A simplicial gauge theory on spacetime Tore G. Halvorsen, NTNU, Trondheim Norway. Joint work with Snorre H. Christiansen, UiO. 1
Abstract In this talk I will introduce the variational form of the SU(N) Yang-Mills equations, and especially mention the continuous symmetry they admit. Then I will show how these equations, or more precisely the action of which they stem from, can be discretized in a compatible way, i.e. symmetry preserving, on spacetime simplicial meshes. The formulation is a generalization of classical lattice gauge theory. 2
Outline Introduction The Yang-Mills equations and symmetries Classical lattice gauge theory Construction of a simplicial gauge theory Conclusion 3
Introduction Fundamental particles are described by gauge theories such as Maxwell s equations and Yang-Mills equations. A gauge theory describes a set of fields through a Lagrangian functional, which is invariant under certain continuous Lie-group transformations of the fields. This so-called gauge symmetry gives rise to conserved quantities, hence it is important to preserve. Physicists have for a long time and with great success used lattice gauge theory to do calculations. 4
Although this is a numerical scheme for the quantum version of YM theories, it is also applicable to the classical theory. We have in a series of papers analyzed lgt, for Maxwell s equations, from a mathematical point of view. The work presented here should we viewed as a prolongation of this. GR is described by similar equations, so it should also be viewed as an approach to this. 5
The Yang-Mills equations Let M = R S be a Riemannian spacetime manifold, where S is a bounded domain in three dimensional Euclidean space, equipped with a Lorentzian signature. Consider the Lie-group SU(n) with associated Lie-algebra su(n). The elements can be represented by matrices. Hermitian conjugate of a matrix g is denoted g H, and the real valued scalar product is g g = Retr(g g H ). 6
The space of smooth k-forms on M are denoted Ω k (M), and Ω k (M) su(n) are identified with the space of smooth su(n) valued k-forms on M. The bracket of Lie algebra valued forms is determined by [u g, v g ]=u v [g, g ], where u, u are k-forms and g, g su(n). A connection one-form on M is an element A =(A 0,A) Ω 1 (M) su(n), A 0 representing the time component and A the space component. 7
The temporal- and spatial- curvature of such a connection one-form are F t (A) =da 0 + d t A +[A 0,A], F s (A) =da + 1 [A, A], 2 where d t and d denote exterior derivative in the temporal and spatial direction respectively. The unknown in SU(n) YM-theory is a connection one-form, and the action describing it is S[A] =S[A] t + S[A] s,where S[A] t = M F t (A) 2, S[A] s = M F s (A) 2. 8
A gauge transformation of A is associated with a choice of G SU(n) such that A 0 G (A 0 + d t ) G 1, A G (A + d) G 1. Theorem 1 The action S[A] is gauge invariant. This gauge invariance gives rise to conservation laws through Noether s theorem, and is the property we wish to preserve at the discrete level. The Euler-Lagrange equation is equivalent to stationary points of the action, i.e. DS[A] A =0, A. 9
Lattice gauge theory In classical lgt, spacetime is discretized by four-dimensional hypercubes. The connection one-forms are lowest order Whitney-elements, with DOF at the edges of the mesh. The fundamental quantity is discrete parallel transport operators, induced by the connection one-form, attached to the edges of the mesh, i.e. U e =exp( e A), U e0 =exp( A 0 ). e 0 Let e = {i, j}. ThenU e should be thought of as parallel transport from i to j. We also notice that U i,j = U 1 j,i. 10
x µ Structure of the LGT lattice x ν n + a µ U n,n+aµ U µν n U n,n+aν n + a ν 11
A face f of the mesh is a square consisting of four vertexes. Given an orientation and an origin these vertexes can be labelled f 0,f 1,f 2,f 3. The discrete curvature is defined by F f = U f0 f 1 U f1 f 2 U f2 f 3 U f3 f 0. Let h be the edge-length, and t the time-step. The action is then defined by S lgt = h 3 t f Re tr (1 F f ), where one sums over faces. 12
A discrete gauge transformation is associated with a choice G i SU(n) for each vertex. One transforms A s.t. U ij G i U ij G 1 j. Under a gauge transformation we observe that F f G f0 F f G 1 f 0. Theorem 1 By the cyclic invariance of the trace the action S lgt is gauge invariant. 13
