y department of physics, king’s college london, university of london, strand… · 2018. 3....

KCL-PH-TH/2016-43

Impact of nonlinear effective interactions on GFT quantum gravity condensates

Andreas G. A. Pithis∗ and Mairi Sakellariadou†

Department of Physics, King’s College London, University of London, Strand, London, WC2R 2LS, U.K., EU

Petar Tomov‡

Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany, EU(Dated: March 19, 2018)

We present the numerical analysis of effectively interacting Group Field Theory (GFT) models inthe context of the GFT quantum gravity condensate analog of the Gross-Pitaevskii equation for realBose-Einstein condensates including combinatorially local interaction terms. Thus we go beyondthe usually considered construction for free models.

More precisely, considering such interactions in a weak regime, we find solutions for which theexpectation value of the number operator N is finite, as in the free case. When tuning the interactionto the strongly nonlinear regime, however, we obtain solutions for which N grows and eventuallyblows up, which is reminiscent of what one observes for real Bose-Einstein condensates, where astrong interaction regime can only be realized at high density. This behavior suggests the breakdownof the Bogoliubov ansatz for quantum gravity condensates and the need for non-Fock representationsto describe the system when the condensate constituents are strongly correlated.

Furthermore, we study the expectation values of certain geometric operators imported from LoopQuantum Gravity in the free and interacting cases. In particular, computing solutions around thenontrivial minima of the interaction potentials, one finds, already in the weakly interacting case, anonvanishing condensate population for which the spectra are dominated by the lowest nontrivialconfiguration of the quantum geometry. This result indicates that the condensate may indeed consistof many smallest building blocks giving rise to an effectively continuous geometry, thus suggestingthe interpretation of the condensate phase to correspond to a geometric phase.

PACS numbers:Keywords:

I. INTRODUCTION

The most difficult problem for all quantum gravity ap-proaches using discrete and quantum pregeometric struc-tures is the recovery of continuum spacetime, its geome-try, diffeomorphism invariance and General Relativity asan effective description for the dynamics of the geometryin an appropriate limit. It has been suggested, a possi-ble way of how continuum spacetime and geometry couldemerge from a quantum gravity substratum in such the-ories is by means of at least one phase transition froma discrete pregeometric to a continuum geometric phase.One refers to such a process as ”geometrogenesis” [1]. Aparticular representative in this class of approaches wheresuch a scenario has been proposed is Group Field The-ory (GFT) [2] where one tries to identify the continuumgeometric phase to a condensate phase of the underlyingquantum gravity system [3] with a tentative cosmologicalinterpretation [4–11].

GFTs are Quantum Field Theories (QFT) defined overgroup manifolds and are characterized by their combina-

∗Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]

torially nonlocal interaction terms. In the perturbativeexpansion it becomes apparent that the Feynman dia-grams of the theory are dual to cellular complexes be-cause of this particular nonlocality. Depending on thedetails of the Feynman amplitudes, the sum over the cel-lular complexes can be interpreted as a possible discretedefinition of the covariant path integral for 4d quantumgravity. The reason for this is that beyond the combina-torial details, GFT Feynman graphs can be dressed bygroup theoretic data of which the function is to encodegeometric information corresponding exactly to the ele-mentary variables of Loop Quantum Gravity (LQG) [12].Using this, it can be shown that GFTs provide a formaland complete definition of spin foam models which givea path integral formulation for LQG [13, 14]. In case theGFTs possess a discrete geometric interpretation, it isalso possible to manifestly relate their partition functionsto (noncommutative) simplicial quantum gravity path in-tegrals [15].

In order to understand the nonperturbative propertiesof particular GFT models, the application of FunctionalRenormalization Group (FRG) techniques is needed [16].In general, these techniques provide the most power-ful theoretical description of thermodynamic phases bymeans of a coarse graining operation that progressivelyeliminates short scale fluctuations. Their successful ap-

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plication to matrix models of 2d quantum gravity [17, 18]serves as an example for the adaption to GFT modelswhich has recently been very actively pursued [19, 21]. Inthis way, the FRG methods enable one to study the con-sistency of GFT models, analyze their continuum limit,chart their phase structure and investigate the possibleoccurrence of phase transitions.

More precisely, standard FRG methodology has beenapplied to a couple of models from a class of group fieldtheories called tensorial GFTs, for which one requiresthe fields to possess tensorial properties under a changeof basis. The common features of the models analyzed sofar are a nontrivial kinetic term of Laplacian type and aquartic combinatorially nonlocal interaction. However,they differ firstly in the size of the rank, secondly inwhether gauge invariance is imposed or not, and thirdlyin the compactness or noncompactness of the used groupmanifold. Remarkably, all these models are shown to beasymptotically free in the UV limit which is deeply rootedin the combinatorial nonlocality of the interaction [20].Furthermore, signs for a phase transition separating asymmetric from a broken/condensate phase were foundas the mass parameter µ tends to negative values in theIR limit analogous to a Wilson-Fisher fixed point in thecorresponding local QFT. To corroborate the existenceof such a phase transition, among others, the theory hasto be studied around the newly assumed ground stateby means of a mean field analysis as noticed in Refs.[19, 21]. One way to check this would amount to findingsolutions to the classical equation of motion in a saddlepoint approximation of the path integral.

The possible occurrence of a phase transition in suchsystems is highly interesting, since it has been suggestedthat phase transitions from a symmetric to a conden-sate phase in GFT models for 4d quantum gravity couldbe a realization of the above-mentioned geometrogene-sis scenario. In such a setting, a pregeometric discretephase, given by an appropriate microscopic GFT model,passes through a phase transition into a continuum geo-metric phase, the dynamics of which is in turn describedby a corresponding effective action. The phase transitionwould then correspond to a RG flow fixed point and couldbe interpreted as the condensation of discrete spacetimebuilding blocks [1, 3].

So far, however, the mentioned FRG results for tenso-rial GFTs can only lend indirect support to the geometro-genesis hypothesis, since a full geometric interpretationof such models is currently lacking. To realize such ahypothesis in this context, one would have to proceedtoward a GFT model enriched with additional geometricdata and an available simplicial quantum gravity inter-pretation that is closely linked to LQG. The applicationof FRG methods to such a model with a combinatoriallynonlocal simplicial interaction term would be needed togive an accurate account of the phase structure of the sys-tem. The hope is that studying its renormalization groupflow will reveal an IR fixed point which marks the phasetransition into a condensate phase ideally corresponding

to a continuum geometric phase. Hence, the aim is togradually increase the sophistication of the studied toymodels to rigorously underpin the GFT condensate as-sumption and connect it to the geometrogenesis hypoth-esis [3].

In this context, the basic aim of GFT Condensate Cos-mology (GFTCC) is to derive an effective dynamics forthe GFT condensate states directly from the microscopicGFT quantum dynamics using mean field theoretic con-siderations, and consequently to extract a cosmologicalinterpretation from them [4–11]. The central assump-tion of GFTCC is that the possible continuum geometricphase of a particular GFT model is ideally approximatedby a condensate state which is suitable to describe spa-tially homogeneous universes. The mean field theoreticconsiderations used so far in Refs. [4–11] to give an ef-fective description of the condensate phase and its dy-namics use techniques which are strongly reminiscent ofthose employed to study the Gross-Pitaevskii equationfor, at most, weakly interacting Bose-Einstein conden-sates [25, 26].

In this article, we will go beyond the analysis of freeGFTCC models and investigate the effect of combinatori-ally local interaction terms (pseudopotentials) for a gaugeinvariant model with Laplacian kinetic term, the meanfield analysis of which was started for the free case in anisotropic restriction1 [5]. We will further elaborate theresults of the free model in this restriction studying theexpectation values of certain geometric operators. Wenote that no additional massless scalar field is added tostudy the evolution of the system in relational terms as inRefs. [7–10]. We choose local interactions for pure prac-tical reasons: in this way the equations of motion take aparticularly simple nonlinear form and to solve them, weemploy numerical methods. Despite the fact that from aphysics viewpoint these models appear as somewhat ar-tificial because such interactions lack a proper discretegeometric interpretation, they have nevertheless a prac-tical utility as simplified versions of more complicatedones, and bring us nearer to the physics which we wantto probe. One might also speculate that the local ef-fective interactions between the condensate constituentscould only be valid on length scales where the true mi-croscopic details of the interaction, namely the combi-natorial nonlocality, are irrelevant. Ultimately, rigorousRG arguments will have the last word on how such termscan or cannot be derived from the fundamental theory;however by adopting a phenomenological point of view,the analysis of the effect of such pseudopotentials mightprove useful to clarify the map between the microscopicand effective macroscopic regimes of the theory.

For a particular choice of the signs of the free pa-

1The isotropic restriction employed in this article differs from theone used in Ref. [7] which renders the interaction term local in thespin label.

2

rameters in the GFT action, we find in the isotropicrestriction solutions which (i) are consistent with thecondensate ansatz, (ii) are normalizable with respect tothe Fock space measure in the weakly nonlinear regime,and (iii) obey a specific condition of which the fulfillmentis required for the interpretation in terms of continuoussmooth manifolds. For such solutions we study the ef-fect of the interactions onto the expectation values ofcertain operators needed for their further geometric in-terpretation. We repeat this analysis for solutions to theequations of motion around the nontrivial minima of theused effective potentials and find that the expectationvalues of the geometric operators are clearly dominatedby low spin modes. In this sense, such solutions can beinterpreted as giving rise to an effectively continuous ge-ometry. Moreover, we discuss the consequences of theinteractions in the strongly nonlinear regime, where so-lutions generally lose their normalizability with respectto the Fock space measure and thus can be interpreted ascorresponding to non-Fock representations of the canon-ical commutation relations.

