The webpages on this site are the work of Diane G. Demers. Here we explain straight and twisted (or even and odd, à la de Rham) multivectors, multicovectors, multivector fields, and differential forms (or collectively, geometric objects) and their three representations, the native, correlated, and extremum, in both geometric and symbolic forms. These webpages also explain the three related algebras: the orientation congruent algebra (or OC algebra for short), and the native exterior and native Clifford algebras; and the two native calculuses: the native exterior calculus of twisted differential forms and the native Clifford calculus of twisted multivectors. Furthermore, we discuss polar and axial vectors, as well as ordinary tensors and tensor densities, and their relationship to the above formalisms. Finally, this website provides my draft papers on these concepts and links related webpages.
Before continuing further new visitors may want to peruse the section below "Helpful Hints for Reading This Website." It links some introductions to Clifford algebra and analysis as well as information on the fonts and web browsers needed to read these webpages.
The material developed on this website will be submitted to scientific journals for peer review and publication. Currently, this website is a frequently-revised work-in-progress with incomplete sections and errors. So every time you visit please reload each page to ensure that you are reading the latest version (rather than your browser's cache of that page).
I have been making some strong claims recently and am aware of the need to justify them mathematically. Constrained by the usual mundane limitations, that goal has been and is currently being given a high priority in my life.
My work on this website mainly to perfect its use of Unicode fonts has recently finished. With this long task now completed I will be adding new substantive material soon.
Because these concepts are fundamental to my thesis, I plan to work next on the orientation congruent contraction operators and their use to derive the OC algebra expression for the Hodge star dual operator. And since I have already done much of the work in this area in LaTeX I plan to post the resulting PDF file on this website when finished.
While twisted geometric objects have profound applications to unoriented and unorientable manifolds as topological and differentiable spaces, detailed consideration of those topics would distract us from our current goal—developing a natural, or native, formalism for twisted objects in physical applications. For physicists and engineers the most common examples of twisted geometric objects occur in electromagnetic theory. There they are represented, for example, by the electric, D, and magnetic, H, excitation fields, respectively, which are more commonly known as the electric displacement and magnetic field intensity vector fields, respectively. Both these fields when natively represented as twisted multivector fields (in a metric setting) or twisted differential forms (in a nonmetric one) have two transversely‑oriented parts: a geometric sign and a geometric magnitude.
The "twistedness" of these fields (considered directly as multivector fields) is reflected in their behavior during integration. It turns out that the dimension of the geometric magnitude is the same as the dimension of the vector space or manifold over which each of these fields is integrated. Hence, the D field is integrated over a surface, while the H field is integrated along a curve. However, in their native representation as twisted multivector fields it is the relationship of the geometric signs of these fields to the geometric sign of each infinitesimal piece of the surface or curve of integration which determines the algebraic sign of its contribution to the integral. The rule is simple: if the two geometric signs are parallel the algebraic sign is positive, while if they are antiparallel the algebraic sign is negative.
The twistedness of these fields (as multivector fields) is also reflected in their operational definition by the processes used to measure them. For example, measurement of the D field requires an instrument known variously as "Mie's double plate" or the "Maxwellian double plates." This instrument is illustrated in the figure below. Its use is described in the quotation following that figure. And the figure after that shows the geometric object—a twisted bivector—that is abstracted from the measurement process of the D field. Note that the geometric sign has no magnitude and the geometric magnitude has no clockwise or counter-clockwise orientation.
Measurement of the Electric Excitation Multivector Field D with the Maxwellian Double Plates. The figure above is taken from page 137 of F. W. Hehl and Yu. N. Obukhov, Foundations of Classical Electrodynamics: Charge, Flux, and Metric, Birkhäuser, Boston, © 2003.
Apparently, the apparatus described by the term "Faraday cage" in this quotation (and not pictured here, but see the next link) would be more accurately described as a "Faraday ice pail." Mr. Faraday's ice pail can be used along with an electrometer to measure charges as described by the PASCO company instruction sheet just linked.
The Geometric Object Abstracted from the Measurement of the Electric Excitation Multivector Field D. The figure above—a twisted bivector—is adapted from Schouten's book Tensor Analysis for Physicists.
To get more of an intuitive feeling for twisted differential forms and multivector fields see Rob Salgado's richly illustrated conference poster "Visualizing Tensors." For more excellent illustrations of these twisted geometric objects see also William Burke's book and papers, Bernard Jancewicz's publications, or Jan Schouten's book Tensor Analysis for Physicists, which, however, uses classical tensor analysis notation exclusively. After a visual introduction the reader may be ready for the more mathematical, but applied, on-line works of Alain Bossavit. More sources are linked below.
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 differ from 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). On this website we freely interchange the two terms twisted and odd, as well as the two terms straight and even.
Twisted multivectors and multicovectors are naturally endowed with two transversely‑oriented parts: a generalized geometric sign and a generalized geometric magnitude. The geometric sign exists in an oriented projective space, while the geometric magnitude exists in an unoriented vector space. The geometric sign and magnitude written together as an ordered pair enclosed in doubled square brackets constitute the native representation of these twisted geometric objects. The auxiliary correlated and extremum representations are also required.
The orientation congruent algebra (or OC algebra for short) is the nonassociative, Clifford‑like algebra that is required to interconvert the symbolic forms of the three representations of twisted and straight objects. It is also essential to the definition of the exterior and Clifford products of twisted multivectors in their natural, or native representation. The resulting two native algebras are called the native exterior algebra and the native Clifford algebra.
Then, just as Élie Cartan's exterior calculus is based on Hermann Grassmann's exterior algebra, the native exterior calculus is based on the native exterior algebra. The native exterior calculus was developed to generalize the exterior calculus by treating twisted differential forms in their native representation; it is the exterior calculus of twisted differential forms in their native representation. Similarly, the native Clifford calculus will provide a foundation for generalizing the geometric calculus of multivectors and Clifford bundle approaches.
The formalisms presented here fully respect the native representation of twisted (or odd) physical quantities. In a sense this work constitutes a manifesto against the prevailing attitude that twisted quantities are of no or secondary importance in physics. As William Burke observes in the preface to his book Applied Differential Geometry: "The importance of twisted tensors in physics has been neglected by nearly everyone."
This attitude causes some authors (for example, da Rocha and Rodrigues) to devise various formal schemes to root out the "dreaded" twisted physical quantities. However, all attempts to sweep twisted quantities under the rug are futile; the effect of their fundamental symmetry properties always leaves an uncomfortable tell-tale bulge under the feet of those who would banish them.
