A simplicial gauge theory
aa r X i v : . [ m a t h . NA ] S e p A simplicial gauge theory
Snorre H. Christiansen, Tore G. Halvorsen
Abstract
We provide an action for gauge theories discretized on simpli-cial meshes, inspired by finite element methods. The action is dis-cretely gauge invariant and we give a proof of consistency. A discreteNoether’s theorem that can be applied to our setting, is also proved.
The Standard Model describes fundamental particles and interactions, exceptgravity, as a quantum field theory with gauge group U (1) × SU (2) × SU (3),see e.g. [33][34]. Lattice Gauge Theory (LGT) [36][21][30] is a computationalapproach with good agreement with experimental data. It has proved par-ticularly useful for the SU (3) sector, concerning quarks and gluons, whereperturbative methods fail.In LGT, the underlying space-time is discretized by a cubical lattice.Scalar fields are assigned degrees of freedom on vertexes. Gauge fields areassigned degrees of freedom on edges, representing parallel transport betweenvertexes (Wilson lines). Curvature is recovered from holonomies aroundsquares (plaquettes) in the lattice (Wilson loops). These quantities are com-bined to form a discrete action (Wilson action) that is gauge invariant, withrespect to gauge transformations at vertexes. The construction of a discretegauge invariant action is a fundamental ingredient in LGT, and is the onlypart of LGT we are concerned with in this paper.There is interest in constructing discrete gauge invariant actions on otherunderlying geometries, in particular simplicial decompositions of space-time.This has already received considerable attention , starting with [4][15]. Fordevelopments motivated by non-commutative differential geometry, see [13].The approach we propose in this paper results in an alternative prescriptionand is inspired by finite element methods, as they have been developed forMaxwell’s equations, corresponding to the U (1) sector. We were blissfully ignorant about these, when submitting the first draft. consistency errors are analysed in the framework of variational crimes [31],see [12] §
26 – 29.For gauge group U (1), a consistent and gauge invariant action on sim-plicial meshes can be obtained through a relatively simple construction andsome applications can be completely analyzed [9]. A key ingredient is a notionof mass lumping [17]. Because of limitations in the scope of mass lumping,we don’t expect this method to extend to the full Yang-Mills action.By a more elaborate procedure, we define here, a consistent and gaugeinvariant discrete action for Yang-Mills theories on simplicial meshes. Thesimplexes are not congruent, so the metric must enter the formulas in a nontrivial way, contrary to the cubical lattices of standard LGT. The metricon a simplex defines a mass matrix for the Whitney two-forms. Wilsonloops associated with the faces of the simplexes are used to represent thecurvature covariantly. Whereas standard LGT sums individual contributionsfrom faces, we sum over simplexes, in which cross terms between the differentfaces appear. These couplings between Wilson loops are weighted by the massmatrix coefficients and made gauge invariant by discrete parallel transportbetween origins.One advantage of the proposed formulas is to allow local mesh refine-ments, useful for instance to efficiently represent singular fields. The con-2istency proof we provide covers such meshes, indicating robustness withrespect to mesh geometry. Another advantage is to accommodate variablemetrics as defined by Regge calculus [29], see Remark 3.This paper serves to introduce the formalism, give the definition, proveconsistency and propose a discrete Noether’s theorem. Preliminary applica-tions to quantum field theory, including numerical results, have been reportedelsewhere [16], while this paper was under review. It is organized as follows.Section 2 contains definitions pertaining to connections and curvature, aswell as Whitney forms. Section 3 includes the definition of the proposeddiscrete Yang-Mills Lagrangian and some comments. Section 4 contains theproof of consistency in a finite element sense. Finally section 5 contains thediscrete Noether’s theorem we introduce. Yang-Mills action
A standard reference for connections and curvature is[20]. Here, we use notations as in [10], which also contains a more com-prehensive presentation of Lie algebra valued differential forms. The mainingredients are as follows.Choose a compact Lie group G with associated Lie algebra g . For simplic-ity we suppose that G is a subgroup of the complex unitary n × n matrices,for some n . Typically an element of G will be denoted G and an element of g will be denoted g . The Hermitian conjugate of a matrix g is denoted g h and the real scalar product is: g · g ′ = Re tr( g h g ′ ) . (1)When no confusion is possible with scalars, the unit matrix is denoted andthe zero matrix is denoted .Let S be a bounded domain in m -dimensional Euclidean space (in ap-plications m = 2 , , k -forms on S is denoted Ω k ( S ). The space Ω k ( S ) ⊗ g can be identified withthe space of smooth g -valued k -forms on S . The bracket of Lie algebra valuedforms is determined by:[ u ⊗ g, u ′ ⊗ g ′ ] = ( u ∧ u ′ ) ⊗ [ g, g ′ ] , (2)where u, u ′ are real valued differential forms and g, g ′ are elements of g .A smooth connection one-form on S is an element A ∈ Ω ( S ) ⊗ g alsocalled gauge field. Its curvature is F ( A ) ∈ Ω ( S ) ⊗ g defined by: F ( A ) = d A + 1 / A, A ] . (3)3e will use such forms with less regularity, typically in some Sobolev space.Gauge transformations of connection one-forms are associated with func-tions Q : S → G , and defined by: G Q ( A ) = QAQ − − (D Q ) Q − . (4)One has: F ( G Q ( A )) = Q F ( A ) Q − . (5)The Yang-Mills action is given by: S ( A ) = Z S |F ( A ) | . (6)Since the adjoint representation is unitary, this action is invariant undergauge transformations. Whitney forms
