21st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 2014. Groningen, The Netherlands

Space-Time Finite-Element Exterior Calculus and Variational Discretizations of Gauge Theories

Joe Salamon1, John Moody2, and Melvin Leok3

Abstract— Many gauge field theories can be described using a the space-time covariant (or multi-Dirac) perspective can be multisymplectic Lagrangian formulation, where the Lagrangian found in [1]. density involves space-time differential forms. While there has An example of a gauge symmetry arises in Maxwell’s been much work on finite-element exterior calculus for spatial and product space-time domains, there has been less equations of electromagnetism, which can be expressed in done from the perspective of space-time simplicial complexes. terms of the scalar potential φ, the vector potential A, the One critical aspect is that the Hodge star is now taken with electric field E, and the magnetic field B. respect to a pseudo-Riemannian metric, and this is most natu- ∂A ∂ rally expressed in space-time adapted coordinates, as opposed E = −∇φ − , ∇2φ + (∇ · A) = 0, to the barycentric coordinates that the Whitney forms (and ∂t ∂t their higher-degree generalizations) are typically expressed in  ∂φ terms of. B = ∇ × A, A + ∇ ∇ · A + = 0,  ∂t We introduce a novel characterization of Whitney forms and their Hodge dual with respect to a pseudo-Riemannian metric where  is the d’Alembert (or wave) operator. The following that is independent of the choice of coordinates, and then apply gauge transformation leaves the equations invariant, it to a variational discretization of the covariant formulation of Maxwell’s equations. Since the Lagrangian density for this ∂f φ → φ − , A → A + ∇f, is expressed in terms of the of the four- ∂t potential, the use of finite-dimensional function spaces that respects the de Rham cohomology results in a discretization that where f is an arbitrary scalar-valued function of space-time. inherits the gauge symmetries of the continuous problem. This Associated with this gauge symmetry is the Cauchy initial then yields a variational discretization that exhibits a discrete data constraint, Noether’s theorem, which implies that an associated multi- momentum is automatically conserved by the discretization. ∇ · B(0) = 0, ∇ · E(0) = 0.

I.INTRODUCTION Typically, the indeterminacy in the equations of motion asso- ciated with the gauge freedom in the field theory is addressed A gauge symmetry is a continuous local transformation by imposing a gauge condition. In electromagnetism, two on the field variables that leaves the system physically commonly imposed gauge conditions are the Lorenz gauge indistinguishable. A consequence of this is that the Euler– ∇ · A = − ∂φ , which yields, Lagrange equations are underdetermined, i.e., the evolu- ∂t tion equations are insufficient to propagate all the fields. φ = 0, A = 0, The fields can be classified into kinetic fields that have no physical significance, and the dynamic fields and their and the Coulomb gauge ∇ · A = 0, which yields, conjugate momenta that have physical significance. The ∂φ ∇2φ = 0, A + ∇ = 0. Euler–Lagrange equations are underdetermined as the gauge  ∂t symmetry implies that there is a functional dependence between the equations, but this also results in a constraint Given different initial and boundary conditions, some prob- (typically elliptic) on the initial data on a Cauchy surface. lems may be easier to solve in certain gauges than others. That is to say the Euler-Lagrange equations derived from Unfortunately, there is no systematic way of deciding which the action are underdetermined due to gauge invariance since gauge to use for a given problem. Additionally, as we will there are more fields than evolution equations to propagate see in section II-D, the gauge group is larger on the field components. However, they are simultaneously with more complicated topology. overdetermined due to the constraints imposed on the Cauchy An important consequence of gauge symmetries is the data. A comprehensive review of gauge field theories from presence of associated conserved quantities. Noether’s first theorem states that for every differentiable, local symmetry * This work was supported in part by NSF Grants CMMI-1029445, DMS- of an action, there exists a Noether current obeying a 1065972, CAREER Award DMS-1010687, and CMMI-1334759. continuity equation. Integrating this current over a spacelike 1Joe Salamon, Physics, University of California at San Diego, La Jolla, surface yields a conserved quantity called a Noether charge. CA 92093, USA [email protected] 2John Moody, , University of California at San Diego, La In electromagnetism, the Noether currents are given by Jolla, CA 92093, USA [email protected] 3 ∂f Melvin Leok, Mathematics, University of California at San Diego, La j0 = E · ∇f, j = −E + (B × ∇)f. Jolla, CA 92093, USA [email protected] ∂t

