arXiv:1201.4151v1 [gr-qc] 19 Jan 2012 ssmtmssi ob n f“hr uniain”This quantization.” “third of scenario one al- a be are such to universe quantized, said single sometimes be is a to rules. eventually of quantum fields, freedom of ready of set degrees a the interact- is Since obeying an envisage scenario it can multiverse one whether topology if ing of words, ask inclusive other in may theory or, begin- quantum change one a the however, build at to general, the least possible In of at construction, topology by ning. the fixed scheme, dynamical, is canonical fully universe any is in correspond- geometry As or while labels). elements by representation group labeled ing with (Lorentz and associated manifold data spatial states a algebraic the network) is in level (spin embedded kinematical graphs of the at space result Hilbert end Ashtekar–Barbero The the in of connection. and written triad are densitized constraints the quantization of the canonical terms where a nonperturba- employed purpose, a is this To scheme in 3]. freedom [2, way of tive degrees gravitational quantize the to aiming framework background-independent a with and theory, field quantum dynamics phenomenology. and effective relativity the with general geo- contact of continuous the the (iii) not when data; are particular metric freedom in in of and dy- description degrees the defined, fundamental effective once somehow geometry, an and is of spacetime namics smooth recovery a control the of full terms the (ii) free- and identify, it; of they over degrees that the spacetime fundamental of of the dom definition of the (i) dynamics concern: quantum These several face challenges. gravity open quantum to approaches independent ∗ ‡ † [email protected] [email protected] [email protected] neapei ie ylo unu rvt (LQG), gravity quantum loop by given is example An background- [1], progress recent much Despite .ITOUTO N MOTIVATION AND INTRODUCTION I. ru edcsooy omlgclfil hoyo quantum of theory field cosmological a cosmology: field Group 2 eiee nttt o hoeia hsc,3 aoieS Caroline 31 Physics, Theoretical for Institute Perimeter ASnmes 88.c 46.s 46.z 98.80.Cq 04.60.Kz, 04.60.Ds, 98.80.Qc, numbers: PACS contribution. reproduce, curvature can a this or interaction, potential, of mixi scalar-field choice term the effective disc on an We by Depending lineari constraint approximation Hamiltonian term. mean-field the interaction menting a and nonlocal propagator, possibly co quantum Hamiltonian and the nonlinear is can fi operator a a of kinetic to wavefunction promoted the variables, is the space cosmology) quantum where loop model or Wheeler–DeWitt minisuperspace a struct olwn h dao edqatzto fgaiya reali as gravity of quantization field a of idea the Following 1 a lnkIsiuefrGaiainlPyis(letEin (Albert Physics Gravitational for Institute Planck Max inuaCalcagni, Gianluca mMulneg1 -47 om emn,EU Germany, Golm, D-14476 M¨uhlenberg 1, Am 1, Dtd aur 9 2012) 19, January (Dated: ∗ tffnGielen, Steffen cee[3 4.Tid h edter rmwr offers framework field-theory the Third, field-quantization local 14]. it of [13, dynamical; sort scheme a made words, naturally other in is topol- provides, topology arbitrary that of such it complexes ogy, forming spin- said, of of vertices, as annihilation) superposition network and all, the (creation as of processes dynamics interaction First such dynamics. defines quantum it the view. Second, of definition of advantages, complete several point a provides has LQG embedding the This spin- from are amplitudes models. Feynman indeed foam whose are and states networks whose quantum manifolds spin are group These on larger theories [16]. turn field models in the tensor [13–15], to into (GFT) related theories strictly states field group LQG LQG of of framework embedding definition covariant by (a 12] dynamics), the [11, via of models obtained, foam is definition states spin complete appro- network spin an but result- of tentative in dynamics the quantum gravity A Einstein’s that limit. to proof priate back the leads of theory and definition ing complete dynamics the quantum is approach the LQG the by faced on oa“hr-uniain etn eeas offered also 10]. were [9, setting in of “third-quantization” favor a in to perspective gravity going quantum canonical a from a eahee,a es tafra ee,b enn a spatial defining given 5]. by for level, [4, geometries, of formal topology space a the at over theory least field at achieved, be can 2 1 r“eod nta f“hr”t niaeti yeo quant of type this indicate to “third” “fiel tion. of adjectives the instead an employ “second” we [6–8] on or now (e.g., from branes confusion, avoid other To into suc or therein). different, vacuum references are into topology and decays geometry brane i final is and strings phenomena initial of certa nonperturbative the field highly via a describe interacting consider to to strings advantages possibility the fie of of particle collection One of a mode tices. collection is “third-quantized” a is a field string of string Polyakov example free an the is While theory field String eieteiseo oooycag,temi difficulty main the change, topology of issue the Beside ,2, 1, .N,Wtro,OtroNL25 Canada 2Y5, N2L Ontario Waterloo, N., t. sritoeao,adteato features action the and operator, nstraint † o xml,acsooia osat a constant, cosmological a example, for ggaiainladmte variables. matter and gravitational ng igteeuto fmto n aug- and motion of equation the zing n ail Oriti Daniele and 2 s refil lsia ouin,the solutions, classical free-field uss e ngopfil hoy econ- we theory, field group in zed nclqatmcsooy(either cosmology quantum onical l,tecodntsaeminisuper- are coordinates the eld, 1 te abi nocuie arguments inconclusive) (albeit Other ti Institute), stein 1, ‡ E-0109 pi-qg-255 AEI-2011-099, geometry nver- in where d,a lds, the n as h iza- d” d l. 2 powerful mathematical and conceptual tools for tackling case is also easily recovered. In Sec. III the field theory the issue of the continuum limit and of the extraction of is defined by promoting the quantum Hamiltonian con- effective dynamics for better contact with phenomenol- straint to the kinetic operator of a (real) scalar field Ψ on ogy. In doing so, however, one has to abandon the fa- minisuperspace. We analyze the relation between differ- miliar framework of canonical quantization of a classical ent kinetic operators and the gauge choice, with particu- (and local) field theory of gravitation, and is forced to lar focus on exactly solvable free theories. We discuss the face new types of conceptual and mathematical difficul- various possible choices for the interaction term. Follow- ties. ing this general definition, we move on to analyze some This program is just as ambitious as the original LQG consequences of the formalism. We analyze the free prop- one, if not more, and is difficult to realize in a complete agator of the theory first, corresponding to the evolution and rigorous way, despite many recent advances. Toy of a single universe (Sec. IV). We show how the embed- models inspired by the full theory then become very im- ding into a field-theory setting has immediate interest- portant. In fact, they fulfill three main purposes: (i) they ing consequences also for the single-universe dynamics. offer a simplified testing ground for ideas and techniques Then, we consider how the presence of interactions affects developed in the full theory; (ii) as such, they also have an this single-universe evolution. Approximating the inter- important pedagogical value; (iii) they may represent, in action as a mean-field term, we find an effective equation principle, an effective, approximate framework to which linear in the field Ψ, correcting the Hamiltonian quantum the full quantum dynamics may reduce, in some limit, constraint equation by an extra term (Sec. V). The lat- and thus they may be directly applicable to phenomeno- ter mixes, in general, gravitational and matter degrees logical studies. Obviously, due to their simplicity, one of freedom, and its exact form depends on the chosen should be cautious in interpreting the result obtained in initial interaction as well as on the mean-field configura- the context of such toy models as truly physical, and their tion considered. We conclude with a discussion of other validity can be assessed only once the relation between possible applications of this formalism. toy model and full theory has been understood.

