in this paper we explore an idea recently introduced by , which incorporates topological information into the existing structures of loop quantum gravity .essentially , in addition to the geometric data contained in the holonomy , this proposal uses the monodromy to encode topology . in this waywe hope to extend loop quantum gravity to the topological realm , so it is clear that background independence " applies not just to a geometric background but a topological one as well . as discussed in , general relativity is only background independent ( _ relational _ ) with respect to the geometry , and not to the topology , dimension , or differential structure . in the approach discussed in this paper it is possible , at least in the quantum case , to extend gravity to be topologically relational as well .there is also a connection between topological relationalism and the `` problem of time '' , which is nicely reviewed in .in addition , this idea may provide a way to take loop quantum gravity to the classical limit , by telling us exactly how to fill in the spaces " of a spin network .the main motivation for this paper is to show that this approach does not modify the basic constructions of loop quantum gravity .a secondary goal is to determine the action of the area and hamiltonian operators on topspin network states .thus , will be only interested in details directly relevant for the construction of the phase space , state space , and quantum operators .the basic results are as follows :  * the structure group will be replaced with for finite group . *the lie algebra will be replaced with the universal enveloping algebra . *the area spectrum is unchanged , but is presented in a form where topological data is now manifest . *the action of the hamiltonian is different due to the equivalent realizations for the topology of the spatial sections .in the rest of this introductory section we will review loop quantum gravity and the topspin network formalism , focusing on the most relevant parts of both . in [ s : variables ] we will discuss the new symmetry which results from the modification of the spin networks and how this changes the phase space of the theory . in [ s : cyl ] we will discuss the state space of the theory , and in [ s : operators ] we present the area operator and explore some differences which result from considering how the hamiltonian acts on topspin network states . [ s : conclusion ] will conclude and interpret some of our results in the context of topology change in quantum gravity .we follow the einstein summation convention , where repeated indices are always summed over .indices that are not to be summed over will be denoted with a bar , such as in the metric .we will begin with a brief review of loop quantum gravity ( lqg ) , highlighting specific constructions which we will be discussing later .there are many nice reviews on the subject , both technical and non - technical .this paper will be borrowing most of the concepts and notations from the texts .specifically , we will mostly follow the canonical approach rather than the newer covariant approach .this is because the topspin construction in lqg follows from using the spin networks as graphs embedded in , rather than lattice graphs without an explicit embedding which is the natural place to start for the covariant approach .the canonical and covariant approaches result in the same hilbert space so our choice should only be convenience .one begins by rewriting general relativity in the connection formalism on a 1-foliated spacetime where the foliations are parametrized by coordinate time .this results in a constrained hamiltonian system .solving these constraints has been the major task of the past twenty years ; the first step was the introduction of the _ ashtekar connection _ .the phase space of the theory is given by this connection and the canonical conjugate momentum .these have a formally singular poisson bracket , and so must be smeared ( in one dimension and two dimensions , respectively ) with test functions on a curve and a surface . in addition , to make it easier to impose gauge invariance , the holonomy of this -connection is used along with the canonically conjugate momentum to define the classical algebra .this technical result provides the background - independent formalism which makes this approach so attractive .the states in lqg are given by _cylindrical functions _ on oriented graphs in the spatial slice .each edge of a graph is labeled with an irreducible representation , and each vertex is labeled with an intertwiner for incoming edges and outgoing edges .the connection on the graph is the restriction of the connection on , and functions of the holonomies can be made gauge invariant by contracting them with the intertwiners .these are the _ spin network states _ and are given by the collection ( one can also consider gauge - dependent states , but we will not be discussing those ) . the hilbert space of cylindrical functions can be defined as a projective limit of the continuous functions on spin networks , with a measure inherited from the haar measure on .  this hilbert space is kinematic only , since we have not solved the hamiltonian constraint .geometric operators such as the area and volume can be defined on this hilbert space , and one of the primary results of lqg is that the area of a surface intersecting a spin network is quantized .the exact results depends on regularization , but the _ restricted spectrum _ of the area operator is  where the surface intersects the edges of the spin network in places . is called the _ immirzi parameter _ , and can be fixed by considering the semiclassical area of a black hole and comparing to the above result . in the canonical approach the dynamics of the theoryare given by the wheeler - dewitt equation , where is the hamiltonian .there are still some ambiguities related to solving the hamiltonian constraint in lqg , but for our purposes we can choose a specific approach and only consider its action on simple spin networks .  besides the quantization of area , other results from lqg include the application to early - universe cosmology , black hole physics , and recent attempts to couple lqg to fermions and yang - mills fields .we will not be discussing these applications further , but the results of this paper could easily be extended to them .lqg can be considered a mature theory ; much of the framework has been made rigorous and a variety of calculations can be performed .however , there are several features which are still unclear , one of which is the topological nature of the spin networks .restricting from a smooth three - manifold to an embedded graph naturally trades continuous " data in favor of discrete " data , and it is unclear how one is to keep track of the topological data from the original manifold .indeed , since a given graph can be embedded in many topologically different three - manifolds it would appear that topology is completely absent from the quantum theory .this is likely why it has been so difficult to find the classical limit of lqg - the question of how to pass from the discrete data of the graph back to the continuous data of the spatial sections has not been answered .topological change in gravity has been studied for quite a while . for smooth manifolds ,morse theory tells us that topological change is associated with critical values of smooth functions .it was also shown by that topology changing manifolds with lorentzian metrics must have closed timelike curves .a similar `` no go '' situation occurs in linearized gravity , where topology change is associated with infinite particle and energy production .these results suggest that classically , topology change may be forbidden . on the other hand, we should expect that in a background - independent formulation of quantum gravity , one should not _ impose _ a topology , in the same way that one should not impose a background field . although background independence is usually formulated in terms of geometric independence ( _ i.e. _ the metric ) , presupposing a topology should be just as offensive .it is sometimes remarked that lqg does not actually describe the original spatial section ( usually taken to be a topological three - sphere ) , but rather a space with a large number of holes given by the topological structure of the spin network .since the hamiltonian operator can add edges to a graph , it can change the hole structure of the graph , and thus of .however , this does not solve the problem of being able to pass into a classical limit to check the results of the theory , since ( as discussed above ) , we can easily embed a graph with a large first betti number into a three - manifold with a much smaller , or possibly trivial , first homology group .an alternative idea comes from the borde - sorkin conjecture , which avoids the creation of infinite energy due to topology change by restricting to only causally continuous spacetimes .this is known as causal set theory , and it allows for topology change by a full discretization of spacetime , which is then built up to be a model for a smooth manifold .this approach brings along many difficulties but one surprising result ; it seems to predict the order of magnitude of the cosmological constant .this approach can be thought of as as discrete as possible " , removing all other structures except for a partial ordering between spacetime points .this paper proposes to incorporate topology change into quantum gravity by adding topological data to the already existing spin networks . in this way we can benefit from the extensive theoretical work that has already been done in lqg without further discretizing spacetime as in the causal set approach .we discuss this in detail in the next subsection .first we recall a few definitions :  a continuous map between -dimensional manifolds and is said to be a * covering map * if for every point there exists a neighborhood such that restriction of to each connected component of is a homeomorphism to . is the * covering space * and is the * base space*. if the fiber of the covering map has points we call an * n - fold cover*. a * branched covering map * is a covering map except at a finite number of points such that is a finite set .the set of points is called the * branch locus * , and its complement is again a covering space .we will call a * branched covering space*.  intuitively , a covering space of is just a discrete set of homeomorphic copies ( _ sheets _ ) of which map back to under .a branched covering space is a covering space where the sheets can collide with each other over the branch locus .topspin networks were first introduced by .that paper was motivated by an interesting mathematical fact , first discovered by alexander :   every orientable closed manifold of dimension is a branched covering of , branched along a subcomplex .  in other words, one can construct any closed -dimensional manifold by gluing together pieces of -spheres along -dimensional submanifolds .we will mostly be interested in the case , where the branching occurs along graphs , but the is very relevant for spinfoams . for three - manifolds , this theorem can be strengthened to three - fold covers along a knot ( the hilden - montesinos theorem ) , although here we consider the general case for a covering of any order .  