Neighbours of Einstein's Equations: Connections and Curvatures

Once the action for Einstein's equations is rewritten as a functional of an SO(3,C) connection and a conformal factor of the metric, it admits a family of ``neighbours'' having the same number of degrees of freedom and a precisely defined metric tensor. This paper analyzes the relation between the Riemann tensor of that metric and the curvature tensor of the SO(3) connection. The relation is in general very complicated. The Einstein case is distinguished by the fact that two natural SO(3) metrics on the GL(3) fibers coincide. In the general case the theory is bimetric on the fibers.Comment: 16 pages, LaTe

MUBs, polytopes, and finite geometries

A complete set of N+1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = p^k, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = p^k. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.Comment: 15 pages; talk at the Vaxjo conference on probability and physic

Geometrical Statistics--Classical and Quantum

This is a review of the ideas behind the Fisher--Rao metric on classical probability distributions, and how they generalize to metrics on density matrices. As is well known, the unique Fisher--Rao metric then becomes a large family of monotone metrics. Finally I focus on the Bures--Uhlmann metric, and discuss a recent result that connects the geometric operator mean to a geodesic billiard on the set of density matrices.Comment: Talk at the third Vaxjo conference on Quantum Theory: Reconsideration of foundation

How much complementarity?

Bohr placed complementary bases at the mathematical centre point of his view of quantum mechanics. On the technical side then my question translates into that of classifying complex Hadamard matrices. Recent work (with Barros e Sa) shows that the answer depends heavily on the prime number decomposition of the Hilbert space. By implication so does the geometry of quantum state space.Comment: 6 pages; talk at the Vaxjo conference on Foundations of Probability and Physics, 201

Client newsletters within Clinical Legal Education and their value to the student participants

The employment law client newsletter project (the Project) runs during each academic year within the Student Law Office (SLO) at Northumbria University. Under the supervision of their clinical supervisor the students research and design a newsletter for distribution to HR professionals employed by an external organisation. The students participate in the Project alongside their live client work. The aim of the Project is to enrich the students’ clinical experience and develop their skills whilst at the same time update and educate the client recipient. Through a pilot study the value of participating in the Project is explored. The findings of the study suggest that the students develop their professional skills from a different perspective, increase their employment law knowledge, gain the commercial awareness of the importance of a well drafted newsletter in practice, and really value the experience