This webpage by Diane G. Demers explains the orientation congruent algebra and the native exterior calculus of twisted differential forms, provides my draft papers on these concepts, and links some related webpages.
The orientation congruent algebra (or OC algebra for short) is a nonassociative, Clifford-like algebra that is essential to the computation of the exterior and Clifford products of twisted multivectors in their natural, or native representation. The native exterior calculus is the exterior calculus of twisted differential forms in their native representation.
Here we use the generic term geometric object, or simply object, to refer to both the twisted and untwisted varieties of multivectors, multicovectors, multivector fields, and differential forms, even though this usage may conflict with that of differential geometry. Following de Rham, twisted objects are also commonly described as impair (French) or odd (English); while ordinary, untwisted (straight) objects are also commonly described as pair (French) or even (English). Polar and axial vectors, as well as ordinary tensors and tensor densities, are related concepts which we will also discuss. To quickly get an intuitive feeling for twisted differential forms see Rob Salgado's richly illustrated conference poster "Visualizing Tensors."
Just as Cartan's exterior calculus is based on Grassmann's exterior algerbra, the native exterior calculus is based on the native exterior algebra which, in turn, is based on the orientation congruent algebra. The native exterior calculus was developed to generalize Cartan's exterior calculus and treat twisted differential forms in their native representation.
This framework of the native represetation of twisted objects using the orientation congruent algebra applies to manifolds of any finite dimension n. It ends the confusing manual sign choices required by, other, previously known approaches. We will see that, by clearing up the many misunderstandings engendered by the usual techniques, this natural and general formulation for twisted geometric objects powerfully resolves many smoldering controversies in the mathematics and physics literature. The latest example of such a controversy is provided by the paper by Roldão da Rocha and Waldyr Alves Rodrigues, Jr., "Pair and impair, even and odd form fields and electromagnetism" vs. the one by Yakov Itin, Yuri N. Obukhov, and Friedrich W. Hehl, "An electric charge has no screw sense—a comment on the twistfree formulation of electrodynamics by da Rocha & Rodrigues." (By the way, this last paper of Itin et al. cites this website and my works on page 5.)
These controversies about twisted physical quantities, although sometimes couched in powerful abstract formalisms, are all directly related to the simple, prototypical clash of two concepts: unoriented space and oriented vectors. Space is, both in physics as well as everyday life and commerce, unoriented. One would not, I think, approach an appliance salesperson and ask to see some refrigerators with a capacity of "546 liters with a right-hand screw sense," rather than simply "546 liters." (This raises the tangential question of whether perhaps some day extremely precise measurements may reveal that the volume of a stationary or moving cube determined by measurements taken in the order of, say, height, width, and then depth is different from its volume determined by measurements taken in a different order.) However, it is meaningful to assign a positive or negative sign to capacity so that gains and losses in volume may be accounted; therefore space may be also be considered as being oriented as what is called an odd or twisted three-vector. On the other hand, vectors and their higher grade analogs, such as bivectors and trivectors, are all patently oriented (in the common sense) and are thus examples of what we call even or straight vectors, bivectors, and trivectors.
The axioms for geometric (Clifford) algebra that Hestenes and Sobczyk give in their book Clifford Algebra to Geometric Calculus were apparently constructed to be directly analogous to those for the real numbers (perhaps, in part, to entice physicists into studying them). We may also recall that there is a long tradition of calling the elements of Clifford algebras Clifford numbers. (Similarly, we have the terms complex numbers, double numbers, and, generally, hypercomplex numbers.)
The next step is to extend the current conception of geometric numbers by considering how sign fits into this analogy. That is, we have in the Clifford numbers a geometric generalization of the real numbers, but only the most primitive generalization of signed real numbers: namely, that in which the geometric sign exactly mirrors (is parallel with) the geometric absolute value or magnitude (which we call the gauge.)
By entertaining the further possibility that the geometric sign of a geometric number may be transverse (in a nonmetric setting) or orthogonally complementary (in a metric setting) to its geometric magnitude we obtain the twisted multivectors. These twisted geometric numbers have both mathematical and physical implications. And this new theory of the native represetation of twisted geometric objects provides an adequate foundation for the investigation of those implications.