specifying now to , a branched covering of order along an embedded graph is completely determined by a representation of the fundamental group .this gives the classes of loops in the graph complement as elements of the permutation group .ambient isotopy classes of these graphs can be represented in two dimensions with a _planar diagram_. this is a choice for a representative of the class where the neighborhood of each node is a set of arcs coming from the node ( see for more background on embedded graphs in three - dimensional space ) .we can use planar diagrams to give a presentation of the branched covering spaces from the above theorem : given a planar diagram , the representation is determined by assigning elements along the arcs of .each tells us how to exchange covers over the branch locus .these elements must obey the _wirtinger relations_. at crossings with an overcrossing edge and undercrossing arcs and , the first relation is for negatively - oriented crossings , the second is for positively - oriented crossings .there are also relations to be satisfied at the vertices , for incoming arcs and outgoing arcs .these relations encode how the sheets of the covering are stitched together so that is appropriately represented .the idea of a topspin network is to identify the branched covers , decorated with permutation labels under the wirtinger relations , with the usual spin networks of lqg , ensuring that the spin and topological labels are compatible . in this waythe spin networks will not only encode all the geometric information of the gravitational field but the topological information as well .  , intertwiners and topological labels .]  a * topspin network * over a compact lie group is a tuple of data consisting of  * a spin network with an embedded graph , * a representation given by an assignment of to each arc in .see figure [ fig : topspin_ex ] for an example topspin network .this definition is identical to the original one of , although we will be considering only irreducible representations since we want to directly compare this theory to lqg . under covering moves ( which we discuss shortly )one can actually get reducible representations starting with irreducible ones .however , as discussed in , there is a trivial equivalence one gets by demanding that the compositions of adjacent intertwiners are the same , which can be removed by considering the representations as irreducible .along with the addition of topological labels , we will have a set of _ covering moves _ which can modify the labeled branch locus but leave the branched covering space unchanged . in addition , we will need the geometric labels to be compatible under the change as well .these _ geometric covering moves _ were found in , and we reproduce them here in figure [ fig : covering_moves ] .these are basically extensions of the reidemeister moves to our decorated graphs , so that under the geometric covering moves the geometry and topology of the spatial sections do not change .it is also easy to see that we can choose that the number of sheets of the cover does not change under these moves ( _ stabilization _ ) , since we can always add another sheet branched over a trivial link with a simple permutation of two sheets .since alexander s theorem can be extended to dimension four as well , this basic construction can also be extended to spin foams .one can then define _ topspin foams _, which are embedded 2-complexes in  ] of sheets that are identified with the one under the action of and extending the coordinates to .in other words , in a neighborhood above an edge we have here one can write in cycle notation .this is generic since elements of can always be written as products of transpositions for .where transpositions are explicitly indicated , a result on any element can be easily found by iteration .this symmetry in the coordinates corresponds to a symmetry of the spatial metric by restriction to each edge .denote points as those which are the restrictions of the coordinates to the branch locus , .these extend to the cover as above , where  ] . for the algebraic structure associated to will take the group algebra .elements can be written like , where and , indexed by .multiplication in this algebra is given by the convolution product a convenient dimensional representation for the symmetric group is the permutation representation which permutes the basis elements of .we will denote for basis elements in this representation .there is also a natural inner product on this algebra given by the choice of scaling is at this point arbitrary ; we will explain our choice when we discuss the area operator in [ s : operators ] .  the algebraic structure for the fields will be the tensor product of these two algebras : in other words , our fields are now -valued one - forms .the poincar - birkoff - witt theorem gives us a basis for , which are polynomials in the basis of , this is an infinite dimensional vector space with basis elements labeled .if we order the basis lexicographically we can label it with a single index and write elements of as .the spatial metric is then the conjugate variables to our coordinate fields will be the fields associated with the extrinsic curvature it is customary to define the scaled coordinate fields as .here we are raising and lowering indices with delta functions and have defined the fields so that , and .we can equip these fields with a poisson bracket by using the commutator on the basis elements : ,\ ] ] where the poisson bracket is defined with functional derivatives .\ ] ] with this we have our phase space equipped with the standard symplectic structure , where is the coupling constant .this is the same symplectic structure as the usual lqg theory except for the extra index , so the adm variables would be reproduced in the same manner ( see for this calculation ) .the enlargement of the algebra to is a direct consequence of using discrete symmetry groups in a physical theory .we can no longer describe the symmetry of the system using lie algebras because they do not `` see '' discrete groups ( we elaborate on this in the next section ) .thus , we are forced to work with associative algebras , and the universal enveloping algebra is a natural way to connect lie algebras and associative algebras . at any stage in our constructionone could restrict to just the first - order elements to recover the original lie algebra , but in doing this one loses all the algebraic structure of the theory - including the bracket ( [ eq : bracket ] ) .so for full consistency and generality we do not want to do this .in fact , this enlargement is completely in line with what is usually done in quantum theory ; that is , promoting the classical fields to operators .this is done in lqg by writing the area operator as a product of the fields , which we will discuss in more detail in [s : operators ] . sincedifferential operators naturally have an algebraic structure isomorphic to a universal enveloping algebra , we are simply promoting our fields to operators immediately and giving them the appropriate algebraic structure .   we will now give a canonical transformation of the poisson algebra ( [ eq : poisson ] ) , following the usual path of canonical quantization of the gravitational field .the spin connection on is the usual one for extended to the infinite polynomial basis , in the usual manner we can rewrite this as ^{jk} ] , the holonomy is defined via this has the formal solution , but it is not clear what the exponential map should be in our case since is no longer a lie algebra . observe that for a general lie group , the exponential map has domain , the tangent space at the identity .the tangent space for our lie group is isomorphic to since is a finite group and has tangent space 0 .this suggests that our exponential map should actually be the identity on and the usual exponential on .this is reasonable because the holonomies ( and derivatives ) should exist in the continuous directions , not in the finite ones since the tangent space there is trivial .  to implement this idea, we can consider the algebra for lie algebra and finite group to be functions with product here we are writing and is a basis element for .now for the connection , write the holonomy equation for an element as and solutions to this equation will be   we are going to define our exponential map as the standard matrix exponential map on a specific representation of on each edge. this will require the exponential map be formally different on each edge .this is exactly what is usually done in lqg since the spin on each edge labels a specific representation of the gauge group .  an irreducible representation is a lie algebra homomorphism on the space of -dimensional anti - hermitian , traceless matrices over a vector field , where the lie bracket is given by the commutator of matrices .we also have the standard injection . by the universality property, there exists a lie algebra homomorphism such that the diagram commutes , .now we have the standard exponential map , defined by we can construct an exponential map by composition ( see also figure [ fig : exp_diagram ] ) : this map is dependent on the representation that we choose .although it would be preferable to define this exponential map in a representation - independent way , it is not at all clear how to do this .a formal exponential map can be defined on the free lie algebra of a set , but the image of this map is inside rather than a lie group on which we can define cylindrical functions in the usual way .this construction can be extended by using the language of model theory and embedding the universal enveloping algebra into an ultraproduct of matrix lie algebras .this gives an exponential map with image in an ultraproduct of representations of the lie group .in fact , the family of exponential maps we have defined above is inspired by those used on in , which they show can be restricted to .we might be able to define cylindrical functions on an ultraproduct of representations of , but we view this construction as not explicit enough for our use . in our approachthe exponential takes values in the locally compact group , and we can still use the haar measure to define the inner product of cylindrical functions on this group .there is an alternative way of defining an exponential map that would work for our purposes , without having to invoke the general model theory result of .this is to consider the standard angular momentum representation of and identify elements in the universal enveloping algebra with the polynomial angular momenta " ( higher order differential operators ) .then the corresponding exponentials are the unitary operators associated to the polynomial angular momenta. one can truncate to a certain order of and obtain a locally compact group .if one restricts to only the first order differential operators , one again obtains and the usual exponential map .this approach is slightly simpler , but we consider the truncation of the spin labels to be undesirable in the present case .  the compactness of the holonomy group is a subtle , yet important element of the usual lqg theory .formulating classical gravity with an connection rather than its complexification is a result of taking the immirzi parameter in ( [ eq : connection ] ) to be real .there have been attempts to construct lqg in the non - compact context , including a recent proposal using projective techniques , but no one has been completely successful in implementing diffeomorphism invariance on such a hilbert space .we avoid this problem in this study by working with a locally compact group and basing our gauge invariance not on the measure , but on the states and operators of the theory .( 0,0)*+()=``u '' ; ( 0,-25)*+=``su'';(25,-25)*+_(v)=``su_l'';(50,-25)*+gl_(v)=``su_l '' ; @<- _ ) `` u'';``su''?*!/_2mm / i ; @- > `` su'';``su_l''?*!/_2mm/ _ ; @- > `` su_l'';``su_l''?*!/_2mm / exp ; @> `` u'';``su_l''?*!