An important type of simplified scenario has been de- II. BRIEF OVERVIEW OF LOOP QUANTUM veloped in the context of LQG, in a symmetry-reduced COSMOLOGY setting of interest for cosmology. In fact, in order to un- derstand certain features of loop quantum gravity, one A. Classical theory often resorts to a minisuperspace model, loop quantum cosmology (LQC), where degrees of freedom are drasti- cally reduced [17, 18]. In a pure Friedmann–Robertson– Our starting point is the description within loop quan- Walker (FRW) universe filled with a massless free scalar tum cosmology of the spatially flat, homogeneous and field, the classical and quantum dynamics of the universe isotropic universe with a massless scalar field as mat- as a whole can be described by the same formalism used ter, which we summarize briefly in this section. In the for a free particle. In particular, the path integral is canonical analysis of dimensionally-reduced general rela- well defined [19, 20] and two-point correlation functions tivity, one restricts integrations to a fixed fiducial three- admit the usual classification [21]. By now, a wealth dimensional cell of comoving volume 0 < , with a flat 0 V ∞ of interesting results have been obtained in this context metric qab which may be taken to be δab in Cartesian [17, 18]. Given this analogy with the free particle, it is coordinates. The four-dimensional metric is then 2 2 2 2 0 a b all the more natural to ask oneself if one can construct ds = N (t) dt + a (t) qab dx dx , (1) a sensible “interaction” among FRW universes and, once − this is done, to change the interpretation of the two-point where a(t) is the scale factor, spatial indices are labeled by Latin indices a, b, =1, 2, 3, and there is a freedom function from particle transition amplitude to field prop- ··· agator as in the usual field quantization. One would then in the choice of the lapse function N(t). Indices i, j, = 1, 2, 3 will denote directions in the tangent space. ··· write down a field theory on minisuperspace, to obtain a 0 a 0 i With a choice of frame ei and dual ea , or- field-quantized LQC framework. Another way to see the 0{ } { } i same field theory would then be as a toy model for group thonormal with respect to qab, the physical triad e = εa 0ei dxa (ε = 1) and e0 = N dt are orthonormal with field theory, in which many of the difficult features of the a ± latter are absent due to the global nature of the formu- respect to (1), and the Levi-Civita connection is lation and to the simplification provided by symmetry a˙ ωi = ε 0ei , ωi =0 , (2) reduction, but where some ideas and techniques can still 0 N j be applied. As with any toy model, one would then use where a dot denotes time derivative. From this one com- it as a pedagogical testing ground and keep it available putes the variables used in loop quantum gravity, the as a possible effective description of the full theory. su i Ashtekar–Barbero (2) connection Aa and the densi- a We propose such a field theory for (loop) quantum cos- tized triad Ei , via mology in this paper, with the above motivations. The Ai = γ ωi , (3a) presentation is organized as follows. We review some ba- a 0 a a a 2 0 0 a sic features of LQC in Sec. II, but the Wheeler–DeWitt Ei = (det e)e
i = a det q ei , (3b) p 3 where γ is the Barbero–Immirzi parameter. improved dynamics scheme [22], where µ(p) p −1/2, A shortcut to the standard canonical analysis is to sub- this is a basis ν of eigenstates of the volume∼ operator | | {| i} stitute the FRW metric and the Ricci scalar stemming ˆ measuring the kinematical volume of the fiducial cell, from (1), V = p 3/2, V | | ˆ a¨ a˙N˙ a˙ 2 ν =2πγG ν ν , (7) R =6 + , (4) V| i | | | i 2 3 2 2 3 aN − aN a N ! where ν = εa 0/(2πγG) has dimensions of length. The states ν canV be normalized to into the Einstein–Hilbert and matter action, which then {| i} ν ν′ = δ . (8) depend on φ, a, and N. The conjugate momenta are h | i ν,ν′ 3 ˙ pa = 3 0 aa/˙ (4πGN) and pφ = 0 a φ/N, and the The basic operators are nowν ˆ, which acts by multipli- conservation− V in time of the primaryV constraint p 0 \ N cation, and exp(iλb), where b = ε (2πγGp )/(3 a2) is (the symbol denotes weak equality) leads to the Fried-≈ a 0 conjugate to ν [and is proportional to the HubbleV pa- mann equation≈ rameter H =a/ ˙ (Na)] and λ = const, which acts as a 2 shift in ν. These satisfy the standard Heisenberg algebra. 2πG p2 p a φ For the matter sector, one chooses the usual Schr¨odinger := 3 0 , (5) K 3 a − 2a ≈ quantization with a natural representation of the Hilbert φ R which should be imposed as a constraint on quantum space kin, the space of square-integrable functions on , 3 H states in quantum cosmology. on which φˆ acts by multiplication andp ˆφ by derivation, and with an orthonormal basis given by φ φ′ = δ(φ φ′) . (9) B. Kinematics h | i − The Hilbert space of the coupled system is then just the Focusing on the gravitational sector for now, the cru- g φ tensor product kin kin. As in traditional approaches cial difference from traditional minisuperspace (Wheeler– to quantum cosmology,H ⊗H the variable N is removed from DeWitt) approaches to quantum cosmology in LQC is the configuration space because the primary constraint that one follows the kinematics of full loop quantum grav- pN 0 would mean that wavefunctions are independent ity, where not the connection but only its holonomies are from≈ N. We note that the full constraint would be a defined as operators [2]. It is convenient to introduce new multiple of N, so that in situations where the resulting conjugate variables c and p, where quantum constraint depends on the choice of lapse func- tion (as below) the choice N = 1 seems more natural γ a˙ 4πGγ p c = ε 1/3 = ε a , p = εa2 2/3 , (6) when considering that N is also originally in the config- V0 N − 2/3 a V0 3 0 uration space. In fact, in cosmology the lapse can be V regarded as a function of the scale factor, N = N(a), i so that Aa only depends on c, and powers of 0 have been which is an independent variable. A choice of the form introduced to make c and p invariant underV the residual N = N(E) in the full theory is somewhat less justified 0 0 2 symmetry a λa, qab qab/λ in Eq. (1). Instead before solving the constraints. → → \ ofc ˆ andp ˆ one now definesp ˆ and exp(iµc) as operators, where µ can be a real parameter or a function of p chosen by means of a suitable procedure. C. Dynamics g The kinematical Hilbert space kin is taken to be the space of square-integrable functionsH on the Bohr com- The quantum analogue of the Friedmann equation (5) pactification of the real line. One can work in a ba- is obtained by starting with the Hamiltonian constraint sis wherep ˆ is diagonal, with orthonormality relation of full general relativity in terms of the variables (3) and ′ expressing the curvature of Ai through the holonomy p p = δp,p′ , so that one is dealing with a nonseparable a hHilbert| i space. In this representation, if µ is taken to be around a loop, taking account of the area gap —the re- a nontrivial function of p the action of the holonomy op- sult in LQG that the area of such a loop cannot assume \ erator exp(iµc) takes a rather complicated form, and it arbitrarily small nonzero values. The Hamiltonian con- is convenient to choose a different representation. In the straint is ˆ ψ(ν, φ) := B(ν) Θ+ ∂2 ψ(ν, φ)=0 , (10) K − φ where ψ is a wavefunction on configuration
space and Θ g 3 K is a difference operator only acting on and of the Throughout the paper we use the symbol for the Hamiltonian Hkin constraint because it will eventually be regarded as a kinetic form operator. Although this differs from the more standard choice of H B(ν)Θψ(ν, φ) := A(ν)ψ(ν + ν , φ)+ C(ν)ψ(ν, φ) symbol H or , it has the further advantage of avoiding confusion − 0 with the Hubble parameter. +D(ν)ψ(ν ν , φ) , (11) − 0 4 where A, B, C, and D are functions which depend on Equation (15) can be shown to agree with the previous the details of the quantization scheme (inter alia, on the expressions (14) in the “semiclassical” limit ν ν . ≫ 0 choice of lapse function) and ν0 is an elementary length In LQC, one normally assumes symmetry of the wave- unit, usually defined by the square root of the area gap function ψ under orientation reversal, (the Planck length up to a numerical factor). The physi- ψ(ν, φ)= ψ( ν, φ) , (16) cal states are the solutions of Eq. (10). Due to the struc- − ture of Eq. (11), in LQC one has an interval’s worth of superselection sectors in g : Θ preserves all subspaces since the kinematical Hilbert space can be split into sym- Hkin metric and antisymmetric subspaces which are superse- spanned by νI + nν0 n Z for some νI . We may restrict ourselves{| to onei | of these∈ } subspaces, i.e., assume lected (in other words, the physics should not depend on that wavefunctions only have support on a discrete lat- the frame orientation). From the field theory perspective, such a requirement is less natural, in particular if inter- tice which we take to be ν0Z [for a generic gauge choice, there may be issues with the definition of (10) at the actions are taken into account; we will allow for general most interesting point ν = 0]. This restriction picks out field configurations without assuming Eq. (16). a separable subspace to which we will limit our analysis. By definition of second quantization, and by construc- tion in our case, the classical solutions of the free field theory will correspond to the quantum solutions of the III. DEFINING THE FIELD THEORY first-quantized model.

