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A Proposal of an Algebra for Vectors and an Application to Electromagnetism

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A Proposal of an Algebra for Vectors and an Application to Electromagnetism A Proposal of an Algebra for Vectors and an Application to Electromagnetism Diego Sa´a 1 Abstract. A new mathematical structure intended to formalize the clas- sical 3D and 4D vectors is briefly described. This structure is offered to the investigators as a tool that bears the potential of being more ap- propriate, for its use in Physics and science in general, than any of the other mathematical structures of geometric origin, such as the Hamilton (or Pauli or Dirac) quaternions, geometric algebra (GA) and space-time algebra (STA). The application of this algebra in electromagnetism is demonstrated, where current concepts are reproduced, in some cases, and modified, in other cases. Several physical variables are proved to satisfy the wave equation. It is suggested the need of an electromagnetic field scalar, with which Maxwell's equations are derived as the result of a simple four-vector product. As a byproduct, new values and units for the dielectric permittivity and magnetic permeability of vacuum are proposed. Mathematics Subject Classification (2010). 02.10.De, 03.50.De, 06.20.fa. Keywords. four-vectors, quaternions, four-vector derivative, electromag- netic theory. 2 Diego Sa´a 1 Contents 1. Introduction 3 2. Four-vectors 8 2.1. Advantages of four-vectors 10 2.2. Complex four-vectors 10 3. Four-vector algebra 11 3.1. Sum, difference and conjugates 11 3.2. Four-vector product 12 3.3. The Magnitude (Absolute Value) and the Norm 13 3.4. Unit four-vector 13 3.5. Identity four-vector 13 3.6. Multiplicative inverse 14 4. Four-vector calculus 14 4.1. Derivatives 15 4.2. Differential of interval 16 4.3. Four-Velocity 16 4.4. Four-gradient 16 5. Four-vectors in electromagnetism 17 5.1. Maxwell's equations from electromagnetic four-vector 18 5.2. Electromagnetic four-vector from potential four-vector 22 5.3. Charge-current four-vector 24 5.4. Generalized charge-current continuity equations 25 5.5. Some electromagnetism in classical Physics 25 5.6. Solution of wave equation 26 5.7. Covariance of physical laws 27 5.8. Electromagnetic forces 27 6. Discussion 29 Acknowledgment 30 References 31 Algebra for Vectors and an Application 3 1. Introduction Four-vectors are regarded as the most proper mathematical structure for the handling of the pervasive four-dimensional variables identified in the Physics of the twentieth-century. In this paper, a new product is defined for four-vectors, which implies a new algebra for the handling of four-vectors. It is a new non-associative algebra that embraces vectors of up to four dimensions that can be extended with- out undue effort to further dimensions. If you are familiarized with vectors, you should find it very easy to work with this new mathematical structure, because it is a rather obvious formalization of vectors. Nevertheless, to the knowledge of the present writer, this algebraic structure has not been discovered before, despite the utmost and acknowledged impor- tance of vectors. The scalar and vector products (or dot and cross products) have not been defined and operated before with a single integrated and co- herent vector structure comparable to the one proposed here. By endowing vectors with the new product, vectors acquire more proper- ties and extend their use for the easier and correct handling of the four- dimensional physical variables. The new four-vector product reveals both the classical dot and cross products, when the first element of their operands, called the \scalar" in quaternionic terminology, is zero. This will be explained later. The author had to decide whether the new mathematical structure should be given a new name, or to maintain the classical one, despite the fact that a new mathematical form for the product is attributed to them. The decision of this author has been to preserve the name for the structure, which also preserves the functionality of three-dimensional vectors. The reader will be able to judge that this decision is justified by the fact that the remaining operations and interpretations continue being the same as what he is used to call simply \four-vectors" or \vectors". In the present paper, the terms \vector product", \four-vector product" or simply \product" will be used to refer to the new product. The classical dot and cross products will be named as such. Later it will be shown that the tensor product of a covariant by its corresponding contravariant four-vector can be reproduced by carrying out the product of the four-vector by itself, since no distinction is found between the covariant and contravariant forms of a four-vector in orthonormal coordinates. The rest of this section is essentially a recount of vector history, so the bored reader might wish to skim to the next section to confront our new proposal. The most similar, to