In their native representation twisted geometric objects are directly decomposable into a generalized, geometric sign, or g-sign for short, and a generalized, geometric measure or content, that we call the gauge. The same two-part decomposition also applies to ordinary (or straight) geometric objects. However, a first explanation of these concepts is more easily given in terms of straight objects.
The g-sign represents a geometric object's orientation only; it is devoid of any measure or content. For example, the g-sign of a vector may be pictured as a ray parallel to it and pointing in the same direction. This ray represents an oriented subspace of the enveloping vector space; it may also be regarded as "living" in the associated oriented projective space where it represents an oriented point. On the other hand, the gauge of an object represents its measure, content, weight, or magnitude combined with its attitude. The attitude of an object, in turn, represents the unoriented line, plane, or higher-dimensional subspace in which it lies. For our vector example, the gauge may be pictured as an unoriented line segment parallel to it.
The native representation of a geometric object can also be given a precise symbolic expression as an equivalence class of ordered pairs determined by calculations in the orientation congruent algebra (and its associated Grassmann or exterior algebra). This section needs more work. However, for the moment, the symbolic form of the native representation of geometric objects is roughly presented in the text file NatRep.txt. The terminology and square bracket notation used in it will be changed in later revisions of this webpage and also in the draft paper on the native exterior calculus.
This section also needs more work. For the moment, we simply state that the author's initial motivating application of the orientation congruent algebra was to directly calculate the exterior product of straight and twisted multivectors in their native representation. This required the development of the native exterior algebra which, in turn, is the algebraic basis of the native exterior calculus.
Properties of the OC algebra include the following:
The most direct understanding of an algebra is obtained by studying its multiplication table. By multiplication table we mean, more formally, the partial Cayley table of the group induced by an associative algebra, or the partial Latin square of the quasigroup induced by a nonassociative algebra. These tables are partial in that they contain only products whose factors have a positive sign in some standard representation.
We next present the multiplication tables of two OC algebras that are particularly important in our exposition. In these tables the maximaum-grade multivector is represented by the symbol for capital Omega (similar to differential geometry's usage) so that the Omega in the table for OC3 is equal to e123 while that in the table for OC5 is equal to e12345. Also note that some symmetries of these tables have been made more visible by tinting red the cells which contain negative entries.
The multiplication table for the orientation congruent algebra OC3 (or OC3,0) as a PNG graphic:
The multiplication table for the orientation congruent algebra OC5 (or OC5,0) as an Adobe Acrobat PDF file (49,525 bytes), version 1.3, June 23, 2005: OC50MultTab013.pdf
The orientation congruent algebra is a member of the large class of algebras known as noncommutative Jordan algebras. I have proved this by a brute force case-by-case examination using Wolfram Research's Mathematica software. Although the code executes in a reasonable time, it would be good to have an analytical proof. The Mathematica code and its explanation are available upon request.
An intriguing aspect of the OC algebra: in contrast to the Clifford algebra, it provides a simple and natural expression for the Hodge star dual operator * and its inverse *-1. Although I have not yet written up my notes for a proof of this fact, the formulas are simply given by
* u = Omega-1 @ u, andHere u is an arbitrary differential form, Omega is the volume form, @ is the OC product (the "at sign" @ is an ASCII substitute for the symbol that I prefer, the AMS LaTeX \circledcirc), and the inverse of Omega (written above as Omega-1) is taken with respect to the OC product. Note that, for pseudo-Riemannian metrics the definition of the Hodge star and its inverse may need to be interchanged depending on which of the two possible conventions is observed. This property of the OC algebra is illustrated in the multiplication tables for OC3 and OC5.
*-1 u = u @ Omega.
Now the Hodge star operator was actually pioneered by Hermann Grassmann as his Ergänzung or complement operator. Grassmann's contribution is acknowledged in the following two quotes from John Browne's Grassmann Algebra book.