We now complete the definition of the field theory on We now define our field theory on (mini)superspace. minisuperspace with the addition of an interaction term The Hamiltonian constraint (10) of the first-quantized for our field. The first-quantized theory, that is (loop) theory is the natural starting point for the free action of quantum cosmology, does not offer indications on how the field theory. We define this action to be this interaction should be defined, so one has to pro- 1 ceed in a rather exploratory way guided only by general S [Ψ] = dφ Ψ(ν, φ) ˆΨ(ν, φ) , (12) f 2 K intuition (and by the results obtained following various ν X Z choices). One could take an arbitrary functional, but we where in the simplest setting we take Ψ to be a real scalar opt for an nth-order polynomial in Ψ not necessarily lo- field. If ˆ is as in Eq. (11), we must assume that the cal in the minisuperspace variables. We will specialize combinationK B(ν)Θ is symmetric in ν, i.e., that to simpler, concrete choices in the following, in order to study some consequences of the formalism. D(ν)= A(ν ν ) (13) − 0 We get the general form for the interacting theory in Eq. (11), in order to reproduce the equation of motion n 1 λ (10). Put differently, for any constraint (11) the action S [Ψ] = dφ Ψ(ν, φ) ˆΨ(ν, φ)+ j (17) i 2 K j! × (12) projects out its self-adjoint part with respect to the ν j=2 X Z X measure given by the kinematical inner product of LQC. j Taking this measure as given, possible manipulations of dφ1 . . . dφj fj (νi, φi) Ψ(νk, φk) , the constraint ˆ are restricted by this requirement. No- ν1...νj Z k=1 tice, however,K that Eq. (13) does hold in LQC for the X Y usual choices of gauge, so we do not need to impose it as where fj(νi, φi) are unspecified functions depending on an additional requirement. To give an example, for the νi, φi i=1,...j . This gives the equation of motion { } preferred lapse choice N = 1 and in improved dynamics, n the functions A, B, C take the form [22] λj ˆΨ(ν, φ)+ dφ . . . dφ − (18) K j! 1 j 1 1 ν ν 3ν j=2 ν1...νj 1 Z A(ν)= ν + 0 ν + 0 ν + 0 , X X− 2 4 − 4 j−1 j 12γ 2√3 Ψ(νl, φl) fˆk(νi, φi; ν, φ)=0, 1 1 3 3p√2 ν0 3 ν0 3 × B(ν)= ν ν + ν , Yl=1 kX=1 | | 4 − − 4 8 √3πγG where

C(ν)= A(ν) A(ν ν0) , (14) −p − − fˆk(νi, φi; ν, φ) := f(µi, ϕi) , (19) where now ν0 = 4λ := 32√3 πγ G, whereas for the 3 solvable “sLQC” model in [23] [which uses N = a and with µ = ν ,...,ν − ,ν,ν ,...,ν and ϕ = p { i} { 1 k 1 k j } { i} the symmetry (16)], φ1,...,φk−1, φ, φk,...,φj . In writing down the field theory{ action we have included} possible nonlocal (in ν) √3 ν 1 A(ν)= ν + 0 , B(ν)= , quadratic terms in the interaction part (specified by the 8γ 2 ν interaction kernel f ) rather than in the kinetic term, in   2 √3 order to emphasize the fact that ˆ is usually chosen to be C(ν)= A(ν) A(ν ν )= ν . (15) K − − − 0 − 4γ a local operator in the geometry in quantum cosmology. 5

