Continuously distributed holonomy-flux algebra
aa r X i v : . [ g r- q c ] J a n Continuously distributed holonomy-flux algebra
Jakub Bilski ∗ Institute for Theoretical Physics and Cosmology,Zhejiang University of Technology, 310023 Hangzhou, China
The procedure of the holonomy-flux algebra construction along a piecewise linear path, which con-sists of a countably infinite number of pieces, is described in this article. The related constructionapproximates the continuous distribution of the holonomy-flux algebra location along a smooth linkarbitrarily well. The presented method requires the densitized dreibein flux and the correspondingoperator redefinition. The derived result allows to formulate the gravitational Hamiltonian con-straint regularization by applying the Thiemann technique adjusted to a piecewise linear lattice.By using the improved Ashtekar connection holonomy representation, which is more accurate thanthe one used in canonical loop quantum gravity, the corrections related to the redefined densi-tized dreibein flux vanish. In this latter case, the Poisson brackets of the continuously distributedholonomy-flux algebra along a link between a pair of nodes are equal to the brackets for thesesmeared variables located at the nodes.
I. INTRODUCTION
Loop quantum gravity (LQG) [1, 2] is a lattice theory ex-pressed in terms of honolomies and fluxes of the Ashtekarvariables [3]. The considered lattice is constructed asa finite collection of links and nodes. The former ob-jects are the oriented graph’s edges with attached SU (2) holonomies. They intersect at the nodes defined as thegraph’s vertices with attached intertwiners. These latterobjects implement the SU (2) invariance on the lattice byaveraging possibly different group elements of adjacent(intersecting) links [4, 5]. It is believed that in the largelinks number limit this discrete system approximates thestandard continuous formulation of the Einstein gravita-tional theory [6]. This limit is controlled by the smallvalues of the parameters that determine the lengths ofthe links (or by the parameter if one assumes an equallength). All the aforementioned structure specifies howthe graph is embedded as the lattice into the spatial man-ifold, the geometry of which it represents.The action of the flux operator in LQG modifies theSU (2) elements at the points where the surface on whichthis operator is defined intersects with links. To preservethe gauge invariance along the links, these points haveto be identified as the nodes with appropriate intertwin-ers attached. It is worth noting that this feature of themodel prevents the values of regulators in the Hamilto-nian constraint from being taken to zero. These valuesdetermine the two points splitting; hence they are as-sociated with the lengths of the links, for instance, byidentifying these lengths with the regulators. If it wouldbe possible to send the regulators to zero, the intersec-tion points of the links and surfaces would need to bedensely-defined on the lattice. This case would lead toa pathological structure. Therefore, the constructionalrestriction that permits taking the value of regulator tozero is introduced by the aforementioned identification of ∗ [email protected] the lengths of the links with the regulators. As a resultof this restriction, the classical limit cannot be directlyrestored, hence the continuous limit of the theory cannotbe reached.In this paper, the flux fields modified by group aver-aging density distributions along links are defined. Al-though these fields act by adding the su (2) generatorsat links in the same manner as the flux operators in theoriginal LQG formulation [2], they do not spoil the SU (2) symmetry. This is possible due to their construction,which connects the separated (by the flux action) pairsof generators (in the holonomy exponential maps to thegroup elements). Moreover, by using the particular func-tionals of holonomies, the constructional modification ofthe flux operators does not contribute to the results oftheir actions. It should be emphasized that these func-tionals reflect the structure of the Ashtekar connectionsinside the graph’s elementary cells [10]. Moreover, theyare the only