18 j q j q − − ,q 9 + 13 S + + 17 5 − 12 q − 4 ,j ,q )( j q + j 10 j → q − + + 12 9 )( − − 4 17 + + 12 4 j q ) q )( j j − 5 ,q i )( 9 4 J,q + + 12 4 − 9 { + q J q J 17 + 5 − 3 ,q i ι ι q i diag ( ( λ + 9 + 3 − − 11 18 10 i q σ j j ( ( – 40 – ) + − 18 − 10 . + + σ q 18 10 q − ) 16 ) j j + ,j 18 + − 10 4 − − SU(2) 8 3 − 5.3 11 q q j ,j j + 16 / ,j ) − 3 j − 11 + 3 + 4 4 + + q 8 11 q j − j 16 ) )( j j ), according to the case that one wants to consider. This ) 3 + 3 8 ,q + )( 11 − 4 J q )( − 15 − 8 C q J ι − 3 + ,q 16 ι − ,j 7 , ,q q ,j + − 4 15 ,j − 9 + 8 + 15 j q )( j + 3 q j + 7 − 15 (SU(2) j − 9 , embedding directly the simple components, identified with )( j + + 9 15 − q 15 )( )( 2 q j j )( S − 3 )( ,q − 7 + + 9 15 − 14 L − − 8 2 − 2 ,q q q − 6 ,q j j + 15 ,j : ,j + 3 − − q ,j 8 2 + + + 7 8 2 q + q q 14 + )) theory to an SU(2) theory, one can start with a group field 2 q j j + )( 6 j j ̟ 7 2 j + + )( 8 2 )( C − q J J q )( 14 )( − )( 2 , ι ι − 6 ,q − 1 − 13 ,q − 5 ,q − − 14 20 ,j ,j + 14 + ,j 2 j j + 6 q q + − − 1 14 20 q + + 5 13 + + j 14 20 q q )( j j )( j j )( )( + + 14 20 )( )( − 1 − 13 − − − 7 1 5 q q − 1 j j − 2 − 4 ,q ,q i ) denotes the product of the fusion coefficients associated to ,q i i − − 7 1 + + 7 1 + 1 + 1 τ q q + + 5 13 + 2 + 4 j j i q ( i i q q 6 1 ( ( + + 7 1 )) . ( ( ( ( J q J q f σ σ ι σ ι C τ for Barrett-Crane-like models), involving some field is either Spin(4) or SL(2 , SU(2), see the discussion in section / Q 4 G ) 0 = C There is a second way to impose the constraints, which makes t This can be done using a suitable map For instance, we can take a simplicial interaction term. For The model then can be constructed in terms of a GFT over SU(2) There are two advantages in this second approach. First, the , In the case of Barrett-Crane models, one needs to replace SU(2) 11 with the spin foamwith geometric models data treated more direct. inthe In the constraints a directly literature certain in sense, more the thisof clea kinetic mig working term, with and, a automati the map like Spin(4) the (or one SL(2 described above, which has t the five tetrahedra ofspecific values the of single theone four-simplex fusion coefficients can associated has then to been tryconfigurations. given, to and t solve this equation, obtaining the effecti where theory defined over four copies of SU(2) and embed it into a cov (or SL(2 the equation in components gives: where SL(2 SU(2)) in terms of thedegrees of geometric freedom, variables whether (i.e.it they makes simple are straightforward bivectors) decoupled the comparison frommodels with the are the geom defined, amplitude i.e. in terms of amplitudes of Spin(4) (SL representations of SU(2), into the full representations of be seen as the transpose of map can be constructed in the same way in which by the insertion of the simplicity constraints. JHEP06(2014)013 (5.41) (5.38) (5.39) (5.40) ) (and related . 5.40 are Spin(4) ) determine our-simplex, I = 0 G 5.29 i ψ Spin(4) | } j ! fore, ) 15 4 ] FT for the full four- ), where { I symbol, with the appro- 1 ook a bit more compact. e solution of ( then the proof that these 1 ̟ϕ G , M J 5 ] the physical states on one es and fusion coefficients. It denotes the five tetrahedra 5 pin(4) representations. The )( ing. The various spin foam , . . . , g J   M he operator appearing in this 8 1 8 f τ modification in the equations. g an explicit solution, since it ̟ϕ, J ̟ϕ g Spin(4) [ ̟ϕ M cific models. All this is indeed ( A . tion itself is rather complicated. } 10 ˆ † V 10 J j ˆ 2 δ ϕ 5 =1 10 i M dan coefficients in ( ̟ϕ, δ Y f  15 [ σ { − f λ   H − f q 10 j V + f 10 M + ! f q J j 2 ) λ f ) + 2 f τ ′ 4 J M J M ( 6 6 J f ] + C M 9 ϕ 9 J  5 4 =1 i M Y − τ , . . . , g ϕ, – 41 – σ [ ,q ′ 1

9 ) is the fusion coefficient associated to the given g + 5 9 ( τ M X J ( 8 ϕ 8 ,...i M,q SU(2) ) ˆ f J M 1 ) according to the signature chosen. ′ 4 3 3 K ,i associated to the faces of each tetrahedron, J = C ) describes the dynamics of condensates as determined by M 7 , 10 X 7 f J f 3 ] = i M A A ...J ¯ 5 ϕ σ 5.40 , . . . , g J 7 ′ 1 7 ϕ, λ M [ J , g 6 6 4 S + J M , we obtain the equation (in Riemannian signature) 5 5 i 4 4 symbol for SU(2), labelled by the representations of Spin(4) J σ M 4 | } 4 ...J J j ...M 1 2 i M 1 , . . . , g J 1 1 σ 15 we denote the Spin(4) invariant with the combinatorics of a f i M g { ( σ × K 4 ) 0 = ′ Spin(4) g } ) group elements: denotes Spin(4) or SL(2 j C (d , 15 H Z { This detailed discussion shows that the true challenges are Once more, the equation ( The operator equation of motion is analogous to what we had be

