Proposal for a New Quantum Theory of Gravity III: Equations for Quantum Gravity, and the Origin of Spontaneous Localisation

We recall a classical theory of torsion gravity with an asymmetric metric, sourced by a Nambu–Goto + Kalb–Ramond string [R. T. Hammond, Rep. Prog. Phys. 65, 599 (2002)]. We explain why this is a significant gravitational theory and in what sense classical general relativity is an approximation to it. We propose that a noncommutative generalisation of this theory (in the sense of Connes’ noncommutative geometry and Adler’s trace dynamics) is a “quantum theory of gravity.” The theory is in fact a classical matrix dynamics with only two fundamental constants – the square of the Planck length and the speed of light, along with the two string tensions as parameters. The guiding symmetry principle is that the theory should be covariant under general coordinate transformations of noncommuting coordinates. The action for this noncommutative torsion gravity can be elegantly expressed as an invariant area integral and represents an atom of space–time–matter. The statistical thermodynamics of a large number of such atoms yields the laws of quantum gravity and quantum field theory, at thermodynamic equilibrium. Spontaneous localisation caused by large fluctuations away from equilibrium is responsible for the emergence of classical space–time and the field equations of classical general relativity. The resolution of the quantum measurement problem by spontaneous collapse is an inevitable consequence of this process. Quantum theory and general relativity are both seen as emergent phenomena, resulting from coarse graining of the underlying noncommutative geometry. We explain the profound role played by entanglement in this theory: entanglement describes interaction between the atoms of space–time–matter, and indeed entanglement appears to be more fundamental than quantum theory or space–time. We also comment on possible implications for black hole entropy and evaporation and for cosmology. We list the intermediate mathematical analysis that remains to be done to complete this programme.

In the first article of this series, we have introduced the concept of an atom of space-time-matter (STM), which is described by the spectral action of noncommutative geometry, corresponding to a classical theory of gravity. In the present work, we use the Connes time parameter, along with the spectral action, to incorporate gravity into trace dynamics. We then derive the spectral equation of motion for the STM atom, which turns out to be the Dirac equation on a noncommutative space.

Quantum Fields on Noncommutative Spacetimes

The Karolyhazy uncertainty relation states that if a device is used to measure a length l, there will be a minimum uncertainty \(\delta l\) in the measurement, given by \((\delta l)^3 \sim L_{\mathrm {P}}^2 l\). This is a consequence of combining the principles of quantum mechanics and general relativity. In this letter we show how this relation arises in our approach to quantum gravity, in a bottom-up fashion, from the matrix dynamics of atoms of space–time–matter. We use this relation to define a space–time–matter (STM) foam at the Planck scale, and to argue that our theory is holographic. By coarse graining over time-scales larger than Planck time, one obtains the laws of quantum gravity. Quantum gravity is not a Planck scale phenomenon; rather it comes into play whenever classical space–time background is not available to describe a quantum system. Space–time and classical general relativity arise from spontaneous localisation in a highly entangled quantum gravitational system. The Karolyhazy relation continues to hold in the emergent theory. An experimental confirmation of this relation will constitute a definitive test of the quantum nature of gravity.

There ought to exist a reformulation of quantum theory which does not depend on classical time. To achieve such a reformulation, we introduce the concept of an atom of space-time-matter (STM). An STM atom is a classical noncommutative geometry (NCG), based on an asymmetric metric, and sourced by a closed string. Different such atoms interact via entanglement. The statistical thermodynamics of a large number of such atoms gives rise, at equilibrium, to a theory of quantum gravity. Far from equilibrium, where statistical fluctuations are large, the emergent theory reduces to classical general relativity. In this theory, classical black holes are far from equilibrium low entropy states, and their Hawking evaporation represents an attempt to return to the [maximum entropy] equilibrium quantum gravitational state.

Noncommutative geometric spaces with boundary: Spectral action

The sixth International Conference on Gravitation & Cosmology (ICGC-2007) was organized at IUCAA, Pune, 17–21 December 2007. This series of international meetings, held every four years under the auspices of the Indian Association for General Relativity and Gravitation (IAGRG), has now spanned two decades. Previous ICGC meetings were held at Cochin University of Science and Technology (2004), Indian Institute of Technology, Kharagpur (2000), IUCAA, Pune (1995), Physical Research Laboratory, Ahmedabad (1991) & Goa (1987). These meetings have broad international participation and feature leading experts in the field of Cosmology, gravitational waves and quantum gravity.

