In this article the holonomy-flux *-algebra, which has been introduced by Lewandowski, Okolow, Sahlmann and Thiemann, is modificated. The new *-algebra is called the holonomy-flux cross-product *-algebra. This algebra is an abstract cross-product *-algebra. It is given by the universal algebra of the algebra of continuous and differentiable functions on the configuration space of generalised connections and the universal enveloping flux algebra associated to a surface set, and some canonical commutator relations. There is a uniqueness result for a certain path- and graph-diffeomorphism invariant state of the holonomy-flux cross-product *-algebra. This new *-algebra is not the only *-algebra, which is generated by the algebra of certain continuous and differentiable functions on the configuration space of generalised connections and the universal enveloping flux algebra associated to a surface set. The theory of abstract cross-product algebras allows to define different new *-algebras. Some of these algebras are presented in this article.
In the project AQV the issue of quantum constraints, KMS-states and algebras of quantum configuration and momentum variables in Loop Quantum Gravity has been argued. There a physical algebra has been required to contain complete observables and the quantum constraints, or at least the quantum constraints are affilliated with this algebra. In this context a first conjecture for a physical algebra is presented in this article. A new *-algebra for LGQ, which is called the localised holonomy-flux cross-product *-algebra, is studied. A suggestion for a physical *-algebra, which contains the localis…
In this article a new C*-algebra derived from the basic quantum variables: holonomies along paths and group-valued quantum flux operators in the framework of Loop Quantum Gravity is constructed. This development is based on the theory of cross-products and C*-dynamical systems. The author has presented a set of actions of the flux group associated to a surface set on the analytic holonomy C*-algebra, which define C*-dynamical systems. These objects are used to define the holonomy-flux cross-product C*-algebra associated to a surface set. Furthermore surface-preserving path- and graph-diffeomor…
The philosophy of the Loop Quantum Gravity approach is to construct the canonical variables by using the duality of infinitesimal connections and holonomies along loops. Based on this fundamental property for example the holonomy-flux *-algebra has been formulated. A generalisation of the one-to-one correspondence between infinitesimal objects: connections and curvature and path based objects: holonomy maps and parallel transports is used to replace the configuration space of the theory. This generalised duality is related to the concept of path connections and holonomy groupoids, which origin…
In this article a new formulation of the Weyl C*-algebra, which has been invented by Fleischhack, in terms of C*-dynamical systems is presented. The quantum configuration variables are given by the holonomies along paths in a graph. Functions depending on these quantum variables form the analytic holonomy C*-algebra. Each classical flux variable is quantised as an element of a flux group associated to a certain surface set and a graph. The quantised spatial diffeomorphisms are elements of the group of bisections of a finite graph system. Then different actions of the flux group associated to s…
Quantum holonomy theory is a candidate for a non-perturbative theory of quantum gravity coupled to fermions. The theory is based on the QHD(M) algebra, which essentially encodes how matter degrees of freedom are moved on a three-dimensional manifold. In this paper we commence the development of a lattice-independent formulation. We first introduce a flow-dependent version of the QHD(M) algebra and formulate necessary conditions for a state to exist hereon. We then use the GNS construction to build a kinematical Hilbert space. Finally we find that operators, that correspond to the Dirac and gra…
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 publ…
A dual holonomy version of operator spin foam models is presented, which is particularly adapted to the notion of coarse graining. We discuss how this leads to a natural way of comparing models on different discretization scales, and a notion of renormalization group flow on the partially ordered set of 2-complexes.