ISBN: 978-90-367-6321-9 743 MTNS 2014 Groningen, The Netherlands Our long-term goal is to develop geometric structure- preserving numerical discretizations that systematically ad- y dresses the issue of gauge symmetries. Eventually, we wish φ to study discretizations of that address the issue of general covariance. Towards this end, we will consider multi-Dirac mechanics based on a Hamilton– Pontryagin variational principle for field theories [2] that is well adapted to degenerate field theories. The issue of general t covariance also leads us to avoid using a discretization that presupposes a slicing of space-time, rather we will consider 4-simplicial complexes in space-time. More generally, we will need to study discretizations that are invariant with respect to some discrete analogue of the gauge x symmetry group. Fig. 1. A section of the configuration bundle: the horizontal axes represent spacetime, and the vertical axis represent dependent field variables. The II.MULTI-DIRAC FORMULATION OF FIELD THEORIES section φ gives the value of the field variables at every point of spacetime. The Dirac [3], [4] and multi-Dirac formulation [2] of me- chanics and field theories can be viewed as a generalization −1 of the Lagrangian and multi-symplectic formulation to the fiber over (t, x), which is ρ ((t, x)). This is illustrated in case whether the Lagrangian is degenerate, i.e., the Legendre Figure 1. transformation is not onto. This approach is critical to gauge The multisymplectic analogue of the tangent bundle is field theories, as the gauge symmetries naturally lead to the first jet bundle J 1Y , which is a fiber bundle over X degenerate Lagrangians. that consists of the configuration bundle Y and the first a a µ partial derivatives vµ = ∂y /∂x of the field variables A. Hamilton–Pontryagin Principle for Mechanics with respect to the independent variables. Given a section Consider a configuration Q with associated tan- φ : X → Y , φ(x0, . . . , xn) = (x0, . . . xn, y1, . . . ym), its gent bundle TQ and phase space T ∗Q. Dirac mechanics is first jet extension j1φ : X → J 1Y is a section of J 1Y over described on the Pontryagin bundle TQ ⊕ T ∗Q, which has X given by position, velocity and momentum (q, v, p) as local coordi- 1 0 n 0 n 1 m 1 m  nates. The dynamics on the Pontryagin bundle is described j φ(x , . . . , x ) = x , . . . , x , y , . . . , y , y ,0, . . . , y ,n . by the Hamilton–Pontryagin variational principle, where the The dual jet bundle J 1Y ? is affine, with fiber coordi- Lagrange multiplier (and momentum) p imposes the second- µ a nates (p, pa ), corresponding to the affine map vµ 7→ (p + order condition v =q ˙, µ a n+1 n+1 1 n 0 pa vµ)d x, where d x = dx ∧ · · · ∧ dx ∧ dx . Z t2 δ L(q, v) − p(q ˙ − v)dt = 0. (1) t1 C. Hamilton–Pontryagin Principle for Classical Fields It provides a variational description of both Lagrangian The (first-order) Lagrangian density is a map L : and , and yields the implicit Euler– J 1Y → Vn+1(X), and let L(j1φ) = L(j1φ) dV = µ a a Vn+1 Lagrange equations, L(x , y , vµ) dV , where (X) is the space of alternat- ∂L ∂L ing (n + 1)-forms over X and L(j1φ) is a scalar function q˙ = v, p˙ = , p = . (2) ∂q ∂v on J 1Y . For field theories, the analogue of the Pontryagin a bundle is J 1Y × J 1Y ?, and the first-jet condition ∂y = va The last equation is the Legendre transform FL :(q, q˙) 7→ Y ∂xµ µ ∂L replaces v =q ˙, so the Hamilton-Pontryagin principle is (q, ∂q˙ ). This is important for degenerate systems as it enforces the primary constraints that arise when the Legendre 0 = δS(ya, ya, pµ) transform is not onto. µ a Z   a   µ ∂y a µ a a n+1 B. Multisymplectic Geometry = δ pa µ − vµ + L(x , y , vµ) d x. (3) U ∂x The geometric setting for Lagrangian PDEs is multisym- a a µ a plectic geometry [5], [6]. The base space X consists of Taking variations with respect to y , vµ and pa (where δy independent variables, denoted by (x0, . . . , xn) ≡ (t, x), vanishes on the boundary ∂U) yields the implicit Euler– where x0 ≡ t is time, and (x1, . . . , xn) ≡ x are space Lagrange equations, variables. The dependent field variables, (y1, . . . , ym) ≡ y, µ a ∂pa ∂L µ ∂L ∂y a form a fiber over each space-time basepoint. The independent µ = a , pa = a , and µ = vµ, (4) ∂x ∂y ∂vµ ∂x and field variables form the configuration bundle, ρ : Y → X. The configuration of the system is specified by a section which generalizes (2) to the case of field theories; see [7] of Y over X, which is a continuous map φ : X → Y , such for more details. As the jet bundle is an affine bundle, the that ρ ◦ φ = 1X , i.e., for every (t, x) ∈ X, φ((t, x)) is in the duality pairing used implicitly in (3) is more complicated.