To this aim, the article is organized as follows. Inthe first part of the second section II A we review theGFT approach to quantum gravity from the classicaland quantum perspective. We then motivate its quan-tum cosmology spinoff called GFTCC in subsection II B.The presentation is kept rather short to motivate the es-sential concepts needed to follow the analysis presentedlater on. We invite the reader familiar with these con-cepts to proceed directly to the third section III where weanalyze in detail the properties of the free and interactingsolutions in subsections III A, III B 1,III B 2 and III B 3,respectively. In the fourth section IV we summarize ourresults, discuss limitations of our analysis and proposefurther studies.

In the appendices A, B and C we supplement themain sections of this article by discussing the notionsof noncommutative Fourier transform, non-Fock repre-sentations and non-Fock coherent states, respectively,needed to allow for a better understanding of the ob-tained results.

II. GROUP FIELD THEORY AND GROUPFIELD THEORY CONDENSATE COSMOLOGY

A. Group Field Theory

GFTs represent a particular class of QFTs which aimat generalizing matrix models for 2d quantum gravity tohigher dimensions. The fields of GFT live on group man-ifolds G or dually on their associated Lie algebras g. Forquantum gravity intended models, G is interpreted as the

local gauge group of gravity.2 The essential idea is thatall data encoded in the fields are solely of combinatorialand algebraic nature thus rendering GFT into a man-ifestly background independent and generally covariantfield theoretic framework [2].

In the following, we introduce aspects of this approachin a shortened manner which are needed for its applica-tion to GFT condensate cosmology in the remainder ofthis article.

1. Classical theory

The classical field theory is specified by choosing a typeof field and an action dictating its dynamics. Most gener-ally, we consider the complex-valued scalar field ϕ livingon d copies of the Lie group G, i.e.,

ϕ(gI) : Gd → C (1)

with I = 1, ..., d. The group elements gI are parallel

transports Pei∫eIA

associated to d links eI and A denotesa gravitational connection 1-form.

Importantly, one demands the invariance under theright diagonal action of G on Gd, i.e.,

ϕ(g1h..., , gdh) = ϕ(g1, ..., gd), ∀h ∈ G (2)

which is a way to guarantee that the parallel transports,emanating from a vertex and terminating at the endpoint of their respective links eI , only encode gauge in-variant data.

For compact G the action is given by

S[ϕ, ϕ̄] =

∫G

(dg)d∫G

(dg′)dϕ̄(gI)K(gI , g′I)ϕ(g′I)+V, (3)

where dg stands for the normalized Haar measure on G.The symbol K denotes the kinetic kernel and V = V[ϕ, ϕ̄]is a nonlinear and in general nonlocal interaction poten-tial. Choices of K, V, d and G define a specific model.The classical equation of motion is then given by∫

(dg′)d K(gI , g′I)ϕ(g′I) +δV

δϕ̄(gI)= 0. (4)

2. Quantum theory: path integral

The quantum theory is defined by the partition func-tion ZGFT . If we write a more general interaction term

2Typically, one chooses G = Spin(4), SL(2,C) or G = SU(2). Thelast is the gauge group of Ashtekar-Barbero gravity lying at theheart of canonical LQG.

3

as a sum of polynomials of degree n, i.e. V =∑n λnVn,

the path integral becomes

ZGFT =

∫[Dϕ][Dϕ̄]e−S[ϕ,ϕ̄] =

∑Γ

∏n λ

Nn(Γ)n

Aut(Γ)AΓ (5)

in the perturbative expansion in terms of the couplingconstants λn. The Feynman diagrams are denoted by Γ,Aut(Γ) is the order of their automorphism group, Nn(Γ)denotes the number of interaction vertices of type n, andAΓ is the Feynman amplitude. Crucially, field argumentsin V are related to each other in a specific combinatori-ally nonlocal pattern which correlates fields among eachother just through some of their arguments. This model-specific combinatorial nonlocality implies that the GFTFeynman diagrams are dual to cellular complexes of ar-bitrary topology [2].

In view of constructing a partition function for 4dquantum gravity, one starts with the GFT quantizationof the Ooguri model [27], a topological BF-theory, whichis based on a real field with G = Spin(4) or SL(2,C) andits Feynman diagrams are dual to simplicial complexes.If d is chosen to equal the dimension of the spacetimeunder construction, the fields are interpreted as (d− 1)-simplices. The d arguments of the fields are then asso-ciated to their (d − 2)-faces. In this case, a particulartype of interaction V describes how d + 1 of these sim-plices are glued together across their faces to constitutethe boundary of a d-simplex. Finally, the kinetic opera-tor K dictates how to glue together two such d-simplicesacross a shared (d− 1)-simplex.

In particular, for the case we are aiming at, the (right-invariant) field is defined over d = 4 copies of G andcorresponds to a quantum tetrahedron or equally a 3-simplex, a choice we further motivate in appendix A.For the construction of the corresponding simplicial pathintegral, the interaction term has five copies of the field.Their arguments are paired in a particular way to forma 4-simplex, given by

V = λ5!

∫(dg)10 ϕ1234ϕ4567ϕ7389ϕ96210ϕ10851 (6)

with ϕ(g1, g2, g3, g4) ≡ ϕ1234 etc. The kinetic term of theaction with kernel K(gI , g′I) = δ(g′Ig

−1I ) is specified by

K =1

2

∫(dg)4 ϕ21234. (7)

The data given so far does not yet permit the reconstruc-tion of a unique geometry for the simplicial complex. In asecond step, one has to impose restrictions which reducethe nongeometric quantum theory to the gravitationalsector.

This can be substantiated by invoking the correspon-dence between GFT and spin foam models. Indeed, anyGFT model defines in its perturbative expansion a spinfoam model [2, 14]. One can then show, that GFTs based

on the Ooguri model, may provide a covariant QFT for-mulation of the dynamics of LQG. In the latter, bound-ary spin network states correspond to discrete quantum3-geometries [12] and transition amplitudes in betweentwo such boundary states are given by appropriate spinfoam amplitudes [13]. A concrete strategy to constructgravitational spin foam models is to start with a spinfoam quantization of the topological BF-theory which isequivalent to setting up its discrete path integral. Im-portantly, it is then turned into a gravitational theory byimposing so-called simplicity constraints. These restrictthe data dressing the spin foam model such that it be-comes equivalent to a discrete path integral for Plebanskigravity. Moreover, the constraints allow one to establishthe link to LQG by restricting the group G to SU(2) [28].

It is precisely in this way, that each so-constructedspin foam amplitude corresponds to a discrete spacetimehistory interpolating in between the boundary configura-tions and is thus identical to a restricted GFT Feynmanamplitude. Therefore, the sum over Feynman diagramsgiven by Eq. (5) can be rewritten as a sum over dia-grams dual to simplicial complexes decorated with quan-tum geometric data which clarifies how the GFT parti-tion function can be intuitively understood to encode thesum-over-histories for 4d quantum gravity.3

3. Quantum theory: 2nd quantized framework

Motivated by the roots of GFT in LQG, it is possibleto construct a 2nd quantized Fock space reformulation ofthe kinematical Hilbert space of LQG of which the statesdescribe discrete quantum 3-geometries. The construc-tion is closely analogous to the one known from ordinarynonrelativistic QFTs [22, 23]. In a nutshell, the construc-tion leads to the reinterpretation of spin network verticesas fundamental quanta which are created or annihilatedby the field operators of GFT. Pictorially seen, exciting aGFT quantum creates an atom of space or a choron andthus GFTs are not QFTs on space but of space itself.

To start with, the GFT Fock space constitutes itselffrom a fundamental single-particle Hilbert space Hv =L2(Gd)

F(Hv) =∞⊕N=0

sym(⊗Ni=1H(i)v

). (8)

3This reasoning could be generalized in different ways. Firstly, thediscussion of the case with noncompact group G can be found inRef. [4]. Secondly, when remaining faithful to simplicial build-ing blocks, one could e.g. consider higher interaction terms whichalso allow for an interpretation in terms of regular simplicial 4-polytopes. Finally, it is in principle possible to go beyond the choiceof simplicial building blocks and define GFTs which are fully com-patible with the combinatorics of LQG. Within this theory, quan-tum states of the 3-geometry are defined on boundary graphs withvertices of arbitrary valence. These correspond to general polyhe-dra and not merely to 3-simplices [29].

4

The symmetrization with respect to the permutationgroup SN is chosen to account for the choice of bosonicstatistics of the field operators and pivotal for the idea ofreinterpreting spacetime as a Bose-Einstein condensate(BEC). Hv is the space of states of a GFT quantum. ForG = SU(2) and the imposition of gauge invariance as inEq. (2), a state represents an open LQG spin networkvertex or its dual quantum polyhedron.4 In the simpli-cial context, when d = 4, a GFT quantum correspondsto a quantum tetrahedron, the Hilbert space of which is

Hv = L2(G4/G) ∼=⊕Ji∈N2

Inv(⊗4i=1HJi

), (9)

with HJi denoting the Hilbert space of an irreducibleunitary representation of G = SU(2).

