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We examine relativistic diffusion through the frame and observer bundles associated with a Lorentzian manifold $(M,g)$. Our focus is on spacetimes with a non-trivial isometry group, and we detail the conditions required to find symmetric solutions of the relativistic diffusion equation. Additionally, we analyze the conservation laws associated with the presence of Killing vector fields on $(M,g)$ and their implications for the expressions of the geodesic spray and the vertical Laplacian on both the frame and the observer bundles. Finally, we present several relevant examples of symmetric spacetimes., Comment: 29 pages. Accepted for publication in Journal of Physics A: Mathematical and Theoretical

The Gotay-Nester-Hinds method is used in this paper to study the Hamiltonian formulation of the Euclidean self-dual action. This action can be used to arrive at the complex Ashtekar formulation of General Relativity or a real connection formulation for Euclidean General Relativity. The main result of the paper is a derivation of the Ashtekar formulation for Euclidean gravity without using any gauge fixing. It is interesting to compare this derivation with the one corresponding to the Holst action. In particular it is worth noting that no "tertiary" constraints appear in the case considered in the present paper., Comment: 29 pages, no figures

The purpose of this paper is to discuss a number of issues that crop up in the computation of Poisson brackets in field theories. This is specially important for the canonical approaches to quantization and, in particular, for loop quantum gravity. We illustrate the main points by working out several examples. Due attention is paid to relevant analytic issues that are unavoidable in order to properly understand how computations should be carried out. Although the functional spaces that we use throughout the paper will likely have to be modified in order to deal with specific physical theories such as general relativity, many of the points that we will raise will also be relevant in that context. The specific example of the mock holonomy-flux algebra will be considered in some detail and used to draw some conclusions regarding the loop quantum gravity formalism., Comment: 37 pages

We study the internally abelianized version of a range of gravitational theories, written in connection tetrad form, and study the possible interaction terms that can be added to them in a consistent way. We do this for 2+1 dimensional and 3+1 dimensional models. In the latter case we show that the Cartan-Palatini and Holst actions are not consistent deformations of their abelianized versions. We also show that the Husain-Kucha\v{r} and Euclidean self-dual actions are consistent deformations of their abelianized counterparts. This suggests that if the latter can be quantized, it could be possible to devise a perturbative scheme leading to the quantization of Euclidean general relativity along the lines put forward by Smolin in the early nineties., Comment: 28 pages

The Hamiltonian description of mechanical or field models defined by singular Lagrangians plays a central role in physics. A number of methods are known for this purpose, the most popular of them being the one developed by Dirac. Here, we discuss other approaches to this problem that rely on the direct use of the equations of motion (and the tangency requirements characteristic of the Gotay, Nester, Hinds method), or are formulated in the tangent bundle of the configuration space. Owing to its interesting relation with general relativity we will use a concrete example as a test bed: an extension of the Pontryagin and Husain-Kucha\v{r} actions to four dimensional manifolds with boundary., Comment: 30 pages, 2 figures