Contrarily, I show that twisted quantities can be fully and completely integrated into the mathematical machinery of physics. My hope is that this complete formal treatment will promote the legitimacy of twisted quantities in physics. Although, F. W. Hehl and Yu. N. Obukhov have already shown how twisted objects are fundamental to the thesis of their book Foundations of Classical Electrodynamics: Charge, Flux, and Metric. This book uses the metric-free (and orientation-invariant) approach.
Setting aside ideology, the adoption of these new formalisms yields practical advantages. In fact, this step is required for the advancement of mathematics, physics, and engineering because it allows equations to be written in sign-invariant form following derivations that are both brief and meaningful. In its own way this advancement is analogous to the introduction of Ricci's tensor analysis or Cartan's exterior calculus into physics.
Continuing in the practical vein, the fundamental understanding provided by this framework resolves past and future controversies and puzzlements in the scientific literature about the sign, orientation, parity, and nature and classification of physical quantities. For example, in the common approaches, when a twisted differential form is pulled back from an n-dimensional manifold to an (n − 1)-dimensional one, a bewildering sign change can occur. (A similar sign change was also observed by da Rocha and Vaz in § 7 of their paper "Extended Grassmann and Clifford algebras.") However, our new theory provides a general, orientation-coherent and sign-invariant formalism for twisted differential forms in differential geometry, and, in particular, for pulling back twisted differential forms between manifolds of any two finite dimensions.
In the engineering literature (see the discussion of references [12, 13] on pages 3–4 in my draft paper below) this case has lead to a confusing dilemma about the discontinuous electromagnetic boundary conditions (the so-called "jump conditions") due to surface charges and currents. In the physics literature (see the discussion of references [20, 21] on pages 3–4 in my draft paper below) it has also lead to an unnecessarily long derivation of the parities of the field quantities of electrodynamics in split space and time from their parities in spacetime. (And the very fact that the authors of these later two cited papers felt it necessary to undertake this derivation highlights the confusion that the conventional approach engenders.)
So far, it appears that the work of mathematical physicists on that aspect of applied Clifford algebra and analysis known as the geometric calculus of multivector manifolds (represented, for example, in the books of David Hestenes and Garret Sobczyk, John Snygg, and Chris Doran and Anthony Lasenby) has remained an isolated branch of the stream of mathematical research. However, the orientation congruent algebra provides simple and natural expressions for the Hodge star dual operator and its inverse. It is thus the key to integrating this and other approaches to conventional Clifford algebra and analysis (as well as my "twisted extension" of them) smoothly with the well-established results of modern tensor analysis and differential geometry.
Also on the theoretical side, but more directly relevant, this work leads to a fundamental change in our understanding of the Clifford algebra-based approaches to physics. These include Hestenes and Sobczyk's geometric calculus of multivectors (Clifford Algebra to Geometric Calculus) and Rodrigues and Capelas de Oliveira's Clifford bundle approach (The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach). Both those approaches can be extended to use the native Clifford product rather than the ordinary Clifford product—with the new formalism subsuming the old ones.
However, the native Clifford product is nonassociative. So that, as it turns out, rather shockingly for those steeped in the current formalisms (including myself), associativity is an unnecessarily too strong a requirement for a Clifford-like algebra that is applied to physics. For further comments see my paper submitted to Annalen der Physik (Berlin).
This framework of the native representation 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 that repeatedly appear in the mathematics and physics literature.
The latest example of such a controversy is provided by the following series of papers (which all cite this website and my works, the first paper in its final journal form only):
My paper, "Reply to da Rocha & Rodrigues' Comments on the Orientation Congruent Algebra & Twisted Forms in Electrodynamics," was submitted to Annalen der Physik (Berlin) on January 7, 2010. It refutes those authors' inaccurate statement in the third paper listed above that the orientation congruent algebra, and by implication the native Clifford algebra, because of their nonassociativity, are incompatible with their Clifford bundle approach. Quite the contrary, the usual Clifford algebra and bundle that is these authors' expertise is subsumed by the new native Clifford algebra and bundle and leads to a new physical principle of "nonassociative irrelevance" for judging whether an equation is physically meaningful. In addition, I argue that the adoption of formalisms that respect the native representation of twisted (or odd) physical quantities is required for the advancement of mathematics, physics, and engineering because their use allows equations to be written in sign‑invariant form and helps to resolve controversies that continue to occur in the literature—such as in the papers listed above.
"Reply to da Rocha & Rodrigues' Comments on the Orientation
Congruent Algebra & Twisted Forms in Electrodynamics," typos corrected
version, January 7, 2010, Adobe Acrobat PDF file (175,888 bytes):
AdPreply_100107.pdf
For more background related to the statements made in the above submitted paper see the section on the native Clifford product of native brackets.
These controversies about twisted physical quantities, although sometimes couched in powerful abstract formalisms, are all rooted in the simple, prototypical clash of two sets of concepts: unoriented and twisted space vs. oriented vectors.
Our first natural conception of space is, both in physics as well as everyday life and commerce, unoriented. For example, one would not, I think, approach an appliance salesperson and ask to see some refrigerators having a capacity of "546 liters with a right‑hand screw sense," rather than simply "546 liters." 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 naturally considered as being oriented as an odd or twisted trivector (also called an odd or twisted three-vector or 3-vector).
On the other hand, vectors and their higher grade multivector analogs, such as bivectors and trivectors, are all patently oriented (in the common sense). They are thus examples of what we call even or straight vectors, bivectors, and trivectors.
(The above discussion suggests the tangential question of whether perhaps extremely precise measurements will someday 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.)
To Be Developed Further
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 (Clifford) numbers by considering how sign fits into this analogy. That is, we have in the geometric (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 has the same attitude as (this phrase is explained shortly below, but for now it may be roughly read as is parallel with) the geometric "absolute value" or magnitude.
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 representation 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 gsign for short, and a generalized, geometric measure or content, which we call the gauge (with apologies to gauge theorists for the use of this very apt word). This same two-part decomposition also applies to ordinary (or straight) geometric objects for which the "native representation" is actually not native.
This situation is analogous to the representation of a rational number as a fraction. Now a fraction is really (the equivalence class represented by) an ordered pair of real numbers consisting of a numerator (analogous to a gauge) and denominator (analogous to a gsign). For rational numbers that are not also integers (analogous to twisted objects) some type of representation equivalent to the two-part fractional notation is required. However, for integers (analogous to straight objects) this is not the case. Nonetheless, integers may always be represented as simpler fractions normalized to have a denominator of 1. Analogously, a straight object has a simpler representation than a twisted one. Therefore, we use the simplest straight object—a vector—as an example in the following first explanation of the native representation.