condensate states canhand, be and used the as development reliable of approximations methods of of approximation for th where or SL(2 similar equations obtained forwork other in condensates), progress. for spe the chosen spin foam/GFT model, encoded in the face amplitud the additional face amplitudes is clear that further simplificationsNonetheless are it needed, as is the useful equa highlights to the display generality it of evenmodels the without proposed procedure proposin so that far are weThe easily Lorentzian are included models propos with lead no as essential well to the same equations. with the interactionsdimensional gauge now group, evaluated being on the embedded the field ( interaction terms for a G Considering again only the caseequation of in the the expectation state value of t priate measure, is left implicit. Finally, the Clebsch-Gor tetrahedron with the given decorationsummation in over terms Spin(4) of representations SU(2) entering and the S analogous to the where we are usingWith a shortened notation to make the equation l associated to the four-simplex, and to the ones of SU(2) by the fusion coefficients. The label JHEP06(2014)013 ) for 5.21 B k (5.42) (5.43) e is sim- A j e jk i ǫ ) which sta- 3 = , AB ensate states from i B . This is the type of ) by group averaging, . 4 Spin(4) e of how one should go ions more concrete in a smological dynamics for ⊂ t action of Spin(4) on the e vertex term, not only in 5.43 d in the literature, leading rections depending on the 0 the dynamics in spin foam or used in the definition of densates in a specific simple equation would result from s that are well-behaved in a X is simple example, which in rees of freedom, and can thus Spin(4) s what one would expect from y constraints as well, to ensure poses of this section we impose ]. ion in our general scheme: take e for which ∈ and taking a weak-coupling limit e dynamics reduces to precisely ), which can be understood most ) and ( 61 h SU(2) σ ∀ ∈ 5.43 5.42 I has to satisfy the linear equation ( ) h h ξ ) can, in a semiclassical (WKB) limit, ∀ 4 ) and ( ) 5.21 4 h 4 5.42 – 42 – h, . . . , g 1 g ( ϕ , . . . , g is really a function on four copies of the coset space 1 h ϕ 1 ) = g 4 ( ). ϕ . When one imposes ( , defined on four copies of Spin(4) and satisfying the gauge 0 5.6 ϕ X , . . . , g ) = 1 4 g . Hence, ( , or equivalently four copies of SU(2) (since Spin(4) itself 3 3 ϕ S SU(2) S / ], by requiring that , . . . , g ∈ ∼ 1 g 0 62 denotes the subgroup of Spin(4) (acting transitively on S 0 ( X X ϕ 0 ). SU(2)); in this section we work with a field on SU(2) X A i e × SU(2) / We have discussed the most general constructions introduce For generic GFT models, we have seen how to construct GFT cond There is an issue with imposing both ( In this section, to make the previous rather general discuss invariance property In order to obtain modelsbe relevant that for can cosmology, include we now gravitationalthat need deg one to describes consider only simplicit geometric configurations (i.e. thos the elementary GFT field some triad simply by noticing thatcoset there is space no Spin(4) natural way to define a righ which is of Wheeler-DeWittconsidering type. the simple condensate Similarly, given the by same the function linear reproduce an effective cosmologicala dynamics that classical resemble gravitational theory;the in classical the (vacuum) Friedmann isotropicchoice equation case, of with th state. quantumfact cor We should will not also bemodels highlight taken and too the group seriously: limitations fieldthe theories of for is kinetic one th supposed term. thing, to most Theabout be of example extracting encoded an has in approximate to th classicalthe be geometric fundamental taken equat GFT/SF only dynamics, assome a and simple templat condensate extract state. an effective co specific model, we show that the equation ( of the non-linear equation ( 5.2 Effective modifiedLet Friedmann us equation: now look a at concrete theexample. resulting example effective We dynamics have for seen GFTthe con that, GFT assuming has that a the nontrivialregime kernel kinetic of and low operat particle that density, one the ‘dipole’ focusses function on state to different models, at length.simplicity as For concreteness, in for [ the pur bilises a fixed where SU(2) Spin(4) ply SU(2) prescription corresponding to the Barrett-Crane model [ JHEP06(2014)013 or- ) = I ction g (5.44) (5.45) ( iven by ξ ], where teraction 61 ) while those 4 X , . ] (5.46) satisfies 1 variance property − ξ , . . . , h 1 SU(2) an be removed from h g h / [ ( † resolved in [ , reducing the Spin(4) Spin(4) Spin(4) 0 ϕ 7→ ) ˆ ∈ ⊂ ] her with invariance under 4 X uce again to a formulation After imposing simplicity, g h space of the dipole function [ X ∀ of a single tetrahedron, but a r geometric significance at the , etation as ‘Hubble parameters’ invariant. In the GFT model y. ) e can always work at the level of , . . . , g X 1 0 · SU(2) X ), the function g h · X ( symmetry, as we do in the following. † ∈ h to a fixed 4 ϕ I ]. The simplicity constraints are then ; 5.43 ) ˆ 1 h 4 X 64 − ∀ SU(2) by rotating around a fixed axis given by h / , h 1 . 3 4 . The map 0 − is just a function of the ‘absolute values’ of 4 ) by ) 63 X X · ξ X h ) are elements of Spin(4) ; 4.10 4 – 43 – , . . . , g which is invariant under separate left actions of h Spin(4) , . . . , g 1 4 5.45 1 SU(2) 0 − / acts on S h → SU(2) that leaves this X h 1 / 0 1 − 0 1 X . It is hence a function on four copies of the space g X g ( 0 X , . . . , g ( , ϕ X 1 ξ i SU(2) h 0 4 | / 1 ) SU(2) g ) =  h / ( ˆ ξ on Spin(4) ϕ SU(2) X (which is simply a compact interval), or alternatively a fun (d  0 ; 4 0 4 ) X ∈ X g Spin(4) ) = ) to gauge-fix the normal ′ . This SU(2) I (d ) exp 0 X . In the case considered here, this action would be trivial, g ∼ ξ ; : Spin(4) 0 X , k ( Z 4 3 , . . . , g I 5.45 X 1 is added as an argument of the GFT field, thus basing the whole f 1 SU(2) k 1 2 − N g / 3 h ( ] there is no explicit coupling of normal vectors in the GFT in S R ϕ := := 61 , . . . , g ∈ ˆ i ξ 1 ξ g | ( X Spin(4) is now the argument of the field and no longer fixed; the gauge in ϕ ) for all \ ′ I elements and take care of the additional SU(2) 0 X k X 3 I Let us now assume the normals have been gauge-fixed and hence c Once simplicity constraints are imposed, the configuration One can use ( g I . We will adopt the second interpretation. k 0 is no longer simply the gauge-invariant configuration space ( the formalism, and define the condensate ( malism explicitly on projected spin networks [ four parallel transports. Theysince do not they admit are a still directlevel subject interpr of simplicial to geometry a is not closurefour obvious S condition, at this and stage. thei On ξ quotient of that by the left actions of SU(2): proposed in [ the right action of SU(2) as previously. Because of the simplicity condition ( invariance to the subgroup SU(2) X between coset spaces is well-defined, as one can easily verif on the right-hand side live in Spin(4) a normal imposed by term, and in this sensewhere they have the no GFT dynamics. field One dependsan can only action then on red of four SU(2) copies of SU(2), toget four copies of SU(2) where on four copies of S ξ one finds that the two operations do not commute. The issue was SU(2) is then This is now consistentthe with arguments the on reduction the left-hand to side the of coset ( space. JHEP06(2014)013 , for 0 (5.50) (5.51) (5.48) (5.47) (5.49) X atically ) for any ′ k I , corresponds al connection kg π to every group ( with additional ξ stabilises π 4 ysis, was missing 3 ) 0 (2) is ) = . I X ariance property; the ) then becomes (setting g π , ocus on models whose ( on S ξ ), ( ξ ng that only three of the lement he effective cosmological 0 f . Since we assume that for α X tself, which is very much SU(2) ∂ ective wave function admit 5.42 ian seems to be required by / nential reduces to the usual α , . SU(2) , together with a ‘mass term’. π lgebra element 1 ) U(2) 4 | ∈ 3 se parallel transports are close ce variables. 1 | , h 1 ~ω − | ≤ | ∀ ~ω ) | ~π | π (Spin(4) ( ) sin( ) = 0 f h ′ I 1 ∼ β g 4 ~ω ∂ ) for the function ( ~π, 4 · α ξ · ∂ ~σ ~σ ! i  5.21 µ β − + i π h, . . . , g + – 44 – 1 α 1 1 ′ I ) g 2 π g | ( ~π 1 ), in coordinates in which the corresponding link has ∆ ϕ − ~ω 1 | − ω I αβ 1 X ) = δ direction. Expanding this in the basis of Pauli matrices, 4