/_2mm/ _  since the holonomy over a single element is naturally an element of a group , the full holonomy over an edge parametrized by will be the direct product this holonomy takes values in where is the index of and is the dimension of the irreducible representation . is locally compact by virtue of the finite index of .this group has an action on it by , however , we have still not taken into account the fact that the holonomy has the same symmetry under deck transformations that the metric does .this can be determined directly by counting the identifications of the sheets by over the edge .write the permutation as product of disjoint cycles , where every element moved by is fixed by .this decomposition is unique up to the ordering of the disjoint cycles .each identifies sheets of the cover with one another , so the number of disjoint cycles is the number of identifications .if we mod out by this action we will get copies of , where is the number of preimages .thus , the holonomy on the edge actually takes values in this also illustrates that the gauge symmetry here is , where is a finite group that interchanges sheets of the cover which are identified over .we give a simple example of the holonomy on a topspin network in [ ap : ex ] .  with these specificationswe can solve the holonomy equation ( [ eq : holo ] ) , and we take the holonomy to be the coordinate variables .it is easy to see this holonomy has the usual local transformation properties under the gauge transformation , because the form of the solution to ( [ eq : holo ] ) is unchanged . specifically , under the gauge transformation , the holonomy transforms like    for the conjugate variable to the holonomy , following the usual approach we will need to smear with an -valued test function over a surface of dimension 2 .thus the classical algebra on which our quantum theory will be based will be given by the holonomy ( [ eq : holo_solution ] ) and the fluxes , which are integrated over a 2-surface :    where .we will now construct the cylindrical functions in this new framework . we will closely follow the approach of , just having to deal with our holonomies living in the locally compact rather than . before doing this however, we will discuss how we can combine the usual notion of a spin network on a spatial section with the topological information given by a branched cover of so that the two structures are mutually compatible .see figure [ fig : branch ] , which shows the spatial section as an embedding in our spacetime manifold and the graph on which we have some spin network labels .we know that this spatial manifold can be described as a branched cover of by specifying an embedded graph and a coloring of the edges of the graph with permutations .( 0,0)*+m=``m '' ; ( 15,0)*+=``s'';(22,0 ) * ; ( 15,-15)*+^3=``s3 '' ; ( 37,-15)*~=p ( ) ; @<-.5ex>@<-^ ) `` m'';``s '' ; `` s'';``s3''?*!/_2mm / p  the image of the spin network in the 3-sphere is , and since the decorated branch locus simply encodes the topological information for , we can adjoin the two graphs and make it into a topspin network by adding trivial representation labels to and trivial permutation labels to . thus the topological information about preserved and the spin network information is contained completely in the trivial part of the branch locus .  to define the hilbert space of cylindrical functions on topspin networks ,we first choose a specific planar diagram for an oriented graph .this diagram contains _ nodes _ , which are either vertices of or oriented crossings of the diagram , and _arcs _ , which are either _ edges _ of ( with two endpoints at vertices ) or curves that end on at least one crossing .hereafter we will suppress writing in for notational simplicity .we have a groupoid of the diagram whose objects are nodes , and whose morphisms are the oriented arcs of .subgroupoids represent subgraphs with the restricted planar diagram .since the holonomy on each arc for is a map to a different matrix group , we will define the direct product there are homomorphisms which are defined on the edge to be the set of all such homomorphisms is .these are homomorphisms in the sense that they preserve the structure of the groupoid .if the edges can be composed then the resulting group elements can be multiplied in the direct product :   we can now follow the usual approach taken in .we have projections which are restrictions of the homomorphism to the subgroupoid .this gives us a projective family .since we have a bijection and is a locally compact group , we can equip with a locally compact topology .we can now form the direct product of these spaces associated to each groupoid , and equip it with the product topology .this space depends on the choice of diagram for each graph .  still following , we define our cylindrical functions as where are functions of finite support on since is only locally compact .if we restrict to first order terms in , we can recover using continuous functions here although we would lose the algebraic structure of the fields .there is an equivalence relation and we define .as discussed in , these cylindrical functions form a * -algebra with the sup norm we illustrate the general form for the cylindrical functions over a simple topspin network in [ ap : ex ] .the cylindrical functions are still defined on an arbitrary diagram , but we can remove the dependence on this diagram in the following manner . we can write elements of in the form , with . we want to define versions of and that will no longer depend on the planar diagram , but only on ( the ambient isotopy class of ) the embedded graph .to this end we consider instead of the subset given by only those of the form with where the are the permutations associated to the oriented arcs of the planar diagram in the wirtinger presentation of associated to the planar diagram , and is a group homomorphism .this restriction has the effect of selecting only those , where the satisfy the wirtinger relations .these can then be viewed as groupoid homomorphisms from to composed with a group homomorphism to .because the wirtinger relations imply the invariance with respect to reidemeister moves , we can then identify with , independently of the choice of a planar diagram .we can then define the total space as we did before , but with now independent of the choice of diagram for each .the next step will be to define a measure for the inner product on our hilbert space .since the cylindrical functions here are defined on finite products of locally compact topological groups , the construction again follows from until we come to the issue of gauge invariance .we can identity the space with the group via the bijection ( [ eq : bijection ] ) . when passing to as above, we can similarly identify it with a subset of , where instead of all set theoretic maps for edges of , we only consider those maps that satisfy the conditions described above , _i.e. _ that implement the wirtinger relations.  the measure on each will be defined in terms of the haar measure on each copy of . the measure on each for edge will be for the holonomy in the copy of .we also need to pull the cylindrical functions back to the group , so we define where . the measure on for a specific subgroupoid  we can then take the product measure on all the possible subgroupoids for our graph and define the hilbert space of cylindrical functions as   the measure constructed in this way on induces a measure on of non - zero total mass , and we obtain corresponding hilbert spaces of cylindrical functions in the measure is constructed by showing that the family of measures can be consistently pulled back to via the projections ( [ eq : projections ] ) .this should work in our case as well , but we do not need the extra technical considerations .  the above measure will not necessarily be gauge invariant ; the gauge invariance in the traditional approach is related to the left - right invariance of the haar measure on  .the holonomy transforms under ( [ eq : holo_transform ] ) , so the cylindrical functions will transform as it follows immediately that the inner product on is invariant if the haar measure is left - right invariant .this would not be true in the current locally compact case , but since there is always a gauge invariant subspace of the hilbert space ( the spin network states ) , this measure will be gauge invariant on those states .we will always restrict to the spin network states when using this inner product .our cylindrical functions now have an extra label to denote the set of permutation labels on the graph .an interesting difference from the usual case is that since we want to consider all possible graphs ( up to ambient isotopy classes , see below ) , the hilbert space now contains cylindrical functions on graphs which describe _ all _ compact oriented three - manifolds , rather than spatial sections of some lorentzian four - manifold . we would then expect the classical limit to somehow select only those networks which lead to spatial sections of a spacetime , or set cylindrical functions on graphs which do not describe such sections to vanishof course , the classical limit is not yet well - defined in the loop framework so we can not confirm this intuition . in the case of riemannian four - manifoldswe should have no problem taking any codimension 1 submanifold we like and representing it as a covering space .the hilbert space of usual lqg can be made separable by considering equivalence classes of under extended diffeomorphisms , which are smooth everywhere except at a finite number of points .then the hilbert space has a basis labeled by the knot class and coloring of the intertwiners and arcs .these states are usually referred to as s - knot states .it is well known from knot theory that the set of ambient isotopy classes of loops ( knots without intersections ) is a countable set , and it can be shown that under the extended diffeomorphism group knots with intersections remain in the class of the underlying loop with vertices removed . therefore the s - knot states form a countable basis for the hilbert space , and in the usual theory the ( extended ) diffeomorphism invariance is what separates the hilbert space .  