We refer to [19][1] for surveys on Whitney forms [32][35]in a finite element guise. For aspects relating to differential geometry andalgebraic topology, one can consult [27]. The following is a summary ofresults needed and serves mainly to introduce notations.Let T be a simplicial complex, spanning the domain S . The set of k -dimensional simplexes in T is denoted T k . Simplexes of dimension 0, 1 and2 are referred to as vertexes, edges, faces respectively. Generic labels foredges and faces will be e and f respectively. The symbol T can be used forsimplexes of any dimension. In the presence of several vertexes we denotethem by i, j, k, l . We suppose an orientation has been chosen for each simplexin T .Let W k ( T ) be the space of Whitney k -forms on T . We also denote by W k ( T ) the space of Whitney k -forms on a simplex T . The canonical basis of W k ( T ) is denoted ( λ T ), T ranging over the set T k of k -dimensional simplexesin T . Explicitly, the 0-forms are spanned by the barycentric coordinate maps.For any vertex i ∈ T , λ i is the piecewise affine map taking the value 1 atvertex i and 0 at the other vertexes. For a k ≥ k -dimensional simplex T ∈ T with vertexes i , . . . , i k , ordered according to the chosen orientationof T , we have: λ i k ··· i = λ T = k ! k X j =0 ( − j λ i j d λ i ∧ . . . d d λ i j . . . ∧ d λ i k . (7)The hat signifies omission of this term. We will only use 0-, 1- and 2-forms.When i, j are vertexes of an edge, the associated Whitney 1-form is: λ ji = λ i d λ j − λ j d λ i . (8)4hen i, j, k are vertexes of a face, the associated Whitney 2-form is: λ kji = 2( λ i d λ j ∧ d λ k − λ j d λ i ∧ d λ k + λ k d λ i ∧ d λ j ) . (9)The space of (real) k -cochains consists of the functions that assign a realnumber to each k -simplex: C k ( T ) = R T k . (10)The coboundary operator δ : C k ( T ) → C k +1 ( T ) is defined as follows. Therelative orientation o ( T, T ′ ) between a simplex T ∈ T k +1 and a simplex T ′ ∈ T k is 0 if T ′ is not in the boundary of T , and ± T ′ is in theboundary of T , the sign depending on whether it is outward oriented or not.For u ∈ C k ( T ), δu ∈ C k +1 ( T ) is defined on any T ∈ T k +1 by:( δu ) T = X T ′ ∈T k o ( T, T ′ ) u T ′ . (11)One has δδ = 0. The de Rham map R k is defined by: R k : (cid:26) Ω k ( S ) → C k ( T ) ,u ( R T u ) T ∈T k . (12)In this formula the k -form u is (pulled back and) integrated on the k -simplexes T , taking into account orientations. By Stokes’ theorem we havecommuting diagrams: Ω k ( S ) d / / R k (cid:15) (cid:15) Ω k +1 ( S ) R k +1 (cid:15) (cid:15) C k ( T ) δ / / C k +1 ( T ) (13)Since for T, T ′ ∈ T k one has: Z T ′ λ T = δ T T ′ (Kronecker delta), (14)the de Rham map induces isomorphisms: R k : W k ( T ) → C k ( T ) , (15)whose inverses are: (cid:26) C k ( T ) → W k ( T ) ,u P T ∈T k u T λ T . . (16)5iven u ∈ W k ( T ) we denote by u • = R k u its associated cochain. Anotheruseful notation is the following. If u ∈ C k ( T ), one defines for any simplex T ∈ T with vertexes i , . . . , i k : u i k ··· i = ± u T , (17)the sign depending on whether the ordering of the vertexes agrees with theorientation of T or not. For instance we can write if u ∈ C ( T ):( δu ) kji = u ik + u kj + u ji . (18)Let I k denote the interpolation operator onto Whitney k -forms – it is theprojection onto W k ( T ) determined by the identity R k I k = R k . Equivalently,for u ∈ Ω k ( S ) one has: I k u = X T ∈T k ( R T u ) λ T ∈ W k ( T ) . (19)Interpolation commutes with the exterior derivative.Whitney forms are not smooth, but have enough regularity for the exteriorderivative, in the sense of distributions/currents of Schwartz and de Rham,to be given by the simplex-wise definition (there are no Dirac measures oninterfaces). We first make some remarks on Lie algebra valued Whitney forms and theirassociated cochains. The de Rham isomorhisms (15) give isomorphisms: R k : W k ( T ) ⊗ g → C k ( T ) ⊗ g , (20)An assignment of a Lie algebra element to each k -simplex will be called aLie algebra k -cochain. For k = 1 this goes as follows. Pick A ∈ W ( T ) ⊗ g .Attached to an edge with vertexes i, j and oriented from i to j , one has anelement A ji ∈ g obtained by writing: A = X ji ∈T A ji λ ji where A ji = Z ji A, (21)Thus ji denotes an edge in T , oriented from the vertex i to the vertex j .With these notations, the cochain associated with A is: A • = ( A ji ) ji ∈T ∈ C ( T ) ⊗ g . (22)6he pullback of the 1-form A ∈ W ( T ) ⊗ g to the edge ji is a constant1-form. Therefore parallel transport from i to j is given simply by: U ji = exp( − A ji ) . (23)We suppose U ji to be close enough to for the logarithm to be unambiguous.Then one is free to think in terms of Lie group