Asymptotically Flat Boundary Conditions for the U(1)

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For the U(1)3 model, there has been much recent progress [5] and working on the latter has begun in [7] where the analysis is restricted to spatial topology of R3 with the asymptotically flat boundary conditions. As asymptotically flat spacetimes are of great importance in GR, this paper is devoted to investigate their properties in U(1)3 model and the results of the present paper were used in [7] which was in fact the main motivation for the present study. To achieve asymptotic symmetry generators [11, 12], we seek for boundary terms to the constraints that produce well-defined phase space functions and Poisson brackets while lapse function and shift vector obey decay behaviours corresponding to asymptotic symmetry transformations. When we are working in the context of Lorenzian or Euclidean GR, one expects these well-defined functions to generate the Poincaréor ISO(4) group respectively depending on the signature. Regarding U(1)3 theory, we examine to what extent we can recover ISO(4) transformations. In fact, the question is whether there are well-defined generators for all generators of ISO(4) in this model or not, and what the main reason for answering yes/no to this question is.

The architecture of this paper is as follows:

In section 2, we briefly review the background needed for the arguments following later. In subsection 2.1, first we express the constraints of GR in terms of both ADM and SU(2) variables. Then observing that they are not well-defined and functionally differentible, we revisit the results and reasoning, presented in [8] and in [9, 10], to improve the constraints to well-defined generators of Poincaréand ISO(4) group for Lorentzian and Euclidean GR respectively. In subsection 2.2, the U(1)3 model will be concisely introduced and its constraints, on the basis of which the rest of the paper is written, are expressed. In section 3, we try to make the constraints well-defined. To do this, first we look for a boundary term destroying differentiability of the constraint and then subtract it from the variation of the constraint. If the boundary term is an exact one form in the field space, it turns out that this expression is functionally differentiable. At this point, there is only one step left for the constraint to be well-defined, and that is to examine its finiteness. we show that in the U(1)3 model, while spacetime translations are allowed asymptotic symmetries, there are no well-defined generators for boosts and rotations. In section 4, we compare the results of sections 2.1 and 3 and exhibit the main source of ill-definedness of boost and rotation generators in the U(1)3 model. In the last section we conclude with a brief summary.

2 Background

2.1 Review of asymptotically flat boundary conditions for the SU(2) case In general, consistent boundary conditions are supposed to provide a well-defined symplectic structure and the finite and integrable charges associated with the asymptotic symmetries. Here integrability means that the variation of the surface charge is an exact one form. In the ADM formulation of asymptotically flat spacetimes, it is assumed that on spatial slices asymptotic spheres are equipped with asymptotically cartesian coordinates xa 2 a at spatial infinity; i.e. r where r = x xa. Taking this as the starting point and seeking for an appropriate →∞ boundary conditions, one sees that on any hypersurface, the fall-off behaviours of the spatial metric qab and its conjugate momentum πab have to be

hab 2 qab =δab + + (r− ) r O ab (2.1) ab p 3 π = + (r− ) r2 O

ab where hab and p are smooth tensor fields on the asymptotic 2-sphere. In (2.1) the first condition follows directly from the form of the spacetime metric for the asymptotically flat case, while the second one is a consequence of demanding a non-vanishing ADM momentum. In order to eliminate the logarithmic singularity existing in the

2 symplectic structure, the leading terms in (2.1) need to admit additional certain parity conditions as

x x ab x ab x hab = hab , p = p (2.2) − r r − r − r ab Indeed, the integral of p h˙ ab over the sphere, which is the coefficient of the singularity, vanishes owing to (2.2). The parity conditions also give rise to the finite and integrable Poincaréor ISO(4) charges. Furthermore, aiming to retain the boundary conditions (2.1) invariant under the hypersurface deformations,

