Differential Forms in Lattice Field Theories: An Overview

Teixeira, F.L.

doi:10.1155/2013/487270

Cited by 24 publications

(37 citation statements)

References 154 publications

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“…where L is the Lagrangian 4-form, F = dA the Faraday 2-form, A the potential 1-form, * J the chargecurrent density 3-form, ∧ the (anti-commutative) exterior product, and d the 4-d exterior derivative [6]. The Hodge star operator * [5][6][7] is a metric-dependent map (isomorphism) from p-forms to (n−p)-forms, p = 0, .…”

Section: Fundamentalsmentioning

confidence: 99%

“…The Hodge star operator * [5][6][7] is a metric-dependent map (isomorphism) from p-forms to (n−p)-forms, p = 0, . .…”

Section: Fundamentalsmentioning

confidence: 99%

“…where G is the Maxwell 2-form, G = * F [1,5,6,8,9]. In addition, since d 2 = 0, the relation F = dA implies dF = 0…”

Section: Fundamentalsmentioning

confidence: 99%

“…Taken together, (3) and (4) constitute Maxwell's equations in 4-d [6,[8][9][10][11]. Apart from its very compact form, these relations show two salient features [5,6].…”

Section: Fundamentalsmentioning

confidence: 99%

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LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Teixeira¹

2014

PIER

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Abstract-We discuss the ab initio rendering four-dimensional (4-d) spacetime of Maxwell's equations on random (irregular) lattices. This rendering is based on casting Maxwell's equations in the framework of the exterior calculus of differential forms, and a translation thereof to a simplicial complex whereby fields and causative sources are represented as differential p-forms and paired with the oriented pdimensional geometrical objects that comprise the set of spacetime lattice cells (simplices). We pay particular attention to the case of simplicial spacetime lattices because these can serve as building blocks of lattices made of more generic cells (polygons). The generalized Stokes' theorem is used to construct discrete calculus operations on the lattice based upon combinatorial relations depending solely on the connectivity and relative orientation among simplices. This rendering provides a natural factorization of (lattice) 4-d spacetime Maxwell's equations into a metric-free part and a metric-dependent part. The latter is encoded by discrete Hodge star operators which are built using Whitney forms, i.e., canonical interpolants for discrete differential forms. The derivation of Whitney forms is illustrated here using a geometrical construction based on the concept of barycentric coordinates to represent a point on a simplex, and the generalization thereof to represent higher-dimensional objects (lines, areas, volumes, and hypervolumes) in 4-d. We stress the role of the primal lattice, the barycentric dual lattice, and the barycentric decomposition lattice in achieving a complete description of the lattice theory. Lattice Maxwell's equations based on the exterior calculus of differential forms and on the use of Whitney forms as field interpolants inherits the symplectic structure and discrete analogues of conservation laws present in the continuum theory, such as energy and charge conservation. It also provides precise localization rules for the degrees of freedom associated with the different fields and sources on the lattice, and design principles for constructing consistent numerical solution methods that are free from spurious modes, spectral pollution, and (unconditional) numerical instabilities. We also briefly consider the relationship between lattice 4-d Maxwell's equations and some incarnations of discretization schemes for Maxwell's equations in (3 + 1)-d, such as finite-differences and finite-elements.

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Section: Fundamentalsmentioning

confidence: 99%

“…The Hodge star operator * [5][6][7] is a metric-dependent map (isomorphism) from p-forms to (n−p)-forms, p = 0, . .…”

Section: Fundamentalsmentioning

confidence: 99%

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LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Teixeira¹

2014

PIER

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“…And they are both based on the vectorial version of Maxwell's equations. As is well known, differential forms can be used to recast Maxwell's equations in a more succinct fashion, which completely separates metric-free and metric-dependent components [4][5][6][7][8][9]. Discrete exterior calculus (DEC), a discrete version of differential geometry, provides a numerical treatment of differential forms directly [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Electromagnetic Theory With Discrete Exterior Calculus

Chen¹,

Chew²

2017

PIER

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Abstract-A self-contained electromagnetic theory is developed on a simplicial lattice. Instead of dealing with vectorial field, discrete exterior calculus (DEC) studies the discrete differential forms of electric and magnetic fields, and circumcenter dual is adopted to achieve diagonal Hodge star operators. In this paper, Gauss' theorem and Stokes' theorem are shown to be satisfied inherently within DEC. Many other electromagnetic theorems, such as Huygens' principle, reciprocity theorem, and Poynting's theorem, can also be derived on this simplicial lattice consistently with an appropriate definition of wedge product between cochains. The preservation of these theorems guarantees that this treatment of Maxwell's equations will not lead to spurious solutions.

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Discrete Field Theory: Symmetries and Conservation Laws

Skopenkov

2023

Math Phys Anal Geom

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Differential Forms in Lattice Field Theories: An Overview

Abstract: We provide an overview on the application of the exterior calculus of differential forms to the ab initio formulation of lattice field theories, with a focus on irregular or "random" lattices.

Cited by 24 publications

References 154 publications

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

Electromagnetic Theory With Discrete Exterior Calculus

Discrete Field Theory: Symmetries and Conservation Laws

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