in principlewe would like our hilbert spaces to also be separable , and it turns out they are .however , due to the addition of the topological labels , we can not use the same set of graphs as in the usual theory .it can be shown that two topologically different three - manifolds can be realized as branched covers over the same graph ( we give such an example in [ ap : knots ] ) .in fact , the hilden - montesinos theorem tells us that this can even be done by restricting to special classes of links .so we will get a separable hilbert space if we consider the discrete set given by the equivalence class of extended diffeomorphisms on graphs and ( for a fixed order of the covering ) a finite number of orientation , spin , and topological labels ( since the topological labels will differentiate the topology ) .there is still the issue of varying the order of the covering , but that just adds a discrete label , so the resulting hilbert space is still separable .one needs to take into account the equivalence relation given by stabilization ( adding trivial coverings ) discussed in .notice that we can also appeal to the hilden - montesinos result to specialize to three - fold covers .the advantage of allowing more general coverings and of working with embedded graphs instead of restricting to knots and links is that one can then implement certain composition operations , as described in .the diffeomorphism invariance of this theory should be investigated in more detail by looking at diffeomorphims ( or extended diffeomorphisms ) on the spatial sections , which are the physical gauge transformations . for a spin network , a spatial diffeomorphism can drag the graph around the manifold as well as change the orientation and coloring of the links . as discussed in , it is possible to project to the diffeomorphism - invariant states such that if the graph changes , the corresponding cylindrical functions are orthogonal .then the action of a diffeomorphism on an invariant cylindrical function is equivalent to the action of a finite group which permutes the labels and orientations of the underlying spin network function .of course , since is supposed to be a diffeomorphism , the topspin network which is the branch locus for must be equivalent under covering moves to the topspin network associated with . thus determining the diffeomorphism invariance of the cylindrical functions would require checking the compatibility of the cylindrical functions under covering moves .we leave this technical result for a later study .to define the operators in this theory we will closely follow the description by .one must regularize these by partitioning the surface over which the phase space variables are smeared into smaller surfaces , and then take .this surface lives in the base space , and is the image of the physical surface under the covering map , .since we are not altering the structure of the spin networks ( only adding extra labels to them ) and acts just like the usual surfaces in lqg , there should be no change in the regularization procedure .therefore we will not discuss it here .the classical operators in our theory are the holonomies and the flux vector fields .let be an oriented , embedded , open , compactly supported surface with embedding from open set .the ( classical ) area of would be    where  is the killing form on and are coordinates on .we would like to use this same approach in our case , but there will be one important difference . in ( [ eq : classical_area ] ) , since the area operator will be proportional to , the ( quadratic ) casimir operator on the lie algebra .in other words , the casimir is the trace of two first - degree differential operators . in our case , since , the inner product will have to be on differential operators of arbitrary degree .thus , to extend this result and retain gauge invariance we need a field that takes values in the center of .this does not in fact change the situation much from the usual case , it just requires a bit more care .for instance , if we have a single second - order operator with basis element of , then would be gauge - invariant ( since it is proportional to an element in the center ) . to clarify this point, we will use the natural grading on , for a polynomial in the universal enveloping algebra with degree in the representation .then we can define a family of area operators , where the first order one will be exactly as ( [ eq : classical_area ] ) with the restriction . to calculate this we split up the surface into surfaces , each intersecting 0 or 1 edges .we also assume that the vertices of the network lie outside .the first order area operator then becomes :=\sum_{i=1}^n \sqrt{{{\vphantom{e^{i\mu}_a}}^{(1)}{e^{i\mu}_a}}(s_i){{\vphantom{e^{aj\nu}}}^{(1)}{e^{aj\nu}}}(s_i)\kappa_{ij}\kappa_{\mu\nu}}\rmd x^1\wedge \rmd x^2.\ ] ] in general we should replace the killing form with an appropriate extension to , but since we have restricted ourselves to first order operators we can use the usual killing form for now .the inner product on implements ( [ eq : cs_product ] ) on the basis elements   the calculation of the area operator exactly follows the usual one ( see ) , so we will simply describe the result . assuming that the vertices of the topspin network do not lie within ( this is the _ restricted spectrum _ ) , the killing form on gives the usual ( since we have restricted to the first - order part ) .however , due to the symmetry on the holonomy elements under the action of ( [ eq : action_finite ] ) from edge , the inner product ( [ eq : cs_product ] ) actually takes place in with as discussed in [ s:2.2 ] .our basis elements would then be in an -dimensional representation .thus , we find that for the restricted spectrum , the area eigenvalues depend on how the cover is stitched together above each edge .the overall result is where we now just sum over the intersections of with .  in the above, we restricted the fields to be first order , which makes the area operator ( [ eq : area ] ) gauge invariant .there are also higher - order gauge invariant operators which lie in the center of .the structure of these can be clarified using the fact that there is a bijection between the universal enveloping algebra and the symmetric algebra  .we can use this bijection to write a basis of in symmetric polynomials of the generators of , these are called _ weyl - ordered polynomials _ ; each of tensors are completely symmetric .such a polynomial will be in the center of the universal enveloping algebra if the adjoint action \ ] ] vanishes for all . by writing out the adjoint action on each term in and using the structure functions defined by =f_{ik}^{~~j}t_j ] ways .the hamiltonian acts on all the vertices , so the full action of on a spin network is to produce a sum of spin networks with an extraordinary edge added at each vertex , chosen in different ways .this motivates the following definition :  [ def : spin - net ] a * spin - net * is a triple , and the set of all such spin - nets we call . for a given spin - net ,there exists a unique source spin - net which is constructed by removing all the extraordinary edges iteratively .now define the sets by iteratively adding extraordinary edges in all topologically inequivalent ways , and set .the details of this definition can be found in . in that same work , it was found that different spin nets are totally disjoint :  [ thm : disjoint ]  * if or .* every has source .since the hamiltonian adds extraordinary vertices , one can use this theorem to find the kernel of the hamiltonian and reduce the problem of finding physical states to one of linear algebra .    to extend the action of the hamiltonian to topspin networks, we consider it as a series of maps between spin networks given by adding an extraordinary vertex to each .specifically , consider the set of all maps which adds an extraordinary edge to the vertex of a spin - net , which has source . in principlethis could be done in several different ways depending on the valence of the vertex , but this consideration can easily be added in .this map changes the spin labeling to be compatible with new intertwiners at and the two new vertices at the ends of the extraordinary vertex .the action of the hamiltonian on a spin network could be decomposed into a composition of such functions , one for each vertex .this construction is naturally a category , with the spin networks as the objects and morphisms . denotes that this category is induced by the action of the hamiltonian , and the represents the choice of structure group for the theory .we would like to extend this category to the case of topspin networks , which will require adding topological labels and enforcing the wirtinger relations .  if we include the permutation labels on each edge in the topspin formalism , the original functions can be extended to include their action on the permutation labels .we thus define * topspin - nets * with sources in exactly the same manner as spin - nets except ensuring that the wirtinger relations are satisfied for the permutation labels at the vertices .this can be easily done when reconstructing the topspin - net from the source as in definition [ def : spin - net ] .the morphisms between topspin - nets are the result of acting with this map on a topspin - net is shown in figure [ fig : h_topspin ] .the relations between the representations on the new edges are given by the wirtinger relations at the vertices .in the first case , we have just that after the action of we have similar relations at the two new nodes , and the new representations are given by there is still some freedom in how one chooses , which is analogous to the freedom one has when choosing the representation in figure [ fig : h_spinnet ] . in the spin network casethe classical operator is unaffected by this , but the quantum operator may be . at this point in the analysiswe have no guiding principal except the analogy to the spin network case , and therefore we choose the simplest non - trivial element for , a transposition of a pair of sheets .this extension of the category for spin networks we denote .again , we can define the hamiltonian as the composition of the functions , one for each vertex .acting on a vertex of a topspin state .the simplest non - trivial choice for is a single transposition .the ( suppressed ) spin labels here are the same as in figure [ fig : h_spinnet ] . ]  by looking at the structure of these two categories , it is easy to see that they are inequivalent ; this means that the action of the hamiltonian in the topspin framework is structurally different than in the usual case .the basic reason is the presence of the geometric covering moves ( figure [ fig : covering_moves ] ) which impose an equivalence relation on the objects of .these moves provide us with a way to `` remove extraordinary edges '' .such things are not possible in the usual lqg because there is not an appropriate notion of edge deletion .we now provide an example of this .consider the pair of vertices given in figure [ fig : v ] .we define the combined geometric covering move .this can change the first diagram to the second , provided the following is satisfied : the result of this move is to destroy a vertex ( the typical vertex associated to the extraordinary edge and make the vertex at the end of the edge that was split into the new typical vertex .note that we must do this in such a way to preserve the properties of the extraordinary edge in definition [ def : extraordinary ] .this move does not , by itself , destroy an extraordinary edge .  .]  