2sN 1 δqab =− (πab πqab)+ ~ qab √q − 2 LN

ab (3) ab 1 ab(3) sN cd 1 2 ab δπ = N√q( R q R) (πcdπ π )q (2.3) − − 2 − 2√q − 2 2sN ac b 1 ab a b ab b ab + (π πc π π)+ √q(D D N q DaD N)+ ~ π , √q − 2 − LN one requires to restrict the fall-oﬀ behaviours of the lapse function, N, and the shift vector, N a. In (2.3), (3) q := det(qab), Rab is the Ricci tensor of the spatial hypersurface, Da is the torsion free, metric compatible connection with respect to qab and s denotes the signature of spacetime mertic, i.e. s = +1 and s = 1 for − Euclidean and Lorentzian spacetimes respectively. It turns out that the most general behaviour of them which also includes the generators of the asymptotic Poincar´eand ISO(4) groups are

a 1 N = βax + α + S + (r− ) O a a b a a 1 (2.4) N = β x + α + S + (r− ), b O a where βa and βab(= βba) are arbitrary constants representing boosts and rotations. Here, if v denotes the − a velocity, then the boost parameter is βa = v =: γva which satisﬁes the identity γ2 + sγ2v2 = 1. This √1+sv2 identity indicates that for a Euclidean boost, when s = 1, the sine and cosine appear in the transformation matrix which says that the Euclidean boost is nothing but a rotation in the x0, ~x plane. In turn, the arbitrary function α, and arbitrary vector αa represent time and spatial translations respectively and S , Sa which are odd functions on the asymptotic S2 correspond to supertranslations. On the other hand, since in vacuum GR the Hamiltonian and diﬀeomorphism constraints, i.e.

3 s 1 ab cd (3) H[N] := d x N − (qacqbd qabqcd)π π √q R √q − 2 − Z (2.5) a 3 a b Ha[N ] := 2 d x N Dbπ , − a Z are the generators of gauge transformations, they have to be finite and functionally differentiable so that their poisson bracket with any function on the phase space can be computed. With regard to (2.1) and (2.4), it is easy to check that the constraints (2.5) are neither finite nor differentiable. To remedy this situation, a surface integral spoiling differentiability should be subtracted from the variation of the constraint functionals. As mentioned before, appropriate boundary conditions result in an exact surface term and thus one can define new expressions for the constraints which now are functionally differentiable. Last step is to examine the convergence of these expressions. Having done this procedure, the authors of [8] obtained the following well-defined constraints

a[b c]d J[N] := H[N] + 2 dSd √qq q [N∂bqca ∂bN(qca δca)] − − I (2.6) a a ab Ja[N ] := Ha[N ] + 2 dSa Nbπ I where is the integration over the asymptotic 2-sphere. H

3 i a This analysis in terms of ADM variables language can be translated to the Ashtekar variables (Aa, Ei ) where i a the connection, Aa, is an su(2)-valued one form and its momentum conjugate, Ei , is a densitized 3-Bein. However, achieving this is challenging since there is an internal su(2) frame whose asymptotic behaviour has to be determined. Accordingly, the boundary conditions (2.1) and (2.2) in terms of the Ashtekar variables can be written as a a a fi 2 E = δ + + (r− ) i i r O i (2.7) i ga 3 A = + (r− ) a r2 O where a 1 if (a, i) = (x, 1), (y, 2), (z, 3), δi = (0 otherwise. a i and fi and ga are tensor fields defined on the asymptotic 2-sphere with the following definite parity conditions x x x x f a = f a , gi = gi . (2.8) i − r i r a − r − a r By the decay conditions (2.7) and (2.8), it is assured that the symplectic structure is well-defined. In Euclidean GR, the constraint surface in the (A, E)-phase space is defined by the vanishing of the following functionals called Gauss, Hamiltonian and diffeomorphism constraints respectively