elements U ji (close to ) orLie algebra elements A ji (close to ). We use the conventions: U ij = U − ji and U ii = , (24)which correspond to: A ij = − A ji and A ii = . (25)A discrete gauge transformation is associated with a choice of Lie groupelements G i ∈ G , one for each vertex i ∈ T . One then transforms A ∈ W ( T ) ⊗ g by operating on its parallel transports. Namely, if U ji are theparallel transports of A , the parallel transports of its image A ′ ∈ W ( T ) ⊗ g will be U ′ ji defined by: U ′ ji = G j U ji G − i . (26)With this choice of gauge transformations, we will construct a gauge invariantapproximation of the “true” Yang-Mills action on simplexes T ∈ T m ofmaximal dimension, which we recall to be defined by: S T ( A ) = Z T |F ( A ) | . (27)In our setting T inherits the Euclidean metric of the ambient space, but asalready indicated one could use a Regge metric instead. The metric enablesintegration of scalar functions on T . It also gives, at each point x of T ,a scalar product on alternating forms above x . The associated norm wasdenoted | · | in (27).Let M be the matrix of the L ( T ) product on W ( T ), equipped with thestandard basis defined by (9). This matrix is indexed by the two-dimensionalfaces of T (and depends on their orientations). Explicitly, for two faces f and f in a simplex T , we put: M f f ( T ) = Z T λ f · λ f . (28)where the scalar product of alternating forms is denoted ( · ). Notice that thismatrix is not diagonal, which can be interpreted as an interaction betweenneighboring faces. 7iven parallel transports U ∈ C ( T , G ), the discrete curvature associatedwith a face f ∈ T with vertexes i, j, k is defined in analogy with squareWilson loops [36] by: F kji = U ik U kj U ji . (29)In other words, one considers the holonomy around the boundary of the face f of the 1-form A ∈ W ( T ) ⊗ g related to U by (23). This formula dependson the ordering of the vertexes and locates the curvature at the vertex i . Thecurvature at vertex j is obtained by permuting indices and satisfies: F ikj = U ji F kji U ij . (30)This gives a formula for parallel transport of curvature from i to j . Con-cerning orientation of a given face, we also notice that the definition (29)implies: F jki = F − kji . (31)Under gauge transformations this curvature behaves as follows. If F ∈C ( T , G ) is the curvature associated with holonomies U ∈ C ( T , G ) by (29),and U is transformed by G ∈ C ( T , G ) to U ′ , according to (26), then thecurvatures F ′ of U ′ are: F ′ kji = G i F kji G − i . (32)When f is a face with vertexes i, j, k , which is ordered as i → j → k and wechoose to locate the curvature at i , we put: F f = F kji . (33)This formula defines the curvature of a pointed oriented face f . For a pointedface f , its distinguished point is denoted ˙ f and called its origin.We now propose the following definition of a discrete action for latticegauge theory on simplexes, as an alternative to (27): Definition 1.
We define: S ′ T ( A ) = X f f M f f ( T ) Re tr (cid:0) U ˙ f ˙ f ( − F h f ) U ˙ f ˙ f ( − F f ) (cid:1) . (34)In this formula we sum over pairs of faces f , f of T , each one having anorientation and a distinguished point. We have incorporated the paralleltransport determined by A , between the distinguished points ˙ f and ˙ f .We can state: Theorem 1.
The action S ′ T is discretely gauge invariant, with respect totransformations of the form (26). T m , so that for A ∈ W ( T ) ⊗ g : S ′ ( A ) = X T ∈T m S ′ T ( A ) . (35) Remark . Define, for any U ∈ C ( T , G ), subject to (24) and any F ∈C ( T , G ) (not necessarily the curvature of U !): L T ( U, F ) = X f f M f f ( T ) Re tr (cid:0) U ˙ f ˙ f (1 − F h f ) U ˙ f ˙ f (1 − F f ) (cid:1) . (36)For fixed U , L T ( U, F ) can be interpreted as an L ( T ) norm squared of F .This norm depends on U , which contrasts with the fact that the usual L ( T )norm, as it appears in (27) is independent of the gauge.However L T ( U, F ) is invariant under transformations (
U, F ) ( U ′ , F ′ ),associated with some G ∈ C ( T , G ) by (26) and (32). Remark . It is most natural to compute a norm in the Lie algebra. Thus inthe definition of the discrete actions, the terms of the form ( F f − ) shouldbe considered as approximations of log F f that are more readily computable.For a general Lie group one could use: S ′ T ( A ) = X f f M f f ( T ) Ad( U ˙ f ˙ f ) log( F f ) · log( F f ) . (37)Here Ad : G → End( g ) is the adjoint representation. Recall that for a given U ∈ G , Ad( U ) : g → g is the tangent map at unity, of the automorphism of G mapping an element G to U GU − . The scalar product on g , denoted herewith ( · ), should make the adjoint representation unitary. Remark . The proposed method can be combined seamlessly with Reggecalculus [29] (see [5][11] for finite element interpretations). In Regge calculusthe metric of a given simplex is determined by the edge lengths and yieldsa mass matrix, as in the adopted setting. It seems that only minor modifi-cations are necessary for the consistency proof we will provide, to cover thecase where the local metrics are Regge metrics interpolating a smooth one.Notice that this action uses just edge lengths for the metric and values ofthe fields at vertexes or edges, never vertex coordinates, so that the methodis ”coordinate free”.
Scalar fields.