In the native representation the gsign represents a geometric object's orientation only; it is devoid of any measure or content. For example, the gsign of a vector may be pictured as a ray parallel to it and pointing in the same direction; in the figure below the gsign of the vector b is the ray r. 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, in the native representation 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; in the figure below the attitude of the vector b is the line l. For our vector example, the gauge may be pictured as an unoriented line segment parallel to it; in the figure below the gauge of the vector b is the line segment s. See also the later figure captioned "The Geometric Pictures of (a) an Even Vector and (b) an Odd Bivector in the Native Representation" in which the gsign and gauge of a vector are directly labeled.
The Finest Geometric Decomposition of a Vector. The figure above illustrates the finest geometric, basis-independent and nonmetric decomposition of a nonzero vector b. This decomposition consists of a line l, a line segment s, and a ray r. The ray r is drawn with a double arrowhead to distinguish it from the vector b. The equal and addition signs are metaphorical.
The native representation (and the two others to be introduced shortly) 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). The symbols for these three representations consist of ordered pairs that are written within doubled brackets of three different "shapes." As suggested by this notation, these three symbolic representations will henceforth also be called simply brackets (of a specified type).
In addition, as just mentioned two other representations will also be needed below. The definitions of gsign and gauge given above for the native representation are different for them; however it is convenient to use the terms geometric sign, gsign, and gauge in a generic sense to apply to all three representations, both geometrically and symbolically.
We write the native bracket using "straight" or "square" doubled brackets as [[sn, gn]]. Here sn is the gsign and gn is the gauge. In this symbolic form of the native representation we have chosen to place the geometric sign first, before the gauge, in analogy with the usual notation for real numbers. Note that this same ordering also occurs in Burke's derivation of the eponymous "William's Twisted Notation" (see page 5 of his draft paper "Twisted Forms: Twisted Differential Forms as They Should Be").
To make our vector example concrete in this notation let b = [[e1, e1]], where e1 is a basis vector. Since the gsign of b may be geometrically represented as a ray parallel to it, any positive multiple of e1 may occupy the gsign position in the native bracket. And thus we have for all real numbers λ > 0
b = [[λe1, e1]].
And we may call the vector λe1 for some λ > 0 a ray representative of the native geometric sign equivalence class.
In preparation for the next series of paragraphs, we may naturally adopt the term ray for, not only the standard geometric concept of an oriented half-line, but also, following Burke (see page 4 of his draft paper "Twisted Forms: Twisted Differential Forms as They Should Be"), for the analogous concept of a (nonzero) oriented subspace that is described by the expression λe1 in the gsign position above. We must say "nonzero" here because the zero vector is not and cannot be part of this set. And, although this set is strictly not a vector subspace, we have used the "space" terminology here in the same spirit as the word space is used in the theory of convex sets to describe a half-space (which is also not a vector space).
We adopt the operator notation Gsnn(b) or Gsnn b to mean the equivalence class of all multivectors that may validly occupy the gsign position of b's native bracket, that is
Gsnn b = {v | b = [[v, e1]]}.
Then we obtain
λe1 ∈ Gsnn b, for all λ > 0,
or, equivalently,
{λe1 | λ > 0} ⊆ Gsnn b.
However, we note that
Gsnn b ≠ {λe1 | λ > 0}.
The last inequality is true because the ray described by λe1 is not large enough to include all elements of the equivalence class that is Gsnn b. However, we give next an expression for the complete native gsign equivalence class.
We may now make the critical observation that a simple (decomposable) multivector occupying a native bracket's geometric sign position actually represents an oriented projective subspace. Therefore, the equivalence class Gsnn b must be invariant under all oriented projective transformations (see Stolfi's book). This explains why Gsnn b is larger than the set of all ray representatives, λe1. Thus, in the following, where λ and μ are real numbers, we have
Gsnn b = {λe1 + u | for all λ > 0 and all u ≠ μb for some μ}.
And this implies that
b = [[λe1 + u, e1]], for all λ > 0 and all u ≠ μb for some μ.
The effect of requiring the native geometric sign equivalence class to be invariant under all oriented projective transformations is to add an extra "degree of freedom" in addition to the variable length of a ray. That extra freedom is the variable attitude of the ray. In terms of the geometry of vector subspaces, instead of a ray we now have an the union of that ray and the unoriented almost-bundle bounded by a critical bounding unoriented subspace. In our example, this bounding unoriented subspace is the "double ray" λe1 for λ ≠ 0. Let us, however, adopt the simple term fan to refer to this concept instead of that long detailed description.
Please note that as used here the meaning of the term "bundle" is more akin to that of projective geometry (see Ziegler's papers), not differential geometry. More specifically, it is not the same kind of bundle as a Clifford bundle. We next motivate this fan terminology using our current example, the even vector b.
However, we may now see in our imagination that the geometric picture associated with this set of rays may be described as a fan. That is because, if the associated example, the vector b, is in a 2-dimensional enveloping vector space, the appearance of this set of rays suggests a hypothetical hand-held Oriental folding fan that has been opened to an almost 360° circular sector. Note that this definition of a fan is applicable to any nonzero even, simple (decomposable) multivector of any grade in an enveloping vector space of any dimension.
However, the geometric picture may not always suggest a familiar physical fan. For example, for an even bivector in 3-space the critical bounding unoriented subspace lies in a plane and the unoriented almost-bundle is, in projective vector subspace terms, a subset of a bundle of planes. This makes the "blade" of the fan occupy almost all the enveloping 3-space, which is clearly not a physical fan (although it suggests a higher-dimensional physical fan—if there were such a thing).
Now, with this conception of the native gsign equivalence class as a fan in mind, one could also imagine ways to illustrate it in figures such as the one below captioned "The Geometric Pictures of (a) an Even Vector and (b) an Odd Bivector in the Native Representation." However, there appears to be no practical advantage in this since the result would seem to become extremely cluttered and confusingly complicated.
For our motivating example, the set of vectors in the native gsign equivalence class may also be described as the set of all vectors in the enveloping vector space V except for the ray consisting of all nonpositive multiples of b = e1, that is,
Gsnn b = {v | v ≠ λe1 for some λ ≤ 0} = V \ {λe1 | for all λ ≤ 0}.
