A manifestly gauge-invariant approach to quantum theories of gauge fields - Ashtekar, Abhay; Lewandowski, Jerzy; Marolf, Donald; Mourao, José Manuel; Thiemann, Thomas (1995)

Coherent state transforms for spaces of connections - Ashtekar, Abhay; Lewandowski, Jerzy; Marolf, Donald; Mourao, José Manuel; Thiemann, Thomas (Journal of Functional Analysis, 1996)
The Segal-Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie group G with its normalized Haar measure mu(H), the Hall transform is an isometric isomorphism hem L(2)(G, mu(H)) to H(G(C)) boolean AND L(2)(G(C), v), where G(C) the complexification of G, H(G(C)) the space of holomorphic functions on G(C), and v an appropriate heat-kernel measure on G(C). We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie group G by (a certain extension of) the space A/g of connections module gauge transformations. The resulting ''coherent state transform'' provides a holomorphic representation of the holonomy C* algebra of real gauge fields. This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4 dimensions. (C) 1996 Academic Press, Inc.

Diffeomorphism-invariant quantum field theories of connections in terms of webs - Lewandowski, Jerzy; Thiemann, Thomas (Classical and Quantum Gravity, 1999)
In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only a finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise-smooth finite paths and loops. In particular, we (a) characterize the spectrum of the Ashtekar-Isham configuration space, (b) introduce spin-web states, a generalization of the spin network states, (c) extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism-invariant states and finally (d) extend the 3-geometry operators and the Hamiltonian operator.

Dressed metric predictions revisited - Kaminski, Wojciech; Kolanowski, Maciej; Lewandowski, Jerzy (Classical and Quantum Gravity, 2020)
It was recently shown that the volume operator of loop quantum cosmology (LQC) and all its positive powers are ill-defined on physical states. In this paper, we investigate how it effects predictions of cosmic microwave background (CMB) power spectra obtained within dressed metric approach for which expectations values of a are the key element. We find that almost every step in the procedure is ill-defined and relies heavily upon a (seemingly premature) numerical truncation. Thus, it suggests that more care is needed in making predictions regarding pre-inflationary physics. We propose a new scheme which contains only well-defined quantities. The surprising agreement of the hitherto models with observational data, especially at low angular momenta l is explained.
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QUANTIZATION OF DIFFEOMORPHISM INVARIANT THEORIES OF CONNECTIONS WITH LOCAL DEGREES OF FREEDOM - Ashtekar, Abhay; Lewandowski, Jerzy; Marolf, Donald; Mourao, José Manuel; Thiemann, Thomas (Journal of Mathematical Physics, 1995)
Quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kuchar model. The main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections. (C) 1995 American Institute of Physics.

SU(N) quantum Yang-Mills theory in two dimensions: A complete solution - Ashtekar, Abhay; Lewandowski, Jerzy; Marolf, Donald; Mourao, José Manuel; Thiemann, Thomas (Journal of Mathematical Physics, 1997)
A closed expression of the Euclidean Wilson-loop functionals is derived for pure Yang-Mills continuum theories with gauge groups SU(N) and U(1) and spacetime topologies R-1 x R-1 and R-1 x S-1. (For the U(1) theory, we also consider the S-1 x S-1 topology.) The treatment is rigorous, manifestly gauge invariant, manifestly invariant under area preserving diffeomorphisms and handles all (piecewise analytic) loops in one stroke. Equivalence between the resulting Euclidean theory and and the Hamiltonian framework is then established. Finally, an extension of the Osterwalder-Schrader axioms for gauge theories is proposed, These axioms are satisfied in the present model. (C) 1997 American Institute of Physics.

Uniqueness of diffeomorphism invariant states on holonomy-flux algebras - Lewandowski, Jerzy; Okolow, Andrzej; Sahlmann, Hanno; Thiemann, Thomas (Communications in Mathematical Physics, 2006)
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.