These could however be also included in the definition of Hamiltonian pφ and νb as also conserved in interac- ˆ. tions, which would then be local in φ and ln(ν/ν0). K For the sLQC model detailed below, the second The above is rather general. Explicit analyses re- quantity would be modified to (2ν/ν0) sin (bν0/2) quire choosing specific interaction terms, that is, func- and we would require locality in the conjugate vari- tions fj (νi, φi). Each of them could correspond to a able ln (ν/ν0)[1 + cos(bν0/2)] . choice of a physical quantity to be conserved through { } the “interaction” of the universes, and of a conjugate In the Wheeler–DeWitt case, we can choose the type physical quantity in terms of which the fields interact, of interaction analogously, since the only difference is in instead, “locally.” That means that the conjugate vari- the kinetic operator (choice of first-quantization scheme) able is identified across “incoming” and “outgoing” uni- and not in the choice of minisuperspace variables. verses. Different choices for the function f(νi, φi), and hence for the quantities conserved in the interaction, can Before starting our analysis of the model, we intro- thus be motivated by different physical considerations. duce a reformulation of the same that is advantageous In particular, the choice will be influenced by the inter- for practical manipulations. Since ˆ is not diagonal in ν, it may be convenient to pretation of such an interaction as true topology change, K that is, splitting/merging of universes, or rather as a use a Fourier transform as in [23], merging/splitting of homogeneous and isotropic patches Ψ(b, φ) := eiνbΨ(ν, φ), (21) within a single inhomogeneous and anisotropic universe. ν The detailed definition of the second scenario in physical X 2π/ν terms is not easy, and we will confine our treatment to 0 ν0 a brief discussion of it at the end. However, it is impor- Ψ(ν, φ)= db e−iνbΨ(b, φ) . (22) 2π tant to keep in mind that its consideration would sensibly Z0 affect the very definition of the field theory to analyze. Examples of interactions are: As Ψ(ν, φ) is real, Ψ(b, φ) = Ψ(2π/ν0 b, φ), and if Eq. (16) holds, then Ψ(b, φ) = Ψ(2π/ν− b, φ). The 0 − If, e.g., only λ3 is nonvanishing and we imple- action (17) becomes • ment locality in (ν, φ) by choosing f(ν , φ ) = i i n ν λ δν1,ν2 δν1,ν3 δ(φ1 φ2)δ(φ1 φ3), we have the field 0 ˜ˆ j − − Si[Ψ] = dφ db Ψ(b, φ) Ψ(b, φ)+ (23) equation 4π K j! j=2 Z Z X j j ˆ λ3 2 ν j Ψ(ν, φ)+ Ψ (ν, φ)=0 . (20) 0 dφ db f¯(b , φ ) Ψ(b , φ ) , K 2 × 2π l l i i l l Z l=1 l=1 Such a potential, which implements locality both   Y Y − ¯ e i k νkbk f(ν , φ ) is the in the 3-volume (i.e., the scale factor) and in the where f(bi, φi) := ν1...νj P i i scalar field φ, implies the existence of conservation (complex conjugate of the) Fourier transform of the func- P laws for the conjugate quantities b and pφ. These ˆ tion f(νi, φi) appearing in (17), and ˜ is an appropriate conservation laws are nothing but the (modified) differential operator. In the above conventions,K second Friedmann and Klein–Gordon equations, re- spectively. Since interactions allow for topology ˆ iν0b −iν0b ˜ = e A ( i∂b)+ C ( i∂b)+ A ( i∂b) e change, in this case for nonzero λ there is a pro- K − − − 3 +B ( i∂ ) ∂2 . (24) cess where two “universes” merge into one, and the − b φ metric in both ingoing patches as well as in the out- The inverse of this operator will give the propagator. going patch is required to be the same (analogous In the sLQC case, Eq. (15), in order to avoid nonlocal to the interaction term proposed in [4]). Because expressions such as 1/( i∂b) in the action we could mul- of the conservation law, there is a discontinuity in tiply the expression for− Θ by ν. However, this would lead the Hubble parameter b H. ∝ to a “nonsymmetric” form such that Eq. (13) is not re- spected [obviously, νD(ν)= νA(ν ν ) = (ν ν )A(ν A different conservation law, namely conservation − 0 6 − 0 − • −3 ν )] and ˆ is not self-adjoint. A symmetrized version of Hubble volume b or locality in the conjugate 0 K 4 of ˆ, which has not been previously considered in the quantity b ν, is suggested if one interprets the in- K teraction as the topology-changing process just de- literature, has scribed with a discontinuity in the causal past of an √3 ν 2 √3 observer passing the “merging point” (this is typ- A(ν)= ν + 0 , B(ν)=1, C(ν)= ν2. ically considered “bad” topology change; see the 8γ 2 − 4γ review [24]).   (25) When the nonsymmetric form resulting from multiplica- Another possibility would be to take the con- tion by B−1(ν) is used in LQC, the kinematical inner • served quantities of the classical (Wheeler–DeWitt) product is modified accordingly to keep the constraint 6 self-adjoint (see, e.g., [25]). The action (12), which in- the theory, see Eq. (51). In the gauge N = a3, consider volves both the constraint and the kinematical inner the modified Hamiltonian product of LQC, is left unchanged by such a redefinition. p (t)2 2 b(t)ν 2 = φ 6πG ν(t) sin 0 . (28) K 2 − ν 2 As mentioned above, the LQC setting is chosen be-  0   cause of the initial motivation to obtain a toy model for Hamilton’s equations give φ˙ = pφ andν ˙ = ∂ /∂b, solved a GFT construction, in turn a field theory formalism for by K LQG states. This is, however, not essential for our gen- 4 eral purposes. One could define an analogous set of mod- b(t)= arctan exp √12πGk(t t0) , (29) ν0 ± − els for Wheeler–DeWitt quantum cosmology, which is ac- h i kν cosh[ √12πG k(t t )] tually simpler from a technical point of view. The usual ν(t)= 0 ± − 0 , (30) ordering for the quantum Friedmann equation is [26] ± √48πG so that the bounce is apparent already in each classical ˆ 4πG 2 2 history (we have treated ν as a continuous parameter := (a ∂a) pφ , (26) K 3 − for the purpose of simplicity here). In fact, Eq. (30) is consistent both with (27) (φ = kt are solutions of the and the scale factor a is now a continuous variable. It is constraint p = const) and with± the expectation value of then convenient to define := 3/(4πG) ln a so that a φ the volume operator (36). is restricted to be nonnegativeN and the constraint is sim- The construction of the Fock space proceeds as usual. ply the Klein–Gordon operatorp∂2 ∂2. One ends up N − φ † with a scalar field theory in 1+1 dimensions with stan- One defines creation operators ak and annihilation oper- dard kinetic operator and an unusual potential term. ators ak from the mode decomposition of generic fields into the above basis, and builds generic elements of the Hilbert space of states from their combined action on the Fock vacuum state 0 . A. Fock space construction This state, as in the| i complete GFT formalism [13, 14], would correspond to a very degenerate “no-space” state, As any ordinary field theory, the above field theory in which no geometric and no topological structure at on minisuperspace can be quantized and a Fock space of all is present. Topological and geometric structures are quantum states can be constructed. The construction of created out of it by the action of the creation operators. the Fock space rests on a choice of a complete basis on A crucial ingredient in the definition of the Fock space the space of fields. is the choice of quantum statistics. In the following, we As customary, we can choose a basis of solutions of † fix the statistics to be bosonic, [ak,a ] δk,k . This the free field theory, that is single-particle states in k k′ ∝ ′ {| i} seems physically natural if quantum states are associated whose elements correspond to classical (expanding or with classical geometries and whole universes. contracting) solutions of the modified Friedmann dynam- In group field theory, where the Fock space would be 4 ics. These solutions are labeled by k, i.e., 3-geometries constructed out of microscopic “building blocks” of space (described by ν) embeddable into a four-dimensional [14], the choice of statistics is less obvious and it is the FRW universe by means of k and a choice of lapse N(t). focus of current research (see for example the discussion In solvable LQC, the deparametrized solutions are of the in [27]). We may also expect the situation to be subtler form if the objects created and annihilated in our field theory 4 are to be interpreted as local homogeneous and isotropic b(φ)= arctan exp √12πG(φ φ0) , (27) patches of a single universe. However, as anticipated, we ν0 ± − h i do not discuss in detail this possibility in this work, and and k is the value of the Dirac observable pφ. Equation thus stick to the simplest choice of statistics. (27), specifying the classical trajectories, will later reap- pear as the “light cone” x(b) = φ of the propagator of IV. THE FREE FIELD THEORY

We now begin the analysis of the field theory we de- 4 We warn the reader about a fine point in terminology. In the fined. We limit most of our considerations to solv- LQC and Wheeler–DeWitt literature, by “classical solutions” one able sLQC, where explicit calculations can be performed. often means the solutions of the unmodified Friedmann equations However, we try to maintain a certain level of generality in classical general relativity. In contrast, by “classical trajecto- in the presentation, to leave room for the study of other ries” we presently mean solutions generated by a Hamiltonian cases. We start with the analysis of the free field theory. that already includes the two effects of replacing the connec- tion by its holonomies and of setting a minimal area for closed We have seen that, already at this level, the field-theory holonomies [such as Eq. (28)]. Both operations can be moti- setting implies certain restrictions on and modifications vated by the first-quantization framework of LQC, but they are of the LQC Hamiltonian constraint operators, and thus performed already at the classical level. of the single-universe dynamics. We focus on these first. 7

A. Equations of motion and Hamiltonian constraint Since χ obeys a Klein–Gordon equation in φ and x and its left and right sectors depend on the combinations φ x, ± For sLQC, the free equation (10) can be solved analyt- it is easy to see that the cosh in the volume operator ically. We reserve the symbol ψ for field solutions of the factorizes as a cosh in the scalar field, so that free theory and the symbol Ψ for the classical field so- χ νˆ χ = ν∗ cosh(√12πGφ) , (36) lutions of the interacting model. Inserting Eq. (15) into h | | i (24), one obtains where the proportionality coefficient ν∗ is the minimum volume at the bounce. Later on we shall derive this result 2 bν 2 i∂ ˜ˆψ = ∂2 12πG ∂ sin 0 ψ . (31) again from the “light-cone” condition of the field-theory bK φ − b ν 2 (  0   ) propagator. For the operator choice (25), the free equation is Setting ψ = ∂bχ and choosing an (irrelevant) integration constant to be zero yields 2 bν 2 ∂2ψ 6πG ∂ sin 0 φ − b ν 2 2 ( 0   ˜ˆ 2 2 bν0 χ = i ∂φ 12πG sin ∂b χ =0 , 2 K − − ν0 2 2 bν0 (     ) + sin ∂b ψ =0 . (37) ν 2 (32)  0    ) with general solution χ = χ+[φ x(b)] + χ−[φ + x(b)], where − One can make the substitution bν 1 z = sin 0 (38) x(b)= ln(tan(ν0b/4)) , 2 √12πG  

