Gravi-Weak Unification

The coupling of chiral fermions to gravity makes use only of the selfdual SU(2) subalgebra of the (complexified) SO(3, 1) algebra. It is possible to identify the antiselfdual subalgebra with the SU(2)L isospin group that appears in the Standard Model, or with its right-handed counterpart SU(2)R that appears in some extensions. Based on this observation, we describe a form of unification of the gravitational and weak interactions. We also discuss models with fermions of both chiralities, the inclusion strong interactions, and the way in which these unified models of gravitational and gauge interactions avoid conflict with the Coleman-Mandula theorem. 1 Approaches to unification In particle physics, the word “unification” is used in a narrow sense to describe the following situation. One starts with two set of phenomena described by gauge theories with gauge groups G1 and G2. A unified description of the two sets of phenomena is given by a gauge theory with a gauge group G containing G1 and G2 as commuting subgroups. In the symmetric phase of the unified theory, the two sets of phenomena are indistinguishable. The subgroups G1 and G2, and hence the distinction between the two sets of phenomena, are selected by the vacuum expectation value (VEV) of an “order parameter”, which is usually a multiplet of scalar fields. The Standard Model (SM) and its grand unified extensions work this way. On the other hand, currently popular theories that claim to provide a unification of gravity with the other interactions do not fit into this general scheme. It is possible to unify gravity and other Yang-Mills interactions in the sense described above, if one allows the order parameter to be a set of one-forms instead of scalars [1]. In order to motivate this, let us begin by considering four one-forms θμ (m = 0, 1, 2, 3) transforming under “internal” global Lorentz transformations θμ → Snθμ, which preserve the internal Minkowski metric η, and linear coordinate transformations θμ → θνΛμ. Let us suppose that the dynamics of the