We consider aspects of the noncommutative approach to the standard model based on the spectral action principle. We show that as a consequence of the incorporation of the Clifford structures in the formalism, the spectral action contains an extended scalar sector, with respect to the minimal standard model. This may have interesting phenomenological consequences. Some of these new scalar fields carry both weak isospin and color indices. We calculate the new terms in spectral action due to the presence of these fields. Our analysis demonstrates that the fermionic doubling in the noncommutative geometry is not just a presence of spurious degrees of freedom, but it is an interesting and peculiar property of the formalism, which leads to physically valuable conclusions. Some of the new fields do not contribute to the physical fermionic action, but they appear in the bosonic spectral action. Their contributions to the Dirac operator correspond to couplings with the spurious fermions, which are projected out.

Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as action functional for the theory. To demonstrate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic expansion in the case of the round three sphere. We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of the product of the three sphere by a circle where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a4 and a6 both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function.

The principal object in noncommutative geometry is the spectral triple consisting of an algebra , a Hilbert space and a Dirac operator . Field theories are incorporated in this approach by the spectral action principle, which sets the field theory action to , where f is a real function such that the trace exists and Λ is a cutoff scale. In the low-energy (weak-field) limit, the spectral action reproduces reasonably well the known physics including the standard model. However, not much is known about the spectral action beyond the low-energy approximation. In this paper, after an extensive introduction to spectral triples and spectral actions, we study various expansions of the spectral actions (exemplified by the heat kernel). We derive the convergence criteria. For a commutative spectral triple, we compute the heat kernel on the torus up to the second order in gauge connection and consider limiting cases.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.

Noncommutative geometry in its many incarnations appears at the crossroad of many researches in theoretical and mathematical physics: from models of quantum spacetime(with or without breaking of Lorentz symmetry) to loop gravity and string theory, from early considerations on UV-divergenciesin quantum field theory to recent models of gauge theories on noncommutatives pacetime, from Connes description of the standard model of elementary particles to recent Pati-Salam like extensions. We list several of these applications, emphasizing also the original point of view brought by noncommutative geometry on the nature of time. This text serves as an introduction to the volume of proceedings of the parallel session “Noncommutative geometry and quantum gravity”, as a part of the conference “Conceptual and technical challenges in quantum gravity” organized at the University of Rome La Sapienza sin September 2014.

The Shortest Vector Problem (SVP) is a cornerstone of lattice-based cryptography, underpinning the security of numerous cryptographic schemes like NTRU. Given its NP-hardness, efficient solutions to SVP have profound implications for both cryptography and computational complexity theory. This paper presents an innovative framework that integrates concepts from quantum gravity, non-commutative geometry, spectral theory, and post-supersymmetry (post-SUSY) particle physics to address SVP. By mapping high-dimensional lattice points to spinfoam networks and by means of Hamiltonian engineering, it is theoretically possible to devise new algorithms that leverage the interactions topologically protected Majorana fermionparticles have with the gravitational field through the spectral action principle to loop through these spinfoam networks where SVP vectors could then be encoded onto the spectrum of the corresponding Dirac-like dilation operators within the system. We establish a novel approach that leverages post-SUSY physics and theories of quantum gravity to achieve algorithmic speedups beyond those expected by conventional quantum computers. This interdisciplinary methodology not only proposes potential polynomial-time algorithms for SVP, but also bridges gaps between theoretical physics and cryptographic applications, providing further insights into the Riemann Hypothesis (RH) and the Hilbert-P ´olya Conjecture. Possible directions for experimental realization through biologically inspired hardware or biological tissues by orchestrated objective reduction (Orch-Or) theory are discussed.

In this talk, I present a theory of quantum gravity beyond Einstein. The theory is established based on spinnic and scaling gauge symmetries by treating the gravitational force on the same footing as the electroweak and strong forces. A bi-frame space-time is initiated to describe the laws of nature. One frame space-time is a globally flat coordinate Minkowski space-time that acts as an inertial reference frame for the motions of fields, the other is a locally flat non-coordinate Gravifield space-time that functions as an interaction representation frame for the degrees of freedom of fields. The Gravifield is sided on both the globally flat coordinate space-time and locally flat non-coordinate space-time and characterizes the gravitational force. Instead of the principle of general coordinate invariance in Einstein theory of general relativity, some underlying principles with the postulates of coordinate independence and gauge invariance are motivated to establish the theory of quantum gravity. When transmuting the Gravifield basis into the coordinate basis in Minkowski space-time, it enables us to obtain equations of motion for all quantum fields and derive basic conservation laws for all symmetries. The gravity equation is found to be governed by the total energy–momentum tensor defined in the flat Minkowski space-time. When the spinnic and scaling gauge symmetries are broken down to a background structure that possesses the global Lorentz and scaling symmetries, we arrive at a Lorentz invariant and conformally flat background Gravifield space-time that is characterized by a cosmic vector with a non-zero cosmological mass scale. We also obtain the massless graviton and massive spinnon. The resulting universe is in general not isotropic in terms of conformal proper time and turns out to be inflationary in light of cosmic proper time. The conformal size of the universe has a singular at the cosmological horizon to which the cosmic proper time must be infinitely large. We show a mechanism for quantum inflation caused by the quantum loop contributions. The Gravifield behaves as a Goldstone-like field that transmutes the local spinnic gauge symmetry into the global Lorentz symmetry, which makes the spinnic gauge field becomes a hidden gauge field. As a consequence, the bosonic gravitational interactions can be described by the Goldstone-like Gravimetric field and space-time gauge field. The Einstein theory of general relativity is expected to be an effective low energy theory. Two types of gravity equation are resulted, one is the extension to Einstein’s equation of general relativity, and the other is a new type of gravitational equation that characterizes the spinnon dynamics.

