Geometry and Physics (Part 2)

Clifford Space: An Extension of Spacetime

An extended object, O, can be be sampled by a finite set of parameters, for instance, by the center of mass coordinates, and by the orientation of its axes of symmetries. Higher multipole deformations, such as the dipole and the quadrupole ones, can also be taken into account. For practical reasons, only a finite number of multipoles can be taken into account. Instead of the infinite number of degrees of freedom, we consider only a finite number of degrees of freedom. We thus perform a mapping from an infinite dimensional configuration space, associated with the object O, to a finite dimensional subspace.

Extended objects of particular interest for theoretical physics are strings and branes. They can be described by coordinate functions X^\mu (\xi^a), \mu=0,1,2,...,N-1, a=0,1,2,...,n-1, where n\le N. Such a description is infinite dimensional. In refs. [1] it was pointed out how one can employ a finite description in terms of a quenched mini superspace.

The idea has been further developed [2]–[6] by means of Clifford algebras, a very useful tool for description of geometry [7].



FiniteDescr3Click on pictures to enlarge them.

Here we are interested in description of spacetime, M_N, and the objects embedded in M_N.Therefore, let us start by considering the squared line element in M_N:

(1)        Q = d s^2 = g_{\mu \nu} d x^\mu d x^\nu,  \quad \quad \mu,\nu=0,1,2,...N-1.

If we take the square root, \sqrt{Q}, we have the following possibilities:

 i) \sqrt{Q} = \sqrt{g_{\mu \nu} {\rm d} x^\mu d x^\nu}\;\;\; scalar

ii)  \sqrt{Q} = \gamma_\mu d x^\mu\;\;\;   vector

Here \gamma_\mu are generators of the Clifford algebra Cl(p,q),
p+q=N,  satisfying

(2)        \gamma_\mu \cdot \gamma_\nu \equiv \frac{1}{2}(\gamma_\mu \gamma_\nu +  \gamma_\nu \gamma_\mu ) = g_{\mu \nu} ,

where g_{\mu \nu} is the metric of M_N.

The generators \gamma_\mu have the role of basis vectors of the spacetime M_N. The symmetric product \gamma_\mu \cdot \gamma_\nu represents the inner product. The antysymmetric (wedge) product of two basis vectors gives a unit bivector:

 (3)        \gamma_\mu \wedge \gamma_\nu \equiv \frac{1}{2}(\gamma_\mu \gamma_\nu -  \gamma_\nu \gamma_\mu )

and has thus the role of outer product. In analogous way we obtain 3-vectors, 4-vectors, etc..

We assume that the signature of an $N$-dimensional spacetime is (1,N-1), i.e., (+ - - - ...). In the case of the 4-dimensional spacetime we thus have the signature (1,3), i.e., (+ - - -). The corresponding Clifford algebra is Cl(1,3).

The basis of Cl(1,N-1) is

(4)        \lbrace 1, \gamma_\mu, \gamma_{\mu_1} \wedge \gamma_{\mu_2},...,  \gamma_{\mu_1} \wedge \gamma_{\mu_2} \wedge ... \wedge\gamma_{\mu_N}\rbrace

A generic element, X \in Cl(1,N-1), is a superposition

(5)        X=\sum_{r=0}^N \frac{1}{r!} X^{\mu_1 \mu_2 ...\mu_r}  \gamma_{\mu_1} \wedge \gamma_{\mu_2}\wedge ... \wedge\gamma_{\mu_r} \equiv X^M \gamma_M

called a Clifford aggregate or polyvector.

In refs. [5,6,8] it has been demonstrated that r-vectors X^{\mu_1 \mu_2 ...\mu_r} can be associated with closed instantonic (r-1)-branes or open instantonic r-branes. A generic polyvector, X=X^M \gamma_M, can be associated with a conglomerate of (instantonic) r-branes for various values of r=0,1,2,...,N.

