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 , , , where . Such a description is infinite dimensional. In refs.  it was pointed out how one can employ a finite description in terms of a quenched mini superspace.
The idea has been further developed – by means of Clifford algebras, a very useful tool for description of geometry .
Here we are interested in description of spacetime, , and the objects embedded in .Therefore, let us start by considering the squared line element in :
If we take the square root, , we have the following possibilities:
Here are generators of the Clifford algebra ,
where is the metric of .
The generators have the role of basis vectors of the spacetime . The symmetric product represents the inner product. The antysymmetric (wedge) product of two basis vectors gives a unit bivector:
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 , i.e., . In the case of the 4-dimensional spacetime we thus have the signature , i.e., . The corresponding Clifford algebra is .
The basis of is
A generic element, , is a superposition
called a Clifford aggregate or polyvector.
In refs. [5,6,8] it has been demonstrated that r-vectors can be associated with closed instantonic (r-1)-branes or open instantonic r-branes. A generic polyvector, , can be associated with a conglomerate of (instantonic) r-branes for various values of .
Our objects are instantonic r-branes, which means that they are localized in spacetime*. They generalize the concept of `event’, a spacetime point, . Instead of an event, we have now an extended event, , described by coordinates , . The space of extended events is called Clifford space, . It is a manifold whose tangent space at any of its points is a Clifford algebra . If is a flat space, then it is isomorphic to the Clifford algebra with elements
In flat -space, the basis vectors are equal to the wedge product
at every point . This not true in curved -space: if we (parallelly) transport a polyvector from a point along a closed path back to the original point, , then the orientation of the polyvector after such transport will not coincide with the initial orientation of . After the transport along a closed path we will obtain a new polyvector . If,in particular, the initial polyvector is one of the Clifford algebra basis elements, , i.e., an object with definite grade, then the final polyvector will be , which is an object with mixed grade. A consequence is that in curved Clifford space , basis vectors cannot have definite grade at all points of .
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 . In such a space, afer a parallel transport along a closed path, the vector part changes into , the bivector part changes into , 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 , where is a curved lower dimensional subspace of , and where only those (parallel) transports are allowed that bring tangent vectors of into another tangent vectors of the same subspace .
The squared line element in Clifford space, , is
Here , and , where denotes the operation of inversion: . The metric of C is
where means the scalar part. A Clifford space with such a metric has signature  (8,8), i.e., . This is ultrahyperbolic space with neutral signature.
In the paper “Quantum Field Theories in Spaces with Neutral Signatures”[http://arxiv.org/abs/arXiv:1210.6820] it is shown that, contrary to the wide spread belief, the physics in spaces with signature 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.}
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