We include a definition of a discrete action for certain so-called scalar fields. We will not prove consistency for it here.9et V be an inner product space on which G acts unitarily. The actionis denoted simply ( G, v ) Gv . Likewise the associated action of g on V isdenoted ( g, v ) gv .Let ∇ denote the canonical flat connection acting on sections Φ : S → V .Given A , the action to approximate is: S T ( A, Φ) = Z T |∇ Φ + A Φ | . (38)Let then Φ ∈ W ( T ) ⊗ V be a discrete scalar field. We can write:Φ = X i ∈T Φ i λ i with Φ i = Φ( i ) . (39)Concerning the cochains associated with Φ and ∇ Φ ∈ W ( T ) ⊗ V we have,by (13): ∇ Φ = X ji ∈T (Φ j − Φ i ) λ ji . (40)The mass matrix for Whitney 1-forms is also denoted by M and is indexedby oriented edges. Thus: M e e ( T ) = Z T λ e · λ e . (41)For an oriented edge e we denote its origin by ˙ e and its target by ¨ e .As a discrete action we propose to use: S ′ T ( A, Φ) = X e e M e e ( T ) U ¨ e ¨ e (Φ ¨ e − U e Φ ˙ e ) · (Φ ¨ e − U e Φ ˙ e (cid:1) , (42)where the scalar product is that of V .Recall that under discrete gauge transformations associated with G i ∈ G ,the parallel transports U transform by (26). The corresponding transforma-tion of Φ is, at the level of cochains:Φ i G i Φ i . (43)It is readily checked that S ′ ( A, Φ) is discretely gauge invariant.
Conventional LGT
For comparison, we recall the usual definition of LGTon cubical meshes. One attaches a discrete parallel transport U ji to any twovertexes i, j of the grid linked by an edge, with the preceding constraint U ji = U − ij . A face f of this mesh is then a square with four vertexes, called10laquette. Given a choice of orientation and origin, these four vertexes canbe labelled f , f , f , f . Barring the coupling constant, the action is thendefined by a sum over faces: X f Re tr( − U f f U f f U f f U f f ) , (44)Remark that the action is independent of the choice of origin and orientationof the faces.Thus in standard LGT, one sums over faces, and the contribution ofeach face is discretely gauge invariant, under transformations (26). On theother hand, for simplicial meshes, we propose to sum over maximal simplexes(tetrahedrons in dimension 3), with gauge invariant cross terms betweenneighboring faces. The counterpart for cubical meshes would be to sum overcubes, inside which plaquettes are coupled two by two. From this point ofview, standard LGT uses just diagonal terms. This is also why the scalarproduct (1) is more apparent in (34) than in (44). One can compare with thefact that the Yee scheme [37] can be deduced from a finite element schemevia mass lumping [24] (whereby mass matrices are approximated by diagonalmatrices in a consistent way).That the continuum limit of the LGT action is the Yang Mills action, isusually argued on the basis of Taylor expansions and a couple of terms in theBCH formula, e.g. [21] p. 786. To a numerical analyst these arguments wouldprove consistency in the finite difference sense. Consistency in the finiteelement sense is related of course, but not identical, putting emphasis on thechoice of function theoretic norms, typically Sobolev norms. Transposingthe finite element arguments we will give here for the simplicial case, tothe cubical case (with tensor-product Whitney forms) would give a novelproof on the convergence of the Wilson action to the Yang-Mills action, inthe continuum limit. We don’t expect consistency proofs based on Taylorexpansion techniques to carry over directly to the simplicial setting, since itis difficult for them to take into account mesh geometry. In this section we want to study the error committed, when approximating(27) by (34). For this purpose we introduce two more discrete actions definedfor A ∈ W ( T ) ⊗ g , for simplexes T ∈ T m . These discrete actions serve onlyto provide intermediate steps between S T ( A ) and S ′ T ( A ), aiming at clarifyingthe analysis. 11irst we define: S T ( A ) = Z T | I F ( A ) | , (45)and remark that, by (1), (19) and (28): S T ( A ) = X f f M f f ( T ) Re tr (cid:0) R f F ( A ) h R f F ( A ) (cid:1) . (46)The sum extends over pairs of faces f , f of T . Second, given also a choiceof origins of the faces, we define: S T ( A ) = X f f M f f ( T ) Re tr (cid:0) ( − F f ) h ( − F f ) (cid:1) . (47)We now evaluate the error in each of the approximations: S T → S T → S T → S ′ T . (48)The reasoning is partly similar to [9], but we need to extend to non-commutativegauge group and abandon the convenience of mass-lumping. For definitenesswe restrict attention to dimension m = 3 for the ambient space S . Maximalsimplexes are then tetrahedrons.We suppose that we have a regular sequence of simplicial meshes T n ofthe domain S . The diameter of a simplex T is denoted h T , and the biggest h T when T is in T n is denoted h n . We suppose that the sequence ( h n ) n ∈ N converges to 0. Let I kn denote the interpolant onto Whitney k -forms associ-ated with the mesh T n . Let X n denote the space W ( T n ) ⊗ g . For ease ofnotation put also I n = I n and J n = I n .The following definition is the natural extension of [12] §
28 to a non-linearsetting.
Definition 2.