In this picture, the no-space state in F(Hv) is devoidof any topological and quantum geometric information.It corresponds to the Fock vacuum |∅〉 defined by

ϕ̂(gI)|∅〉 = 0. (10)

By convention, it holds that 〈∅|∅〉 = 1. Exciting a one-particle GFT state over the Fock vacuum is expressedby

|gI〉 = ϕ̂†(gI)|∅〉 (11)

and understood as the creation of a single open 4-valentLQG spin network vertex or of its dual tetrahedron.

The GFT field operators obey the Canonical Commu-tation Relations (CCR)[ϕ̂(g), ϕ̂†(g′)

]= 1G(g, g

′) and[ϕ̂(†)(g), ϕ̂(†)(g′)

]= 0.

(12)The delta distribution 1G(g, g′) =

∫Gdh∏I δ(gIhg

′−1I )

on the space Gd/G is compatible with the imposition ofgauge invariance at the level of the fields as in Eq. (2).5

Using this, properly symmetrized many particle statescan be constructed over the Fock space by

|ψ〉 = 1√N !

∑P∈SN

P

∫(dg)dNψ(g1I , ..., g

NI )

N∏i=1

ϕ̂†(giI)|∅〉,

(13)with the wave functions ψ(g1I , ..., g

NI ) = 〈g1I , ..., gNI |ψ〉.

Such states correspond to the excitation of N open dis-connected spin network vertices. The contruction of suchmultiparticle states is needed for the description of ex-tended quantum 3-geometries.

Using this language, one can set up second-quantizedHermitian operators to encode quantum geometric ob-servable data. In particular, an arbitrary one-body op-erator assumes the form

Ô =∫

(dg)d∫

(dg′)d ϕ̂†(gI)O(gI , g′I)ϕ̂(g′I), (14)

4This also holds true for G = SL(2,C) and G = Spin(4) when gaugeinvariance and simplicity constraints are properly imposed.

5The case of noncompact group G is discussed in Ref. [4].

with O(gI , g′I) = 〈gI |ô|g′I〉 given in terms of the matrix

elements of the first-quantized operators ô. For Hermi-tian operators, these have to suffice of course O(gI , g′I) =(O(g′I , gI))∗. For example, the number operator is givenby

N̂ =

∫(dg)dϕ̂†(gI)ϕ̂(gI). (15)

Strictly speaking, N exists only in the zero-interactionrepresentation which is when all representations of theCCRs are equivalent to the Fock representation, as iswell known within the context local QFTs [34]. Anotherrelevant operator encoding geometric information is thevertex volume operator

V̂ =

∫(dg)d

∫(dg′)d ϕ̂†(gI)V (gI , g

′I)ϕ̂(g

′I), (16)

wherein V (gI , g′I) is given in terms of the LQG volume

operator between two single-vertex spin networks and ananalogous expression holds for the LQG area operator[4, 11, 24].

B. Group Field Theory Condensate Cosmology

The cosmology of the very early universe provides anatural setting in which quantum gravity effects can beexpected to have played a decisive role. The GFTCC re-search program attempts to describe cosmologically rel-evant geometries by applying the previously summarizedtechniques.

In this context, the goal is to model homogeneous con-tinuum 3-geometries and their cosmological evolution bymeans of particular multiparticle GFT states, i.e. con-densate states, and their effective dynamics. A possiblemechanism which could lead to such condensate states, issuggested by the concept of phase transitions in GFT. Asexplained in the Introduction, the FRG analysis of spe-cific GFT models has found IR fixed points in all casesinvestigated so far, suggesting a phase transition from asymmetric to a broken/condensate phase [19, 21]. Thecondensate corresponds to a nonperturbative vacuum ofa GFT model described by a large sample of bosonic GFTquanta which all settle into a common ground state awayfrom the Fock vacuum. To confirm the occurrence of sucha transition a mean field analysis of the broken phase hasto be undertaken. Ideally, the effective dynamics of theresulting effective geometry should admit a descriptionin terms of the one given by General Relativity for thecorresponding classical geometry, perhaps up to modifi-cations [4, 11].

1. Condensate states

In the following, we briefly recapitulate the motivationfor why GFT condensate states serve as a good ansatz

5

to effectively capture the physics of homogeneous contin-uum spacetimes following Refs. [4, 11] and review impor-tant aspects of their construction. We turn then to theextraction of the effective dynamics from the microscopicGFT action.

In the case of spatial homogeneity, which is relevant tous, it is possible to reconstruct the geometry from anypoint as the metric is the same everywhere.6 This homo-geneity criterion translates on the level of GFT states tothe requirement that all quanta occupy the same quan-tum geometric state. This is the reason for choosing GFTcondensate states as the main ingredient for GFTCCin close analogy to the theory of real Bose condensates[25]. Furthermore, for a state to encode in some ade-quate limit information allowing for the description ofa smooth metric 3-geometry, one assumes that a largeconstituent number N will lead to a good approxima-tion of the continuum. Moreover, the simplicial buildingblocks are required to be almost flat. This near flat-ness condition translates on the level of the states tothe requirement that the probability density is concen-trated around small values of the curvature. Finally, fora classical cosmological spacetime to emerge from a givenquantum state, it should exhibit semiclassical properties.Crucially, condensate states automatically fulfill such adesirable feature because they are coherent states andas such exhibit, in a certain sense, ultraclassical behav-ior by saturating the number-phase uncertainty relationand are thus the quantum states which are the closest toclassical waves. We will discuss the construction of suchstates and their properties in the following.

Using the Fock representation of GFT as recapitulatedin appendix B, we decompose the field operator ϕ̂(gI)in terms of annihilation operators {ĉi} of single-particlequantum geometry states {|i〉} yielding

ϕ̂(gI) =∑i

ψi(gI)ĉi. (17)

Following the logic of the Bogoliubov approximationvalid for ultracold, non- to weakly interacting and di-lute Bose condensates [25, 26], if the ground state i = 0has a macroscopic occupation, one separates this expres-sion into a condensate term and one for all the remainingnoncondensate components. This yields

ϕ̂(gI) = ψ0(gI)c0 +∑i 6=0

ψiĉi, (18)

where one replaces the operator ĉ0 by the c-number c0 sothat the average occupation number of the ground state

is given by N = 〈ĉ†0ĉ0〉. In the next step one redefinesσ ≡√Nψ0 as well as δϕ̂ ≡

∑i 6=0 ψiĉi giving rise to

ϕ̂(gI) = σ(gI) + δϕ̂(gI), (19)

6The procedure to reconstruct the spatial metric by means of theinformation encoded in the quantum state is briefly adumbrated inappendix A.

where ψ0 is normalized to 1. This ansatz is only justi-fied if the ground state is macroscopically occupied, i.e.,N � 1 and the fluctuations δϕ̂ are regarded as small.One calls the classical field σ(gI) the mean field of thecondensate which assumes the role of an order parame-ter. Making use of the particle density n(gI) = |σ(gI)|2and a phase characterizing the coherence properties ofthe condensate, we write the mean field in polar form as

σ(gI) =√n eiθ(gI). (20)

This illustrates that the order parameter can always bemultiplied by an arbitrary phase factor without affectingthe physical measurement. This behavior is identified asa global U(1)-symmetry of the system which is associatedwith the conservation of the total particle number. UponBEC phase transition a particular phase is chosen whichamounts to the spontaneous breaking of this symmetry.

By construction, the Bogoliubov ansatz (19) gives riseto a nonzero expectation value of the field operator, i.e.,〈ϕ̂(gI)〉 6= 0, indicating that the condensate state is in,or rather close to, a coherent state.

Concretely, the simplest choice for the order parameteris provided by a condensate state

|σ〉 = A eσ̂|∅〉, σ̂ =∫

(dg)4 σ(gI)ϕ̂†(gI), (21)

which is constructed from quantum tetrahedra all en-coding the same discrete geometric data.7 It defines anonpeturbative vacuum over the Fock space. The nor-malization factor is given by

A = e−12

∫(dg)4 |σ(g)|2 . (22)

We require in addition to the right invariance as in Eq.(2) invariance under the left diagonal action of G, i.e.,σ(kgI) = σ(gI) for all k ∈ G. The latter encodes theinvariance under local frame rotations.

Such states are coherent because they are eigenstatesof the field operator,

ϕ̂(gI)|σ〉 = σ(gI)|σ〉, (23)

such that indeed 〈ϕ̂(gI)〉 = σ(gI) 6= 0 holds (as long as|σ〉 is not the Fock vacuum). Due to this property theexpectation value of the number operator immediatelyyields the average particle number

N =

∫(dg)4 |σ(gI)|2

the number operator is well defined and its expectationvalue is finite, the states used here are Fock coherentstates (cf. appendices B and C). By construction, sucha description is only valid for noninteracting or weaklyinteracting condensates. Toward the strongly interactingregime, it has to be replaced by one given in terms of non-Fock coherent states, as the appendices B and C suggest.

2. Effective dynamics

After having discussed the construction of suitablestates, let us briefly summarize how the effective con-densate dynamics can be obtained from the underlyingGFT dynamics as in Refs. [4, 11].

