Covariant Quantum Gravity Research Articles

Planck Length Emerging as the Invariant Quantum Minimum Effective Length Determined by the Heisenberg Uncertainty Principle in Manifestly Covariant Quantum Gravity Theory

The meaning of the quantum minimum effective length that should distinguish the quantum nature of a gravitational field is investigated in the context of manifestly covariant quantum gravity theory (CQG-theory). In such a framework, the possible occurrence of a non-vanishing minimum length requires one to identify it necessarily with a 4-scalar proper length s.It is shown that the latter must be treated in a statistical way and associated with a lower bound in the error measurement of distance, namely to be identified with a standard deviation. In this reference, the existence of a minimum length is proven based on a canonical form of Heisenberg inequality that is peculiar to CQG-theory in predicting massive quantum gravitons with finite path-length trajectories. As a notable outcome, it is found that, apart from a numerical factor of O1, the invariant minimum length is realized by the Planck length, which, therefore, arises as a constitutive element of quantum gravity phenomenology. This theoretical result permits one to establish the intrinsic minimum-length character of CQG-theory, which emerges consistently with manifest covariance as one of its foundational properties and is rooted both on the mathematical structure of canonical Hamiltonian quantization, as well as on the logic underlying the Heisenberg uncertainty principle.

On the Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity

The theoretical prediction of the stochastic property of the quantum cosmological constant and the quantum stochastic nature of event horizons has crucial implications on the physics of space-time and black holes in particular. One of these consequences concerns a new mechanism, which is investigated here, for the stochastic regularization of singular black-hole solutions of classical general relativity. The problem is posed in the context of the theory of covariant quantum gravity (CQG-theory), namely the manifestly covariant, constraint-free and finite graviton-mass quantum Hamiltonian approach developed by Cremaschini and Tessarotto (2015–2022), which permits to cast the theory in a frame-independent setting. It is precisely the trajectory-dependence feature of the theory and the intrinsic stochastic property of quantum gravity which turn out to be crucial properties for reaching quantum regularization of classical singular solutions.

Background Independence and Gauge Invariance in General Relativity Part 2—Covariant Quantum Gravity

Background independence is often being claimed as the characteristic property of several current and past models of Quantum Gravity. In actual fact, such a notion has a wider connotation and must be rooted into the validity of the general covariance principle, demanding its logical connection with the notions of manifest covariance and (quantum) gauge invariance. In fact, as we intend to show here, it involves (a) the existence of a well-defined, albeit arbitrary, classical background space-time; and (b) the suitable realization of a dynamical equation for the related background metric field tensor, referred to as quantum-modified Einstein tensor field equation, which actually determines it in a suitable functional setting. Remarkably, it is proved that in the context of the theory of Covariant Quantum Gravity (CQG-theory), recently developed by Cremaschini and Tessarotto (2015–2022), background independence implies that such an equation “emerges” rigorously from the same CQG-theory. This follows in terms of a stochastic quantum expectation value evaluated with respect to the corresponding characteristic quantum PDE. It is shown that an analogous emergence property applies also to the background metric field tensor in terms of stochastic fluctuations of the corresponding underlying quantum tensor of gravitational field. These results warrant the consistent validity of background independence for the prescription of the space-time metric tensor in CQG-theory.

Physical Properties of Schwarzschild-deSitter Event Horizon Induced by Stochastic Quantum Gravity.

A new type of quantum correction to the structure of classical black holes is investigated. This concerns the physics of event horizons induced by the occurrence of stochastic quantum gravitational fields. The theoretical framework is provided by the theory of manifestly covariant quantum gravity and the related prediction of an exclusively quantum-produced stochastic cosmological constant. The specific example case of the Schwarzschild–deSitter geometry is looked at, analyzing the consequent stochastic modifications of the Einstein field equations. It is proved that, in such a setting, the black hole event horizon no longer identifies a classical (i.e., deterministic) two-dimensional surface. On the contrary, it acquires a quantum stochastic character, giving rise to a frame-dependent transition region of radial width between internal and external subdomains. It is found that: (a) the radial size of the stochastic region depends parametrically on the central mass M of the black hole, scaling as ; (b) for supermassive black holes is typically orders of magnitude larger than the Planck length . Instead, for typical stellar-mass black holes, may drop well below . The outcome provides new insight into the quantum properties of black holes, with implications for the physics of quantum tunneling phenomena expected to arise across stochastic event horizons.

