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Commutative Hypercomplex Mathematics Hypercomplex Algebra 4-D Function Theory Hypercomplex Analysis Links Home Clyde Davenport's Commutative Hypercomplex Math Page Page 1 of 11 Commutative Hypercomplex Mathematics Hypercomplex Algebra 4-D Function Theory Hypercomplex Analysis Links Home Introduction Below, we shall give a comprehensive sketch of the 4-D commutative hypercomplex algebra ( not quaternions) and its associated function theory and analysis. The great advantage is a complete, classical 4-D function theory, something that is impossible with quaternions and other noncommutative or nonassociative systems. Fortunately, a wide audience should be able to follow the discussion, because the commutative hypercomplex math is derived directly from well-known, fundamental concepts, such as groups, rings, calculus, complex variables, matrices, complex function theory, and vector analysis. For a discussion of elementary group and ring theory, see any good introductory text on abstract algebra, such as Herstein, 1986 . There will be no occasion for deep theorems and complicated proofs. Top First, some background. How many of us know that Sir William Rowan Hamilton developed the quaternions (4-D numbers similar to those that we will discuss here, except noncommutative) in the 1830s specifically for field calculations? And that scientists and engineers of the time vehemently resisted their use? Something in the mindset at the time simply could not accept the notion that there could be a "fourth dimension," especially if it was claimed to be time. They clung tenaciously to a primitive combination of component- by-component calculation and extensive use of geometry. By the late 1800s, their calculations were accompanied by elaborate geometrical figures that looked like the 2 X 4 framing of a house. To allay their aversions, around 1880, J. W. Gibbs in America and O. W. Heaviside in Britain reformulated quaternion analysis so that all expressions would be constrained to three dimensions or less. For example, in the cross product of two three- dimensional vectors they arbitrarily set i×i=j×j=k×k=i×j×k=0 so that the result would come out as another 3-D vector. The quaternion product of two three-dimensional vectors is ab = - a·b+a×b, which has a scalar part and a 3-D vector part (i.e., is 4-D). Therefore, Gibbs and Heaviside avoided the quaternion product notation, and used only the dot and [modified] cross product components in what they cleverly renamed as vector analysis . Scientists and engineers accepted this subterfuge because it met their prejudice about 3-D being inviolate and it did not have the word "quaternion" mentioned anywhere. Nevertheless, vector analysis is a form of quaternion analysis [Crowe, 1967 ]. Top Hamilton developed the quaternion algebra by trial and error. Apparently, he had a prejudice of his own: that every nonzero element (i.e., having at least one nonzero component) should have a multiplicative inverse. By adopting this view, he was led directly to quaternions, because we now know that the quaternions are the only 4-D division algebra. What he didn't realize was that the quaternions form a group ring [i.e., the 1,i,j,k elements and their negatives form a group of order eight (the quaternion group, of course), and elements of the form 1x+iy+jz+kw, with x,y,z,w real, form a ring]. He didn't realize it because the notions of group and ring hadn't been developed at that time. We now know that there are exactly five distinct groups of order eight upon which group rings of 4-D elements may be constructed. The fact that we exclusively use the quaternion case (vector analysis) in science and engineering apparently stems from the fact that it was discovered first and the others were not examined for potential application when they were eventually uncovered. For a timeline on the development of quaternion analysis, see Jeff Biggus' quaternion history page. Top Group algebras, including those mentioned here, were first studied and described over one hundred years ago [ Peirce, 1881 ], [ Study, 1889 ]. No less than Dedekind published a paper [Dedekind, 1885 ] describing algebras that are direct sums of copies of the complex field, including the commutative hypercomplex algebra that will be described below. Accordingly, I do not claim original discovery of the commutative hypercomplex algebra [Davenport(1), 1991], but do claim origination of certain of its representations, http://home.usit.net/~cmdaven/hyprcplx.htm 25/12/2006 Clyde Davenport's Commutative Hypercomplex Math Page Page 2 of 11 interpretations, and the formulation of the function theory and analysis which is constructed upon it. Top Commutative Hypercomplex Algebra Basis Group In order to keep this manageable for an Internet reader, I will merely sketch the line of reasoning and the main results. For convenience, I will use D