Many of the most exciting open problems in high-energy physics are related to the behavior and ultimate nature of gravity and spacetime. In this dissertation, several categories of new results in quantum and classical gravity are presented, with applications to our understanding of both quantum field theory and cosmology. A fundamental open question in quantum field theory is related to ultraviolet completion: Which low-energy effective field theories can be consistently combined with quantum gravity? A celebrated example of the swampland program---the investigation of this question---is the weak gravity conjecture, which mandates, for a U(1) gauge field coupled consistently to gravity, the existence of a state with charge-to-mass ratio greater than unity. In this thesis, we demonstrate the tension between the weak gravity conjecture and the naturalness principle in quantum field theory, generalize the weak gravity conjecture to multiple gauge fields, and exhibit a model in which the weak gravity conjecture solves the standard model hierarchy problem. Next, we demonstrate that gravitational effective field theories can be constrained by infrared physics principles alone, namely, analyticity, unitarity, and causality. In particular, we derive bounds related to the weak gravity conjecture by placing such infrared constraints on higher-dimension operators in a photon-graviton effective theory. We furthermore place bounds on higher-curvature corrections to the Einstein equations, first using analyticity of graviton scattering amplitudes and later using unitarity of an arbitrary tree-level completion, as well as constrain the couplings in models of massive gravity. Completing our treatment of perturbative quantum gravity, outside of the swampland program, we also reformulate graviton perturbation theory itself, finding a field redefinition and gauge-fixing of the Einstein-Hilbert action that drastically simplifies the Feynman diagram expansion. Furthermore, our reformulation also exhibits a hidden symmetry of general relativity that corresponds to the double copy relations equating gravity amplitudes to sums of squares of gluon amplitudes in Yang-Mills theory, a surprising correspondence that yields insights into the structure of quantum field theories. Moving beyond perturbation theory into nonperturbative questions in quantum gravity, we consider the deep relation between spacetime geometry and properties of the quantum state. In the context of holography and the anti-de Sitter/conformal field theory correspondence, we test the proposed ER=EPR correspondence equating quantum entanglement with wormholes in spacetime. In particular, we demonstrate that the no-cloning theorem in quantum mechanics and the no-go theorem for topology change of spacetime are dual under the ER=EPR correspondence. Furthermore, we prove that the presence of a wormhole is not an observable in quantum gravity, rescuing ER=EPR from potential violation of linearity of quantum mechanics. Excitingly, we also prove a new area theorem within classical general relativity for arbitrary dynamics of two collections of wormholes and black holes; this area theorem is the ER=EPR analogue of entanglement conservation. We next turn our attention to the emergence of spacetime itself, placing consistency conditions on the proposed correspondence between anti-de Sitter space and the Multiscale Entanglement Renormalization Ansatz, a special tensor network that constitutes a computational tool for finding the ground state of certain quantum systems. Further examining the role of quantum entanglement entropy in the emergence of general relativity, we ask whether there is a consistent microscopic formulation of the entropy in theories of entropic gravity; we find that our results weaken equation-of-state proposals for entropic gravity while strengthening those more akin to holography, guiding future investigation of theories of emergent gravity. Finally, we examine the consequences of the Hamiltonian constraint in classical gravity for the early universe. The Hamiltonian constraint allows for the Liouville measure on the phase space of cosmological parameters for homogeneous, isotropic universes to be converted into a probability distribution on trajectories, or equivalently, on initial conditions. However, this measure diverges on the set of spacetimes that are spatially flat, like the observable universe. In this thesis, we derive the unique, classical, Hamiltonian-conserved measure for the subset of flat universes. This result allows for distinction between different models of cosmic inflation with similar observable predictions; for example, we find that the measure favors models of large-scale inflation, as such potentials more naturally produce the number of e-folds necessary to match cosmological observations.