Recent work on canonical transformations in quantum mechanics is applied to transform between the Moncrief metric formulation and the Witten-Carlip holonomy formulation of 2+1-dimensional quantum gravity on the torus. A non-polynomial factor ordering of the classical canonical transformation between the metric and holonomy variables is constructed which preserves their classical modular transformation properties. An extension of the definition of a unitary transformation is briefly discussed and is used to find the inner product in the holonomy variables which makes the canonical transformatio…
This paper investigates the structure of gravitational singularities at the level of the connection. We show in particular that for FLRW spacetimes with particle horizons a local holonomy, which is related to a gravitational energy, becomes unbounded near the big-bang singularity. This implies the $C^{0,1}_{\mathrm{loc}}$-inextendibility of such FLRW spacetimes. Again using an unbounded local holonomy we also give a general theorem establishing the $C^{0,1}_{\mathrm{loc}}$-inextendibility of spherically symmetric weak null singularities which arise at the Cauchy horizon in the interior of blac…
Parametrization of holonomy-flux phase space in the Hamiltonian formulation of $SO(N)$ gauge field theory with $SO(D+1)$ loop quantum gravity as an exemplification
We use vielbein bundle's horizontal lift path integral formulation and gauge theory's holonomy map to compactly describe parallel transport and geodesic equations on a manifold. This is first applied to the geometry of general relativistic Schwarzschild as a review. The Newton-Cartan take of the Schwarzschild metric derived by previous literature is then adopted, and the analysis is repeated for both non-torsional and torsional geometry. Transport curves considered include a circular timeless loop, circular geodesic loop, radial geodesic loop, and a stationary loop. The findings on the three g…
We propose a new holonomy formulation for spin foams, which naturally extends the theory space of lattice gauge theories. This allows current spin foam models to be defined on arbitrary two-complexes as well as to generalize current spin foam models to arbitrary, in particular finite groups. The similarity with standard lattice gauge theories allows to apply standard coarse graining methods, which for finite groups can now be easily considered numerically. We will summarize other holonomy and spin network formulations of spin foams and group field theories and explain how the different represe…
Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms. While this result is particularly imp…
In this and the companion paper a novel holonomy formulation of so called Spin Foam models of lattice gauge gravity are explored. After giving a natural basis for the space of simplicity constraints we define a universal boundary Hilbert space, on which the imposition of different forms of the simplicity constraints can be studied. We detail under which conditions this Hilbert space can be mapped to a Hilbert space of projected spin networks or an ordinary spin network space. These considerations allow to derive the general form of the transfer operators which generates discrete time evolution…
Using self dual Ashtekar variables, we investigate (at the effective level) the spherically symmetry reduced model of loop quantum gravity, both in vacuum and when coupled to a scalar field. Within the real Ashtekar-Barbero formulation, the system scalar field coupled to spherically symmetric gravity is known to possess a non closed (quantum) algebra of constraints once the holonomy corrections are introduced, which forbids the loop quantization of the model. Moreover, the vacuum case, while not anomalous, introduces modifications which are usually interpreted as an effective signature change …
Inspired by the low wave-length limit of topological M-theory, which re-constructs the theory of $3+1$D gravity in the self-dual variables' formulation, and by the realization that in Loop Quantum Gravity the holonomy of a flat connection can be non-trivial if and only if a non-trivial (space-like) line defect is localized inside the loop, we argue that non-trivial gravitational holonomies can be put in correspondence with space-like M-branes. This suggests the existence of a new duality, which we call $H$ duality, interconnecting topological M-theory with Loop Quantum Gravity. We spell some a…
We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gra…
We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is peaked on flat connections, and states are built upon it by creating local curvature excitations. The inner product induces a discrete topology on the gauge group, which turns out to be an essential ingredient for the construction of a continuum limit Hilbert space. This leads to a representation of the full holonomy-flux algebra of loop quantum gravity which is unitarily-inequivalent to the one ba…
The central object of this paper is a holonomy formulation for spin foams. Within this new represen- tation, we analyze three general requirements: locality, composition law, cylindrical consistency. In particular, cylindrical consistency is shown to fix the arbitrary normalization of the vertex amplitude in the case of Euclidean signature.
This article describes the regularization of the generally relativistic gauge field representation of gravity on a piecewise linear lattice. It is a part of the program concerning the classical relativistic theory of fundamental interactions, represented by minimally coupled gauge vector field densities and half-densities. The correspondence between the local Darboux coordinates on phase space and the local structure of the links of the lattice, embedded in the spatial manifold, is demonstrated. Thus, the canonical coordinates are replaceable by links-related quantities. This idea and the sign…