p  1 x = cos( = : if we assume that the gravitational connection is constant ]. The equation ( g g g ]) = ) = exp( g , . . . , g 69 [ ω 1 is a function on SU(2) – are now in SU(2). Concretely, we impose π g R ( ( I ξ 65 f ϕ g g and are independent. This condition is the one we would get autom exp( ∆ 1 ξ ~ω P · ~σ = satisfied an SU(2) closure condition of the form ( g = i ϕ 1 . We can associate this Lie algebra element with a gravitation are the Pauli matrices, the Laplace-Beltrami operator on SU g ω i SU(2). Note that this is not a special case of the previous inv which are arbitrary equal to the identity) σ ].) This parametrisation of SU(2) associates a Lie algebra e ′′ I ∈ After all these preliminaries, we can now proceed to derive t To proceed, we assume that there is a closure condition meani g ′ 13 (Note that thein second [ term, which will drop out of the WKB anal dynamics from GFT modelskinetic with operator is a the particular Laplace-Beltrami kinetic operator term. on SU(2) We f A motivation for this choiceGFT is renormalisation [ that the presence of the Laplac to the “sinereminiscent of of the what happens connection”, inour rather loop configurations quantum than cosmology all (LQC) the gauge-invariant combinations connection of i the symmetries. Using the parametrisation for SU(2) given by unit coordinate length in the we have where In this sense, the fundamental dynamical variable, the Lie a element where we recall that exponential, over the dual link we are considering, the path-ordered expo whose parallel transport is four arguments of the if the field k, k where the arguments instance. With allnow this the being required done, geometric interpretation the as arguments minisuperspa of the coll action on SU(2) on itself is transitive, while the action of S JHEP06(2014)013 , (2). O ~π ]) = su I (5.55) (5.56) (5.54) (5.52) (5.53) g → [ . First, ase, and of SU(2) ≃ I with the ~π π 3 ( 3 , ξ R S ~π , the variables ∈ S , bstitute the co- 0  ] are proportional as X I I π [ B . ), does not appear. In ), rewrite S 2 I I I ~π as rotations of the three- I κ ~π or this scheme to be self- ~π ~ ∇ 0 and 5.48 × X × ( nical pairs of dynamical variables I , ′ × × π ) ~κ ~κ oordinate vector I ~η κ ar-Barbero variables in a Bianchi IX ~π nection at leading order. is the momentum conjugate to ( − + +  2 2 I O · ions (acting on all four arguments ′ 2 ). Identifying ~κ I I ~κ ) contains only the leading term in 2 ~π = ~τ , and the corresponding Lie algebra − | Z −
1 / ] vanish; − 2 1 I 1 SU(2). The transformation property of 5.52 I ) ~ω X ∂S/∂π 1 I p , and hence the function p ∈ I ,B ξ B I q ′ ~π ≈ | I ] + ~π · := ~ǫ SU(2) I π ) I I | + + ~π is the standard cross product in + π [ k, k 1 . Eq. ( 2 ( B I ≃ 4 2 = [ ∀ S ~ω – 45 – ~π I × | I ~π , I − ~π , being of higher order ( ≃ )] µ I = B ′ ] has to vary rapidly compared to the modulus, which − − k I I B × 1 ξ I · 1 0. The equation we obtain is ~π I SO(3) δ ~π where I q ~π (2) and kg q × ′ ( → B ~κ + I ~π su ) becomes the Hamilton-Jacobi equation for the classical π ~κ → ~τ [ κ I ~π × S I ) = + X I ~τ ) = 5.52 I ′ 7→ under separate left actions of SU(2) k ~π kg + [ I I under a left multiplication is )] = ( ξ ~π g I S ~π π ( π g ( π π Second, we have an invariance under the simultaneous action → ] = [ I ) are not all independent. Let us consider this in more detail S ) in terms of slowly varying amplitude and rapidly varying ph ~π 12 ~π [ . Similarly, under right multiplication we have under these transformations tells us that /κ S ] ~κ 5.52 I (2). S π [ su S ]. ) =: k 70 is the Killing form on . ( S · π ] exp(i Because of the symmetries of the function In order to take the semiclassical (WKB) limit, we can then su ) can be parametrised by I In class A Bianchi models, this condition can be satisfied for the cano S π , a bivector associated to one of the faces of a tetrahedron. F 12 [ I the SU(2) coordinates Infinitesimally, the six independent possibleof transformat where the WKB expansion, and the term in on all arguments, action there is an invariance of A by choosing them asmodel diagonal in in [ a given fiducial basis, e.g. for Ashtek to the identity, we can approximate sin( take the (formal) eikonal limit the WKB approximation, ( sphere (using the map SU(2) elements of consistent, the phase of the function π is itself peaked near the identity in SU(2) identity in SU(2), this action corresponds to rotating the c or infinitesimally elements can be interpreted directly as a gravitational con ordinate expression of the Laplace-Beltrami operator into appearing in ( where i.e. the WKB “angular momenta” Invariance of JHEP06(2014)013 ) 4 = 1. G B ors i ± and 5.59 A i to set (5.57) (5.58) (5.60) = erpret i , which under a i S V · p i S T are pairwise , . i and )  T ] ) is the effective i κ means that I ( A π ~ǫ = SU(2), where ) (5.59) [ O κ 5.59 S ( G = 0 and therefore we I = O 4 j ~ ∇ ~π iables 2 I B = · ~π a geometric interpretation 2 j i set of the function ), just as in loop quantum p − p onnections that were flat on B to be chosen to interpret the − 1 is a dimensionless normalised ) then becomes (indices (2) element 2 j hree-sphere. Invariance under mics. Before discussing ( 1 i ~π ometry. Here we can set q infinitesimal parallel transports, su V , this is the classical Friedmann q −  5.52 κ 2 i , with the modification of replacing 1 p k I , and X q − = 0. This can be solved to express = 0 (meaning that all by an · 2  i with cosmological variables, we follow i 1 I 2 ~π ~ǫ T I i κ ij a g B π q γ −  2 I ] + ij 1 depend on the state, and we used ~π I γ O j j π q [ − T = A · S i – 46 – 1 i k i,j T A X ) becomes ≃ q j − I ) vanish:  = + , and we further assume that the state is such that 6= 2 i I i X
, where each ν p 5.52 ij P i ~π 2 . Furthermore, we can use the invariance of i γ ) + V 5.59 i i + and conjugate B
p . At leading order in B 2 2 i I and
i · p ~π 2 i = i i B 0, this interpretation is consistent when ν µ − π i − 2 i ( π 1 µ 1 i 1 for our setting to be consistent, which also allows us to int − k > is an area element which we identify with the usual scale fact 2 i q i i P ~ǫ A and ≪ B  A 2 · i i / S i p T
i i 2 i B X µ A ] = (2)). We then obtain i I i for constants su π = and cyclically. Then ( [ X P p i 3 S i a B ν 2 = a = k i = 1 In order to identify the At the level of the WKB approximation that we have employed, ( We now recall that our reconstruction procedure, providing = 0 by a suitable left multiplication. The equation ( A A, p run from 1 to 3) i 4 the anisotropic contributions to ( µ can be interpreted in terms of classical gravitational dyna which implies the conservation law orthogonal in in terms of the other momenta where the dimensionless quantities j where in full generality, let us first specialise to an isotropic ge ‘Hamiltonian constraint’ satisfied by the WKB phase space var simultaneous translations of the group elements corresponding to translations androtations rotations has acting on already the been t used previously. The invariance their geometric interpretation as bivectorsand and write conjugate the connection by its sine (here represented by the variable to the variables appearingthe in scale our of the quantum tetrahedra. states, We required have used c the gauge freedom to equation for an empty universe with spatial curvature should assume that cosmology (LQC). Since as is the group ofcondensate isometries states. of spatial hypersurfaces that has Lie algebra element. ~π JHEP06(2014)013 e . 2 i A . For (5.62) p i X 1 2 ) at lowest − ) (5.61) 2 3 i 5.60 .  3 2 3 κ, p foliated by round A ( isfied by the GFT A 1 are the scale factors dependent of ). Since we assume 4 O 2 A i R a e this as an indication s universes in General  orrections coming from 5.59 ) = + 1 was not needed for the 2 j ction procedure needs to 1 condition does not come of left-invariant one forms quation so obtained, and 2 n the group of isometries p . The terms neglected can } i g at. The GFT dynamics  ve to be improved to fully ≪ i ur variables, i.e. for relating le of various distance scales ≪ the above effective equation. p + ravitational dynamics which 3 ical Friedmann equation, has e e interaction term), quantum i s. 2 umes it to be the Levi-Civita 0 2 i p p A ase seems to suggest that this { p e domain in which our classical A 1 ibed in section ( 2 he A ij ), including both the subdominant γ  j + A i 2 5.61 A  , 3 j i 1 p 6= A i X and cyclically, where A 2 2 3 1 2 A a 2  − – 47 – = SU(2), which we derive for the convenience of a ) for a given basis 2 i + ). The assumption k A e G = ˙ 3 a N 2 i 0 p ) is of order one, so that the condition ( 1 p 2 ⊗ A ∝ p k i 3 e p 5.60 X ) in powers of A 0 ), nor for the definition of the quantum states, but it is ( 2 2 2 k A a − = constant would correspond to flat 5.59 5.60 k + ij p 3 γ . In metric variables, one finds a kinetic term quadratic in th P p j B 1 A p i 3 A A appearing in ( 1 j k 6= A i X + + 2 2 i plus a potential representing spatial curvature which is in p 1 A 1, in order to interpret the effective classical dynamics sat i p p 2 i ≪ X A 1 i 2 p A Let us now return to the general anisotropic case given by ( The constant In the general anisotropic case, the dynamics of homogeneou = directly as the gravitational connection (which, if one ass H from the dynamics of theMoreover, theory, we nor is have it already neededbe pointed for further out deriving developed, how the in(as same particular in reconstru loop for quantum whatthat cosmology). concerns our choices the However, of at ro GFTensembles, the dynamics (here condensate moment totally we states, neglecting tak reproduce th and General approximation Relativity in scheme a ha cosmological setting. spheres of varying radius.)assume its validity One and could geometricreconstruction of interpretation procedure beyond would course allow th it. stick As to said, in the fact, t e the reader in appendix terms in the WKBthe approximation of higher the order terms above equation in and the effective the cosmological c dynamic Relativity is given byof one spatial of hypersurfaces. theshould Consistency Bianchi be with models, the the Bianchi depending IX isotropic o model, c with momenta imposed for the geometricdiscrete interpretation to we want continuum to variables, using give the to procedure o descr order in the WKB approximation,no while consistent identical to solutions. the class ( that derivation of the equation ( condensate, we can expand ( p instance, for the choice of variables on SU(2), the (Riemannian) Hamiltonian constraint is where we are only looking at terms up to quadratic order in appearing in the 3-metric be viewed as higherbecome derivative negligible corrections in toallows the the also to low-curvature effective compute regime g explicitly the we corrections are to ( lookin connection of an FRW metric, is JHEP06(2014)013 (5.64) ments. , ) 4 . ) (5.63) . ) cal Bianchi IX 3 3 , h , h 2 2 , . . . , h 1 , h ) where the previous , h h 1 a tuning of the coeffi- 1 ( icular, from the LQG ‘anisotropy potential’, C h † h , ( gration. Repeating our ] does not contribute to ( ϕ † † ¯ ) ˆ ϕ ˆ have made certain choices ame definition. We could ψ ˆ 4 ψ ) ) es shows that these choices sate states in GFT. is group. 3 ϕ, neglects higher order multi- 3 [ hich only the kinetic term in the derivation of an effective cs. Nevertheless, the example to be improved in order to be ainly have to involve the GFT , g V rahedra are glued to simplices , g for GFT models corresponding 2 2 f GFT dynamics, which is in the , . . . , g is redundant, and so, using the , g 1 , g 1 g 1 5.2 4 g ( ) and change variables to go to a g ( h † 1 ( , the only additional difficulty is that † † ϕ − 4 ˆ A ψ ) ˆ ˆ g ψ ) 4 3 ) h and , e 1 , g , e 3 1 3 4 − 4 h g h − 4 1 1 g ) giving the dynamics of our GFT condensate − 3 2 − 3 – 48 – , g , g , g 2 , . . . , g 1 ) for the case of gauge group Spin(4), 2 5.61 1 h − 4 h 1 h g 1 1 − 1 2 − 2 4.10 − g 1 with ( , g g , g 1 ( ψ 1 ξ i h h 1 0 4 1 | ) − 1 − 1  g h g ) = ˆ ( ξ ( 4 ξ (d  ξ under simultaneous left or right multiplication of its argu 4 4 ) 3 ) ) ξ g h h (d ) exp , . . . , g (d ) for non-compact gauge groups such as SL(2 (d ξ 1 4 3 ( Z ) g ) ( g ), none of the possible choices exactly reproduce the classi 2 1 g N , that should be interpreted as a non-zero spatial curvature 5.64 ϕ p (d (d := := 5.61 Z Z ˆ i ξ 1 2 1 2 ξ | in ( = = ij ˆ ξ ˆ ξ γ While there is a freedom in the choice of state that amounts to Again, the semiclassical analysis of anisotropic condensat cients normalisation of the Haar measure that Vol(Spin(4)) = 1, definition would be ill-defined due to the infinite volume of th we can replace formulation without redundant integrations. One obtains symmetries of the GFT field can lead to divergences under inte to Lorentzian signature. As we discuss in appendix definition of the dipole condensate ( states has the general form of a quadratic kinetic term and an independent of dynamics in the truncation toin quadratic obtaining order this in momenta. result:particle We correlations a and choice leads ofthe to condensate GFT an state action effective which dynamics contributes,perspective to and an w a approximation in choice which of the GFT GFT model. potential In part the effective equations cannot capture anpotential essential and aspect its o prescriptionor for other determining building blocks. the A wayspotential better tet in approximation some would way. cert The effective ‘Hamiltonian constraint’ ( It is now explicit that the integration over using the invariance of (reduction to kinetic termable only, quantum to states, reproduce etc.) Generaldemonstrates Relativity have in the the general effective applicability dynami semiclassical of dynamics our from procedure the quantum for dynamics of5.3 conden Effective modifiedWe Friedmann can equation: now Lorentzian try case to repeat the constructions of section For the compact groupnow Spin(4) try this to is use just ( a rewriting of the s JHEP06(2014)013 , ξ A 5.2 (5.67) (5.65) (5.68) (5.66) (5.69) ) which , a group ogeneous 5.52 , ) en gauge-fixed . In Lorentzian Spin(4) C ϕ , . This means we ∈ ϕ sists of a Laplacian k ∀ d SL(2 . However, this is just is now interpreted as a ) ⊂ ξ , e o an extended formalism 0 diag straints, which, as we saw , are still infinite for gauge 1 nce, we do not work with ) 2 X , − | scription used in section . C ) ) k entzian signature gravity. i , ) 3 π h ( mvented with a suitable choice κ h 1 ( f 1 , − SU(2) i β − O 3 when compared with ( SL(2 g ∂ ( ∈ / α π = ξ 4 i | I , kg ∂ )
3 ) = 0 1 h 2  ) ′ C I − → ∀ ) , β g h I k ( π ). The function π 2 ξ (d B α ) h . For the purposes of extending the example 3 · 4 that are the analogue of the coordinates on 1 π ) 3 ! h I g − 2 4.10 µ 3 4 + π (d + – 49 – , g , kg R 3 αβ ′ I + ( 1 is not normalisable. As we show in appendix δ g h I − 3  i ∆ k B ξ 1 , g | · ) reveals that, due to the invariance of the function SU(2), separately invariant under left actions of SU(2) 2 I h I ) = / 1 h X ) 2 π − B 1

