DERIVING MULTIPRODUCT OF MULTIVECTORS

The polynomial is perhaps the PRIME STRUCTURE of NUMALGEBRA. For example:
a0 + a1x + a2x2 + a3x3 + ... + anxn. [1] Or:
a1x1 + a2x2 + a3x3 + a4x4 + b1x22 + b2x2 + b3x22 + c1x33 + c2x33 + dx44 + ex55. [2]

What's a significant difference between polynomials [1] and [2]? Answer: Every "unknown" (x) in polynomial [1] differs from every other one by degree, so [1] is "thoroughly heterogeneous". But polynomial [2] exhibits subhomogeneity:

  1. Subpolynomial a1x1 + a2x2 + a3x3 + a4x4 has 4 terms of linear or first degree;
  2. Subpolynomial b1x22 + b2x2 + b3x22 has 3 terms of second degree;
  3. Subpolynomial c1x33 + c2x33 has 2 terms of third degree;
  4. So polynomial [2] has 3 homogenous subpolynomials, but, in general, is heterogeneous. (Note: I'm avoiding "inhomogeneous" because of its special usage "elsewhere".)

Thus, a significant difference between polynomials arises in homogeneity and heterogeneity. Now, my DERIVATIONS of INNER and OUTER products show that NEW OPERATIONS and NEW TYPES of ALGEBRA arise out of the familiar NUMALGEBRA, which goes back to the ancient Babylonians and Egyptians. A more general polynomial MIXES significant amounts of homogeneity and heterogeneity:

  1. We write two linear trinomials in formal product: (a1x1 + a2x2 + a3x3) (b1x1 + b2x2 + b3x3).[3A]
  2. We expand that product: (a1b1x1x1 + a2b1x2x1 + a3b1x3x1 + a1b2x1x2 + a2b2x2x2 + a3b2x3x2 + a1b3x1x3 + a2b3x2x3 + a3b3x3x3).[3B]
  3. We recast that product into two parenthetical groupings:
    1. the first grouping is HOMOGENEOUS (subscripts the same in each term);
    2. the second grouping is HETEROGENEOUS (subscript differ in each term): (a1b1x1x1 + a2b2x2x2 + a3b3x3x3) + (a1b2x1x2 + a2b1x2x1 + a1b3x1x3 + a3b1x3x1 + a2b3x2x3 + a3b2x3x2). [3C]

But is the homogenous-heterogenous distinction also present in vector "polynomials"? Let's see.

Please recall that, in DERIVING OUTER PRODUCT, we considered three vectors:
a = a1d1 + a2d2 + a3d3.|| b = b1d1 + b2d2 + b3d3.|| c = c1d1 + c2d2 + c3d3.[4]

These are certainly homogeneous, of the same degree for each basis vector. Let's see what happens when we take inner product of two of them, say, the first two (noting that di · di = 1):
a · b = a1b1d1 · d1 + a2b2d2 · d2+ a3b3d3 · d3. [5]

Again, HOMOGENEOUS. Furthermore, SYMMETRICAL.

Now, let's take outer product of the same vectors, showing what happens to their basis vectors:
aÙb = (a1d1 + a2d2 + a3d3) (b1d1 + b2d2 + b3d3) = a1b1d1Ùd1 + a2b1d2Ùd1 + a3b1d3Ùd1 + a1b2d1Ùd2 + a2b2d2Ùd2 + a3b2d3Ùd2 + a1b1d1Ùd1 + a1b3d1Ùd3 + a2b3d2Ùd3 + a3b3d3Ùd3. [6]

The TERMS in BLACK are NULLED OUT because a VECTOR IS NOT ORTHOGONAL TO ITSELF. If we omit them and re-order, we have:

aÙb = a1b2d1Ùd2 + a2b1d2Ùd1 + a1b3d1Ùd3 + a3b1d3Ùd1 + a2b3d2Ùd3 + a3b2d3Ùd2. [6']

Behold! HETEROGENEITY and ANTISYMMETRY in [6'] -- in opposition to the HOMOGENEITY and SYMMETRY in [3].

Well, it's obvious from this that THE MIXTURE OF HOMOGENEITY AND HETEROGENEITY in the MORE GENERAL NUMALGEBRAIC POLYNOMIAL can be ACHIEVED VIA A VECTOR OPERATION, provided we also bring along MIXED SYMMETRY and ANTISYMMETRY. How? Simply ADD THE TWO OPERATIONS. (And this is what is done apparently everywhere in THE LITERATURE -- although I don't know where you will find the "POLYNOMIAL" MOTIVATION I've introduced.)

But I avoid this simplistic step for the sake of DISCOVERING MORE from THE END RESULT.

Let's build a new MULTIVECTOR PRODUCT:

  1. Consider a polynomial-style product, denoted by "#", of two of these vectors, say, a, b.
  2. Assume only that, under sharp-product, SCALARS COMMUTE WITH SCALARS AND VECTORS, but VECTORS DO NOT NECESSARILY COMMUTE.
  3. Require that terms of left-multiplier be listed first; those of right-muliplier, second.