This is done by using the infinite tower of Schwinger-Dyson equations

0 = δϕ̄〈O[ϕ, ϕ̄]〉 =〈δO[ϕ, ϕ̄]δϕ̄(gI)

−O[ϕ, ϕ̄]δS[ϕ, ϕ̄]δϕ̄(gI)

〉, (25)

where O is a functional of the fields. One extracts anexpression for the effective dynamics by setting O equalto the identity. This leads to〈

δS[ϕ, ϕ̄]

δϕ̄(gI)

〉= 0 (26)

with the action S[ϕ, ϕ̄] as in Eq. (3). When the expecta-tion value is taken with respect to the condensate state|σ〉, one obtains the analog of the Gross-Pitaevskii (GP)equation for real Bose condensates∫

(dg′)4K(gI , g′I)σ(g′I) +δV

δσ̄(gI)= 0. (27)

This is in general a nonlinear and nonlocal equation forthe dynamics of the mean field σ and is interpreted asa quantum cosmology equation. In analogy to the GPequation, it has no direct probabilistic interpretation.These features might appear as a problem when trying torelate the GFTCC framework to LQC [30] or Wheeler-DeWitt (WdW) quantum cosmology [31]. However, theydo not pose a problem for the direct extraction of cos-mological predictions from the full theory. We refer toRefs. [4, 5, 7, 8, 10], where it has been demonstrated howa Friedmann-like evolution equation can be derived fromsuch an effective dynamics of specific GFT condensates.

III. TOWARD THE MEAN FIELD ANALYSISOF AN INTERACTING GFTCC MODEL

In the following, we proceed with analyzing the quan-tum dynamics of a particular GFT/GFTCC model in thefree and interacting cases. The larger scope of such ananalysis is to see whether one can construct particularcondensate solutions which admit e.g. an interpretationin terms of smooth continuous 3-geometries and are inline with the geometrogenesis picture.

We first review a free model and discuss how the gen-eral solution for an isotropic condensate is obtained fromthe equation of motion of the mean field σ in subsectionIII A. By doing so, we follow closely Ref. [5] and furtherelaborate special solutions. We extensively discuss thegeometric interpretation of such solutions by analyzingtheir curvature properties and by computing the expec-tation values of the volume and area operators importedfrom LQG. In subsection III B we then introduce twotypes of combinatorially local interaction terms in sub-section III B 1 and firstly treat them in subsection III B 2as perturbations of the aforementioned free solutions. Westudy solutions around the nontrivial minima of the re-sulting effective potentials in subsection III B 3 and dis-cuss the expectation values of the LQG volume and areaoperators in this case. Finally, we conclude the analysisby interpreting the obtained results.

In order to study the quantum dynamics of the meanfield σ as in Eq. (27), at first we have to specify thedetails of the action

S[ϕ, ϕ̄] =

∫(dg)4(dg′)4ϕ̄(gI)K(gI , g′I)ϕ(g′I) + V[ϕ, ϕ̄].

(28)Most generally, one could study the evolution in rela-tional terms by adding a free massless scalar field φ intothe action. For the GFT field one would then haveϕ = ϕ(gI , φ) where φ ∈ R accounts for the relationalclock as discussed in Ref. [7]. In these terms, the localkinetic operator is given by

K = δ(g′Ig−1I )δ(φ′ − φ)

[−(τ∂2φ +

4∑I=1

∆gI ) +m2

]. (29)

The signs of the terms appearing in K are chosen suchthat the functional S in the partition function Z isbounded from below. For τ > 0, the operator −(τ∂2φ +∑4I=1 ∆gI ) is positive and also this choice accounts for

the correct coupling of matter to gravity as noticed inRef. [4]. The Laplacian on the group manifold is mo-tivated by the renormalization group analysis of GFTmodels where one can show that it is generated by radia-tive corrections (cf. [21]). The ”mass term” is related tothe GFT/spin foam correspondence, as it corresponds tothe spin foam edge weights.8 Throughout the remainderof this article, we will focus on ”static” mean fields, i.e.σ(gI , φ) = σ(gI). The choice of action is then finalizedby selecting the interaction term V.

8Freezing the kinetic operator to the identity, would then lead tothe ultralocal truncation of the model and establish the above-discussed correspondence between certain GFT and spin foam mod-els [14] for an appropriate choice of interaction term.

7

A. A static free model

In a first approximation, we neglect all interactions andset V = 0. Using Eqs. (27) and (29), this yields[

−4∑I=1

∆gI +m2

]σ(gI) = 0. (30)

To find solutions to this dynamical equation, we intro-duce coordinates on the SU(2) group manifold, use in-variance properties of σ(gI) and apply symmetry reduc-tions, where we closely follow the results of Ref. [5] andelaborate them where needed.

To this aim, assume that the connection in the holon-omy g = Pei

∫eA remains approximately constant along

the link e with length `0 in the x-direction, which yieldsg ≈ ei`0Ax . In the polar decomposition, this gives

g = cos(`0|| ~Ax||)1 + i~σ~Ax

|| ~Ax||sin(`0|| ~Ax||), (31)

with the su(2)-connection Ax = ~Ax · ~σ and the Paulimatrices {σi}i=1...3. In the next step, we introduce thecoordinates (π0, ..., π3) together with π

20 + ... + π

23 = 1

which specifies an embedding of SU(2) ∼= S3 into R4.Due to the isomorphism SO(3) ∼= SU(2)/Z2, the choiceof sign in π0 = ±

√1− ~π2 corresponds to working on one

hemisphere of S3. With the identification

~π =~Ax

|| ~Ax||sin(`0|| ~Ax||), (32)

we can parametrize the holonomies as

g(~π) =√

1− ~π21 + i~σ · ~π, ||~π|| ≤ 1, (33)

where ||~π|| = 0 corresponds to the pole of the hemisphereand ||~π|| = 1 marks the equator. In these coordinates theHaar measure becomes

dg =d~π√

1− ~π2. (34)

Using the Lie derivative on the group manifold acting ona function f , one has for the Lie algebra elements

~Bf(g) ≡ i ddtf(e

i2~σtg)|t=0. (35)

With this the Laplace-Beltrami operator ~B2 = −∆g interms of the coordinates ~π on SU(2) is given by

−∆gf(g) = −[(δij − πiπj)∂i∂j − 3πi∂i]f(~π). (36)

This applies to all group elements gI , I = 1, ..., 4 dressingthe spin network vertex dual to the quantum tetrahedron.

In the most general case, the left and right invarianceimplies that σ(gI) lives on the six-dimensional domainspace SU(2)\SU(2)4/SU(2). It is thus parametrized by

six invariant coordinates πIJ = ~πI ·~πJ , with I, J = 1, 2, 3and 0 ≤ |πIJ | ≤ 1.

Using the above, Eq. (30) gives rise to a rather compli-cated partial differential equation. To find solutions, oneimposes a symmetry reduction by considering functionsσ which only depend on the diagonal components πIIand, furthermore, are assumed to be all equal. Togetherwith Eq. (32), this yields

p ≡ πII = sin2(`0|| ~Ax||). (37)

Using this, one can rewrite Eq. (30) as

−[2p(1− p) d

2

dp2+ (3− 4p) d

dp

]σ(p) + µσ(p) = 0, (38)

with µ ≡ m2

12 and p ∈ [0, 1] for which analytic solutionscan be found [5].9

Indeed, it makes sense to refer to this symmetry reduc-tion (to just one variable p) as an isotropization. Retro-spectively, this can be seen when rewriting Eq. (38) usingp ≡ sin2(ψ). With this we obtain

− [ d2

dψ2+ 2 cot(ψ)

d

dψ]σ(ψ) + 2µσ(ψ) = 0, ψ ∈ [0, π/2],

(39)which can be compared to the Laplacian on a hemisphereof S3 acting on a function σ(φ, θ, ψ), given by

−∆σ(φ, θ, ψ) = − 1sin2(ψ)

[∂

∂ψ(sin2(ψ)

∂

∂ψσ) + ∆S2σ

],

(40)with φ ∈ [0, 2π], θ ∈ [0, π] and ψ ∈ [0, π/2]. The functionσ is called isotropic or zonal if it is independent of φ andθ [32]. These are spherically symmetric eigenfunctions of−∆S2 for which Eq. (39) is equal to Eq. (40). Hence, thesymmetry reduction can be seen as explicitly restrictingthe rather general class of condensates to a representativewith a clearer geometric interpretation.The general solution to Eq. (38) is given by

σ(p) = 4√

1− pp

[a P1212

√1−2µ−1(2p− 1) +

b Q1212 (√

1−2µ−1)(2p− 1)], (41)

with a, b ∈ C and P,Q are associated Legendre functionsof the first and second kinds, respectively. With respectto the measure induced from the full Fock space, oneyields for the average particle number

N =

∫(dg)3|σ(g1, g2, g3)|2

= 2π

∫dp

√p

1− p|σ(p)|2

In the following, we want to specify the possiblevalues of µ in the symmetry reduced case by meansof discussing the spectrum of the operator −

∑I ∆gI .

Its self-adjointness and positivity imply that its eigen-values {m2} lie in R+0 . The compactness of the do-main space SU(2)\SU(2)4/SU(2) entails that the spec-trum is discrete and the respective eigenspaces are finite-dimensional. This also holds for the symmetry reducedcase.