Let k · k denote the norm of some Banach space in whichΩ ( S ) ⊗ g is dense, and containing each X n . We say that two actions S n and S ′ n defined on X n are consistent with each other, with respect to k · k , if forall smooth A we have:sup A ′ ∈ X n | D S n ( I n A ) A ′ − D S ′ n ( I n A ) A ′ | / k A ′ k → n → ∞ . (49)More precisely if the above expression is O ( ǫ n ), for some sequence ǫ = ( ǫ n ) →
0, we speak of consistency of order ǫ .12f there is a constant C > A and the sequence( T n ) but not on n ) such that quantities a n and b n satisfy a n ≤ Cb n for all n ,we write a n (cid:22) b n or a n = O ( b n ).Consider a simplex T of dimension d and let Φ : ˆ T → T be a scaling mapof the form Φ( x ) = h T x + y . We have, for any u ∈ Ω k ( T ): k u k L p ( T ) = h − k + d/pT k Φ ⋆ u k L p ( ˆ T ) . (50)Arguments based on this identity will be referred to as scaling arguments.For instance if we have a sequence of elements u n ∈ X n which is boundedin L ( S ) and e n is an edge in T n we deduce by scaling that we have bounds: | Z e n u n | (cid:22) h / e n . (51)which gives: | u n • | ℓ ∞ = max e ∈T n | u ne | (cid:22) h / n . (52)This is enough to guarantee that the logarithm is unambiguous as requiredinitially.On cochains, we consider norms of the following form (with no coeffi-cients): for u ∈ C k ( T ) | u | = X T ∈T k | u T | . (53)We will frequently use that on reference simplexes ˆ T , the cochain norm isequivalent to functional norms, as they appear in the right hand side of (50).We let A ∈ Ω ( S ) ⊗ g be smooth. We put A n = I n A . Remark that foredges e in T n we have: A ne = ( I n A ) e = Z e A = O ( h e ) , (54)and for faces f in T n we have:( δA n • ) f = (d I n A ) f = Z f d A = O ( h f ) . (55) Step one
We compare S and S . We first remark: Lemma 1.
We have: kF ( A n ) − J n F ( A n ) k L ( T ) = O ( h T kF ( A n ) k L ( T ) ) , (56) and: k J n F ( A n ) k L ( T ) (cid:22) kF ( A n ) k L ( T ) . (57)13 roof. By scaling, knowing that F ( A n ) lives in the space of Whitney formsof maximal polynomial order 2 ([10] § | A F ( A ) A ′ = d A ′ + [ A, A ′ ] . (58)Since Whitney forms are stable under the exterior derivative, we have:D F ( A n ) A ′ − J n D F ( A n ) A ′ = [ A n , A ′ ] − J n [ A n , A ′ ] . (59) Lemma 2.
We have: k [ A n , A ′ ] − J n [ A n , A ′ ] k L ( T ) = O ( h T k A ′ k L ( T ) ) . (60) Proof.
Remark that the interpolation is exact on a tetrahedron if A n is con-stant on it. On a reference tetrahedron we can therefore write: k [ A n , A ′ ] − J n [ A n , A ′ ] k L ( ˆ T ) (cid:22) k∇ A n k L ∞ ( ˆ T ) k A ′ k L ( ˆ T ) . (61)The estimate on T then follows by scaling. Proposition 1.
The actions S and S are consistent of order h for the L norm.Proof. We have: D S T ( A n ) A ′ = Z T F ( A ) · D F ( A ) A ′ , (62)and: D S T ( A n ) A ′ = Z T J n F ( A ) · D J n F ( A ) A ′ . (63)With a more compact notation we can evaluate the difference: | R ( F − J F ) · D F A ′ + R J F · (D F − J D F ) A ′ | (cid:22) h T kF k L ( T ) k A ′ k L ( T ) . (64)Summing these estimates for all the tetrahedrons and applying the Cauchy-Schwarz inequality gives the result. 14 tep two We compare now S and S . Lemma 3.
Let a face f have vertexes i, j, k . We have: Z f F ( A n ) = Z f d A n + 12 [ A n , A n ] , (65)= A nik + A nkj + A nji + 16 (cid:0) [ A nji , A nkj ] + [ A nkj , A nik ] + [ A nik , A nji ] (cid:1) . (66) Proof.
Remark that: Z f d A n = A nik + A nkj + A nji , (67)Using formulas of the type: Z f λ ji ∧ λ kj = 1 / , (68)one gets the second term. Proposition 2.
We have: − F f ( A n ) = Z f F ( A n ) + O ( h f ) . (69) Proof. F f ( A n ) can be estimated with the help of the BCH formula and com-pared with the formula previously obtained for the right hand side.Using the same arguments as in the proof of Lemma 3, we get: Lemma 4.
We have: D | A ( R f F ( A )) A ′ = R f d A ′ + [ A, A ′ ] (70)= A ′ ik + 1 / A nji , A ′ ik ] − [ A nkj , A ′ ik ])+ (71) A ′ kj + 1 / A nik , A ′ kj ] − [ A nji , A ′ kj ])+ (72) A ′ ji + 1 / A nkj , A ′ ji ] − [ A nik , A ′ ji ]) . (73) Proposition 3.
We have: − D | A F f ( A ) A ′ = D | A R f F ( A ) A ′ + O ( h f | δA ′ • | + h f | A ′ • | ) . (74)15 roof. Define an entire function φ , by setting, for z = 0: φ ( z ) = (1 − e − z ) /z. (75)Recall that: F kji = exp( − A ik ) exp( − A kj ) exp( − A ji ) , (76)so that: − D F kji ( A ) A ′ = exp( − A ik ) φ (ad( − A ik )) A ′ ik exp( − A kj ) exp( − A ji )+ (77)exp( − A ik ) exp( − A kj ) φ (ad( − A kj )) A ′ kj exp( − A ji )+ (78)exp( − A ik ) exp( − A kj ) exp( − A ji ) φ (ad( − A ji )) A ′ ji . (79)Expand using φ ′ (0) = − / Proposition 4.
