ar X iv : h ep - t h / 99 10 04 4 v 1 6 O ct 1 99 9 GRAVITY AND INSTANTONS

Conventional non-Abelian SO(4) gauge theory is able to describe gravity provided the gauge field possesses a specific polarized vacuum state in which the instantons have a preferred orientation. Their orientation plays the role of the order parameter for the polarized phase of the gauge field. The interaction of a weak and smooth gauge field with the polarized vacuum is described by an effective long-range action which is identical to the Hilbert action of general relativity. In the classical limit this action results in the Einstein equations of general relativity. Gravitons appear as the mode describing propagation of the gauge field which strongly interacts with the oriented instantons. The Newton gravitational constant describes the density of the considered phase of the gauge field. The radius of the instantons under consideration is comparable with the Planck radius. 1 Instantons in an external field This work reviews the idea proposed in 1,2 which suggests a new explanation for the origin of gravitational forces. In the proposed scenario gravity arises as a particular effect in the conventional Yang-Mills gauge theory 3 formulated in flat space-time. All geometrical objects necessary for the gravitational phenomenon can originate from the gauge degrees of freedom if a particular nontrivial vacuum state, which we will call the vacuum with polarized instan-tons develops in the SO(4) gauge theory. Consider Euclidean formulation of the SO(4) gauge theory. The gauge algebra for SO(4) gauge group consists of two su(2) gauge subalgebras, so(4) = su(2)+su(2). The instantons and antiinstantons can belong to any one of these two available su(2) gauge subalgebras. It is convenient to choose the generators for one su(2) gauge subalgebra to be (−1/2)η aij and the generators for the other one to be (−1/2)¯ η aij. To distinguish between these two subalgebras we will refer to them as su(2)η and su(2)¯ η. Symbols η aij , ¯ η aij are the usual 't Hooft symbols, a = 1, 2, 3, i, j = 1, · · · , 4. In this notation the gauge potential and the strength of the gauge field are A ij µ = −