To finally concretize the spectrum, we have to intro-duce boundary conditions, which we infer from physicalassumptions. For this we can exploit that we are look-ing for solutions to the equation of motion which admitan interpretation in terms of smooth metric 3-geometriesand thus obey the above-mentioned near flatness con-dition. In the group representation, this condition con-cretely translates into demanding that the character ofthe group elements decorating the quantum tetrahedraare close to χ(1Ji) = 2Ji + 1 according to Refs. [4, 11].On the level of the mean field this leads to the require-ment that the probability density is concentrated aroundsmall values of the connection or its curvature. In thesymmetry reduced case this condition holds for σ(p) if theprobability density |σ(p)|2 is concentrated around smallvalues of the variable p and tends to zero at the equatortraced out at p = 1. The latter translates into a Dirichletboundary condition on the equator,

σ(p)|p=1 = 0, (43)

which is only obeyed by the Q-branch of the generalsolution Eq. (41). Using this, the spectrum of theDirichlet Laplacian is given by µ = −2n(n + 1) withn ∈ (2N0 + 1)/2.10,11

Equivalently, these solutions correspond to the eigen-solutions of Eq. (39) obeying the boundary conditionσ(π2 ) = 0. They are given by

σj(ψ) =sin((2j + 1)ψ)

sin(ψ), ψ ∈ [0, π

2] (45)

with j ∈ 2N0+12 corresponding to the eigenvaluesµ = −2j(j + 1). On the interval [0, π2 ] these solutions

10The only eigenfunction of the Dirichlet Laplacian to the eigenvalueµ = 0 is the trivial function. In the mean field analysis of phasetransitions the mean field is supposed to vanish if the driving pa-rameter µ turns to 0. Here, consistency with the flatness conditionimplies the vanishing of the mean field σ for µ = 0.

11Due to the linear character of the free problem, the solutions havea rescaling invariance with respect to the chosen boundary condi-tions. This means that two solutions for different boundary condi-tions σ′(1) can be rescaled into one another according to

N [σ′1(1)]

N [σ′2(1)]=|σ′1(1)|2

|σ′2(1)|2, (44)

which obscures the interpretation of the quantity N and otherobservables in the free case. This rescaling property is lost once(strong) nonlinear interactions are considered as in the next sub-section.

are exactly equal to those hyperspherically symmetriceigenfunctions of the Laplacian on S3 which vanish onthe equator. Furthermore, observe that these are justthe characters χj(ψ) of the respective representation forj.

In view of the geometric interpretation of these solu-tions, we want to illustrate and then discuss the behaviorof the first few eigensolutions by plotting their probabil-ity density |σ(p)|2 in Fig. 1 or |σ(ψ)|2 in Fig. 2, re-spectively. The plot illustrates that the probability den-sity is concentrated around small values of the variablep or ψ, respectively. In general, eigensolutions remainfinitely peaked around p = 0 or ψ = 0. Solutions forslightly perturbed eigenvalues µ are infinitely peaked aslimp→0 |σ(p)|2 ∼ 1/p.

j=1/2j=3/2j=5/2j=7/2j=9/2

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

p

σ j(p)2

Figure 1: Probability density of the free mean field over p.

j=1/2j=3/2j=5/2j=7/2j=9/2

0.0 0.5 1.0 1.50

2

4

6

8

10

12

14

ψ

σ j(ψ)2

Figure 2: Probability density of the free mean field over ψ.

A concentration of the probability density aroundsmall p corresponds to a concentration around small cur-vature values. This is because small p, itself directlyproportional to the gravitational connection A, impliessmall field strength via F = DAA. Naively, in turn thisleads to a small 3-curvature R, as is known from thefirst-order formalism for gravity. This is important for

9

consistency matters, meaning that the building blocksof the geometry are indeed almost flat which is neededto approximate a smooth continuum 3-space. Aroundp = 1 or ψ = π2 , tracing out the equator of S

3, the so-lutions vanish. The occurrence of the finite number ofoscillatory maxima does not a priori pose a problem tothe fulfillment of the near flatness condition since theeigensolutions are indeed concentrated around small val-ues of p or angles ψ, far away from the equator. Forthe characters of the corresponding representations, thenear flatness condition means that they should be closeto χ(1j) = 2j + 1 [4, 11]. Our solutions obey this re-quirement since in Eq. (45) limψ→0 σj(ψ) exactly yields2j + 1. In this light, using the solutions σj(ψ), we cancompute the average of the field strength12 F i ∼ p givenby

〈F̂ i〉N∼∫ π/2

0

dψ sin2(ψ) |σj(ψ)|2 F̂ i / N > 0, (46)

which is illustrated in Fig. 3. The dots indicate the dis-crete contributions to the field strength for a particular j-mean field and show a dominance of the 1/2-eigensolutionover the others on which we comment below. In light of

●

● ● ● ● ● ● ● ● ● ●0 1/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 19/2 21/20.00

0.05

0.10

0.15

0.20

j

F(σ(j))

Figure 3: Un-normalized spectrum of the field strength withrespect to the eigensolutions σj(ψ) in arbitrary units.

the previous discussion, it may seem a bit surprising thatthe expectation value of the field strength is nonzero de-spite the fact that p = 0 is the most probable value ofthe corresponding mean field. However, the extendedtail of the probability density with the finite oscillatory

12Relating p to the field strength is justified when considering a pla-quette 2 in a face of a tetrahedron so that we can make use of thewell known expression

Fkab(A) =1

Tr(τkτk)lim

Area2→0Trj

(τk

hol2ij (A)− 1Area2

)δiaδ

jb ,

where a, b ∈ {1, 2} and for the su(2)-algebra elements τk = − i2σkthe relation Tr(τkτk) = − 1

3j(j + 1)(2j + 1) holds [30]. This yields

Fk ∼ sin2(ψ) = p.

maxima accounts for the average being bigger than themost probable value. The finite value indicates that thespace described by the condensate is of finite size. Wewill come back to this point at the end of this section.13

In the last step, we want to transform our nearlyflat solutions to the spin-representation which facilitatesmost directly the extraction of information about theLQG volume and area operators and is crucial for thegeometric interpretation of the solutions. To this aim,notice that due to the left- and right-invariance of σ(gI),the mean field is in particular a central function on thedomain space, i.e., σ(hgIh

−1) = σ(gI) for all h ∈ SU(2).This holds for the isotropic function σ(p) or σ(ψ), analo-gously. In this case isotropy coincides with the notion ofcentrality. Using the Fourier series of a central functionon SU(2) [32], the Fourier series for the mean field in theangle parametrization is given by

σj(ψ) =∑

m∈N0/2

(2m+ 1) χm(ψ) σj;m, (47)

with the ”plane waves” given by the characters χm(ψ) =sin((2m+1)ψ)

sin(ψ) . The Fourier coefficients are then obtained

via

σj;m =2

π

1

2m+ 1

∫ π/20

dψ sin2(ψ) χm(ψ) σj(ψ), (48)

and m ∈ N02 . Using this, the Fourier coefficients of thesolutions σj(ψ) (cf. Eq. (45)) yield

σj;m =2

π

1

2m+ 1

(−1)2j−1

2 (2j + 1) cos(

2mπ2

)(2m− 2j) (2m+ 2j + 2)

, (49)

with j ∈ 2N0+12 . In the spin-representation, the expec-tation value of the volume operator with respect to themean field is decomposed as

〈V̂ 〉 ≡ V = V0∑

m∈N0/2

|σj;m|2Vm with Vm ∼ m3/2 (50)

and V0 ∼ `3p.14 The normalized volume V/V0 is shownin Fig. 4 for different values of j. The dots indicatethe discrete contributions to the volume for a particularj. Eigensolutions for smaller j or |µ| have a bigger vol-ume in comparison to those with larger j, especially thej = 3/2 eigensolution has the relatively biggest volume.Importantly, the volume is finite for all j indicating thatthe space which the condensate approximates must be of

13The last word on the flatness behavior of such solutions also in theinteracting case, however, lies with the analysis (of the expectationvalue) of a currently lacking GFT-curvature operator, as alreadynoticed in Ref. [11].

14As a side remark, notice that if µ > 0, naively j would be complexand thus also the spectra of the geometric operators such as thevolume V̂ .

10

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j=1/2j=3/2j=5/2j=7/2j=9/2j=11/2j=13/2j=15/2j=17/2j=19/2

01/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 19/2 21/20.0000.005

0.010

0.015

0.020

0.025

0.030

m

Vσ j(m

)V0

Figure 4: Normalized spectrum of the volume operator withrespect to the eigensolutions σj(ψ) in arbitrary units.

finite size. Hence, a general solution which can be de-composed in terms of eigensolutions, describes a finitelysized space of which the largest contributions arise fromlow spin modes. Finally, Fig. 5 illustrates the uncer-tainty of the volume operator, which is monotonouslyincreasing in j and indicates that its expectation valueassumes a sharper value if the condensate resides in lowerj-modes.

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●

0 1/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 19/20.00.2

0.4

0.6

0.8

1.0

1.2

j

ΔV(σ(j)) V 0

Figure 5: Standard deviation of the volume operator over j.

Analogously, the expectation value of the area operatorfor an individual face of a quantum tetrahedron in thecondensate is given as

〈Â〉 ≡ A = A0∑

m∈N0/2

|σj;m|2Am (51)

with Am ∼ (m(m + 1))1/2 and A0 ∼ `2p. Dependingon the solution σj , the spectrum of the normalized areaA/A0 is illustrated in Fig. 6 showing a dominance of the1/2-representation and otherwise with a similar interpre-tation as in the case of the volume operator.