′ −1 to bring this to the form so that x (b) = [(2√12πG/ν0) sin(ν0b/2)] . Note that 1 2 2 3 b parametrizes the circle S from which the point b = 0 ∂φψ 6πG (1 2z )+(4z 6z )∂z (or b =2π/ν ) must be removed to make the coordinate − − − 0 +2z2(1 z2)∂2 ψ =0 . (39) transformation (33) well defined. As b 0, we see that  − z → dx/db . We will encounter this singular behavior By separation of variables and an Ansatz ψ(z, φ) = when giving→ ∞ the expression of the propagator of the the- eikφzµ g(z2), one obtains (y := z2) ory in terms of b, see Eq. (51) below. 1 k2 1 We may expand the general solution in Fourier modes, 1+ + 2(µ + µ2) (1 + µ)2 g(y) (40) 8y 6πG − 4 +∞     3 ′ ′′ χ(b, φ)= dk A˜ (k) eik[φ−x(b)] + A˜ (k) eik[φ+x(b)] , + µ + y(µ + 2) g (y)+ y(1 y)g (y)=0 , L R 2 − −    −∞Z n o (33) which the reader will recognize as a special case of Euler’s and hence hypergeometric differential equation [28, Eq. 9.151] ′′ ′ 2π/ν0 +∞ y(1 y)g + [c (a + b + 1)y]g abg =0 , (41) ν − − − ψ(ν, φ)= 0 db e−iνb dk x′(b) 2π provided that the coefficient of g(y)/y is made to vanish: Z0 −∞Z ik[φ−x(b)] ik[φ+x(b)] 1 i k2 AL(k) e + AR(k) e µ± = 1+ . (42) × −2 ± 2 3πG n +∞ +∞ o r ν = 0 dx e−iνb(x) dk (34) Then, Eq. (40) is (41) with a = b = (µ + 1)/2 and c = 2π µ +3/2; two independent solutions around y = 0 are −∞Z −∞Z given in terms of Gaussian hypergeometric functions ik(φ−x) ik(φ+x) AL(k) e + AR(k) e , × ikφ µ µ± +1 µ± +1 3 2 ψµ = e z(b) ± 2F1 , ; µ± + ; z(b) , h i ± 2 2 2 where b(x) = (4/ν0) arctanexp(√12πGx) and AL :=   (43) ikA˜ and A := ikA˜ are chosen so that ψ is real. L R R where z(b) is given by Eq. (38) and µ are the two roots − In LQC (first-quantized theory), the existence of a ± (42). In terms of the variables ν and φ, the general solu- bounce can be proven analytically with the state χ. Not- tion can then be written as ing that the volume operator in x representation acts as 2π/ν0 +∞ νˆ ∂b cosh(√12πGx)∂x, one can compute its expec- ∝ ∝ ν0 tation value on the state (33), with the scalar product ψ(ν, φ)= db e−iνb dk (44) 2π Z0 −∞Z ∗ ∗ χ νˆ χ = i dx(χ ∂φ νˆ χ νˆ χ∂φχ ) . (35) h | | i − | | − | | A+(k) ψµ+(k)(b, φ)+ A−(k) ψµ (k)(b, φ) Z × −   8 for some functions A+(k) and A−(k) (the interpretation function whose z and φ dependence are completely un- of which as left- and right-moving modes is less obvi- correlated. Figure 1 suggests that also here the wave −λk2 ous). Taking A+ = e and A− = 0 we obtain a packet follows the classical trajectories [Eq. (30)] which wave packet, plotted in Fig. 1 (using Mathematica) as contain a bounce, but we must leave a detailed numerical a function of b and φ. The wave packet is peaked on investigation of the existence of a bounce to future work. both “expanding” and “contracting” classical solutions −1/2 φ = (12πG) ln tan(ν0b/4). This can be compared For Wheeler–DeWitt quantum cosmology, the free field with± a wave packet composed out of right-moving modes equation is just the (1 + 1)-dimensional wave equation x′(b)eik[φ+x(b)] in solvable LQC, sharply peaked on the with general solution ψ(a, φ)= ψ [φ 3/(4πG) ln a]+ + − “contracting” branch (see Fig. 2). ψ [φ + 3/(4πG) ln a], decomposed into Fourier modes − p as p +∞ ik[φ−N (a)] ik[φ+N (a)] ψ(a, φ)= dk fL(k) e + fR(k) e . −∞Z n o (45)

B. Propagator

Having fixed a self-adjoint operator ˆ in the action (12), one needs to invert it to obtain theK propagator of the theory. In the generic case (10), we can give a “spin foam” series expansion by using the results given in [21] (extending those of [19, 20]), where we determined the Feynman propagator corresponding to a constraint of the form (Θ + ∂2) [that is, for B(ν)=1] to be − φ ∞ ψ b,φ FIG. 1. Wave packet | ( )| formed out of the solutions iGF = Θν νM 1 ... Θν2 ν1 Θν1 ν′ (46) − (43), with λ = 0.01 and ν0 = 3πG = 1, so that b ∈ [0, 2π). νM 1,...,ν1 M=0 − νm=ν X X6 m+1 p 1 ∂ nk−1

× (nk 1)! ∂Θwkwk kY=1 −   p e−i√Θwmwm (φ−φ′) .  p  m=1 2 Θwmwm j=1 (Θwmwm Θwj wj ) j=m X 6 −  p Q  We refer to [21] for notation and details of the calculation. From this it follows that a Green’s function (right inverse) for ˆ with general B(ν) is K G (ν, φ; ν′, φ′) G (ν, φ; ν′, φ′) := F , (47) R B(ν′)

