Reformulation of Electromagnetism with Differential Forms

Abstract

Japan IntroductionClassical electromagnetism is a well-established discipline. However, there remains some confusions and misunderstandings with respect to its basic structures and interpretations. For example, there is a long-lasting controversy on the choice of unit systems. There are also the intricate disputes over the so-called EH or EB formulations. In some textbooks, the authors respect the fields E and B as fundamental quantities and understate D and H as auxiliary quantities. Sometimes the roles of D and H in a vacuum are totally neglected.These confusions mainly come from the conventional formalism of electromagnetism and also from the use of the old unit systems, in which distinction between E and D,o rB and H is blurred, especially in vacuum. The standard scalar-vector formalism, mainly due to Heaviside, greatly simplifies the electromagnetic (EM) theory compared with the original formalism developed by Maxwell. There, the field quantities are classified according to the number of components: vectors with three components and scalars with single component. But this classification is rather superficial. From a modern mathematical point of view, the field quantities must be classified according to the tensorial order. The field quantities D and B are the 2nd-order tensors (or 2 forms), while E and H are the 1st-order tensors (1 forms). (The anti-symmetric tensors of order n are called n-forms.)The constitutive relations are usually considered as simple proportional relations between E and D, and between B and H. But in terms of differential forms, they associate the conversion of tensorial order, which is known as the Hodge dual operation. In spite of the simple appearance, the constitutive relations, even for the case of vacuum, are the non-trivial part of the EM theory. By introducing relativistic field variables and the vacuum impedance, the constitutive relation can be unified into a single equation.The EM theory has the symmetry with respect to the space inversion, therefore, each field quantity has a definite parity, even or odd. In the conventional scalar-vector notation, the parity is assigned rather by hand not from the first principle: the odd vectors E and D are named the polar vectors and the even vectors B and H are named the axial vectors. With respect to differential forms, the parity is determined by the tensorial order and the pseudoness (twisted or untwisted). The pseudoness is flipped under the Hodge dual operation. The way of parity assignment in the framework of differential forms is quite natural in geometrical point of view.

Dimensional scaffolding of electromagnetism using geometric algebra

“…Formulations of EM in the language of p-forms [44,45] have shown that there is an intrinsic geometric structure in the three-dimensional case. This is related to the Hodge dualities between the EM field magnitudes [46][47][48][49][50].…”

Improving student understanding of electrodynamics: the case for differential forms

“…It generalizes the notion of total differential of a scalar function (9). Unlike the ordinary derivative, it is a dimensionless operator.…”

“…Another intriguing point is that E and H obey a similar transformation law (as D and B do), suggesting the pair share a common mathematical nature. Sometimes, an additional clarification is introduced based on mirror symmetries, dividing the fields into polar (or true) vectors such as E and D, and axial vectors (or pseudovectors) such as B and H. 3 However, it is obviously not enough to enlighten the form of all ( 7)-( 8) and as remarked by Kitano,9 "in spite of the simple appearance, the constitutive relations, even for the case of vacuum, are the non-trivial part of the EM theory. ".…”

“…Exterior calculus originates from the pioneering works of Grassmann and Cartan and it is concerned with the properties of mathematical objects called differential forms. These latter have raised considerable attention because of their many applications in physics, as testified by the number of articles [9][10][11][12][13][14][15][16][17] and books [18][19][20][21][22][23][24][25][26][27] dedicated to differential forms, but they are usually not introduced at an undergraduate level and as far as we know, only more advanced textbooks make abundant use of this formalism (see e.g. 19,25,28,29 ).…”

Geometric Structure behind Duality and Manifestation of Self-Duality from Electrical Circuits to Metamaterials

“…Maxwell's electromagnetic theory in a four-dimensional spacetime has an analogous structure to the sheet problem discussed in Section 4. To see this analogy, we use the differential-form approach to Maxwell's equations [62,[89][90][91][92]. The wedge product, exterior derivative, integral, and Hodge star are naturally extended in dimensions greater than two.…”