The discrete actions S and S are consistent of order h with respect to the norm defined by: k A k = k d A k L + k A k L . (80) Proof.
Let f and f be faces of a tetrahedron T in T n . Define: L n A ′ = D | A = A n (cid:0) R f F ( A ) h R f F ( A ) (cid:1) A ′ . (81)and: L ′ n A ′ = D | A = A n (cid:0) ( − F f ( A )) h ( − F f ( A )) (cid:1) A ′ . (82)Combining Propositions 2 and 3 gives: L ′ n A ′ = L n A ′ + O ( h T | δA ′ • | + h T | A ′ • | ) . (83)We have, by scaling: | M f f | (cid:22) h − T . (84)Insert it in the error term of (83) and write: h T | δA ′ • | T + h T | A ′ • | T (cid:22) h / T k d A ′ k L ( T ) + h − / T k A ′ k L ( T ) . (85)We sum over all tetrahedrons and apply a Cauchy-Schwarz inequality,remarking that: ( X T ∈T h T ) / (cid:22) h n . (86)This concludes the proof. 16 tep three We compare S and S ′ . Proposition 5.
We have: U li F kji U il = F kji + O ( h T ) , (87) and also: D | A U li ( A ) F kji ( A ) U il ( A ) A ′ = D | A F kji ( A ) A ′ + O ( h T | δA ′ • | + h T | A ′ • | ) . (88) Proof.
For the first assertion we write: U li F kji U il − F kji = U li ( F kji − ) U il − ( F kji − ) , (89)and conclude using: F kji − = O ( h T ) . (90)For the second one we compute:D | A U li F kji U il A ′ = U li D | A F kji ( A ) A ′ U il − U li [ φ (ad( − A li )) A ′ li , F kji ] U il . (91)On the right hand side, in the second term, we can replace F kji by F kji − and use again (90). From this the second assertion follows.As in the previous paragraph we can deduce: Proposition 6.
The discrete actions S and S ′ are consistent of order h with respect to the norm defined by: k A k = k d A k L + k A k L . (92) Conclusion
Adding the three estimates proved in Propositions 1, 4 and 6,we get:
Theorem 2.
The discrete actions S and S ′ are consistent of order h withrespect to the norm defined by: k A k = k d A k L + k A k L . (93)The arguments introduced also immediately show that, concerning theaction itself, we have consistency of order h . That is, if A is a smooth gaugepotential, we have: S n ( I n A ) − S ′ n ( I n A ) = O ( h n ) . (94)17 A discrete Noether’s theorem
In this section we propose an analogue of Noether’s first theorem [26], ex-pressed for discretizations over simplicial complexes, when the group actingon the fields preserves fibers, as defined below. This discrete result provides adiscrete conservation law associated with discrete gauge invariance. While itdoes not capture the full power of the continuous one, it is sufficient to proveconstraint preservation for evolution problems as in [8]. Recall that non-invariance of Lie algebra valued Whitney forms under discrete gauge trans-formations makes the standard action (27) problematic for the simulation ofevolution, because constraints are not preserved [10]. For electromagnetics,the conservation law associated in the continuum with gauge invariance, isnothing but electric charge conservation, of great physical significance.Related discrete Noether’s theorems have been discussed in particular in[23][22].We suppose we have a finite simplicial complex T . The maximal dimen-sion of the simplexes in T is m . We write S ⊳ T to say that S is a subsimplexof T . If S is a simplex and i a vertex not in S , S + i is the simplex obtainedby adjoining the vertex i to S . Conversely, if S is a simplex and i a vertexof S , S − i is the face of S opposite i .We suppose that on each maximal simplex T ∈ T m we have attachedfields Φ T of the form: Φ T = (Φ T ( S )) S⊳T ∈ Y S⊳T V T ( S ) . (95)That is, Φ T attaches a value in some space V T ( S ) to each subsimplex S of T . We call V T ( S ) the fiber above S .We suppose that we have a Lagrangian L T attached to T , which is afunction: L T : Y S⊳T V T ( S ) → R . (96)In the following we fix a simplex T ∈ T m . We suppose we have a oneparameter group action Λ T which acts separately on each fiber V T ( S ):Λ T ( S ) : R → Aut( V T ( S )) , (97)and for t ∈ R : Λ T [ t ]Φ T = (Λ T ( S )[ t ]Φ T ( S )) S⊳T . (98)We suppose that this group action leaves L T invariant: ∀ t ∈ R L T (Λ T [ t ]Φ T ) = L T (Φ T ) . (99)18e define the (local) infinitesimal generators: ξ T ( S ) = ∂ | t =0 Λ T ( S )[ t ]Φ T ( S ) . (100)and the (local) Euler-Lagrange functions: E T ( S ) = ∂ | S L T (Φ T ) , (101)and put: F T ( S ) = E T ( S ) ξ T ( S ) . (102)For each simplex S ⊳ T and each i ∈ T \ S choose a number p T ( i, S )subject to the condition that, for any simplex S ′ of dimension at least 1: X i ∈ S ′ p T ( i, S ′ − i ) = 1 . (103) Proposition 7.
