We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differe... more We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus $\Omega^1$ depending on the Hamiltonian $H=p^2/2m + V(x)$ and a flat quantum connection with torsion on it such that a quantum formulation of autoparallel curves (or `geodesics') reduces to Schr\"odinger's equation. The connection is compatible with a natural quantum symplectic structure and associated generalised quantum metric. A remnant of our approach also works on any symplectic manifold where, by extending the calculus, we can encode any hamiltonian flow as `geodesics' for a certain connection with torsion which is moreover compatible with an extended symplectic structure. Thus we formulate ordinary quantum mechanics in a way that more resembles gravity rather than the more well-studied idea of formulating geometry in a more quantum manner. We then apply the same approach to the Klein Gordon equation on Minkowski space with a background electromagnet...
This paper is devoted to further results on the nontrivially associated categories $\mathcal{C}$ ... more This paper is devoted to further results on the nontrivially associated categories $\mathcal{C}$ and $\mathcal{D}$, which are constructed from a choice of coset representatives for a subgroup of a finite group. We look at the construction of integrals in the algebras $A$ and $D$ in the categories. These integrals are used to construct abstract projection operators to show that general
In this note we show that the single soliton solutions known previously in the $1+1$ dimensional ... more In this note we show that the single soliton solutions known previously in the $1+1$ dimensional affine Toda field theories from a variety of different methods \cite{H1,MM,OTUa,OTUb}, are in fact not the most general single soliton solutions. We exhibit single soliton solutions with additional small parameters which reduce to the previously known solutions when these extra parameters are set to zero. The new solution has the same mass and topological charges as the standard solution when these parameters are set to zero. However we cannot yet completely rule out the possibility that other solutions with larger values of these extra parameters are non-singular, in the cases where the number of extra parameters is greater than one, and if so their topological charges would most likely be different.
New single soliton solutions to the affine Toda field theories are constructed, exhibiting previo... more New single soliton solutions to the affine Toda field theories are constructed, exhibiting previously unobserved topological charges. This goes some of the way in filling the weights of the fundamental representations, but nevertheless holes in the representations remain. We use the group double-cross product form of the inverse scattering method, and restrict ourselves to the rank-one solutions.
Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) t... more Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We extend the cochain twist framework to connections and Riemannian structures and provide
In this paper we extend the standard differential geometric theory of Hamiltonian dynamics to non... more In this paper we extend the standard differential geometric theory of Hamiltonian dynamics to noncommutative spaces, beginning with symplectic forms. Derivations on the algebra are used instead of vector fields, and interior products and Lie derivatives with respect to derivations are discussed. Then the Poisson bracket of certain algebra elements can be defined by a choice of closed 2-form. Examples
The purpose of this paper is to put into a noncommutative context basic notions related to vector... more The purpose of this paper is to put into a noncommutative context basic notions related to vector fields from classical differential geometry. The manner of exposition is an attempt to make the material as accessible as possible to classical geometers. The definition of vector field used is a specialisation of the Cartan pair definition, and the paper relies on the
We pose the following question: If a physical experiment were to be completely controlled by an a... more We pose the following question: If a physical experiment were to be completely controlled by an algorithm, what effect would the algorithm have on the physical measurements made possible by the experiment? In a programme to study the nature of computation possible by physical systems, and by algorithms coupled with physical systems, we have begun to analyse (i) the algorithmic
... physical oracle can compute in polynomial time - the class BPP//log*. Joint work with John Tu... more ... physical oracle can compute in polynomial time - the class BPP//log*. Joint work with John Tucker and Jose Felix Costa. CS Calude et al. (Eds.): UC 2009, LNCS5715, p. 1, 2009. cO Springer-Verlag Berlin Heidelberg 2009
Causality, Meaningful Complexity and Embodied Cognition, 2009
We have developed a mathematical theory about using physical experiments as oracles to Turing mac... more We have developed a mathematical theory about using physical experiments as oracles to Turing machines. We suppose that an experiment makes measurements according to a physical theory and that the queries to the oracle allow the Turing machine to read the value being measured bit by bit. Using this theory of physical oracles, an experimenter performing an experiment can be modelled as a Turing machine governing an oracle that is the experiment. We consider this computational model of physical measurement in terms of the theory of measurement of Hempel and Carnap (see ). We note that once a physical quantity is given a real value, Hempel's axioms of measurement involve undecidabilities. To solve this problem, we introduce time into Hempel's axiomatization. Focussing on a dynamical experiment for measuring mass, as in , we show that the computational model of measurement satisfies our generalization of Hempel's axioms. Our analysis also explains undecidability in measurement and that quantities are not always measurable.
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Papers by Edwin Beggs