i 3 i a j a Gi[Λ ]= d x Λ ∂aEi + ǫijkAaEk Z 3 i a b H[N]= d xNǫijkFabEj Ek (2.9) Z a 3 a j b j Ha[N ]= d x N F E A Gj . ab j − a Z Here i i i i j k F = ∂aA ∂bA + ǫ jkA A , (2.10) ab b − a a b i Λ is the Lagrange multiplier associated with Gi and N is the densitized lapse function with weight 1. It is − desired to attain well-defined form of these functionals with the smearing functions including ISO(4) generators i (2.4). To do this, first one has to ascertain an appropriate decay behaviour for Λ . Since the leading term of Gi 2 i is (r ) odd, convergence of Gi[Λ ] requires the following fall-off condition O − i i λ 2 Λ = + (r− ) (2.11) r O where λi are even functions defined on the asymptotic 2-sphere. It is straightforward to verify that (2.11) also i ensures the differentiability of Gi[Λ ]. Even after subtracting the surface integral destroying differentiability of the Hamiltonian and diffeomorphism constraints (2.9), it turns out that they are convergent only for translations and not for boosts and rotations. This situation should be cured in a way that 1) the generators stay functionally differentiable and 2) the well-defined generator for translations which is already available can be recovered up to a pure gauge. As shown in [9, 10], ultimately the well-defined forms of the symmetry generators are

i a b i a i J[N]= H[N] dSa NǫijkA E E Gi[Λ ]+ dSa E Λ¯ − b j k − B i B I I (2.12) a a a i b i a i Ja[N ]= Ha[N ] dSa N A E Gi[Λ ]+ dSa E Λ¯ − b i − R i R I I i i i i 1 j b a i i i i a where Λ = Λ + Λ¯ = Λ ǫijkδaδ β and Λ = Λ + Λ¯ = Λ + δ βa. The second term appearing in either R R − 2 k b B B i expressions in (2.12) is the surface term subtracted to make the original functionals (2.9) differentiable. The third term is subtracted to get rid of the source of divergence for boosts and rotations but puts them again in the status of non-differentiability which is modified by adding the last term. As expected, the volume terms added to the constraints are proportional to the Gauss constraint and thus do not change the translation generator on the constraint surface of the Gauss constraint.

4 2.2 Review of U(1)3 model for Euclidean quantum gravity

In [4], Smolin introduced the weak ﬁeld limit of the Euclidean gravity GN 0, where GN is the Newtonian → gravitational constant, by expanding the phase space variables (A, E) as

2 E = E0 + GN E1 + GN E2 + ... 2 (2.13) A = A0 + GN A1 + GN A2 + ..., at the level of the action. The resulting theory is not to be confused with standard perturbation theory. More precisely, consider the Hamiltonian for Euclidean gravity

1 3 a i [E, A]= d x N Ha + NH + Λ Gi (2.14) H G N Z i i Rescaling the dimensionful quantities in (2.18) by GN , namely the connection Aa GN Aa and the Lagrange i i → multiplier Λ GN Λ , the Gauss constraint of (2.9) and (2.10) become → a a k j a Gi = aE = ∂aE + ǫij GN A E D i i a k i i i i j k (2.15) F =∂aA ∂bA + ǫ jkGN A A . ab b − a a b respectively. From (2.15), it is obvious that the internal gauge symmetry is still SU(2). However, in the limit GN 0, the second term which brings in the self-interaction is switched oﬀ. The Poisson bracket of a pair of → Gauss constraints then commutes as one has

i j Gi[Λ ], Gj [∆ ] GN , (2.16) { }∝ and the symmetry group contracts from SU(2) to three independent Abelian internal gauge symmetry U(1), namely U(1)3, each of which corresponds to one of the gauge ﬁelds Ai (i = 1, 2, 3). Consequently, the constraints remain ﬁrst class and have the following simpler forms

j 3 j a Cj[λ ]= d x Λ ∂aEj Z a 3 a j b j b Ca[N ]= d x N F E A ∂bE (2.17) ab j − a j Z 3 j a b C[N]= d x NFabEk El ǫjkl Z 3 where Cj, Ca and C are the Gauss, diﬀeomorphism and Hamiltonian constraints for the U(1) model respectively j j j and F = ∂aA ∂bAa is the corresponding curvature. The Hamiltonian then reads as ab b −

1 3 a i [E, A]= d x N Ca + NC + Λ Ci (2.18) H G N Z and the only non-vanishing Poisson brackets of the pair of the constraints in the algebra will be

a b a Ca[N ],Cb[M ] = Ca[ M ] { } L−→N a Ca[N ],C[N] = C[ N] (2.19) { } L−→N ia b C[N],C[M] = Ca[E E (N bM M bN)]. { } i D − D

The algebra, except for the vanishing Poisson bracket of a pair of Gauss constraints, Ci’s, is isomorphic to the algebra of GR as it can be easily seen. To see some work done in this context, we refer the reader to [5].

5 3 Generators of Asymptotic symmetries for U(1)3 model

As the U(1)3 model is a test laboratory for GR, it is of interest to know whether boundary conditions and asymptotic symmetries of these two theories are identical. The question to be answered in this section is whether the ISO(4) group can be considered as the asymptotic symmetries of the model. In other words, is there a way to construct well-deﬁned functionals from the constraints (2.17) while the lapse and the shift include the ISO(4) generators? Although, the model being persued is not Euclidean GR and we don’t presume that the ISO(4) group is the asymptotic symmetry, it is appealing to investigate to what extent the model admits a subgroup of the ISO(4) group. In what follows we examine the constraints (2.17) and try to make them well-deﬁned.

Gauss constraint The action of the Gauss constraint on the phase space variables is

j i j j δ A = Ci[Λ ], A = ∂aΛ Λ a { a} − a i a (3.1) δ E = Ci[Λ ], E = 0 Λ j { j } Thus one sees that

j 3 j a j a 3 j a 3 j a δCj [Λ ]= d x Λ ∂aδE = dSa Λ δE d x (∂aΛ )δE = d x (∂aΛ )δE j j − j − j Z I Z Z (3.2) = d3x (δ Aj )δEa (δ Ea)δAj Λ a j − Λ j a Z a 2 j 1 is functionally diﬀerentiable. Here the surface term has been dropped because δEj = (r− ) and Λ = (r− ). a 2 j 3 O O As ∂aE = O(r ) odd, the integrand of Cj[Λ ] is (r ) odd and hence the constraint is also ﬁnite. j − O −

Scalar constraint It is straightforward to see that

i i j a δN A (x)= C[N], A (x) = 2NǫijkF E c { c } − ac k c c c b (3.3) δN E (x)= C[N], E (x) = 2ǫikl∂b(NE E ) i { i } k l

Thus the variation of this constraint is

3 j a b j b a δC[N]= d xǫjklN δFabEk El + 2FabEl δEk Z 3 j j a b j b a = d xǫjklN (∂aδA ∂bδA )E E + 2F E δE b − a k l ab l k Z 3 a b j j a b j b a = 2 d xǫjkl ∂a(NE E δA ) δA ∂a(NE E )+ NF E δE (3.4) k l b − b k l ab l k Z 3 j b a j a b a b j = d x 2ǫjkl NF E δE δA ∂a(NE E ) + 2 dSa(NE E δA ) ab l k − b k l k l b Z Z 3 j b k a a b j = d x δA (δN E ) (δN A )δE + 2δ dSa ǫjkl(NE E A ) b j − a k k l b Z I We have pulled the variation out of the surface integral in (3.4) because the correction terms are (r 1) even O − for a translation and (1) odd for a boost. Now we deﬁne the new generator as O

j a b C′[N] := C[N] 2 dSa NǫjklA E E (3.5) − b k l I

6 which is functionally diﬀerentiable. At this step one is supposed to check whether it is ﬁnite.

3 j a b j a b C′[N]= d xǫjkl F E E N 2∂a(A E E N) ab k l − b k l Z 3 1 j a b a b j j b a j a b j a b =2 d xǫjkl F E E N E E N∂aA A E N∂aE A E N∂aE A E E ∂aN (3.6) 2 ab k l − k l b − b l k − b k l − b k l Z 3 j b a j a b j a b = 2 d xǫjkl A E N∂aE + A E N∂aE + A E E ∂aN − b l k b k l b k l Z Here terms of the form AEN∂E are (r 4) even for a translation and (r 3) odd for a boost. Thus they are O − O − convergent. The last term which is of the form AEE∂N vanishes for a translation but is divergent for a boost. Therefore, C′[N] is well-deﬁned for a translation and the source of its divergence for a boost is

3 j a b 3 1 j a b 3 j a b 2 d xǫjkl(A E E βa)= d x (βaǫjklg δ δ ) d x βaǫjklA δ E + finite − b k l − r2 b k l − b k l Z Z Z (3.7) 3 j a b = d x βaǫjklA δ E + finite − b k l Z j where in going from the first line to the second one we have used the parity of gb to drop the linear singularity and so we are left with the logarithmic singularity. Wishing to get rid of the divergence, one has to subtract (3.7) from (3.5) but this would play an undesirable role on the constraint surface since (3.7) is proportional neither to the constraints nor to a part of them. Consequently, time translations have a well-defined generator (3.5), but boosts do not! A more detailed argument is given in section 4.

Vector constraint The vector constraint acts on the canonical variables as follows

i a i i δN~ Ac(x)= Ca[N ], Ac(x) = N~ Ac c { a c } −L c (3.8) δ ~ E (x)= Ca[N ], E (x) = ~ E N i { i } −LN i So, variation of the constraint is

a 3 a j b j b j b j b δCa[N ]= d x N δF E + F δE δA ∂bE A ∂bδE ab j ab j − a j − a j Z 3 a j b j b j b j b j b j b = d x N ∂aδA E ∂bδA E + ∂aA δE ∂bA δE δA ∂bE A ∂bδE b j − a j b j − a j − a j − a j Z 3 j a b j a b a j b a j b = d x δA ∂b(N E ) δA ∂a(N E )+ N ∂aA δE N ∂bA δE a j − b j b j − a j Z a j b a j b a b j b a j b j a N δA ∂bE + ∂b(N A )δE + dSa (N E δA N E δA N A δE ) − a j a j j b − j b − b j I (3.9) 3 j b a b a b a j j a = d x δA E ∂bN ∂b(N E ) + δE N ∂aA + A ∂bN a j − j j b a Z h i h i a b j b a j + dSa (N E δA N E δA ) j b − j b I 3 j a b j a b b b a j = d x δA ~ E + δE ~ A + dSa (N E δA N E δA ) a −LN j j LN b j j − j b Z h i I 3 j a b j a b j b a j = d x δA δ ~ E δE δ ~ A + δ dSa (N E A N E A ) a N j − j N b j b − j b Z I h i Here the third term of the surface integral in the fourth line is (1) odd for a rotation and O(r 1) even for a O − translation and so can be put away. Furthermore, in the last step one can pull the variation out of the surface integral since the correction terms are (r 1) even for a translation and (1) odd for a rotation. O − O

7 So the new generator should be deﬁned as

a a a b b a j C′ [N ] := Ca[N ] dSa N E N E A (3.10) a − j − j b I which is functionally diﬀerentiable. To check its ﬁnitess, we rewrite (3.10) as a volume integral