since B(ν) ˆ G (ν, φ; ν′, φ′)= ( Θ ∂2)G (ν, φ; ν′, φ′) Kν,φ R B(ν′) − ν − φ F = δ δ(φ φ′) , (48) ν,ν′ − FIG. 2. Wave packet |ψ(b,φ)| in solvable LQC, with AL(k)= where the subscript of ˆ indicates that differential and −0.01k2 K 0,AR(k)= e in (34), where again ν0 = 3πG = 1. difference operators act on the first argument only. Since ˆ is self-adjoint, a left inverse is obtained by tak- K ′ ′ The bounce picture is not as clear, analytically, as that ing the adjoint of the right inverse, GL(ν, φ; ν , φ ) = ′ ′ in sLQC. Equation (39) is not a Klein–Gordon equation, GR(ν , φ ; ν, φ). derivatives in φ do not map directly into derivatives in For the sLQC model, we can give a more explicit ex- z, and the volume operatorν ˆ √1 z2∂ acts on a pression for the propagator, exploiting the relation to ∝ − z 9 the Klein–Gordon equation; a Green’s function for (31) one has to resort to approximation methods and various is given by truncations. One is of course the perturbative expansion of the field theory around the Fock vacuum and the study dx Gs (b, φ; b′, φ′)= i∂ G [x(b), φ; x(b′), φ′] of the corresponding Feynman diagrams and amplitudes. R db b { KG } b=b′ This is, for example, the level at which the current un- ′ ′ derstanding of full GFT’s (and of the corresponding spin GKG[x(b), φ; x(b ), φ ] = i∂b ν0 , (49) foam models) stands. Another type of technique, aim- 4√3πG sin( ν0b′ ) ( 2 ) ing at an approximate understanding of nonperturbative features of the theory and at the extraction of effective where GKG is the Feynman propagator for the Klein– Gordon equation. Explicitly [29], dynamics from the “fundamental” one, is mean field the- ory. The application of such technique in the full GFT 1 framework has just started [30], and a similar study of G = ln µ2[(φ φ′)2 (x x′)2 iǫ] , (50) KG −4π − − − − the toy model we defined here is what we focus on in the following. where the usual iǫ prescription cancels the singularities on the “light cone,” so that As said, a first approximation to the dynamics of an 2 −1 2 ′ interacting system is obtained if one assumes that, in s (96π G) iν0 [x(b) x(b )] GR = − . an appropriate quantum state ξ , the system fluctuates sin ν0b sin ν0b′ (φ φ′)2 [x(b) x(b′)]2 iǫ 2 2 around a configuration with nonvanishing| i expectation { − − − − (51)} s value of the field operator Ψˆ and replaces Ψ=ˆ Ψˆ + δΨˆ G blows up as b 0 or b′ 0 because the coordinate R in the quantum (operator) version of the fieldh equationi transformation from→ b,b′ to x,→ x′ becomes singular there, (18). A similar expansion can already be done at the as we noted below (33). Hence, these are not physical classical level by writing the field Ψ appearing in the ac- singularities. To obtain a Green’s function which can tion(17)asΨ = Ψ +δΨ, that is, when studying the field be extended to those values, one would have to solve 0 theory dynamics around a different (nontrivial) vacuum Eq. (48) with appropriate boundary conditions. In the Ψ . The resulting effective action from such an expansion above example, the b representation elects ν as the mo- 0 [starting from Eq. (17)] is the following: menta; swapping representation, the physical interpreta- tion of the light-cone poles as classical trajectories would be unchanged. S = S[Ψ = Ψ + δΨ] S[Ψ = Ψ ] Notice that the choice of the Feynman propagator for eff 0 − 0 the particular Green function to use can be justified for- 1 = dφ Ψ0(ν, φ)+ δΨ(ν, φ) ˆδΨ(ν, φ) mally by the analogy with the free particle case, and thus 2 K ν Z   by a definition of “time-ordering” and thus causality con- Xn λ ditions with respect to the values of the scalar field used + j dφ . . . dφ f (ν , φ ) j! 1 j j i i as internal time for the system. Another justification, j=2 ν1...νj X X Z possibly more satisfactory, for the same choice of ana- j j−m j lytic continuation in the complex plane is the fact that j Ψ0(νk, φk) δΨ(νl, φl) , this choice makes not only the propagator itself but also × m m=1   k=1 l>j−m the formal field-theory path integral well defined, at least X Y Y as far as the free theory is concerned. (52)

In comparison, we note that for the Hamilto- where we have assumed that the functions nian constraint (26) of standard Wheeler–DeWitt fj(ν1,...,νj, φi,...,φj ) are symmetric under any quantum cosmology one just considers the Klein– permutations of their j pairs of variables. This assump- tion, needed only to have a more compact expression for Gordon equation, and the propagator of the theory is 5 G [ 3/(4πG) ln a, φ; 3/(4πG) ln a′, φ′]. the result, can be lifted straightforwardly. KG We can then isolate the terms that are linear, quadratic p p and higher-than-quadratic in the dynamical field δΨ to V. TAKING INTERACTIONS INTO ACCOUNT: obtain (using the standard convention that a sum run- MEAN-FIELD APPROXIMATION ning from n to n 1 is empty) −

We now want to extend our analysis to the field in- teractions, i.e., to the effects of topology change on the 5 dynamics of a single universe (or, in the alternative inter- When the assumption is not satisfied, the expression replacing the third and fourth lines in (52), for given j, is obtained by: (i) pretation we suggested, of the interaction of the various choosing m ordered elements out of the ordered set of j variables homogeneous patches of the universe on the dynamics of the functions fj ; (ii) convoluting m fields δΨ and (j − m) of each of them). As in any nontrivial field theory, the fields Ψ0 with the same functions fj , with respect to the chosen exact solution of the dynamics is beyond question, and variables; (iii) summing over m from 1 to j. 10

with j Ψ0,fj ,λj ˆ = dφ3 . . . dφj Ψ0(νk, φk) ˆ Kj Seff = dφ Ψ0(ν, φ) δΨ(ν, φ)+ ν3...νj K X Z kY=3 " ν X Z fj(ν1,ν2,...,νj, φ1, φ2,...,φj ) . (55) n × λj + dφ1 . . . dφj It depends on the original coupling constants λ , on (j 1)! j j=2 ν1...νj X − X Z the original interaction kernels fj , and, crucially, on the j−1 mean-field configuration Ψ0 chosen as new vacuum. f (ν , φ ) Ψ (ν , φ ) δΨ(ν , φ ) × j i i 0 k k j j k=1 ! # The new effective interactions Veff for δΨ0 depend on Y the same data, and are given by the last term in Eq. (53): 1 + dφ δΨ(ν, φ) ˆδΨ(ν, φ) 2 K n " ν λj Z Veff = dφ1 . . . dφj fj (νi, φi) (56) nX j! λj j=3 ν1...νj Z + dφ1 . . . dφj fj (νi, φi) X X (j 2)! j j−m j j=2 − ν1...νj Z j X X Ψ (ν , φ ) δΨ(ν , φ ) , j × m 0 k k l l m=1 − Ψ (ν , φ ) δΨ(ν , φ )δΨ(ν , φ ) X   kY=1 l>jYm × 0 k k 1 1 2 2 k=3 ! # from which one can read out the new interaction kernels. Y n λj + dφ1 . . . dφj fj (νi, φi) The above is totally general. The simplest case is when  j! the original field theory contains only interactions of the j=2 ν1...νj Z X X lowest (nonquadratic) order, that is, when n = 3. Then,  j j−m j j one obtains the effective action (now assuming the linear Ψ (ν , φ ) δΨ(ν , φ ) . × m 0 k k l l  term vanishes) m=3   k=1 l>j−m X Y Y 1 (53) S [δΨ] = dφ dφ δΨ(ν , φ ) ˆ δΨ(ν , φ ) eff 2 1 2 1 1 Keff 2 2 ν ,ν X1 2 Z λ The term that is linear in δΨ vanishes if the mean-field + 3 dφ dφ dφ f (ν , φ ) 3! 1 2 3 3 i i configuration Ψ0 is chosen to satisfy the classical equa- ν ,ν ,ν 1X2 3Z tion of motion of the original field theory. In practice, it 3 is extremely hard to find a classical solution of the inter- δΨ(νk, φk) , (57) acting theory, so, as is the case in some condensed matter × k=1 systems, we resort to a further approximation valid in the Y limit of small coupling constants. That is, we will choose with the effective Hamiltonian constraint operator solutions of the free field theory as our mean-field vacua ˆ ˆ eff = δ(φ1 φ2)δν1,ν2 + λ2 f2(ν1, φ1; ν2, φ2) Ψ0, either exact (when possible) or approximate, and K − K then assume that the coupling constants λ are very small j +λ3 dφ3 f3(νi, φi)Ψ0(ν3, φ3) . (58) (that is, that topology change is strongly suppressed in ν3 Z this cosmological second-quantized toy model) and that, X because of this, the same vacua represent approximate In this simple case, one can also easily give the general- solutions of the full equation of motion. Under these as- ization to nonsymmetric interaction kernels: sumptions, we can neglect the linear terms in the above 1 effective action. ˆ = ˆ(ν , φ ; ν , φ )+ λ f (ν ,ν , φ , φ ) Keff K 1 1 2 2 2 2 2 1 2 1 2  λ The quadratic term in δΨ defines an effective + 3 dφ [f (ν ,ν ,ν , φ , φ , φ ) 3 3 3 1 2 3 1 2 3 Hamiltonian constraint for the cosmological second- ν3 X Z quantized model, taking into account the small pro- +f3(ν1,ν3,ν2, φ1, φ3, φ2) cesses of merging/splitting of homogeneous isotropic uni- + f (ν ,ν ,ν , φ , φ φ )] Ψ (ν , φ ) verses/patches. This effective Hamiltonian constraint op- 3 3 2 1 3 2 1 0 3 3 erator is given by: + (ν ν ) . (59) 1 ↔ 2  n λ ˆΨ0,fj ,λj = δ(φ φ )δ ˆ + j ˆΨ0,fj ,λj It is clear that the main issue in this approach, for the Keff 1 − 2 ν1,ν2 K (j 2)! Kj j=2 extraction of effective single-universe dynamics from the X − (54) initial field-quantized model, is the choice of mean field 11

Ψ0. We have already anticipated the general requirement where χk(ν, φ) is the solution to Eq. (10) labeled by k. of choosing (approximate) solutions of the original field Clearly, this expectation value can then be equivalently theory equations, and the difficulty involved in doing so. viewed as a first-quantized (real) wavefunction ψ(ν, φ), We will now work out a choice of mean-field vacuum. Eq. (60). In both viewpoints there is a normalization Having done so, we will turn to the role of the original condition: For ξ to be in the Fock space, one must | i2 interaction kernels fj and the consequences, at the level have d̺(k) ξ(k) < , while ψ(ν, φ) defines a first- of the effective Hamiltonian constraint, of some interest- quantized state| if| it has∞ finite norm in the appropriate ing choices of the same. We keep this final discussion physicalR inner product, e.g., on the possible resulting dynamics rather brief, leaving a more thorough analysis to the future.6 ψ 2 = B(ν) ψ(ν, φ ) 2 , (63) k k | 0 | ν One can follow two conceptually different but mathe- X matically similar approaches to the choice of mean-field at some fixed φ0 [22]. vacuum, yielding, eventually, the same result. The first For the solvable sLQC model, where the general so- would be to take a known physical state from the first- lution is Eq. (34), we have to choose appropriate func- quantized theory and write tions AL(k) and AR(k) in the mean-field approximation. If we consider only a single right-moving mode k0 > 0 Ψ0(ν, φ)= ψ(ν, φ) . (60) [A (k) = 0, A (k)= δ(k k )], L R − 0 In the second, one takes a second-quantized coherent +∞ ν state in a Fock space picture [31], ψ (ν, φ)= 0 dx e−iνb(x)eik0(φ+x) +c.c. k0 2π −∞Z ξ = exp d̺(k) ξ(k) a† 0 , (61) | i k | i k 2√12πGν π Z  cos 0 ln where k labels classical solutions to the constraint, ∼ "√12πG k0ν0 ! − 4 # d̺(k) is some measure determined by the normaliza- 2ν0 † 3 2 1/4 cos(k0φ) . (64) tion of single-particle states k = ak 0 [by imposing ×(3π Gk0) d̺(k) k k′ = 1], and 0 is| thei Fock| vacuum.i These h | i | i states are eigenstates of the annihilation operators ak Here we have used a stationary-phase approximation withR eigenvalue ξ(k), so that the field operator Ψ(ˆ ν, φ) has expectation value ig(x) ig(x0) 2π dx e e ′′ ∼ sig (x0) Z g′(x0)=0 ξ Ψ(ˆ ν, φ) ξ = d̺(k) ξ(k)χ (ν, φ)+c.c. ξ 2 , X h | | i k k k Z  (62) in the limit ν ν0, in which we find that ψ does not decay and, in general,≫ does not satisfy a normalization condition. Alternatively, we might consider a Gaussian, giving a wave packet centered around a classical trajec- 6 −1/2 A remark is in order. The logic of a mean-field approximation tory φ = x = (12πG) ln tan(ν0b/4), similar to is the following. One starts from a given dynamical model of the wave packets− studied− by Kiefer for Wheeler–DeWitt the universe in second quantization through some functions fj quantum cosmology [32]. The stationary-phase method (in addition to a given free theory dynamics). One aims at ob- taining an effective free theory, taking into account some effects then gives of the presence of interactions, around a new vacuum. One has +∞ then to identify what this relevant new vacuum is, and extract ν0 2 the effective dynamics around it (which will of course depend on ψ(ν, φ)= dx dk e−iνb(x)−λ(k−k0) +ik(φ+x) +c.c. 2π the original choice of interactions fj ). In our presentation we will Z be forced to follow a different logic. We will first discuss choices −∞ of mean field, then consider interesting possibilities for effective +∞ 1 2 Hamiltonian dynamics one may want to obtain as a result of the ν0 ik0(φ+x)− (φ+x) −iνb(x) = dx e 4λ +c.c. mean-field approximation of the second-quantized dynamics, and 2√πλ finally we will discuss which initial model (functions fj ) would −∞Z lead to the effective Hamiltonian considered, given a mean-field √ − 1 ln 2ν 12πG configuration. The reason for this line of argument is twofold. 2ν√12πG 48πGλ  k0ν0  First, given the progress in the context of LQC and the nov- ψk0 (ν, φ) , elty of our second-quantized reformulation, we have some con- ∼ k0ν0 ! trol over the possible forms of mean-field configurations but very little constraints on the “correct” choice of interactions. Second, (65) in the present paper, whose main goal is to introduce the new second-quantized framework, we are more interested in explor- where we pick up an extra factor from the Gaussian. For ing the available possibilities, the outcomes of various choices of large ν, the field now falls off faster than any power of models, and the ways to deal with them, rather than analyzing ν. The scalar field φ not only acquires an effective pe- the properties of one specific model. riodic potential [V f(a) cos(k φ) for n = 3], but it ∼ 0 12 also becomes nonminimally coupled with gravity via a for example of the type that would be needed for inflation nontrivial function f(a). in the early universe. Take, again, n = 3 and Eq. (58). The contribution in We can redo the analysis for Wheeler–DeWitt cosmol- f2 is nontrivial only if the effective Hamiltonian is nonlo- ogy using a wave packet of the usual form, cal, otherwise it would just be an extra piece defining the +∞ initial Hamiltonian constraint. Since we are interested in − − 2 −N a local constraint, we can ignore it, set λ = 0, and ab- ψ(a, φ)= dk e λ(k k0) eik[φ (a)] 2 sorb λ3 in the potential V (φ). Then, a function f3 that −∞Z happens to match the behavior of the mean-field vacuum π ik [φ−N (a)] − 1 [φ−N (a)]2 and contains the appropriate dependence on ν and φ on = e 0 e 4λ , (66) λ top of it would be r so that for large a (and at fixed φ) we find again a fall-off behavior f3(νi, φi)= δ(φ1 φ2)δ(φ2 φ3)δν1,ν2 δν2,ν3 − 2− −1 φ V (φ ) B(ν )ν [Ψ (ν , φ )] . (69) 3 − 3 ln a 3 3 3 0 3 3 ψ(a, φ) a√ 4πG 2λ 16πGλ (67) × ∼ faster than any power of a. Notice that, by construction, f3 is symmetric with re- spect to all its arguments. In general, any statement about the explicit form of Clearly, this is rather ad hoc and one should have an the contribution to ˆ will strongly depend not only on K independent justification (and possibly a full derivation the chosen field configuration Ψ0, but also on the form of from a fundamental GFT) for a given choice of interac- the interactions in our model, which are not strongly con- tions f3, such that the wished-for effective Hamiltonian strained. Let us discuss possible results for this effective constraint comes out, for a reasonable choice of nonper- Hamiltonian dynamics, and how to obtain them. turbative vacuum which should also be independently jus- A possibility which has been suggested by studies in tified. LQC [19, 20] is that, for a “monomial” interaction, the GFT coupling constant λ is related to the cosmological However, the above derivation proves an intriguing possibility, to be explored further: an underlying, more constant Λ. This possibility had been also considered previously [13, 14], but, in analogy with matrix mod- fundamental dynamics of creation/annihilation of uni- els and tensor models and with the “third-quantization” verses, i.e., topology change, or of merging/splitting of model of [4], the coupling λ would be expected to be homogeneous and isotropic patches within a single uni- related to the exponential of the cosmological constant verse could result, at an effective level, in a nontrivial po- rather than Λ itself. For example, the formal arguments tential term for the (homogeneous and isotropic) scalar of [4] suggest that a second-quantized field-theory Feyn- field, and a cosmological constant term. man amplitude should correspond to eiS evaluated on the Notice also that, in the first-quantized LQC and classical spacetime represented by the Feynman diagram. Wheeler–DeWitt frameworks, the presence of a potential Our simplified scheme suggests another way to obtain spoils the separation of positive-frequency and negative- a relation between a fundamental field-quantized cou- frequency sectors, as recalled, e.g., in [21]. This is not an pling and an effective cosmological constant. Under the issue in our second-quantized model (being a field the- assumption that our toy cosmological model comes out ory), and we are able to generate a scalar potential (non- from some more fundamental GFT dynamics, and thus trivial in the inflationary early universe) without formally that the coupling constant of the toy model can be re- changing the structure of the free theory. lated to the fundamental GFT interactions, the mean With the same procedure, we can obtain also other field approximation can relate very directly the coupling types of effective contributions, for instance a nonvanish- constant of the cosmological model to an effective cos- ing curvature k = 1 (closed and open universe, respec- mological constant in the single-universe dynamics (free tively). It is sufficient± to replace theory), that is, in an effective Hamiltonian constraint. Let us see how this can happen. 3k The effective contribution to ˆ should be of the form νV (φ) νV (φ) ν1/3 (70) 2 K → − 8πG Λ = Λ B(ν) ν . This term grows with large ν and, for aK trivial polynomial interaction (that is, trivial interac- tion kernels fj ), presumably it could not come from a in Eq. (69) to obtain an effective spatial curvature term of normalizable Ψ. However, a general choice of interaction the form one finds in the classical Friedmann equation. It functional, such as that in Eq. (17), can accommodate is also straightforward to obtain an effective Hamiltonian an effective cosmological constant. Actually, it can even constraint corresponding to, e.g., the k = 1 model in reproduce a nonconstant scalar-field potential term, LQC [33]. In fact also in that case the term corresponding to spatial curvature only acts by multiplication with a = V (φ) B(ν) ν2 , (68) function of ν. KV 13