The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity.

An excruciating issue that arises in mathematical, theoretical and astro-physics concerns the possibility of regularizing classical singular black hole solutions of general relativity by means of quantum theory. The problem is posed here in the context of a manifestly covariant approach to quantum gravity. Provided a non-vanishing quantum cosmological constant is present, here it is proved how a regular background space-time metric tensor can be obtained starting from a singular one. This is obtained by constructing suitable scale-transformed and conformal solutions for the metric tensor in which the conformal scale form factor is determined uniquely by the quantum Hamilton equations underlying the quantum gravitational field dynamics.

Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model

We discuss a (3+1)-dimensional covariant quantum space-time describing a FLRW cosmology with Big Bounce, obtained by a projection of the fuzzy hyperboloid Hn4. This provides a background solution of the IKKT matrix model with mass term. We characterize the bosonic fluctuation spectrum, which consists of a tower of higher-spin modes, truncated at n. The modes are organized in terms of an underlying SO(4, 2) structure group, which is broken to the SO(3, 1) isometry of the background. The resulting higher-spin gauge theory includes all degrees of freedom required for gravity, and should be well suited for quantization. All modes propagate with the same speed of light, even though local boost invariance is not manifest. The propagating metric perturbation modes comprise those of a massless graviton, as well as a scalar mode. Gauge invariance allows to obtain the analog of the linearized Einstein-Hilbert action, which is expected to be induced upon quantization.

Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity

The axiomatic geometric structure which lays at the basis of Covariant Classical and Quantum Gravity Theory is investigated. This refers specifically to fundamental aspects of the manifestly-covariant Hamiltonian representation of General Relativity which has recently been developed in the framework of a synchronous deDonder–Weyl variational formulation (2015–2019). In such a setting, the canonical variables defining the canonical state acquire different tensorial orders, with the momentum conjugate to the field variable g μ ν being realized by the third-order 4-tensor Π μ ν α . It is shown that this generates a corresponding Hamilton–Jacobi theory in which the Hamilton principal function is a 4-tensor S α . However, in order to express the Hamilton equations as evolution equations and apply standard quantization methods, the canonical variables must have the same tensorial dimension. This can be achieved by projection of the canonical momentum field along prescribed tensorial directions associated with geodesic trajectories defined with respect to the background space-time for either classical test particles or raylights. It is proved that this permits to recover a Hamilton principal function in the appropriate form of 4-scalar type. The corresponding Hamilton–Jacobi wave theory is studied and implications for the manifestly-covariant quantum gravity theory are discussed. This concerns in particular the possibility of achieving at quantum level physical solutions describing massive or massless quanta of the gravitational field.

$\operatorname{SL}(2,\mathbb{C})$ Chern-Simons theory, flat connections, and four-dimensional quantum geometry

A correspondence between three-dimensional flat connections and constant curvature four-dimensional simplices is used to give a novel quantization of geometry via complex SL(2,C) Chern-Simons theory. The resulting quantum geometrical states are hence represented by the 3d blocks of analytically continued Chern-Simons theory. In the semiclassical limit of this quantization the three-dimensional Chern-Simons action, remarkably, becomes the discrete Einstein-Hilbert action of a 4-simplex, featuring the appropriate boundary terms as well as the essential cosmological term proportional to the simplex's curved 4-volume. Both signs of the curvature and associated cosmological constant are present in the class of flat connections that give rise to this correspondence. We provide a Wilson graph operator that picks out this class of connections. We discuss how to promote these results to a model of Lorentzian covariant quantum gravity encompassing both signs of the cosmological constant. This paper presents the details for the results reported in reference [1] (arXiv:1509.00458).

