AA covariant polymerized scalar ﬁeld in loop quantum gravity
Florencia Ben´ıtez , Rodolfo Gambini and Jorge Pullin
1. Instituto de F´ısica, Facultad de Ciencias,Igu´a 4225, esq. Mataojo, 11400 Montevideo, Uruguay.2. Department of Physics and Astronomy,Louisiana State University, Baton Rouge, LA 70803-4001, USA.3. Instituto de F´ısica, Facultad de Ingenier´ıa,Julio Herrera y Reissig 565, 11300 Montevideo, Uruguay.
We propose a new polymerization scheme for scalar ﬁelds coupled to gravity. It hasthe advantage of being a (non-bijective) canonical transformation of the ﬁelds andtherefore ensures the covariance of the theory. We study it in detail in sphericallysymmetric situations and compare to other approaches.
I. INTRODUCTION “Polymerization” is a common procedure for constructing candidates for semi-classicaltheories stemming from loop quantum gravity. The idea is that in the Hilbert space commonly used in loop representations of diﬀeomorphism invariant theories like generalrelativity, the connection is not a well deﬁned variable. However, its holonomy is. Whenmatter ﬁelds are present, or when one is working in situations with reduced symmetry, notall the variables present may be connections anymore. Yet, to try to mimic the behaviorsone would see in the full theory, or to make matter compatible with the Hilbert spaces ofinterest, one usually considers inner products in which certain variables are not well deﬁnedalthough their exponentials are. To construct the equations of the theory one thereforereplaces the variables in question by their exponentials, or, since one is interested in realvariables, the substitution tends to be of the form x → sin( kx ) /k where x is the variablein question and k is known as the polymerization parameter. It is clear that in the limit k → k plays the role of the lengthof the loop along which one would compute a holonomy if the variable in question had beena connection. Sometimes the exponentiated quantity, in the case where the variable is nota connection, is known as “point holonomy”. The use of this type of construction has beenused widely in loop quantum cosmology  and also in spherically symmetric situations. It isalso understood as a non-standard representation of the canonical commutation relations andcan be applied in ordinary systems in quantum mechanics . The use of point holonomieshas also been advocated in the full theory to represent scalar ﬁelds .The use of polymerizations has been criticized for the lack ofcovariance. Since one istypically working in a canonical context the polymerization alters spatial variables. Theconstruction is generically slicing-dependent .In this paper we would like to propose a new polymerization procedure for scalar variablesand apply it to the case of a scalar ﬁeld coupled to gravity in spherical symmetry. Thenovelty is that the polymerization procedure is a non-bijective canonical transformationon the variables of classical general relativity coupled to a scalar ﬁeld. As a result, itsapplicaiton does not interfere with covariance. At ﬁrst this may appear surprising. If it a r X i v : . [ g r- q c ] F e b is a canonical transformation one is essentially dealing with the same theory. How couldthis be capturing non-trivial quantum corrections to the theory? The answer is that thecanonical transformation is non-invertible and therefore the resulting theory is not unitarilyequivalent to the non-holonomized one [7–9]. The canonical transformation leads to a theorythat can be viewed as the semiclassical theory stemming from a non-standard representationof the Weyl algebra. As we shall see, the construction for the scalar ﬁeld is general, but forthe geometrical variables we do not know a general polymerization procedure yet. We willrestrict to study in detail the polymerization in the context of eﬀective spherically symmetricgravity to make it concrete.Recently, the eﬀective theory resulting from a standard polymerization of a gauge ﬁxedtheory of gravity coupled to a scalar ﬁeld in spherical symmetry was studied  to addressthe semiclassical corrections of gravitational collapse. This system had been studied inclassical general relativity by Choptuik . We will show that the results of that analysisdo not change signiﬁcantly with the polymerization proposed in this paper. II. SPHERICALLY SYMMETRIC GENERAL RELATIVITY IN NEWVARIABLES
We recall the basics of gravity coupled to a scalar ﬁeld in spherical symmetry . Theclassical variables are the triads in the radial and transverse directions and their canonicallyconjugate momenta E x , E ϕ , K x , K ϕ . Their relations to the usual metric variables are, ds = − α dt + Λ dx + R d Ω (2.1)with, Λ = E ϕ /x , R = | E x | and to the extrinsic curvatures K xx = − sign( E x ) ( E ϕ ) K x / (cid:112) | E x | and K θθ = − (cid:112) | E x | A ϕ / (2 γ ) with γ the Immirzi parameter.The total Hamiltonian is, H T = (cid:90) dx (cid:8) N x (cid:0) ( E x ) (cid:48) K x − E ϕ ( K ϕ ) (cid:48) − πP φ φ (cid:48) (cid:1) + N (cid:32) − E ϕ √ E x − √ E x K ϕ K x − K ϕ E ϕ √ E x + (cid:0) ( E x ) (cid:48) (cid:1) √ E x E ϕ − √ E x ( E x ) (cid:48) ( E ϕ ) (cid:48) E ϕ ) + √ E x ( E x ) (cid:48)(cid:48) E ϕ + 2 πP φ √ E x E ϕ + 2 π √ E x E x ( φ (cid:48) ) E ϕ (cid:33)(cid:41) . (2.2)where prime denotes ∂/∂x .The Hamiltonian and diﬀeomorphism constraint satisfy the usual algebra with structurefunctions. This makes quantization diﬃcult. However, redeﬁning the shift and lapse in thefollowing way, ¯ N x = N x + 2 N K ϕ √ E x ( E x ) (cid:48) (2.3)¯ N = E ϕ N ( E x ) (cid:48) (2.4)one has that, H T = (cid:90) dx (cid:8) ¯ N x (cid:2) ( E x ) (cid:48) K x − E ϕ ( K ϕ ) (cid:48) − πP φ φ (cid:48) (cid:3) + ¯ N (cid:34) √ E x (cid:32) (cid:0) ( E x ) (cid:48) (cid:1) E ϕ ) − − K ϕ (cid:33)(cid:35) (cid:48) − K ϕ √ E x φ (cid:48) P φ E ϕ + 2 π ( E x ) (cid:48) P φ √ E x ( E ϕ ) + 2 π √ E x E x ( E x ) (cid:48) ( φ (cid:48) ) ( E ϕ ) . (2.5)With the diﬀeomorphism constraint, D = ¯ N x (cid:2) ( E x ) (cid:48) K x − E ϕ ( K ϕ ) (cid:48) − πP φ φ (cid:48) (cid:3) (2.6)and Hamiltonian constraint H = ¯ N (cid:34) √ E x (cid:32) (cid:0) ( E x ) (cid:48) (cid:1) E ϕ ) − − K ϕ (cid:33)(cid:35) (cid:48) − K ϕ √ E x φ (cid:48) P φ E ϕ + 2 π ( E x ) (cid:48) P φ √ E x ( E ϕ ) + 2 π √ E x E x ( E x ) (cid:48) ( φ (cid:48) ) ( E ϕ ) (2.7)such that the latter has an Abelian algebra with itself. In the vacuum case this paved theway for the complete quantization of the model. With the scalar ﬁeld present, the terminvolving the derivative complicates the Abelianization involved in promoting the algebra toa consistent quantum one. This led us in a previous paper  to consider the polymerizedversion of a totally gauge ﬁxed form of the theory following closely what was done byChoptuik. The choice E x = x , K ϕ = 0 leads to a reduced theory where only the scalar ﬁeldneeds to be polymerized. A question remains when one follows this quantization procedure.Will diﬀerent gauge and polymerizations choices lead to an equivalent quantum theory? III. A NEW COVARIANT POLYMERIZATION
We will apply the following canonical transformations for the scalar ﬁeld, φ , the curvature, K ϕ , and their canonical momenta, P φ y E ϕ , φ (cid:55)→ sin( kϕ ) k , P φ (cid:55)→ P ϕ cos( kϕ ) K ϕ (cid:55)→ sin( ρK ϕ ) ρ , E ϕ (cid:55)→ E ϕ cos( ρK ϕ ) (3.1)where k and ρ are the polymerization parameters for the ﬁeld and curvature, respectively.Canonical transformations in the spherically symmetric context have also been consideredin .Applying the canonical transformation we have that, H (cid:0) ¯ N (cid:1) = ¯ N √ E x ( E x ) (cid:48) ( E x ) (cid:48)(cid:48) cos ( ρK ϕ )2 ( E ϕ ) + ¯ N (cid:0) ( E x ) (cid:48) (cid:1) cos ( ρK ϕ )8 ( E ϕ ) √ E x − ¯ N √ E x (cid:0) ( E x ) (cid:48) (cid:1) ( E ϕ ) (cid:48) cos ( ρK ϕ )2 ( E ϕ ) − ¯ N √ E x (cid:0) ( E x ) (cid:48) (cid:1) ρ ( K ϕ ) (cid:48) sin ( ρK ϕ ) cos ( ρK ϕ )2 ( E ϕ ) − ¯ N ( E x ) (cid:48) √ E x (cid:18) (cid:18) sin ( ρK ϕ ) ρ (cid:19)(cid:19) − N √ E x sin ( ρK ϕ ) cos ( ρK ϕ ) ( K ϕ ) (cid:48) ρ − N sin ( ρK ϕ ) cos ( ρK ϕ ) √ E x P ϕ ϕ (cid:48) ρE ϕ + 2 π ¯ N ( E x ) (cid:48) cos ( ρK ϕ ) P ϕ √ E x ( E ϕ ) cos ( kϕ )+ 2 π ¯ N √ E x E x ( E x ) (cid:48) ( ϕ (cid:48) ) cos ( kϕ ) cos ( ρK ϕ )( E ϕ ) . (3.2)The diﬀeomorphism constraint is unchanged, therefore so is its algebra with itself and theHamiltonian. Moreover, one can check that the Hamiltonian remains Abelian with itself, (cid:2) H (cid:0) ¯ N ( x ) (cid:1) ; H (cid:0) ¯ M ( y ) (cid:1)(cid:3) ≈ . (3.3)So the resulting theory has the same constraint algebra as before. IV. RELATION TO OTHER POLYMERIZED APPROACHES
As we mentioned, gravity with spherical symmetry coupled to a scalar ﬁeld was recentlystudied in the context of a semi-classical loop quantum gravity analysis of the phenomenadiscussed by Choptuik . In that work, we used the same gauge ﬁxing as Choptuik had considered in his original analysis in classical general relativity. In it one takes the usualSchwarzschild coordinates for the exterior of a black hole, plus an additional condition onthe extrinsic curvature. In terms of our variables, this corresponds to E x = x , K ϕ = 0. Letus see how the equations with the new polymerization we present in this paper look like inthat gauge choice: N (cid:48) N − ( E ϕ ) (cid:48) E ϕ + 2 x − ( E ϕ ) x = 0 (4.1)( E ϕ ) (cid:48) E ϕ − x + ( E ϕ ) x − πx (cid:32) ( P ϕ ) x cos ( kϕ ) + ( ϕ (cid:48) ) cos ( kϕ ) (cid:33) = 0 (4.2)˙ ϕ = 4 πN P ϕ E ϕ x cos ( kϕ ) (4.3)˙ P ϕ = − πN P ϕ E ϕ x k sin [ kϕ ]cos ( kϕ ) + 4 πx E ϕ (cid:20)(cid:18) N E ϕ − xN ( E ϕ ) (cid:48) + N (cid:48) E ϕ xE ϕ (cid:19) ϕ (cid:48) cos ( kϕ )+ xN ϕ (cid:48)(cid:48) cos ( kϕ ) − xN k ( ϕ (cid:48) ) cos ( kϕ ) sin ( kϕ ) (cid:105) (4.4)
14 12 10 8 log|( p p * )| l o g M Traditional polym = 0.384
Covariant polym = 0.401
FIG. 1: The scaling of the black hole mass observed ﬁrst by Choptuik  for the collapse ofspherically symmetric massless scalar ﬁeld. Here we depict it for the traditionally polymerizedtheory and the covariant one. Graphs are for the polymerization parameter value k = 1, which inpractice is a very exaggerated value, in order to make discrepancies with classical general relativitymore visible. As can be seen both polymerized theories diﬀer little, the exponents γ diﬀer by lessthan 5%. K x = − πP ϕ ϕ (cid:48) x (4.5)Comparing with the more traditional polymerization we considered in , the last threeequations are modiﬁed. The ﬁrst term in the last equation is new and the cosines in thedenominator of the second and third equations were absent. These terms can potentiallymake a signiﬁcant diﬀerence in some regions of phase space.To try to test this, we conducted numerical simulations like the ones in our previous paper to determine the scaling law of the mass of the ﬁnal black hole formed by the collapse of ascalar ﬁeld as a function of a parameter in the initial data. The ﬁgure shows the comparisonfor the rather unnaturally large value k = 1 of the polymerization parameter. As can beseen, the diﬀerences are very small. Of course, since the Choptuik phenomena is determinedby the exterior geometry of the black hole where the ﬁelds are weak, it is perhaps notsurprising that the two polymerizations yield essentially the same result. But on the otherhand, it provides evidence that the results found do not have signiﬁcant slicing dependence,as the new polymerization does not depend on them. It should be noted that, for small blackholes, the Choptuik phenomena do involve regions of large curvature immediately outsidethe horizon and yet the results appear largely unchanged, as discussed in the ﬁgure. V. DISCUSSION
We have introduced a new polymerization for scalar ﬁelds coupled to gravity. This can beviewed as a non-standard representation of the Weyl algebra diﬀerent than the one usuallyconsidered in loop quantum gravity coupled to scalar ﬁelds. It has the advantage that itis a canonical transformation from the original variables. That means that it preserves theconstraint algebra and the covariance of the theory, which previous choices did not. As it isnot-invertible in the whole of phase space it still allows to have the usual novel phenomenathat loop quantizations introduce in regions where one expects general relativity not to bevalid, like close to singularities. In particular it will admit a representation on a Hilbert spacedeﬁned in terms of the Ashtekar Lawandowski measure . Although we have only exploredthe implications of the covariant polymerization in the context of spherically symmetricscalar ﬁelds, it is possible that an analogue could be found for the full theory. This willrequire further investigations.
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