,-lass Quantum Grav 5 (1988) 491-506 Printed in the UK The equation of deviation in a conformally invariant theory of gravitation and electromagnetism Y Q Cai and G Papini Department of Physics and Astronomy, University of Regina, Regina, Saskatchewan, S4S OA2 Canada Received 22 June 1987, in final form 29 October 1987 Abstract. The equation of deviation IS derived for a conformally invariant theory of gravitation and electromagnetism in which scale is generated by a topological constraint Compared with its counterpart in the Einstein-Maxwell theory, the equation contains some extra terms proportional to the electromagnetic potential indicating the existence of linear and non-linear Bohm-Aharonov effects These are present also in the related theories of Weyl and Dirac From the same equation one can derive an expression for the quantisation of geometry which further illustrates the topological mechanism for the genesis of scale As a by-product, the magnitudes of some Einstein-Maxwell corrections in Weber-type experiments are estimated 1. Introduction In 1918, Weyl [l] proposed a unified theory of gravitation and electromagnetism, in which the length of a vector is assumed to be non-integrable. This means that if a vector has length I at a point P, 1 changes by SI= /K,,8Xp (1.1) after a parallel displacement axp The vector K,, so introduced is interpreted as the electromagnetic field It follows from (1.1) that comparing the lengths of two vectors which are not separated by an infinitesimal distance is a path-dependent operation. The comparison is therefore possible only if standards of length are itrbitrarily set up at each point in spacetime A change in these standards can be parametrised by means of an arbitrary function A(x) of the coordinates, such that a length 1 is transformed into K; = K,, +(In A(x)).,, (1.3) the total change in the length 1 of a vector displaced around a small closed loop is SI = IF,,,SS’” (1.4)