In lgt only the diagonal terms contribute, each term being gauge invariant, comparable to the Yee-scheme, a mass-lumped FEM scheme. Conisistency follows from numerical integration estimates. For simplicial meshes, we propose instead to sum over tetrahedra in which faces interact two by two, in a gauge invariant fashion. 14
Construction of a simplicial gauge theory Assume T is a simplicial complex spanning the spatial domain S, and let i, e, f, and T denote vertexes, edges, faces, and tetrahedrons respectively. T will also be used for simplexes of any dimension. Time is discretized with a time-step t, and T is repeated at every time step, resulting in a spacetime simplicial complex T. The basic building block is T I τ, I τ =[τ, τ + t]. 15
Let W k (T ) be the space of Whitney k-forms, with canonical basis (λ T ). The 0-forms λ i are the barycentric coordinate maps, taking the value 1 at vertex i and 0 at the others. e = {i, j}: λ e = λ i dλ j λ j dλ i. f = {i, j, k}: λ f =2(λ i dλ j dλ k +c.p.). These k-forms have to be extended to k-forms on T. In addition, we have to define temporal edgeand face- basis functions. The spatial Whitney k-forms are extended to be piecewise affine in time. The extended basis are denoted (Λ T (τ) ). 16
The temporal edge basis functions are constructed as follows To every vertex i in the spatial mesh, there is a temporal edge e t (τ) ={i τ,i τ+ t }, i τ = i(τ). The temporal basis function attached to e t is defined as Λ et (τ)(t) = λ i π 1 t dt, t I τ 0, otherwise, where π : M = R S S. 17
The temporal face basis functions are constructed as follows To every edge e in the spatial mesh, there is a temporal face f t (τ) =e(τ) t. The temporal basis function attached to f t is defined as Λ ft (τ)(t) = π (λ e ) 1 t dt, t I τ 0, otherwise, where π : Ω(S) Ω(M). This construction ensures that the temporal basis are orthogonal to the spatial basis. The space spanned by (Λ T, Λ Tt ) is denoted W k (T). 18
The interpolated action Let A =(A 0,A) W 1 su(n), i.e. A = e t A 0,et Λ et + e A e Λ e, where the summations are over oriented edges, and note A 0,et = A 0, A e = A. e t e The curvature of such a one-form is F t (A) = e A e d t Λ e + e t A 0,et dλ et + e,e t [A 0,et,A e ]Λ et Λ e, F s (A) = e A e dλ e + 1 2 e,e [A e,a e ]Λ e Λ e. 19
Remark: Λ e Λ e / W 2 (T). Let (I t,i s ) and (J t,j s ) denote interpolation operators onto the temporal- and spatial- Whitney one- and two-forms, respectively. They are projection operators defined by I t u = ( u)λ et, e t e t I s u = e ( e u)λ e, J t u = ( u)λ ft, e t f t J s u = f ( f u)λ f, and are well defined as maps : Ω k W k, k =1, 2. 20
Define I := I t + I s, J := J t + J s, and I = J, := (d t,d), then by Stokes theorem. In particular d I s = J s d, consistent with classical Whitney elements. The interpolated FEM action is then S = S J t S J [A] t = M = Re f t,ft S J [A] s = M J t F t (A) 2 = M ft,f t tr J t f t (F t )J t f t (F t ) H, M ft,f t = J s F s (A) 2 = = Re f,f M f,f tr J s f (F s )J s f (F s ) H, M f,f = + S J s,where M M Λ ft Λ f t, Λ f Λ f. By inspiration from lgt we will construct an approximation to this action. 21
The spatial part Let T be a tetrahedra with vertices {i, j, k, l}, and choose A W 1 (T ) su(n). Attached to an edge e = {i, j}, oriented i j, one has an element A e = A ij su(n), A e = A, and parallel transport e i j is given by U e = U ij =exp(a ij ). Sign convention A ij = A ji implying U ij = U 1 ji. We also assume U is close enough to 1 so that the logarithm is unambiguous. The discrete spatial curvature associated with f = {i, j, k} is defined by F s ijk = U ij U jk U ki, and is in analogy with square Wilson loops. 22
The formula locates the curvature at i. The curvature at j is F s jki = U ji F s ijku ij, and gives a formula for parallel transport of curvature. Also note that F s ikj =(F s ijk ) 1. When f is oriented i j k, and the curvature is localized in i, wewritef s f = F s ijk. This defines a pointed oriented face, and the distinguished point of f is denoted f. 23