To remind you of what "a polynomial-style product" is, I perform this product:
(j1x1 + j2x2) (k1x1 + k2x2) = j1k1x1x1 + j2k1x2x1 + j1k2x1x2 + j2k2x2x2. [7]

We model the NEW MULTIVECTOR PRODUCT below, constrained by conditions (2), (3), stated above:
a#b = (a1d1 + a2d2 + a3d3)# (b1d1 + b2d2 + b3d3) = a1b1d1#d1 + a2b1d2#d1 + a3b1d3#d1 + a1b2d1#d2 + a2b2d2#d2 + a3b2d3#d2 + a1b3d1#d3 + a2b3d2#d3 + a3b3d3#d3. [8]

Please note that the formal product and its expansion in [8] resembles the formal product in [3A] and its expansion in [3B]. Can we separate HOMOGENOUS from HETEROGENEOUS, as in [3C]?

Please recall that I formulated both [3] and [8] so that HOMOGENEITY OF A TERM is easy to detect. A TERM IS HOMOGENEOUSLY COMPOSED if, and only if, ALL ITS SUBSCRIPTS AGREE; else, HETEROGENEOUSLY COMPOSED. So we can reformulate [8] by COLLECTING ALL HOMOGENEOUS TERMS IN ONE PARENTHETICAL GROUP, ALL HETEROGENEOUS TERMS IN A SECOND PARENTHETICAL GROUP.

a#b = (a1b1d1#d1 + a2b2d2#d2 + a3b3d3#d3) + (a1b2d1#d2 + a2b1d2#d1 + a1b3d1#d3 + a3b1d3#d1 + a2b3d2#d3+ a3b2d3#d2). [8']

[8'] is now like the GROUPINGS in [3C]. Furthermore, The FIRST PARENTHETICAL GROUP IS EQUIVALENT TO THE RIGHT-HAND SIDE OF [5], while the SECOND PARENTHETICAL GROUPING IS EQUIVALENT TO THE RIGHT-HAND SIDE OF [6'] if you CHANGE Ù TO #.

Hence, we can write:
a # b = a · b + a Ù b. [9]

MATHEMATICS is about PATTERNS, either TRANSFORMING them or CONSERVING them. We can say that, pattern-wise, INNER PRODUCT is MATCH (of subscripts or types); OUTER PRODUCT is MIX; MULTIPRODUCT is MIX-and-MATCH.

I introduced "sharp product" tentatively to see the result. Unlike the usual treatment in books and papers, we did not simply ADD INNER AND OUT PRODUCTS. Rather, IN CASTING THE VECTOR-NOMIALS in THE PATTERN OF ALGEBRAIC POLYNOMIALS AND EMULATING THE LATTER PRODUCT, we DISCOVERED A PRODUCT THAT IS THIS SUM. But WE KNOW SOMETHING that was not KNOWN before: THIS PRODUCT DERIVES FROM DISTINGUISHING HOMOGENEITY-SYMMETRY from HETEROGENEITY-ANTISYMMETRY. AND PLEASE NOTE: ALL THREE MULTIVECTOR PRODUCTS -- INNER PRODUCT, OUTER PRODUCT, MULTIPRODUCT -- WERE DERIVED (NOT "COMMANDED") BY MEANS OF "HIGH SCHOOL ALGEBRA"! (Ignore the RECONDITE JAZZ OF THE ELITISTS!)

Let's change from the tentative symbol, "#". Many texts and papers simply write MULTIPRODUCT by JUXTAPOSITION. But this looks too much like ORDINARY ALGEBRAIC MULTIPLICATION, and can confuse. Let's use Ä as the OPERATIONAL SYMBOL for MULTIPRODUCT.

MULTIPRODUCT OF VECTORS: aÄb = a · b + a Ù b. [9']

To cite "precedent" for the above formulation, consider the Kronecker Symbol (which I denote by "S", rather than the standard d, used for a BASIS UNIT): Sij = 1, for i = j; or 0, otherwise. Hey! This yields "1" for THE HOMOGENOUS (i = j); yields "0" for THE HETEROGENEOUS. But the Kronecker Symbol OPERATES ON OPERANDS. Consider a "super-operator", which OPERATES ON OPERATORS: mijº·, if i = j (INNER: HOMOGENEOUS); mijºÙ, if i ¹j (OUTER: HETEROGENEOUS).

Please note THE PATTERN OF THE MULTIPRODUCT: SCALAR PLUS BIVECTOR. We'll see that PATTERN turn up "elsewhere", often in MATH going back centuries!

It seems to be "dimension-nonsense". But I derived it from another PATTERN: POLYNOMIAL SEPARATED INTO HOMOGENEOUS AND HETEROGENOUS SUBPOLYNOMIALS. You can't quarrel with that!

Apparently, Grassmann hit on this, late in his career, by finding it was the only way to derive Hamilton's QUATERNIONS. And Hamilton found it. But neither developed it -- only Clifford.