Space-Time Second-Quantization Effects and the Quantum Origin of Cosmological Constant in Covariant Quantum Gravity

Space-time quantum contributions to the classical Einstein equations of General Relativity are determined. The theoretical background is provided by the non-perturbative theory of manifestly-covariant quantum gravity and the trajectory-based representation of the related quantum wave equation in terms of the Generalized Lagrangian path formalism. To reach the target an extended functional setting is introduced, permitting the treatment of a non-stationary background metric tensor allowed to depend on both space-time coordinates and a suitably-defined invariant proper-time parameter. Based on the Hamiltonian representation of the corresponding quantum hydrodynamic equations occurring in such a context, the quantum-modified Einstein field equations are obtained. As an application, the quantum origin of the cosmological constant is investigated. This is shown to be ascribed to the non-linear Bohm quantum interaction of the gravitational field with itself in vacuum and to depend generally also on the realization of the quantum probability density for the quantum gravitational field tensor. The emerging physical picture predicts a generally non-stationary quantum cosmological constant which originates from fluctuations (i.e., gradients) of vacuum quantum gravitational energy density and is consistent with the existence of quantum massive gravitons.

What is gravity?

We review the historical pursuits and recent developments on the origin of gravity. We start with the Newtonian gravity and Einstein gravity to introduce the two major leaps of the historical pursuits on the origin of gravity. Then we introduce the recent developments on the origin of gravity from the point of view of modified gravity, quantum gravity and holographic gravity. For Newtonian gravity, we start with Keplers law on planetary motion to introduce the Newtons universal law of gravitation, and recent developments on the modified Newtonian dynamics and dark matter; for Einstein gravity, we explain the geometrization of gravity, and introduce the application of Einstein gravity on cosmology and gravitational waves; for modified gravity, we introduce it from the point of view of extra gravitational degrees of freedom, higher derivative gravity and higher dimensional gravity; for quantum gravity, we introduce it from the point of view of covariant quantum gravity, canonical quantum gravity and other scenarios of quantum gravity; for holographic gravity, we introduce its holographic picture, emergent properties and its connection with quantum information. However, the answer to the question of gravity origin still remains unknown. We point out here that three possible roads to approach the nature of gravity can be taken in future. The first one is the coming observations of gravitational waves from both ground-borne and space-borne gravitational-wave detectors. A system involving either black hole or neutron star has certainly run into the regime of strong gravity, which is quite different from previous tests for general relativity that is in the regime of weak gravity. The gravitational-wave observation will most likely provide us with clues on classical gravity with any possible deviations if Einstein gravity does modify itself at classical level. The second one is the understanding of the cosmic acceleration at both early-time and late-time. The cosmic acceleration at early-time is supposed to be driven by cosmic inflation, and the cosmic acceleration at late-time is supposed to be driven by the cosmological constant or alternatively dark energy. Future precise observations of radiation fluctuations and matter fluctuations will show us the way approach the nature of quantum gravity. The third one is the cognition of the relation of quantum gravity and quantum information. The past fifty years or so of pursuit on quantum gravity has leaded us to the path on the conjecture that quantum gravity might be emerged from quantum information. Although the perspective of this idea is not clear, it might be our best shot for now.

More on Weinberg’s no-go theorem in quantum gravity

We complement Weinberg's no-go theorem on the cosmological constant problem in quantum gravity by generalizing it to the case of a scale-invariant theory. Our analysis makes use of the effective action and the BRST symmetry in a manifestly covariant quantum gravity instead of the classical Lagrangian density and the $GL(4)$ symmetry in classical gravity. In this sense, our proof is very general since it does not depend on details of quantum gravity and holds true for general gravitational theories which are invariant under diffeomorphisms. As an application of our theorem, we comment on an idea that in the asymptotic safety scenario the functional renormalization flow drives a cosmological constant to zero, solving the cosmological constant problem without reference to fine tuning of parameters. Finally, we also comment on the possibility of extending the Weinberg theorem in quantum gravity to the case where the translational invariance is spontaneously broken.

Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity

Key aspects of the manifestly-covariant theory of quantum gravity (Cremaschini and Tessarotto 2015–2017) are investigated. These refer, first, to the establishment of the four-scalar, manifestly-covariant evolution quantum wave equation, denoted as covariant quantum gravity (CQG) wave equation, which advances the quantum state ψ associated with a prescribed background space-time. In this paper, the CQG-wave equation is proved to follow at once by means of a Hamilton–Jacobi quantization of the classical variational tensor field g ≡ g μ ν and its conjugate momentum, referred to as (canonical) g-quantization. The same equation is also shown to be variational and to follow from a synchronous variational principle identified here with the quantum Hamilton variational principle. The corresponding quantum hydrodynamic equations are then obtained upon introducing the Madelung representation for ψ , which provides an equivalent statistical interpretation of the CQG-wave equation. Finally, the quantum state ψ is proven to fulfill generalized Heisenberg inequalities, relating the statistical measurement errors of quantum observables. These are shown to be represented in terms of the standard deviations of the metric tensor g ≡ g μ ν and its quantum conjugate momentum operator.

Hamiltonian approach to GR \u2013 Part 2: covariant theory of quantum gravity

A non-perturbative quantum field theory of General Relativity is presented which leads to a new realization of the theory of covariant quantum gravity (CQG-theory). The treatment is founded on the recently identified Hamiltonian structure associated with the classical space-time, i.e., the corresponding manifestly covariant Hamilton equations and the related Hamilton–Jacobi theory. The quantum Hamiltonian operator and the CQG-wave equation for the corresponding CQG-state and wave function are realized in 4-scalar form. The new quantum wave equation is shown to be equivalent to a set of quantum hydrodynamic equations which warrant the consistency with the classical GR Hamilton–Jacobi equation in the semiclassical limit. A perturbative approximation scheme is developed, which permits the adoption of the harmonic oscillator approximation for the treatment of the Hamiltonian potential. As an application of the theory, the stationary vacuum CQG-wave equation is studied, yielding a stationary equation for the CQG-state in terms of the 4-scalar invariant-energy eigenvalue associated with the corresponding approximate quantum Hamiltonian operator. The conditions for the existence of a discrete invariant-energy spectrum are pointed out. This yields a possible estimate for the graviton mass together with a new interpretation about the quantum origin of the cosmological constant.

Ground state of the universe in quantum cosmology

We find a physical state of a closed universe with the minimal excitation of the universe expansion energy in quantum gravity. It is an analog of the vacuum state of the ordinary quantum field theory in the Minkowsky space, but in our approach an energy of space of a closed universe together with the energy of its matter content are minimized. This ground state is chosen among an enlarged set of physical states, compared with the ordinary covariant quantum gravity. In our approach, physical states are determined by weak constraints: quantum mechanical averages of gravitational constraint operators equal zero. As a result, they appear to be non-static in such a modification of quantum gravity. Quantum dynamics of the universe is described by Schrödinger equation with a cosmic time determined by weak gravitational constraints. In order to obtain the observed megascopic universe with the inflation stage just after its quantum beginning, a lot of the energy in the form of the inflaton scalar field condensate is prescribed to the initial state. Parameters of the initial state for a homogeneous model of the universe are calculated.

The volume operator in covariant quantum gravity

A covariant spin-foam formulation of quantum gravity has recently been developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In particular, the geometrical observable giving the area of a surface has been shown to be the same as the one in loop quantum gravity. Here we discuss the volume observable. We derive the volume operator in the covariant theory and show that it matches the one of loop quantum gravity, as does the area. We also reconsider the implementation of the constraints that define the model: we derive in a simple way the boundary Hilbert space of the theory from a suitable form of the classical constraints and show directly that all constraints vanish weakly on this space.

Wheeler-DeWitt equation and Feynman diagrams

We present a systematic expansion of all constraint equations in canonical quantum gravity up to the order of the inverse Planck mass squared. It is demonstrated that this method generates the Feynman diagrammatic technique involving graviton loops and vertices. It also reveals explicitly the back-reaction effects of quantized matter and graviton vacuum polarization. This provides an explicit correspondence between the frameworks of canonical and covariant quantum gravity in the semiclassical limit.

On the general covariance and strong equivalence principles in quantum general relativity

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport in classical general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincare as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity.