Gauge Theory of Gravitation

Diaa A Ahmed

e-mail: diahmed@yahoo.com

(Rotterdam November 1997)

Gravitation represent quantum topological phenomenon on a large scale. Introducing the action of gravitation into functional quantum space -through a phase angle- requires a change in our understanding of both gauge theory and general relativity. This display its unity with the rest of gauge interactions and demonstrate that the manifold is compact.

I. INTRODUCTION

The dimensionless coupling G (x) = k m / r , plays the role of phase angle of cosmological origin- coupling between dipole moments. We can derive its action in functional space by introducing it into the group structure, exp { i g T^a . w^a (x)}. We may use the notation {^a_b} = g^am g_mb , and its contracted form { } , to represent this phase angle and to facilitate the calculations in terms of basis { w^a (p)}, { e_b (p)} , and phase angles < w^a | e_b >. The phase angles exist in functional space and its contractions equal in the geometric language to its projection on the metric. Action of the phase angle reconstructs the wave front and this determines the geometric structure of the manifold and the fields. We will now consider that these phase angles perform rotations that are subject to the same algebraic scheme as the fields W. This sort of rotations will mix up fields in order to bring in the transformations on these fields. Furthermore the transformations will not change the structure of physical law. This means that the phase w^a (x) , that is involved is a general one and G (x) , is associated with the same algebraic scheme of T^a. This will permit these transformations to induce the appropriate transformations on the fields. Therefore we will assume,

The infinitesimal transformations,

Where,

Where,

These expressions replaces D_l , W^a_l , w^a , in the standard Yang Mills formalism,

Points:

¶_l w^a : Topological nature of the source.

f^abg w^b W^g_l : Topological nature of the transformations, performed by the functional w^a , and isovectors and Riemannian manifold.

II. YANG MILLS EINSTEIN GAUGE FIELD

From the commutator [ D_l , D_m ] , F^a_{l m} =

Conformity of this structure with Yang Mills is brought about by complexity of the transformations that G^a_b , performs, in conformity with the structure of the space itself.

The linear terms in lines 3 and 4,

Lines 2 , 3 and 4, show some new properties of {^a_b} , as they mediate the way between the linear and the bilinear terms in

F^a_{l m}. This can be interpreted as w^a- wave equation as distinct from W^a_l-wave equation.

Note: Further work on,

[1] Quantum Topology, and the references cited therein.

[2] J C Taylor, Gauge Theories of Weak Interactions, Cambridge 1976.

[3] J L Lopes, Gauge Field Theories, Pergamon Press 1981.

[4] MTW, Gravitation, Freeman 1973.