The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.
There is a consistent and simple interpretation of the quantum theory of isolated systems. The interpretation suffers no measurement problem and provides a quantum explanation of state reduction, which is usually postulated. Quantum entanglement plays an essential role in the construction of the interpretation.
Entanglement has been called the most important new feature of the quantum world. It is expressed in the quantum formalism by the joint measurement formula. We prove the formula for projection valued observables from a plausible assumption, which for spacelike separated measurements is an expression of relativistic causality. The state reduction formula is simply a way to express the joint measurement formula after one measurement has been made, and its result known.
This textbook for the second year undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering most of the usual linear algebra topics. -/- Geometric algebra and its extension to geometric calculus simplify, unify, and generalize vast areas of mathematics that involve geometric ideas. Geometric algebra is an extension of linear algebra. The treatment of many linear algebra topics is enhanced by geometric algebra, for example, determinants and orthogonal transformations. And geometric algebra does much (...) more, as it incorporates the complex, quaternion, and exterior algebras, among others. (shrink)
This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus. It is a sequel to my Linear and Geometric Algebra. That text is a prerequisite for this one. -/- Linear algebra and vector calculus have provided the basic vocabulary of mathematics in dimensions greater than one for the past one hundred years. Just as geometric algebra generalizes linear algebra in powerful ways, geometric calculus generalizes vector calculus in powerful ways.
This paper gives two complete and elementary proofs that if the speed of light over closed paths has a universal value c, then it is possible to synchronize clocks in such a way that the one-way speed of light is c. The ﬁrst proof is an elementary version of a recent proof. The second provides high precision experimental evidence that it is possible to synchronize clocks in such a way that the one-way speed of light has a universal value. We (...) also discuss an old incomplete proof by Weyl which is important from an historical perspective. (shrink)
This paper is about representations for Artificial Intelligence systems. All of the results described in it involve engineering the representation to make AI systems more effective. The main AI techniques studied here are varieties of search: path-finding in graphs, and probablilistic searching via simulated annealing and genetic algorithms. The main results are empirical findings about the granularity of representation in implementations of genetic algorithms. We conclude by proposing a new algorithm, called “Long-Term Evolution,” which is a genetic algorithm running on (...) an evolving problem description. We see this as modelling the evolution of a species from simpler (more coarsely described— fewer genes) types of organisms to more complex ones. The results, which are reported here of our experiments with the algorithm make it seem a promising optimisation technique. (shrink)
In general relativity, a spacetime and a gravitational field form an indivisible unit: no field, no spacetime. This is a lesson of Einstein's hole argument. We use a simple transformation in a Schwartzschild pacetime to illustrate this.
This paper (i) gives necessary and sufficient conditions that clocks in an inertial lattice can be synchronized, (ii) shows that these conditions do not imply a universal light speed, and (iii) shows that the terrestrial redshift experiment provides evidence that clocks in a small inertial lattice in a gravitational field can be synchronized.
This paper gives two complete and elementary proofs that if the speed of light over closed paths has a universal value c, then it is possible to synchronize clocks in such a way that the one-way speed of light is c. The first proof is an elementary version of a recent proof. The second provides high precision experimental evidence that it is possible to synchronize clocks in such a way that the one-way speed of light has a universal value. We (...) also discuss an old incomplete proof by Weyl which is important from an historical perspective. (shrink)