a 3 a j b a j b a b j b a j C′ [N ]= d x N F E N A ∂bE ∂a(N E A N E A ) a ab j − a j − j b − j b Z h i 3 a j b a j b a b j a b b a j = d x N F E N A ∂bE N E F ∂a(N E N E )A ab j − a j − j ab − j − j b Z h i (3.11) 3 a j b a b b a j = d x N A ∂bE + ∂a(N E N E )A − a j j − j b Z h i 3 j b a a b a b 3 j b = d x A E ∂aN + N ∂aE E ∂aN = d x A ~ E − b j j − j − bLN j Z h i Z 4 3 In the last line of (3.11), the term AN∂E is (r− ) even for a translation and (r− ) odd for a rotation which j b a O O a a means it is convergent. AbEj ∂aN vanishes for both translation and rotation because α is a constant and βb is 2 antisymmetric. On the other hand, the other term of the form AE∂N is (r− ) odd for a rotation and vanishes a O for a translation. Thus, Ca′ [N ] is well-defined for a translation and the source of its divergence for a rotation is f a d3x AjEaβb = d3x βbAj δa + j + . . . b j a a b j r Z Z f a = d3x βbAjδa + d3x βbAj j + finite (3.12) a b j a b r Z Z 3 b j a = d x βaAbδj + finite Z where the second integral in the second line is convergent since its integrand is (r 3) odd. Again, (3.12) can O − not be written in terms of the constraints and thus subtracting the volume integral from (3.10) is not allowed. Consequently, spatial translations have a well-defined generator (3.10), but rotations do not!

4 Comparison with the SU(2) case

In this section, we intend to place the situation under scrutiny and see what exactly causes the diﬀerence between U(1)3 and SU(2) cases that the former does not admit generators for boosts and rotations, but the latter does. i i i+ i + First, we split Fab and Gi into its Abelian and non-Abelian parts, i.e. Fab = Fab + Fab− and Gi = Gi + Gi− i+ i i i j k + a j a where Fab = ∂aAb ∂bAa, Fab− = ǫijkAaAb , Gi = ∂aEi and Gi− = ǫijkAaEk ; accordingly the Hamiltonian − i and diﬀeomorphism constraints have also two parts corresponding to the plus and minus pieces of Fab and Gi, + a + a a namely H[N]= H [N]+ H−[N] and Ha[N ]= Ha [N ]+ Ha−[N ], where

+ 3 j+ a b 3 j a b H [N]= d xNǫjklFab Ek El ,H−[N]= d xNǫjklFab−Ek El (4.1) Z Z + 3 a j+ b j + 3 a j b j H [N]= d x N (F E A G ),H−[N]= d x N (F −E A G−) = 0. (4.2) a ab j − a j a ab j − a j Z Z i 4 2 Due to the boundary conditions, F − = (r ) even and G− = (r ) odd. Hence, the integrand of H [N] ab O − i O − − is (r 4) even for a translation and (r 3) odd for a boost. Therefore the minus parts of these constraints are O − O − already ﬁnite. We show that H−[N] is also functionally diﬀerentiable. Its action on the canonical variables is

l l m n b δ − A := H−[N], A (x) = 2NǫilkǫimnA A E N c { c } − c b k c c c b n (4.3) δ − E := H−[N], E (x) = 2NǫijkǫilnE E A N l { l } j k b

8 Using (4.3), one observes

3 j a b j a b j a b δH−[N]= d x Nǫjkl(δFab−Ek El + Fab−δEk El + Fab−Ek δEl ) Z 3 l c b n c m n b = d x (δAc(2NǫijkǫilnEj EkAb )+ δEl (2NǫilkǫimnAc Ab Ek)) (4.4) Z 3 l c c l = d x (δA (δ − E ) δE (δ − A )) c N l − l N c Z + + a which means H−[N] is differentiable. Consequently, what need to be modified are H [N]= C[N] and Ha [N ]= a 3 Ca[N ], thus all failure to be well-defined is rooted in the U(1) part of the Hamiltonian and diffeomorphism constraints existing in the SU(2) case. So, as long as finding the source of divergence and non-differentiability is concerned calculations are the same in both cases. This brings us back to (3.7) and (3.12) for boosts and rotations respectively. For a boost the source of divergence can be expressed as

