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Hoop Algebras

Roger Beresford
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Hoops are "Symmetry-conserving Partial-fraction-Division Algebras" with symmetries or "sizes" that are conserved on vector multiplication and division. Conserved symmetries lead, via Noether's theorem, to forces and to particles. Consequently, Hoops subsume all the algebras (including Real, Complex, Quaternion, Octonion, Clifford and Wedge) relevant to particle physics. They may lead to a new paradigm for physics (possibly related to M-theory), based on finite Moufang Loops (so vectors have multiplicative inverses) with "Frobenius conservation" (so the factors of the symbolic multiplication table determinant are conserved symmetries). Do not confuse hoop multiplication (which multiplies two vectors to give a product and two, possibly null, remainders) with the use of groups to transform a single vector from one set of coordinates to another - as in the "standard model".

Hoops involve some neglected and some new mathematical concepts:-
  1. "Primal" unsigned continuous numbers (the half-line, 0 U R+) can be developed from set theory without introducing negation or subtraction.
  2. "Folding" sets of primal numbers introduces signs via equivalence relationships. 2-folding creates Integers and Reals with the signs {+, -}. Folding destroys some information.
  3. "Generalized signs" are primitive (NOT cyclotomic) roots of unity s^j, with s^r = +1. They are created by r-fold equivalence relationships. Complex numbers are one case of r=4, with signs {+,i,-,-i}. r=3 gives "terplex" signs {+, J, J^2} with ternary symmetries relevant to quarks. Algebraic loops with r-fold symmetry fold to become algebras with generalized signs.
  4. Algebraic loops with the division property (zx).(yz)=z.(xy).z are Moufang Loops; this ensures that every vector has a multiplicative inverse in Hoop vector multiplication. Groups are associative loops and have the Moufang property.
  5. Real factors of the determinant of the inverse symbolic multiplication table are "conserved symmetries" or "sizes" for all group tables and for a few non-associative Moufang Loops. Sizes are conserved on multiplication, Det[AB]=Det[A] Det[B] (up to a sign); they provide denominators for partial-fraction formulations of the inverse.
  6. "Hoop Algebras" are r-folded from (m.r).(m.r) conservative tables. They are division algebras because they possess addition and generalized subtraction using r-fold signs together with the Hoop multiplication-division property. All groups, together with octonions, split-octonions and some other non-associative Moufang loops, are hoops because they are folded versions of larger conservative loops.
  7. "Signed hoop tables" may result from folding. They are not loops because their products include signed elements that are not members of the defining set. Examples - the complex set is {1,i} but the product i.i is -1. The quaternion set is {1,i,j,ij}, with i.j=ij but j.i=-ij. Pauli-\[Sigma] algebra is {t,x,y,z} with z.y=ix etc.
  8. "Remainders" are created to maintain size conservation on multiplication or division of vectors when some sizes are zero. Hoop operations then "project" the results into constrained sub-algebras (with the same zeroed sizes) and "eject" remainders. This maintains size conservation and eliminates division-by-zero. The Hoops.m package includes over 80 hoops with this property. Remainders have analogies with particle interactions, where properties are conserved by ejecting particles of lower symmetry.
  9. "Polar-Cartesian duals" (generalizing the {r,theta} dual of the x+i y complex plane) exist when quadratic sizes occur. Cartesian-Polar interconversions are provided for 44 hoops in Hoops.m. Abelian duals have angles that add on vector multiplication. This generalises powers and roots.
  10. "Multi-phase sinusoidal orbits" are duals with sizes that multiply to one (ignoring any "factored-out" zeroes). Some stable orbits resemble multi-phase de-Broglie waves with fundamental sizes resembling Planck areas and elements that act as quantum operators. This may determine which particles are stable. Higher frequency resonances may relate to unstable particle resonances.
  11. "Dozal" multiplication rules (for vectors with 12 elements) may correspond to four types of interaction between particles, with different size combinations corresponding to particles with different conserved properties. Half-spin quantum operators are fermionic Dozal elements. Their three-phase orbits have reciprocal squared radii, giving three size regimes (point-particle scale, Planck-scale, cosmic scale), related to T-duality in M-theory and the "large numbers hypothesis".
  12. Bosonic "Hexal" is supersymmetric to (i.e. folded from) Dozal, but has fewer symmetries, so supersymmetry gives fewer partner particles. This is, so far, the only prediction arising from the hoop concept. The Pauli exclusion principle may not apply to bosons because they are in equivalence relationships to many fermions.
The real division algebras R, C, H, & O are degenerate (monosized) hoops; complex functions are specialisations of more general functions. Hoops subsume, but do not invalidate, complex mathematics. Many standard mathematical operations are specialisations of general hoop operations; many "obvious mathematical truths" are special cases restricted to the "real algebras without divisors of zero".

I am still seeking solutions to outstanding problems:
  1. Non-abelian "pseudo-polar" duals do not have conserved radii, so they only provide "pseudo-powers" and "pseudo-roots". They may introduce uncertainty via non-commutative operations that produce rotated vectors.
  2. Dozal hoops conserve many functions resembling the conserved properties of particles; I cannot correlate them with particles.
  3. The Schrodinger equation describes "information about particles" and leads to dispersing wave-packets. Replacing the 4 directions implied by "i" with the 12 directions of Dozal, and possibly eliminating dispersion with non-linear terms (as in the KdV equation) could lead to a description of stable particles as multi-phase wave-packets.
  4. I still hope to bring space-time (or perhaps a different description as "time-space", in which time enters the metric as an un-squared primal term) into the system. One possibility is to combine particle interactions (described by hoops) with the Lorentz transformation of coordinates (described by matrix multiplication using the P16 group or Clifford(4) ).
  5. Testable predictions are needed, beyond "most supersymmetric particles do not exist".

    Updated by the author to Version 1c, February 21, 2007. OK with Mathematica 4.2 or later

*Mathematics > Algebra > Group Theory

Moufang loops, conservative algebras, continuous orbits, determinants, directors, Frobenius-conservation, generalized signs, hoops, Primal numbers, remainders, shapes & sizes (of vectorss), signed-tables, partial-division-by-zero, quasigroups, renormalization, roots of unity, vector division, univectors.
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*Groups, Loops, & Hoop Algebras   [in MathSource: Packages and Programs]
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ALGHCAtest.nb (447.9 KB) - Validation of ALGHCA.nb
HoopAbstract.doc (19.5 KB) - Microsoft Word document
HoopAlfigs.doc (45 KB) - Microsoft Word document
HoopAlg&Physics.nb (147 KB) - Mathematica Notebook [for Mathematica 5.2]
HoopAlgebraSupplement.doc (174.5 KB) - Microsoft Word document
HoopFundamentals.nb (233.7 KB) - Mathematica Notebook [for Mathematica 5.2]
HoopSup.nb (204.6 KB) - Mathematica Notebook [for Mathematica 5.2]
Hoops&Physics.doc (59 KB) - Microsoft Word document
Hoops.nb (175.9 KB) - Mathematica Notebook [for Mathematica 5.2]
HoopsAll.zip (236.6 KB) - Zip of all files