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Fourvector Algebra Control [28] Fourvector Algebra Diego Sa´a 1 (1) Escuela Polit´ecnica Nacional. Quito – Ecuador. e-mail: [email protected] Abstract The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors in a natural form. It is shown the excellent ability to perform rotations with the use of fourvectors, as well as their use in relativity for producing Lorentz boosts, which are understood as simple rotations. PACS: 02.10.Vr Key words: fourvectors, division algebra, 3D-rotations, 4D-rotations Contents 3.5 Scalar multiplication . 14 3.6 Unitfourvector . 14 1Introduction 2 3.7 Fourvector division . 14 1.1 General ............ 2 3.8 Right factor of a fourvector . 14 1.2 The Hamilton quaternions . 5 3.9 Left factorofafourvector . 15 1.3 The Pauli quaternions . 6 3.10 Solution of quadratic fourvec- 1.4 Dirac matrices . 6 tor polynomials . 16 arXiv:0711.3220v1 [math-ph] 20 Nov 2007 2 Thefourvectors 7 4 Fourvectorrotation 17 2.1 Discussion . 8 4.1 Composition of rotations . 20 2.2 Complexfourvectors . 9 4.2 Reflections........... 21 3 Fourvector algebra 10 5 Conclusions 22 3.1 Product with matrices . 12 3.2 Thenorm ........... 13 3.3 Identity fourvector . 13 3.4 Multiplicative inverse . 13 1 1 Introduction Heaviside was expressing when he wrote that there is much more thinking to be done [to set 1.1 General up quaternion equations]. In principle, most everything done with the new system of vec- In this paper it is suggested the use of tors could be done with quaternions, but the fourvectors with the purpose of replacing the operations required to make the quaternions 3D vectors and the quaternions. Because behave like vectors added difficulty to using the fourvectors contain the three dimensional them and provided little benefit to the physi- vectors and can be considered a formalization cist.” of them. “Gibbs was acutely aware that quater- The discovery of the quaternions is attributed nionic methods contained the most important to the Irish mathematician William Rowan pieces of his vector methods.” [32] Hamilton in 1843 and they have been used for After a heated debate, “by 1894 the debate the study of several areas of Physics, such as had largely been settled in favor of modern mechanics, electromagnetism, rotations and vectors” [32]. relativity [26], [20], [6], [2], [23], [13]. James Alexander MacFarlane was one of the Clerk Maxwell used the quaternion calculus debaters and seems to have been one of the in his Treatise on Electricity and Magnetism, few in realizing what the real problem with published in 1873 [22]. An extensive bibliog- the quaternions was. “MacFarlane’s attitude raphy of more than one thousand references was intermediate - between the position of about Quaternions in mathematical physics the defenders of the Gibbs–Heaviside system has been compiled by Gsponer and Hurni and that of the quaternionists. He supported [14]. the use of the complete quaternionic product The modern vectors were discovered by of two vectors, but he accepted that the the Americans Gibbs and Heaviside between scalar part of this product should have a 1888 and 1894. Their work may be consid- positive sign. According to MacFarlane the ered a sort of combination of quaternions and equation j k = i was a convention that ideas developed around 1840 by the German should be interpreted in a geometrical way, Hermann Grassman. The notation was pri- but he did not accept that it implied the marily borrowed from quaternions but the negative sign of the scalar product”. [25] geometric interpretation was borrowed from (The emphases are mine). Grassman’s system. He incorrectly attributed the problem to a By the end of the nineteenth century secondary and superficial matter of repre- the mathematicians and physicists were hav- sentation of symbols, instead of blaming to ing difficulty in applying the quaternions to the more profound definition of the quater- Physics. nion product. “MacFarlane credited the Ryan J. Wisnesky [32] explains that “The controversy concerning the sign of the scalar difficulty was a purely pragmatic one, which product to the conceptual mixture done by 2 Hamilton and Tait. He made clear that the ford. According to Gaston Casanova [4] “It negative sign came from the use of the same was the English Clifford who carried out the symbol to represent both a quadrantal versor decisive path of abandoning all the commu- and a unitary vector. His view was that tativity for the vectors but conserving their different symbols should be used to represent associativity.” “This algebra absorbs the those different entities.” [25] (The emphasis Hamilton quaternions, the Girard’s complex is mine). quaternions, the cross product and the com- plex numbers, the hyperbolic numbers and At the beginning of the twenty century, the dual numbers.” [4]. Also the Hestenes’ Physics in general, and relativity theory “geometric product” conserves associativity in particular, was