contributions of the links-defined holonomiesthat are present in the lattice-smeared Hamiltonian con-straint [7, 10]. Furthermore, the preserved gauge invari-ance should not be surprising, because the presence ofintertwiners on the lattice is the result of solving thefirst-class constraint (known as the Gauss constraint) in-dependently of the fluxes-consisting scalar constraint.The introduced modification allows then to derive thecontinuous and classical limits of the lattice gravity ex-actly. In this case, although the lengths of the links re-main finite, the regulators in the scalar constraint can beformally taken to zero. II. ALGEBRA AT THE BOUNDARY
In this article, a simple toy model consisting of a singlepiecewise linear link l h v i := l h v, v + l i := l h v, v + l/N i◦h v + l/N, v + 2 l/N i◦ ... ◦h v + ( N − /N, v + l i (1)is going to be considered. In the limit when the numberof pieces N goes to infinity, this structure approximatesa smooth link arbitrarily well. Let the endpoints of thelink be called the boundary and all the structure inside— the interior.By considering the limit N → ∞ in (1), the piecewiselinear framework acquires the physical applicability, cf. [2]. The lattice composed of the related piecewise linkslattice is the quantity on which the gravitational theory isconstruable [7]. Moreover, the diffeomorphism invarianceon this structure is relatively easily implementable [8, 9];thus, this latter issue is not going to be discussed in thisanalysis.The canonical fields A a := A ja τ j and E a := E aj τ j ,where τ j := − i σ j / is the generator of the su (2) algebra( σ j represents a Pauli matrix), are regulated differently.Let h − l h v i := h − l h v,v + l i := Q j =1 (cid:0) h ( j ) l h v i (cid:1) − denote the SU (2) holonomy of the su (2) -valued connection A a , where theindices in brackets are not implicitly summed. The re-ciprocal factors are defined by h ( j ) l h v i := P exp (cid:18)Z l h v i ds ˙ ℓ a h v i ( s ) A ( j ) a (cid:0) l ( s ) (cid:1) τ ( j ) (cid:19) , (2)where l h v i := L ε h v i is the length of the (piecewise lin-ear) path, L represents a fiducial length scale, and ε h v i denotes the path-related dimensionless parameter.The holonomy expansion around the parameter ε h v i can be expressed as a polynomial of the connection, h ± l h v i = ± l h v i (cid:0) A l h v + l i − A l h v i (cid:1) + 12 l h v i A l h v i + O ( ε ) . (3)This formula is obtained by applying the derivative ex-pansion (cid:0) A l h v + l i − A l h v i (cid:1) /l h v i = ∂ l A l h v i + O ( ε ) andleads to the following relation between the connectionsand holonomies, h l h v i − h − l h v i = l h v i (cid:0) A l h v + l i − A l h v i (cid:1) + O ( ε ) . (4)This link-smeared representation of the connections lo-cated at nodes supports the construction of the latticegravity in [7, 10].The lattice quantity that represents the densitizeddreibein degree of freedom is the flux through the sur-face orthogonal to the spatial direction of this densi-tized dreibein. Let S l h v i denotes the surface of area l ⊥ h v i := ǫ ll ⊥ l ′⊥ L ε ⊥ h v i ε ′⊥ h v i normal to the link l h v i atthe node v . The related flux is defined as f i (cid:0) S l h v i (cid:1) := Z S l h v i n a E ai := Z S l h v i dy dz ǫ abc E ai ∂ y x b ∂ z x c ≈ ε ⊥ h v i ε ′⊥ h v i L E li h v i , (5)where n a projects E ai into the directions normal to thesurface at each point. The approximation becomes theequality in the flat limit. Next, the smearing of the densitized dreibein along thelink l h v i is defined. The continuous probability distribu-tion of its homogeneous linear density E i ( l h v i ) reads ¯ E l (cid:0) R h v i (cid:1) := Z l h v i E i ( l h v i )= 1 L ε h v i Z dl h v i E i ( l ) ≈ mean E li (cid:0) l h v i (cid:1) , (6)where ¯ E l (cid:0) R h v i (cid:1) := ¯ E l (cid:0) l h v i∧ S l h v i (cid:1) = ¯ E l (cid:0) l h v i∧ l ⊥ h v i∧ l ′⊥ h v i (cid:1) .The last relation in (6) is exact regarding the linear path,where mean E li (cid:0) l h v i (cid:1) := 12 (cid:0) E li h v i + E li h v + l i (cid:1) . (7)This quantity, which replaces the probability distributionin the linear path approximation, equals to the densitizeddreibein that is averaged at the boundary.