744 MTNS 2014 Groningen, The Netherlands The second equation of (4) yields the covariant Legendre Type I generating function of a symplectic map, which is transform, FL : J 1Y → J 1Y ?, intended to approximate the exact discrete Lagrangian, ∂L ∂L Z h pµ = , p = L − va. (5) exact a ∂va ∂va µ Ld (q0, q1) ≡ L (q0,1(t), q˙0,1(t)) dt, µ µ 0

This unifies the two aspects of the Legendre transform by where q0,1(t) satisfies the Euler–Lagrange boundary-value combining the definitions of the momenta and the Hamilto- problem. There are systematic methods of constructing nian into a single covariant entity. computable approximations of the discrete Lagrangian [9], [10], and it can be shown that if the computable discrete D. Multi-Dirac formulation of Maxwell’s equations Lagrangian approximates the exact discrete Lagrangian to The electromagnetic Lagrangian density is given by a given order of accuracy, then the resulting variational integrator exhibits the same order of accuracy [11], [12]. 1 1 1 µν L(A, j A) = − dA ∧ ?dA = − Fµν F , (6) In the case of field theories, the boundary Lagrangian [13], 4 4 which is a scalar-valued function on the space of boundary where Fµν = Aµ,ν − Aν,µ, d is the exterior derivative, and data, plays a similar role, and the exact boundary Lagrangian Vk Vn−k ? is the Minkowski Hodge star ? : (M) → (M) has the form defined uniquely by the identity, Z Lexact(ϕ| ) ≡ L(j1ϕ˜) k k k k ∂U ∂U hhα , β iiv = α ∧ ?β , U where hh , ii is the Minkowski metric on differential forms, where ϕ˜ satisfies the boundary conditions ϕ˜|∂U = ϕ|∂U , and v is the volume form. For example, for standard and ϕ˜ satisfies the Euler–Lagrange equation in the interior Minkowski spacetime with metric signature (+ − −−), the of U. As with the case for Lagrangian ODEs, a computable Hodge star acts on 2-forms as follows: approximation of the boundary Lagrangian can be obtained by replacing the space-time integral with a quadrature rule, ?dt ∧ dx = dz ∧ dy, ?dy ∧ dz = dt ∧ dx, and considering a finite-element approximation of the con- ?dt ∧ dy = dx ∧ dz, ?dz ∧ dx = dt ∧ dy, figuration bundle. As we will see, in the case of Maxwell’s equations, this involves the use of Whitney forms as the ?dt ∧ dz = dy ∧ dx, ?dx ∧ dy = dt ∧ dz, finite-dimensional configuration bundle. For a more in-depth discussion, see, for example, page 411 III.SPACE-TIME WHITNEY FORMS of [8]. The Hamilton-Pontryagin action principle is given in co- The Whitney k-forms are a finite-dimensional subspace ordinates by of k-differential forms, and they are dual to k-simplices via integration pairing. They were introduced by Whitney in Z     µ,ν ∂Aµ 1 µν 4 [14], and they are typically expressed in terms of barycentric S = p ν − Aµ,ν − Fµν F d x, U ∂x 4 coordinates. The barycentric coordinates λi are defined on a where U is an open subset of X. The implicit Euler– k-simplex with vertex vectors v0, v1, . . . vk as functions of Lagrange equations are given by the position vector x such that µ,ν Xk Xk µ,ν µν ∂Aµ ∂p λivi = x λi = 1. p = F ,A = , = 0, i=0 i=0 µ,ν ∂xν ∂xν µ,ν Then, on a k-simplex ρ := [v0, v1, . . . vk], the Whitney k- and by eliminating p lead to Maxwell’s equations: k µν form wρ is ∂ν F = 0. k Note that the gauge symmetry of the action given in (6) k X i wρ = k! (−1) λidλ0 ∧ dλ1 ∧ ... dλci ∧ · · · ∧ dλk, is more general than what is typically considered in the i=0 standard formulation of electromagnetism. Since the action where the hat indicates an omitted term and the superscript only depends on dA, then the Lagrangian density is invariant k is usually dropped when the order of the form is clear. under shifts of A by any closed 1-form, in other words, 1- Whitney forms are a crucial ingredient in Finite Element forms ω such that dω = 0. Contrast this with the standard Exterior Calculus (FEEC), a finite element framework that formulation shown in section I, which implies that only exact encompasses all standard and mixed finite element formula- 1-forms ω = df leave the dynamics invariant. tions through the use of the de Rham complex formed by the exterior derivative d and the Koszul operator κ. The E. Discrete Multi-Dirac Variational Integrators k framework is described in terms of the PrΛ spaces, which The theory of variational integrators provide a way of are the spaces of order r polynomials on differential k-forms. − k k discretizing Lagrangian mechanical systems so as to ob- Indeed, Whitney forms characterize Pr Λ ⊆ PrΛ . For tain numerical integration schemes that are automatically more details, see [15], [16]. symplectic. They are based on the concept of a discrete The problem with the representation of Whitney forms Lagrangian Ld : Q × Q → R, which can be viewed as a in terms of barycentric coordinates is that the Hodge star