now consider the effect of this move on a specific topspin network , figure [ fig : example ] .the topspin - net has two extraordinary edges on the sides and the center line has the specific representation denoted in the diagram .we then use the move on the right typical vertex , making the two extraordinary edges share a typical vertex . now perform the move on this typical vertex , removing it .the definition of extraordinary edges requires that edges that come from the extraordinary vertices of the extraordinary edges must intersect at least once , at most twice .but in figure [ fig : example ] they _ never _ intersect , and so we have removed the extraordinary edges .so we have explicitly shown that at least in some cases , there can be two different topspin networks which are equivalent under geometric covering moves but which are not of the same level .in fact , it would also appear they are from different source networks as well . in figure[ fig : example ] , is a source topspin - net ( no extraordinary edges ) but appears to have a source net like the -graph . in other words , we have shown theorem [ thm : disjoint ] does not hold for topspin - nets .the statement is now that the categories can not be the same since it is clearly not possible to find a structure - preserving functor between them . even at a very abstract level ,the structure of topspin networks under the action of the hamiltonian is quite different than in the usual case of spin networks .the most obvious consequence that follows from this observation is that the kernel of the wheeler - dewitt operator no longer reduces to a linear problem .we should also note that the category is _ not the same _ as the loop quantum gravity 2-category constructed in .however , the 1-morphisms in that category are in fact the equivalence relations we imposed on our category . in principle, one could check to see if these two categories are compatible with each other and enlarge to include the action of the hamiltonian .however , this construction would play no part in our current analysis so we leave it to a later work .in this paper we have expanded on the proposal that topological information can be included in the framework of loop quantum gravity by using topspin networks .this idea is based on the fact the spatial sections of a foliated spacetime can be realized as a branched covering space .if we identify the branch locus with a spin network and add topological labels , we should be able to study both the topology and the geometry which results from the canonical quantization of gravity .we implemented this by requiring that the metric be invariant under a deck transformation over the branch locus .this extends the structure group of gravity to a gauge theory .this symmetry naturally applies to the phase space as well , the only change being that the fields now take values in the universal enveloping algebra , rather than .this is naturally seen as a generalization of the usual lqg case , since there is an injection .it can also be viewed as promoting the fields to operators from the outset and allowing them to take polynomial values .we found that this change does not greatly impact the structure of the theory by explicitly showing the canonical transformations of the phase space and the hilbert space of cylindrical functions retain most of their structure .thus , all of the techniques known to the lqg community about quantizing background - independent theories should be applicable in this case as well .the inner product is not gauge invariant on the entire hilbert space , but it is on a subset containing the gauge invariant spin networks .we took the familiar case of the area operator and showed that in our construction , the usual result must be multiplied by the order of the symmetry group over the branch locus .one could write this as a set of integers for arcs of the graph , where .this does not alter the qualitative features of the theory ; the area operator still measures the geometric area by counting the spin labels .we also discussed the action of the hamiltonian on topspin networks , and how it would be different on cylindrical functions of these networks . using the category - theoretic viewpoint ,it is clear that the two situations are inequivalent , and the inclusion of topological information complicates the situation .this is naturally due to the fact that there may be more than one topologically equivalent way to specify the spatial sections as branched covering spaces . in the same way that consideration of gauge symmetry complicates quantum field theory, this topological symmetry naturally complicates this analysis . at this pointwe do not discuss what would have to be done to solve the hamiltonian constraint in this approach , but hopefully we have displayed the key differences between using spin networks and topspin networks in lqg .we close with a few words on the status of topology change in lqg . now that we have determined how the area operator and the hamiltonian operator act on topspin networks, we can try to answer the question , `` does topology change occur in lqg ? '' at the level of this analysis , the answer is yes .it is clear from [ s : area_operator ] that the area operator is capable of tracking some changes in the topology , because it explicitly contains information regarding the branch locus .this is not a complete specification , since there is apparently nothing to prevent other branch loci from having the same identifications but different topologies - identifying different sheets or even having a different covering degree .in fact , the area operator is of somewhat limited use when studying topology change , since the area of a surface intersecting the spin network might not be preserved under a change of topology anyway .however , it does illustrate the relationship between topology and geometry ; the area of a surface can depend on the topology if you change the topology in such a way that the surface changes .this would occur , for instance , in the well - known trouser topology . as the spatial section passed from the trunk of the pants to the legs ,if the surface was punctured by the critical point in the crotch , the area of the surface near the critical point could clearly change .on the other hand , the hamiltonian generally appears to directly change the spatial topology by adding edges which might not be removable with geometric covering moves .of course , we have not included the complete dynamics in our analysis , so this statement is only true at the level of kinematics .the question also appears to be difficult to answer completely since it would require a complete characterization of the topspin networks from the perspective of geometric covering moves .showing two topspin networks are equivalent reduces to just finding one set of covering moves ; showing they are inequivalent is showing there is _none_. such partial answers are a common feature in topology , but there is certainly nothing to suggest one will _ always _ be able to find a covering move to undo the action of the hamiltonian .thus , at the level of kinematics this approach admits topology change .this is clearly not the only proposal in the literature which attempts to track topological information in quantum gravity , but we believe it represents an interesting one .the construction naturally follows from alexander s theorem and nicely fits into the existing theory of loop quantum gravity .the connection to the topology through the fields and operators allows one to formulate questions regarding it , which is not generally true in lqg since the reduction of the 3-manifold to a graph destroys the topological structure . if we had the full dynamical theory of lqg one could begin to carefully study the inclusion of topology by using the methods discussed herethis could lead to a theory of quantum gravity which is topologically relative as well as geometrically relative , and could give us some insight into the topology of our spacetime models .i would like to acknowledge m. marcolli for her great assistance with this project . without herthis work would not have been possible .the comments from the two reviewers were also very valuable in clarifying several important points .this example comes from the book .specialize to _ cyclic _ branched coverings .thinking of the 3-sphere as , the cyclic action is obtained by identifying points of a rotation by about a line .the rotation group here is , and by performing surgery on a curve in the base space and the preimage of that curve in the branched surface we can construct two inequivalent topologies branched over a common locus .first we recall the definition of a framed knot .a * framed knot * is an embedding together with a specification of the trivialization of the normal bundle .this framing can be completely described with an integer which tells us how many times ( and in what direction ) the normal bundle wraps around the knot .it turns out that using the belt trick the framing of a knot can be described by twists given by the basic reidemeister move .figure [ fig : twist ] shows with a framing , and figure [ fig : crossing ] shows the sign convention for the framing of the normal bundle .specifically , take the knot in figure [ fig : base ] , which has framing + 1 : two positive crossings on the right and one negative crossing on the left . in each of the cycliccovers , the inverse image of this knot has copies of the double crossing ( see figure [ fig : cover ] for case ) .it is pretty easy to see that in the -fold cover the preimage will be curves with framing -1 linked as shown .this framing is all that is required to define _ rational surgery _ ( or dehn surgery ) of a 3-sphere , in which a torus is removed and then reattached by twisting a number of times around the meridian given by the framing . by the hilden - montesinos theorem , any closed oriented 3-manifold can be obtained in this way .the linking with actually describes the whitehead link , which is a trefoil ( ) linked with the unknot ( ) with framing .we now perform this surgery in the base space by taking an -neighborhood about the knot from figure [ fig : base ] .since the 3-sphere can be described by the gluing of two solid tori , where the parallel of one is the meridian of the other , this surgery returns our base space to .next , it is pretty easy to see that the link in ( figure [ fig : cover ] ) is isomorphic ( through reidemeister moves ) to with framing + 3 .this is a generalization of the above process , and it creates the lens space ( note ) .  by proposition 22.9 of ,the case is actually a poincar sphere .these two spaces are not homeomorphic , although they are both branched over the trefoil in .this can be seen from their homology groups .the poincar sphere has the homology of a sphere , however , the lens space has a nontrivial middle homology group which arises from the action of on the unknot : thus we have two inequivalent topological spaces which have the same branch locus in .to illustrate some of the ideas presented in this paper we will give an example which shows the form of the holonomy and the cylindrical functions on a simple topspin network .consider the `` eyeglasses '' graph ( figure [ fig : eyeglass ] ) , which consists of two arcs , and one vertex with appropriate coloring for a topspin network .we assume this is the branch locus of an order 3 covering .we take as generators of , with a dimension representation , and as elements of .the connection is a function of the elements of , taking values in -valued one - forms , there is also a dimension representation of the universal enveloping algebra , . over the edgethe holonomy as a function of is here and we have the constant ( with respect to ) integral . now using the definition of our exponential map ( [ eq : exp_l ] ) we have where would be the element in the matrix lie algebra representing the appropriate polynomial of basis elements of .for example , in the spin-1/2 representation with generators ( diagonal ) , the element would be represented by ^ 2\\ & = \rmi\left(\begin{array}{cc } 1&0\\ 0&-1\end{array}\right)=i\pi_1(\sigma_3).\end{aligned}\ ] ] over the first edge in figure [ fig : eyeglass ] the holonomy would be now enforcing the symmetry given by the permutation label on , we reduce the group from and find where .the edge has trivial symmetry label , so the holonomy would be and .the entire group is .the groupoid for this graph contains one object , the vertex and two morphisms , with source and target maps this groupoid does not have any non - trivial subgroupoids .the cylindrical functions are given by maps on the elements of :

Abstract: we discuss the extension of loop quantum gravity to topspin networks , a proposal which allows topological information to be encoded in spin networks . we will show that this requires minimal changes to the phase space , c*-algebra and hilbert space of cylindrical functions . we will also discuss the area and hamiltonian operators , and show how they depend on the topology . this extends the idea of background independence " in loop quantum gravity to include topology as well as geometry . it is hoped this work will confirm the usefulness of the topspin network formalism and open up several new avenues for research into quantum gravity .