Then following the oriented subspace sense of the word ray introduced above, we may say that the set described by the expression λe1 + u used above for the native gsign equivalence class of b is the (nonpositive) ray-deleted space associated with b—another straightforward but cumbersome phrase. Note that we must write "nonpositive" rather than "negative" here because the zero vector is also removed.
Since the geometric sign alone determines the direction of b, the following equality shows how a reversal of direction is reflected in the geometric sign equivalence class:
Gsnn −b = {−(λe1 + u) | for all λ > 0 and all u ≠ μb for some μ},
which implies that
−b = [[−(λe1 + u), e1]], for all λ > 0 and all u ≠ μb for some μ.
We also have, more generally, for any (even or odd) simple (decomposable) multivector b
Gsnn −b = −Gsnn b,
if we make the natural interpretation that the negative sign on the right-hand side distributes over all elements in the equivalence class.
Similar to what we have done for the geometric sign, we adopt the operator notation Gaun(b) or Gaun b to mean the equivalence class of all multivectors that may validly occupy the gauge position of b's native bracket, that is
Gaun b = {v | b = [[e1, v]]}.
Recall now that the gauge of b is geometrically represented as a line segment of the same length parallel to it. This implies that the algebraic sign of the gauge is irrelevant. Then we have
b = [[e1, ±e1]].
And we obtain simply
Gaun b = {±e1}.
We also have, more generally, for any (even or odd) simple (decomposable) multivector b
Gaun b = Gaun −b = −Gaun b,
if we again make the natural interpretation that the negative sign in the rightmost expression distributes over all elements in the equivalence class.
UNDER REVISION
So far we have only defined the equivalence class operators Gsnn b and Gaun b related to the
geometric sign and gauge of the native bracket. The real number analogs of
these operators are not commonly defined or used. However, with an analysis
based on the OC algebra contraction operator we may also define two
direct analogs to the signum (or
sign) function, sgn, and the absolute value
function, abs, of the real numbers. These are
the functions gsnn and gaun which, just as sgn and
abs do, return real numbers for their arguments.
UNDER REVISION
Note that, if we adopt the operator notation gaun(b), or gaun b, to mean the gauge of
b in the native bracket, we may then write
gaun b = ±e1,
or, more generally, for any (even or odd) simple (decomposable) multivector b
gaun b = gaun ±b = ±gaun b.
Therefore, by combining all the above equations as well as the gsgn and gaun notations, we may write
b = [[λe1 + u, ±e1]],
or, more generally, for any (even or odd) simple (decomposable) multivector b
b = [[gsnn b , gaun b]].
To reinforce the above discussion we now exhibit the real number analog of the native decomposition of geometric numbers. Let r and λ be real numbers with λ > 0. Then we have the standard multiplicative decomposition of r in terms of the two nonlinear functions sgn, the signum function, and abs, the absolute value function, as follows:
r = sgn (r) abs (r) = sgn (λr) abs (±r).
Because the concept of a twisted geometric object is dependent on the concept of a complementary subspace, a twisted object is only defined if the dimension of the manifold or vector space in which it exists is specified. Therefore, in the following examples we take that dimension to be 3, that is, the dimension of space.
In the native representation the gsign of a twisted bivector may be pictured as curvy arrow. The curviness of this arrow may be taken as an indication that it is actually an oriented vector subspace (which may also be given an oriented projective space interpretation as an oriented point).
The Geometric Pictures of (a) an Even Vector and (b) an Odd Bivector in the Native Representation. The figure above shows side-by-side the geometric pictures of a nonzero even vector b and a nonzero odd bivector C. The ray representing the gsign of b is drawn with a double arrowhead in conformance with the previous figure. However, the image of the odd bivector in this figure is adapted from Schouten's book Tensor Analysis for Physicists, in which he draws the ray representing its gsign as a curvy arrow with a single arrowhead. The curviness of this arrow may be interpreted as an indication that it actually represents one of the oriented vector subspaces in the equivalence class of those associated with the gsign of the odd bivector.
EDITING FRAGMENT from figure caption
is used to represent one of the oriented vector subspaces in the equivalence
class of those associated with the gsigns in the native and correlated
representations
UNDER REVISION
UNDER REVISION
The native representation of a twisted bivector can also be written in bracket
notation. For example, let the twisted bivector C be
[[e1, e23]].
UNDER REVISION
The native representation of a twisted bivector can also be written in bracket
notation. As a specific example, let the twisted bivector C = [[e1, e23]], where e1 and e23 are basis multivectors. Since the gsign of C may be geometrically represented as a ray parallel to e1, any positive multiple of e1 may occupy the gsign position in the native bracket. And thus we have for all real numbers λ > 0
C = [[λe1, e23]].
However, we also have the inequality
gsnn C ≠ λe1
because the ray described by λe1 is not large enough to include all elements of the equivalence class that is the native gsign of C.
UNDER REVISION
We may now make one critical addition to the above equation to reflect the fact
that the geometric sign actually represents an even larger equivalence class of
oriented subspaces than that described by its ray representative, λe1. As was true for an even vector, the oriented
subspace that is gsnn C may also be interpreted as an element of an oriented
projective space. However, now this element must be transverse to or
not incident with the basis multivector in the gauge position of the
native bracket for C, that is e1. Under this
interpretation the gsign must be invariant under all oriented projective
transformations (see Stolfi's book). Thus, we have for
all vectors u such that u
∧ e1 = λe1 ∧ λe23 for some λ > 0
C = [[u, e23]].
In the expression λe1 + u appearing above in the gsign position, the term λe1 represents one of the oriented vector subspaces in the equivalence class of those associated with this gsign. The set of all oriented subspaces in this equivalence class is, of course, represented by the full gsign expression.
However, the geometric picture may not always suggest a physical fan. For example, when we discuss the native gsign equivalence class of an odd bivector in 3-space later below, the critical bounding subspace is a plane. This makes the fan appear more like an oriented half-space of 3-space, thus suggesting a higher-dimensional half-open physical fan (if there were such a thing).
Although there is only one truly native representation of odd objects, in practice two types of expressions are required to carry out calculations: the native and the correlated. This is analogous to the representation of complex numbers in both polar and rectangular forms with the polar form most suited to the multiplication of complex numbers and the rectangular form to their addition. Note, however, that the rectangular form of complex numbers is relatively facilitative of their multiplication as well.