A discrete gauge transformation is associated with a choice G i SU(n) for each vertex. One then transforms A by which implies U ij G i U ij G 1 j, F s f G i F s f G 1 i. The temporal part Let f t (τ) ={i τ,j τ,j τ+ t,i τ+ t } be a temporal face, and pick A =(A 0,A) W 1 (T) su(n). Attached to a spatial edge e(τ) ={i τ,j τ }, one has as before an element A e(τ) = A iτ j τ su(n), and parallel transport from i τ to j τ is again given by U iτ j τ =exp(a iτ j τ ). 24
Attached to a temporal edge e t (τ) ={i τ,i τ+ t }, one has an element A 0,et (τ) = A 0,iτ i τ+ t su(n), A 0,et (τ) = e t A 0, and parallel transport i τ i τ+ t is given by U 0,et (τ) =exp(a 0,et (τ)) We use the sign convention A 0,iτ i τ+ t = A 0,iτ+ t i τ which corresponds to U 0,iτ i τ+ t = U 1 0,i τ+ t i τ. Again we assume U and U 0 to be close enough to 1 so that the logarithm of them is unambiguous. The discrete temporal curvature associated with f t (τ), located at i τ is defined by F t i τ j τ j τ+ t i τ+ t = U iτ j τ U 0,jτ j τ+ t U jτ+ t i τ+ t U 0,iτ+ t i τ. Parallel transport from i τ to i τ+ t is given by F t i τ+ t i τ j τ j τ+ t = U 0,iτ+ t i τ F t i τ j τ j τ+ t i τ+ t U 0,iτ i τ+ t. 25
When f t (τ) is oriented i τ j τ j τ+ t i τ+ t, and the curvature is localized in i τ,wewritef t f t (τ) = F t i τ j τ j τ+ t i τ+ t. The distinguished point of f t (τ) is denoted f t (τ). Under a discrete gauge transformation, one transforms A 0 such that U 0,iτ i τ+ t G iτ U 0,iτ i τ+ t G 1 i τ+ t, which implies F t f t (τ) G i τ F t f t (τ) G 1 i τ. 26
It is F 1 that approximates the continuous curvature, and a possible, but not gauge invariant discrete action is S = S t + S s S[A] t = Re f t,f t M ft,f t tr (F t f t 1)(F t f t 1)H, S[A] s = Re f,f M f,f tr (F s f 1)(F s f 1) H. We make this action gauge invariant by parallel transport. The spatial part Let f and f be two spatial faces of a tetrahedron T. The curvature at f and f can be connected by Uf f. However, the curvature associated to f at time τ, will interact with the curvature at f at times τ and τ + t. 27
This is resolved by U 0. We approximate the spatial part of the action as S L [A] s = Re f(τ),f (τ ) M f(τ),f (τ ) tr Uf (τ) f(τ) (F f(τ) s 1)U f(τ) f (τ) U 0, f (τ) f (τ ) (F f s (τ ) 1)H U 0, f (τ ) f (τ). The temporal part By properties of the canonical basis (Λ ft ), the interactions between the temporal curvature occur only at equal time intervals, i.e. τ, τ + t. Thus, we localize the curvature at both f t and f t at time τ. The curvature at f t (τ) and ft(τ) can be connected by Uf t (τ) f t (τ). 28
We approximate the temporal part of the action as S L [A] t = Re M ft (τ),ft (τ) tr f t (τ),ft (τ) U f t (τ) f t (τ) (F t f t (τ) 1) U f t (τ) f t (τ)(f t f t (τ) 1)H. Theorem 1 The action S L [A] =S L [A] t + S L [A] s is discretely gauge invariant. It is also possible to construct a gauge invariant action for a complex scalar field. This can be found in the preprint http://arxiv.org/abs/1006.2059 There we have also stated and proved a discrete Noether s theorem, applicable in the FEM setting. 29
Consistency Definition 1 Two actions S m and S m defined on X m are consistent with each other, w.r.t. a norm, iffor any A X m we have DS m [A]A DS sup m[a]a A X m A 0, m where X m = W 1 (T m ) su(n), T m is a sequence of simplicial meshes of M, and such that the discretization parameters converge to 0 as m. We proved consistence w.r.t. the norm A := A L (H 1 ) + t A L (L 2 ). 30
Techniques used Stability of the interpolation operators L 2 L 2. Scaling arguments. Inverse estimates. The result relies also on a CFL condition, i.e. there exists a constant C such that 0 < 1 C t m h m C m. 31
References Rothe, H. J. (2005), Lattice Gauge Theories, An Introduction, Vol.74 of World Scientific Lecture Notes in Physics, 3. edn, World Scientific. Creutz, M. (1986), Quarks, gluons and lattices, Cambridge Monographs on Mathematical Physics, reprinted (with corrections) edn, Cambridge. Christiansen, S. H. and Halvorsen, T. G. (2011), Discretizing the Maxwell-Klein-Gordon equation by the lattice gauge theory formalism, IMA Journal of Numerical Analysis, 31: 1-24. Christiansen, S. H. and Halvorsen, T. G. (2009), Convergence of lattice gauge theory for Maxwell s equations, BIT, 49(4): 645-667. Christiansen, S. H. and Halvorsen, T. G. (2010), A gauge invariant discretization on simplicial grids of the Schrödinger eigenvalue problem in an electromagnetic field, SIAM Journal on Numerical Analysis. 32