Generalized GFT Condensates and Cosmology

Transcription

1 Generalized GFT Condensates and Cosmology Lorenzo Sindoni MPI für Gravitationsphysik (Albert Einstein Institute) Potsdam/Golm, Germany in collaboration with D. Oriti, D. Pranzetti and J. Ryan ILQGS,

2 Derivation from GFT of models for cosmology: a map Dynamics: Field equations (encode the sum over triangulations) GFT: microscopic theory for spacetime Identify special class of states (LQG data) Effective (quantum?) cosmology/ Friedmann equation Establish a correspondence with minisuperspace sector (FRW, Bianchi) 2

3 Simple condensates Gielen, Oriti, LS, Phys. Rev. Lett. 111 (2013) 3, & JHEP 1406 (2014) 013 Calcagni, Gielen, Oriti, LS Context: Group Field Theory (GFT), a field theory on an auxiliary (group) manifold. It incorporates many ideas and structures from LQG (and spinfoam models) in a second quantized language. Spacetime should emerge from the collective dynamics of the microscopic degrees of freedom. Condensates: all the quanta in the same state. Simple quantum states of the full theory, can be put in correspondence with Bianchi cosmologies: symmetry reduction at the quantum level. Effective dynamics for cosmology: contact with LQC and Friedmann equations? Significance: identify states that, for the number of d.o.f. involved, have an hydrodynamical interpretation. ILQGS of S. Gielen 3

4 First proposals: field coherent states (and squeezed states) Dynamics: Field equations (encode the sum over triangulations) GFT: microscopic theory for spacetime Identify special class of states (LQG data) Effective (quantum?) cosmology/ Friedmann equation 4 Establish a correspondence with minisuperspace sector (FRW, Bianchi) Can we improve the states to include connectivity information? Yes: introducing refinement moves

5 Plan of the talk Brief review of the basic ideas in GFT Connectivity, relation with LQG and spinfoams Brief review of previous work on condensates (ILQGS by S. Gielen) Construction: vertex homogeneity and refinement moves Remarks and conclusions 5

6 Group Field Theories: second quantization language for discrete geometry Group field theories are (quantum) field theories over a group manifold (NOT space/spacetime) Notation Basic definitions (g 1,...,g D ) g v C g v = g (v,1),...,g (v,d) ϕ : G G... G D The theory is formulated in terms of a Fock space. Bosonic statistics is used. [ˆϕ(g v ), ˆϕ (g w )] = R (g v,g w ) Gauge invariance on the right is required: ˆϕ(g 1,g 2,g 3,g 4 )= ˆϕ(g 1 h, g 2 h, g 3 h, g 4 h), h SU(2) Work in progress with C. Tomlin to understand these requirements from the canonical quantization of a certain field theory, as well as the relation with the functional formulation. SU(2) dγ 4 δ(g (v,i) γ(g (w,i) ) 1 ) i=1 6

7 GFT quanta: spin network vertices... Consider D=4, G=SU(2). These quanta have a natural interpretation in terms of spin-network vertices (4-valent) h 4 h 1 h 2 h 3 7

8 ...and quantum tetrahedra Via a noncommutative Fourier transform it can be formulated in group variables (extrinsic geometry). Consider SU(2). 4 ˆϕ(g 1,g 2,g 3,g 4 )= db 4 e gi (B I ) ˆ ϕ(b 1,B 2,B 3,B 4 ) B I su(2) B 1 + B 2 + B 3 + B 4 =0 i=1 h 1 h 4 h 2 h 3 We have a second quantized theory that creates (quantum) tetrahedra ˆ ϕ (B 1,B 2,B 3,B 4 ) 0 1 8

9 Playing with the wavefunctions we can build states associated to connected graphs We can generate states that can be used as boundary states (=spatial geometries) for transition amplitudes. (dg) 4 f(g 1,g 2,g 3,g 4 )ˆϕ (g 1,g 2,g 3,g 4 ) 0 One can create states with given wavefunctions. According to the particular dependence of the wavefunction we attach to the state a graph. g1 1 g 1 g 1 g 1 Convolution f 1 (g 1 )f 2 (g1) dh 1 f 1 (h 1 g 1 )f 2 (h 1 g1) dg v dg w dh v ψ v (h v g v )ψ w (h v g w )ˆϕ (g v )ˆϕ (g w ) 0 9