In our case, it is the native representation that is used to multiply both even and odd objects, while the correlated representation is used to add, multiply, and differentiate them. This last use is most important since we aim to construct a calculus. And, just as one would not dream of using the polar form to do complex analysis, so too we must use the correlated representation to do the native exterior calculus.
The name of the correlated representation is so derived because its two components' grades are correlated: either directly (for even objects) so that they are equal, or inversely (for odd objects) so that they sum to n, the dimension of the manifold or space in which they lie. Of course, the native representation also shares this property. However, the native and correlated representations can be distinguished by the different fundamental sign rules that they obey.
We have seen above that the symbolic description of an odd geometric object in the native representation is directly evocative of that object's corresponding natural image, but what geometric picture is associated with the correlated representation of a geometric object?
Here we introduce the correlated bracket as the symbolic form of the correlated representation. ⟪s, g⟫.
Finally, to connect with the usual differential geometry notation for odd differential forms we need one more representation, the extremum. The extremum representation is so named because its first component's grade is always either the smallest possible grade, 0 (for even objects), or the largest possible grade, n (for odd objects). That is, the first component of the extremum representation is always an arbitrary non-zero scalar multiple of either 1 or Ω±. For an explanation of the meaning and significance of the symbols Ω± , Ω+ , and Ω− used here and throughout this website, please see the section on the two types of OC algebras.
What geometric picture is associated with the extremum representation of a geometric object?
Here we introduce the extremum bracket as the symbolic form of the extremum representation.
The Geometric Pictures of an Even Vector in the (a) Native, (b) Correlated, and (c) Extremum Representations. The figure above shows side-by-side the geometric pictures of a nonzero even vector b in the native, correlated, and extremum representations. In this figure a double-headed arrow is used to represent one of the oriented vector subspaces in the equivalence class of those associated with the gsigns in the native and correlated representations. This is a different convention than that used by Schouten in his book Tensor Analysis for Physicists.
The Geometric Pictures of an Odd Bivector in the (a) Native, (b) Correlated, and (c) Extremum Representations. The figure above shows side-by-side the geometric pictures of a nonzero odd bivector C in the native, correlated, and extremum representations. The above image of an odd bivector in the native representation is adapted from Schouten's book Tensor Analysis for Physicists.
The symbols for these three representations consist of ordered pairs that are written within doubled brackets. As suggested by this notation, these three representations will henceforth also be called the native, correlated, and extremum brackets (rather than representations). They are given in the following table along with some examples of each that represent our familiar even vector b and odd bivector C. These examples are all from a manifold or space of dimension 3. The first component of all three brackets in this table is the gsign, written generically as s; the second component is the gauge, written generically as g.
| Type | General Symbol | Examples | |
|---|---|---|---|
| Even Vector b | Odd Bivector C | ||
| Native | [[sn, gn]] | [[e1, e1]] | [[e1, e23]] |
| Correlated | ⟪sc, gc⟫ | ⟪e1, e1⟫ | ⟪e1, e23⟫ |
| Extremum | ((se, ge)) | ((1, e1)) | ((Ω+, e23)) |
An extremum value is patently associated with any extremum bracket as its gsign. In this case the extremum value may be either an arbitrary non-zero scalar multiple of 1 (for even objects) or an arbitrary non-zero scalar multiple of Ω± (for odd objects). However, a similar extremum value may also be associated with the native and correlated brackets. The extremum value of a native bracket is defined to be sn ⊚ gn, where the Unicode character "CIRCLED RING OPERATOR" ⊚ symbolizes the orientation congruent product. For a correlated bracket it is defined to be sc ⊚ gc. However, for a correlated bracket that represents an even object the extremum value is naturally restricted to be a positive (rather than non-zero) scalar multiple of 1.
In addition to the relationship of the grades of their gsign and gauge, the native, correlated, and extremum brackets are further distinguished by the fundamental sign rules given in the following table. These fundamental sign rules are divided into two types: migratory and unbinding. A migratory sign rule describes how a negative sign prepended to the outside of a bracket is distributed within (or, metaphorically speaking, migrates into) the bracket to its gsign and gauge. On the other hand, an unbinding sign rule describes how the sign of a bracket's extremum value may be inverted. The origin of the term unbinding is made clear in the next paragraph.
Note that there is no unbinding sign rule for the correlated bracket since there is no way to change its extremum value without also reversing its orientation. In practice this is no impediment to the use of the correlated bracket in calculations. We only need consider its extremum value to be bound (at least temporarily during any one calculation) to some particular sign‑wise, but not magnitude‑wise, fixed value of an arbitrary positive scalar multiple of 1 (for even objects) or an arbitrary non-zero scalar multiple of Ω± (for odd objects). In addition, all correlated brackets must be bound to the same extremum value during a calculation.
| Native | Correlated | Extremum | |
|---|---|---|---|
| Migratory | −[[sn, gn]] = [[−sn, −gn]] | −⟪sc, gc⟫ = ⟪−sc, −gc⟫ | −((se, ge)) = ((se, −ge)) |
| Unbinding | [[sn, gn]] = [[sn, −gn]] | ―― | ((se, ge)) = ((−se, −ge)) |
The symbolic correlated and extremum representations of geometric objects were originally presented on this website in a text file that I continue to include here for historical purposes: NatRep.txt. As is evident, the terminology and square bracket notation used in this file has been changed to that used above in the current version of this website and also in the draft paper on the native exterior calculus.
In the native representation the definitions of both the Clifford and exterior products of multivectors are almost trivial (although perhaps conceptually challenging to those accustomed to the standard formalisms). In light of this near triviality, one may ask why should we bother to express these operations in the correlated representation? The answer is that the process of exterior differentiation requires the simultaneous use of multiplication and addition. And, since the addition of multivectors is most naturally expressed in the correlated representation, we must work in the "lowest common denominator"—the correlated representation—when we take derivatives.
To Be Developed Further
We now provide the almost trivial definition of the native Clifford product for multivectors in the native representation. Let u and v, respectively, be twisted or straight simple (decomposable) multivectors with representations in native brackets given by [[su , gu]] and [[sv , gv]], respectively. Then the native Clifford product of u and v is defined as
Here we have symbolized the native Clifford product as ∘̃, the Unicode "RING OPERATOR" with a "COMBINING TILDE" (the Unicode equivalent of my preferred AMS LaTeX operator symbol \tilde{\circ}), and the orientation congruent product as ⊚, the Unicode character "CIRCLED RING OPERATOR"; see the Symbol Test Page. We have also used s and g, respectively, as generic symbols for the gsign and gauge, respectively, of a native bracket.
