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William Kingdon Clifford: British philosopher and mathematician who, influenced by the non-Euclidean geometries of Bernhard Riemann and Nikolay Lobachevsky, wrote “On the Space-Theory of Matter” (1876). He presented the idea that matter and energy are simply different types of curvature of space, thus foreshadowing Albert Einstein's GR. http://www.britannica.com/eb/article?tocId=9024381 [Broken] . Also: http://en.wikipedia.org/wiki/William_Kingdon_Clifford

It was William Kingdon Clifford who was responsible for the first translation into English of Riemann’s 1854 paper on the new non-Euclidean geometries. He was much influenced by that work and in 1870 his address to the Cambridge Philosophical Society included these points Riemann has shown that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space we live in belongs...

"I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact,

1. That small portions of space are in fact analogous to little hills on a surface which is on average flat; namely that the ordinary laws of geometry are not valid in them.

2. That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.

3. That this variation of the curvature of space is what really happens in that phenomenon that we call the motion of matter, whether ponderable or ethereal.

4. That in the physical world nothing else takes place but this variation, subject (possibly) to the laws of continuity."

Did Clifford actually anticipate Einstein as some people have suggested? We shall never know. Historians of mathematics and science disagree about this, but Clifford’s

powers of imagination about space were undoubtedly great. In Volume One of Lectures and Essays (edited by FR. Pollock and L. Stephen and published by Macmillan in 1879) in one of several references, Clifford ‘anticipates’ Einstein’s curvature of physical space in these words

"I am supposed to know that the three angles of a rectilinear triangle are exactly two right angles. Now suppose that three points are taken in space, distant from one another as far as the Sun is from Alpha Centauri, and the shortest distances between these points are drawn so as to form a triangle... Then I do not know that this sum would differ at all from two right angles; but also I do not know that the difference would be less than ten degrees."

Also in Volume One (pp. 237-238) he writes

"Now, whatever may turn out to be the ultimate nature of the ether and of molecules, we know that to some extent at least they obey the same dynamic laws, and that they act upon one another in accordance with these laws. Until, therefore, it is absolutely disproved, it must remain the simplest and most probable assumption that they are finally made of the same stuff – that the material molecule is some kind of knot or coagulation of ether."

Clifford Space as a Generalization of Spacetime: Prospects for Unification in Physicshttp://lanl.arxiv.org/abs/hep-th/0411053

Author: Matej Pavsic

Comments: 12 pages; Talk presented at {\it 4th Vigier Symposium: The Search For Unity in Physics}, September 15th--19th, 2003,

The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold ($C$-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. We assume that $C$-space is the true space in which physics takes place and that physical quantities are Clifford algebra valued objects, namely, superpositions of multivectors, called Clifford aggregates or polyvectors. We explore some very promising features of physics in Clifford space, in particular those related to a consistent construction of string theory and quantum field theory.

Kaluza-Klein Theory without Extra Dimensions: Curved Clifford Space

http://xxx.lanl.gov/abs/hep-th/0412255

Authors: M. Pavsic

Comments: 15 pages; References added, typos corrected

A theory in which 16-dimensional curved Clifford space (C-space) provides realization of Kaluza-Klein theory is investigated. No extra dimensions of spacetime are needed: "extra dimensions" are in C-space. It is shown that the covariant Dirac equation in C-space contains Yang-Mills fields of the U(1)xSU(2)xSU(3) group as parts of the generalized spin connection of the C-space.

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# Clifford Space and String/M-theory?

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