whether the first direct gravitational wave ( gw ) detection will be made during the forthcoming joint ligo - virgo run , or rather will be claimed by the next generation of interferometers , the intriguing era of gravitational wave astronomy is in any case expected to begin .  in this perspective ,spherical resonant detectors , having the capability of locating on their own the gravitational wave source in the sky , may represent an important resource for this arising discipline .contrarily to an interferometer , a sphere is a multichannel detector capable in principle to determine all the four parameters characterizing a gw ( two angles for the arrival direction , and the amplitudes of the two polarizations ) ; in order to deal with this distinguishing feature , several data analysis issues must be however tackled ( see - for an incomplete but elucidating historical perspective ) .  in and a coherent burst search algorithm has been proposed , that fully exploits the multichannel capabilities of the sphere by transforming the detector s outputs into a five - dimensional time series corresponding to the spheroidal components of the gravitational wave .this time series is in turn processed by a coherent , wavelet - based module ( henceforth cws , as coherent waveburst for the sphere ) generating a list of candidate burst events endowed with all the relevant parameters ( arrival time , arrival direction , amplitude , etc ... ) and is also able , to some extent , to reject non - transverse excitations ( which are surely not due to gw s ) , thus reducing the false alarm rate by a factor of about 10 .the reason of using wavelets rather than the traditional matched filter(mf)-ing technique is two - fold .mf - based searches are computationally expensive , as they require the exploration of a parameter space which is made extremely large by the presence of several channels ( for example , the multichannel mf is direction dependent , differently from the traditional single channel one ) .moreover wavelet - based searches , contrarily to a mf one , do not require any detailed prior knowledge of a signal , a very welcome feature for an unmodeled burst search such as the one developed in .+ on the other hand , as the use of the mf is known to give the optimal signal - to - noise ratio ( snr ) , the wavelets flexibility has a cost in term of accuracy in the determination of the burst parameters .  to this respect ,the present work can be considered as a natural evolution of in that it implements a multidimensional coherent mf as a follow - up of the method cited above .this hierarchical strategy is crucial in reducing the parameters space because for each trigger only a little region is explored around the values indicated by the cws module , thus making the whole procedure reasonably fast .the plan of the paper is the following : in section [ apip ] the whole pipeline is briefly described , showing in particular how the mf module is interfaced to the previous part of the analysis chain ; then in section [ resu ] the improved method is tested against the same mock data sets ( software injections of signals on a background endowed with both gaussian and non - gaussian noise , generated on the basis of a numerical simulation of the relevant components of the detector ) , thus showing the improvements obtained in the parameters estimation .finally , section [ conc ] contains our conclusions .the first part of the pipeline , which is exhaustively discussed in and , will be just briefly summarized here , leaving aside many technical details .the first step consists in converting the six transducers outputs into five channels corresponding to the 5 quadrupolar modes of the gw tensor : with where labels the components of the real spherical harmonics ( defined and normalized as in ) , denotes the cartesian component of the tensor spherical harmonics and is the versor of the arrival direction of the gw .this conversion allows to extract the information about the gw from the detector s outputs ( or , equivalently , to express its noisy content in term of its gw - equivalent " ) and is possible once a mathematical model for the detector is available . in the case at hand ,the model consists of a set of coupled oscillators describing the dynamics of the relevant sphere vibrational modes , of the transducers , and of the electrical circuit that are at the core of the readout devices ( see for details ) .the next step is to combine the fourier transforms of the five channels into a single channel : this process is performed by extracting from the dataset the spectral density matrix , defined from and then by building the following quantity :   this quantity is then fed into the cws algorithm , which is a modified version of the coherent waveburst algorithm developed for ligo ( - ) , that has been adapted to our computational needs ( a standard macpro machine with two 3ghz processors ) and to our detector configuration .this algorithm provides a list of triggers containing the various event features ( central time of events , amplitude , arrival direction , central spectral frequency and spectral frequency spread ) .the trigger list is finally further refined through an independent determination of the arrival direction ( the so - called determinant method ) , and a trigger is vetoed when the two reconstructed directions are not compatible .to summarize :    if the time dependence , arrival direction and polarization of the gw signal were known , that is if the expression for the five functions were available , the optimal detection strategy would be to perform a multidimensional matched filter . as explained in , the multichannel mf consists in analyzing the following scalar channel : and is the strategy which gives the optimal snr for the given signal .the integrals are done on both positive and negative frequencies . however , in a real experimental situation is not known a priori and to employ efficiently a mf one should explore a huge number of points in a parameter space describing the temporal profile of the signal , as well as the arrival direction and polarization . thus the use of a mf in such a context is computationally very demanding .  the cws method exposed in the previous paragraph has been shown to find candidate events with a good efficiency / reliability ratio , without need to know the form of the signal .this allows to keep under control the computation time but of course it comes at a price for what concerns parameter estimation .the improvement we are introducing in this work consists in performing a multidimensional mf only in the time segments where candidates events have been selected by the cws , and to restrict the analysis to a small portion of parameter space centered around the parameters provided by the cws triggers .  specifically , we consider a two parameter family of sine - gaussian shapes to describe the time profile of the signal {m_n}^{(2)}(\theta,\phi)\,,\end{aligned}\ ] ] impinging from the generic direction characterized by polar angles .since we have verified that the unmodeled search alone already gives a good estimation of the frequency spread ( an estimation which is not improved by the mf procedure ) , we kept it as fixed , in such a way that , for each candidate event we are basically exploring a small portion of the 4-dimensional parameter space .such a space is explored through a maximum - seeking algorithm , which searches for the point characterized by the maximum snr .we maximized first on the arrival time , as the peak of the snr is very sharp against this parameter , then on the central frequency and finally over the two angles of the arrival direction together , keeping at each stage the other parameters fixed at the value indicated by the cws triggers .this is a simplified choice of course , with respect to building a full parameter template bank , for instance , but it allows to obtain good results while keeping a low computational burden.  once found , the maximum - snr point in the parameter space defines the improved parameters of the candidate event , and if they happen to be too much different from the ones given by the unmodeled search , the event is discarded as spurious .this means that the mf follow - up procedure , as well as improving the events parameters , acts as a further veto against non - gw triggers , i.e. against signals which are not compatible with the transverse - traceless nature of gw s .such false triggers can come from gaussian noise or from non - gaussian disturbancies .the cws method alone is already optimized to reduce the latter s , and we will show that the application of the mf follow - up improves the performance with respect to the rejection of false triggers due to gaussian noise .we have tested the mf method on the triggers obtained in , i.e. 50 gw - injections , ( 25 for each polarization ) for each of the four amplitudes : and also on an equal number of non - gw injections ( signals whose power has been distributed randomly among the five sphere channels ) with the same amplitude .the injected signals are sine - gaussian of the same type of the matching function given in ( [ singau ] ) , this allows us to assess the precision in parameter estimation irrespectively of a mismatch between the filter and the signal in the data .an eventual mismatch could of course worsen the estimations , but we do not investigate here on that .  for the sake of clarity , we have reported in table [ gwnongw ] the detection efficiency for both gw and non gw - injections obtained in ; these numbers , which show that the cws alone already reduces by a factor of 5 - 10 the number of non - gaussian false triggers , have not changed in the present analysis.  .detection efficiency for injected gw signals and non - gw disturbances . adapted from .[ cols="^,^,^",options="header " , ]matched filtering techniques are widely used in gravitational wave analysis . howeverthey can have a high computational cost , because of the possibly large volume of parameter space required by a comprehensive search .the case of spherical detector is even worse as its multimode feature makes it highly sensitive to the arrival direction , adding the source position in the sky as further parameter .+ here we have shown how , in the case of burst searches , combining an unmodeled search with a matched one can exploit the goodies of both while taming the baddies of both .the unmodeled makes no assumption on the type of signal , but gives a non - optimal estimation of the parameters of the burst events triggered .on the other hand the matched filtered search is optimal once the shape of the signal is known . herewe use the output triggers of the unmodeled search to massively restrict the search in the parameter of the best matching template ; more specifically we used a 4-dimensional parameter space ( arrival time , central frequency and source angles in the sky ) to present an example of matched filtering search .our search is very much simplified with respect to a general one as the parameter space is not fully explored .+ to make a comprehensive analysis one would need template banks and systematic scanning of the parameter space , but even with this primitive attempt we have shown how the first estimate made by the unmodeled search can be improved by a very light matched filter search .it is a pleasure to thank m. maggiore and m. pohl for useful discussions and encouragement .the work of fs is supported by the fond national suisse .99    c. z. zhou and p. f. michelson , phys. rev .d * 51 * ( 1995 ) 2517 .e. coccia , j. a. lobo and j. a. ortega , phys .d * 52 * , 3735 ( 1995 ) .j. a. lobo and m. a. serrano , europhys .lett .  * 35 * , 253 ( 1996 ) .t. r. stevenson , phys .d * 56 * ( 1997 ) 564 .s. m. merkowitz and w. w. johnson , phys .d * 56 * ( 1997 ) 7513 [ arxiv : gr - 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Abstract: we study the performance of a multidimensional matched filter as a follow - up module of the coherent method recently developed by two of us for the detection of gravitational wave bursts by spherical resonant detectors . we have tested this strategy on the same set of injections used for the coherent method and found that the matched filter sensibly improves the determination of relevant parameters as the arrival time , amplitude , central frequency and arrival direction of the signal . the matched filter also improves the false alarm rate , reducing it roughly by a factor of 3 . the hierarchical structure of the whole analysis pipeline allows to obtain these results without a significant increase of the computation time .   dpartment de physique nuclaire et corpusculaire , universit de genve + dpartment de physique thorique , universit de genve , ch-1211 geneva , switzerland + istituto di fisica , universit di urbino , i-61029 urbino , italy + infn , sez . firenze , sesto fiorentino , i-50019 italy