745 MTNS 2014 Groningen, The Netherlands of a is significantly easier to compute in is given by space-time adapted coordinates, and the Hodge star shows k! up in the Lagrangian density for electromagnetism. Even kw [W ] = sgn(ρ ∪ τ) ρ k n! though this does not present much of an obstacle for vacuum D E Vn (v − v ), (V (v − x)) ∧ W electromagnetism, an explicit characterization of the Hodge i=1 i 0 vj ∈τ j k star simplifies the calculations when matter sources and · Vn Vn . h i=1 (vi − v0), i=1 (vi − v0)i material properties are added into the dynamics. This is Then, one can show that the characterization of Whitney perhaps best seen through the permittivity  and permeability k-forms presented in Proposition 2 and Theorem 1 are equiv- µ ; any variation in their values mimic the effects of alent. Since the Hodge star applied twice is the identity map a varying . In particular, the electric field E (up to a sign), the coordinate-independent characterization and electric displacement field D are related by the Hodge of Whitney forms given in Theorem 1 provide an explicit star with respect to a metric induced by the permittivity, and characterization of the Hodge dual of the space of Whitney the magnetic induction B and the magnetic intensity H are k-forms. related by the the Hodge star with respect to a metric induced by the permeability. B. Space-time Whitney forms in Electromagnetism We now apply our space-time FEEC discretization to the vacuum electromagnetic action A. Space-time Whitney Forms 1 Z S = − dA ∧ ?dA. To simplify the task of using Whitney forms in space-time 4 M multi-Dirac discretizations of Maxwell’s equations, we intro- Assume the manifold M has a simplicial triangulation into duce a characterization of Whitney forms that is coordinate simplices σp. Discretizing the vector potential A to linear P p p independent. The proofs of these results can be found in [17]. order yields Ap = i