Note that the native Clifford product is inherently nonassociative since the sign‑determining multiplication of the gsigns is effected by the nonassociative orientation congruent product. We are, in fact, forced to adopt this surprising definition to maintain consistency with the rest of the native representation and algebra framework.
Also note that if each gauge part, gu or gv , is reducible to a scalar multiple of a single basis blade, then the product of the these parts may also be computed with the ordinary Clifford product (or any other Clifford‑like product) rather than the OC product since the sign of a single scalar-multiplied basis blade in a native bracket's gauge position is irrelevant. However, when these parts are not single scalar multiples of basis blades then the OC product (or its negative) must be used since the relative sign relationship among the terms which constitute the linear combination of basis blades that is the simple multivector in each gauge position must be that determined by the orientation congruent product.
For those practitioners of the more or less standard Clifford algebra‑based theories, such as Hestenes and Sobczyk's geometric calculus of multivectors or Rodrigues and Capelas de Oliveira's Clifford bundle approach, the inherent nonassociativity of the native Clifford product may present a challenging shock. It implies that the set of physically meaningful equations expressed in these formalisms includes only those that can be reformulated into an equivalent native Clifford algebra version. This statement may be dubbed the principle of nonassociative irrelevance, or, perhaps, algebraic indifference. The native Clifford algebra also seems to be the first nonassociative algebra that is naturally and widely applicable to many areas of physics. For related commentary see my paper submitted to Annalen der Physik (Berlin).
The definition of the native Clifford product for multivectors in the correlated representation is more involved than that for multivectors in the native representation. Nonetheless, it is simplest for odd‑dimensional base spaces; in this case the more familiar orientation congruent algebra of a symmetric bilinear form plays a role. However, for even‑dimensional base spaces the OC algebra of a symplectic bilinear form is required.
The addition of multivectors is much more involved than their Clifford multiplication. It requires the use of the correlated representation.
To Be Developed Further
Similar to the case of the native Clifford product discussed above, the native exterior product of multivectors in the native representation is also rather simply defined.
Let u and v, respectively, be twisted or straight simple (decomposable) multivectors with representations in native brackets given by [[su, gu ]] and [[sv, gv ]], respectively. Then the native exterior product of u and v is defined as
Here we have symbolized the native exterior product as ∧̃, the Unicode "LOGICAL AND" with a "COMBINING TILDE" (the Unicode equivalent of my preferred AMS LaTeX operator symbol \tilde{\wedge}), and the orientation congruent product as ⊚, the Unicode character "CIRCLED RING OPERATOR"; see the Symbol Test Page. We have also used s and g, respectively, as generic symbols for the gsign and gauge, respectively, of a native bracket.
The native exterior product of native brackets is associative. A naive and incorrect "proof" of this theorem runs this way: Associativity follows because [[sn, gn ]] = 0 if gn = 0. However, we will provide a correct proof at later time.
In the correlated representation there is one more complication compared to the native Clifford product of multivectors in this representation: The dual space of multicovectors must also be part of the process of native exterior multiplication. In fact, we require the orientation congruent algebra of multivecfors. However, similarly to the native Clifford product of multivectors in the correlated representation, if the base space is even‑dimensional the orientation congruent algebra of multivecfors with a symplectic, rather than symmetric, bilinear form must be used for the native exterior multiplication of multivectors in the correlated representation.
The addition of multivectors is much more involved than their exterior multiplication. It requires the use of the correlated representation.
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.
To Be Added
In this section I present as much information as possible about the orientation congruent algebra. It should become clear to the reader of this section that this algebra has a life of its own outside of its application to the native representation of twisted geometric objects.
The Unicode character "CIRCLED RING OPERATOR" ⊚ is the symbol for the OC product. It is the Unicode equivalent of the symbol that I use in AMS LaTeX \circledcirc, or, in plain text files, the "at sign" @; for details and variations see the Symbol Test Page. Although, the OC product is nonassociative, for multivectors whose exterior product is nonzero the orientation congruent product agrees with the exterior product. It turns out that this behavior contributes to just the right amount of associativity needed in applications.
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 orientation congruent algebras naturally divide into two classes depending on the commutation properties of the unit‑magnitude, maximum‑grade multivector. This division is exactly mirrored by the parity of the underlying vector space of grade 1 multivectors, or base space. The unit‑magnitude, maximum‑grade multivector of an OC algebra with an odd‑dimensional base space commutes with all other multivectors just as the identity or unit element 1 does. On the other hand, the unit‑magnitude, maximum‑grade multivector of an OC algebra with an even‑dimensional base space does not possess this property.
In both types of orientation congruent algebras the unit‑magnitude, maximum‑grade multivector has a grade which is complementary to that of the unit element since n + 0 = n. And so, to distinguish it from the pseudoscalar of Hestenes' geometric algebra, we call it a counit as derived from "COmplementary grade UNIT." (Apologies to coalgebra and Hopf algebra experts: this name fits too well to not use.) For OC algebras with an odd‑dimensional base space we call it a perfect counit, while for OC algebras with an even‑dimensional base space we call it an imperfect counit. An orientation congruent algebra with a perfect (resp., imperfect) counit is called a perfect (resp., imperfect) OC algebra.
These facts are reflected by the multiplication tables presented in the section below: The most symmetrical multiplication tables are those for which the OC algebras have odd‑dimensional base spaces and perfect counits.
In these tables the perfect counit for those orientation congruent algebras with an odd‑dimensional base space is represented by the symbol bold capital Omega Ω (similar to differential geometry's usage). Thus the bold capital Omega in the table for OC3 is equal to e123 while that in the table for OC5 is equal to e12345. On the other hand, the imperfect counit for those orientation congruent algebras with an even‑dimensional base space is represented by the symbol upside‑down bold capital Omega ℧ (\mho in AMS LaTeX). Thus the inverted bold Omega in the table for OC4 is equal to e1234. An inverted bold Omega is used to alert the reader that, as explained just above, the counit in OC algebras with an even‑dimensional base space, unlike the odd‑dimensional case, is imperfect, and so does not generally commute with other multivectors under OC multiplication.
In contrast to these tables (for which the niceties of LaTeX typography are available) in the text of these webpages (and perhaps also on the blackboard) we modify the above conventions for counit symbols. Thus we also use a bold capital Omega with a plus sign subscript Ω+ for the perfect counit of an orientation congruent algebra with an odd‑dimensional base space but a not necessarily specified dimension or signature. On the other hand, we use a bold capital Omega with a negative sign subscript Ω− for the imperfect counit of an orientation congruent algebra with an even‑dimensional base space but a not necessarily specified dimension or signature.