From the above, it is not clear whether a certain eigen-solution could be dynamically preferred over others. The

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01/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 19/2 21/20.000.01

0.02

0.03

0.04

0.05

0.06

m

Aσ j(m

)A0

Figure 6: Normalized spectrum of the area operator with re-spect to the eigensolutions σj(ψ) in arbitrary units.

near flatness condition seems to be better fulfilled bylower eigenmodes, that means for those solutions witha lower number of oscillatory maxima. These are the so-lutions which are mostly concentrated around small con-nection or curvature values. This might be connectedto the recent findings of a dynamically reached low spinphase in a similar GFT condensate cosmology model[9]. In this light, it is striking that the computation ofthe expectation values of the volume, area and the fieldstrength operators all display the dominance of low j-modes. This can be seen to be in favor of the condensatepicture where the field quanta tend to condense into thesame simple quantum geometric state. Below we explorethe case of interacting models which is pivotal for the ge-ometric interpretation of the solutions and the extractionof phenomenology.

We want to make a final remark about restricting ourattention solely to the those solutions obeying the nearflatness condition. Of course one could consider moregeneral solutions to Eq. (38) which are not necessarilypeaked around p = 0 as in Ref. [5]. Despite the fact, thatsuch solutions cannot be interpreted in terms of smoothcontinuous 3-geometries according to the near flatnesscondition proposed in Ref. [4], their properties couldnevertheless be studied in a similar manner which willbe done elsewhere.

B. Static interacting models

In this subsection we add to the above consideredfree model two different combinatorially local interactionterms, i.e. pseudopotentials, and analyze their effect onthe behavior of the solutions and the expectation valuesof releavant operators. This enables us to study aspectsof the resulting effective quantum geometries.

One might speculate that such simplified interactions

11

between the condensate constituents are only relevant ina continuum and large scale limit, where the true com-binatorial nonlocality of the fundamental theory couldbe effectively hidden. This idea can perhaps be moti-vated by speculating that while the occurrence of UVfixed points in tensorial GFTs is deeply rooted in theircombinatorial nonlocality (cf. [19, 20]), the occurrence ofIR fixed points, akin to Wilson-Fisher fixed points in thecorresponding local QFTs, seems to be unaffected by thisfeature. Ultimately, rigorous RG arguments will have thedecisive word whether combinatorially local interactionterms may be derived from the fundamental theory. Inthis way, studying the effect of pseudopotentials and try-ing to extract physics from the solutions can be usefulto clarify the map between the microscopic and effectivemacroscopic dynamics of the theory and is instructive togain experience for the treatment of the correspondingnonlocal terms which have a clearer discrete geometric in-terpretation. In this light, we will consider two classes oflocal interactions, mimicking the so-called tensorial andthe above-introduced simplicial interactions.

1. General setup of the interacting GFTCC models

The models on which we built our analysis assume anaction of the form

S[ϕ, ϕ̄] =

∫(dg)4(dg′)4ϕ̄(gI)K(gI , g′I)ϕ(g′I) + V[ϕ, ϕ̄],

(52)with the kinetic operator

K = δ(g′Ig−1I )[−

4∑I=1

∆gI +m2

](53)

and the general pseudopotential mimicking so-called ten-sorial interactions

VT [ϕ] =∑n≥2

κnn

∫(dg)4(|ϕ(gI)|2)n (54)

which is even powered in the modulus of the field. Oneobtains for the equation of motion of the mean field[−

4∑I=1

∆gI +m2

]σ(gI) + σ(gI)

∑n=2

κn(|σ(gI)|2)n−1 = 0.

(55)Observe that the combinatorial locality implies that wedo not make use of any nontrivial pairing pattern for thefields and when applying the same symmetry assump-tions as above one has σ(g1, g2, g3, g4) = σ(g, g, g, g) =σ(p). Considering only one summand for the interaction,we yield

− [2p(1− p) d2

dp2+ (3− 4p) d

dp]σ(p) + µσ(p) +

κσ(p)(|σ(p)|2)n−1 = 0, (56)

with n = 2, 3, 4, ....

In the following, we focus on the case of real-valuedGFT fields and set n = 2, for which the equation ofmotion reads

− [2p(1− p) d2

dp2+ (3− 4p) d

dp]σ(p) +µσ(p) +κσ(p)3 = 0,

(57)with the effective potential

Veff [σ] =µ

2σ2 +

κ

4σ4. (58)

The signs of the coupling constants determine the struc-ture of the ground state of the theory. For appropriatelychosen signs of µ and κ the potential, and thus the spec-trum of the theory, is bounded from below. However,only for µ < 0 and κ > 0 one can have a nontrivial(nonperturbative) vacuum with

〈σ〉 6= 0, (59)

which is needed to be in agreement with the condensatestate ansatz. The two distinct minima of the potentialare located at 〈σ0〉 = ±

√−µ/κ where the potential has

strength V0 = −µ2/4κ, which is lower than the value forthe excited configuration σ = 0. The system would thussettle into one of the minima as its equilibrium config-uration and could be used to describe a condensate.15

This potential is illustrated in Fig. 7 and contrasted tothe case where µ > 0 for which the potential is a con-vex function of σ with minimum at 〈σ〉 = 0. The lat-ter setting cannot be used to describe a condensate withN 6= 0. For other choices of signs, the equilibrium con-figuration 〈σ〉 = 0 is unstable or metastable and shouldbe dismissed. The upshot of this discussion is that if theeffective action is to represent a stable system and a con-densate of GFT quanta, one must choose the signs of thecoupling constants accordingly.16

15A sign change of the driving parameter µ from positive to nega-tive values induces a spontaneous symmetry breaking of the globalZ2-symmetry of the action specified in Eq. (52). This symme-try would have guaranteed the conservation of oddness or evennessof the number of GFT quanta as it corresponds to the conserveddiscrete quantity (−1)N . For complex-valued GFT fields the ana-loguous situation would correspond to the spontaneous breaking ofthe global U(1)-symmetry of the action which would have guaran-teed the conservation of the particle number N .

16For real BECs, κ < 0 gives an attractive interaction and only a largeenough kinetic term can prevent the condensate from collapsing. Inthe opposite case where κ > 0, the interaction is repulsive and ifit dominates over the kinetic term the condensate is well describedin terms of the so-called Thomas-Fermi approximation [25].

12

σ

V(σ)

0

μ>0, κ>0 μ0Figure 7: Plot of the effective potential Veff [σ] =

µ2σ2 + κ

4σ4.

Similarly, when considering the local pseudopotentialmimicking the above-introduced simplicial interaction forreal-valued GFT fields

VS [ϕ] =κ

5

∫(dg)4ϕ(gI)

5, (60)

one has

− [2p(1− p) d2

dp2+ (3− 4p) d

dp]σ(p) +µσ(p) +κσ(p)4 = 0.

(61)For such a model the effective potential reads

Veff [σ] =µ

2σ2 +

κ

5σ5. (62)

We will ignore here that this potential is unboundedfrom below to one side.17 Only for (µ < 0, κ > 0) or(µ < 0, κ < 0) one can have a nontrivial (nonpertur-bative) vacuum in agreement with the condensate stateansatz and the discussion of the choice of signs is sim-ilar to the firstly considered potential. Classically, thecorresponding minima of the potential are then locatedat σ0 = ± 3

√∓µ/κ where the potential has strength

V0 = (∓µ/κ)2/3(3µ/10). This is illustrated in Fig. 8for one case and contrasted to the situation where µ > 0which would lead to 〈σ〉 = 0.

2. Perturbation of the free case

In a first step, we consider the interaction term as aperturbation of the free case discussed in subsection III A

17Notice that when using four arguments in the group field ϕ, highersimplicial interaction terms known to be e.g. of power 16 or 500would lead in the local point of view, adopted here, to boundedeffective potentials Veff like (58) and the discussion of their effectswould be rather analogous.

σ

V(σ) 0

μ>0, κ>0 μ0Figure 8: Plot of the effective potential Veff [σ] =

µ2σ2 + κ

5σ5.

using the same boundary conditions σ(1) = 0 and dif-ferent σ′(1) to solve numerically the nonlinear differen-tial equations (57) and (61), respectively. By followingclosely the procedure adopted in the free case, we com-pute the effect of perturbations onto the probability den-sities and the spectra of geometric operators. In this waywe obtain a clear qualitative picture of the effect of in-teractions by comparing the results to the ones obtainedfor the free case.

In the following, we discuss the behavior of solutionsfor the pseudotensorial potential (58) with µ < 0 andκ > 0 and where the qualitative results differ also forthe pseudosimplicial potential (62) with µ < 0 and κ > 0(or κ < 0) so that the potentials would possess nontrivialminima. The effect of weak nonlinearities in the equationof motion onto the solutions is illustrated respectively inFig. 9 and Fig. 10 in the p- and ψ-parametrizationsand is contrasted to the behavior of the free solutionsof subsection III A. In general, the finiteness of the freesolutions at the origin is lost due to the interactions.Crucially, the concentration of the probability densitiesaround the origin can still be maintained giving rise tonearly flat solutions, as long as |κ| does not become toobig. For larger j, i.e. larger |µ|, one sees that the depar-ture from the free solutions is less pronounced becausethe µ-term of the potential dominates longer over theκ-term.

