From Quantum Gravity to Quantum Field Theory via Noncommutative Geometry

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.Comment: 34 pages, 3 figure

On the Fermionic Sector of Quantum Holonomy Theory

In this paper we continue the development of quantum holonomy theory, which is a candidate for a fundamental theory based on gauge fields and non-commutative geometry. The theory is build around the QHD(M) algebra, which is generated by parallel transports along flows of vector fields and translation operators on an underlying configuration space of connections, and involves a semi-final spectral triple with an infinite-dimensional Bott-Dirac operator. Previously we have proven that the square of the Bott-Dirac operator gives the free Hamilton operator of a Yang-Mills theory coupled to a fermionic sector in a flat and local limit. In this paper we show that the Hilbert space representation, that forms the backbone in this construction, can be extended to include many-particle states.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1712.0593

Quantum Gravity and the Emergence of Matter

In this paper we establish the existence of the non-perturbative theory of quantum gravity known as quantum holonomy theory by showing that a Hilbert space representation of the QHD(M) algebra, which is an algebra generated by holonomy-diffeomorphisms and by translation operators on an underlying configuration space of Ashtekar connections, exist. We construct operators, which correspond to the Hamiltonian of general relativity and the Dirac Hamiltonian, and show that they give rise to their classical counterparts in a classical limit. We also find that the structure of an almost-commutative spectral triple emerge in the same limit. The Hilbert space representation, that we find, is inherently non-local, which appears to rule out spacial singularities such as the big bang and black hole singularities. Finally, the framework also permits an interpretation in terms of non-perturbative Yang-Mills theory as well as other non-perturbative quantum field theories. This paper is the first of two, where the second paper contains mathematical details and proofs.Comment: 37 pages, 2 figure

Quantum Holonomy Theory

We present quantum holonomy theory, which is a non-perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a C*-algebra that involves holonomy-diffeomorphisms on a 3-dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi-classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi-classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi-classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost-commutative algebra emerges from the holonomy-diffeomorphism algebra in the same limit.Comment: 76 pages, 6 figure

Representations of the Quantum Holonomy-Diffeomorphism Algebra

In this paper we continue the development of Quantum Holonomy Theory, which is a candidate for a fundamental theory, by constructing separable strongly continuous representations of its algebraic foundation, the quantum holonomy-diffeomorphism algebra. Since the quantum holonomy-diffeomorphism algebra encodes the canonical commutation relations of a gauge theory these representations provide a possible framework for the kinematical sector of a quantum gauge theory. Furthermore, we device a method of constructing physically interesting operators such as the Yang-Mills Hamilton operator. This establishes the existence of a general non-perturbative framework of quantum gauge theories on a curved backgrounds. Questions concerning gauge-invariance are left open.Comment: 28 page

Quantum Holonomy Theory and Hilbert Space Representations

We present a new formulation of quantum holonomy theory, which is a candidate for a non-perturbative and background independent theory of quantum gravity coupled to matter and gauge degrees of freedom. The new formulation is based on a Hilbert space representation of the QHD(M) algebra, which is generated by holonomy-diffeomorphisms on a 3-dimensional manifold and by canonical translation operators on the underlying configuration space over which the holonomy-diffeomorphisms form a non-commutative C*-algebra. A proof that the state that generates the representation exist is left for later publications.Comment: 13 page

Nonperturbative Quantum Field Theory and Noncommutative Geometry

A general framework of non-perturbative quantum field theory on a curved background is presented. A quantum field theory is in this setting characterised by an embedding of a space of field configurations into a Hilbert space over $\mathbb{R}^\infty$. This embedding, which is only local up to a scale that we interpret as the Planck scale, coincides in the local and flat limit with the plane wave expansion known from canonical quantisation. We identify a universal Bott-Dirac operator acting in the Hilbert space over $\mathbb{R}^\infty$ and show that it gives rise to the free Hamiltonian both in the case of a scalar field theory and in the case of a Yang-Mills theory. These theories come with a canonical fermionic sector for which the Bott-Dirac operator also provides the Hamiltonian. We prove that these quantum field theories exist non-perturbatively for an interacting real scalar theory and for a general Yang-Mills theory, both with or without the fermionic sectors, and show that the free theories are given by semi-finite spectral triples over the respective configuration spaces. Finally, we propose a class of quantum field theories whose interactions are generated by inner fluctuations of the Bott-Dirac operator.Comment: 36 page

C*-algebras of Holonomy-Diffeomorphisms & Quantum Gravity II

We introduce the holonomy-diffeomorphism algebra, a C*-algebra generated by flows of vectorfields and the compactly supported smooth functions on a manifold. We show that the separable representations of the holonomy-diffeomorphism algebra are given by measurable connections, and that the unitary equivalence of the representations corresponds to measured gauge equivalence of the measurable connections. We compare the setup to Loop Quantum Gravity and show that the generalized connections found there are not contained in the spectrum of the holonomy-diffeomorphism algebra in dimensions higher than one. This is the second paper of two, where the prequel gives an exposition of a framework of quantum gravity based on the holonomy-diffeomorphism algebra.Comment: 18 pages, reference added, minor correctio

The Seiberg-Witten Map on the Fuzzy Sphere

We construct covariant coordinate transformations on the fuzzy sphere and utilize these to construct a covariant map from a gauge theory on the fuzzy sphere to a gauge theory on the ordinary sphere. We show that this construction coincides with the Seiberg-Witten map on the Moyal plane in the appropriate limit. The analysis takes place in the algebra and is independent of any star-product representation.Comment: 30 pages, references adde

Emergent Dirac Hamiltonians in Quantum Gravity

We modify the construction of the spectral triple over an algebra of holonomy loops by introducing additional parameters in form of families of matrices. These matrices generalize the already constructed Euler-Dirac type operator over a space of connections. We show that these families of matrices can naturally be interpreted as parameterizing foliations of 4-manifolds. The corresponding Euler-Dirac type operators then induce Dirac Hamiltonians associated to the corresponding foliation, in the previously constructed semi-classical states.Comment: one figur