12 + 12
FIG. 1. Flux average at the boundary
Then, the Poissson algebra regarding the canonicalADM fields [11] q ab and p ab can be derived. Consid-ering the lattice variables intersecting at the boundary(see FIG. 1), one obtains n h l h v i , mean f i (cid:0) S l h v i (cid:1)o = 12 (cid:0) τ i h l h v i + h l h v i τ i (cid:1) . (8)The su (2) generators appeared at the nodes, i.e. at thepoints, where the intertwiners implement group averag-ing of generators transformations. Therefore, the link(with the holonomy) remained SU (2) -invariant. III. GAUGE SYMMETRY ALONG THE LINK
The analysis of the algebra in the interior requires moreprecision. Considering the flux of the ¯ E l (cid:0) R h v i (cid:1) proba-bility distribution along l h v i , the Poisson brackets (seeFIG. 2) read n h l h v i , ¯ f li (cid:0) R h v i (cid:1)o = lim N →∞ N τ i h l h v i + h l h v i τ i + 2 N − X n =1 h l h v,v + nl/N i τ i h l h v + nl/N,v + l i ! . (9)Here, the trapezoidal rule was applied to the integrationalong the piecewise linear path in (1). Each of the el-ements in the sum in the lower line of (9) violates thegauge symmetry.One can demonstrate this problem explicitly. Let apoint w ∈ l h v i in the link’s interior be selected, hence v = w = v + l . Next, it is worth to define the followingauxiliary group elements, + h := gh = hg = h , − h := g − h = hg − = h . (10)The SU (2) -invariant holonomy satisfies gh l h v,v + l i g − = + − h l h v,v + l i = − + h l h v,v + l i = h l h v,v + l i = h l h v,w i h l h w,v + l i = + h l h v,w i − h l h w,v + l i , (11)for any pair g, g − ∈ SU (2) . Then, for the su (2) -modifiedinterior elements in the sum in the lower line of (9), onefinds g h l h v,w i τ i h l h w,v + l i g − = + h l h v,w i τ i h − l h w,v + l i = h l h v,w i τ i h l h w,v + l i . (12)By adding an intertwiner at W would restore the gaugesymmetry of the system; this object is a group averag-ing projector. It, however, becomes positioned at eachnode after solving one of the first-class constraints, calledthe Gauss constraint. The action of holonomies and fluxes is given by another one, which has to be imple-mented only after the Gauss constraint. This restric-tion occurs because the linear combination of the latteris included in the former Hamiltonian constraint. There-fore, this combination vanishes identically only after theGauss constraint is solved. This procedure determinesthe constraints implementation order by the LQG tech-nique [1, 2]. As a result, holonomies and fluxes act onthe gauge-invariant lattice. Thus the action of the fluxprobability distribution ¯ f li (cid:0) R h v i (cid:1) is ill-defined (in the in-terior), in the sense that it breaks the gauge symmetry.To restore the symmetry, the following modification ofthe densitized dreiben probability distribution in (6) isintroduced, ¯ G l (cid:0) R h v i (cid:1) := 1 L ε h v i Z v + lv ds ˙ l ( s ) g (cid:0) l ( s ) (cid:1) E i (cid:0) l ( s ) (cid:1) g − (cid:0) l ( s ) (cid:1) ≈ mean E li (cid:0) l h v i (cid:1) . (13)The last approximation remains correct, because mean E li (cid:0) l h v i (cid:1) acts only at the nodes, i.e. at the points, wherethe invariance is controlled by the intertwiners. The lat-ter objects are adjusted for any SU (2) elements at theintersecting links; hence also for the elements modifiedby the g or g − multiplication. FIG. 2. Flux probability distribution along the link
Analogously, one defines the gauge-invariant flux prob-ability distribution ¯ g li (cid:0) R h v i (cid:1) := 1 L ε h v i Z v + lv ds ˙ l ( s ) g − (cid:0) l ( s ) (cid:1)Z S l ( s ) n a E ai (cid:0) l ( s ) (cid:1) g (cid:0) l ( s ) (cid:1) . (14)This leads to the following improvement of the algebrain (9) (see also FIG. 2), n h l h v i , ¯ g li (cid:0) R h v i (cid:1)o = lim N →∞ N τ i h l h v i + h l h v i τ i + 2 N − X n =1 3 Y j =1 h ( j ) l h v,v + nl/N i τ i h ( j ) l h v + nl/N,v + l i ! . (15)The product in the second line expresses the trans-fer of the su (2) generators between l h v, v + n l/N i and l h v + n l/N, v + l i in the exponential maps from represen-tations to group elements. The gauge symmetry of each element is preserved along the whole link l h v i , exceptthe point v + n l/N . By using the abstract notation, thisissue can be simply expressed in the relation g h l h v,w i g − τ i gh l h w,v + l i g − = Y j =1 h ( j ) l h v,w i τ i h ( j ) l h w,v + l i . (16) IV. ALGEBRA ALONG THE LINK