optical networks are important for a wide variety of applications , from conventional optical routers and classical optical computing to quantum communication networks , linear optical quantum computing and optical metrology .the physical system that underpins all of these networks is the _ multi - mode interferometer_. it is a collection of passive optical elements such as beam splitters , phase shifters and polarisers , as well as active elements such as optical squeezers , photodetectors and switches .these networks can be implemented in bulk optics , fibre optics , or on chip .the typical applications such as optical quantum computing , quantum imaging and quantum metrology all require an extremely high precision in the optical elements. however , in any practical implementation there will be significant variations in the interferometer elements ( sometimes exceeding 10% of the specified value ) .depending on the application , such variations may be critically detrimental to the operation of the interferometer .furthermore , the variations in some elements will have a much greater effect on the functionality of the interferometer than those of others .it is therefore key to improve our understanding of element sensitivity of optical circuits .  in this paper, we explore the sensitivity of the nonlinear sign ( ns ) gate used in linear optical quantum computing as an example of circuit variation analysis .we consider two versions of this gate with identical circuit complexity in terms of the number of optical elements , input states and detection devices , and operating at the same success probability ( see fig .[ fig : ns ] ) .we will find that the design of the gate has significant implications for the process fidelity s sensitivity on variations in the components .this means that any optical interferometer design will have to be tested against alternative designs for the best performance given realistic components . in section [ sec: ns ] we present two different version of the ns gate and in section [ sec : components ] we study the effect of components variations .section [ sec : multi ] looks at the broader connection to multi - parameter estimation theory , and we conclude with a brief discussion in section [ sec : disc ] .  ) .however , the two circuits differ dramatically in the way they respond to variations in their component characteristics.,width=321 ]the ns gate is a key component in the original proposal by knill , laflamme and milburn ( klm ) for a quantum computer constructed from linear optical elements , single photon sources and photodetection .the main operation of the gate is to induce a nonlinear phase shift on an optical mode defined by where is the state of photons in the input mode and are complex amplitudes normalised to 1 .the action of the ns gate on photon number states with is not defined , and allows for additional freedom in the construction of .  the optical circuit for the original ns gate is shown in fig . [ fig : ns]a .it is a simple circuit that lends itself well to analysis .however , the circuit is not unique . we can define a `` reverse '' ns gate , shown in fig . [fig : ns]b , that achieves the same transformation on the space of zero , one and two photons , with the same probability of success .however , the two implementations do differ in the way they respond to variations in the transmission coefficients on the beam splitters and path length deviations . in this paperwe analyse the difference in performance under these systematic errors of the two incarnations of the ns gate. we will refer to the original gate in fig .[ fig : ns]a as the klm ns gate , and to fig .[ fig : ns]b as the reverse ns gate .  the ns gate is _ nonlinear _ in the sense that no combination of linear optical elements can implement the transformation in eq .( [ eq : ns ] ) .the gate is induced by interference with ancilla photons in additional modes , followed by post - selection on a particular measurement outcome in the ancilla modes .this implies that the ns gate is successfully implemented with a probability smaller than one .the traditional implementation employs a single ancilla photon and two extra optical modes .the maximum success probability of any ns gate is  , which is achieved by both implementations in fig .[ fig : ns ] .  next, we establish our conventions in describing the ns gates .we define the action of a beam splitter as a matrix transformation on the mode operators and of the two input modes and , such that and the mode operators are defined by the usual commutation relations = \delta_{jk}\ , , \end{aligned}\ ] ] with the kronecker symbol .all other commutators are zero .the klm ns gate in fig .[ fig : ns]a has beam splitter angles where the phases in fig .[ fig : ns]a are all zero .we determine the beam splitter angles for the reverse ns gate by requiring that the success probability of the ideal gate is again one quarter , and that the ancilla state and detection signature is the same as the klm ns gate .since we are primarily interested in finding an alternative gate and at this point do not wish to generate a complete family of ns gates , this construction suffices .we construct the mode transformations from eq .( [ eq : bs ] ) and collect the terms that have a single creation operator and no mode operators , corresponding to the post - selection on a detected photon in mode and no detected photons in mode .we then obtain coefficients , , and for the zero , one , and two - photon terms in the output state , respectively . solving for , we obtain where {8 } \qquad\text{and}\qquad  \chi_2 = \frac{\sqrt{16\sqrt{2}-13}}{7 } \ , , \end{aligned}\ ] ] with all phases in fig .[ fig : ns]b equal to zero .post - selection is implemented by projecting the three - mode output state onto the state , which has exactly one photon in mode 2 , and zero photons in mode 3 . for these values ,the coefficients , yielding an overall success probability of a quarter , independent of the input state in mode .  in the next sectionwe consider imperfections in the beam splitter transmission coefficients and the path lengths in the interferometer .we do not consider the effect of imperfect photodetection and ancilla preparation here , since this has already been considered in detail before by lund _et al . _in any practical implementation of the ns gate , there will be variations in the components , such as beam splitter reflectivities and path length differences that introduce unwanted phases in the optical modes .these are systematic errors that must be overcome by calibration of some sort , rather than quantum error correction codes . in this sectionwe will determine the sensitivity of each of the ns gate to variations in the three beam splitters and five path lengths . to this endwe use the gate fidelity as a figure of merit . we find that given a specified target gate fidelity , the tolerances of the optical components vary significantly .let denote a trace preserving quantum process on a density operator .we define the gate fidelity of relative to an ideal ( unitary ) gate as the quantity where is the uniform ( haar ) measure over the quantum state space .unfortunately , the ns gate is a non - trace - preserving quantum process , since the probability of success for the gate is less than one .moreover , the success probability of the gate changes significantly with variations in the optical components .the probabilities of finding the detector signature that heralds success for the two ns gates as a function of variations in the beam splitter angles are shown in fig .[ fig : nsprobs ] .while the theoretical maximum success probability of the ideal ns gate is one quarter , larger probabilities of finding the right detector outcomes are possible when the beam splitter coefficients change and the implemented gate deviates significantly from the ideal ns gate .we note that the curves for the first and third beam splitters are mirror images of each other in both the klm and reverse ns gate .this is explained by the time - reversal symmetric nature of the gates , keeping in mind that time - reversed detectors are sources , and vice versa .  .in the top graph , is the variation away from the ideal value in the klm ns gate , while in the bottom graph is the variation away from the ideal value in the reverse ns gate .the black line indicates the ideal success probability of one quarter .success probabilities larger that 0.25 are allowed , and indicate a significant departure from the ideal ns gate.,title="fig:",width=226 ] + . in the top graph , is the variation away from the ideal value in the klm ns gate , while in the bottom graph is the variation away from the ideal value in the reverse ns gate .the black line indicates the ideal success probability of one quarter .success probabilities larger that 0.25 are allowed , and indicate a significant departure from the ideal ns gate.,title="fig:",width=226 ]  the variation in success probability means that we can not use eq .( [ eq : gf ] ) in a straightforward manner .the process must be normalised , but this means that eq .( [ eq : gf ] ) can no longer be evaluated analytically for the ns gates .instead , we average the gate fidelity over 10000 random uniformly sampled input states for each value of gate component variations .since our input state consists of a linear superposition of the first three fock states ( and ignoring a global phase ) , the state space is given by the unit sphere in a three - dimensional complex hilbert space , where is the complex plane and is the usual complex vector norm .any state in can be obtained by applying a suitable matrix to an initial state , and finding a uniform distribution over the state space reduces to finding a uniform distribution over the set of unitary matrices acting on with respect to the haar measure .this is accomplished using the complex normal distribution on  .  in the remainder of this section, we study the effect of beam splitter variations and path length differences individually , and calculate the minimum , maximum and mean gate fidelity for a distribution of variations across the circuit .the gate fidelities for the klm and reverse ns gate with imperfect beam splitters are shown in fig .[ fig : bsvariations ] .the klm ns gate is particularly sensitive to variations in the second beam splitter , which is the one that directly interacts with the signal mode .the gate is significantly less sensitive to variations in the two other beam splitters .for example , at a fixed gate fidelity of , the tolerance in the first and third beam splitters , ( and , respectively ) is more than three times larger than .the curves for the first and third beam splitters are again mirror images of each other , as expected .the reverse ns gate shows a similar range of sensitivities to beam splitter variations .again , the two beam splitters in the direct signal path have the greatest effect on the gate fidelity , and the shape of these curves are very similar to the curve for the klm ns gate . again, we have mirror symmetry for the first and third beam splitters .   