# Clifford Space and String/M-theory?

1. Jan 18, 2005

### pelastration

A relation between Clifford Space and String?

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 Physics http://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

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.

Last edited by a moderator: May 1, 2017
2. Jan 18, 2005

### arivero

Clifford algebra is basically a way to do differential geometry. If you devise a method (and there are some in the market) to quotient from the clifford DX.DX=1 to external dx^dx=0, you are done. Nothing to do with strings in principle. Some relation no non commutative geometry.

3. Jan 20, 2005

### pavsic

I
In fact, Clifford algebra is basically a way to do differential geometry. Basically. But if one employs not only basis vectors, but the entire Clifford algebra which is generated from the basis vectors, one obtains much more.

Clifford algebra can be used to describe a geometric structure which is much richer than the ordinary geometry of spacetime. A space (in particular, spacetime) consists of points (events). But besides points there are also lines, surfaces, volumes, etc.. Description of such geometric objects has turned out to be very elegant if one employs multivectors which are the outer products of vectors. Multivectors and their linear combinations, polyvectors, are elements of Clifford algebras. Instead of spacetime we can thus consider a more general space, the so called Clifford space.
This is a space of the oriented $(r+1)$-dimensional areas, i.e., of point, lines, areas,
volumes, etc., described by multivector coordinates x^{\mu_1 \mu_2 ... \mu_r}.

At the first step we may take Clifford space as the arena for physics. As in the ordinary relativity we consider a motion of a point particle with coordinates x^\mu (or better an extended object whose center of mass coordinates are x^\mu), so in the relativity extended to Clifford space (C-space) we consider a motion of an object whose description is given in terms of 16 multivector coordinates s,x^\mu, x^{\mu_1 \mu_2}, x^{\mu_1 \mu_2 \mu_3} x^{\mu_1 \mu_2 \mu_3 \mu_4}. Those coordinates are an extension of the center of mass coordinates. An extended object, e.g., a closed string, is description in the first approximation by its center of mass coordinates x^\mu. A better approximation is to describe a closed string not only in terms of x^\mu, but also in terms of the corresponding bivector coordinates x^{\mu \nu}. And analogously for closed higher dimensional objects, p-branes.

The next step is to consider the dynamics of the "arena" as well. We consider curved dynamical C-space. Since it has 16 dimensions, it provides a possible realization of Kaluza-Klein theory. No extra dimensions of spacetime are needed: "extra dimensions" are in C-space.

Matej Pavsic

Some references

1) Matej Pavsic, "The Landscape of Theoretical Physics: A Global View;
From Point Particles to the Brane Wordl and Beyond, in Search of a
Unifying Principles", (Kluwer Academic Publishers, Dordrecht 2001).
It discusses - among others- Clifford Algebra and its usefulness in
the description of geometry and physics.
http://www-f1.ijs.si/~pavsic/

2) Matej Pavsic, "Clifford Algebra Based Polydimensional Relativity and Relativistic Dynamics",
Talk given at IARD 2000 Conference: 2nd Biennial Meeting, Ramat Gan, Israel, 26-28 June 2000
Foundations of Physics 31 (2001) 1185-1209,
http://xxx.lanl.gov/abs/hep-th/0011216

3) C.Castro, M. Pavsic, "Higher Derivative Gravity and Torsion from the Geometry
of C-spaces, Physics Letters B 539 (2002) 133-142,
http://xxx.lanl.gov/abs/hep-th/0110079

4) Matej Pavsic, "Clifford Space as the Arena for Physics",
Presented at IARD 2002 Conference: 3rd Biennial Meeting, Washington, D.C., 24-26 Jun 2002,
Found of Physics 33 (2003) 1277-1306,
http://xxx.lanl.gov/abs/gr-qc/0211085

5) Matej Pavsic, "Kaluza-Klein Theory without Extra Dimensions: Curved Clifford Space"
http://xxx.lanl.gov/abs/hep-th/0412255

6) S.~Ansoldi, A.~Aurilia, C.~Castro and E.~Spallucci, Phys.\ Rev.\ D {\bf 64}, 026003 (2001),
http://xxx.lanl.gov/abs/hep-th/0105027.

7) A. Aurilia, S. Ansoldi and E. Spallucci, Class.\ Quant.\ Grav.\ {\bf 19} (2002) 3207
http://xxx.lanl.gov/abs/hep-th/0205028.

4. Jan 20, 2005

### pavsic

I mean:

An extended object, e.g., a closed string, is described in the first approximation by its center of mass coordinates x^\mu. A better approximation is to describe a closed string not only in terms of x^\mu, but also in terms of the corresponding bivector coordinates x^{\mu \nu}. And analogously for closed higher dimensional objects, p-branes.

The bivector coordinates describe an oriented area associated with a closed string, and can be straightforwardly calculated.

5. Jan 21, 2005

### arivero

Hello pavsic. What about Nottale's idea of getting the successive doubled algebras (complex, cuaternions, biquaternions) from the ambiguity between forward and backward derivation? Could this ambiguity benefit form a formulation with Clifford algebra?

6. Nov 13, 2008

### MTd2

Can I still trust the 2 articles, by Pavsic, mentioned in the 1st post and do research based on them?

7. Nov 13, 2008

### garrett

I find Matej's work to be unusual but reasonable.

8. Nov 14, 2008

### MTd2

Is that "unusual" bad or good? And why? Is his calculations wrong or assumptions unreasonable?

Last edited: Nov 14, 2008