Conservative multiplication involves remainders.

From: Roger Beresford (mail_at_beresford22.freeserve.co.uk)
Date: 02/18/05

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    Date: 18 Feb 2005 16:15:00 -0500

    Many algebras, (R, C, H, O,, Clifford, Davenport, Dirac, Lie, Study,
    Pauli, Wedge, etc,) are "Conservative Division Algebras" because their
    multiplication tables have the Frobenius determinant conservation
    property and the Moufang division property. Every vector
    A={a(1),a(2)...} has a multiplicative inverse Ai and has one or more
    size (factors of the determinant of the multiplication table, mapped
    with the vector coefficients). Sizes are conserved on
    multiplication/division, Det[A] Det[B] = Det[AB] (up to a sign). Apart
    from R, C, H, O over the real field, non-trivial vectors can have
    sizes that are zero. Unless A & B have the same zeroes, the product
    (or dividend if B is an inverse) will be "projected" onto a
    sub-algebra with fewer non-zero sizes. Conservation can be retained if
    the "lost" sizes are "ejected" as remainders. In Natural Number
    division, A/B = C+R with 0<=R<B, and CB+R = A. In conservative
    algebras, A/B = C+Rl+Rr and CB+Rl = A. The remainders carry the sizes
    of A that are less than those (effectively infinite) sizes of Bi that
    are zero in B. As multiplication and division are the same operation,
    multiplication also has remainders, AB=C+Rl+Rr, CBi+Rl = A, CAi+Rr =
    B.

    I call these algebras "Hoops" because they are loops or rings that
    conserve their "shape" (the list of sizes). MathSource/4894 includes a
    HoopAlgebra package and HoopDemo, a Mathematica notebook arranged to
    be accessible to non-users of Mathematica. The following transcript of
    Ex 5. demonstrates remainders using the abelian C3C2 hoop, the
    non-abelian D3 hoop, and the monosized Pauli-sigma hoop. The first two
    are over the real field, and the last over the complex field. All
    hoops work over both the real & the complex field. In each case,
    vectors have been chosen to have different zeroes. The calculations
    are not shown here, but use the "genTimes" procedure for
    multiplication (giving the product and remainders), "genInverse" to
    calculate the multiplicative inverse, and "shape" to give the
    appropriate shape for each vector.
     "genInverse" projects Ai (etc) onto the appropriate sub-algebra by
    giving a vector with the same zeroes as the uninverted vector. (This
    is only shown explicitly for C3C2)

    (*Example 5a*) muldiv[{2,1,-2,-2,3,-2},{3,1,1,3,1,1},"C3C2"];

    A = 2 1 -2 -2 3 -2
    B = 3 1 1 3 1 1
    AB = 0 8 -8 0 8 -8
    Rl = 2 -1 0 -2 1 0
    Rr = 5/3 5/3 5/3 5/3 5/3 5/3
    ShapeA = 0 2 48 28
    ShapeAi= 0 1/2 1/48 1/28
    ShapeB = 10 0 16 0
    ShapeBi=1/10 0 1/16 0
    ShapeAB= 0 0 768 0
    ShapeRl= 0 2 0 28
    ShapeRr= 10 0 0 0
    AB/B+Rl=A, {2,1,-2,-2,3,2}
    AB/B+Rr=B, {3,1,1,3,1,1}

    Only one size survives in the product; the remainders carry the lost
    sizes, and so allow the original vectors to be recovered as AB/B+Rl
    etc.

    (*Example 5b*) muldiv[{2,1,-2,2,3,-2},{3,1,1,3,1,1},"D3"];

    A = 2 1 -2 2 3 -2
    B = 3 1 1 3 1 1
    AB = 12 12 -4 12 12 -4
    Rl = 4/3 1/3 -8/3 4/3 7/3 -8/3
    Rr = 0 0 0 0 0 0
    ShapeA = 2 4 8
    ShapeB = 0 10 0
    ShapeAB= 0 40 0
    ShapeRl= 2 0 8
    ShapeRr= 0 0 0
    AB/B+Rl=A, {2,1,-2,2,3,-2}
    AB/B+Rr=B, {3,1,1,3,1,1}

    Non-abelian hoops have repeated sizes, so D3 has only 3 sizes, in
    contrast to the four for C3C2.

    (*Example 5c*) muldiv[{5,4,0,3},{2,i,4-i,3},"P4"];

    A = 5 4 0 3
    B = 2 i 4-i 3
    AB = 19+4i 11+17i 23+7i 17-16i
    Rl = 0 0 0 0
    Rr = 2 i 4-i 3
    ShapeA = 0
    ShapeB = -19+8i
    ShapeAB= 0
    ShapeRl= 0
    ShapeRr= -19+8i
    AB/B+Rl=A, {5,4,0,3}
    AB/B+Rr=B, {2,i,4-i,3}

    In this case, there is only one size, a1^2-a2^2-a3^2-a4^2, and A is
    chosen to make this zero. Consequently there is no left remainder and
    AB/B=A, whilst the right remainder is all of B.

    Conservative multiplication/division uses remainders to eliminate
    division by zero, by first subtracting (ejecting) those sizes that
    would lead to infinities.

    Can anyone provide an interpretation of this in terms of the light
    cone for the P4 algebra?

    Roger Beresford.

    ".. we must be willing to question basic, seemingly self-evident,
    elements of reality." (Brian Greene, The Fabric of the Cosmos" 2004,
    p426.)

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