3 j a b 3 a 3 k + d x βaǫjklA δ E = d x (βaδ )G− = d x Λ¯ (Gk G ) − b k l k k B − k Z Z Z k 3 k b = Gk[Λ¯ ] d x ∂b(Λ¯ E ) (4.5) B − B k Z k k b = Gk[Λ¯ ] dSbΛ¯ E B − B k I ¯ k where we used ∂bΛB = 0. As expected, at the end the volume term is proportional to Gk and does not have any impact on constraint surface. One can verify that J[N] in (2.12) is the final well-defined generator. Actually 3 the subtle point is exactly here. In the U(1) case, the absence of Gk− is responsible for excluding a well-defined boost generator. Going to (3.12) and trying to get rid of it, one sees

3 j b a 3 j b ¯ i d x Abβaδj = d x Ab(ǫijkδkΛR) Z Z b 3 j b i 3 j fk i = d x A (ǫijkE Λ¯ ) d x A (ǫijk Λ¯ )+ finite b k R − b r R Z Z 3 ¯ i = d x ΛRGi− + finite (4.6) Z 3 i + = d x Λ¯ (Gi G )+ finite R − i Z i i a = Gi[Λ¯ ] dSa Λ¯ E + finite R − R i I i 3 where ∂aΛ¯ = 0 has been used and the second integral in the second line is dropped since it is (r ) odd. R O − Again the volume term is proportional to the Gauss constraint as desired. It is straightforward to investigate that a Ja[N ] is the well-defined generator for the spatial translations and rotations. Here, just like as (4.5), presence 3 of Gi− (which is zero in the U(1) case) plays a crucial role to obtain the generator. In addition to the technical reasoning one can add a further intuitive one: The SU(2) Gauss constraint generates rotations on the internal tangent space associated with the internal indices j,k,l,... while asymptotic rotations act on the spatial tangent space corresponding to the indices a, b, c, . . . . Due to the boundary conditions Ea δa these tangent spaces get identified in leading order so that it is not surprising that one can “undo” j ∝ j an unwanted asymptotic rotation by an internal one. This cannot work in the U(1)3 case because the Gauss constraint does not generate internal rotations.

9 5 Conclusion

Motivated by the fact that the GN 0 limit of Euclidian general relativity is an interesting toy model for GR, → this paper is devoted to study its boundary conditions yielding well-defined symplectic structure and finite and integrable charges associated with the asymptotic symmetries. We have demonstrated that in U(1)3 model, the boundary terms spoiling functionally differentiability of the constraints are exact one-forms, and therefore all constraints can be improved to differentiable functionals. However, these functionals are not finite for boosts and j a rotations and we have shown that the reason is the lack of the non-Abelian term, that is ǫijkAaEk , in the Gauss constraint of this model compared to that of general relativity. The other motivation for this paper came from our companion paper [7] where we needed the decay behaviour of the phase space variables in order to select appropriate gauge fixings and Green functions.

Acknowledgements

S.B. thanks the Ministry of Science, Research and Technology of Iran and FAU Erlangen-N¨urnberg for ﬁnan- cial support.

