Andrzej Okolow

Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

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 important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.

Diffeomorphism covariant representations of the holonomy-flux ⋆-algebra

Recently, Sahlmann [1] proposed a new, algebraic point of view on the loop quantization. He brought up the issue of a ⋆-algebra underlying that framework, studied the algebra consisting of the fluxes and holonomies and characterized its representations. We define the diffeo-morphism covariance of a representation of the Sahlmann algebra and study the diffeomorphism covariant representations. We prove they are all given by Sahlmanns decomposition into the cyclic representations of the sub-algebra of the holonomies by using a single state only. The state corresponds to the natural measure defined on the space of the generalized connections. This result is a generalization of Sahlmanns result [2] concerning the U (1) case.

Construction of spaces of kinematic quantum states for field theories via projective techniques

We present a method of constructing a space of quantum states for a field theory: given phase space of a theory, we define a family of physical systems each possessing a finite number of degrees of freedom, next we define a space of quantum states for each finite system, finally using projective techniques we organize all these spaces into a space of quantum states which corresponds to the original phase space. This construction is kinematic in this sense that it bases merely on the structure of the phase space and does not take into account possible constraints on the space. The construction is a generalization of a construction by Kijowski - the latter one is limited to theories of linear phase spaces, while the former one is free of this limitation.

Kinematic quantum states for the Teleparallel Equivalent of General Relativity

A space of kinematic quantum states for the Teleparallel Equivalent of General Relativity is constructed by means of projective techniques. The states are kinematic in this sense that their construction bases merely on the structure of the phase space of the theory and does not take into account constraints on it. The space of quantum states is meant to serve as an element of a canonical background independent quantization of the theory.

Quantization of Diffeomorphism Invariant Theories of Connections with a Non-Compact Structure Group—an Example

A simple diffeomorphism invariant theory of connections with the non-compact structure group

ADM-like Hamiltonian formulation of gravity in the teleparallel geometry

{\mathbb {R}}

New diffeomorphism invariant states on a holonomy-flux algebra

of real numbers is quantized. The theory is defined on a four-dimensional ‘space-time’ by an action resembling closely the self-dual Plebański action for general relativity. The space of quantum states is constructed by means of projective techniques by Kijowski [1]. Except for this point the applied quantization procedure is based on Loop Quantum Gravity methods.

Quantum group connections

We present the second (and final) part of an analysis aimed at introducing variables which are suitable for constructing a space of quantum states for the Teleparallel Equivalent of General Relativity. In the first part of the analysis we introduced a family of variables on the “position” sector of the phase space. In this paper we distinguish differentiable variables in the family. Then we define momenta conjugate to the differentiable variables and express constraints of the theory in terms of the variables and the momenta. Finally, we exclude variables which generate an obstacle for further steps of the Dirac’s procedure of canonical quantization of constrained systems we are going to apply to the theory. As a result we obtain two collections of variables on the phase space which will be used (in a subsequent paper) to construct the desired space of quantum states.