Define, for any vertex i ∈ T : W T ( i ) = F T ( i ) + X S⊳T : i S p T ( i, S ) F T ( S + i ) , (104) and for any two distinct vertexes i, j ∈ T : V T ( i, j ) = F T ( i ) − F T ( j )+ X S⊳T : i,j S p T ( i, S ) F T ( S + i ) − p T ( j, S ) F T ( S + j ) . (105) Then we have: ( m + 1) W T ( i ) = X j : j = i V T ( i, j ) . (106) Proof.
In this proof, in which T is fixed, we drop the index T . The summationvariables S, S ′ are subsimplexes of T . First we remark: X j : j = i (cid:0) F ( i ) + X S : i,j S p ( i, S ) F ( S + i ) (cid:1) (107)= mW ( i ) − X j : j = i X S : i Sj ∈ S p ( i, S ) F ( S + i ) . (108)then we remark: X j : j = i (cid:0) F ( j ) + X S : i,j S p ( j, S ) F ( S + j ) + X S : i Sj ∈ S p ( i, S ) F ( S + i ) (cid:1) (109)= X j : j = i (cid:0) F ( j ) + X S ′ : j ∈ S ′ p ( j, S ′ − j ) F ( S ′ ) (cid:1) , (110)= X S ′ F ( S ′ ) − W ( i ) . (111)19rom invariance of the Lagrangian we get : X S ′ F ( S ′ ) = 0 , (112)and this concludes the proof.In the applications we have in mind, if a simplex S ∈ T is included intwo maximal simplexes T, T ′ ∈ T m we have V T ( S ) = V T ′ ( S ), and the globalvariable Φ has the property Φ T ( S ) = Φ T ′ ( S ). When this happens for allchoices S, T, T ′ such that S ⊳ T, T ′ ∈ T m , we have a well defined fiber V S above each S ∈ T and the action S will be of the form: S = X T ∈T m L T : Y S ∈T V ( S ) → R . (113)Moreover we suppose that the group action Λ acts separately on the fibers V ( S ), independently of any embedding into a maximal simplex T . In thissetting we define the (global) infinitesimal generators: ξ ( S ) = ∂ | t =0 Λ( S )[ t ]Φ( S ) , (114)the (global) Euler Lagrange functions: E ( S ) = ∂ | S L (Φ) = X T ∈T m : S⊳T E T ( S ) , (115)and put: F ( S ) = E ( S ) ξ ( S ) . (116)We suppose finally that we have chosen the numbers p T ( i, S ) independentlyof T containing i and S . When i and S are not included in any simplex of T we set p ( i, S ) = 0. The preceding Proposition gives, by adding contributionsfrom all maximal simplexes T : Proposition 8.
Define, for any vertex i ∈ T : W ( i ) = F ( i ) + X S ∈T : S + i ∈T i S p ( i, S ) F ( S + i ) , (117) and for any two distinct vertexes i, j ∈ T linked by an edge: V ( i, j ) = X T ∈T m : i,j ∈ T V T ( i, j ) . (118) Then we have: ( m + 1) W ( i ) = X j : i + j ∈T V ( i, j ) . (119)20n brief, equation (119) expresses a weighted sum of (global) Euler-Lagrangefunctions applied to infinitesimal generators, as a discrete divergence. In-deed it is natural to think of V ( i, j ) as degrees of freedom of a vectorfield V .Choose any cellular complex dual to T , so that, in particular, the domain iscovered by cells dual to the vertexes i ∈ T . Then V ( i, j ) is the flux fromthe cell dual to i into the cell dual to j , through the dual face of the edge ij ∈ T . The right hand side of (119) is then the total flux leaving the celldual to i , which is the natural degree of freedom for the divergence of V . Theessential antisymmetry property V ( i, j ) = − V ( j, i ) guarantees that summingthe discrete divergence over a union of top-dimensional dual cells, leaves onlya boundary term.These considerations apply directly to the proposed simplicial gauge the-ory, for which moreover we have variables attached only to 0- and 1- sim-plexes. Acknowledgments
We are grateful to Elizabeth Mansfield for helpful comments on Noether’stheorems.This work, conducted as part of the award “Numerical analysis and simu-lations of geometric wave equations” made under the European Heads of Re-search Councils and European Science Foundation EURYI (European YoungInvestigator) Awards scheme, was supported by funds from the ParticipatingOrganizations of EURYI and the EC Sixth Framework Program.
References [1] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exteriorcalculus, homological techniques, and applications.
Acta Numer. , 15:1–155, 2006.[2] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exteriorcalculus: from Hodge theory to numerical stability.
Bull. Amer. Math.Soc. (N.S.) , 47(2):281–354, 2010.[3] A. Bossavit. Mixed finite elements and the complex of Whitney forms.In
The mathematics of finite elements and applications, VI (Uxbridge,1987) , pages 137–144. Academic Press, London, 1988.214] N.H. Christ, R. Friedberg, and T.D. Lee. Random lattice field theory:General formulation.
Nuclear Physics, Section B , 202(1):89–125, 1982.cited By (since 1996) 71.[5] S. H. Christiansen. A characterization of second-order differential op-erators on finite element spaces.
Math. Models Methods Appl. Sci. ,14(12):1881–1892, 2004.[6] S. H. Christiansen. Stability of Hodge decompositions in finite ele-ment spaces of differential forms in arbitrary dimension.
Numer. Math. ,107(1):87–106, 2007.[7] S. H. Christiansen and T. G. Halvorsen. Convergence of lattice gaugetheory for Maxwell’s equations.
