Generalised effective cosmology from group field theory

Abstract

We propose a new method for obtaining an effective Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology from the quantum gravity dynamics of group field theory (GFT), based on the idea that an FLRW universe is characterised by a few macroscopic observables.Rather than relying on assuming a particular type of quantum state and computing expectation values in such a state, here we directly start from relations between macroscopic observables (defined as one-body operators) and formulate dynamics only for those observables. We apply the effective approach to constrained quantum systems (as developed by Bojowald and collaborators) to GFT, providing a systematic expansion in powers of . We obtain a kinematical phase space of expectation values and moments, which does not require an a priori choice of clock variable. We identify a gauge fixing of the system which corresponds to choosing one of the cosmological variables (with the role of extrinsic curvature) as a clock and which allows us to rewrite the effective dynamics in relational form. We show necessary and sufficient conditions for the resulting dynamics of expectation values to be compatible with those of classical FLRW cosmology and discuss the impact of quantum fluctuations.

Hamiltonian group field theory with multiple scalar matter fields

“…The connection of this toy model to full GFT was then clarified in [36] where a deparametrised approach was adopted for GFT by choosing the scalar field argument as a time variable from the beginning, and the Legendre transform was computed leading to a Hamiltonian for GFT. In the truncation to quadratic terms the resulting dynamics were very similar to that of [35], again leading to LQC-like cosmological dynamics (see also [37] for more details of the effective cosmology of this model). The method of deparametrisation using scalar fields had previously appeared in canonical LQG [38][39][40], where it also leads to a different Hilbert space which is not subject to (Hamiltonian or diffeomorphism) constraints.…”

“…We will now proceed with the main application of the canonical formulation of GFT, namely the extraction of an effective cosmological dynamics, similarly to what has been done in numerous examples before [17,22,[32][33][34][35]37]. The main new element in the derivation we give in this paper is the use of a deparametrised form of GFT dynamics in a theory describing quantum gravity coupled to multiple scalar matter fields.…”

“…They can, however, provide reliable approximations for a period of time which may be sufficiently long for practical purposes. These are well-known issues in the study of cosmological dynamics from GFT [22,23,37]. All proposals for GFT models for quantum gravity are interacting field theories, presumably only well-defined in the path integral setting.…”

“…It is then very natural to interpret the conserved energyĤ and momentumP in the GFT as corresponding to the conserved canonical momenta (π χ ) 0 and π χ associated to the D scalar matter fields, and we will use this interpretation in what follows. In the case where a GFT state can be interpreted as a spatially homogeneous geometry, this interpretation was already used in previous work such as [35,37] where the Hamiltonian was interpreted as the conjugate momentum of a single scalar matter field. Note however that these conserved quantities exist both in the GFT formulated here and in usual scalar field theory on a fixed background regardless of whether the geometry is spatially homogeneous.…”

“…A particularly interesting class of states are the (Fock) coherent states which can be defined as being the eigenstates of the annihilation operator at the initial time when Heisenberg and Schrödinger picture agree. These states have semiclassical behaviour and can be constructed so that relative uncertainties in energy and volume become very small at late times, which can be interpreted as the emergence of a classical universe in GFT [37].…”