References [1] A. Ashtekar: “New Hamiltonian formulation of general relativity”, Phys. Rev. D36 (1987), 1587 A. Ashtekar, “New Variables for Classical and Quantum Gravity”, Physical Review Letters 57 (1986) 2244–2247. J. Barbero, “Real Ashtekar variables for Lorentzian signature space-times”, Physical Review D 51 (1995) 5507–5510, [gr-qc/9410014] [2] T. Thiemann, “Quantum Spin Dynamics (QSD)”, Class. Quantum Grav. 15 (1998), 839-873. [gr-qc/9606089] [3] “Loop Quantum Gravity - The First 30 Years”, A. Ashtekar, J. Pullin (eds.), World Scientific, Singapore, 2017. J. Pullin, R. Gambini, “A First Course in Loop Quantum Gravity”, Oxford University Press, Oxford, 2011. C. Rovelli, “Quantum Gravity”. Cambridge University Press, Cambridge, 2008. T. Thiemann, “Modern Canonical Quantum General Relativity”, Cambridge University Press, Cambridge, 2007 [4] L. Smolin, “The GNewton 0 limit of Euclidean quantum gravity”, Class.Quant.Grav. 9 (1992) 883-894, → [hep-th/9202076] [5] C. Tomlin, M. Varadarajan, “Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity” Phys. Rev.D87 (2013) no.4, 044039, [gr-qc/1210.6869] A. Laddha, “Hamiltonian constraint in Euclidean LQG revisited: First hints of off-shell Closure”, [gr- qc/1401.0931] A. Henderson, A. Laddha, C. Tomlin, “Constraint algebra in loop quantum gravity reloaded. II. Toy model of an Abelian gauge theory: Spatial diffeomorphisms” Phys.Rev. D88 (2013) no.4, 044029, [gr-qc/1210.3960] A. Henderson, A. Laddha, C. Tomlin, “Constraint algebra in loop quantum gravity reloaded. I. Toy model of a U(1)3 gauge theory”, Phys.Rev. D88 (2013) 4, 044028, [gr-qc/1204.0211] M. Varadarajan, “On quantum propagation in Smolin’s weak coupling limit of 4d Euclidean Gravity”, Phys. Rev. D 100, 066018 (2019), [gr-qc/1904.02247] [6] P. A. M. Dirac: “Lectures on quantum mechanics”, J. Math. Phys. 1 (1960), 434. [7] S. Bakhoda, T. Thiemann: “Reduced Phase Space Approach to the U(1)3 model for Euclidean Quantum Gravity” [8] R. Beig and N. óMurchadha: “The Poincarégroup as the symmetry group of canonical general relativity, Annals Phys. 174 (1987), 463. [9] T. Thiemann: “Generalized boundary conditions for general relativity for the asymptotically flat case in terms of Ashtekar’s variables” Class. Quant. Grav. 12 (1995), 181. arXiv:9910008 [gr-qc].

10 [10] M. Campiglia: “Note on the phase space of asymptotically flat gravity in Ashtekar-Barbero variables”, Class. Quant. Grav. 32 (2015), 14. arXiv:1412.5531 [gr-qc]. [11] T. Regge and C. Teitelboim, “Role of Surface Integrals in the Hamiltonian Formulation of General Relativity”, Annals Phys. 88 (1974) 286 R. Sachs, “Asymptotic symmetries in gravitational theory”, Phys. Rev. 128 (1962) 2851 E. T. Newman and T. W. J. Unti, “Behavior of Asymptotically Flat Empty Spaces”, J. Math. Phys. 3 (1962) no.5, 891 R. Penrose, “Asymptotic properties of fields and space-times”, Phys. Rev. Lett. 10 (1963) 66. R. Penrose, “Zero rest mass fields including gravitation: Asymptotic behavior”, Proc. Roy. Soc. Lond. A 284 (1965) 159 [12] G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited”, Phys. Rev. Lett. 105 (2010) 111103 [gr-qc/0909.2617] M. Campiglia and A. Laddha, “Asymptotic symmetries and subleading soft graviton theorem”, Phys. Rev. D 90 (2014) 12, 124028 [hep-th/1408.2228] M. Campiglia and A. Laddha, “New symmetries for the Gravitational S-matrix” JHEP 1504 (2015) 076 [hep-th/1502.02318] J. Corvino and R. M. Schoen, “On the asymptotics for the vacuum Einstein constraint equations”, J. Diff. Geom. 73 (2006) 2, 185 [gr-qc/0301071] L. H. Huang, “Solutions of special asymptotics to the Einstein constraint equations”, Class. Quant. Grav. 27 (2010) 245002 [gr-qc/1002.1472] M. Henneaux, C. Troessaert, “BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach”, JHEP 03, 147 (2018), [gr-qc/1801.03718 ]