lacking an appropriate [18]. In this sense, the associativity of the mathematical formalism to represent the product is finally abandoned in the fourvector new physical quantities that were being algebra proposed in the present paper. This discovered. But, despite the fact that all means that the fourvectors do not constitute physical variables such as space-time points, a Clifford Algebra [2] or a Geometric Alge- velocities, potentials, currents, etc., were rec- bra [1]. This is a collateral effect of the pro- ognized that must be represented with four posed algebra, and constitutes a hint about values, the quaternions were not being used the form the fourvectors should handle, for to represent and manipulate them. It was example, a sequence of rotations; besides, the necessary to develop some new mathematical complex numbers are not handled as in the devices to manipulate such variables. Be- Hamilton quaternions, where the real number sides vectors, other systems such as tensors, is put in the scalar part and the imaginary in spinors, matrices and geometric algebra were the vector part, but a whole complex number developed or used to handle the physical can be put in each component, so it is pos- variables. sible to have up to four complex numbers in each fourvector. But, what is more impor- During the twenty century we have wit- tant, it is known that in quantum mechanics, nessed further efforts to overcome the diffi- observables do not form an associative alge- culties remaining, with the development of bra, so this could be the natural algebra for other algebras, which recast several of the Physics. ideas of Grassman, Hamilton and Clifford in The proposed algebra could have been a slightly different framework. An example in already developed, around 1900, under this direction is Hestenes’ Geometric Algebra the name of hyperbolic quaternion, which in three dimensions and Space Time Algebra is a mathematical concept introduced by in four dimensions. [16], [17], [21], [8] [18] Alexander MacFarlane of Chatham, Ontario. The commutativity of the product was The idea was dismissed for its failure to abandoned in all the previous quaternions conform to associativity of multiplication, and in some algebras, such as the one of Clif- but it has a legacy in Minkowski space and 3 as an extension of “split-complex numbers”. Physics using quaternions. But this attempt Like the quaternions, it is a vector space has been plagued with recurring pitfalls for over the real numbers of dimension 4. There reasons until now unknown to both physi- is only the mention to such quaternions but cists and mathematicians. The quaternions no accessible references to confirm if those have not been making problem solving easier quaternions satisfy the same algebraic rules or simplifying the equations. Very often the given in the following for the fourvectors. Hamilton quaternions require an extreme ha- However our intent is to convince the reader bility to guess when and where a quaternion that the presented here is one of the most needs to be conjugated, in order to obtain important mathematical tools for Physics. some particular result. I believe that this has been due to an The fourvector representation is without a internal problem in the mathematical struc- doubt a more unified theory in comparison to ture of the Hamilton quaternions, which I the classical vector or matrix representations. will try to reveal in the present paper. The correction of such problem constitutes a new The use of fourvectors allows discerning non-commutative, non-associative normed constants, variables and relations, previously algebraic structure with which it is possible unknown to Physics, which are needed to to work with fourvectors in an improved complete and make coherent the theory. way relative to the Hamilton, Pauli or Dirac quaternions, geometric algebra, space–time The vectors have lost some ground in algebra and other formalisms. favor of the Hamilton quaternions due to the lack of an appropriate 4D-algebra. For In the present paper, in particular, the example, Douglas Sweetser, who has worked present author exhibits the application of the extensively in the application of Hamilton fourvectors to 3D and 4D rotations, which quaternions to many possible physical areas, requires a reformulation of the Hamiltonian in general with very little success, sustains mathematics. these opinions: “Today, quaternions are of interest to historians of mathematics. Vector The powerful Mathematica R package analysis performs the daily mathematical includes a standard algebra package for the routine that could also be done with quater- manipulation of the Hamilton quaternions. nions. I personally think that there may be I have borrowed from that package the 4D roads in physics that can be efficiently symbol, as double asterisk, to represent the traveled only by quaternions.” [29] fourvector product.
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