A Comment on the Degrees of Freedom in the Ashtekar Formulation for 2+1 Gravity

We show that the recent claim that the 2+1 dimensional Ashtekar formulation for General Relativity has a finite number of physical degrees of freedom is not correct. PACS number(s): 04.20.Cv, 04.20.Fy In a recent paper [1] Manojlović and Miković claim that the number of degrees of freedom in Ashtekar’s formulation for 2+1 dimensional General Relativity is finite at variance with previous results of the authors [2]. We stand by the results of that paper wherein we were able to prove that, in spite of having the same number of first class constraints as phase space variables per point, the number of degrees of freedom in this formulation is infinite. In this comment we show that [1] is incorrect on several counts. (i) The statement appearing in page 3034: “..., by performing a gauge transformation on a null connection, one can always reach a non-null connection ...” is not true. This statement is ‘proved’ by (25) of [1]. However, (25) of that paper is incorrect because the gauge transformations of the connection that the authors have used are wrong. Specifically, eq.(17) should say δA1 = − dǫ dθ + (E2f3 −E3f2)ǫ 3 and eq.(19) should be δA3 = −A2ǫ 1 + f3ǫ 2 + E1f2ǫ 3 The origin of the error in the last equation can be traced back to the use of an incorrect symplectic structure to derive it from the constraints. (E I Ȧ I a) evaluated on (9), (10) of [1] is E1Ȧ1 +E2Ȧ2 −E3Ȧ3 not E1Ȧ1+E2Ȧ2 +E3Ȧ3 as in (14) of [1]. This is because I is an SO(2,1) index and tIt I = −1 not +1. Correction of these errors leads to the following version of equation (25) δ(f2 ± f3) = −ǫ 1(f3 ± f2) + [ǫ 2(f2 ± f3)] ′ − ǫ2A1(f3 ± f2) (1) +[ǫ3E1(f3 ± f2)] ′ − ǫ3A1E1(f2 ± f3)− ǫ 3(E2f3 − E3f2)(A3 ±A2), where f2 ≡ dA2 dθ −A1A3, f3 ≡ dA3 dθ − A1A2.