Our objects are instantonic r-branes, which means that they are localized in spacetime*. They generalize  the concept of `event’, a spacetime point, x^\mu,~\mu=0,1,2,3. Instead of an event, we have now an extended event, {\cal E},  described by coordinates X^{\mu_1 \mu_2 ...\mu_r}, r=0,1,2,3,4. The space of extended events is called Clifford space, C. It is a manifold whose tangent space at any of its points is a Clifford algebra Cl(1,3). If C is a flat space, then it is isomorphic to the Clifford algebra Cl(1,3) with elements

(6)        X=\sum_{r=0}^4  \frac{1}{4!} X^{\mu_1 \mu_2 ...\mu_r}  \gamma_{\mu_1 \mu_2 ...\mu_r}\equiv X^M \gamma_M.

In flat C-space, the basis vectors are equal to the wedge product

(7)        \gamma_M = \gamma_{\mu_1} \wedge \gamma_{\mu_2}\wedge ... \wedge \gamma_{\mu_r}

at every point {\cal E} \in C. This not true in curved C-space: if  we (parallelly) transport a polyvector A=A^M \gamma_Mfrom a point {\cal E} \in C along a closed path back to the original point, {\cal E}, then the orientation of the polyvector A after such transport will not coincide with the initial orientation of A. After the transport along a closed path we will obtain a new polyvector A'=A'^M \gamma_M. If,in particular, the initial polyvector is one of the Clifford algebra basis elements, A=\gamma_M, i.e., an object with definite grade, then the final polyvector will be A'=A'^M \gamma_M, which is an object with mixed grade. A consequence is that in curved Clifford space C, basis vectors cannot have definite grade at all points of C.

The situation in a curved Clifford space, C,  is analogous to that in a usual curved space, where after the (parallel) transport along a closed path, a vector changes its orientation. In Clifford space, a change of orientation in general implies a change of a polyvector’s grade, so that, e.g., a definite grade polyvector changes into a mixed grade polyvector.

However, if we impose a condition that, under parallel transport, the grade of a polyvector does not change, then one has a very special kind of curved Clifford space [9]. In such a space, afer a parallel transport along a closed path, the vector part \langle A \rangle_1 = a^\mu \gamma_\mu changes into \langle A' \rangle_1 =a'^\mu \gamma_\mu, the bivector part \langle A \rangle_2 =a^{\mu \nu} \gamma_\mu\wedge \gamma_\nu changes into \langle A' \rangle_2 =a'^{\mu \nu} \gamma_\mu \wedge \gamma_\nu, etc., but one grade does not change into another grade. Such special Clifford space, in which the consequences of curvature manifest themselves within each of the subspaces with definite grade separately, but not between those subspaces, is very complicated. We will not consider such special Clifford spaces, because they are analogous to the usual curved spaces of the product form M= M_1 \times M_2 \times...M_n, where M_i \subset M is a curved lower dimensional subspace of M, and where only those (parallel) transports are allowed that bring tangent vectors of M_i into another tangent vectors of the same subspace M_i.

The squared line element in Clifford space, C, is

(8)        {\rm d} S^2 = G_{M N} {\rm d} x^M x^N = {\rm d} X^\dagger {\rm d} X = \langle {\rm d} X^\ddagger {\rm d} X \rangle_0 .

Here {\rm d} X = {\rm d} x^M \gamma_M, and {\rm d} X^\ddagger = {\rm d} x^M \gamma_M^\ddagger, where \ddagger denotes the operation of inversion: (\gamma_{\mu_1}\gamma_{\mu_2} ... \gamma_{\mu_r})^\ddagger = \gamma_{\mu_r}\gamma_{\mu_{r-1}} ...\gamma_{\mu_r}. The metric of  C is

(9)        G_{MN} = \gamma_M^\ddagger * \gamma_N = \langle \gamma^\ddagger \gamma_N \rangle_0 ,

where \langle ~~\rangle_0 means the scalar part. A Clifford space with such a metric has signature [6] (8,8), i.e., (++++++++--------). This is ultrahyperbolic space with neutral signature.