and in the three beam splitters , respectively .the reverse ns gate is most sensitive to the first and third beam splitter , while the klm ns gate is sensitive only to the second.,title="fig:",height=170 ]  and in the three beam splitters , respectively .the reverse ns gate is most sensitive to the first and third beam splitter , while the klm ns gate is sensitive only to the second.,title="fig:",height=170 ]  both gates have a success probability of , but the sensitivity of the reverse ns gate of variations in two beam splitters instead of one makes the klm ns gate the preferable circuit for practical implementations . in implementations that require high precision , the circuit design may require a variable beam splitter such as a directional coupler that is tuneable either mechanically , electrically , or thermally . since these structures are likely costly , and will introduce other imperfections ( such as drift ) , the fewer beam splitters that need to be corrected in this way , the better .generally , there will be room to optimise the circuit design based on the tolerances of the circuit on the variations in its elements .in practice , variations in beam splitters can be quite large in bulk optics ( on the order of 10% ) .a similar analysis can be performed for the various path length differences in the circuit .the klm ns gate is completely insensitive to path length variations encoded in the phases and , as expected .similarly , the reverse ns gate is insensitive to variations in ( see fig . [fig : phase ] ) . for the remaining phasesthere is a marked difference in the two gates .the klm ns gate loses only about a percent in fidelity when the path lengths associated with and varies by half a wavelength , while the reverse gate sees a significant drop in average gate fidelity .compared to the beam splitter variations we can say that the klm ns gate is effectively insensitive to path length differences , while the reverse ns gate is very sensitive to path length differences .this is another reason to strongly prefer the klm ns gate over the reverse ns gate , and underlines the importance of circuit analysis for component variations .   and in the five relevant path lengths , respectively .the klm ns gate is again far less sensitive to path length variations than the reverse ns gate.,title="fig:",height=170 ]  and in the five relevant path lengths , respectively .the klm ns gate is again far less sensitive to path length variations than the reverse ns gate.,title="fig:",height=170 ]   in addition to individual errors in components , we may consider compound errors in all beam splitters and path lengths .this describes the more realistic behaviour where all components are subject to small variations .we define an error vector with eight components ( three for the beam splitters and five for the path lengths ) and magnitude . for a given total error distributed among the components , all possible error configurations are described by a 7-sphere of radius .we randomly sample this error space and calculate the gate fidelity .the results are shown in fig .[ fig : compoundall ] .  for each value of generate 50000 random vectors . the gate fidelity for each is then calculated again using 10000 random input states into the ns gate . in fig .[ fig : compoundall ] we plot the minimum , maximum and mean gate infidelity as a function of .the maximum gate infidelity is the worst case scenario for a given .it reaches a maximum of approximately 0.76 with increasing for both the klm and reverse ns gate .this value represents the limit where the circuit is so badly constructed that it no longer outperforms a randomly constructed circuit , and we call this the randomisation limit .the gate infidelity of the reverse ns gate reaches this value much faster than the klm ns gate , which is consistent with our earlier observation that the reverse ns gate is more sensitive to variations in the two in - line beam splitters , compared to the klm ns gate that is sensitive to variations in the single in - line beam splitter . in both casesthe effect of path length variations is much less important than the beam splitter variations .   for the klm ( top ) and reverse ( bottom ) ns gates .the minimum gate fidelity exhibits strong statistical fluctuations due to the relatively long tail in the distribution of gate fidelities for a given .both circuits reach the same randomisation limit of 0.76.,title="fig:",width=321 ] + -3 mm  for the klm ( top ) and reverse ( bottom ) ns gates .the minimum gate fidelity exhibits strong statistical fluctuations due to the relatively long tail in the distribution of gate fidelities for a given .both circuits reach the same randomisation limit of 0.76.,title="fig:",width=321 ]  the minimum gate infidelity exhibits strong statistical fluctuations ( the green lines in fig . [fig : compoundall ] ) .this is due to the long tail in the fidelity distribution for given .only relatively few circuit configurations will give a high gate fidelity , and the finite number of samples ( 50000 ) is unlikely to hit upon the true minimum gate infidelity .this line is therefore more accurately characterised as a lower bound on the maximum gate fidelity ( or , equivalently , an upper bound on the minimum gate infidelity ) .the method used to analyse compound errors can be generalised for any system , but in practice this is a computationally intensive process .generating error vectors over input states requires evaluations of the gate fidelity . for 3-port optical networksthis is still tractable , but for general -port networks the number of optical elements and therefore the dimension of the error vector scales as .we will need more sophisticated theoretical methods to analyse complex optical networks .one potentially fruitful approach towards analysing the effect of component variations on the gate fidelity in linear optical networks is to use the theory of multi - parameter quantum estimation .when we consider the components of the multi - mode interferometer , the variations away from the ideal designed value become a vector of random variables in a parameter estimation problem .we can then use techniques from quantum metrology , information geometry , and the theory of the dynamical evolution of quantum states to shed light on the sensitivity of an interferometer on its elements .is taken just before the component of interest ( dashed line ) , and it is the state that maximises the quantum fisher information ( evaluated by the variance of the generator of with respect to ) , while still being capable of triggering the detector array in the required way.,width=226 ]  the quantum fisher information ( qfi ) is a metric in the state space that is parametrised by the random variables .it is a special case of the bures metric .intuitively , the quantum fisher information is the amount of information about that is contained in the state .however , for our purposes it is sufficient to note that a large qfi means that we can detect small variations in .therefore , is also a metric for the sensitivity of on .let a unitary transformation of an optical circuit be denoted by , and deviations in the characteristics of a component ( such as a beam splitter or phase shifter ) are generated by .the unitary transformation corresponding to an inaccurate component is then where is a component of the vector .the qfi for is bounded by the variance of with respect to the optical quantum state immediately prior to the component  . by studying with respect to a variety of quantum states ( average , best and worst case scenario ) we can estimate the effect of variations of that component on the total gate .post - selection on a particular detection signature ( as in the case of the ns gate ) will typically exclude certain states , and the most informative average qfi will no longer be due to a uniform distribution of in the quantum state space . instead , the set of that are to be averaged over should be constructed from a uniform distribution of input states over all the non - ancilla input modes , tensored with the ancilla input states and transformed to the state just before the component .the above procedure still requires averaging over a large number of states . to circumvent this lengthy process, we need a way to determine the optimal conditional state that maximises the qfi as evaluated by , while still being capable of triggering the detectors according to the required signature ( see fig .[ fig : qfi ] ) . the variance must be evaluated with respect to this optimal state .again , this may be a computationally difficult problem .finally , we can calculate the weighted average over the variance as a measure of the sensitivity of a component to variations : where is the probability that the state leads to the required detector signature , and is the haar measure over the entire multi - mode optical state just before entering component . for the ns gatethis is a three - mode state .we also explicitly included the subscript on to remind ourselves that the variance depends on the input state. which of these approaches is most suitable likely depends on the specifics of the optical circuit under consideration , and the exact relation between circuit analysis and quantum parameter estimation will be the subject of future studies .we have shown that the construction of optical networks that implement a given unitary transformation is generally not unique , and that variations in the components of the network can have dramatically different effects on the network ( gate ) fidelity . moreover , different network topologies for the same transformation may place very different precision requirements on the components , and any practical implementation should involve a circuit analysis on how to best implement the optical network . for small networks , a simple numerical calculation of the average gate fidelity may be tractable , butlarger networks require more sophisticated methods .one such method is the quantum fisher information , which can be calculated efficiently for optical components by considering the variance of the generator of translations .our findings prompt a number of important questions for future research into the practical construction of optical networks : ( i ) why do some elements in a network require much more precise fabrication than others ? ( ii ) how can we design optical networks that minimise the number of sensitive elements ? ( iii ) how can we determine the component characteristics _ in situ _ , after the network has been fabricated ?these questions will be studied further in future work .pk thanks mercedes gimeno - segovia for stimulating discussions on alternative implementations of the ns gate .18ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] link:\doibase 10.1038/nphoton.2010.94 [ * * , ( ) ] link:\doibase 10.1038/35051009 [ * * ,  ( ) ] link:\doibase  10.1103/revmodphys.79.135 [ * * ,  ( ) ] link:\doibase 10.1017/cbo9781139193658 [ _ _ ] ( , ) link:\doibase 10.1088/1367 - 2630/6/1/051 [ * * ,  ( ) ] \doibase http://dx.doi.org/10.1103/physrevlett.95.040502 [ * * ,  ( ) ] link:\doibase 10.1103/physreva.68.022313 [ * * ,  ( ) ] link:\doibase  10.1016/s0375 - 9601(02)00069 - 5 [ * * ,  ( ) ] link:\doibase 10.1016/s0375 - 9601(02)01272 - 0 [ * * ,  ( ) ] link:\doibase 0.1007/s00023 - 007 - 0345 - 5 [ * * ,  ( ) ] link:\doibase 10.1038/nphoton.2009.116 [ * * ,  ( ) ] link:\doibase  10.1038/ncomms6337 [ * * ,  ( ) ] link:\doibase 10.1038/nphoton.2009.93 [ * * ,  ( ) ] link:\doibase 10.1038/nphoton.2011.35 [ * * ,  ( ) ] link:\doibase 10.1109/tit.2009.2016067 [ _ _ ] ( , ) link:\doibase 10.1103/physreva.82.022107 [ * * ,  ( ) ] link:\doibase 10.1016/0375 - 9601(67)90366 - 0 [ * * ,  ( ) ] link:\doibase 10.1006/aphy.1996.0040 [ * * , ( ) ]

Abstract: we investigate the effect of variations in beam splitter transmissions and path length differences in the nonlinear sign gate that is used for linear optical quantum computing . we identify two implementations of the gate , and show that the sensitivity to variations in their components differs significantly between them . therefore , circuits that require a precision implementation may benefit from additional circuit analysis of component variations to identify the most practical implementation . we suggest possible routes to efficient circuit analysis in terms of quantum parameter estimation .