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.

Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism-invariant context - Ashtekar, Abhay; Marolf, Donald; Mourao, José Manuel; Thiemann, Thomas (Classical and Quantum Gravity, 2000)
Osterwalder and Schrader introduced a procedure to obtain a (Lorentzian) Hamiltonian quantum theory starting from a measure on the space of (Euclidean) histories of a scalar quantum field. In this paper, we extend that construction to more general theories which do not refer to any background, spacetime metric (and in which the space of histories does not admit a natural linear structure). Examples include certain gauge theories, topological field theories and relativistic gravitational theories. The treatment is self-contained in the sense that an a priori knowledge of the Osterwalder-Schrader theorem is not assumed.

Physical properties of quantum field theory measures - Mourao, José Manuel; Thiemann, Thomas; Velhinho, Jose (Journal of Mathematical Physics, 1999)
Well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory. The difference of typical configurations of free massive scalar field theories with different masses is studied. We apply the same methods to study the Ashtekar-Lewandowski (AL) measure on spaces of connections. In particular we prove that the diffeomorphism group acts ergodically, with respect to the AL measure, on the Ashtekar-Isham space of quantum connections modulo gauge transformations. We also prove that a typical, with respect to the AL measure, quantum connection restricted to a (piecewise analytic) curve leads to a parallel transport discontinuous at every point of the curve. (C) 1999 American Institute of Physics. [S0022-2488(99)00404-1].

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.

The status of diffeomorphism superselection in euclidean 2+1 gravity - Marolf, Donald; Mourao, José Manuel; Thiemann, Thomas (Journal of Mathematical Physics, 1997)
This work addresses a specific technical question of relevance to canonical quantization of gravity using the so-called new variables and loop-based techniques of Ashtekar, Rovelli, and Smolin. In particular, certain ''superselection laws'' that arise in current applications of these techniques to solving the diffeomorphism constraint are considered, Their status is elucidated by studying an analogous system: 2 + 1 Euclidean gravity, For that system, these superselection laws are shown to be spurious. This, however, is only a technical difficulty. The usual quantum theory may still be obtained from a loop representation and the technique known as ''Refined Algebraic Quantization.'' (C) 1997 American Institute of Physics.