It is convenient to introduce the auxiliary quantity thatrepresents the l/N -short interval ∆ l h v n i := l h ¯ v n − l/ (2 N ) , ¯ v n + l/ (2 N ) i (17)centered at ¯ v n := v + ( n − / l/N . Thus, the lower lineof (15) takes the form N − X n =1 3 Y j =1 n Y p =1 N Y q = n +1 h ∆ l h v p i τ i h ∆ l h v q i . (18)In the limit N → ∞ , the connection in the exponent ofeach h ∆ l h v p i is arbitrarily well approximated by the con-stant connection A jl h ¯ v p i that is located at the interval’scenter. Consequently, each holonomy can be expressedby h ∆ l h v p i = cos (cid:0) α ( j ) p (cid:1) + 2 sin (cid:0) α ( j ) p (cid:1) τ ( j ) , (19)where the notation was simplified by introducing anotherauxiliary variable α jp := ∆ l h v ( p ) i A jl h ¯ v ( p ) i . As a result,the formula in (18) takes the form ( N − (cid:0) τ i h l h v i + h l h v i τ i (cid:1) + ( N − − cos (cid:18) l h v i X j =1 A ( j ) l (cid:0) l h v i (cid:1)(cid:19)! τ i − N − X n =1 sin n X p =1 α ip − N X q = n +1 α iq ! , (20)where A ( j ) l ( l h v i ) := P Np =1 A jl h ¯ v p i . For detailed derivationsee [7]. The last term in (20) vanishes in the limit N → ∞ and only the first pair of elements contributes to (15), n h l h v i , ¯ g li (cid:0) R h v i (cid:1)o = 12 τ i h l h v i + h l h v i τ i + 2 sin (cid:18) L ε l h v i X j =1 A ( j ) l (cid:0) l h v i (cid:1)(cid:19) τ i ! . (21)If the O ( ε ) corrections are neglected, this result coin-cides with the gauge-invariant expression at the bound-ary, given in (8). Moreover, despite its complicated struc-ture, this outcome preserves the symmetry even if thesecorrections are included.Eventually, one should consider the symmetric distri-bution of the Ashtekar connections introduced in (4).This distribution coincides with the only structure ofholonomies, which is present in the Poisson bracketsin the Hamiltonian constraint of the lattice gravity in[7, 10]. In this case, in which the regularization lattice isquadrilaterally-hexahedral, the O ( ε ) corrections vanishand the final result becomes n(cid:16) h l h v i − h − l h v i (cid:17) , ¯ g li (cid:0) R h v i (cid:1)o = 12 (cid:18) τ i (cid:16) h l h v i − h − l h v i (cid:17) + (cid:16) h l h v i − h − l h v i (cid:17) τ i (cid:19) . (22)It is easy to see that the same outcome is obtained con-cerning the connections-holonomies relation in (4) andthe fluxes mean f i (cid:0) S l h v i (cid:1) acting only at the boundary. V. CONCLUSIONS
In the analysis in this article, it was demonstrated howthe specific modification of the densitized dreibein flux representation leads to its well-defined action on links’interiors. Moreover, this result coincides with the sameaction at nodes if one considers the representation in (4).This latter representation appears to be physically fa-vored due to the geometrical arguments. It is the link-defined holonomy functional in the symmetry-preservingmap from the set of the continuous variables forming theHamiltonian constraint into its lattice-smeared equiva-lents [10]. The densitized dreibein flux functional inthis map is given by the probability distribution in (14).Thus, the whole symmetry-preserving map from the con-tinuous into the lattice variables is self-consistent [7].This article provided another argument to favor theconnections-holonomies relation in (4).Taking the limit ε H → , while keeping ε Γ small butfinite becomes possible. Here, ε H is the Hamiltonian-related and ε Γ is the lattice-related regularization pa-rameter, respectively. However, the removal of these tworegulators identification is just a formal manipulation.The spectrum of the scalar constraint operator is not zeroonly on the states on which it acts, thus on the existinglinks. Therefore, the release of the regulators identifi-cation is only a formal method to obtain, by an exactprocedure, the classical Hamiltonian from the operator.One more comment is worth to be added. It is possible todefine the vacuum states for the zero-valued connections A = 0 that lead to the vanishing holonomies by using theholonomy distribution in (4) (see discussions in [7, 10]).The related links, however, are not going to be removed.This comment clarifies the restriction of the Hamiltonianconstraint operator domain to the existing links.Finally, in the Abelian case, the result in (22) equalsthe analogous cosmologically-reduced derivation in [12].However, it is worth noting that this latter outcomeis not calculated by using the expansion in (3). It isbased on the explicit treatment of a holonomy as anSU (2) group element and the holonymy’s exponent asthe representation of this Lie group [13]. ACKNOWLEDGMENTS
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