Here we also introduce the term indefinite counit for the counit of
an OC algebra with both a not necessarily specified base space parity
and a not necessarily specified signature.
In documents composed with LaTeX we indicate an
indefinite counit with the symbol
,
a stylized superimposition of a noninverted and inverted bold capital Omega.
(This symbol may be pronounced as "Omho.")
However, generally in these webpages we
indicate an indefinite counit with a bold capital Omega with a "plus
or minus sign" subscript Ω±.
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 three OC algebras that are particularly important in our exposition. Note that in the multiplication tables (displayed or downloadable below) for the orientation congruent algebras OC3 and OC5, but not OC4, the products which have a sign which is opposite to that of the corresponding Clifford algebra products have been underlined. Also note that some symmetries of these tables have been made more visible by tinting red the cells which contain negative entries.
For an explanation of the meaning and significance of the two symbols in these tables, bold capital Omega Ω and upside‑down bold capital Omega ℧, please see the above section on the two types of OC algebras.
The multiplication table for the orientation congruent algebra OC3 (or OC3,0) displayed as a PNG graphic:
The multiplication table for the orientation congruent algebra OC4 (or OC4,0) downloadable as an Adobe Acrobat PDF file (22,652 bytes), January 3, 2010: OC40MultTab1-2.pdf
The multiplication table for the orientation congruent algebra OC5 (or OC5,0) downloadable as an Adobe Acrobat PDF file (49,525 bytes), June 23, 2005: OC50MultTab1-3.pdf
The orientation congruent algebra is a Clifford-like algebra. Clifford-likeness, in the sense that I use this term, may be defined by considering products of the basis multivectors that are derived from an orthonormal set of basis vectors. Then the Clifford-likeness of some algebra means that the product of two given basis multivectors in that algebra is same as their Clifford product up to a ± difference in sign. In other words, the OC product is a ± sign deformation (or mutation) of the Clifford product. For a further discussion of this terminology see page 2 of my draft paper submitted to Annalen der Physik (Berlin).
The orientation congruent algebra is a member of the large class of nonassociative algebras known as noncommutative Jordan algebras (see p. 69-14 of the book chapter "Nonassociative Algebras" by Murray R. Bremner, Lúcia I. Murakami, and Ivan P. Shestakov, p. 18 of the preprint version of this chapter, or pp. 140 ff. of the book An Introduction to Nonassociative Algebras by Richard D. Schafer). Further investigation of the properties of the orientation congruent algebra as a nonassociative algebra and a possibly more specific classification of it within this realm is a subject for future research.
I have proved that the OC algebra is a noncommutative Jordan algebra 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 an explanation of it are available upon request.
An intriguing and extremely important 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. This property of naturally expressing the Hodge star operator and its inverse is illustrated in the multiplication tables for the orientation congruent algebras OC3, OC4, and OC5.
Although I am in the process of writing up my notes for a proof of this fact, the formulas for the Hodge star dual operator ∗ and its inverse ∗−1 are
Here lower case alpha α is an arbitrary straight (not twisted) differential form in its ordinary (not native) representation, bold capital Omega with a plus or minus sign subscript Ω± is the volume form, and the Unicode character "CIRCLED RING OPERATOR" ⊚ is the OC product. The inverse of the volume form Ω± , written above as (Ω±)−1, is with respect to the OC product.
Here also we have used the symbol convention laid down in the above section on the two types of OC algebras to represent the volume form by a bold capital Omega with a plus or minus sign subscript Ω± . Distinguishing between a perfect and imperfect counit, respectively, by following our convention of using a bold capital Omega with a plus sign subscript Ω+ and one with a negative sign subscript Ω− , respectively, is not necessary in these formulas since they are generally valid for both odd- and even‑dimensional manifolds.
These formulas are also valid for both Riemannian and pseudo‑Riemannian metrics. Note, however, that for pseudo‑Riemannian metrics the formulas for the Hodge star and its inverse may need to be interchanged depending on which of the two possible conventions is observed. Furthermore, these formulas may be rather trivially extended to twisted differential forms represented either in the standard notation of differential geometry or in the equivalent extremum bracket notation of the native framework.
Please note that the Hodge star operator just discussed is not the orientation‑invariant one since it depends on the specific orientation of the volume form Ω± . (The orientation-invariant Hodge star operator is also described by Burke as twisted or by da Rocha and Rodrigues as impair.) In fact, the sign of the Hodge star dual of a differential form will be inverted if the orientation of the volume form is changed. The orientation-invariant Hodge star requires the use of the twisted, rather than the straight, volume form in its definition.
It is certainly possible (and necessary for this new theory) to express the orientation‑invariant Hodge star operator and its inverse (just as naturally as in the above formulas) using the native bracket representations for straight and twisted differential forms. However, we delay the completion of this project until later.
As suggested by the above OC algebra expression for the Hodge star operator, the fact is that the orientation congruent algebra is intimately and directly connected with the conventions of tensor analysis and differential geometry. However, these conventions are distorted by the common Clifford algebra-based approaches based on the geometric calculus of multivector manifolds and the Clifford bundle as they are currently understood and practiced. Nonetheless, this fundamental aspect of the orientation congruent algebra promises to be the springboard from which to launch a program to unify conventional differential geometry with the just-mentioned Clifford algebra-based approaches. For a more remarks in this vein see the section of this website titled "Integrating Clifford Analysis with Tensor Analysis and Differential Geometry."
At this point I quote some statements of John Snygg that illustrate the common geometric calculus attitude towards the Hodge star operator.
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 that we 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," Proc. Royal Society of Edinburgh 23:169–181 (1989–1900), 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 3-dimensional 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 quaternion."
The series of orientation congruent algebras OCn starting with n = 1 and increasing are isomorphic to 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 on-line. 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.
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 (the boosts of special relativity) in the prototypical spaces with Lorentzian signature (n − 1, 0). 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 an exclamation written into an early paper by the knot theorist Louis H. Kauffman. A particular formula in that early paper of Kauffman 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 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 Rafał Abłamowicz 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 Rafał Abłamowicz 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.