In the paper “Quantum Field Theories in Spaces with Neutral Signatures”[]  it is shown that, contrary to the wide spread belief, the physics in spaces with signature (n,n) makes sense.

*The usual p-branes are localized in space, but they are infinitely extended into a time-like direction, so that they are (p+1)-dimensional worldsheets in spacetime.}


[1] Ansoldi S, Aurilia A, Castro C and Spallucci E 2001 Phys. Rev. D 64 026003 [arXiv:hep-th/0105027]; Aurilia A, Ansoldi S and Spallucci E 2002 Class.   Quant.    Grav. 19 3207  [arXiv:hep-th/0205028].
[2] Castro C 1999 Chaos, Solitons and Fractals 10 295 Chaos, Solitons and Fractals   12 (2001) 1585; Castro C and Pavšič M 2002; Phys. Lett. B 539 133 [arXiv:hep-th/0110079]; Castro C and Pavšič M 2005 Prog. Phys. 1 31
[3] Pavšič 2001 The Landscape of Theoretical Physics: A Global View; From Point       Particles to the Brane World and Beyond, in Search of a Unifying Principle (Dordrecht:   Kluwer)
[4] Pavšič M 2001 Found. Phys. 31 1185 [arXiv: hep-th/0011216].
[5] Pavšič M 2003 Found. Phys. 33 1277 [arXiv: gr-qc/0211085].
[6] Pavšič M 2007 Found. Phys. 37 1197 [arXiv: hep-th/0605126].
[7] Hestenes D 1966 Space-Time Algebra (New York:Gordon and Breach)
Hestenes D and Sobcyk G 1984 Cliff ord Algebra to Geometric Calculus (Dordrecht:
[8] Pavšič M 2012 Localized Propagating Tachyons in Extended Relativity Theories, arXiv: 1201.5755 [hep-th].
[9] Castro C 2012 Int. J. Theor. Phys. DOI 10.1007/s10773-012-1295-3
Castro C 2012 Adv. Appl. Cli ord Algebras DOI 10.1007/s00006-012-0370-4

About Matej Pavšič

I am theoretical physicist at Jožef Stefan Institiute
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1 Response to Geometry and Physics (Part 2)


    What is the large and small scale structure of space-time – it’s very geometry?


    Neither, large or small scale structure, have been adequately answered by the various models.


    For over a thousand years debate on Euclid’s 5th postulate, the parallel postulate, that holds in Euclidian Geometry was the world that most thought they lived in.


    We don’t live in ‘flatland’ – and few see beyond.


    The few that did, including Lobachevsky, Poincaré, Riemann imagined affine, projective, hyperbolic and spherical geometry, where Euclid’s 5th postulate does not hold.


    In a watershed, Eugenio Beltrami in 1868 proved the 5th postulate is independent from others.


    Gregorio ‘Ricci’-Curbastro and his protégé, Tullio Levi-Civita, invented and fostered Tensor Analysis, that has enabled the depiction of large scale structure of space-time.


    General relativity (GR), our present model for the large scale structure, is dependent on ‘parallel transport’, metric, and the covariant derivative – as defined in Tensor (Ricci) Analysis and Riemannian Geometry.


    Tensor (Ricci) Analysis while eminently useful for GR, as the equivalent of the ‘hammer’, it fails as a complete tool to define many of the contemporary models of small scale structure.


    So many tools have been taken out of the ‘algebraic toolshed’, that Physics is ‘troubled’.


    Quantum Physics had rediscovered heavy artillery in the algebra of Élie Cartan, Hilbert and Marius Sophus Lie.


    Geometric Algebra (GA) intiated by Hermann Grassmann and William Kingdon Clifford was left on the byways, as Tensor (Ricci) Analysis replaced GA around the turn of the century.


    For those in Physics today, GA resurgence is due to David Hestenes, who has demonstrated William Kingdon Clifford’s original geometric interpretation.


    Fig. 1

    David Hestenes – ‘Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics’:



    Let’s get back to geometry – perhaps Geometric Algebra, as a tool, and Intuitionism, as a Philosophy.

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