We show that Euclidean three-dimensional gravity coupled to a Gaussian scalar massive matter field in the first-order dreibein formalism gives a quantum theory which has a finite perturbative expansion around a nonvanishing background. We also discuss a possible mechanism to generate a nontrivial background metric starting from Rovelli-Smolin loop observables.PACS number(s): 04.60. +n, 12.25. +e Some time ago, Witten [I] showed that pure threedimensional (3D) quantum gravity (QG) in the first-order dreibein formalism is a finite off-shell (topological) theory when expanding around a vanishing background, i.e., (e; ) =O. This result comes from the fact that Einstein 3 D theory is off shell and hence it is dependent on the variables that represent the gravitational field. Later, Deser et al. [2] extended Witten's result showing that the theory remains finite even when expanding around a flat background gravitational field ( e ; ) zS; (in Euclidean space). Of course, we now have that ~e t ( e; )#O. In this work, we shall demonstrate that, as far as a perturbative theory is concerned, Euclidean 3 D gravity coupled to a Gaussian scalar massive matter field still yields to a finite quantum theory in the first-order dreibein formalism.In the end, we shall discuss a possible "quantum" mechanism for generating as an "order parameter" a nontrivial metric background from some gauge-invariant and diffeomorphism-invariant nonlocal observables of the pure topological theory, i.e. of 3 D Q G itself. These ob-servable~ and their algebra were first introduced by Rovelli and Smolin [3] in the frame of Ashtekar's reformulation of canonical 4 D general relativity [4] recently specialized to the case of ( 2 + 1 )-dimensional Einstein gravity [5]. PATH-INTEGRAL FORMULATION OF THE THEORYFirst-order dreibein gravity with a Euclidean signature is described by the action where we have absorbed a k -' factor into the dreibein e; and the spin connection W~= E "~~W , , , is an independent variable. In the following we shall consider the coupling of Eq. (1) to a real scalar massive matter field which has the first-order action [6] I M =~ S d 3 x [ @ a d~e~a , q + + ( @ a ) 2 + e m 2 p 2 ]where @ is a Lagrange multiplier, ef is formally the inverse matrix [ e i ]-', e ~~e t -'( ef ) and, of course, we assume that Det(e,, )#O. Notice that the Euclidean metric gpv is given by g,v=e;e:Sab.This equation can be rewritten in the form after u-sing the equation of motion of @ and setting k?f=deef, where @f are tensor densities of weight +. Now, we need to fix a gauge. We choose a Landau-type gauge: ape; =0=apo; .(3)The resultant ghost and gauge-fixing action is then I where C?, and a ) , are Lagrange multipliers, c, , z b and responding Euclidean path integral has the form za ,d are Faddeev-Popov ghosts. The sum of Eq. (I), Eq.(2), and Eq. (4) gives the total quantum action I. The cor-~a ) e~a ) w~~a )~~a ) c b~a a a ) d e-' . ( 5 )The action I is invariant under the following nilpotent (on-shell) Becchi-Rouet-Stora-Tyutin (BRST) transforma-'permane...
We show that 2+1-dimensional Euclidean quantum gravity is equivalent, under some mild topological assumptions, to a Gaussian fermionic system. In particular, for manifolds topologically equivalent to Σ g × IR with Σ g a closed and oriented Riemann surface of genus g, the corresponding 2+1-dimensional Euclidean quantum gravity may be related to the 3D-lattice Ising model before its thermodynamic limit.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.