A good concise introduction to the necessary results and proofs of applied Clifford algebra and geometric calculus is found in Harke's paper "An Introduction to the Mathematics of the Space-Time Algebra." For a good on-line summary see Ian C. G. Bell's "Multivector Methods" which is section 5 of his "Maths for (Games) Programmers" webpages. One cannot neglect David Hestenes' introductory (and advanced) papers found on his website. Also, Alan Macdonald provides the extended paper "A Survey of Geometric Algebra and Geometric Calculus." Finally, Wikipedia provides the articles "Clifford algebra" and "Geometric algebra."
The pages on this website are intended to be displayed on your video monitor at a resolution of at least 800 ⨉ 600 pixels. However, these webpages can be read more easily when they look more like the printed page. This occurs at resolutions of 1024 ⨉ 768 pixels and higher. The need for higher video resolutions is also due in part to the way mathematics is displayed on this website. Here we use special fonts containing mathematical symbols (see the next paragraphs) that may not display well at lower video resolutions.
There are several methods for displaying mathematics on the web—but they are all somewhat problematic. The author of this website takes what seems to be a middle ground by balancing compatibility with the computer systems of potential readers and the ease of authoring. Currently this website uses a combination of HTML code sent in UTF-8 encoding together with some conventional and Unicode fonts that are assumed to be already installed on the reader's computer. Therefore, some adjustments to your system may be required to view the mathematics on this website. You may simply need to change your browser's settings; however, you may also need to install new fonts or a new web browser. However, I cannot, am not, and will not be legally or morally responsible for any or all possible harmful effects of implementing any part of or all of the instructions that follow—you follow them entirely AT YOUR OWN RISK.
Symptoms of a font or browser problem may include, instead of the proper symbol, the display of something else—perhaps a blank space, or some combination of one or more nonsensical symbols, question marks, or boxes containing whatever. To test your system's ability to display some of the symbols used on this website please look at the Symbol Test Page.
Some fonts that can display mathematics are usually already installed on modern Microsoft Windows or Apple Macintosh computers. However, the display of mathematics on these webpages makes very severe demands on fonts. It appears that none of the commonly preinstalled fonts functions well enough to be generally useful throughout this website. However, as a compromise, I have instructed your web browser to use certain preinstalled fonts if they are available. Also, many browsers will by default substitute some already installed fonts for the fonts they are told to use but which are not currently available. Therefore, without installing the fonts I recommend below you may see something that, in many cases, may be meaningful. However, what you see may not be presented in the nicest mathematical typography, or perhaps, even correctly.
The math on this website is designed to be displayed as I intend by using fonts from three font "families." All these font families are available for downloading from other websites essentially as "freeware." Certain fonts from two of the font families are required; however, the fonts from the third family only improve the typography in a nonessential way.
It happens that some combinations of fonts, font file formats, web browsers, and operating systems may not be compatible. Indeed, there is some danger that installing fonts and/or web browsers may cause undesirable changes in or damage to your computer system. Although, I believe that for most readers following the suggestions given below will not be harmful, I have to advise you of the potential risks (which could be serious). Always consult your own information technology expert before making these or any other changes to your computer system. And DON'T SUE ME!
WARNING: Instructions for modifying your computer system by installing or removing fonts and installing web browsers is about to be given. I am not legally or morally responsible for any or all possible harmful effects of implementing any part of or all of the instructions that follow—you follow them entirely AT YOUR OWN RISK.
IMPLEMENTING ANY PART OF OR ALL OF THE FOLLOWING INSTRUCTIONS IS DONE ENTIRELY AT YOUR OWN RISK OF DAMAGE TO YOUR COMPUTER SYSTEM INCLUDING ANY AND ALL INCIDENTAL OR CONSEQUENTIAL DAMAGES.
The required fonts that are not customarily preinstalled are from the DejaVu and GNU FreeFont families. The freeware DejaVu fonts can be downloaded in TrueType format from "Download - DejaVu." Currently only the Sans part of this font family is required for this website. The freeware GNU FreeFont (also known as the Free UCS Outline Fonts) font family is available at "Free UCS Outline Fonts - Summary [Savannah]." At this time only the FreeSerif variety of this font family is used here.
James Kass' shareware Code2000 font provides a simpler alternative to installing the above set of required fonts. (Although, for most symbols, the above freeware fonts' rendering appears to be superior.) The single user fee for this font is only US$5.00! It is available at "Download Code2000."
These webpages also use some parts of the TeX Computer Modern (CM) family of fonts. But these fonts are used more for typographical niceties, rather than for semantic necessity. So installing the CM fonts is not essential to reading the mathematics on this website.
There are many fonts in the Computer Modern family, but at the moment only the following ones are used: cmmi12, cmmib10, cmr12, and cmb10. These can be downloaded in several different formats from specific directories (folders) in the CTAN TeX archive: the CM PostScript Type 1 PFM directory, the CM PostScript Type 1 PFB directory, the CM TrueType directory, and the CM OpenType directory. Note that currently you only need a tiny few (four) of the many fonts in these directories. These fonts are also available from the American Mathematical Society in PFM and PFB format at "AMSFonts."
I have found that on my Windows 2000 system the PostScript Type 1 formats of these CM fonts are compatible with most web browsers. Be sure to install at least the PFM subformat of these fonts. But I believe it is best to install the PFB subformat along with the PFM one with both present simultaneously in the same install folder. It may also be necessary to remove any of these fonts already installed on your system that are in the TrueType or OpenType format. Removal of the later format seems to be necessary in particular for current versions of the Mozilla Firefox web browser. However, for older versions of the K-Meleon browser it is the OpenType format alone that works. Be sure to save copies of any font files before you remove them in case you need to reinstall them later for another application that needs them.
Another version of the TeX Computer Modern font family is available in both PFB and OTF formats at "Computer Modern Unicode fonts." The OTF file format of this CM Unicode font family is best for my Windows 2000 computer system because its installation does not interfere with my current CM font installation in the PostScript Type 1 PFM file format (as part of the MikTeX typesetting system). Currently only the CMU Serif Roman (filename cmunrm.xxx), CMU Serif Roman Bold Nonextended (filename cmunrb.xxx), and CMU Serif Italic (filename cmunti.xxx) parts of this family are used on this website.
For information on some good web browsers for mathematical symbols see the webpage "Mathematical Symbols on the Web." If you prefer Microsoft Internet Explorer, you should use version 6 or higher. And no matter what your browser is, you may need to manually change the setting under the menu bar: View/Encoding to Unicode (UTF-8).
These webpages 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.
These webpages 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.
Last modified: Wednesday, November 9, 2011 07:40 -0500 (EST)
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