In loop quantum cosmology, Friedmann-LeMaitre-Robertson-Walker (FLRW) space-times arise as well-defined approximations to specific \emph{quantum} geometries. We initiate the development of a quantum theory of test scalar fields on these quantum geometries. Emphasis is on the new conceptual ingredients required in the transition from classical space-time backgrounds to quantum space-times. These include a `relational time' a la Leibnitz, the emergence of the Hamiltonian operator of the test field from the quantum constraint equation, and ramifications of the quantum fluctuations of the background geometry on the resulting dynamics. The familiar quantum field theory on classical FLRW models arises as a well-defined reduction of this more fundamental theory.Comment: 19 pages, no figures; A reference and footnote added, PACs and minor typos corrected; a significant technical clarification added to section IV.A; minor typos corrected; two references adde

The simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations). In particular the notion of embedded spin-foam we use allows to consider knotting or linking spin-foam histories. Also the main tools as the vertex structure and the vertex amplitude are naturally generalized to arbitrary valency case. The correspondence between all the SU(2) intertwiners and the SU(2)×SU(2) EPRL intertwiners is proved to be 1-1 in the case of the Barbero-Immirzi parameter |γ| ≥ 1, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.

but we do not have quantum gravity." This phrase is often used when analysis of a physical problem enters the regime in which quantum gravity effects should be taken into account. In fact, there are several models of the gravitational field coupled to (scalar) fields for which the quantization procedure can be completed using loop quantum gravity techniques. The model we present in this paper consist of the gravitational field coupled to a scalar field. The result has similar structure to the loop quantum cosmology models, except for that it involves all the local degrees of freedom of the gravitational field because no symmetry reduction has been performed at the classical level. PACS numbers: 4.60.Pp; 04.60.-m; 03.65.Ta; 04.62.+v I. INTRODUCTIONThe recent advances in loop quantum gravity (LQG) [1][2][3][4], strongly suggest that the goal of constructing a candidate for quantum theory of gravity and the Standard Model is within reach. Remarkably, that goal can be addressed within the canonical formulation of the original Einstein's general relativity in four dimensional spacetime. A way to define 'physical' dynamics in a background independent theory, where spacetime diffeomorphisms are treated as a gauge symmetry, is the framework of relational Dirac observables (often also called "partial" observables [5], [6,7],[8] section I.2 of [2]). The main idea is, that part of the fields adopt the role of a dynamically coupled observer, with respect to which the physics of the remaining degrees of freedom in the system is formulated. In this framework the emergence of the dynamics, time and space can be explained as an effect of the relations between the fields. As far as technical issues of a corresponding quantum theory are concerned, the most powerful example of the relational observables framework is the deparametrization technique [9][10][11][12]. This allows to map canonical General Relativity into a theory with a (true) non-vanishing Hamiltonian, that is independent of the (emergent) time provided by the observer fields. All this can be achieved at the classical level, the framework of Loop Quantum Gravity (LQG) itself, provides then the tools of the quantum theory like quantum states, the Hilbert spaces, quantum operators of the geometry and fields and well defined quantum operators for the classical constraints of General Relativity (see [2],[4] and references therein). The combination of LQG with the relational observables and deparametrization framework makes it possible to construct general relativistic quantum models. Applying LQG techniques to perform the quantization step has the consequence that the quantum fields of the Standard Model have to be reintroduced within the scheme of LQG. This is due to the reason that the standard quantum field theory (QFT) defined on the Minkowski (or even ADS) background is incompatible with quantization approach used in LQG. Therefore, the resulting quantum theory of gravity cannot be just coupled to the Standard Model in it's present form. The formulation of the full St...