BIT , 49(4):645–667, 2009.[8] S. H. Christiansen and T. G. Halvorsen. Discretizing the Maxwell-Klein-Gordon equation by the lattice gauge theory formalism.
IMA J. Numer.Anal. , 31(1):1–24, 2011.[9] S. H. Christiansen and T. G. Halvorsen. A gauge invariant discretizationon simplicial grids of the Schr¨odinger eigenvalue problem in an electro-magnetic field.
SIAM Journal on Numerical Analysis , 49(1):331–345,2011.[10] S. H. Christiansen and R. Winther. On constraint preservation in nu-merical simulations of Yang-Mills equations.
SIAM J. Sci. Comput. ,28(1):75–101 (electronic), 2006.[11] S.H. Christiansen. On the linearization of regge calculus.
NumerischeMathematik , pages 1–28. 10.1007/s00211-011-0394-z.[12] P. G. Ciarlet. Basic error estimates for elliptic problems. In
Handbookof numerical analysis, Vol. II , Handb. Numer. Anal., II, pages 17–351.North-Holland, Amsterdam, 1991.[13] A. Dimakis and F. M¨uller-Hoissen. Discrete differential calculus:Graphs, topologies, and gauge theory.
J. Math. Phys. , 35(12):6703–6735, 1994.[14] J. Dodziuk and V. K. Patodi. Riemannian structures and triangulationsof manifolds.
J. Indian Math. Soc. (N.S.) , 40(1-4):1–52 (1977), 1976.[15] J.M. Drouffe and K.J.M. Moriarty. Gauge theories on a simplicial lattice.
Nuclear Physics, Section B , 220(3):253–268, 1983. cited By (since 1996)5. 2216] T.G. Halvorsen and Macdonald Sørensen T. Simplicial gauge the-ory and quantum gauge theory simulation.
Nuclear Physics B , doi :10.1016/j.nuclphysb.2011.08.016, 2011.[17] Y. Haugazeau and P. Lacoste. Condensation de la matrice masse pourles ´el´ements finis mixtes de H (rot). C. R. Acad. Sci. Paris S´er. I Math. ,316(5):509–512, 1993.[18] R. Hiptmair. Canonical construction of finite elements.
Math. Comp. ,68(228):1325–1346, 1999.[19] R. Hiptmair. Finite elements in computational electromagnetism.
ActaNumer. , 11:237–339, 2002.[20] S. Kobayashi and K. Nomizu.
Foundations of differential geometry. VolI . Interscience Publishers, a division of John Wiley & Sons, New York-Lond on, 1963.[21] J.B. Kogut. The lattice gauge theory approach to quantum chromo-dynamics.
Reviews of Modern Physics , 55(3):775–836, 1983. cited By(since 1996) 90.[22] E. L. Mansfield. Noether’s theorem for smooth, difference and finiteelement systems. In
Foundations of computational mathematics, San-tander 2005 , volume 331 of
London Math. Soc. Lecture Note Ser. , pages230–254. Cambridge Univ. Press, Cambridge, 2006.[23] E. L. Mansfield and G. R. W. Quispel. Towards a variational complexfor the finite element method. In
Group theory and numerical analysis ,volume 39 of
CRM Proc. Lecture Notes , pages 207–232. Amer. Math.Soc., Providence, RI, 2005.[24] P. Monk. An analysis of N´ed´elec’s method for the spatial discretizationof Maxwell’s equations.
J. Comput. Appl. Math. , 47(1):101–121, 1993.[25] J.-C. N´ed´elec. Mixed finite elements in R . Numer. Math. , 35(3):315–341, 1980.[26] P. J. Olver.
Applications of Lie groups to differential equations , volume107 of
Graduate Texts in Mathematics . Springer-Verlag, New York,second edition, 1993.[27] V. V. Prasolov.
Elements of homology theory , volume 81 of
GraduateStudies in Mathematics . American Mathematical Society, Providence,RI, 2007. Translated from the 2005 Russian original by Olga Sipacheva.2328] P.-A. Raviart and J. M. Thomas. A mixed finite element method for2nd order elliptic problems. In
Mathematical aspects of finite elementmethods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome,1975) , pages 292–315. Lecture Notes in Math., Vol. 606. Springer, Berlin,1977.[29] T. Regge. General relativity without coordinates.
Nuovo Cimento (10) ,19:558–571, 1961.[30] H. J. Rothe.
Lattice gauge theories , volume 74 of
World Scientific LectureNotes in Physics . World Scientific Publishing Co. Pte. Ltd., Hackensack,NJ, third edition, 2005. An introduction.[31] G. Strang. Variational crimes in the finite element method. In
Themathematical foundations of the finite element method with applicationsto partial differential equations (Proc. Sympos., Univ. Maryland, Balti-more, Md., 1972) , pages 689–710. Academic Press, New York, 1972.[32] A. Weil. Sur les th´eor`emes de de Rham.
Comment. Math. Helv. , 26:119–145, 1952.[33] S. Weinberg.
The quantum theory of fields. Vol. I . Cambridge UniversityPress, Cambridge, 2005. Foundations.[34] S. Weinberg.
The quantum theory of fields. Vol. II . Cambridge Univer-sity Press, Cambridge, 2005. Modern applications.[35] H. Whitney.
Geometric integration theory . Princeton University Press,Princeton, N. J., 1957.[36] K. G. Wilson. Confinement of quarks.
Phys. Rev. D , 10(8):2445–2459,1974.[37] K.S. Yee. Numerical solution of initial boundary value problems involv-ing maxwells equations in isotropic media.