On page 750 of the Grassmann Algebra book is the citation:
Hodge W V D 1952
Theory and Applications of Harmonic Integrals
Cambridge
The "star" operator defined by Hodge is of similar nature to Grassmann's complement operator.
And on page 341 Browne makes this allusion:
In more modern works the Ergänzung for general metrics has become known as the Hodge Star operator. But we do not use the term here since our aim is to develop the concept by showing how it can be built straightforwardly on the foundations laid by Grassmann.
Along this line we also provide this quotation from page 7, section 6, "Grassmann-Hodge stars (Kocik 1979)," of the paper "Isometry from reflections versus isometry from bivector" by Zbigniew Oziewicz (2009, Online First, Adv. Appl. Clifford Alg.):
The concept of a ‘Hodge-star’ was introduced by Hermann Grassmann in his second monograph in 1862, under the name ‘Ergänzung’ (supplement) [Grassmann 1862, § 3.4 ]. The present-day star-notation, *, was introduced by Hermann Weyl in 1945.
Based on this precedence I suppose then that one could call the orientation congruent algebra the Clifford-Grassmann algebra (or CG algebra for short). But that is perhaps too confusing given current terminology. The possibility is also raised that the full orientation congruent algebra is also cryptically or, at least, implicitly contained in Grassmann's work.
The OC2 algebra is isomorphic to the hyperbolic quaternions of Alexander Macfarlane. Macfarlane published a paper "Hyperbolic Quaternions," (1989-1900, Proc. Royal Society of Edinburgh 23:169-181) on his discovery of "a system of quaternions complementary to that of Hamilton, which is capable of expressing trigonometry on the surface of the equilateral hyperboloids."
According to this brief biographical essay on Macfarlane, Emil Borel later observed that the 3D hyperboloid model of hyperbolic space describes the kinematic velocity space of special relativity. Also see the webpage "Alexander Macfarlane and the Ring of Hyperbolic Quaternions" and the Wikipedia webpage "Hyperbolic Quaternions."
The series of orientation congruent algebras OCn starting with n = 1 and increasing are isomorphic to the the double or split-complex numbers for n = 1, the hyperbolic quaternions for n = 2, and their generalizations to higher dimensions for n > 2. These generalizations are part of the sequence of structurally-hyperbolic Cayley-Dickson algebras that are dual in a natural sense to the usual Cayley-Dickson algebras, which may then be called structurally-elliptic. See the section on the structurally-hyperbolic algebras.
Title:
The Orientation Congruent Algebra: A Nonassociative Clifford-Like Algebra
Abstract.
The correlated grade form of twisted blades (twisted simple multivectors)
faithfully renders in symbols their native geometric structure. The discovery
of this paper's nonassociative Clifford-like algebra was driven by trying to calculate
exterior products of straight and twisted multivectors directly in a basis of
this form. The key was found to be the orientation congruent (OC) algebra. This
paper is being published electronically in about ten sections, each offered as soon
as written. In this first section we axiomatize the orientation congruent algebra
by generators and relations. The next section derives the sign factor function sigma and proves that the Clifford product times it is the multiplication of an explicitly Clifford-like algebra isomorphic to the orientation congruent algebra. Later
sections are planned to show how to calculate the OC product in Mathematica
and Clical; to define the orientation congruent contraction operators, deduce
their properties, derive other expressions for them, and use them to compute the
OC product within the exterior algebra using a modified Cartan decomposition
formula; to develop the algebra's product sequence graph with labeled edges; to
derive a predictor of a null associator as a function of the grades of the three
elements in it; to prove the associomediative property of the algebra's counit; to develop matrix representations (under a nonassociative matrix product) of the orientation congruent product; and to discuss the motivating application per se and as inspiration for the first set of axioms.
This paper is available for downloading or reading online. It will be split by section into separate PDF files. Each file will be offered as soon as it is written. As the paper approaches completion the finished sections may be combined into one file. Revisions may occur so be sure to get the latest versions of each section.
