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

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Space-Time Finite-Element Exterior Calculus and Variational Discretizations of Gauge Field Theories 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 Field 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 tensor 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 − ; r2φ + (r · 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 = r × A; A + r r · 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 + rf; is expressed in terms of the exterior derivative 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. r · B(0) = 0; r · 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– r · 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 r · A = 0, which yields, conjugate momenta that have physical significance. The @φ r2φ = 0; A + r = 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 manifolds 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, Mathematics, 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 · rf; j = −E + (B × r)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 general relativity 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 tensor product 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 manifold 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 Hamiltonian mechanics, 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.
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