Pair and impair, even and odd form fields, and electromagnetism

Abstract

In this paper after reviewing the Schouten and de Rham definition of impair and pair differential form fields (not to be confused with differential form fields of even and odd grades) we prove that in a relativistic spacetime it is possible (despite claims in contrary) to coherently formulate electromagnetism (and we believe any other physical theory) using only pair form fields or, if one wishes, using pair and impair form fields together, in an appropriate way. Those two distinct descriptions involve only a mathematical choice and do not seem to lead to any observable physical consequence if due care is taken. Moreover, we show in details that a formulation of electromagnetic theory in the Clifford bundle formalism of differential forms where the two Maxwell equations of the so called free metric approach becomes a single equation is compatible with both formulations of electromagnetism just mentioned above. In addition we derive directly from Maxwell equation the density of force (coupling of the electromagnetic field with the charge current) that is a postulate in the free metric approach to electromagnetism. We recall also a formulation of the engineering version of Maxwell equations using electric and magnetic fields as objects of the same nature, i.e., without using polar and axial vectors.

Reply to da Rocha and Rodrigues' comments on the orientation congruent algebra and twisted forms in electrodynamics

“…However, this difference is no impediment to the integration of the OC algebra per se into the Clifford bundle formalism used by da Rocha and Rodrigues [1,8]. That is because the result of a calculation can always be interconverted between the two products by carrying along with the result of a Clifford multiplication the sign factor σ that transforms it into the result of an orientation congruent multiplication.…”

“…The OC algebra is used to define both the native exterior algebra and the native Clifford algebra 1 . To understand the remaining issue, how and why the native Clifford and ordinary Clifford algebras (and their bundle-theoretic extensions) are compatible, we need to discuss the orientation congruent algebra's role in the calculation of native Clifford products.…”

Some Differential Geometry

“…A thoughtful discussion of this issue may be found in [5]. Solution We recall (see Table 3.1 in Chap.…”

Gravitation as Plastic Distortion of the Lorentz Vacuum

“…Many tentatives have been done in order to establish energy-momentum and angular momentum conservation laws in GRT. All tentatives are based on the fact that although the field g is part of the spacetime structure the 'geometry' must also have an object that makes the role of its 'energy-momentum tensor' in order to warrant the conservation of the energy-momentum of the gravitational and matter fields However all tentatives resulted (until now) deceptive (unless some additional hypothesis are postulated for the Lorentzian spacetime structure, as e.g., that such structure is parallelizable) because all that have been attained was the association of a series of pseudo-energy momentum tensors to the gravitational field 14 and without additional commentaries from our part we quote here what the authors of [81] have to say on this issue: "It is a shame to loose the special relativistic total energy conservation in General Relativity. Many of the tentatives to resurrect it are quite interesting, many are simply garbage".…”

“…However, the formulation of such a theory using as unique ingredients the geometric objects of Minkowski spacetime and the correct representative of the gravitational field (the distortion field h) living in that spacetime and in such way that the resulting 14 The first one to introduce an energy-momentum pseudo-tensor for the gravitational field was [21]. However, it is possible to introduce an infinity of distinct energy-momentum pseudotensors for the gravitational field in GRT.…”