On Representations of the Quantum Holonomy Diffeomorphism Algebra

Abstract

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 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 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.

On nonperturbative quantum field theory and noncommutative geometry

“…We need to show that the operators M f exist on H scalar . This question is similar to the case considered in [1]. We begin by showing…”

“…Proof. In [1] we prove that (31) and (32) give rise to a strongly continuous Hilbert space representation of the QHD(M ) algebra in the special case where s i = 1 for all i ∈ N. This proof can be straight forwardly adopted to the case where {s i } i∈N is a monotonously increasing sequence and we leave it to the reader to check this.…”

“…It follows from the definition of the integrals, that the numerical value of the limit of the integrals is bounded by f ∞ . Like in [1] it follows…”

“…In this paper we present both a general method of constructing nonperturbative quantum field theories as well as two concrete examples. First we consider the case of a real scalar field and second we revisit the case of a gauge field, which was first developed in [1,2]. In both cases we find that the operator expansion in terms of weighted eigenvectors of a Laplace operator coincide in a local and flat limit with the plane wave expansion of a canonically quantised field.…”

“…The idea to construct a Dirac-type operator over a space of field configurations and to let it interact with an algebra of field operators was first proposed in [9] and later developed in [10]- [16]. The HD(M ) algebra was first introduced in [4,8] and further studied in [1,2,5,17,18,19]. After completing the papers [1,2] we became aware of the work by Higson and Kasparov [3], where a similar Hilbert space construction was used and where the infinite-dimensional Bott-Dirac operator was first introduced -the operator, which is the central object in the present paper.…”