10 Important remarks The graphs are not embedded into a manifold. They are abstract graphs. Data associated to the states are given a geometric interpretation in terms of conventional parallel transports and fluxes. Spin 0 still corresponds to trivial geometry, but it is counted by the number operator. The Bosonic statistics implies that wavefunctions are automatically symmetrized w.r.t. the permutation of vertices. The inner product of the Fock space does not guarantee that states associated to different graphs are orthogonal (they are if they contain a different number of vertices). The geometry of the states should be inferred by inspection of the action of (second quantized) operators associated to geometric observables. Understood for 1-body operators, not for 2-body or more. 10

11 The perturbative dynamics generate a 4D sum over geometries approach to QG (spinfoams) The goal is achieved incorporating the ideas and methods of the spinfoam approach Perez Liv. Rev. Rel Idea: look for equations of motion for the quantum field which lead to a perturbative expansion around the Fock vacuum (no geometry state) dh J K(g I ; h J )ˆϕ(h J )+ dg 16 V(g I,g I2,g I3,g I4,g I4 )ˆϕ(g I2 )ˆϕ(g I3 )ˆϕ(g I4 )ˆϕ(g I5 )+ dg 16 U(g, g I2,g I3,g I4,g I5 )ˆϕ (g I2 )ˆϕ (g I3 )ˆϕ (g I4 )ˆϕ (g I5 )+... ψ = 0 Feynman rules are such that each contribution is a spinfoam amplitude (achieved by controlling the combinatorics and the kernels). Problem: try to make sense of the perturbative expansion. Oriti

12 Correlation functions of GFT and spinfoams C(Γ 2 Γ 1 ) dg internal When computing the correlation functions between boundary states the Feynman rules glue tetrahedra into 4-simplices. This is controlled by the combinatorics of the interaction term. The amplitude is designed to match spinfoam amplitudes. For example, the interaction kernel can be chosen to be the EPRL vertex in a group representation. 12

13 The dynamics can be designed to give rise to the transition amplitudes with sum over 4d geometries included (discrete path integral for gravity) Bonus: QFT language and tools for the dynamics, still at a background independent level (still no spacetime!) Question: what can we say of the solution of the equation of motion? Idea: proceed in analogy with condensed matter physics and design trial states, parametrised by relatively few variables, and deduce from the dynamics of the fundamental model the optimal induced dynamics. 13

14 We need some trial states to get effective continuum dynamics We need trial states that contain the relevant information about the regime that we want to explore. Fock space give us several interesting possibilities. Simple case: field coherent states σ = N [σ]exp(ˆσ) 0 ˆσ = d 4 gσ(g I )ˆϕ (g I ) Simple state, but not a state with an exact finite number of particles Might be inspired by the idea that spacetime is a sort of condensate (Volovik, Hu) Can be generalized to other states (squeezed, multimode...) 14

15 Condensates can be naturally interpreted as homogeneous (but anisotropic)cosmologies ) 3 2 σ(b I ) g ij ds 2 = g ij ω i ω j 4 P ( B2 IJ B4 IJ 3 P P 5 P 3 Elementary quanta possessing the same wavefunction: the metric tensor in the frame of the tetrahedron is the same everywhere. Vertex/wavefunction homogeneity! Can be interpreted in terms of homogeneous (anisotropic) cosmologies, once a reconstruction procedure into a 3D group manifold has been specified. The reconstruction procedure is based on the idea that each of these tetrahedra is embedded into a background manifold: the edges are aligned with a basis of left invariant vector fields. 15

