43 research outputs found
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 . 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
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