This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper."The principal argument for the adoption of geometric algebra is that it provides a single, simple mathematical framework which eliminates the plethora of diverse mathematical descriptions and techniques it would otherwise be necessary to learn." [15]

In a recent letter, 1 Pitowsky has given a model of electron spin in which "Every electron at each given moment has a definite spin in all directions", but which, he claims, does not imply Bell's inequality. A non-Kolmogorov probability theory in the model prevents the usual proofs of Bell's inequality from going through. I give here a very simple proof of a Bell-type inequality from the quoted statement. The inequality shows that the statement is inconsistent with quantum mechanics.Consider N pairs of electrons in the singlet state. One member of each pair moves to the left and the other to the right. Let N (A + : C + ) be the number of pairs in which the left member has spin up in the A direction and the right member has spin up in the C direction. Let N (A + C − :) be the number in which the left member has spin up in the A direction and spin down in the C direction. According to the quoted statement, these are meaningful quantities. ThenQuantum mechanics predicts that if N (A + : C + ) is measured, then, where θ AC is the angle between A and C. According to the quoted statement N (A + : C + ) exists independently of whether it is measured or not and so the approximation holds whether it is measured or not. The above inequality is inconsistent with the approximation for θ AB = θ BC = 60 • and θ AC = 120 • 1 I. Pitowsky, Phys. Rev. Lett. 48, 1299Lett. 48, (1982.

We give a simple, elementary, direct, and motivated construction of the geometric algebra over R n .Mathematics Subject Classification 2000. Primary 15A66.

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.The central, and revolutionary, postulate of special relativity is that the speed of light between two points in an inertial frame is a universal constant. The speed is defined in terms of synchronized clocks at the points. (See below.) But the Hafele-Keating experiment [1] and muon decay experiments which measure time dilation [2] show that a universal time does not exist, and so the notion of separated synchronized clocks can have no a priori meaning. It follows that the speed of light can have no meaning until a definition of synchronized clocks is given. It is not simply that the speed cannot be measured; it can have no meaning.The purpose of this paper is not to enter into the debate surrounding the above argument, for I believe that its logic is sound. Instead, the purpose is to accept the conclusion of the argument and provide a proper foundation for the universal light speed postulate by giving a simple account of the physical principles involved in clock synchronization.Some of the ideas developed here could profitably be used by teachers of relativity. I know of no special relativity text which discusses clock synchronization before the speed of light.

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.