16 We can exploit smart states in the Fock space to model homogeneous spatial geometries (in principle all the Bianchi types), and their extrinsic curvature. These states might be controlled by a simple collective wavefunction, but they have some notion of hydrodynamic regime in them. They are states of the full quantum theory, and they might be seen as homogeneous. Next step: add connectivity (and use refinement moves) to introduce more features implied by the dynamics (correlations) and reduce the need for a reconstruction procedure. 16

17 Some considerations on states for homogeneous cosmologies Homogeneous cosmologies are highly symmetric geometries, controlled by a small number of parameters (and an isometry group). One option would be to construct states associated to graphs adapted to the particular isometry group, keeping wavefunction homogeneity, by enforcing then some combinatorial regularity. Regular tessellations. Another option would be to relax wavefunction homogeneity and combinatorial regularity, consider more general triangulations, obtained, for instance, by random sampling of homogeneous spaces. Proliferation of (mostly redundant) microscopic data. Both these approaches translate directly homogeneity of the macroscopic configuration into a microscopic homogeneity. Furthermore, one has to write the state explicitly, a problem if we want to keep track of a large number of d.o.f.. 17

18 The requirements for our trial states Our states should be able to involve an infinite number of degrees of freedom Possibly, they should not depend on a special triangulation/graph. Generic superpositions should be allowed, to keep at least part of the sum over geometries. We do not necessarily require that each element of the superposition is in itself associated to a homogeneous geometry. Homogeneity should be only a property of the macroscopic level of the description. The states should be parametrized by only the data associated to continuum geometries. The condensates should be as simple as possible, but be as faithful as possible to the properties of the exact state as implied by the equations of motion. Analysis of the dynamics. 18

19 The ideas behind the construction Generalize the simple condensate keeping wavefunction homogeneity as a basic ingredient. The wavefunction of a single vertex will still have a hydrodynamic nature: controls the macroscopic state. Define a modular construction, with the replication of the same basic unit, encoding now the gluing (i.e. including the necessary convolutions). Start from a seed state, representing the simplest triangulation for the desired geometry and act on it with refinement moves that allow the growth of the state with the addition of new quanta to the preexisting ones. 19

20 Implementation: wavefunction homogeneity Wavefunction homogeneity: use operators that create vertices with a given wavefunction ˆσ (h v ) dg v σ(h v g v )ˆϕ (g v ) To ensure that only the data of a tetrahedron are stored up to a global rotation, impose gauge invariance on the left. σ(γ L g v γ R )=σ(g v ), γ L,γ R SU(2) The group elements h are not associated to parallel transports. They are the ones used for the gluing. g1 1 g 1 g 1 g 1 S 3 dh v1 dh v2 dh v3 dh v4 dh v5 δ(h (v1,1)h 1 (v 5,4) )δ(h (v 1,2)h 1 (v 4,3) )δ(h (v 1,3)h 1 (v 3,2) )δ(h (v 1,4)h 1 (v 2,1) )δ(h (v 2,2)h 1 (v 5,3) ) δ(h (v2,3)h 1 (v 4,2) )δ(h (v 2,4)h 1 (v 3,1) )δ(h (v 3,3)h 1 (v 5,2) )δ(h (v 3,4)h 1 (v 4,1) )δ(h (v 4,4)h 1 (v 5,1) ) ˆσ (h v1 )ˆσ (h v2 )ˆσ (h v3 )ˆσ (h v4 )ˆσ (h v5 ) 0 20

21 Implementation: refinement moves Using the previous creation operators, define refinement moves that allow us to grow the state in a systematic way First condition: topology preserving Second condition: respect vertex homogeneity Natural candidates for a simplicial context: Pachner moves, removing a certain set of tetrahedra from the state and replacing them with a different set. Only the (1,4) is easily written in terms of second quantized operators: destroys only one tetrahedron. All the others would destroy more than one: not known how to keep the action local (we are working with identical particles, and we are not labeling the vertices) 21