When |κ| and |σ′(1)| are small, solutions will remainnormalizable with respect to the Fock space measure, i.e.

N =

∫(dg)3|σ(g1, g2, g3)|2

j=1/2j=1/2, κ=0.1j=3/2j=3/2, κ=0.1j=5/2j=5/2, κ=0.1

0.0 0.5 1.0 1.50

2

4

6

8

10

12

14

ψ

σ j(ψ)2

Figure 9: Probability density of the interacting mean fieldover ψ for Veff [σ] =

µ2σ2 + κ

4σ4.

j=1/2j=1/2, κ=0.1j=1/2, κ=1j=3/2j=3/2, κ=0.1j=3/2, κ=1

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

p

σ j(p)2

Figure 10: Probability density of the interacting mean fieldover p for Veff [σ] =

µ2σ2 + κ

4σ4.

in the strongly nonlinear regime goes in hand with thebreaking of the rescaling invariance expressed by Eq. (44)and signals the breakdown of the ansatz used here. Suchbehavior is not surprising, as it is well known within thecontext of local QFTs that the proper treatment of inter-actions necessitates the use of non-Fock representationsfor which N is infinite (cf. appendix B and [34]). We willget back to this point below.

With regard to the average of the field strength, oneobserves that κ > 0 increases 〈F̂ i〉/N for some j in com-parison to the free case, whereas for negative κ the ex-pectation value decreases. This behavior is reminiscent ofthe effect of similar interactions onto the effective curva-ture of the space described by the condensate in Ref. [10],where it was shown that a bounded interaction potentialgenerically leads to recollapsing condensate solutions.

By means of the numerically computed solutions, one

which N →∞ depend on the numerical accuracy of the used solver.In this sense the observation of such behavior is a qualitative result.

can obtain their corresponding Fourier components andwith these one yields in close analogy to the free casethe modified spectra of the volume and area operators,illustrated in Fig. 11 and Fig. 12. The plots clearly

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Figure 11: Normalized spectrum of the volume operator withrespect to the interacting mean field σj(ψ) for κ = 0.22 (tri-angles) compared to the respective free solutions (dots).

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A(σ j(m

))A0

Figure 12: Normalized spectrum of the area operator withrespect to the interacting mean field σj(ψ) for κ = 0.22 (tri-angles) compared to the respective free solutions (dots).

indicate that perturbations for κ > 0 increase both thevolume and the area, however, in the weakly nonlinearregime they remain finite. More specifically, one observesthat the effect of the perturbations upon the spectra ofthe volume and area are more pronounced for small j,i.e. small |µ|, since for these the nonlinearity dominatesquickly over the µ-term of the potential. Moreover, onenotices that when pushing κ to larger values as a conse-quence V and A quickly blow up in the same way as Ndoes, whereas 〈F̂ i〉/N remains finite.

14

For the pseudosimplicial potential one obtains qualita-tively analogous results with the differences to the freesolutions being more emphasized since the nonlinearityis stronger.

3. Solutions around the nontrivial minima

To chart the condensate phase and understand itsproperties, it is necessary to study numerically the so-lutions to the nonlinear differential equation (57) aroundthe nontrivial minima. To this aim, we choose for thecoupling constants in Eq. (58) in such a manner that thepotential forms a Mexican hat, as in Fig. 7, and selectthe position of the minimum σ0 as well as σ

′(1) as theboundary condition in order to find solutions numerically.

Without any loss of generality, we will use the samevalues for µ as in the previous subsections. Apart fromthe requirement that they assume negative values theycould be completely arbitrary since here we do not studyeigensolutions to the Dirichlet Laplacian as in subsectionIII A.

Figure 13 and Fig. 14 show the resulting probabilitydensity and the potential over p and ψ computed for anexemplary choice for the values of the free parameters.Depending on the sign of σ′(1) or σ′(π/2) the solutioneither climbs over the local maximum at σ = 0, thenreaches the other minimum after which it ascends the leftbranch of the potential or directly climbs up the rightbranch shown in Fig. 7. For the choice of parametersleading to Fig. 13 and 14, the solutions are normalizable.In general, for small σ′(1) the solutions crawl slowly outof the minima and if σ′(1) is almost zero, the solutionsremain almost constant up to p = 0, where the regularsingularity of the differential equation finally kicks in.The contribution of the Laplacian term is less pronouncedfor smaller µ than for larger ones, as Fig. 14 in the ψ-parametrization illustrates. Similar results are obtainedwhen one keeps µ fixed while decreasing |σ′(π/2)|.

It is clear that as long as for the boundary conditionσ′ ≈ 0 holds, this is equivalent to neglecting the Lapla-cian part of the kinetic term K in the equation of motion.The solutions, exemplified by Figs. 13 and 14, show thatthe properties of the nontrivial ground state are then de-fined by the ultralocal action. Solving the equation ofmotion starting at the minima of the effective potentialsgives rise to almost constant, i.e. homogeneous, functionson the domain.19

The geometric interpretation of such solutions which”sit” in the equilibrium position is slightly obstructed.This is due to the fact that the above-used near flatness

19Completely neglecting the Laplacian from the onset, is only jus-tified when the interaction is dominant which corresponds to theregime of large ground state condensate ”density”, i.e. κN � 1.In the context of real BECs this is known as the Thomas-Fermiapproximation [25].

Figure 13: Semilog plot of the probability density and po-tential for solutions σ(p) with µ = −1.5, κ = 0.01, σ(1) =12.2474 and σ′(1) = ±100 for the potential Veff [σ] =µ2σ2 + κ

4σ4. Solutions were computed by means of MAT-

LAB’s ODE45 solver which is based on an explicit Runge-Kutta (4, 5) formula. Output was generated for 105 points onthe interval [0, 1] while making use of highly stringent errortolerances.

μ=-1.5, σ'(π/2)>0μ=-1.5, σ'(π/2)

means of a well defined GFT-operator capturing the av-erage curvature of the 3-space described by means of thecondensate state.

In spite of the current lack of such an operator to deter-mine the curvature information stored in the mean field,it is possible to obtain from exemplary numerical solu-tions the spectrum of the volume and area operators asillustrated in Figs. 15, 16 and 17. Solutions which arecomputed around the nontrivial minima give rise to adifferent qualitative form of the spectrum of the volumeand area as compared to the ones obtained in subsec-tions III A and III B 2; nevertheless we emphasize againthe relevance of low spin modes. In general, for differentµ the dominant contribution to the volume V and areaA comes from the Fourier coefficients with m = 1/2,whereas in the case discussed in the previous subsectionsthe predominant contribution comes from the Fourier co-efficients with m = j. This is due to the fact that σ re-mains mostly constant and is thus best approximated bythe simplest nontrivial mode for m = 1/2. In particular,one can check that the contributions to V and A com-ing from the other modes, are exponentially suppressedwhen σ′(1) ≈ 0.

Moreover, the volume and area remain finite in theweakly nonlinear case and when the boundary conditionσ′(1) is relatively small. The use of weak interactions isthus instructive in order to understand the qualitativebehavior of the solutions in particular with regard to theexpectation values of the geometric operators. Since thesize of κ has only a quantitative impact on the spectrum,as Fig. 16 suggests, an analogous form of the spectracan be expected also in the strongly nonlinear regime.Furthermore, for bigger values of |σ′(1)| and/or stronglynonlinear interaction terms, the volume and area, as wellas the expectation value of the number operator N̂ blowup quickly. As noticed above, this signals the break-down of the simple condensate state ansatz used hereand suggests the need for non-Fock coherent states oncethe strongly correlated regime is explored (cf. appendixC).20

Such solutions yield for all choices of µ < 0 and κ > 0for the averaged observables

〈Ô〉N≈ const., (64)

since σ is approximately constant. This naturally applies

to the averaged field strength 〈F̂i〉N which is larger than

in the corresponding free case. This indicates that thechosen effective GFT interactions have the effect of pos-itively curving the effective geometry described by thecondensate state. This is again reminiscent of similarfindings in Ref. [10] where it was shown that relationally

20Figure 16 also seems to suggest that V is ever increasing for κ→ 0.However, in such a limit, it is more appropriate to treat the systemas in the free case, which we discussed in subsection III A.

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m

V(σ μ(m

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Figure 15: Normalized discrete spectrum of the volume oper-ator (in arbitrary units) with respect to the interacting meanfield σµ(ψ): Solutions σµ(ψ) were obtained with κ = 0.01but boundary conditions differ for each µ to solve arounda nontrivial minimum of the respective potential Veff [σ] =µ2σ2 + κ

4σ4.

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150

200

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m

V(σ μ(m

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Figure 16: Normalized discrete spectrum of the volume oper-ator (in arbitrary units) with respect to the interacting meanfield σµ(ψ): Solutions σµ(ψ) were obtained for µ = −1.5, dif-ferent κ and the same boundary conditions σ′(π

2) to solve

around the respective nontrivial minima of the potentialVeff [σ] =

µ2σ2 + κ

4σ4.

evolving and effectively interacting GFTCC models dis-play recollapsing solutions when the interaction potentialis bounded from below as here.