to refer to the commutative hypercomplex algebra. This choice alludes to "duplex space," which significance will become evident below. Readers wanting more detail may refer to my monograph [Davenport(2), 1991 ] or my contribution to the book by Ablamowicz(1), et al, 1996 . My monograph is out of publication, but a Web search of major science library online catalogs should turn up a copy. Top We will be aiming our formulation at physics applications, so we will use the notation Z=1x+iy+jz+kct , with x,y,z,ct real, for an element of the algebra D. In the fourth component, t represents time, and c is a scale factor depending upon the medium in which we are working; in a vacuum, it would be the speed of light. What we really desire for these elements is that they would form a field , but it was long ago proved that no such field could exist [Frobenius, 1877 ]. Nevertheless, we shall see that something analogous to a field is possible and will satisfy all our requirements attendant upon creating a function theory and analysis and applying them to physics applications. Top We start by establishing a group upon the basis elements 1,i,j,k . It must be Abelian because we ultimately want multiplication of elements of D to be commutative. This is a small enough problem that we can simply draw an 8 X 8 table of the basis elements and their negatives, then fill in the products by trial and error in such a way as to get uniqueness of products within each column and each row, and diagonal symmetry of the table. The summarized result is ij=ji= k jk=kj= -i ki=ik= -j ii= jj= -kk= -1 ijk= 1 Top The group identity element is 1. The second line indicates that every element has a multiplicative inverse. Associativity is immediately proved if we can find a real matrix representation, which we shall do below, hence we have a group. The fact that the group is Abelian assures that the ring that we will construct upon it will be commutative. With only a little manipulation, one may verify that the group is C 2 X C 4, where C n is the cyclic group of order n. [ASIDE: Thanks to Peter Jack, who pointed out that it is not the same as the dihedral group of symmetries of the square.] [NOTE: There are two other eighth-order commutative groups but neither has an element of cyclic order 4, which is necessary for complex-like behavior.] A matrix representation of the basis elements will prove to be very useful, but it is not intuitively obvious how to construct the same. I happened upon the following while constructing 4-D Cauchy-Riemann conditions by trial and error: Top http://home.usit.net/~cmdaven/hyprcplx.htm 25/12/2006 Clyde Davenport's Commutative Hypercomplex Math Page Page 3 of 11 Top The fact that this is a faithful representation may be verified by simple matrix multiplication and comparison with the multiplication table given earlier. These matrices are orthogonal, with determinant +1. The basis elements also have a 2 X 2 complex matrix representation; it is: These are the commutative counterparts of the Pauli spin matrices of physics. If one were to recast quantum mechanics using commutative hypercomplex mathematics (and I have no doubt that it could be done), then these matrices would play an important role. Top Commutative Ring We now have everything that we need to establish a ring over the basis group. The ring elements have the form Z=1x+iy+jz+ kct . Addition and subtraction are performed term-by- term, the same as for vectors. Exactly as for the complex variable case, multiplication of two elements is done by multiplying each term of the second element by every term of the first element, with reduction of the 1,i,j,k cross products by use of the group multiplication table, followed by collection of like terms. The result is: Top Multiplication of 4-D elements is commutative because multiplication is commutative in the basis group. Next, we need a definition of multiplicative inverse, or division by, elements of the form Z=1x+iy+jz+kct . It is not intuitively obvious what form it might take, and that might be a serious problem, except that we can construct a matrix representation of the element, then take the inverse of that. We do so as follows: We have a matrix representation of the basis elements 1,i,j,k . We substitute the matrices into the Z=1x+iy+jz+kct form and perform simple matrix addition to telescope the element into the form of a single matrix: Top which has the usual matrix inverse, itself expandable into the vector form. It is remarkable http://home.usit.net/~cmdaven/hyprcplx.htm 25/12/2006 Clyde Davenport's Commutative Hypercomplex Math Page Page 4 of 11 that the matrix inverse of the typical matrix element of D is another matrix having the same distinctive pattern of entries: Top The reader may verify that, in the vector form, if one multiplies Z by Z-1 (or vice-versa), one obtains a result of unity.
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