( C 5.64 , g ), such as , f kg 1 3 ( I h H ξ X 1 for a discussion of this group and its geometry). As before, 4.11 g ∆ ( C ) = ϕ , e 3 ) = h 4 1 − 3 , g 3 SU(2), which is 3-dimensional hyperbolic space, or Hom(2) as , g . The GFT field then becomes a function on four copies of the hom / 2 ); the resulting state , g to Lorentzian signature, we shall assume the normals have be 3 ) , affect the gauge invariance property imposed on the field h 2 ) reduces to C C 1 H , , , g − 2 5.2 5.2 ∈ 1 g 5.21 , g 0 ( 1 ϕ X h , and there is no further gauge invariance property of the fiel 1 0 Again, we can assume that the kinetic term of the GFT model con This discussion ignores the imposition of the simplicity con However, a closer look at ( − 1 chosen above, the Laplacian is X g ( 3 ξ so that ( S and a mass term. For the coordinates on H on the four arguments. the integrals appearing in ( a technical, not a conceptualof problem, gauge which fixing. is easily circu function on four copies of SL(2 signature, imposing them as we did for the Barrett-Crane pre group SL(2 is precisely the transformation between Riemannian and Lor in order toincluding impose a normal gauge that invariance is now properly, an element one of H should go t under conjugation, this state of affairsfunctions defined is over due the only correct to space, i.e. the SL(2 fact that, for convenie of section means that we now require the WKB approximation will give in section where This corresponds to the “analytic continuation” to space SL(2 can proceed as before, using the definition ( manifold (see appendix JHEP06(2014)013 ) h ). O ~π ) we 5.69 5.69 (5.70) (5.72) (5.71) (5.73) (5.74) (5.75) → 5.69 ~π , namely , ξ 5.2 . The most , and ( , ξ i . V )  i ] , κ p I ( . π  [ ) O ] = I κ S ( I π = i [ in terms of the other π O j ~ ∇ S 4 I 2 I B I ), which we were not able = B · ~π ~π ondition on normalisation of the state ed in section ~ s not correspond precisely ∇ 2 j i and p condensate state where the ly to the change of signature thing seems consistent as we × × B ons. For the purposes of this 5.59 i 1 + 2 on hyperbolic space, I j larisation explicitly. We then ~η , the structure of the equation T e. This is again a non-compact ~π n ~π i 0 1 + q +  X A  · 2 q I 2 i 1 + I ~π I p = ~τ , X q ) on all four arguments of i · 2  i I C 2 B X ~π ~ǫ 1 + 1 + κ , a = 0 to express q q  ] + ] + ~ǫ I 1 + ij I I O γ B π + ~π j q [ [ 2 I = I A S S – 50 – ~π ~π i = 0 by a suitable translation. From ( k i,j X A ≃ ≃ 4 = + ] j give constraints on the variables appearing in ( ~π  + 1 + I I 2 6= I i
X ~π p S ) is cut off at some ‘maximal boost’. This regulator ~π q 2 δ ~π ) C I × + i + , + I , means that
non-normalisable. However, the linear effective equation B 2 I P and 2 i ~τ I ~π · ~π ~π i p ξ i ξ + × | π I 7→ ~τ 1 + 1 + ~π [ I + ( + ~π q i 2 i S ~ǫ ~π B  A · 2 under separate left actions of SU(2) S ] = i → = 0. To identify the quanta that make up the GFT condensate wit I i S B I ~π ~π X [ ] = B S I i π × [ X . We can then also set I i S ~π B Since this is simply the analytically continued version of ( Again, the symmetries of natural choice is to use the condition analogous to what is us and again geometric tetrahedra, we then also need to impose a closure c and so, as before, we can use which are the translations andsymmetry rotations which of will hyperbolic spac make can be taken to infinity without affecting the effective equati calculation, we can continuefind without that considering this regu momenta we have derived for the dipole condensate does not depend the explicitly. One canredundant hence integration define over a SL(2 regularised normalisable Again, we can introduce cosmological variables becomes to match to any Bianchito model any in Bianchi model Riemannian signature, inshows that Lorentzian it changing signature. doe the GFT Nevertheless gaugein group the corresponds metric precise formulation thatargued. one would In expect, particular, and going so to every the isotropic limit we obtai thus obtain The invariance of invariance under the simultaneous action of SL(2 or infinitesimally JHEP06(2014)013 (6.1) . It is s in this dom in the ]. 75 should be related – κ 73 y the mathematical e eld. In fact, here the eory of pure quantum in cosmology. A scalar dimensional reasons, we essary in our context. We to do what one would do can be seen as a regulator iable used in the quantum needs to be careful in order e obtained indications that ompactify the scalar degree of the positive curvature of erspace to include not only models captures the degrees ζ of a spin network, or in the ogy, it is essential to be able so followed in loop quantum of a classical theory of gravity. tions in the gravitational field eresting result which suggests ron. s of the GFT dynamics, i.e. look lly closed universes, and that the , )) x ( ζφ with dimensions of length, so that the new ζ – 51 – ) := exp(i φ ( . Note that the only dimensionful parameter that has are related. We will see that in the effective Friedmann ζ,x ζφ U κ := ˜ φ and ζ which is negatively curved. 3 (which has dimensions of area) used in the WKB approximation . This last route has been tentatively explored in [ ]). An alternative would be to look for matter degrees of free κ 72 , ] and can be adapted easily to the GFT setting (for matter field 71 16 determines the coupling of matter to gravity, and so just lik – ζ 14 Let us consider a (real) scalar field, the most natural matter In continuum loop quantum gravity, in order to be able to appl The restriction to compact groups does not seem strictly nec emergent matter context, see [ effective theory of perturbations overfor background solution under diffeomorphisms is naturallydual associated simplicial geometry to picture the to vertex an elementary tetrahed framework of compact groupstheory to is a a scalar ‘point field, holonomy’ the physical var where the spatial curvaturethat is the models still we positive. investigatespatial generically This curvature describe we is spatia obtained anthe before int gauge was group not simply SU(2)configuration a used space result is as H configuration space for the GFT fi appeared so far is 6 Beyond vacuum andThe homogeneity: GFT matter models and we perturbations have discussed so far are candidates for a th geometry. We have seen thatof a freedom certain class of of spatiallytheir states homogeneous dynamics in can cosmologies, these reproduce, and under aHowever, we few in assumptions, hav order that to connectto to describe any two realistic more ingredients: modelcorresponding of matter to cosmol fields inhomogeneities. and perturba 6.1 Adding matter:The a most scalar direct field wayin of Wheeler-DeWitt including geometrodynamics, matter that degreesthe is of gravitational to freedom but enlarge is alsogravity sup matter [ fields. This approach is al have seen that Lorentzian modelsto can avoid be divergences. defined, although It one isof then freedom not at clear the whethermust we vertex in need by either to introducing case c a introduce point a holonomy. parameter For not clear at this stage how equation variable for the GFT field is an element of U(1) associated to the vertex. The parameter to be taken to zero after the quantum theory has been defined. to Newton’s constant. JHEP06(2014)013 , ) ) ′ ˜ φ =0 6.6 , (6.5) (6.6) (6.9) (6.7) (6.4) (6.2) (6.3) (6.8) i ′ 4 (6.10) ) , ′ ˜ ) φ , ˜ , φ i ′ I otherwise. , ). g 4 , . . . , g . ( ˜ Φ) R ′ 1 † , 4.4 g ˆ ′ I ϕ ( , h = 0 ϕ ) ( ) ′ U(1), in the case ˜ ′ , . . . , h φ  ϕ ξ 1 ˜ , ) i φ µ I × ) h ˜ by a kinetic term φ g ( ′′ ˜ , 4 φ, + ( † ˜ ′ φ I , xtended domain space, † to obtain in the WKB ˜ φ ϕ , ′ g 4 ˆ ϕ ) ˆ , simplicity. There is no ( ′′ 4 , ∆ h ˜ φ ϕ ) , h 5.2 τ ing the weak-coupling limit = o extend the domain space 4 κ ( i ˜ + densate defined by Φ) ) , . . . , g ) = 0 ′ , uantum field are then O ′ ′ rresponding to the kinetic and 1 I , . . . , g ˜ ′ I ˜ φ g φ ′′ or Spin(4) 1 , h , , g = go through just as before. Again, g ′ ∆ I 1 ′ 4 I , . . . , g ( ˜ R 2 φ g 5 1 g − I ϕ ( 1 g I ) ˆ h ˆ ( × ′ ϕ X ( − τ p † , defined in analogy to ( ˜ , ) φ ξ 4 ′ between the gravitational and matter ) ϕ ,  ′′ ) I + σ , . . . , g ˜ ) ˆ ) ′ 4 φ ˜ g φ ′ τ 1 ˜
, φ , ˜ φ g (( 2 I , ′ I ( ) ξ g 4 g I ) ( − K 1 ′ h (0) = 0) ˆ ) 1 ϕ ˜ B ˜ − φ I φ , . . . , g f ˜ h ) for a point holonomy and in all of φ · ( − 4 ′ g 1 since the scalar is left invariant by local Spin(4) , – 52 – ( δ ˜ π g φ, I ˜ , we can focus on the class of solutions to ( 4 Φ) ) , ξ ( 2 ; ϕ π 5 ′ I ′ , ˆ ) I ( ϕ 4 ′ g − h ) ˜ 1 ′ φ ) , g ˜ − ′ φ , . . . , g − I I h ( ˜ 1 I − φ g g , . . . , g δ ( ( h (d ˜ ) 1 φ B ˜ 1 δ 4 φ, ( ˜ g · K φ ) ; − ( 1 δ ′ , ′ I I 4 ) g ¯ g I ϕ ) ′ I ( ′ B ) = ′ h ′ with , g ′ (d g 1 ˜ I φ , g ˜ φ g − I I d ˜ (d φ ξ Z ( i I g g ˜ ˜ φ, ) = 0 implies that d φ ( 0 ( X K | ; Z + x ξ d G 4 ′ takes values in [0 I ′ ( 4 I )  ˜ ′ δ ) φ = ˜ φ h ′ ˆ , g ) ξ = d i I g x  i (d g 4 ( ) 4 ( ) (d ˜ ′ Φ) f ′ ) 4 ˜ , φ K g g ) I , g ′ ) exp I h (d (d ′ ( g (d ξ ( ϕ ( Z Z † ) Z ˆ ϕ ˜ 2 1 φ N , , ) ]= I ˜ ¯ φ g ϕ := := , ( ′ I i If we now focus on models with a Laplacian kinetic term on the e The extended GFT model including a scalar field is then defined ϕ ˆ ′ ϕ, ξ g h [ ξ ( | k ˆ ϕ S the effective dynamics splits intopotential two terms, separate the equations former co being The two-point function can be computed to be where we allow for a nontrivial relative weight and following the discussion in section parts of the kinetic term, we can redo the analysis of section plus a potential.of At the this GFT abstract from level, four all copies we have of done Spin(4) is to t Spin(4) that satisfy limit (using that if we assume an interaction of odd order, for the ‘dipole’ con where the coordinate which is the same equationof that the we effective would obtain equation from for consider the single condensate of Riemannian signatureadditional that gauge we invariance property now for transformations. restrict The to basich for commutation relations technical for the q It should then be clear that the constructions of section JHEP06(2014)013 Since (6.11) 13 0 in the τ < tic operator can be viewed ion of GFT models for er can be included into he perturbations above ar field is compactified he treatment, not least GFT kinetic term to the ll be able to describe the ric, one has to find first a of the perturbations. For ented earlier in the paper. ) for some state-dependent ng whether matter degrees nsiderable extension of the ogeneities will be encoded in 2 the various points that need s free scalar field if the sign ion into a state that stores it above the GFT condensates, p language. In order to encode cs of gravity coupled to such a n terms of the Fourier modes Friedmann equation obtained ut adding them by hand. Riemannian signature gravity) gravity) on superspace. − is natural to look for the physics . k this metric and its Lorentzian signature, ( ) 4 κ a . For isotropic states, we saw before ( ˜ φ O + is chosen appropriately and physical units ˜ 2 φ 6 ζ a τ p , we obtain the same Friedmann-type equation − – 53 – R = 2 p 2 0 for Riemannian signature, it would be a − k τ > is the momentum conjugate to ˜ φ . 0, so that we finally get ∂S/∂ := phonons k > ˜ φ p ]. is chosen appropriately ( These calculations show that a very natural extension of the The deviation from homogeneity complicates significantly t Since condensates are candidates for quantum states that wi 76 In quantum cosmology, just like in our discussion in GFT, the choice of kine τ 13 constant scalar field is described byand a a positive wave operator definite of Laplacian Lorentzian (for signature (for Lorentzian are restored. Just as in usual quantum cosmology, the dynami because everything has toin be a expressed GFT in state acomplete perturbations coordinate-free set of an of otherwise invariantsinstance, homogeneous that one met can could be give usedon the to a spectrum spatial store of slice. the thein shape perturbations Second, terms one i of has correlations to among translate GFT this quanta. informat Lorentzian case), the dimensionful parameter that the gravitational part of this can be written as This is just theof classical Friedmann equation with a massles the Laplacian on U(1)from is the the same WKB asusing approximation, on point independent holonomies of in the whether definition the of the scal GFT model. cosmological sector of GFT, it issmall deviations natural from to condensate assume states. that inhom of In other cosmological words, perturbations it ini.e. the GFT regime of fluctuations as a choice of metricsee on [ superspace. For a geometric discussion of where matter sector leads to thein correct the coupling WKB to limit. gravitythe in Clearly, GFT the they models. are just Morequantum an work gravity example is models of needed with howof in matter matt freedom the fields, are fundamental or already definit present in in understandi the existing6.2 models, witho Perturbations/inhomogeneities An important point remainsa to spatially be homogeneous addressed:analysis universe. of the the dynamics This solutionsNonetheless, of subject to we t the can requires provide GFT co ato equations rough be of and addressed. motion tentative picture pres of JHEP06(2014)013 , i of Ψ δ | (6.14) (6.13) (6.15) (6.12) with a 1, that i is a state Ψ ≪ , making a | i i ǫ σ Ψ | | on ted information . We will adapt is case. This will } with µ mogeneities has to i ˆ O s of the theory, which Ψ { | o keep under control. ion problem that is found e states. If tate is specified. them solving the equations ables place ion parameter, hen the equations of motion s) can be understood in terms ity, and that we have discussed h this inequality is violated, and cuss inhomogeneous cosmologies , at we are going to develop. There- . . = 0 i i i < η . Ψ Ψ