MULTIPRODUCT: HOMOGENIZER

As a first application of MULTIPRODUCT, look back at [8'] and [9']:
aÄb = a · b + a Ù b. [10]

Suppose we set a = b. [11]

Since di · di = 1, the a · b in [8'] becomes a · a = a12 + a22 + a32 [12],
that is, THE HOMOGENEOUS TERM OF [8'] beccomes SIMPLIFIED.

And, since di Ù dt = 0 for i ¹ j, then a Ù b in [9] becomes a Ù a = 0 [13],
that is, THE HETEROGENOUS TERM IN [8'] VANISHES.

THUS, MULTIPRODUCT IS THE GREAT HOMOGENIZER! And what is [12]? It's THE PYTHAGOREAN FORM, which PROVIDES A METRIC -- THE STRUCTURE ARTICULATING THE GEOMETRY WITH A TOPOLOGY!

FURTHERMORE, THE METRICS OF SOME GEOMETRIES (e.g. SPACE-TIME) MUST MIX POSITIVE AND NEGATIVE TERMS, which is easily ACCOMODATED VIA MULTIPRODUCTS. Hence, the more general labelling of CLIFFORD ALGEBRAS or CLIFFORD NUMBERS is Ci Þ C(p,n) , p + n = i, where p denotes the number of positive terms in the METRIC, n the number of negative terms.

Thus, the PAULI ALGEBRA used for quantum theory is labelled C(3,0) because it has 3 positive space-coordinated terms and 0 negative time-coordinated term, with the BASIS: [1, d1, d2, d3]. The DIRAC ALGEBRA, used in the relativistic theory of the electron, hence, also for special theory of relativity, is labelled C(3,1) because it has 3 positive space-coordinated terms and 1 negative time-coordinated term, with the BASIS: [1, d1, d2, d3, d4].

In 1984, physicist K. R. Greider showed that all relativistic quantum fields of spins 0, 1/2, 1, can be developed with the single DIRAC formalism, whereas the standard treatment is a patchwork of several formalisms. This is just one among many reasons I can cite for calling MAINSTREAM MATH a "babel" math for ignoring MULTIVECTOR THEORY and so many other resources -- some even going back centuries or thousands of years -- making the task of students unnecessarily difficult!

MULTIPRODUCT: UNIFIER

In a sense, MULTIVECTOR THEORY IS BUILT AROUND A SINGLE EQUATION, [9], and its EXTENSIONS to 3-D, etc. In MULTIVECTOR CALCULUS, MULTIPRODUCT takes the individual OPERATORS of GRAD (ÑA), DIV (Ñ · A), CURL (Ñ Ù A), of STANDARD VECTOR CALCULUS and UNIFIES THEM INTO A SINGLE OPERATOR:
ÑA = Ñ A + Ñ Ù A. [14]

Whereas, in standard vector calculus, we usually find the gradient only of a scalar, the operation of multiproduct allows us to find the gradient of any multivector!

For, more UNIFIFICATION RESULTS IN DIRECTED INTEGRATION. Given a vector-valued function, f, of a vector, x, we can write (noting separability and rewriting MULTIPRODUCT as JUXTAPOSITION):
òfÄdx = òfdx = ò(f · dx + f Ù dx) = òf · dx + òf Ù dx. [15]

Hestenes has finds that THE INVERSE OF GRADIENT INVOKES THE CAUCHY INTEGRAL FORMULA, and has sketched a THEORY OF THE DIRECTED INTEGRAL, both in RIEMANNIAN and LEBESGUE form.

Hestenes argues that "many notions of homology theory can more readily be expressed by directed integrals than by the scalar-valued integrals of cohomology.... It remains to be seen to what degree homology theory can be regarded as a theory of directed integrals."

A MATHEMATICAL consequence of this UNIFICATION is that REAL AND COMPLEX ANALYSIS ARE UNIFIED AND COMPLEX ANALYSIS IS EXTENDED IN WAYS IT CANNOT BE IN ITS STANDARD FORM!

A consequence in PHYSICS is that MAXWELL'S ELECTROMAGNETIC FIELD EQUATIONS BECOME A SINGLE EQUATION (FOR A SINGLE PROCESS!), VIA MULTIPRODUCT!

MULTIPRODUCT: SUPERFACTOR

Please note that we can write
x12 + x22 + ... + xn2 = (x1w + x2w + ... + xnw)2 [16],
where w is any unit vector such that wi·wj = 1, for i = j; or 0, for i ¹j.

CONVERSELY, we can FACTOR: (x12 + x22 + ... + xn2)1/2 = ±(x1w + x2w + ... + xnw). [17]

(In his "Quantum Theory of the Electron", Dirac applied a MATRIX form of this factoring to the Klein-Gordon equational replacement of Schrödinger's Equation, leading to discovery of ANTIMATTER! But his derivation takes 4 pages. MULTIPRODUCT takes only 4 lines: FACTORPOWER! Incidentally, it was a comment similar to [16], [17] above which I found on p. 61 of 100 Years of Mathematics by British mathematician, George Temple, that gave me the idea for my MULTIPRODUCT DERIVATION.)

And MULTIPRODUCT has many other fascinating properties.