Towards a Path Integral for Pure Spin Connection Formulation of Gravity

A proposal for the path-integral of pure-spin-connection formulation of gravity is described, based on the two-form formulation of Capovilla et. al. It is shown that the resulting effective-action for the spin-connection, upon functional integration of the two-form field Σ and the auxiliary matrix field ψ is non-polynomial, even for the case of vanishing cosmological constant and absence of any matter couplings. Further, a diagramatic evaluation is proposed for the contribution of the matrix-field to the pure spin connection action. PACS numbers :04.20.Cv, 04.20.Fy, 2.40.+m † e-mail address : [email protected] 1 In the past few years, it has become increasingly evident that the description of gravity in terms of connection variables instead of the metric, originally due to Ashtekar [1] is well-tailored for the discussion of quantum aspects of the theory. This has been attributed to the close parallel between this description and Yang-Mills theories, topological solutions thereof as well as the invention of loop variables. Motivation to obtain a natural covariantization of Ashtekar theory, led Capovilla et al to introduce a classical action for gravity (and a one-parameter family of generally covariant gauge-theories) purely in terms of a spin-connection [2]. This action is obtained by solving the classical equation of motion for the ‘metric-variable’ Σ, from the self-dual two form action for a SL(2, C) connection A, a non-dynamical matrix field ψ and two-forms Σ [3]. The equivalence of the pure-connection theory to that of Ashtekar can be shown by a 3 + 1 decomposition [4], as well as by comparing the constraints arising due to diffeomorphism and gauge invariances of the theory in the two formulations [5]. In an interesting alternative approach, Peldán [6] performed inverse Legendre transform on the Hamiltonian comprising purely of constraints (characteristic of diffeomorphism invariant theories) and obtained a pure-spin connection action. The apparent discrepancy between the actions of Peldán and Capovilla et al, can be removed by rewriting the tracelessness condition on ψ in the pure-spin connection action [7]. With an overall agreement on the consistency of the pure-spin-connection formulation of gravity at the classical level at hand, it is only natural now to start exploring the quantum properties of it. In a recent paper, Smolin [8] has furnished a path integral for Euclidean case, starting from the Hamiltonian of the ‘googly’ theory. For eliminating the Gauss law and diffeomorphism constraints from the integrand, he uses the ‘time’ component of the gauge field as a Lagrange multiplier and solves the diffeomorphism constraints explicitly using the classical solution of Capovilla et al [2]. He further proposes to choose gauge-fixing conditions for the A field as linear expressions, so that the gauge field action remains at most quadratic (this happens only in the limit GN → 0, and hence for the ‘googly’ theory alone) and the path integral can, be evaluated exactly, producing an ‘effective action’ for 2 the matrix field and the ghost fields introduced by the Faddeev-Popov determinant. Although motivated in part by Smolin’s paper, we wish to make a different proposal for the path integral. We begin with the two-form action for the metric variable Σ, coupled to the gauge-field A and the auxiliary matrix field ψ, as in reference [3]. As in any quantum theory of gravity, the path integral must include fluctuations of the metric, we functionally integrate over Σ first to obtain the effective action for the gauge-field A and ψ. Throughout the discussion the integral over A is only in a formal sense, since we do not display the gauge-fixing terms and the F − P determinant, those will be discussed in a future publication as work is still in progress on these issues. Consider then the following formal definition of the Euclidean path integral : Z = ∫