We argue that refining, coarse graining and entangling operators can be obtained from time evolution operators. This applies in particular to geometric theories, such as spin foams. We point out that this provides a construction principle for the physical vacuum in quantum gravity theories and more generally allows construction of a (cylindrically) consistent continuum limit of the theory.In this paper we point out that time evolution maps, which appear in simplicial discretizations [13,14], can also be interpreted as refining and coarse graining maps. As we will argue here, this applies in particular to gravitational dynamics, e.g. spin foams [15][16][17][18].One reason why the appearance of time evolution as coarse graining or refining maps applies in particular to gravitational or other diffeomorphism invariant systems is the following: As argued in [19][20][21][22][23], diffeomorphism symmetry in discrete systems translates to a symmetry, which can be interpreted as moving vertices in the discrete space time described by the dynamical variables of the theory. These vertex translations can also be understood as time evolution. Now, vertices can be even moved on top of each other, which gives a coarse graining of the underlying state. Alternatively, vertices can split into two and in this way give a refinement. Indeed, this argument was used in [23] to show that diffeomorphism symmetry implies discretization independence.More generally, diffeomorphism invariant systems are totally constrained, i.e. the Hamiltonian is given by a combination of constraints. In the case of a totally constrained system, the time evolution operator should be a projection operator [24,25], projecting onto socalled physical states. Thus physical states should not evolve 1 .For discrete time evolutions that change the number of degrees of freedom, this leads to the puzzle of how to identify states from Hilbert spaces of 'different size' 2 . We will argue that such states describe indeed the same physical state, expressed, however on two different discretizations. The equivalence relation is provided by the refining time evolution operator. We will explain how this notion can be formalized into the construction of an inductive limit Hilbert space. Such an inductive limit construction is also used for the (kinematical) Hilbert space of loop quantum gravity [27,28].The inductive limit Hilbert spaces, however, which are defined via an equivalence relation between states from Hilbert spaces based on different discretizations, require (so-called cylindrical) consistency conditions: Physical observables should not depend on which representative they have been determined on. Indeed, we will connect these consistency conditions with a notion of path independence for (refining) time evolution. This relates, then, to the requirement of diffeomorphism invariance.Discrete (non-topological) theories typically break the diffeomorphism symmetry [22]. The hope, however, is that diffeomorphism symmetry can be recovered in the continuum limit. We will ex...

To obtain a well defined path integral one often employs discretizations. In the case of gravity and reparametrization invariant systems, the latter of which we consider here as a toy example, discretizations generically break diffeomorphism and reparametrization symmetry, respectively. This has severe implications, as these symmetries determine the dynamics of the corresponding system. Indeed we will show that a discretized path integral with reparametrization invariance is necessarily also discretization independent and therefore uniquely determined by the corresponding continuum quantum mechanical propagator. We use this insight to develop an iterative method for constructing such a discretized path integral, akin to a Wilsonian RG flow. This allows us to address the problem of discretization ambiguities and of an anomalyfree path integral measure for such systems. The latter is needed to obtain a path integral, that can act as a projector onto the physical states, satisfying the quantum constraints. We will comment on implications for discrete quantum gravity models, such as spin foams.

In this work, we investigate the 4d path integral for Euclidean quantum gravity on a hypercubic lattice, as given by the Spin Foam model by Engle, Pereira, Rovelli, Livine, Freidel and Krasnov (EPRL-FK). To tackle the problem, we restrict to a set of quantum geometries that reflects the large amount of lattice symmetries. In particular, the sum over intertwiners is restricted to quantum cuboids, i.e. coherent intertwiners which describe a cuboidal geometry in the large-j limit.Using asymptotic expressions for the vertex amplitude, we find several interesting properties of the state sum. First of all, the value of coupling constants in the amplitude functions determines whether geometric or non-geometric configurations dominate the path integral. Secondly, there is a critical value of the coupling constant α, which separates two phases. In both phases, the diffeomorphism symmetry appears to be broken. In one, the dominant contribution comes from highly irregular, in the other from highly regular configurations, both describing flat Euclidean space with small quantum fluctuations around them, viewed in different coordinate systems. On the critical point diffeomorphism symmetry is nearly restored, however.Thirdly, we use the state sum to compute the physical norm of kinematical states, i.e. their norm in the physical Hilbert space. We find that states which describe boundary geometry with high torsion have exponentially suppressed physical norm. We argue that this allows one to exclude them from the state sum in calculations. I. MOTIVATIONThe spin foam approach has been developed to give a rigorous meaning to the path integral for quantum gravity (see [1] for a review). Its central idea rests on the observation that the first order formalism of GR can be rewritten as a certain constrained topological theory [2,3], dubbed "BF theory". A quantization choice for these constraints is what specifies the spin foam model, and in recent years there have been several proposals [4][5][6][7]. A popular choice has emerged in the so-called EPRL-FK model [5,8,9], which possesses quite useful properties. In particular, the resulting amplitude has an asymptotic expression for large quantum numbers which reproduces the Regge action [10][11][12]. Furthermore, the resulting path integral for the EPRL-FK model naturally has boundary states which resemble the spin network states from canonical loop quantum gravity [13][14][15], which is why it has been coined "covariant loop quantum gravity" ([16], see also [17,18]).There are several open questions, however. While there are numerous results available which elucidate the property of a single vertex amplitude, very little is known about the behaviour of the whole path integral.1 In particular, it is the realm of many building blocks which is of utmost importance if one wants to understand the * benjamin.bahr@desy.de † sebastian.steinhaus@desy.de 1 There are some results on the asymptotic expression for more than one vertex [19,20], as well as self-energy calculations [21].continuum limi...

Tensor network techniques have proved to be powerful tools that can be employed to explore the large scale dynamics of lattice systems. Nonetheless, the redundancy of degrees of freedom in lattice gauge theories (and related models) poses a challenge for standard tensor network algorithms. We accommodate for such systems by introducing an additional structure decorating the tensor network. This allows to explicitly preserve the gauge symmetry of the system under coarse graining and straightforwardly interpret the fixed point tensors. We propose and test (for models with finite Abelian groups) a coarse graining algorithm for lattice gauge theories based on decorated tensor networks. We also point out that decorated tensor networks are applicable to other models as well, where they provide the advantage to give immediate access to certain expectation values and correlation functions. OPEN ACCESS RECEIVED

So far spin foam models are hardly understood beyond a few of their basic building blocks. To make progress on this question, we define analogue spin foam models, so called spin nets, for quantum groups SU(2) k and examine their effective continuum dynamics via tensor network renormalization. In the refinement limit of this coarse graining procedure, we find a vast non-trivial fixed point structure beyond the degenerate and the BF phase. In comparison to previous work, we use fixed point intertwiners, inspired by Reisenberger's construction principle [1] and the recent work [2], as the initial parametrization. In this new parametrization fine tuning is not required in order to flow to these new fixed points. Encouragingly, each fixed point has an associated extended phase, which allows for the study of phase transitions in the future. Finally we also present an interpretation of spin nets in terms of melonic spin foams. The coarse graining flow of spin nets can thus be interpreted as describing the effective coupling between two spin foam vertices or space time atoms.