Section 1: An Axiom System for the OC Algebra
Version 1.3, June 27, 2005: Adobe Acrobat PDF file (295,623 bytes),
OriCon013ch01.pdf
Note: Contrary to the above plan, this paper will not be posted section-by-section and may possibly not be developed further beyond version 1.5 below. But some essential material from this paper will incorporated into the newer paper on the native exterior calculus
Title:
The Orientation Congruent Algebra
Part I: A Nonassociative Clifford-Like Algebra
Abstract. Similar to Version 1.3.
Table of Contents
Complete paper: The Orientation Congruent Algebra: A Nonassociative Clifford-Like Algebra
Version 1.5, December 4, 2005: Adobe Acrobat PDF file (865,403 bytes),
OriCon015ch01-09.pdf
Note: There are few errors in this draft; but none of them is fatal to the thrust of this research. An errata sheet or a revised draft paper will eventually be posted here.
This section added rather hastily on December 9, 2009 puts the cart before the horse since the fundamental sections on the native Clifford and native exterior algebras are currently empty. However, my research has picked up dramatically in the last 2-3 months. And although at present this work resides only in my notes and memory, I will here describe some more recent investigations.
I now know how to use the orientation congruent algebra in even-dimensional spaces—which has been a problem since the beginning many years ago. It requires in part using a "nonmetric" approach.
I put quotation marks around the word nonmetric above because in this approach we apply a peculiar kind of Clifford algebra (or, actually, its related orientation congruent algebra) which may be called variously a nonmetric Clifford algebra, or a formally metric Clifford algebra or, using perhaps the best choice of words, a Clifford algebra of multivecfors (please note, not multivectors) following the terminology of the paper by W. A. Rodrigues, Jr. and Q. A. G. de Souza at http://arxiv.org/abs/math-ph/0703053.
This type of Clifford algebra has also been developed for manifolds using a bundle-theoretic approach by M. P. Burlakov in the very last section of his paper "Clifford structures on manifolds." The original Russian version is at http://mi.mathnet.ru/eng/into27 and an English translation is at http://dx.doi.org/10.1007/BF02414874.
Reviews are at http://www.ams.org/mathscinet-getitem?mr=1619716 and http://www.zentralblatt-math.org/zmath/en/search/?q=an:0930.53030&format=complete.
In addition, a further interesting complication is that while odd-dimensional spaces require the orientation congruent algebra derived from the (rather more standard) orthogonal version of the Clifford algebra of multivecfors (that is, with the split-Euclidean metric of signature (n, n), for an n-dimensional space), on the other hand, the even-dimensional spaces require a Clifford algebra of an arbitrary, nonsymmetric bilinear form in the terminology of Bertfried Fauser and Rafal Ablamowicz in http://arxiv.org/abs/math.QA/9911180
This is also known as a quantum Clifford algebra to use the terminology of Bertfried Fauser in http://arxiv.org/abs/math.QA/0202059
See also Rafal Ablamowicz and Pertti Lounesto's book chapter "On Clifford algebras of a bilinear form with an antisymmetric part" at http://books.google.com/books?id=OpbY_abijtwC&pg=PA167
I have seen the need to use this generalized type of Clifford algebra several times over the years. However, that was still within a metrical context. And then the bilinear form involved was truly a generally nonsymmetric bilinear form, that is, neither purely symmetric nor antisymmetric. However, the situation is much simplified by using the nonmetric Clifford algebra of multivecfors which combines the space of multivectors with the dual space of multicovectors into one space.
Then the nonsymmetric bilinear form involved is not so arbitrary, but is properly the symplectic bilinear form associated with a symplectic vector space. For an n-dimensional physical space this symplectic vector space requires a 2n-dimensional (Wick) basis of vectors and covectors. Then, for example, the matrix representation of the symplectic bilinear form for a 2-dimensional physical space may be written in the standard form
[ 0 1 0 0]
[-1 0 0 0]
[ 0 0 0 1]
[0 0 -1 0].
Then it is the case that for an even-dimensional physical space (e.g., 4-dimensional spacetime), the final algebra that is used is the orientation congruent analog of the Clifford algebra of multivecfors, which, in turn, is the analog of the Clifford algebra of a symplectic bilinear form.
