G. Shipov in "A Theory of Physical Vacuum" (IITAPRANS 1998) presents an interesting concept for treating the Vacuum as an A4 space of Absolute Parallelism. Euclidean spaces are included in such A4 spaces, but an A4 space can be much more complex than a simple euclidean 4 dimensional space.

Spaces of absolute parallelism support a global Basis Frame matrix [F] of functions that have a non-zero determinant, and therefor a global inverse [G]. Such domains induce a linear differential connection between the vectors that make up the basis frame: From [F][G] = [1], there are two points of view.

d[F] + [Delta][F]=0.
or
d[F] - [F][C]=0.

[C] is the usual Cartan (right) matrix of connection 1-forms, and [Delta] is the Cartan (left) matrix of connection 1-forms as used by Shipov.

A4 spaces are 4 dimensional spaces for which both the matrix of Cartan Curvature 2-forms and the vector of Cartan Torsion 2-forms generated by exterior differentiation of the Basis Frame are globally zero. However certain 3D subspaces of A4 may or may not exhibit zero Cartan curvature and Cartan torsion. Note that if the spaces are constrained to support an equivalence class of metrics, then the curvature induced by the Christoffel symmetric portions of the connection can be cancelled out by interactions between the metric and the anti-symmetric components of the connection. It is these features that give the A4 spaces interesting structure beyond that of a simple void.

Note that the concept of Cartan Torsion 2-forms is not the same as the concept of torsion due to an asymmetric "affine" connection. Affine torsion is related to certain anti-symmetries in the partial derivative coefficients that make up the Cartan connection [C]. These anti-symmetries are different from the anti-symmetries generated from the Shipov connection [Delta]. Moreover, the A4 spaces can be defined in a manner such that the direction fields (Vierbeins) generated by the Inverse Frame matrix [G] are not integrable in the sense of Frobenius. In other words, spaces of absolute parallelism can support topological torsion.

The A4 spaces focus attention on several different types of Torsion.

1. The Frenet Torsion of a space curve.
2. The Torsion 2-forms of a Cartan connection.
3. The Torsion of an asymmetric Affine connection.
4. The Topological Torsion of the Vierbein fields that can be constructed from the Inverse of the Basis Frame that is used to define the A4 space.

All of these forms of torsion are distinct.

The theory is still not complete, but the following results are well established:

Definition: A variety {x} is a space of Absolute Parallelism if the matrix of Cartan curvature 2-forms and the vector of Cartan torsion 2-forms defined on the variety vanish.

Theorem 1: Consider an arbitrary matrix [F] of n x n functions on the variety {x] where the variety is now restricted such that det [F ] <> 0. Then the restricted domain is a space of Absolute Parallelism

Shipov Conjecture: The physical vacuum is a 4 dimensional space of Absolute Parallelism , further restricted to the Lorentz equivalence class of Basis Frame matrices, F, such that [transposeF][M][F]=[M]. The matrix [M] is the Minkowski index matrix.

Theorem 2: The domain of support for an arbitrary 1-form of Action on a variety generates a space of Absolute Parallelism. (This method of generating a basis frame connects the Calculus of variations based upon Action 1-forms to spaces of absolute parallelism)

Theorem 3: The domain of support of a projective variety defines a space of absolute parallelism.

Theorem 4: A Basis Frame generated by the Jacobian of a parametric map is free from affine torsion. The Cartan matrix [C] is equivalent to the matrix constructed from the Christoffel symbols of the induced metric on the space of parameters.

A short (and incomplete) book report on Shipov's text, other remarks about Absolute Parallelism and proofs of the theorems stated above are given in the download pdf files below.

If a space of absolute parallelism is further constrained to those matrices of Basis Frames which are symmetric (such as metrics) then the projective frames (defined as det [F] <>0) produce what are called polarities. Polarities are special projective transformations that establish duality relationships in projective geometry.

With non-constant generators (a 6 parameter infinity that preserve the Minkowski metric), and det[F] non - zero, the equivalence class of Lorentz Basis Frames produces an A4 space, but the space may have a 3D subspaces with non-zero Cartan Curvature 2-forms and Cartan Torsion 2-forms. Some of these Lorentz Basis Frames have non-zero Cartan Torsion 2-forms on the 3D subspace, and some do not. Similarly, some of these Lorentz Basis Frames have non-zero Cartan Curvature 2-forms on the 3D subspace, and some do not.

	Downloads
	shipov.pdf 400k	.tex NA