Ψ δ | δ | σ | ) ǫ i . If this is the case, for the observables ǫ is given by the structure of the state I ǫ µ g ǫ + + ˆ ( is a solution of the equations of motion, so is O i ˆ σ i C ≪ i i ] σ Ψ † µ – 54 – − h Ψ ˆ η ˆ ϕ | O Ψ h i ϕ, [ ˆ i → | µ i → | ˆ ˆ , for example, the split of the state into homogeneous . In order to be consistent with our replacement of the σ O O Ψ | i η | h | , and one can obtain a set of equations involving the σ

i | Ψ h Ψ | , the quantum equations of motion can be turned into an 5 as well as the various quantities used in the parametrisation σ instead of with the state i i σ , | η Ψ | in such a way that we reproduce the results of the exact soluti , at least in the approximate sense that we consider only limi i . Thus, the reconstruction procedure has to be extended to th i σ 3 | Ψ δ | ǫ This is a subproblem of the general sampling and reconstruct At this point, for the restricted set of observables we can re The assumption that we are dealing with a small amount of inho A second task will have to do with solving the quantum dynamic As discussed in section we can use + i µ Ψ in the construction of semiclassicalin section states in quantum grav infinite set of equations for all the correlation functions, In general it will bethey possible will to set choose the theoretical observables for errorfore, of whic the the preliminary effective step theory before th usingis our the machinery to enumeration dis of the observables of the GFT that we want t and can be understood only once the geometric content of the s We stress that the meaning of the parameter controls the deviation from the perfect condensate state, exact solution condensate wavefunction relative error at most of magnitude given accuracy and in general we will be interested in a certain set of observ O be translated into the appearance of a dimensionless expans the state part and inhomogeneities requires that be a first task ahead. inhomogeneities. we are not able to solve exactly even for the simple condensat solving the equations offor motion any in small the deviation homogeneous fromof sector, it perturbation t (not theory, necessarily homogeneou about the quantum states andof ask motion. that this is compatible with The perturbation has to be such that, if | JHEP06(2014)013 of the ds, it is reasonable to ould be to simply work to concrete calculations, btain an analogue of the effective field theories for theory on minisuperspace. the background GFT con- n be described in terms of ynamics directly from the e useful, in addition to the by all approaches to quan- ondensate) solutions of the he GFT configuration space ly homogeneous) spacetime, ective dynamics for the GFT eference frame could be used e throwing away information he localisation and the state equations of motion, special bed by effective “phononic” rs in terms of which GFT is indeed geometric observables the homogeneous backgrounds of phase transitions, as it can states describing homogeneous ates. It is also worth stressing ld be extracted from the latter. Indeed, this analysis has been urbations can be localised. One sults should now be improved, . a Bogoliubov transformation). s over the gauge-invariant GFT ]. This line of work goes then in ed in this paper. 75 – 73 configurations, invariant under local actions from collective excitations of the very same – 55 – generally encoded in a GFT quantum state. When going be- gauge-invariant emergent matter ). While this restriction is motivated by the interpretation C , reference frame Last comes the issue of re-interpreting and rewriting the eff On the basis of these considerations, then, we can expect to o These considerations are so far very general. To turn them in To achieve this, it could be crucial to use the directions in t An approximate way to study this dynamics of perturbations w the direction of identifying degrees of freedom constitutinggeneralised, spacetime and itself. reanalysed in These light of re the results present Bogoliubov-de Gennes formalism forthat Bose-Einstein this condens method, going beyondanalysis a of mean-field deviation approach, from mightprovide b homogeneity, a in first the estimate description a of semiclassical the background breakdown geometry. of the regime that ca phonons as an effective fielddefined by theory the on background GFT the condensate, continuum as (spatial opposed to a field that we have so far neglected:cosmologies our was analysis restricted of the to condensate of Spin(4) or SL(2 fundamental degrees of freedom incan only discrete depend geometry, on where gauge-invariantabout quantities, it a means local we ar yond the homogeneous condensate, the informationfor the in construction such of a a r coordinatewould system then in which generically the look pert forcondensate. gauge-variant perturbation All the geometricof information motion of needed the to GFT definedensate perturbations t state on (assumed the to spacetime be defined also by a semiclassical state) shou 7 Conclusions In this paper, wetum have addressed gravity: one to fundamentalfundamental extract issue microscopic faced an quantum effective dynamics of macroscopic the continuum theory. d one would need to identify thein elementary terms excitations above ofconjecture operators. that In the analogyfields elementary whose with excitations relationship the might withformulated case be the might of not fundamental descri be quantum field a simple flui operato linear transformation (e.g at the classical GFTclassical level GFT and equations, study obtainingalready perturbations their carried around effective out (c action. intypes simple of cases perturbations (simple aroundscalar them) solutions fields and of over shown non-commutative the flat to spacetimes give [ rise to JHEP06(2014)013 ]. 78 quantum d ressing one is to un- ation enforcing semi- nterpreted as macro- ]. Understanding this by something else; the e are considering. It is hat the basic geometric 79 mogeneous and isotropic wards approaching a Big oth metric geometry — a ation and in the isotropic , Planckian) regime cannot dynamic approximation of tum gravity and quantum ample can be extended to s of cosmological variables, erality, for a general choice yond the ones used in this n, presumably a transition t approximations involved. er corrections. s. Just like in the physics es and follows the evolution cular choice of kinetic term ] and more generally in [ e mean field whose effective 77 ry. Our construction is thus ubble rates) corresponded to ly considered in cosmology. [ unctions’ used to define the ak down. As we have argued, e has made for the quantum cause these condensate states acting building blocks of ge- res the relevant quantum dy- um fluid or condensate, advo- 77 ributions from arbitrary num- as we have shown. in foam models of 4 le of the same building blocks, ed nonlinear and nonlocal quan- age which of the approximations -point functions can be computed and n geometrogenesis – 56 – We have constructed a class of condensate states that can be i There are many directions for future work. Perhaps the most p More fundamentally, as one extrapolates to higher curvatur deep quantum-gravity regime will require methods that go be scopic, spatially homogeneousThese geometries states of are non-perturbative thebers in type that of they excitations usual contain ofa cont the concrete (no-space) realisation Fock of vacuum ofcated the previously the picture in theo of the spacetime contextThis as of picture group a can field quant then theory be (GFT) investigatedare [ dynamically analogous in to GFT: coherent be or squeezed states, their states. These effective equations taketum the cosmology form equations. of generalis We haveof shown GFT all model of this andeasy in for to full specialise the gen the different effective types equations to of specific condensate GFT/sp states w used to derive effective equations for the ‘condensate wavef gravity. We have investigated aleads simple to example an where effective equationcase, a that parti to reduces, the in classicalLorentzian a signature Friedmann WKB and equation approxim to in include vacuum. a massless This scalar ex field, derstand more carefully theThe nature picture of and spacetime regime asthe of a fundamental GFT the quantum condensate differen involves dynamics.building a hydro blocks It describe required near-flatand us configurations specific to at approximations assume the to the t sca we full are GFT working dynamics. in Incase, a term where regime in our outside example of thewhat lowest high one order would curvature. in expect momenta In (i.e. fromclassicality H GR, the on we our ho also quantum employed states. awould It WKB have is approxim to not be clear at considered this first st in computing theof leading the ord universe backwardsCrunch), in one time expects the towards hydrodynamic thethis approximation Big to would bre Bang be (or signalleddynamics for by large is quantum described fluctuations byof over th our Bose-Einstein condensates, quantum this cosmologicalstate means equation is that no the longer ansatzGross-Pitaevskii on equation a for good the approximationnamics. mean and field If no has longer we tocosmology, captu take be it this replaced would analogy mean seriouslybe that for described a the by high-curvature case aometry. (presumably of state Instead, quan consisting one of mightfrom near-flat, expect a weakly pre-geometric a phase inter quantumscenario to phase that a transitio phase often of goes an under approximately the smo name of JHEP06(2014)013 ) , e 1 − ) ′ 4 g ( ). For the ′ 3 C , g , 1 tative Fourier − ) ′ 4 g ( ′ 2 ) (A.1) Lorentzian signature , g , e 1 ′ 3 ressed. No significant − ˜oMagueijo,o˜ao Roberto ince of Ontario through ) that has to be recovered , h ′ cial support from the