Also, apparently, the Clifford algebra of multivecfors can be nicely explained in the abstract language of mathematical category theory and it appears to be based on the biproduct of a module. See the books reviewed at http://www.ams.org/mathscinet-getitem?mr=941522 and http://www.ams.org/mathscinet-getitem?mr=1712872.
I could go further here and tie the notion of a correlation in projective geometry into this phenomenon that odd- and even-dimensional spaces, respectively, must be treated by algebras associated with bilinear forms that are either split-Euclidean or canonically symplectic, respectively. However, I will only add that these new concepts create a naturally nonmetrical setting for projective geometry. They then provide a true nonmetrical foundation for projective geometry which the cobasis used by John Browne in his Grassmann Algebra book does not.
I also think that implied in this new work may be a foundation for Hestenes and Sobczyk's geometric calculus of multivectors. But that is a long-term project. I am now busy writing up this new material and integrating it with my earlier results.
To be added.
To be added.
I am currently writing a paper on the native exterior calculus of twisted differential forms. As mentioned above in the introduction to this webpage, the orientation congruent algebra is integral to developing the native exterior calculus. This paper, "Exterior Calculus in the Image of Odd Forms with the Orientation Congruent Algebra," will treat two problems in electrodynamics as examples.
An abstract of this draft paper: ExtCalc.pdf.
As mentioned in the above section on the properties of the OC algebra, the orientation congruent algebra may be viewed as a structurally-hyperbolic Clifford algebra, a generalization of the usual Clifford algebra. That is because the foundation of the orientation congruent algebra lies within the new algebraic theory of structurally-elliptic and structurally-hyperbolic algebras.
In this framework, the usual Clifford algebra, which we call structurally-elliptic, is the dual of the orientation congruent algebra, which we call structurally-hyperbolic. This terminology has its origin in the fact that the Clifford algebras are naturally suited to represent elliptic (ordinary) rotations in the prototypical spaces with Euclidean signature (n, 0) while the orientation congruent algebras are naturally suited to represent hyperbolic rotations (boosts) in the prototypical spaces with Lorentzian signature (n-1, 1). More about this will be added later.
This Clifford algebra dualization leads naturally to the dualization of the structurally-elliptic Cayley-Dickson algebras (i.e., the usual Cayley-Dickson algebras) to the structurally-hyperbolic Cayley-Dickson algebras. These new algebras are partly illustrated in the section on Macfarlane's hyperbolic quaternions (which we may more systematically call the structurally-hyperbolic quaternions).
The loops (quasigroups with identity) induced by the structurally-hyperbolic Cayley-Dickson algebras are characterized by some interesting new identities. In particular, it turns out that the structurally-hyperbolic octonions are neither Bol nor Moufang loops, but a kind of hybrid of the two. More details on this will be forthcoming as time permits.
"Structurally-Hyperbolic Algebras Dual to the Cayley-Dickson and Clifford Algebras or Nested Snakes Bite Their Tails" is the title of a draft paper on this theory. The unusual subtitle is an allusion to a phrase used by the knot theorist Louis H. Kauffman in an early paper of his. A particular formula in Kauffman's paper inspired a more general formula for the product of the structurally-hyperbolic Cayley-Dickson algebras. In fact, this draft paper mostly establishes basic terminology and the fundamental formulas relating the two kinds of Cayley-Dickson algebra products—structurally-elliptic and structurally-hyperbolic.
An abstract of this draft paper:
Nested_080715_abs.pdf.
The latest draft of this paper from July 15, 2008:
Nested_080715.pdf.
This webpage may at times be under revision with a high frequency—sometimes changing every few minutes. Please use your web browser's refresh or reload button (keyboard shortcut usually CONTROL + R) whenever you visit to ensure that you are seeing the latest version.
Graphic info for contacting me:
Last modified: Sunday, December 20, 2009
All Breakout™ puzzles, physics and math papers, and all other content of this website including the design of this website are the work of Diane G. Demers and are copyrighted (©) material: Copyright © 2002-2009 Diane G. Demers. All rights reserved. The Breakout™ name and logo are trademarks (™) owned by Diane G. Demers.