Towards a unification of gravity and Yang-Mills theory.

We introduce a gauge and diffeomorphism invariant theory on Yang-Mills phase space. The theory is well defined for an arbitrary gauge group with an invariant bilinear form, it contains only first class constraints, and the spacetime metric has a simple form in terms of the phase space variables. With gauge group SO(3, C), the theory equals the Ashtekar formulation of gravity with a cosmological constant. For Lorentzian signature, the theory is complex, and we have not found any good reality conditions. In the Euclidean signature case, everything is real. In a weak field expansion around de Sitter spacetime, the theory is shown to give the conventional Yang-Mills theory to the lowest order in the fields. PACS numbers: 04.60.+n, 04.50.+h, 04.20.Fy Email address: [email protected] Comparing the two dominating Hamiltonian formulations of (3+1)-dimensional Einstein gravity; the ADM-formulation [1] and the Ashtekar formulation [2], it is quite clear that gravity in (3+1)-dimensions (and (2+1)-dimensions) seems to prefer the Yang-Mills phase-space compared to the geometrodynamical phase-space. This is mainly due to the fact that the constraints in the theory simplify a lot in the transition to the Ashtekar formulation. There is, however, an ugly spot in this otherwise very beautiful formulation: the need for complex fields and complicated reality conditions. The Ashtekar Hamiltonian for Einstein gravity is known to exist for (2+1)and (3+1)-dimensional Lorentzian and Euclidean gravity, but it is only for the (3+1)-dimensional Lorentzian case (the theory we are most interested in) that we need complex fields! Another ”disadvantage” of the Ashtekar variables can be found in the matter couplings: although both scalar fields and spinor fields can be beautifully incorporated into the theory, the coupling to the spin-1 Yang-Mills fields destroys the simplicity of the constraints. See [3]. Either the coupling is non-polynomial or one has to rescale the pure gravity part of the Hamiltonian constraint by multiplying it with the determinant of the metric. Both choices will probably severly complicate the canonical quantization of this theory. These three facts taken together – (1) gravity prefers the Yang-Mills phase-space, (2) the theory is complex, and (3) the coupling to spin-1 Yang-Mills fields does not seem natural – could indicate that there exists another underlying more beautiful theory. This would presumably be a real (non-complex) unified theory of gravity and Yang-Mills theory, for some gauge group G, such that, when the larger symmetry is broken down to G ∼ SO(3)×GYM , the need for complex fields appears. In this letter, we will describe a candidate theory for this unification. We have, however, not been able to find a real theory for the case of Lorentzian signature of the metric. Otherwise, the theory fulfills the requirements put on it so far: it is a diffeomorphism and gauge invariant theory valid for any gauge group which has a non-degenerate invariant bilinear form. Furthermore, our model reduces to conventional Einstein gravity with a cosmological constant if one uses the gauge group SO(3, C), and in an expansion for weak fields around de-Sitter spacetime, the theory agrees with conventional Yang-Mills theory to lowest order. Here is the Hamiltonian for the unified theory: Htot = NH +NHa + ΛiG (1) H := 1 4 ǫabcǫijk(E)E E(B + 2iλ 3 E) ≈ 0 (2) Ha := 1 2 ǫabcE B i ≈ 0 (3) Gi := DaE i = ∂aE i + fijkAjaEak ≈ 0 (4) The index-conventions are: a, b, c, ..... are spatial indices on the three dimensional hypersurface, and i, j, k, ....... are gauge-indices in the vector representation, and therefore take Compared to [4] and [6], the fields are rescaled as follows: E → −iEai, Aai → iAai, F i ab → iF i ab