four-vectors, mathematical structure proposed to rep- resent the physical variables has been the quaternions. Quaternions were known by Gauss by 1819 or 1820, but unpublished. Their official discovery is attributed to the Irish mathematician William Rowan Hamilton in 1843 and they have been used for the study of several areas of 4 Diego Sa´a 1 Physics, such as mechanics, electromagnetism, rotations and relativity [1], [2], [3], [4], [5], [6]. James Clerk Maxwell used the quaternion calculus in his Treatise on Electricity and Magnetism, published in 1873 [7]. An extensive bibliography of more than one thousand references about Quaternions in mathematical physics has been compiled by Gsponer and Hurni [8]. The Americans Gibbs and Heaviside discovered the modern vectors be- tween 1888 and 1894. Their work may be considered a sort of combination of quaternions and ideas developed around 1840 by the German physicist Hermann Grassman. The notation was primarily borrowed from quaternions but the geometric interpretation was borrowed from Grassman's system. Hamilton and his followers, such as Tait, considered quaternions as a math- ematical structure with great potential to represent physical variables. Nev- ertheless, they have not lived up to the expectations of physicists. By the end of the nineteenth century both mathematicians and physi- cists were having difficulty applying quaternions to Physics. The authors of reference [9] explain that quaternions constituted an interme- diate step between a plane geometric calculus (represented by the complex numbers) and the contemporary vector analysis. They allowed to simplify the writing of a problem and, under certain conditions, allowed the geomet- ric interpretation of the problem. Their multiplication has two physically meaningful products, but, supposedly, the presence of two parts in the same number complicated the direct handling of the quaternions. Maxwell, Heaviside, Gibbs and others noted the problems with the quater- nions and began a heated debate, with Tait and other advocates of quater- nions, which by 1894 had largely been settled in favor of modern vectors. Gibbs was acutely aware that quaternionic methods contained the most im- portant pieces of his vector methods. However, in the preface of Gibbs' book, Dr. Edwin Wilson affirms that \Notwithstanding the efforts which have been made during more than a half century to introduce Quaternions into physics the fact remains that they have not found wide favor." [10] Crowe comments that \Maxwell in general disliked quaternion methods (as opposed to quaternion ideas); thus for example he was troubled by the non-homogeneity of the quaternion or full vector product and by the fact that the square of a vector was negative which in the case of velocity vec- tor made the kinetic energy negative. The aspects that Maxwell liked were clearly brought in his great work on electricity; the aspects he did not like were indicated only by the fact that Maxwell did not include them." [11] Heaviside was aware of the several difficulties caused by quaternions. He wrote, for example, \Another difficulty is in the scalar product of Quater- nions being always the negative of the quantity practically concerned. Yet another is the unreal nature of quaternionic formulae" [12]. The difficulty was a purely pragmatic one, which Heaviside expressed saying that \there Algebra for Vectors and an Application 5 is much more thinking to be done [to set up quaternion equations], for the mind has to do what in scalar algebra is done almost mechanically" [12]. \There is great advantage in most practical work in ignoring the quaternion altogether, and also the double signification of a vector above referred to, and in abolishing the quaternionic minus sign." (Heaviside's emphasis) [12]. In principle, most everything done with the new system of vectors could be done with quaternions, but the operations required to make quaternions be- have like vectors added difficulty to using them and provided little benefit to the physicist. Precisely Crowe quotes the following paragraph attributed to Heaviside: \But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vecto- rial aspects antiphysical and unnatural, and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalars and vectors, using a very simple vectorial algebra in my papers from 1883 onward." [11] Alexander MacFarlane was one of the debaters and seems to have been another of the few in realizing what the real problem with the quaternions was. \MacFarlane's attitude was intermediate - between the position of the defenders of the Gibbs{Heaviside system and that of the quaternionists. He supported the use of the complete quaternionic product of two vectors, but he accepted that the scalar part of this product should have a positive sign. According to MacFarlane the equation j k = i was a convention that should be interpreted in a geometrical way, but he did not accept that it implied the negative sign of the scalar product".
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