The Hamiltonian formalism of the generalized unimodular gravity theory, which was recently suggested as a model of dark energy, is shown to be a complicated example of constrained dynamical system. The set of its canonical constraints has a bifurcation -splitting of the theory into two branches differing by the number and type of these constraints, one of the branches effectively describing a gravitating perfect fluid with the time-dependent equation of state, which can potentially play the role of dark energy in cosmology. The first class constraints in this branch generate local gauge symmetries of the Lagrangian action -two spatial diffeomorphisms -and rule out the temporal diffeomorphism which does not have a realization in the form of the canonical transformation on phase space of the theory and turns out to be either nonlocal in time or violating boundary conditions at spatial infinity. As a consequence, the Hamiltonian reduction of the model enlarges its physical sector from two general relativistic modes to three degrees of freedom including the scalar graviton. This scalar mode is free from ghost and gradient instabilities on the Friedmann background in a wide class of models subject to a certain restriction on time-dependent parameter w of the dark fluid equation of state, p = wε. For a special family of models this scalar mode can be ruled out even below the phantom divide line w = −1, but this line cannot be crossed in the course of the cosmological expansion. This is likely to disable the generalized unimodular gravity as a model of the phenomenologically consistent dark energy scenario, but opens the prospects in inflation theory with a scalar graviton playing the role of inflaton.

We use a functional renormalization group equation tailored to the Arnowitt–Deser–Misner formulation of gravity to study the scale dependence of Newton’s coupling and the cosmological constant on a background spacetime with topology . The resulting beta functions possess a non-trivial renormalization group fixed point, which may provide the high-energy completion of the theory through the asymptotic safety mechanism. The fixed point is robust with respect to changing the parametrization of the metric fluctuations and regulator scheme. The phase diagrams show that this fixed point is connected to a classical regime through a crossover. In addition the flow may exhibit a regime of “gravitational instability”, modifying the theory in the deep infrared. Our work complements earlier studies of the gravitational renormalization group flow on a background topology (Biemans et al. Phys Rev D 95:086013, 2017, Biemans et al. arXiv:1702.06539, 2017) and establishes that the flow is essentially independent of the background topology.

The Asymptotic Safety hypothesis states that the high-energy completion of gravity is provided by an interacting renormalization group fixed point. This implies nontrivial quantum corrections to the scaling dimensions of operators and correlation functions which are characteristic for the corresponding universality class. We use the composite operator formalism for the effective average action to derive an analytic expression for the scaling dimension of an infinite family of geometric operators d d x √ gR n . We demonstrate that the anomalous dimensions interpolate continuously between their fixed point value and zero when evaluated along renormalization group trajectories approximating classical general relativity at low energy. Thus classical geometry emerges when quantum fluctuations are integrated out. We also compare our results to the stability properties of the interacting renormalization group fixed point projected to f (R)-gravity, showing that the composite operator formalism in the single-operator approximation cannot be used to reliably determine the number of relevant parameters of the theory.

Higgs fields are attributes of classical gauge theory on a principal bundle P → X whose structure Lie group G if is reducible to a closed subgroup H. They are represented by sections of the quotient bundle P/H → X. A problem lies in description of matter fields with an exact symmetry group H. They are represented by sections of a composite bundle which is associated to an H-principal bundle P → P/H. It is essential that they admit an action of a gauge group G.Higgs fields are attributes of classical gauge theory on a principal bundle P → X if its symmetries are spontaneously broken [2,6,7]. Spontaneous symmetry breaking is a quantum phenomenon, but it is characterized by a classical background Higgs field [8]. Therefore, such a phenomenon also is considered in the framework of classical field theory when a structure Lie group G of a principal bundle P is reduced to a closed subgroup H of exact symmetries.One says that a structure Lie group G of a principal bundle P is reduced to its closed subgroup H if the following equivalent conditions hold:• a principal bundle P admits a bundle atlas with H-valued transition functions,• there exists a principal reduced subbundle of P with a structure group H.A key point is the following.Theorem 1. There is one-to-one correspondence between the reduced H-principal subbundles P h of P and the global sections h of the quotient bundle P/H → X possessing a typical fibre G/H.In classical field theory, global sections of a quotient bundle P/H → X are treated as classical Higgs fields [2,7].In general, there is topological obstruction to reduction of a structure group of a principal bundle to its subgroup. In particular, such a reduction occurs if the quotientGiven a principal bundle P → X whose structure group is reducible to a closed subgroup H, one meets a problem of description of matter fields which admit only an exact symmetry subgroup H. Here, we aim to show that they adequately are represented by sections of the composite bundle Y (5) which is associated to an H-principal bundle P → P/H. A key point is that Y also is a P -associated bundle (Theorem 3), and it admits the action (15) of a gauge group G. In the case of a pseudo-orthogonal group G = SO(1, m) and its maximal compact subgroup H = SO(m), we obtain this action in the explicit form (24).Forthcoming Part II of our work is devoted to Lagrangians of these matter fields and Higgs fields.