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IV. CONCLUSIONS [3] H. Yoshimura and J. Marsden, “Dirac structures in Lagrangian me- chanics Part I: Implicit Lagrangian systems,” J. Geom. Phys., vol. 57, In summary, gauge field theories exhibit gauge symmetries no. 1, pp. 133–156, 2006. that impose Cauchy initial value constraints, and are also [4] ——, “Dirac structures in Part II: Variational underdetermined. These result in degenerate field theories structures,” J. Geom. Phys., vol. 57, no. 1, pp. 209–250, 2006. [5] J. Marsden, G. Patrick, and S. Shkoller, “Multisymplectic geometry, that can be described using multi-Dirac mechanics. There variational integrators, and nonlinear PDEs,” Commun. Math. Phys., is a systematic framework for constructing and analyzing vol. 199, no. 2, pp. 351–395, 1998. Galerkin variational integrators for Hamiltonian PDEs based [6] J. Marsden, S. Pekarsky, S. Shkoller, and M. West, “Variational methods, multisymplectic geometry and ,” J. on a suitable choice of numerical quadrature and finite- Geom. Phys., vol. 38, no. 3-4, pp. 253–284, 2001. element approximation of the configuration bundle. In order [7] J. Vankerschaver, H. Yoshimura, and M. Leok, “The Hamilton– to develop a discretization of Maxwell’s equations, which in- Pontryagin principle and multi-Dirac structures for classical field theories,” J. Math. Phys., vol. 53, no. 7, p. 072903 (25 pages), 2012. volves a space-time variational principle on differential forms [8] R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, that involves the Hodge star with respect to a Minkowski and applications, 2nd ed., ser. Applied Mathematical Sciences. New metric, we introduced a coordinate-independent expression York: Springer-Verlag, 1988, vol. 75. [9] M. Leok and T. Shingel, “General techniques for constructing vari- for Whitney forms that provided an explicit characterization ational integrators,” Front. Math. China, vol. 7, no. 2, pp. 273–303, for the Hodge dual of Whitney forms. Finally, space-time 2012, (Special issue on computational mathematics, invited paper). Whitney forms provide a method for preserving the gauge [10] J. Hall and M. Leok, “Spectral variational integrators,” Numer. Math., 2012, (submitted, arXiv:1211.4534 [math.NA]). symmetry of electromagnetism at a discrete level. [11] J. Marsden and M. West, “Discrete mechanics and variational integra- We are currently exploring extensions of our space-time tors,” Acta Numer., vol. 10, pp. 317–514, 2001. characterization of Whitney forms to higher-degree approx- [12] G. Patrick and C. Cuell, “Error analysis of variational integrators of unconstrained Lagrangian systems,” Numer. Math., vol. 113, no. 2, pp. imation spaces by considering the construction introduced 243–264, 2009. by Rapetti and Bossavit in [18] that is related to geometric [13] J. Vankerschaver, C. Liao, and M. Leok, “Generating functionals and subdivision and rescalings of the simplices in a simplicial Lagrangian partial differential equations,” J. Math. Phys., vol. 54, no. 8, p. 082901 (22 pages), 2013. complex. In addition, we are investigating extending this [14] H. Whitney, Geometric integration theory. Princeton, N. J.: Princeton construction to include both material properties and matter University Press, 1957. sources as to simulate the full theory of electrodynamics. [15] D. N. Arnold, R. S. Falk, and R. Winther, “Finite element exterior calculus: from Hodge theory to numerical stability,” Bull. Amer. Math. Soc. (N.S.), vol. 47, pp. 281–354, 2010. REFERENCES [16] ——, “Finite element exterior calculus: homological techniques, and applications,” Acta Numer., vol. 15, pp. 1–155, 2006. [1] M. Gotay, J. Isenberg, J. Marsden, and R. Montgomery, “Momentum [17] J. Salamon, J. Moody, and M. Leok, “Geometric representations of maps and classical relativistic fields. Part I: Covariant field theory,” Whitney forms and their generalization to Minkowski spacetime,” 1998, (preprint, arXiv:physics/9801019 [math-ph]). 2013, (in preparation). [2] J. Vankerschaver, H. Yoshimura, and M. Leok, “The Hamilton- [18] F. Rapetti and A. Bossavit, “Whitney forms of higher degree,” SIAM Pontryagin principle and multi-Dirac structures for classical field J. Numer. Anal., vol. 47, no. 3, pp. 2369–2386, 2009. theories,” J. Math. Phys., vol. 53, no. 7, pp. 072 903, 25, 2012.

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