A. 4 ′ 2 arious quantities. In this GFT field defined on four opies of SL(2 g field theory over the back- tes. The ultimate goal, of rs of the quantum gravity ( ′ 1 , h ′ 1 g els for Lorentzian spacetimes he paper arise, and there are ( over the condensate states we h n case to Lorentzian signature ode, we conjecture, the physics er Institute is supported by the n important direction of future ( ϕ m gravity effects in cosmology. hould try to recast the effective ) ) nt as an invariance property of ϕ ′ ′ 4 4 ) d from the fundamental quantum ′ 3 , h ′ 2 has the same symmetries as the GFT , . . . , g , . . . , g ). The noncompactness of the group , h ′ ′ 1 1 ′ 1 g C K ( , g , , h 4 ϕ 3 ) ′ 4 , h 2 , . . . , g – 57 – , h 1 1 g , . . . , g h ( ′ 1 ( K , g K ) ) 4 , e , e 1 3 − 4 g , h 3 , . . . , g 2 1 , g g , h 1 ( 1 − 4 K h g ) ( 2 4 ¯ ϕ , g 4 1 h − 4 d g , . . . , g 4 1 1 g g g ( d ), e.g. when defining integrals and structures as a non-commu ( ¯ 3 ϕ ¯ ϕ C ) 4 ′ , ) 4 ′ h ) g ′ (d g (d 3 ) 4 (d ) 4 h g ) , this is straightforward to see, as g (d (d ϕ In the models we consider, the signature of the metric tensor A primary concern is that already the classical GFT action in (d Z Z = = Z leads to a number of technical difficulties when working with a copies of SL(2 is encoded in the choicehave of to local gauge be group. based, Therefore, mod in four dimensions, on SL(2 paper, and that will beresearch, explored as in we future work. tried tohave Similarly, considered. a discuss, The is physics the of studyof such fluctuations of cosmological should fluctuations perturbations enc (inhomogeneities)dynamics and of one such s GFT perturbationsground in homogeneous the geometries form of definedcourse, an by is effective our to use condensate such sta gravity effective dynamics, dynamics, directly to extracte obtain predictions of testable quantu Acknowledgments We thank Emanuele Alesci,Percacci, Abhay Carlo Ashtekar, Rovelli, Frank Edgroup Hellmann, at Wilson-Ewing, J the and AEI for all usefulGovernment discussions. the of Canada Research membe at through Perimet Industrythe Canada Ministry and of by the Researchvon Prov & Humboldt Innovation. Stiftung with DO a acknowledges Sofja finan Kovalevskaja Award. A Regularisation of LorentzianThe models generalisation of the resultsrequires obtained some in care the in Riemannia appendix the we definition consider andmodifications briefly manipulation in the of the key essence theonly points of v some the that small procedure need technical introduced adjustments. to in t be add is ill-defined, since thethe imposition GFT field of leads the to closure spurious constrai integrations over one or more c transform on the group. kinetic term, assuming thatfield the kinetic operator JHEP06(2014)013 . . 4 3 ) ) ) 1 C C , × , g , (A.5) (A.2) (A.8) (A.7) ) 5 1 ) (A.6) , g − 9 3 8 g SL(2 8 , , h , g ) (A.3) 2 ) , ′ 3 now leads to ∼ , g 4 10 1 , h g 4 from SL(2 , h − , g 9 ( 1 h viz. ′ 2 , ) (A.4) 4 g h ϕ diag 1 ) g 3 ( ) ) 1 3 , h ′− 3 ψ ′ 1 , g C 10 ) ′− g 4 and , g 1 , h 3 3 g up manifold without 4 ( , g − 9 4 4 g 2 g ψ g , g , h 2 7 ) 1 2 infinite, and so does not , g he Haar measure, but in duced dependence of the ′ 3 g SL(2 guri model for BF theory equivalent to a restriction , g 6 ( ′− 2 tual difficulty in rewriting / n be finite. The quantum 4 , h g , h 4 ψ 1 g , g ) 2 ′ 2 ntegrations, ) , . . . , g f compact groups this is just 1 9 h 1 1 g ( C g to define the models, one can , g ction term, given by , h ( − ( 7 ¯ , ′− 1 ψ 1 ′ 1 g ϕ ϕ 3 g . 6 ) ′− ) 1 1 , h has no gauge invariance property ) 9 g g h 3 , g 1 1 ( 1 ) = , g 3 g ψ − 1 (d ) 8 , h ( − 7 g , e 2 C 3 g Z 5 , ) 3 , g 5 ) there will be several different parametrisations, C  3 , h , , g g , g C . Integration over 1 , g 1 2 , d , g SL(2 1 , this becomes 1 , h − 1 7 δ ( − − 7 ψ g , g ϕ SL(2 g Z ) g 8 1 δ ( K ′ 4 4  g ) g ϕ g ( , g 3 ( ) )] = – 58 – ( ′ 1 i 7 ϕ ′ 4 )= g ψ , h − 1 )] = 4 ) 2 ′ , g g 3 1 6 ≡ , g ) = 10 − , g , h 4 ′ i 3 3 g ′ , g 2 1 g ( 5 3 , . . . , g h , g , g ′ , g 1 ( , h ψ 2 2 ′ , g 1 ) , g g 1 ¯ ψ 4 ( g 1 1 − 4 , g , g † ( g 3 − − 10 4 1 g 1 † ( ˆ ) ϕ i g g g ′ g ˆ ϕ , ψ g 14 ( 2 2 ( h ) ) , ϕ 4 4 ) , g , g ≡ ) (d 3 1 1 4 3 i , g − − ) 10 4 3 , g h , ψ , g g g 2 h by the gauge-fixed field 3 6 1 , g C g , . . . , g 2 , g (d , g ( ϕ , g 1 1 2 1 g → , g g ψ Z ( ( 1 − 10 3 , g ˆ g 10 g ϕ ψ ) 1 ( [ ˆ ) [ denotes the Dirac delta over the homogeneous space obtained 9 g ] = C g g ϕ ( 3 ¯ , ( ) ψ ¯ ϕ 10 (d C ψ , 4 ) ) ψ, g × Z [ g (d SL(2 satisfies standard commutation relations, : SL(2 = reg (d k δ Z ˆ S ψ Z ψ Let us work out explicitly the regularised version for the Oo One way out is the observation that the closure constraint is = Oo Notice that in terms of the original four copies of SL(2 = V k 14 Oo S V after imposing gauge invariance. corresponding to closure andfield so the action defined this way ca consistent with the commutation relations of ˆ factors proportional to the volumea of finite the constant group. which Inthe can the Lorentzian be case case set o it todefine means one a that by variational the a principle. action normalisation can of only t be zero or in four dimensions. Here the kinetic term is where One can rewritedivergences the arising theory from in redundantintroduce terms integrations. a of gauge-fixed In field a order field on this second gro which will be equivalent representations of the same model. for the kinetic termany we action looked in at. this There way,fields once is on we of the keep course track arguments. no in concep each By term its of definition, the the re field of the domain of the field to the homogeneous space SL(2 where we have defined with a single redundant integration. There is also an intera and rewrite the theory in terms of this, removing redundant i Replacing the field JHEP06(2014)013 . i e ) 0 1 , for , g (B.2) (B.1) (A.9) e 5 , since (A.11) (A.10) ϕ = , g , and 8 i . , 0 ) g 1 , g 1 0 ex − 7 − 1 g g ( h 5 1 g ϕ ) − 4 0 h = 5 , g 5 × 2 , h ) rns out that one can 1 8 , g egration explicit. The − 1 6 , h , , h ularised. h 1 1 7 terms of the homogeneous , g 1 erive the dynamics of the 9 − − h te but non-zero action, can 7 10 − 4 with spatial slices given by g s, etc.) appearing in GFTs 4 g g h e presentation of the various ( , and one can set 4 9 SU(2) satisfying the Maurer- h form. 1 g g ϕ 3 ϕ relativity from scratch. Similar he same redundant integration, − 7 N dt , ) eliminating the redundant inte- action. This ansatz corresponds ≃ 9 h , h = = 8 = 3 7 . , g 4 9 h S 0 k 8 , h ( e 1 ]. 0 , g ψ − 1 ) 3 , h , h 80 6 ∧ h or in terms of the original field , e 1 1 1 , , g j 7 , h − − ψ − 4 4 9 e 3) g 5 34 0 g g , h ( 3 8 1 2 jk g g ϕ , h is equal to , i − 7 ) 4 ) ǫ h 7 0 h = = 1 2 1 – 59 – ( reg Oo which is not integrated over, but is in fact inde- = 1 , g 3 8 − 9 , g ψ − V 6 i 3 0 h ) ( ( g 3 , g = + , g ) ψ 5 i 2 i , h , h ( ) , h e e 9 2 1 1 , g reg k , g 0 h 4 − − 4 9 1 S d ) 0 7 g , h g g i g ( ( 1 h ( 2 7 = 4 a h ϕ g g ϕ ( ) h ) 4 2 0 = = = ψ reg Oo 9 i , g 2 7 S ) , g , h e 3 3 9 h h , g , g 7 (d 2 2 , h , h h 1 1 6 Z , g , g 1 − − 4 7 1  g , h g g g ( g 9 ( 1 6 are functions of time only, there is no summation over the ind d ϕ ¯ g g h ϕ ( 9 because of the gauge invariance property of ) 3 N = = ψ Z ) 0 g ) becomes g g 1 6 ×  (d h h (d and = A.8 Z i Z + a Oo = A regularised version of these models, with potentially fini Although this is in general not a consistent procedure, it tu The various quantities (actions, convolutions of operator V reg Oo Notice that this interaction term andwhich the can kinetic term be have then t factorised and removed when the model is reg Cartan relations substitute the ansatz of3-spheres with a the spatially round metric, homogeneousto into a geometry, the tetrad Einstein-Hilbert given by where define a (fiducial) basis of left-invariant one forms on We can now change variables to be given either in terms of the gauge-fixed field so that ( S all we have doneregularised Ooguri is action change variables to make the redundant int for noncompact groups then have tospace be obtained understood after as imposing defined thegrations. in closure However, constraint, for thus conveniencestructures, in we the are notation not and going in to th write them explicitlyB in this Dynamics of theFor Bianchi completeness IX and model toanisotropic but clarify homogeneous Bianchi our IX universe choicederivations in general can of be variables, found in we textbooks d such as [ instance. pendent of This now appears to be a function of JHEP06(2014)013 , 2 e are  . ∧ is (B.4) (B.8) (B.6) (B.5) (B.3) (B.9) (B.7) 1 e . R 3 | 2 e + 2 more !! | 2 . 2 . x ) , ω  3 3 2 2 1 ) 3 a d 3 3 ( 1 a ω a R 2 a ( + cyclic perm a − 2 = . ) 3 2  2 ) a L 1 a onstant coming from 2 1 4 2 2 a a a ) etc., the Hamiltonian ( 3 2 − + ( + cyclic perm which simplifies the form a 2 ) + 2 − 3 ) 3  he literature. If one sticks a 2 3 one has to keep in mind that a p 3 3 3 ) 2 a ) cyclically for 2 rangian a + ˙ 1 a 2 ( 1 1 a p , 2 2 2 for the Levi-Civita connection a 2 3 1 a a JK 3 al (this would vanish for Bianchi ( a ) a ˙ ( a + cyclic perm a , a 1 ω 2 J 2 2 + 2 more 3 + a e  ∧ a  a = e 2 2 ( + 3 J ∧ 4  2 N I 1  + a + ( 3 N N 3 1 N 2 ˙ a 2 ) J 2 2 ω 3 2 a a 3 a 1 1 I a − 3 + a ) a p 2 1 ˙ a a a + 2 1 a 2 1 ω 1 a a ( a = ! a a p − IJ ( 2 i 3 +  − 1 2 p a + + p ) 3 3 2 i  3 1 = dω 1 ) , 3 a a a 1 i ! a − 2 a 4 2 I N ( = p a 2 – 60 – 2 1 a a i + ( ) a  i de X 2 + − 1 IJ 3 a − p 2 a − 3 4 1 R  2( 1 ) a a p 3 3 a 2 2 4 1 a a ˙ N + N a a , 2 a  1 1 3 a 0 ˙ a + ˙ a + e + ( ( N 2 − N i 3 . The Ricci scalar is 4 i + ∧ ˙  2 1 a ) a  a i 23 contribute to the Ricci scalar. The relevant contributions + 2 more + 2 2 3 4 3 e ˙ 1 2 1 N a a a − a 2 R − 1 2 a 2 p IJ 2 i ! 2 ˙ a a = 1 2 p 3 + ( p ˙ IJ ˙ 2 i i N 2  − Na 2 N N 4 and ¨ a a 1 N i ) R 1 )

