Comment on "Gravity and the Poincaré group"

Following the approach of Grignani and Nardelli [1], we show how to cast the two–dimensional model L ∼ curv + torsion + cosm.const — and in fact any theory of gravity — into the form of a Poincaré gauge theory. By means of the above example we then clarify the limitations of this approach: The diffeomorphism invariance of the action still leads to a nasty constraint algebra. Moreover, by simple changes of variables (e.g. in a path integral) one can reabsorb all the modifications of the original theory. Vienna, January 1993 ∗e-mail: [email protected] The similarities of Cartan’s formulation of gravity to the gauge theories responsible for the remaining interactions has again and again lead to attempts of reformulating gravity as a gauge theory (cf. e.g. [2]). The reformulation of pure 2+1 dim gravity as a Poincaré gauge theory with Chern–Simons action [3] and of 1+1 dim Liouville gravity as a SO(2,1) (Λ 6= 0) resp. ISO(1,1) (Λ = 0) gauge theory of the BF–type [4] certainly spurred such endeavors, all the more since in these cases it was crucial for the successful quantization. By introducing the so–called Poincaré coordinates q(x) as auxiliary fields, Grignani and Nardelli formulated several gravitational theories as Poincaré gauge theories [1]. Although their gauge theoretical formulation is equivalent to the original theories, it, to our mind, misses the decisive advantage for quantization present in the above mentioned works, i.e. the ability to ’eat up’ the diffeomorphism invariance of the respective gravitational theory by gauge transformations and, correlated to that, to have to deal with the quite well–known space of flat connections (cf. also [5]). Moreover, as we shall illustrate at the 2 dim model of NonEinsteinian Gravity given by [6] L = e (− γ 4 R + β 2 T 2 − λ), (1) it is not only possible to formulate any gravitational theory as a Poincaré gauge theory along the lines of [1], but all of these formulations are trivially equivalent to the original ones after an appropriate shift of variables so that the Poincaré coordinates drop out completely. The basic quantities in (1) are the orthonormal one–forms e, e ≡ det eμ , the SO(1,1) connection ω, the Ricci scalar R = 2 ∗ dω, and T a = ∗Θ with the torsion two–form Θ ≡ De. The first order (or Hamiltonian) form of (1) is LH = −e( π2 2 R + πaT a −E), (2) with E ≡ 1 4γ (π2) 2 − 1 2β π − λ, (3) as is most easily seen [9] by plugging the field equations for the momenta πA ≡ (πa, π2) back into (2). The first two terms of LHd x can be rewritten in a standard manner as πAF A in which F is the curvature two–form of the appropriate Poincaré group ISO(1,1): F ≡ dA+ A ∧ A = ΘPa + dωJ. This follows by making use of the iso(1,1) Lie Algebra [Pa, Pb] = 0, [Pa, J ] = εa Pb and setting A = ePa + ωJ. (4) The reader who is further interested in the classical and quantum mechanical aspects of the integrable model (1) shall be refered to the literature [7], [8], [9], [10], as well as references therein. Here (1) serves only as a nontrivial two–dimensional example for the present considerations.