VI. DISCUSSION AND OUTLOOK In our field-theory picture, the present-day universe would resemble some configuration of many particles (re- −1 In this paper we outlined the construction of a field the- gions of linear size b ) that looks homogeneous to high ory of universes drawing inspiration from the perspective precision at large scales. This could be a condensate advanced in quantum gravity by group field theory, and phase of the theory where discrete translation invariance by the general idea of “third quantization of gravity,” in the ν variable is spontaneously broken, as is known that had been advanced in the early days of the subject for systems in condensed matter physics. Then, the chal- [4, 9, 10]. We have also studied various aspects of the lenge would be to define a model whose collective behav- formalism, in particular the consequences it has for the ior agrees with the standard cosmological perturbation single-universe dynamics, that is, for the standard (loop) theory. quantum cosmology setting. These come already from However, the resulting action would be nonlocal in the embedding of the canonical dynamics within a field order to accommodate the infinite multiplicity of spa- theory, as encoded in the free field theory. More interest- tial points into a finite-dimensional, minisuperspace-like ing consequences, of course, come from the existence of phase space. For instance, following the above-mentioned interactions, which, we showed, can be taken into account gradient expansion, a linear perturbation would be de- via mean field approximation. Given the subject, and the fined via the “interaction” of two patches in a nonlocal current level of understanding of the fundamental theory quadratic term: (in either the LQG or in the GFT formulation), our goal was then in many respects necessarily of an exploratory da dφ da′ dφ′ f(a,a′, φ, φ′)Ψ(a, φ)Ψ(a′, φ′) , nature. In particular, the least developed point of the Z Z discussion concerns the choice of interaction. As in all where f is a function which should encode the correct simplified models for cosmology, however, the main task dynamics to match with the perturbed Hamiltonian con- will be to better justify the assumptions and the dynam- straint. Just as we did for the effective Hamiltonian ics chosen from the fundamental theory, and to possibly constraint in the previous section, one could explicitly show how such a simplified model can emerge naturally calculate f from this perturbed Hamiltonian constraint. in some sectors of the full theory. Understanding this The main difficulty to overcome, in developing properly issue will also clarify the role and physical significance of the separate universe perspective of our second-quantized conserved currents within the model. cosmology, is to develop first a proper quantum cosmol- Before concluding, we should mention an alternative ogy version of the separate universe approach to cosmo- view of the physics of this “group field cosmology,” that logical perturbations. Admittedly, it is unclear at the we anticipated in passing in the course of our presenta- present stage whether applying our framework to the tion. While loop quantum or Wheeler–DeWitt cosmol- problem of cosmological perturbations has practical ad- ogy describe a single evolving quantum universe, in the vantages over conventional strategies, unless some con- field cosmological model one has many-particle interact- servation law be implemented. It is yet another possi- ing states. Instead of interpreting them as n distinct uni- bility worth exploring, however. On the other hand, the verses merging and splitting in topology-changing “scat- implementation of the separate universe idea within a tering” processes, one could think of them as n FRW field-theoretic formalism like the one we propose could patches which, collected together, approximate a sin- have a better chance of being derived from fundamen- gle inhomogeneous universe. This is reminiscent of the tal formulations of quantum gravity such as GFT. This separate universe approach [34–36], where inflationary would give a more solid ground to this way of dealing with large-wavelength perturbations are represented as spa- cosmological perturbations and also offer a way to test tial gradients among homogeneous patches of Hubble fundamental models via their cosmological predictions. size.7 In each patch centered at some spatial point x, one has a “local” scale factor a(t, x), Hubble parame- ter H(t, x), and so on. In particular, the local scale ACKNOWLEDGMENTS factor a(t, x) = a(t) exp[ Φ (t, x)] encodes both the − NL minisuperspace variable a and the nonlinear scalar per- This research was supported by Perimeter Institute for turbation ΦNL. Linear perturbations can then be iden- Theoretical Physics. Research at Perimeter Institute is tified with gradients. At the linear level, a(t, x) supported by the Government of Canada through Indus- x x ≈ a(t)[1 ΦNL(t, )]; call δa = a(t)ΦNL(t, ). One has try Canada and by the Province of Ontario through the x − x i i − x [a(t, 1) a(t, 2)]/(x x ) ∂ia(t, ), so that, up to a Ministry of Research & Innovation. Support from the A. − 1 − 2 ∼ numerical factor, for a perturbation of wavelength λ we von Humboldt Stiftung, via a Sofja Kovalevskaja Award, x get δa λ∂ia(t, ). is gratefully acknowledged. ∼

7 For other work in the quantum cosmology context which is sim- ilar to the spirit of the separate universe approach, see, e.g., [37] 14

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