a ) i ˙ a 1 i = a ( 1 ( i 1 4 31 ˙ a a a a

( X L 2 R − , ω + 1 4 + − i ) − 3 i 2 a X (

2 is 2 2 = e N ) ˙ 3 N a a = i 2 N and their conjugate momenta ( 1 1 a ) a ) N H i ˙ 2 ¨ ¨ a a i a i 1 ( ) ( IJ a − L i a a a ˙ N a ( 1 i = IJ a a p

i 4 R L greatly: a = ⊃ ⊃ = = 0 0 H = ˙ i i 12 gives R R ω H R of which takes the form of kinetic termI). plus The anisotropy potenti conventional choice for the lapse function is H or after integration by parts, discarding the boundary term where the “2integration more” over terms the three-sphere denote that cyclic we permutations. set to one, Up the to Lag a c When computing the Riemann tensor ω Various choices for thewith canonical the variables can be found in t and cyclically for Solving Cartan’s equation of structure only the components JHEP06(2014)013 . 2 ]) )
i 3 80 A h ( (C.1) (C.2) 2 (B.10) (B.11) (B.12) (B.13) h i . X + 3 2 1 h , ! 1 . − h 2 ! i  + 2 2 − , 3 and cyclically. In h A i 0 0 1 A 3 1 ! 2 complex traceless h 2 , a