Analogously, such a discussion can be repeated for thepseudosimplicial potential, where the solutions to thenonlinear equation of motion are illustrated in Fig. 18.The resulting behavior of the relevant operators is sim-ilar and will not be repeated here, though it should be

16

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0 1/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 19/20

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40

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Figure 17: Normalized discrete spectrum of the area opera-tor (in arbitrary units) with respect to the interacting meanfield σµ(ψ): Solutions σµ(ψ) were obtained with κ = 0.01but boundary conditions differ for each µ to solve arounda nontrivial minimum of the respective potential Veff [σ] =µ2σ2 + κ

4σ4.

kept in mind that only such interaction terms can bemore closely related to models with a simplicial quan-tum gravity interpretation.

To summarize the main points of this subsection, wenote that we have computed static condensate solutionsaround the nontrivial minima of the interaction poten-tials of which the essential features can be defined bymeans of the ultralocal action. We found that the con-densate consists of many GFT quanta residing in the lowspin mode m = 1/2. This is indicated by the analysis ofthe discrete spectra of the geometric operators. Such lowspins actually correspond to the IR regime of the theory.Hence, these results fit well into the picture suggested bythe above- mentioned FRG analyses which find IR fixedpoints in all GFT models considered so far marking theformation of a condensate phase of which main featuresare supposed to be captured by means of the employedcondensate state. In this sense, one can understand thecondensate phase as to describe an effectively continu-ous homogeneous and isotropic 3-space built from manysmall building blocks of the quantum geometry.

IV. DISCUSSION AND CONCLUSION

The purpose of this article was to investigate and in-terpret the impact of simplified interactions onto staticGFT quantum gravity condensate systems describing ef-fective 3-geometries with a tentative cosmological inter-pretation. To this aim, we extensively examined the geo-metric properties of a free system in an isotropic restric-tion by studying the spectra of the volume and area op-erators imported from LQG and comparing the results to

Figure 18: Semilog plot of the probability density and poten-tial for solutions σ(p) with µ = −1.5, κ = 0.01, σ(1) = 5.3132and σ′(1) = ±100 for the potential Veff [σ] = µ2 σ

2 + κ5σ5. So-

lutions were computed by means of MATLAB’s ODE113 pro-cedure which is a variable order ’Adams-Bashforth-Moultonpredictor-corrector’ solver. Output was generated for 105

points on the interval [0, 1] while making use of highly strin-gent error tolerances.

the perturbed case. In a last step, we studied the featuresof the GFT condensate when the system sits in the non-trivial minima of the effective interaction potentials. Themain result of this study is then that the condensate con-sists of many discrete building blocks predominantly ofthe smallest nontrivial size encoded by the quantum num-ber m = 1/2 – which supports the idea that an effectivelycontinuous geometry can emerge from the collective be-havior of a discrete pregeometric GFT substratum [3].In this sense, our results also strengthen the connectionwith LQC where the typically used quantum states areconstructed from the assumption that the quanta of thegeometry all reside in the same lowest nontrivial config-uration [30]. Together with the recently obtained resultswhich show how free GFT condensate models dynami-cally reach a low spin phase [9], this lends strong sup-port to the idea that condensate states are appropriatefor studying the cosmological sector of LQG.

The results of this article can also be seen as a sup-port of the idea proposed in Ref. [36]: The Laplacianin the kinetic operator K, originally motivated by fieldtheoretic arguments to guarantee the consistent imple-mentation of a renormalization scheme, might only be aproperty of the UV completed GFT without a significantphysical effect in the effectively continuous region whichis expected to correspond to the small spin (IR) regime

17

together with many building blocks of the quantum ge-ometry. In this regime, the kinetic term is then suggestedto become ultralocal, thus allowing for a straightforwardinterpretation of the GFT amplitudes in terms of spinfoam amplitudes for quantum gravity. The numericalanalysis done here indeed suggests that from the ultralo-cal action alone one can find that the condensate consistsof many GFT quanta residing in the low spin configura-tion.

In the following we want to comment on the limita-tions of our discussion. We implicitly assumed that thecondensate ansatz is trustworthy for any µ ≤ 0, whereµ = 0 marks the critical value at which the phase tran-sition from the unbroken into the condensate phase issupposed to take place [19, 21]. With respect to thesefindings, our analysis should be complemented by inves-tigating whether indications for a phase transition intoa condensate phase can be observed with the mean fieldtechniques employed here, e.g., by means of the analyt-icity properties of the partition function, and whethertheir possible absence might be related to the expectationthat true phase transitions are only realized for GFTs onnoncompact manifolds, like Lorentzian quantum gravitymodels, as noticed in Ref. [19].

In this light, it is worth noting that in the context ofweakly interacting, diluted and ultracold nonrelativisticBECs [25] it is well understood that Bogoliubov’s meanfield and perturbation theory [26] becomes invalid andbreaks down in the vicinity of the critical point of thephase transition because quantum fluctuations becomeimportant. Of course, as is generally known today, meanfield approaches work only accurately as effective descrip-tions of thermodynamic phases well away from criticalpoints. A satisfactory description for such systems whichsystematically extends Bogoliubov theory and cures itsinfrared problems has been given in terms of FRG tech-niques [33]. The example of real BECs suggests that theanalog of the Bogoliubov ansatz for quantum gravity con-densates should be similarly extended by means of FRGmethods at the critical point. This could be relevant forbetter understanding the nature of the phase transitionwhich is possibly related to the ”geometrogenesis” sce-nario.

It is also well understood that Bogoliubov theoryfor real BECs breaks down, when considering conden-sates with rather strongly interacting constituents. Like-wise, FRG techniques can systematically implement non-perturbative extensions to Bogoliubov’s approximation.These suggest that for Bose condensates with approxi-mately pointlike interactions like in superfluid 4He, it isonly possible to realize a strongly interacting regime fora very dense condensate [33]. This example could indi-cate a similar failure of the quantum gravity condensateansatz when considering the strongly interacting regime.Indeed, when increasing the coupling constant κ in thissector, the average particle number N grows. If κ is toolarge, we find that solutions are generally not normaliz-able with respect to the Fock space measure. The regime

of large number of quanta N and the eventual failureof |σ〉 to be normalizable in this sector certainly markthe breakdown of the Gross-Pitaevskii approximation tothe dynamics (27) for the simple condensate state con-structed with Bogoliubov’s ansatz (19). In this regime,quantum fluctuations and correlations among the con-densate quanta become relevant and only solutions to thefull quantum dynamics together with FRG techniqueswould be capable of capturing adequately their impact.This entails that the approximation used here should onlybe trusted in a mesoscopic regime where N is not toolarge, as already noticed in Ref. [7]. Nevertheless, thefinding of solutions corresponding to non-Fock represen-tations gives a forecast on what should be found whenconsidering nonperturbative extensions of the techniquesused here.

In fact, the loss of normalizability is not too surpris-ing because it is a generic feature of massless or inter-acting (local) QFTs according to Haag’s theorem whichrequire the use of non-Fock representations [34]. How-ever, finding such solutions is first of all intriguing as amatter of consistency because non-Fock representationsare also required in order to describe many particle sys-tems in the thermodynamic limit. It is only in this limitthat inequivalent irreducible representations of the CCRsbecome available which is a prerequisite for the occur-rence of nonunique equilibrium states, in turn essentialto consistently describing phase transitions [34]. It isalso interesting for a second reason, since in the contextof quantum optics it was understood that such non-Fockcoherent states with an infinite number of (soft) photonscan be described in terms of a classical radiation field[35]. Hence, the occurrence of non-Fock coherent statesolutions in our context might also play a role in theclassicalization of the system and could be important toconsistently capture continuum macroscopic informationof the GFT system. This would intuitively make sense,because one would expect to look for the physics of con-tinuum spacetimes in the regime far from the perturba-tive Fock vacuum corresponding to the no-space state.

To fully extract the geometric information encoded bysuch solutions, it would then also be necessary to go be-yond the use of the simplified local interactions and ex-plore the effect of the proper combinatorially nonlocal in-teractions encountered in the GFT literature (e.g. on theexpectation values of the geometric operators) in orderto compare it to the results obtained here. Additionally,it is worth mentioning that only for proper simplicial in-teraction terms the quantum geometric interpretation israther straightforward while for the others a full geomet-ric interpretation is currently lacking.

In a next step, the time evolution of the condensatewith respect to a relational clock could be studied. Thiswould be in the spirit of [7], where for an isotropic andfree condensate configuration a Friedmann-like evolutionwas found. It would allow for the comparison between thetwo settings and the extraction of further phenomenolog-ical consequences from our model.

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Another, perhaps more consistent way to properly re-late the quantum geometric information stored in thecondensate to a classical counterpart, could be to recon-struct the metric from the mean field as reviewed in ap-pendix A and investigate its isometries. Should the con-densate approximate a continuous homogeneous geome-try, one would expect that diffeomorphism invariance berestored as highlighted in a related context in Ref. [37].This could be helpful when comparing the isotropic re-striction employed in Ref. [7] to the one used here andalso when anisotropic condensates are studied.

Finally, we remarked in our analysis that the notionof near flatness used in the previous subsections shouldbe reconsidered for the condensate solutions around thenontrivial minima since it cannot be straightforwardlyapplied then. Statements regarding the flatness prop-erty of such solutions can only be

y department of physics, king’s college london, university of london, strand… · 2018. 3....

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