Quantum gravity coupled to matter via noncommutative geometry

2010

The structure of the Dirac Hamiltonian in 3+1 dimensions is shown to emerge in a semi-classical approximation from a abstract spectral triple construction. The spectral triple is constructed over an algebra of holonomy loops, corresponding to a configuration space of connections, and encodes information of the kinematics of General Relativity. The emergence of the Dirac Hamiltonian follows from the observation that the algebra of loops comes with a dependency on a choice of base-point. The elimination of this dependency entails spinor fields and, in the semi-classical approximation, the structure of the Dirac Hamiltonian.

Communications in Mathematical Physics, 2011

We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom.

Symmetry, Integrability and Geometry: Methods and Applications, 2012

We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a natural class of semiclassical states which entail emerging fermionic degrees of freedom. In the particular semiclassical approximation where all gravitational degrees of freedom are turned off, a free fermionic quantum field theory emerges. We end the paper with an extended outlook section.

Classical and Quantum Gravity, 2014

A link between canonical quantum gravity and fermionic quantum field theory is established in this paper. From a spectral triple construction which encodes the kinematics of quantum gravity semi-classical states are constructed which, in a semi-classical limit, give a system of interacting fermions in an ambient gravitational field. The interaction involves flux tubes of the gravitational field. In the additional limit where all gravitational degrees of freedom are turned off, a free fermionic quantum field theory emerges. *

Classical and Quantum Gravity, 2009

This paper establishes a link between Noncommutative Geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac type operator which resembles a global functional derivation operator. The interaction between the Dirac operator and the algebra reproduces the Poisson structure of General Relativity. Moreover, the associated Hilbert space corresponds, up to a discrete symmetry group, to the Hilbert space of diffeomorphism invariant states known from Loop Quantum Gravity. Correspondingly, the square of the Dirac operator has, in terms of canonical quantum gravity, the form of a global area-squared operator. Furthermore, the spectral action resembles a partition function of Quantum Gravity. The construction is background independent and is based on an inductive system of triangulations. This paper is the first of two papers on the subject. 1

J.Korean Phys.Soc. 65 (2014) 1754-1798

We review a novel and authentic way to quantize gravity. This novel approach is based on the fact that Einstein gravity can be formulated in terms of symplectic geometry rather than Riemannian geometry in the context of emergent gravity. An essential step for emergent gravity is to realize the equivalence principle, the most important property in the theory of gravity (general relativity), from U(1) gauge theory on a symplectic or Poisson manifold. Through the realization of the equivalence principle which is an intrinsic property in symplectic geometry known as the Darboux theorem or the Moser lemma, one can understand how diffeomorphism symmetry arises from noncommutative U(1) gauge theory and so gravity can emerge from the noncommutative electromagnetism, which is also an interacting theory. As a consequence, it is feasible to formulate a background independent quatum gravity where the prior existence of any spacetime structure is not a priori assumed but defined by fundamental ingredients in quantum gravity theory. This scheme for quantum gravity resolves many notorious problems in theoretical physics, for example, to resolve the cosmological constant problem, to understand the nature of dark energy and to explain why gravity is so weak compared to other forces. In particular, it leads to a remarkable picture for what matter is. A matter field such as leptons and quarks simply arises as a stable localized geometry, which is a topological object in the defining algebra (noncommutative ⋆-algebra) of quantum gravity.

2011

These notes summarise a talk surveying the combinatorial or Hamiltonian quantisation of three dimensional gravity in the Chern-Simons formulation, with an emphasis on the role of quantum groups and on the way the various physical constants (c,G,\Lambda,\hbar) enter as deformation parameters. The classical situation is summarised, where solutions can be characterised in terms of model spacetimes (which depend on c and \Lambda), together with global identifications via elements of the corresponding isometry groups. The quantum theory may be viewed as a deformation of this picture, with quantum groups replacing the local isometry groups, and non-commutative spacetimes replacing the classical model spacetimes. This point of view is explained, and open issues are sketched.

Quantum Gravity Mathematical Models and Experimental Bounds, 2007

This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that arises naturally as the classical limit; a theory with nonsymmetric metric and a skew version of metric compatibilty. Meanwhile, in quantum gravity a key ingredient of our approach is the proposal that the differential structure of spacetime is something that itself must be summed over or 'quantised' as a physical degree of freedom. We illustrate such a scheme for quantum gravity on small finite sets.

International Journal of Modern Physics A, 2007

Gravity is proposed. Alain Connes' Noncommutative Geometry provides a framework in which the Standard Model of particle physics coupled to general relativity is formulated as a unified, gravitational theory. However, to this day no quantization procedure compatible with this framework is known. In this paper we consider the noncommutative algebra of holonomy loops on a functional space of certain spin-connections. The construction of a spectral triple is outlined and ideas on interpretation and classical limit are presented.

2005

Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a nonlinearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.