. 1 A 3 2 +cyclic perm 2 × a − = 1 2  A T ! 2 appears as the space of rent anisotropy potential 2 3 1 0 0 + 2 = so that the spatial metric − 3 2 A σ / + cyclic perm : − one might choose different 1 3 2 1 2  ], the Lagrangian is A 3 ) ) ) 2 2 3 i A i A T 1 = = i 3 2 h 80 he same kinetic term. Going a l sign of the kinetic term. h A A ] = 1 h A √ ven by 3 6 ( A 1 3 h √ 2 2( A i = = ( and ,T X r i 2  2 + N h T 4 1 [ are generators of an SU(2) subgroup; + 2 ,T 2 + 2 / 6 2 , such as 1 + ,T 3  i / 2 ) T 3
T , 5 3 a / 3 3 1 h ) ! 3 1 p A ) consists of all 2 2 1 ) A T A 2 2 . 1 h 2 SU(2) C i A 2 2 and p 1 0 h , A ( / 3 A 0 1 √ ( A 5 − ( ) − h ,T  (2 2 − C ,T

− sl – 61 – h 4 ! ,

] = )+

T 3 = 3 + 4 p 4 3 2 N 0 i 0 0 N ,T 2 p σ 1 p 1

+ 3 T p Lorentzian geometry. [ 3 ! = = i A we obtain the Hamiltonian (compare section 4.1.2 in [ h 2 2 d 2 5 2 3 1 ˙ , A A h h A 1 2 1 ˙ + + h A 3 A . 2 1 2 i p p + 2 more + h ] = 0 ). In these variables, also used in [ 1 h 1 i 2 p − ,T 2 e p √ 3 2 0 2 3 h ,T  1 A h ! h = 1 ⊗ 1 1 1 1 h T ˙ i [ h A A A i 0 N 0 i e ,T r 0 + 2  2

( 2 ˙ i the Lagrangian becomes ! h

p 1 − h = i 3 1 ˙ i h 2 i 1 p 0 1 0 0 A A p 2 σ 2 P N 2 1

) A 4 A N i 1 are now the conjugate momenta to are the Pauli matrices. Clearly 1 = = i 4 h − now conjugate to i i A ( A 1 4 ( p σ i = T T i √ p − A different common choice of variables is given by Choosing a different Bianchi model would correspond to a diffe First, we note that the Lie algebra X L = = = L and the Hamiltonian is, with the same choice of lapse where H H and again choosing (essentially the same termsto with Riemannian different signature coefficients) corresponds but to t changing the overal C The homogeneous space SL(2 For the geometric variables appearingvariables in which are the quadratic GFT in Fock space, the scale factors terms of the with Here we discuss theunit geometry timelike normal of vectors this in homogeneous 4 space which is given by matrices. A convenient basis for this (real) Lie algebra is gi there is also a Bianchi V subalgebra generated by where JHEP06(2014)013 .
1 . )) 2 ) π 2 3
π 6 π 2 1 τ (C.7) +i = 1 (C.4) (C.6) (C.3) (C.9) (C.8) π ~π + 2 ⊗ 2 − π 2 − 1 | 6 π y 2 1 τ | ) one can = ~π 2( p + + ). The first y 1 − 2 − 5 | π C.4 1 τ on SU(2) that x can be seen as C.3 | + 2 p )(1 ⊗ , ~π 3 ω 3 . 1 β β . 5 π π 3 ω C τ + 2 2 ω ) π + i 2 + d ∈ − β + as in ( π ; (C.5) + ) + ) λ α 1 ) element can then be 4 2 2
β = d ~π ˆ g π τ 2 π C + d ) d = β d term in ( , h ~π , · β 66 s space from left-invariant ⊗ λ − + d j otheties of the plane, which d w − 4 2 1 = ~π K ) is non-diagonal; its (nor- π λ τ π β 2 1 + (2 , x, y, w + d d d C g = ~π
)( i , sponding to the compact and λ 2 ( ˆ β p g R 2 ) is hence i β π d 1 π (2 − 55 h (2 − C π d π ∈ 1 β , d 3 sl 1 K π  − + d τ + 1 )) + 2(2 2 √
, λ 3 λ + − 2 3 = ⊗ 2 ~π π d h 2 π ) left-invariant forms to get 3 ! ~π + 1 2 ) τ ~π 1 β 44 − ) + 2(2 C i π + ¯ x − 2 1 − , SU(2). The coordinates π 1 − 2 ) 2 π g 1 / j d w ¯ π x 1 α ) ). A general SL(2 + (2 )+2 π 2 λ − p + ˆ i 2 2( ,K 2 C + C p 2 ~π − 2 2 τ π , g , 3 π (2) Killing form gives the bi-invariant metric + d π y π − 1 π dˆ − ( ⊗ λ + π – 62 – su )( − 1 λ = 2 4 ) ) d 1 α e 3 τ )(1 − ij α α α π δ g 33 α p ) + 1  ¯ y + d 4 π + d + d = ˆ + ¯ τ y e + d λ = w + (2 ) λ λ k λ g d ⊗
λ α j d d ,K π d − − 2 2 d α ω e d 1 1 α α τ ~π x α ( j − + d − ( − ⊗ π − g λ 2(2 − λ i + (2 e 1 = (and which cover only half of SU(2)) correspond to ) d ω
+ (2 + 2(2 − 1 ijk

2 α p τ ǫ


ij and the real and imaginary parts of 24 2 ~π 2 2 2 δ 5.2 = π ⊗ π K = ~π ~π λ − . 5 3 ! i − 3 τ 1 π − − ), and we are using the decomposition ¯ = ), obtained from expanding x ω π 3 )) + (2 1 1 C + C p − π 2 ¯ 2 y , 15 2 5 , 1 x y 2 π p p + i τ π − π 2 1 (2 K ~π 2 + π π ⊗ ) + sl ~π − 2 1 1 β 3 + − ! τ π 1 π − ( 3 2 λ w 1 2( π π p + d 1 − − π λ 1 3 1 e e − λ π π π √ λ = d λ (1 0 ) λ λ λ = β e 2d λ C , − x

One can construct a left-invariant metric on the homogeneou In the basis we have chosen, the Killing form on = = 4d = = 2d = d = 4d = (2 SL(2 1 5 4 3 2 6 g τ τ τ τ τ τ in the basis of and one-forms on SL(2 on SU(2), contracting these with a multiple of the term alone gives the left-invariant forms on SU(2), compute the remaining contributions to the SL(2 This corresponds to the Lie algebrahere of appears the group as Hom(2) the of Borel hom subgroup of SL(2 which is the round metric on the three-sphere. From the secon In this decomposition, written as malised) non-zero elements are g coordinates on the homogeneous space SL(2 we introduced in section It has three positivenoncompact directions. and The three bi-invariant negative metric on eigenvalues, SL(2 corre JHEP06(2014)013 ) 1 , 2, 3 √ R C.11 α/ (C.15) (C.14) (C.11) (C.10) (C.16) (C.12) (C.17) by − 1 , 3 = R SU(2) which y . / 2 ) , 2 . τ . λ . C 2 2 1 2 2 , 2 z ) . z √ β + oup of the group of L(2  2 + d 1 = + d Kaluza-Klein: τ 2 + d 2 2 1 y now obtained by orthog- ω y λ d + v, y, z is not unimodular, and its terms of the left-invariant mputing the right-invariant `ala β . right-invariant (with respect + d 2 β 3 + d d − τ rbolic space into z (2 drops out. The metric ( z λ 2 d 2 o d 2 1 z gives another metric of constant not . e π ω + 2 1 (C.13) v j d + x + 2 d 6 ) is then = − υ v τ z 2 2 2 z 4 3 − ) 2 2 y v ⊗ = 2 0 if one considers the hyperboloid of t α i 2 − − C.11 υ − z = 2 y 1 + ij + d y d ). Note that already the left-invariant and  δ + v > α , υ d 1 v, λ C ω 2 d τ v d , y d + 1 2 λ d 2 α y ) to more familiar coordinate systems on hyper- 2 + e y – 63 – , v 2 + v z (2 2 2 5 x = 2 1 − x τ + − 2 2 2 v + 2  + + 2 4 ω λ, α, β y t 2 2 v 2 d t d λ (notice that
= − 2 2 − , describing hyperbolic space as the submanifold of 1 + 2 ω λ , υ z u z u e   2 + + = 2 d τ 2 = d = 2 2 v 2 1 y y 1 ) = to the orbits of the action of SU(2), v ) for υ + ) SU(2) C 4 / 1 + 1 + , ) τ C.13 C ,  = = SL(2 t, x, y, z 2 3 ). ( 3 g H − SL(2 C H g g , g = ) viewed as a metric on the group Hom(2) is ) C , 2. C.11 √ SL(2 g β/ Eq. ( A natural left-invariant metric on the homogeneous space is The relation of the coordinates ( − = The induced metric is found to be given by One can now solve ( As expected from the construction, any dependence on z has constant negativeforms curvature, on and Hom(2). isisometries SL(2 explicitly The given left in action of Hom(2) on itself is a subgr onally projecting The last three terms then give a metric on the quotient space S is simply Changing coordinates to bolic space is the following: consider the embedding of hype The right-invariant metric on Hom(2) given by The identification with the coordinates used in ( future timelike vectors in Minkowski space), this reduces t to the right action offorms Hom(2) on on Hom(2), itself), as can be seen from co and choose null coordinates Killing form is degenerate. right-invariant volume elements on Hom(2) differ; the group negative curvature with isometry group SL(2 JHEP06(2014)013 , ]. ] SPIRE ] ommons ]. ] IN ][ , ]. 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