22 The equation for the refinement move operator Graphical representation of the action on a state h 4 v 3 h 7 v 4 h 4 M v h 3 h 2 Γ \v h 9 h 6 h 8 h 5 h 3 h 2 Γ \v h 1 v 2 h 10 v 1 h 1 Equation for the operator: M, ˆσ (h 1,h 2,h 3,h 4 ) = (dh) 6ˆσ (h 4,h 5,h 6,h 7 )ˆσ (h 7,h 3,h 8,h 9 )ˆσ (h 9,h 6,h 2,h 10 )ˆσ (h 10,h 8,h 5,h 1 ) 22

23 A solution We do not know the full space of solutions for the equation for the refinement move (abelian case helps, but it is qualitatively different). We can get a special solution if we impose a (nontrivial) condition ˆσ(hv ), ˆσ (h w ) = L (h v,h w ) SU(2) dγ 4 δ(γh (v,i) (h (w,i) ) 1 ) i=1 M = (dh) 10ˆσ (h 4,h 5,h 6,h 7 )ˆσ (h 7,h 3,h 8,h 9 )ˆσ (h 9,h 6,h 2,h 10 )ˆσ (h 10,h 8,h 5,h 1 )ˆσ(h 1,h 2,h 3,h 4 ) 23

24 The states The basic principles are easy: choose a wavefunction encoding the macroscopic data, choose a seed state, act with the refinement moves. Ψ F = F ( M) seed Ψ F = c Γ (F ) Γ,σ Γ(seed, M) The state has still a large amount of freedom in specifying the linear superposition, even though it is not ergodically exploring the space of triangulations at fixed topology. The same ideas can be generalized easily to cases in which the field theory is able to store more data. We examined the case of the colored GFTs required to have a better control on states with additional structures (e.g. boundaries). Melonic moves are more natural, in these cases. 24

25 Some remarks We have not addressed the dynamics. The major difference w.r.t. the case of the coherent state should be some nontrivial effect due to the connectivity. The state has still a large amount of freedom in specifying the linear superposition: ambiguity/flexibility in the sum over triangulations that are allowed. The states implement a notion of wavefunction homogeneity. Not clear if it is possible to obtain homogeneity w.r.t. a discrete isometry group, to mimic closely the classical counterpart. Geometric operators exploring the multiparticle structure of the state are needed. For example: the definition of a holonomy operator for a loop incident on several vertices (preliminary work with V. Skrinjar) Due to the commutation relations between the modified operators, the states are not normalisable. However, one can still compute expectation values of certain one body operators, for which the normalisability becomes immaterial. 25

26 One body observables Despite the nontrivial structure of the states, we can say something about the expectation values of one body operators, and verify whether the vertex wavefunction stores the relevant geometric information. A dg v dg w A(g v,g w )ˆϕ (g v )ˆϕ(g w ) A(g v,g w )=(g v  g w) It turns out that normalizability is not really an issue, when we consider these observables. Symmetrization of the wavefunction is helping us. By inspection of the matrix elements:  = ˆN dh v dg v dg w σ(h v g v )A(g v,g w )σ(h v g w ) The vertex wavefunction controls the expectation values, as for simple condensates, but the expression of the expectation values includes a modification which includes the effect of the gluing. 26

27 Conclusions We have extended the family of states encoding some specific notion of homogeneity to states possessing nontrivial structure, in terms of correlations between quanta. Using refinement moves, we can have a relatively compact expression for the state, with a minimal number of variables for the parametrization of the state. We can perform simple calculations, despite the presence of connectivity. The procedure can be generalized to more complicated situations (spherical symmetry). These states are another step towards the hydrodynamics of GFT quanta encoding the idea of spacetime as a condensate with states encoding also topological data. 27

28 Thank you 28

29 The general picture for effective equations Impose the EOM of GFT: ask that the state are physical The equations for the mean field: Wheeler-deWitt equation? among σ Ô[ϕ, ϕ]ĉ σ =0 other things H ( Ĉ) σ =0 Depends on the GFT model Nonlinear!! To get an effective cosmological dynamics, we need to translate the equations for the mean field into equations relating these expectation values (~ Ehrenfest theorem) 0=F(a ψ, H ψ, ) ȧ 2 + k a a 2 + corrections 29